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Endogenous comparative advantage, gains from trade and symmetry-breaking
Endogenous Comparative Advantage, Gains From Trade and SymmetryBreaking Arpita Chatterjee PII: DOI: Reference:
S0022-1996(17)30113-7 doi: 10.1016/j.jinteco.2017.08.009 INEC 3078
To appear in:
Journal of International Economics
Received date: Revised date: Accepted date:
1 December 2013 21 June 2017 29 August 2017
Please cite this article as: Chatterjee, Arpita, Endogenous Comparative Advantage, Gains From Trade and Symmetry-Breaking, Journal of International Economics (2017), doi: 10.1016/j.jinteco.2017.08.009
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Symmetry-Breaking
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Endogenous Comparative Advantage, Gains From Trade and
Arpita Chatterjee∗ University of New South Wales
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June 2017
Abstract
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Similar countries often choose very different policies and specialize in very distinct industries. This paper proposes a novel mechanism to explain policy diversity between similar countries from an open economy perspective. I study optimal policies in a two country model where policies affect the determinants of trade patterns. Symmetric countries choose asymmetric policies in equilibrium if aggregate income is sufficiently convex in own policy. Such high social increasing return to policy, despite constant returns to scale in production, generate enough gains from trade to sustain policy diversity in equilibrium. Any asymmetric equilibrium is welfare-improving compared to the common autarky optimum. Similarly, a more asymmetric Nash equilibrium Pareto dominates a less asymmetric one. As an application, I consider a model in which the government’s choice of education policy affects skill distribution and comparative advantage. In this application, the existence of an asymmetric Nash equilibrium in education policy requires that the elasticity of the education cost of agents with respect to skill is relatively low. (JEL Classification: F11, E62.) Keywords: Symmetry-breaking, Endogenous comparative advantage, Gains from trade, Education policy.
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School of Economics, UNSW Business School, University of New South Wales, Sydney, Australia. (Email : [email protected], Tel : +61293854314).
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Introduction
Why do similar countries choose very different domestic policies? 1 Existing theories (Benabou and Tirole, 2006; Alesina and Angeletos, 2005) explain this by modeling the optimal policy problem in a closed economy setting. They rely on coordination failure as the reason for policy diversity.
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However, studying each country separately, coordination failure cannot rule out the similarity of
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equilibrium policies as an equally plausible outcome. This is why Matsuyama (2002, page 241) notes that “coordination failure offers no compelling reason” for why we should expect to observe diversity across space, time, or groups.
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Matsuyama (2002, 2004) instead proposes symmetry-breaking – the existence of an asymmetric equilibrium in a symmetric setting– as an explanation for observed diversity. The logic of symmetrybreaking relies on modeling interdependence between the relevant spaces, times, or groups. In is the only stable equilibrium outcome.
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Matsuyama’s (2002, 2004) symmetric strategic setup, symmetry-breaking implies that asymmetry In this paper, I apply the logic of symmetry-breaking to explain why similar countries may
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choose different domestic policies in equilibrium. I model the policy problem in an open economy where two symmetric countries choose a domestic policy that affects their comparative advantage. I show that diversity in equilibrium policy arises because it allows countries to gain from international
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specialization and trade. 2 This provides an alternative answer for why similar countries choose different policies, without relying on coordination failure. In the optimal policy problem, however, any symmetric equilibrium is also stable. Hence, I
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cannot rely on Matsuyama’s (2002) argument of instability in symmetric equilibrium to ensure equilibrium asymmetry. Instead, I offer necessary and sufficient conditions for symmetry-breaking in equilibrium policy by relying on social increasing returns to policy. Under these conditions, the
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open economy equilibrium exhibits a stronger form of symmetry-breaking by guaranteeing that no symmetric Nash equilibrium exists, not just that any symmetric equilibrium is unstable. This sufficient condition is a close parallel to the sufficient condition in Amir, Garcia, and Knauff (2010) which similarly ruled out the existence of any symmetric equilibrium. Thus, in this symmetric setup, asymmetry is the only possible equilibrium outcome in the open economy. What kind of policies might affect international comparative advantage? International trade literature predicts comparative advantage on the basis of differences in domestic factor endowments (Heckscher-Ohlin), skill distribution (Grossman and Maggi, 2000; Bougheas and Riezman, 2007), sector-specific technologies (Ricardo) or institutions (Costinot, 2009; Levchenko, 2007). Typically, these differences are treated as entirely exogenous. 3 However, these determinants of trade pattern 1 Bosworth and Collins (2008), Panagariya (2006), and He and Kuijis (2007) highlight important policy differences between India and China. Arora and Gambardella (2005, Chapter 3) discuss differences in economic policies between Ireland and Greece. Important differences in economic policies between the US and Europe have received considerable attention in the literature (Alesina, Glaeser and Sacerdote, 2001; Krueger and Kumar, 2004). 2 Similar countries specialize in distinct industries in the open economy. This observation is the key motivation behind Grossman and Maggi (2000). Baumol and Gomory (2000) document that for the largest economies (Germany, Japan, and the US) the cross-industry pattern of specialization is remarkably stable. 3 Notable exceptions are Clarida and Findlay (1991, 1992) and Deardorff (1997).
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ACCEPTED MANUSCRIPT are affected by the choice of domestic education policy, national policies on savings and capital accumulation, sector-specific R&D policies, and labor-market policies or credit market reforms. 4 In this paper, I endogenize the source of comparative advantage by allowing domestic policy to affect comparative advantage.5
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My general setup consists of a two-good, two-factor, two-country model in which both the goods
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and factor markets are perfectly competitive. The planner in each country chooses a single policy that affects productivity differently for different goods; for example, the policy in question affects the relative technological progress or skill composition of a country. Because the two countries are otherwise identical, a difference in government policy is the only potential source of comparative
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advantage. Both policies affect the equilibrium price under free trade. Hence to formulate an optimal policy the policy makers in each country need to take the relevant policies of their trade
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partners into consideration. Rather than each country choosing its policy in isolation, countries interact in the open economy to achieve the optimal design of the national policies that impact international comparative advantage.
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I model the optimal policy problem as a non-cooperative optimization in which each country chooses a policy to maximize its own aggregate welfare. The optimal policy problem in this setting is a symmetric game in which each player’s best response is decreasing in the action of the other player. Can these symmetric countries choose asymmetric policies in a pure strategy Nash equilibrium
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(PSNE)? I show that the countries face a clear trade-off. An asymmetric policy choice gives rise to endogenous comparative advantage and gains from trade, but also means that at least one of the identical countries is choosing a policy that would be suboptimal in the absence of trade
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opportunities. The gains from trade thus need to be weighed against the loss in welfare due to a suboptimal domestic policy.
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If an asymmetric equilibrium exists, it is associated with higher aggregate welfare for both countries compared to the common autarky optimum. In an asymmetric equilibrium, ex-ante identical countries are ex-post different with endogenous comparative advantage in different industries. This leads to welfare gains in the asymmetric equilibrium. More generally, a more asymmetric PSNE Pareto dominates a less asymmetric PSNE. When are the gains from trade due to asymmetric policy choice large enough to ensure symmetrybreaking in optimal policy? In the optimal policy problem, a symmetric equilibrium is always stable, thus Matsuyama’s (2002) symmetry-breaking condition does not apply in this setting. I rely on 4
Policymakers in many countries emphasize the international comparative advantage they derive from domestic policies in an increasingly integrated world. For example, international competitiveness within the knowledgeintensive service sector is an important factor in formulating education policies in service-exporting nations like India or Ireland. The Indian task force report on Human Resource Development in Information Technology (2000) makes 47 specific recommendations with a view ”to create a sustainable competitive advantage” in knowledge-led businesses. A very similar picture of the education policy in Ireland emerges from the Human Capital Priority program of Ireland’s National Development Plan (2007). 5 Note here that I am restricting attention to a single, national policy that affects the entire population of a country. If a government can choose different policies for different entities within a country (across different provinces or states, for example), a similar type of asymmetric equilibrium and endogenous internal trade may arise within the homogeneous population of a single country.
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ACCEPTED MANUSCRIPT quasiconvexity in the welfare function with respect to the country’s own policy, as used in Amir, Garcia and Knauff (2010) to rule out any symmetric equilibrium. 6 This quasiconvexity in welfare in own policy arises as a result of social increasing returns to policy. I show that sufficiently large social increasing return to policy, defined as convexity in aggregate income with respect to own
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policy, can ensure the existence of an asymmetric equilibrium. In fact, these conditions ensure Nash equilibrium in the open economy is asymmetric.
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a stronger form of symmetry-breaking compared to Matsuyama (2002) by guaranteeing that any I illustrate the existence of social increasing returns to policy in a specific application of education policy. In the application, the social planner allocates a fixed education budget between two
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categories of education, – higher and basic. 7 The agents have heterogeneous ability, and choose skill by incurring an education cost. The total and marginal costs of education increase in skill
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and decrease in ability. The education choice is intermediated by the government’s allocation of an education budget. Apart from the endogenous skill choice by agents and education policy choice by the government, the model is a standard Heckscher-Ohlin economy.
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I show that the endogenous skill choice of agents implies an increasing return to education policy. The degree of the social increasing return is governed by the elasticity of the education cost in skill. Previous literature on education policy (Benabou, 2002) identifies this elasticity as the determinant of progressiveness of the education system. This elasticity is the key institutional
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parameter for symmetry-breaking in this application, provided that the technologies of the two sectors are sufficiently different and consumer preference is reasonably diversified. To summarize, this paper makes three contributions. First, it provides a new answer, along
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the line of Matsuyama (2002), to the well-researched question (Benabou and Tirole, 2006; Alesina and Angeletos, 2005): ”Why do similar countries choose very different policies?”. This study shows
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that when comparative advantage is endogenous to domestic policy, gains from trade can be an explanation for equilibrium policy diversity between similar countries. 8 Second, I exploit the prop6
This approach to symmetry-breaking has many applications in the industrial organization literature. For example Amir (1996, 2000), Amir and Wooders (2000), Amir, Garcia and Knauff (2008), in the context of R&D investment and capacity choice under demand uncertainty by ex-ante identical firms. In this paper, I apply this approach to symmetry-breaking in a novel context: an optimal policy problem in which domestic policy affects international comparative advantage. It is also novel methodologically, since Amir, Garcia and Knauff’s (2010) condition ensures that the necessary condition of symmetric equilibrium is never satisfied. In this setting, however, the necessary condition for symmetric Nash equilibrium is satisfied if both countries choose their common autarky optimum policy. Instead, I rely on ruling out sufficient second order conditions of symmetric equilibrium. 7 Climent and Mukhopadhyay (2013) study the implications of a similar budget allocation problem for the Indian economy. 8 In a related work, Acemoglu, Robinson and Verdier (2015) argue that in a technologically interconnected world, the world equilibrium may be asymmetric, involving different economic institutions and technology levels for different countries. In their framework, in which the world technology frontier is affected by all countries, an asymmetric world equilibrium emerges provided the function aggregating the innovation decisions of all countries into the world technology frontier is sufficiently convex. This resonates with the sufficient condition for symmetry-breaking in my framework in terms of sufficient convexity of the aggregate income function with respect to own policy. However, the global linkages in their model are purely technological and in my model the global linkages are purely pecuniary and arise from world goods market equilibrium. Also, Acemoglu, Robinson and Verdier (2014) analyze properties of the world equilibrium, while I analyze both the non-cooperative Nash equilibrium and cooperative world equilibrium and show that the cooperative world equilibrium is more likely to be asymmetric. A related argument is developed in Aoki (2001), who also notes the possibility that international interconnections might be at the root of this type of
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ACCEPTED MANUSCRIPT erties of a standard general equilibrium model of international trade to characterize the conditions for, and welfare-implications of, an asymmetric Nash equilibrium in a symmetric non-cooperative policy game. I show that the conditions of symmetry-breaking that rely on payoff nonconcavities in a submodular game (Amir, Garcia and Knauff, 2010) are related to the concept of social increasing
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returns to policy, defined as convexity in aggregate income with respect to policy. I also estab-
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lish that greater equilibrium asymmetry gives rise to larger welfare gains. This naturally implies a Pareto ranking of asymmetric equilibria – a more asymmetric Nash equilibrium Pareto dominates a less asymmetric Nash equilibrium. Third, I contribute to the trade and education policy literature (Chang and Huang, 2012; Bougheas, Kneller and Riezman, 2009) by showing that endogenous skill
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choice by agents, in the presence of simple, linear education subsidies, can be a source of social increasing return to policy and symmetry-breaking in optimal policy. 9
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The rest of the paper is simply structured. In Section 2, I present the general framework and the optimal policy problem, and establish the conditions for symmetry-breaking in optimal policy and its consequent welfare implications. I analyze the application to education policy in Section 3.
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Section 4 concludes.
Optimal Policy in a General Framework
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I consider a two-good, two-factor, perfectly competitive world comprising two large, identical countries. I denote autarky variables using a superscript A and trade variables using a superscript T. All foreign country variables are designated by an asterisk. Let Fi denote the aggregate production
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of good i and Ci denote the aggregate demand of good i. In equilibrium, both production and consumption are functions of equilibrium price and the domestic policy, γ.
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Let u(C1 , C2 ) denote the direct utility function and I denote the aggregate income in a country. The equilibrium relative price of good 1 is pj , {j = A, T }, where j denotes the state of the economy (autarky A or free trade T ). Aggregate welfare, a function of equilibrium price and income, is denoted by V j , j = A, T. I assume homothetic preferences which imply that the aggregate demand for each good and indirect utility is linearly homogenous in income, institutional diversity (see also Rodrik, 2008). 9 Note that I derive conditions for symmetry-breaking and the consequent welfare implications in an optimal policy problem in a two country, two good economy. In this framework domestic policy is the only determinant of international comparative advantage and I do not impose any further structure on how the policy affects the economy. This flexibility implies that the framework can allow for many different types of policy problems. But without further structure on the economy, the sufficient conditions for symmetry-breaking rely on the convexity of the aggregate income function, an equilibrium object. In different policy problems, different restrictions on economic fundamentals can satisfy the requirement of sufficient convexity of the aggregate income function. The application to education policy highlights one such example. In this application, the sufficient condition relies on elasticity of the education cost function in a Hecksher-Ohlin economy with endogenous skill choice. Alternatively, in an economy with Grossman and Maggi (2000) production structure, submodularity in production can give rise to social increasing return to education policy and hence equilibrium policy diversity. This resonates with the key result of Chang and Huang (2012).
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Ci = fi (pj )I
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Vj
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= u(f1 (pj )I, f2 (pj )I) j
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= f (p )I where f (pj ) ≡ u(f1 (pj ), f2 (pj )).
Each government chooses a domestic policy γ and that is the only source of comparative advantage in the open economy. Without loss of generality, let an increase in γ confer a comparative
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advantage to good 1. I interpret γ as a policy that affects sector-specific technologies or the relative factor endowments. Government policy γ affects equilibrium quantities directly through its effects
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on economic fundamentals and indirectly through the equilibrium price. Profit maximization by competitive final good producers ensures that at any point (Q1 , Q2 ) on the production possibility frontier aggregate value of production (pj Q1 + Q2 ) is maximized (for any
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γ and any pj ) such that pj equals the marginal rate of transformation, dQ2 . p = dQ1 j
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In competitive equilibrium,
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Q1 = F1 (pj , γ) Q2 = F2 (pj , γ).
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Aggregate income in equilibrium is denoted by Y (pj , γ). From the equality of aggregate value of production and aggregate income, I = Y (pj , γ) = pj F1 (pj , γ) + F2 (pj , γ).
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Thus, Y (pj , γ) is the same as the national income function in Copeland and Taylor (2003) or the revenue function in Dixit and Norman (1980). In the competitive equilibrium of the closed economy, Ci (pA , γ) = Fi (pA , γ) , i = 1, 2,
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and hence the equilibrium price, pA , depends only on own policy γ. In the open economy, both countries’ policies determine the equilibrium terms-of-trade pT (.) from the goods market clearing of the world, Ci (pT , γ) + Ci∗ (pT , γ ∗ ) = Fi (pT , γ) + Fi∗ (pT , γ ∗ )
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and the foreign policy γ ∗ affects domestic welfare through the terms-of-trade externality. The 6
ACCEPTED MANUSCRIPT economy, for any given γ, is a standard Walrasian economy. Below, I define the competitive equilibrium. Definition of Competitive Equilibrium For any given policy γ, the competitive equilibrium is defined by a sequence of equilibrium price and quantities {pj , Ci , Qi , Ci∗ ,
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Q∗i , Y (pj , γ), Y ∗ (pj , γ ∗ ), i = 1, 2, j = A, T } such that
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1. C1 , C2 solve the consumer’s utility maximization problem in the home country max u(C1 , C2 ) s.t. pj C1 + C2 = I.
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C1 ,C2
Similarly, Ci∗ solves the consumer optimization problem in the foreign country. In equilibrium
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Ci = fi (pj )I
2. Q1 , Q2 solve the producer’s profit maximization problem, which implies that aggregate income
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is maximized subject to the technology and factor endowment constraints of the economy. We denote the feasible set of (Q1 , Q2 ), given the technological and endowment constraints and a given policy γ, by P P F (γ).
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max I = pj Q1 + Q2 s.t. (Q1 , Q2 ) ∈ P P F (γ).
Q1 ,Q2
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Similarly, Q∗i solves the producer optimization problem in the foreign country. In equilibrium
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Q1 = F1 (pj , γ) Q2 = F2 (pj , γ), I = Y (pj , γ) = pj F1 (pj , γ) + F2 (pj , γ).
3. Markets clear.
Autarky: fi (pA )Y (pA , γ) = Fi (pA , γ), i = 1, 2.
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Open economy: fi (pT )Y (pT , γ)+fi (p T )Y (p T , γ ) = F i (p T , γ) + F ∗i (p T , γ ∗ ). I denote home aggregate welfare under autarky by U A (γ) and home aggregate welfare under ∗
trade as U T (γ, γ ∗ ). Similarly, Foreign aggregate welfare under autarky is U A (γ ∗ ) and Foreign aggregate welfare under trade is U T (γ ∗ , γ)− thus, for the welfare function for any country, the first variable in the argument refers to the country in question. Note that because the two countries are symmetric, welfare is the same function of policy in both countries. By definition of the indirect utility function,
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U A (γ) ≡ f (pA (γ))Y (pA (γ), γ),
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U T (γ, γ ∗ ) ≡ f (pT (γ, γ ∗ ))Y (pT (γ, γ ∗ ), γ). I assume that U A (γ) and U T (γ, γ ∗ ) are both differentiable up to the second order with respect to
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all the arguments. Similarly, one can define U A (γ ∗ ) and U T (γ ∗ , γ). The world welfare is denoted by W (.),
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W (γ, γ ∗ ) ≡ U T (γ, γ ∗ ) + U T (γ ∗ , γ).
Let us consider a bounded policy space γ ≤ γ ≤ γ. I also restrict attention to pure strategy Nash equilibrium (PSNE) in the non-cooperative game. 10 In addition, an assumption is that every
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individual residing within the national boundaries of a country is subject to the same policy, even though policies may differ across national boundaries. 11 Governments’ optimization problem is defined below:
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Definition of the Policy Problem In the closed economy, each government chooses γ to maximize aggregate welfare, U A (γ). The autarky optimal policy is therefore defined as γ A = arg max U A (γ).
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γ∈[γ,γ]
In the non-cooperative policy problem in the open economy, the home government chooses γ to
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maximize U T (γ, γ ∗ ) taking γ ∗ as given. I focus on pure strategy Nash equilibrium as the solution
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concept in this non-cooperative policy game. The best response functions are defined as BR(γ ∗ ) = arg maxU T (γ, γ ∗ ),
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γ∈[γ,γ]
BR(γ) = arg maxU T (γ ∗ , γ). γ ∗ ∈[γ,γ]
In the cooperative policy problem in the open economy, the social planner chooses (γ, γ ∗ ) to maximize world welfare, W (γ, γ ∗ )
(γ P O , γ ∗P O ) = arg max {U T (γ, γ ∗ ) + U T (γ ∗ , γ)} (γ,γ ∗ )∈[γ,γ]2
The Nash equilibrium solution is denoted by (γ N E , γ ∗N E ) where the first argument refers to the 10
Note that in this setup a symmetric mixed strategy Nash equilibrium (MSNE) always exists (Dasgupta and Maskin 1986). In a game characterized by strategic substitutability, MSNE is usually unstable, and countries attain greater welfare in an asymmetric PSNE than in a symmetric MSNE. Moreover, it is standard in the policy literature to focus on the PSNE as the relevant solution concept. 11 Note that if the government of any country is allowed to impose different policies on different sections of its citizens, even within a country endogenous asymmetry, production specialization and internal trade may arise. Analysis of the policy problem of regional asymmetry within the domestic economy is similar to the cooperative policy problem in the world economy.
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ACCEPTED MANUSCRIPT equilibrium choice of the home country and the second argument refers to the equilibrium choice of the Foreign country. If there are more than one pairs of Nash equilibria, k-th Nash equilibrium is denoted as (γ N Ek , γ ∗N Ek ), k = 1, 2. Let us similarly define (γ P O , γ ∗P O ) as the Pareto optimum solution for the cooperative policy problem.
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The countries are identical in terms of all economic fundamentals. When both countries choose
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the same policy (γ = γ ∗ ), they are endogenously in autarky. This gives us the first insightful property of the welfare function which I use throughout the paper:
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U T (γ, γ) = U A (γ).
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A similar property applies to the equilibrium price under trade and autarky:
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pT (γ, γ) = pA (γ).
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A very important difference between trade and autarky is that the equilibrium price is less sensitive
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to each country’s policy γ in symmetric trade equilibrium compared to the corresponding autarky: ∂pT (γ, γ) 1 dpA (γ) = . ∂γ 2 dγ
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This difference in the first order derivative of the equilibrium price helps us to derive an important difference between the second order derivative of the aggregate welfare function under trade and autarky.
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The aggregate welfare function in the open economy satisfies the gains from trade property. By this property, a country, for any given own policy, can gain in welfare terms by trading with a
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partner with a different policy,
U T (γ, γ ∗ ) U T (γ, γ) ∀γ ∗ , ∀γ,
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with equality at γ = γ ∗ . Hence, the relative welfare of choosing γ i over γ j , γ i %= γ j , improves in open economy over the relative welfare in autarky, conditional on the trading partner choosing γ j .12 In this sense, trade favors asymmetry in policy. Below I summarize these properties of the welfare function. Comparing Welfare under Autarky and Trade at Symmetric Policies By symmetry 12
From (11), we know that U T (γ i , γ j ) U A (γ i ) . > U T (γ j , γ j ) U A (γ j )
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pT (γ, γ) = pA (γ), ∂pT (γ, γ) 1 dpA (γ) = . ∂γ 2 dγ
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U T (γ, γ) = U A (γ),
Gains from Trade Property of the Welfare Function under Trade By gains from trade property of the welfare function
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U T (γ, γ ∗ ) U T (γ, γ)∀γ ∗ , ∀γ, with equality at γ = γ ∗ .
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When the autarky optimum policy is not unique, it is straightforward to show that at least one asymmetric equilibrium exists in both the non-cooperative and cooperative policy problems of the
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open economy using (11). I therefore restrict attention to the case in which the autarky problem has a unique optimum, γ A . In this game, the unique autarky optimum policy γ A is also a unique candidate for a symmetric Nash equilibrium (in which γ N E =γ ∗N E = γ A ) . When do these symmetric countries choose asymmetric policies in the open economy if the
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autarky problem has a unique optimum? To answer this question, it is important to understand an ′
important property of asymmetric Nash equilibrium. Suppose that (γ ′N E , γ N∗ E ) is an asymmetric
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PSNE. By gains from trade (11),
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U T (γ A , γ A ) ≤ U T (γ A , γ N∗ E ). ′
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Since γ ′N E is the best response to γ N∗ E , ′
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U T (γ A , γ N∗ E ) ≤ U T (γ ′N E , γ N∗ E ).
(12)
Thus, at any asymmetric PSNE, both countries gain in aggregate welfare compared to the common autarky optimum. This result is a powerful one. It implies that in an open economy Nash equilibrium, symmetric countries will necessarily benefit from a strategic choice of different policies. This simple, intuitive result is used to identify necessary and sufficient conditions for the existence of equilibrium policy diversity. 13 Next, I show that the convexity of the aggregate income function with regard to own policy is crucial for equilibrium policy diversity for a bounded policy space γ ≤ γ ≤ γ. If the aggregate income function, Y (pT , γ), is concave in policy γ, an asymmetric NE doesn’t exist. If the aggregate income Y (pT , γ), is sufficiently convex in policy γ, all NE in the open economy are asymmetric. Next, I relate the conditions for equilibrium asymmetry relying on convexities in the aggregate income 13
Note that the welfare gain in the asymmetric equilibrium is therefore due to an increase in production specialization which allows for an expansion of the consumption possibility frontier through trade, and not necessarily because of price difference between autarky and free trade equilibrium.
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ACCEPTED MANUSCRIPT function to the conditions for symmetry-breaking in the literature (Matsuyama, 2002; Amir, Garcia and Knauff, 2010), and also to establish the welfare properties of asymmetric equilibria. In the non-cooperative policy problem, mild conditions ensure that the policy game is submodular which implies decreasing best responses for both players. This strategic substitutability
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implies that at least one PSNE exists in the non-cooperative policy game, according to Topkis
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(1978).14 The Pareto optimum is asymmetric whenever an asymmetric PSNE exists, given (12). It is straightforward to show that γ P O = γ ∗P O = γ A is the only candidate for a symmetric Pareto optimum. Alternatively, if γ P O = γ ∗P O = γ A is the unique Pareto optimum, a Pareto-improving asymmetric equilibrium cannot exist. If the world welfare W (., .) is strictly concave, the symmetric
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strategy profile at (γ A , γ A ) is the unique Pareto optimum, and an asymmetric PSNE does not exist. If the aggregate income function, Y (pT , γ), is concave in policy γ, the world welfare W (., .)
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is strictly concave. Thus, convexity in Y (pT , γ) in γ is necessary for the existence of an asymmetric PSNE.
To derive the properties of the welfare function and the best replies sufficient for the existence of
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an asymmetric PSNE, first note that the policy game does not satisfy the sufficient conditions for the existence of an asymmetric PSNE in a symmetric game in the literature. In Matsuyama (2002) an asymmetric PSNE exists if the symmetric PSNE is Cournot unstable. In this two-good model with terms-of-trade externality as the only source of strategic interaction, the symmetric equilibrium, if it
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exists, is stable. The stability of the symmetric equilibrium follows from the substitutability of the two goods in consumption. Amir, Garcia and Knauff (2010) rule out any symmetric equilibrium since in their game the payoff function does not satisfy the necessary condition for an interior
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optimum at any interior point of symmetry. In this case, the welfare function U T (γ, γ ∗ ) has slope 0 at (γ A , γ A ) as a result of the gains from trade property,
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hence the exact sufficient condition
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outlined in Amir, Garcia and Knauff (2010) is not satisfied in my framework. Similar to Amir, Garcia and Knauff (2010), however, I rely on some form of payoff nonconcavity to rule out a symmetric PSNE in this case. The problem of ruling out any symmetric PSNE is technically equivalent to ensuring that ( γ A , γ A ) is a local minimum of U T (γ, γ A ), even though γ A is a local maximum of U A (γ). The sufficient condition for symmetry-breaking requires that the welfare function in the open economy, U T (γ, γ A ), is quasi-convex in own policy, γ, at (γ A , γ A ). Because (γ A , γ A ) is the only possible symmetric NE, this sufficient condition rules out symmetric equilibrium in this game by imposing a robust jump of the best replies across the diagonal of the strategy space, as illustrated in Figure 1a and similar to Amir, Garcia, Knauff (2010). 16 However, unlike Amir, Garcia, Knauff (2010), I rule out 2
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(γ,γ ) 0. The condition for guranteeing submodularity of the non-cooperative policy game is ∂ U∂γ∂γ ∗ 15 T A A The necessary first order condition of maximization of U (γ, γ ) is satisfied at γ since by gains from trade and endogenous autarky properties of the welfare function, U T (γ, γ A ) is an upper envelope of U A (γ) with equality γ = γ A , where γ A is the critical point of U A (γ). 16 Because the game is symmetric, any asymmetric PSNE appears in pairs. When there are only asymmetric PSNEs in this game, the total number of PSNEs is even. In a game with continuum action space, there are usually an odd number of PSNEs by Wilson’s Oddness Theorem (1971). This result is based on degree theory and requires continuity of the best response form. Ruling out symmetric equilibrium in this game involves a robust jump of the best replies 14
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satisfied at a point of symmetry, even though the first order condition is satisfied.
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Figure 1a: Discontinuous Best Response Curves and Multiple Asymmetric Equilibria
across the diagonal of the strategy space as in Amir, Garcia, Knauff (2010), hence my results are consistent with Wilson’s Oddness Theorem (1971).
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Figure 1b: Discontinuous Best Response Curves and Extreme Asymmetric Equilibrium This quasiconvexity in the welfare function arises in the presence of social increasing return to policy, i.e. if aggregate income is convex in government policy. If
∂ 2 Y (pT ,γ) ∂γ 2
< 0, W (γ, γ ∗ ) is strictly
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concave and there is no asymmetric equilibrium in the open economy. On the other hand, U T (γ, γ ∗ ) is quasiconvex in γ at (γ A , γ A ), even though U A (γ) is quasiconcave at γ A , if
∂ 2 Y (pT ,γ) |(γ A ,γ A ) ∂γ 2
is
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positive and sufficiently large. The condition needed for the existence of a unique interior autarky optimum which is not a symmetric Nash Equilibrium is that 2 ∂ Y (pT , γ) ∂pT ∂p∂γ ∂γ
(γ A ,γ A )
∂ 2 Y (pT ,γ) |(γ A ,γ A ) ∂γ 2
2 ∂ Y (pT , γ) dpA ∂ 2 Y (pT , γ) < |(γ A ,γ A ) < . dγ (γ A ,γ A ) ∂γ 2 ∂p∂γ
2 T (p ,γ) dpA < ∂ Y∂p∂γ dγ
ensures that a unique interior optimum 2 T (p ,γ) ∂pT (γ A ) exists under autarky, and sufficient convexity in aggregate income ( ∂ Y∂p∂γ ∂γ A A < Here, the condition
∂ 2 Y (pT ,γ) |(γ A ,γ A ) ) ∂γ 2
(γ A ,γ A )
ensures that the symmetric policy choice at the autarky optimum,
(γ ,γ ) (γ A , γ A ),
is
not a Nash equilibrium in the open economy. It is possible to satisfy this condition because of the differences in slope of the price functions at a point of symmetry: 1 dpA (γ) ∂pT (γ, γ) = . ∂γ 2 dγ This implies that all PSNEs in the open economy are asymmetric, since the only candidate for a symmetric PSNE, (γ A , γ A ) is ruled out–γ A is a local minimum of U T (γ, γ ∗ ) if γ ∗ = γ A . This
13
ACCEPTED MANUSCRIPT sufficient condition for symmetry-breaking, which relies on sufficient convexity of the aggregate income function is similar in spirit to Acemoglu, Robinson and Verdier’s (2015) sufficient condition for world asymmetric equilibrium relying on sufficiently convex world technology frontier. I
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summarize these conditions for the existence of an asymmetric equilibrium in the next proposition. Proposition 1 (Symmetry-Breaking and Social Increasing Return) If aggregate income is 2Y
(pT ,γ) ∂γ 2
< 0), no asymmetric PSNE exists in the optimal policy problem. If
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concave in policy ( ∂
2
T
(p ,γ) aggregate income is sufficiently convex in own policy (given an interior γ A , if ∂ Y∂γ |(γ A ,γ A ) is 2 ∂ 2 Y (pT ,γ) ∂pT ∂ 2 Y (pT ,γ) |(γ A ,γ A ) > ∂p∂γ ∂γ A A ), all PSNEs in the positive and sufficiently large, namely ∂γ 2 (γ ,γ )
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open economy are asymmetric. Whenever an asymmetric PSNE exists, the Pareto optimum is also asymmetric.
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Proof. See Appendix.
If U T (γ, γ ∗ ) is strictly quasiconvex in own policy over the entire policy space and γ A is an
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interior optimum in autarky, a unique pair of asymmetric NE is given by the extremes ( γ, γ).and (γ, γ). This extreme case of equilibrium policy diversity is demonstrated in Figure 1b, in which the
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best response of either country is to choose one of the extreme policies:
BR(γ ∗ ) = γ if γ A < γ ∗
(13)
γ
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BR(γ ∗ ) = γ if γ ≦ γ A
γ
This extreme case deserves a special mention since among all asymmetric PSNEs, both countries gain maximum aggregate welfare at the asymmetric PSNE that exhibits maximum policy difference.
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The welfare properties of different asymmetric PSNEs follow from a simple generalization of the gains from trade property (11). A country not only gains in welfare by trading with a partner who has a different policy, but it gains even more from trade as the difference between its own policy and the policy of its trading partner grows. 17 Thus, for any given own policy, the welfare of a country increases with an increase in foreign policy if the foreign policy is greater than the own policy:
∂U T (γ, γ ∗ ) ∂γ ∗ T ∂U (γ, γ ∗ ) ∂γ ∗
> 0 f or γ ∗ > γ, <
(14)
0 f or γ ∗ < γ.
This property of comparative-advantage driven trade and the definition of the Nash equilibrium allow us to rank various asymmetric PSNEs in terms of associated welfare. Consider two asymmetric PSNEs (γ N E1 , γ ∗N E1 ) and (γ N E2 , γ ∗N E2 ), as illustrated in Figure 1a, such that the home country 17 Ethier (2008) highlights a similar result, ”the greater the differences, the greater the gains” in comparative advantage-driven trade.
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ACCEPTED MANUSCRIPT is an exporter of good 1 in both these equilibria (γ N EK > γ ∗N EK , k = 1, 2). The countries are more different in the first PSNE than in the second, γ ∗N E1 < γ ∗N E2 < γ N E2 < γ N E1 . Both countries attain a higher welfare in (γ N E1 , γ ∗N E1 ) than in (γ N E2 , γ ∗N E2 ). Hence, an equilibrium with greater asymmetry increases the welfare of both countries. 18 I describe this property in the
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next proposition. Proposition 2 provides a welfare-ranking of multiple asymmetric PSNEs on any
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given side of the diagonal of the strategy space.
Proposition 2 (Welfare Ranking of Asymmetric PSNEs) Consider two asymmetric PSNEs – (γ N E1 , γ ∗N E1 ) and (γ N E2 , γ ∗N E2 ) such that γ ∗N E1 < γ ∗N E2 < γ N E2 < γ N E1 . Both countries have
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a higher welfare at (γ N E1 , γ ∗N E1 ) than at (γ N E2 , γ ∗N E2 ). Proof. See Appendix.
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The analysis so far provides a characterization of the existence conditions for, and welfare implications of, symmetry-breaking in a non-cooperative policy game in the open economy. However, this analysis has an important shortcoming. Here the necessary and sufficient conditions for the
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existence of an asymmetric PSNE crucially depend on convexity of the aggregate income function with respect to policy, and the aggregate income function is an equilibrium object. In the absence of any more structure on the economy, it is difficult to pinpoint the restrictions on economic fundamentals that ensure convexity of the aggregate income function.
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In order to address this, I next consider an application to education policy in a standard Heckscher-Ohlin model, in which the agents optimally choose their skill. I show that social increas-
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ing return in aggregate production arises from the optimality of agents’ skill choice, even though the technologies are constant returns to scale. This ensures that the necessary condition for the existence of an asymmetric PSNE is always satisfied. I further characterize the parameter restrictions
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sufficient to guarantee the existence of an asymmetric PSNE. In this application, I also show that if the countries are initially asymmetric, such initial asymmetry in autarkic welfare is amplified under free trade, given simple parameter restrictions.
3
An Application to Education Policy
I consider an application of the abstract general policy problem to education policy. Each Government allocates a fixed education budget, financed by lumpsum taxation, between higher and 18
While any asymmetric NE is always better for both countries compared to the autarky optimum, the two countries are not necessarily equally well-off. To see this, consider a common equilibrium price under free trade. Note that at (γ N E , γ ∗N E ) the welfare of the two countries is given by: f (pT )Y (pT , γ N E ) and f (pT )Y (pT , γ ∗N E ). Thus, asymmetry of welfare would require that Y (pT , γ N E ) = Y (pT , γ ∗N E ). While the two countries face the same equilibrium price in free trade, their production is subject to different production possibility constraints under the asymmetric NE. Hence, it is possible that the maximum value of income differs between the two countries.
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ACCEPTED MANUSCRIPT primary education. I consider a simple model of endogenous skill choice by heterogenous agents. There are two types of labor in each country. High types are born with the ability h and low types are born with the ability l, h > l. There are nL low type workers and nH high type workers. A positively skewed skill distribution implies nL > nH .
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Each type of worker chooses a skill level that incurs a cost of education (measured in welfare
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terms). Both the total and marginal cost of education rise in the level of skill and decrease in ability. The cost of education is affected by the allocation of education budget by the government. Let he denote the level of skill chosen by high type workers and le denote the level of skill chosen by low type workers. The total effective endowment of high-skilled and low-skilled labor is given
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by
H e = he nH and Le = le nL .
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As before, u denotes the direct utility function,
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u(C1 , C2 ) = C1µ C21−µ , where µ is the expenditure share of good 1 and Ci is the aggregate consumption of good i. Here, Ci is given by
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L nH cH i + nL ci
L where cH i and ci denote individual consumption of high and low type agents. Good 2 is the
numeraire and pj is the relative price of good 1 in state j = A, T . I assume that factor price
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equalization holds over the entire strategy space at the equilibrium price for the specified set of parameters and functional forms.
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H High types choose skill, he , and consumption, cH 1 and c2 , to maximize utility
he ǫ ) , β h > 0, 1 < ǫ < 2, h H s.t. pj cH ≤ wH he , 1 + c2
H u(cH 1 , c2 ) − β H (
given equilibrium wage, price and educational institutional parameters. Here, wH (wL ) refers to the wage of high and low type workers. Low type workers solve a similar optimization problem. e
The welfare cost of higher education is captured by β H ( hh )ǫ . The total and marginal costs of higher education rise in the skill that the agent acquires, he , and falls in the ability of the agent, h. Note that a fall in β t , t = H, L reduces the cost of education. I define β t as the quality of educational institutions, thus ability of the agents is complementary to the quality of educational institutions in the acquisition of skill. Here, ǫ is the elasticity of the cost of education with respect ′′
to skill. Note that given an ability h, the relative cost of acquiring higher skill
h′′
>
β ( h )ǫ h′ , H hh′ , β H ( h )ǫ
increases in ǫ. I define ǫ as the progressivity of the education system, following the interpretation
in Benabou (2002). The condition ǫ > 1 ensures that the second order condition of optimality of the agents’ skill choice is satisfied.
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ACCEPTED MANUSCRIPT Producers maximize profit given the equilibrium wage, price and effective endowments of skills . Producers of the two goods employ high and low skilled labor with different intensities eη
e1−η i
Fi (Hie , Lei ) = Hi i Li
T
where η i denotes the share of high-skilled labor per unit cost of good i. Good 1 is relatively more
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intensive in high-skilled labor, η 1 > η 2.
The optimal skill choice function, incorporating the equilibrium wage rates obtained from the
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zero profit conditions, is given by
1−a
L
e
1 H ǫ p η1 −η2 ǫ−1 = c h′ ( ) , ch′ = f (η 1 , η 2 , µ, ǫ), a = µη 1 + (1 − µ)η 2 , nH β H
(15)
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H
e
−a
1 Lǫ p η1 −η2 ǫ−1 = cl ′ ( ) , cl′ = f (η 1 , η 2 , µ, ǫ). nL β L
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where H (= hnH ) and L (= lnL ) denote the original endowments of the skills in the economy. In competitive equilibrium all the optimization conditions hold and both labor and goods markets clear as in Section 2.
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Government policy choice, γ, affects the cost of education and hence optimal skill choice (15), as specified below, thus, domestic education policy affects aggregate welfare both directly, via its effect on factor distributions of the economy, and indirectly via its effect on the equilibrium price. As in
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Section 2, aggregate indirect utility in the competitive equilibrium of the economy is a function of
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the equilibrium price and the aggregate income, cpj
−µ
Y (pj , γ).
Aggregate income, Y (pj , γ), is given by wH (pj )H e (γ) + wL (pj )Le (γ), where wL (pj ) = cl pj
η 2 /(η 2 −η 1 )
, wH (p) = ch pj
−(1−η 2 )/(η 2 −η 1 )
.
Here c, ch , cl , and cp denote constants that depend on the economic fundamentals. The equilibrium price, pj , depends only on domestic policy in autarky (pA (γ)) and on both countries’ policy in the open economy (pT (γ, γ ∗ )). This economy is thus an example of the general framework outlined in Section 2. The government has a total education budget T = 1, and chooses a fraction γ ∈ [0, 1] to spend on higher education. The remainder goes to primary education. The government expenditure on higher education, γ, improves the quality of higher education institutions, β H = g(γ), g ′ < 0.
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ACCEPTED MANUSCRIPT Similarly, β L = g(1−γ).19 Higher education and basic education institutions have a baseline quality level (bH and bL ,respectively), and that government policy affects the institutions in a simple linear fashion, = bH − cγ, bH > c > 0,
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β L = bL − c(1 − γ), bL > c > 0.
T
βH
Note that the government always has the option of improving both types of institution equally (γ = .5), but may choose to attach different priorities to different institutions. From the optimal
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skill choice function, (15), the condition ǫ > 1 ensures that H e is a convex function of γ, given pj . Thus when agents choose their skill levels optimally, optimal skill function in the economy is convex in government policy, even though government policy affects education costs in a simple
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linear fashion. This ensures that the necessary condition for symmetry-breaking is satisfied in this framework.
The government takes the optimal response of the agents and market clearing conditions as
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given and allocates the education budget to maximize the aggregate indirect utility by choosing γ. As before, each government in the closed economy chooses γ to maximize aggregate welfare, U A (γ). In the non-cooperative policy problem in the open economy, each government maximizes U T (γ, γ ∗ )
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taking the other country’s policy as given. In the cooperative policy problem in the open economy, the social planner maximizes the world welfare, W (γ, γ ∗ ).
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Note that in this model,
1
1
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1−ǫ ∂β H ∂β L1−ǫ ∂ 2 Y (., γ) = wH (p)sH (p) + wL (p)sL (p) , ∂γ 2 ∂γ 2 ∂γ 2
(16)
where st (p), t = H, L, are given from (15) as: 1−α
sH (p) = p (η1 −η2 )(ǫ−1) , −α
st (p) = p (η1 −η2 )(ǫ−1) .
Given 1 < ǫ < 2, a fall in ǫ increases the convexity of (16) provided bH − c > 1 and bL − c > 1. Hence, if ǫ is sufficiently low, (η 1 − η 2 ) is sufficiently high and consumers do not strongly prefer either of the goods, all PSNE in the open economy are asymmetric. 20 The following proposition summarizes the condition for equilibrium policy diversity in the open economy Proposition 3 If 1 < ǫ ≤ǫ < 2 and bH − c > 1 and bL − c > 1, (η 1 − η 2 ) is sufficiently high and consumers do not strongly prefer either of the goods, only asymmetric PSNEs exist in the open 19
The education system assumptions essentially mean that education is publicly funded and that government decides the type of institution to emphasize. 20 I have formulated the application in terms of a subsidy to education policy, with the government choosing to allocate fixed resources between different categories of education. Equivalently, I can formulate the education problem as the allocation of one unit of lumpsum tax between different skill groups in the form of a wage subsidy, and this alterative formulation leaves the analysis essentially unchanged. In this formulation, the optimization problem of the
18
ACCEPTED MANUSCRIPT economy optimal policy problem. Proof. See Appendix. In this asymmetric Nash equilibrium, these symmetric countries may attain different welfare levels under trade, both of which are higher than their autarky optimum welfare. Thus, while both
T
countries are better off in an asymmetric Nash equilibrium compared to their common autarky
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optimum, the ex-post gains from trade are not necessarily identical among the ex-ante identical countries. I illustrate this in an example in the context of the education budget allocation problem. Suppose that γ A = 1 implying that both countries find it optimal to invest all their resources
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into higher education in autarky, and suppose in an asymmetric Nash equilibrium γ N E1 = 1 and γ ∗N E1 = 0. Thus, in the open economy in a Nash equilibrium one country invests all its resources into higher education and the other country invests all its resources into basic education. To illustrate
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that the ex-post gains from trade are not necessarily identical among the ex-ante identical countries, one needs to find parameter restrictions such that
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U T (γ = 1, γ ∗ = 0) > U T ∗ (γ ∗ = 0, γ = 1) > U A (γ A = 1). I show in the appendix that this is more likely if (η 1 − η 2 ) is sufficiently high and consumers do not strongly prefer either of the goods, similar conditions under which existence of an asymmetric
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Nash equilibrium is more likely. Thus, parameter restrictions that make an asymmetric Nash equilibrium more likely, also increases the possibility of asymmetric welfare outcomes at such a
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Nash equilibrium. If gains from trade are large, a country may prefer to choose the ”bad” policy– here, investing only in basic education–to benefit from such gains, even though it would have preferred to be the country with the ”good” policy–here, investing only in higher education. 21
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A related question to ask in this context is: if these countries are initially asymmetric can they attain proportionally more (than their initial asymmetry) different welfare under asymmetric Nash equilibrium than under autarky? To answer this question, I model the initial asymmetry of countries in the traditional Heckscher-Ohlin way: the countries are otherwise identical except for their initial endowment of skilled labor H. Suppose that the home country is less skill abundant high skill type is reduced to
he ǫ ) }, β h h
>
0, ǫ > 1,
H s.t. pcH 1 + c2
≤
wH g(γ)he ,
H max {u(cH 1 , c2 ) − β H (
H e cH 1 ,c2 ,h
given equilibrium wage, price and educational institutional parameters. Low type workers solve a similar optimization problem. I formulate the wage subsidy schedules as follows: g(γ)
=
b + cγ, b > c > 0,
g(1 − γ)
=
b + c(1 − γ).
In this case, the optimality of skill choice also ensures that the necessary condition for the existence of an asymmetric PSNE is satisfied. The endogenous skill functions (15) and the sufficient conditions for the existence of an asymmetric PSNE, as illustrated in proposition 3, essentially remain unchanged in this alternative formulation of the policy problem. 21 I sincerely thank an anonymous referee for pointing out this possibility to me.
19
ACCEPTED MANUSCRIPT implying that the relative inherent ability (or productivity) of high-skill labour in the home country ∗
( hl ) is lower than that in the foreign country ( hl∗ ) : < 1.
T
κ=
h l h∗ l∗
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This is the typical nature of factor endowment differences assumed in a Heckscher-Ohlin economy with pre-existing comparative advantage. The skill distributions in the two countries are assumed to be the same (nH = n∗H < nL = n∗L ). It is straightforward to show that the optimal choice of policy in autarky, γ A , does not depend on this factor endowment difference. Hence, γ A = γ A∗ , ∗
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though welfare at the autarky optimum is different, U A (γ A ) %= U A (γ A∗ ). Suppose initial welfare asymmetry (at the autarky equilibrium) is captured by
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aǫ U A (γ A ) = κ ǫ−1 < 1. ∗ A A∗ U (γ )
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Let us consider the case where sufficient conditions for asymmetric NE are satisfied and ( γ N E = 0, γ ∗N E = 1) is the NE. If (η 1 − η 2 ) is sufficiently large and
nH nL
>
bH −c bL ,
the relative welfare at the
asymmetric NE is magnified compared to the autarky welfare asymmetry,
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U A (γ A ) U T (γ = 0, γ ∗ = 1) < . U T ∗ (γ ∗ = 1, γ = 0) U A∗ (γ A )
PT
The application to education policy in this section provides a setting in which aggregate income is convex in government policy. This leads to an asymmetric equilibrium policy choice for symmetric countries to encourage trade. In the presence of standard constant returns to scale technology in a
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Heckscher-Ohlin set-up, this social increasing return to policy arises as a result of the endogenous skill choice of agents. Other examples of frameworks and policy problems that may generate sufficient convexity in the aggregate income function and equilibrium policy diversity include an economy with Grossman-Maggi (2000) production structure.
4
22
Conclusion
In this paper, I show that similar countries may choose different policies because these policies allow them to specialize in different industries, and therefore gain from international trade. I find that even identical countries may optimally choose different policies in an open economy, and b oth 22
In Chatterjee (2012) I consider an alternative specification which shares the production structure of Grossman and Maggi (2000) but has a similar specification of consumer preferences and government policy as before. For simplicity, I assume that allocation of education budget by the government directly affects the effective endowment of skill. The submodular production technology implies that even when government policy affects skills in a linear fashion, aggregate production and income can be sufficiently convex in policy, and this gives rise to symmetrybreaking in optimal education policy in the open economy. Thus, in this case an additional convexity introduced by the endogenous skill choice problem of agents, is not necessary for convexity of aggregate income with respect to policy.
20
ACCEPTED MANUSCRIPT of these symmetric countries may gain in aggregate welfare in any asymmetric Nash equilibrium compared to the autarky optimum. In an application in the competitive economy, an asymmetric equilibrium arises if education policy affects the determinants of trade, namely effective endowments of skill, in a strong convex fashion. I construct a model in which agents optimally choose their skill
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levels given government policy, and show that the optimal skill is a convex function of government
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policy giving rise to social increasing returns to the policy.
A natural extension of the pure welfare maximizing optimal policy problem is to allow governments to also care about redistributive equity. In the presence of redistributive concerns, it is no longer true that both countries attain higher social welfare at an asymmetric PSNE compared to
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any symmetric PSNE. Also, allowing for endogenous comparative advantage in a political economy framework, the political preferences of the government can be a source of comparative advantage
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in an open economy. Exploring the implications for an asymmetric policy choice among nearly identical economies in a more complete political economy setting is an important part of my future research agenda.
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This paper outlines a general mechanism that applies to many different policies which potentially affect comparative advantage in open economies. For any particular application, it is important to model the domestic economic environment carefully. For example, education policy is an important policy for encouraging trade, but there are also several reasons why education policy is important
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for the domestic economy. Since future human capital is typically not accepted as collateral, the availability of credit for financing educational expenditure is limited. This aspect of the education policy has received attention in both trade and macroeconomic policy literature (Benabou, 2002;
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Ranjan, 2001; Chesnokova and Krishna, 2009). Also, skills learned in the earlier stages of academic development are complementary to the acquisition of advanced skills. In future work, I intend
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to incorporate these aspects of education in a more complete application, to study the interplay between the domestic and international motives of optimal education policy.
21
ACCEPTED MANUSCRIPT Acknowledgement This paper has benefitted immensely from the very helpful comments of the Editor, Giovanni Maggi, and two anonymous referees. This paper is based on the first chapter of my Ph.D. thesis, ”Why Do Similar Countries Choose Different Policies? Endogenous Comparative Advantage and Welfare Gains.” I am indebted to Gene Grossman and Esteban Rossi-Hansberg for
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invaluable guidance and encouragement. I would like to thank Rabah Amir, Roland Benabou, Saroj
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Bhattarai, Varanya Chaubey, Arnaud Costinot, Avinash Dixit, Arghya Ghosh, Richard Holden, Kala Krishna, Devashish Mitra, Vinayak Tripathi, Robert Staiger, Alan Woodland, and participants of various workshops and seminars for their comments and suggestions. I thank the International Economics Section (Princeton University), Social Policy Program (Woodrow Wilson School), and
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UNSW Business School (University of New South Wales) for their financial support, and Indian Statistical Institute (Calcutta) for their hospitality during my research visits. All remaining errors
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PT
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are mine.
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ACCEPTED MANUSCRIPT References [1] Acemoglu, D., J. Robinson and T. Verdier, ”Asymmetric growth and institutions in an interdependent world (2015), forthcoming Journal of Political Economy.
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[2] Alesina, A., and G. M., Angeletos, ”Fairness and redistribution”, The American Economic
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ACCEPTED MANUSCRIPT [15] Bougheas, S., R. Kneller and R. Riezman, ”Optimal education policies and comparative advantage”, CESifo Working Paper No. 2631 [16] Bougheas, S., and R. Riezman, ”Trade and the distribution of human capital”, Journal of
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ACCEPTED MANUSCRIPT [29] Deardorff, A., ”International externalities in the use of domestic policies to redistribute income ”, (1997), Discussion Paper, vol.405. University of Michigan. SSRN Electronic Journal. doi:10.2139/ssrn.46825 [30] Deardorff, A., ”Growth or decline of comparative advantage”, Journal of Macroeconomics 38
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[31] Dixit, A. K., and V., Norman, 1980. Theory of International Trade: A Dual, General Equilibrium Approach. Cambridge Economic Handbook Series, Cambridge University Press.
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[32] Ethier, W., ”The greater the differences, the greater the gains?”, University of Pennsylvania, mimeo (2007).
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[33] Ethier, W., ”National and international returns to scale in the modern theory of international trade”, American Economic Review, Vol 72, No 3 (1982a), pp. 389-405. [34] Ethier, W., ”Decreasing costs in international trade and Frank Graham’s argument for pro-
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tection”, Econometrica, Vol 50, No 5 (1982b), pp. 1243-1268. doi:10.2307/1911872 [35] Grossman, G., and G., Maggi, ”Diversity and trade”, American Economic Review, Vol. 90,
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No. 5, (2000), pp. 1255-1275. doi:10.1257/aer.90.5.1255 [36] Grossman, G., and E., Rossi-Hansberg, ”External economies and international trade redux”, Quarterly Journal of Economics, (2010), pp. 1255-1275. doi:10.1162/qjec.2010.125.2.829
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[37] He, J., and L., Kuijis, ”Rebalancing China’s economy– modeling a policy package”, World Bank China Research Paper No. 7, (2007).
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[38] Human Resource Development in Information Technology (2000), challenge
and
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of
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http://www.education.nic.in/uhe/uhe-reports.asp. [39] Krueger, D., and K. B., Kumar, ”Skill-specific rather than general education: A reason for USEurope growth differences?”, Journal of Economic Growth, Vol. 9, No. 2, (2004), pp. 167-207. doi:10.1023/b:joeg.0000031426.09886.bd [40] Levchenko, A., ”Institutional quality and international trade”, Review of Economic Studies, Vol. 74, Issue 3, (2007), pp. 791 - 819. doi:10.1111/j.1467-937x.2007.00435.x [41] Lyn, G., and A. Rodriguez-Clare, ”External economies and international trade redux: Comment”, Quarterly Journal of Economics (2013), 1895-1905. doi:10.1093/qje/qjt017 [42] Matsuyama, K., ”Explaining diversity: symmetry-Breaking in complementarity games”, American Economic Review,
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ACCEPTED MANUSCRIPT [43] Matsuyama, K., ”Financial market globalization, symmetry-breaking, and endogenous Inequality of Nations”, Econometrica, Vol. 72, No. 3, (2004), pp. 853-884. doi:10.1111/j.14680262.2004.00514.x [44] Human Capital Priority Program of Ireland’s National Development Plan (2007), National
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Skills Strategy Research Report of Ireland (http://www.skillsstrategy.ie/). [45] Panagariya, A., ”India and China: Trade and foreign investment”, presented at the Pan Asia 2006 Conference, Stanford Center for International Development, June 1-3, (2006).
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[46] Ranjan, P., ”Dynamic evolution of income distribution and credit-constrained human capital investment in open economies”, Journal of International Economics, Vol. 55, Issue 2, (2001),
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pp. 329-358. doi:10.1016/s0022-1996(01)00103-9
[47] Rodrik, D., ”Second-best institutions”, American Economic Review, 98(2), (2008), 100-104. doi:10.1257/aer.98.2.100
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[48] Topkis, D., ”Minimizing a submodular function on a lattice”, Operations Research, Vol. 26,
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PT
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No. 2, (1978), pp. 305-321. doi:10.1287/opre.26.2.305
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ACCEPTED MANUSCRIPT 5
Appendix
5.1
Derivation and Proofs for Section 2
T
From homothetic demand, aggregate demand for good i, Ci , is linearly homogenous in income,
Here, I is aggregate income,
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I = pQ1 + Q2 ,
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Ci = fi (pj ) ∗ I.
where Qi stands for aggregate production of good i. The aggregate indirect utility is given by
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V = u(f1 (pj ) ∗ I, f2 (pj ) ∗ I) = If (pj ),
MA
where f (p) = u(f1 (p), f2 (p)). From Roy’s identity
f ′ (pj )I = −C1 = −f1 (p)I. f (pj )
(17)
ED
The marginal effect of p on indirect utility is given by
PT
f ′ (pj )I + f (pj )Q1 + f (pj )(pj
From (17),
−dQ2 dQ1
AC CE
The condition p = M RT =
f (pj )(Q1 +
dQ1 dQ2 + ). dp dp
implies that the marginal effect of p on indirect utility is f (pj )(Q1 +
f ′ (pj ) )I). f (pj )
f ′ (pj ) I) = f (pj ) ∗ (Q1 − f1 (pj )I) = f (pj ) ∗ (Q1 − C1 ). f (pj )
Hence from (5), the marginal effect of p on indirect utility is zero in autarky. Assume that the policy instrument γ provides absolute advantage in good 1 and absolute disadvantage in good 2. Since the total value of consumption equals income, f1 (pj ) ∗ pj + f2 (pj ) = 1.
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Proof of Proposition 1. This proof is completed in three steps. Step 1: Uniqueness and first order optimality of (possible) symmetric Pareto optimum and Nash equilibrium: The symmetric strategy profile in which both countries choose the autarky optimum, (γ A , γ A ), is the only candidate for a symmetric Pareto optimum and for a symmetric PSNE. If γ A is an interior autarky optimum, (γ A , γ A ) satisfies the first order neces27
ACCEPTED MANUSCRIPT sary condition of local optimality for W (γ, γ ∗ ), and γ A satisfies the first order necessary condition ∗
of local optimality for U T (γ, γ A ) w.r.t. γ and for U T (γ A , γ ∗ ) w.r.t. γ ∗ . There are three results to prove in this step. First, (γ A , γ A ) is the only possible symmetric Pareto optimum.The result that (γ A , γ A ) is the
T
only candidate for a symmetric Pareto optimum directly follows from the symmetry of the setup
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which ensures that W (γ, γ) = 2U A (γ), and the optimality of γ A in autarky. Second, we need to prove that (γ A , γ A ), is the only candidate for a symmetric PSNE. Suppose ! ) by deviating to γ A , that (! γ, γ !), γ ! %= γ A is a PSNE. The home country attains a payoff of U T (γ A , γ given that the foreign country chooses γ !. The resulting change in payoff (U T (γ A , γ ! ) − U T (! γ, γ !))
SC
! ) − U A (γ A )) and an increase in autarky utility consists of a gain from trade component ((U T (γ A , γ
(U A (γ A ) − U A (! γ )). Hence, the autarky optimum, γ A , is a profitable unilateral deviation from γ !, if the other country chooses γ !. Thus (! γ, γ !), γ ! %= γ A cannot be a PSNE.
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Third, if γ A is an interior autarky optimum, (γ A , γ A ) satisfies the first order necessary con-
dition of local optimality for W (γ, γ ∗ ), and γ A satisfies the first order necessary condition of local ∗
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optimality for U T (γ, γ A ) w.r.t. γ and for U T (γ A , γ ∗ ) w.r.t. γ ∗ . To see this, note that if γ A is an interior autarky optimum, the first order condition (FOC) of maximization must be satisfied at γ A . Note that in equilibrium
From the definition of U A (.),
ED
I = Y (pj , γ) = pj F1 (pj , γ) + F2 (pj , γ).
AC CE
From Roy’s identity,
PT
dpA f ′ (pA ) ∂Y dU A (.) )Y ) + f (pA ) = f (pA )(F1 + . dγ dγ f (pA ) ∂γ
f (pA )(F1 +
f ′ (pA ) )Y ) = f (pA )(F1 (pA , γ) − C1 ), f (pA )
which equals zero from (5). Therefore, dU A (.) ∂Y = f (pA ) = 0 at γ A from the FOC. dγ ∂γ
Similarly, from the definition of W (.) and using (6), ∂Y ∂W (.) = f (pA ) . ∂γ ∂γ Therefore, the FOC of optimization of the world welfare is satisfied at (γ A , γ A ). If γ A is an interior autarky optimum, γ A satisfies the first order necessary condition of local
28
ACCEPTED MANUSCRIPT ∗
optimality for U T (γ, γ A ) w.r.t. γ and for U T (γ A , γ ∗ ) w.r.t. γ ∗ . The expression for f (pT )(F1 ((pT , γ)) − C1 )
∂U T (γ, γ A ) ∂γ
is
∂pT ∂Y + f (pT ) ∂γ ∂γ
T
At a symmetric strategy, excess demand of good 1, (F1 − C1 ), is zero and the equilibrium free trade economy is essentially autarky. Hence, at (γ A , γ A ) pT (γ, γ) = pA (γ)
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price is the same as the autarky equilibrium price, since a symmetric strategy choice in the open
NU
SC
F1 (pT (γ, γ), γ) − C1 = 0, and ∂U T (γ, γ A ) ∂Y ∂W (γ, γ A ) = = f (pj ) . ∂γ ∂γ ∂γ Step 2: Existence of an asymmetric Nash equilibrium and the second order properties of the payoff functions: If γ A is an interior autarky optimum and if W (γ, γ ∗ ) is globally
MA
strictly concave, (γ A , γ A ) is the unique maximum of W (γ, γ ∗ ) and an asymmetric PSNE does not exist. If γ A is an interior autarky optimum and U T (γ, γ ∗ ) is quasiconvex in its own strategy ( γ) at (γ A , γ A ), there is only asymmetric Nash equilibrium in the open economy . If U T (γ, γ ∗ ) is
ED
strictly quasiconvex in own strategy ( γ), a unique pair of asymmetric NE is given by the extremes (γ, γ).and (γ, γ).
There are two results to prove in this step.
PT
First, if γ A is an interior autarky optimum and if W (γ, γ ∗ ) is globally strictly concave , (γ A , γ A ) is the unique maximum of W (γ, γ ∗ ). In that case, there is a unique symmetric Pareto optimum in which both countries choose the autarky optimal policy, γ A . If an asymmetric PSNE exists, it
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is a Pareto improvement over the autarky optimum, thus if W (γ, γ ∗ ) is strictly concave, a Pareto improvement over (γ A , γ A ) is not possible. If W (γ, γ ∗ ) is strictly concave, an asymmetric PSNE does not exist.
Second, if U T (γ, γ ∗ ) is quasiconvex in own strategy ( γ) at (γ A , γ A ), the symmetric strategy (γ A , γ A ) is a local minimum of U T (γ, γ ∗ ). This implies that the home country government has a ′
unilateral profitable deviation to a γ %= γ A if the Foreign government chooses γ A . Hence, in this case, (γ A , γ A ) is not a PSNE and a symmetric PSNE does not exist. Any Nash equilibrium, in this case, must be asymmetric. If U T (γ, γ ∗ ) is strictly quasiconvex in own strategy ( γ), there is a unique pair of asymmetric NE given by the extremes (γ, γ).and (γ, γ). From the definition of strict quasiconvexity, U T (λγ 1 + (1 − λ)γ 2 ) < max{U T (γ 1 ), U T (γ 2 )}, ∀γ 1 , γ 2 ∈ [γ, γ]2 . where γ 1 = (γ, γ) and γ 2 = (γ, γ). Any γ ′ ∈ (γ, γ) yields a lower pay-off than γ ′ = γ or γ ′ = γ, for any given foreign policy γ ∈ [γ, γ]. Hence, ∀γ ∈ [γ, γ], the best response is either γ or γ. Neither γ nor γ is an autarky optimum. Therefore, γ (γ) is not the best response to γ (γ), and the best
29
ACCEPTED MANUSCRIPT response to γ is γ and vice versa. The only two PSNEs are (γ, γ) and (γ, γ). Step 3: Existence of asymmetric Nash equilibrium and social increasing return : 2 (pT ,γ) If ∂ Y∂γ < 0, W (γ, γ ∗ ) 2 2 T ∂ Y (p ,γ) |(γ A ,γ A ) is positive ∂γ 2
is strictly concave in (γ, γ ∗ ) and an asymmetric NE does not exist. if 2 T (p ,γ) ∂pT ∂ 2 Y (pT ,γ) and sufficiently large ( ∂ Y∂p∂γ |(γ A ,γ A ) ), U T (γ, ∂γ A A < ∂γ 2 (γ ,γ )
T
γ ∗ ) is quasiconvex in γ at (γ A , γ A ) even though U A (γ) is quasiconcave at γ A , and all Nash equilibria
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in the open economy are asymmetric.
The sufficient condition for equilibrium asymmetry in step 2 requires U T (γ, γ ∗ ) to be quasiconvex in γ at (γ A , γ A ), even though γ A is an interior autarky optimum. Let us consider the difference
SC
between the curvatures of U T (γ, γ ∗ ) at (γ A , γ A ) and U A (γ) at γ A . Note that the curvature of U A (γ) is given by,
2 2 A dpA ′ A ∂Y d2 U A (γ) A ∂ Y A ∂ Y (p , γ) (f = (p ) ) ) . + f (p ) + f (p dγ dγ 2 ∂γ ∂p∂γ ∂γ 2
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First, note that,
MA
∂ 2 Y (pA , γ) ∂F1 . = ∂γ ∂γ∂p
Since γ gives absolute advantage in good 1 and absolute disadvantage in good 2,
Hence,
∂ 2 Y (pA ,γ) ∂γ∂p
ED
δF2 /δγ < 0, δF1 /δγ > 0.
0.
d2 U A (γ) dγ 2
PT
At the interior autarky optimum γ A ,
∂Y ∂γ
∂ 2 Y (pA ,γ) |γ A < 0, ∂γ 2 2 Also, ∂∂γY2 < 0 implies
= 0. Hence, if
|γ A < 0 and γ A is the unique interior optimum.
it implies that that the world
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welfare function is strictly concave and that there is a unique Pareto optimum, and there is no asymmetric PSNE in the non-cooperative policy game. This proves the necessary condition for the existence of an asymmetric PSNE. In fact, even if
∂ 2 Y (pA ,γ) ∂γ 2
> 0, as long as
∂ 2 Y (pT ,γ) |(γ A ,γ A ) ∂γ 2
2 T (p ,γ) dpA < ∂ Y∂p∂γ dγ
(γ A ,γ A )
γ A is the
unique interior optimum. In the non-cooperative policy problem, at (γ A , γ A )
At (γ A , γ A ), if
∂ 2 Y (pT ,γ) |(γ A ,γ A ) ∂γ 2
∂2U T ∂ 2 Y ∂pT ∂2Y j + = f (p )( ). ∂p∂γ ∂γ ∂γ 2 ∂γ 2 2 T (p ,γ) ∂pT > ∂ Y∂p∂γ ∂γ
(γ A ,γ A )
ensures that
(20) ∂2U T ∂γ 2
is positive. This ensures
that (γ A , γ A ) is a local minimum. Hence, an asymmetric PSNE must exist. Note that at (γ A , γ A ), ∂pT ∂γ
A
T A = .5 dp dγ and p (γ, γ) = p (γ). It is possible that
2 ∂ Y (pT , γ) ∂pT ∂p∂γ ∂γ
(γ A , γ A )
∂ 2 Y (pT , γ) < |(γ A , ∂γ 2
30
γA)
2 ∂ Y (pT , γ) dpA < ∂p∂γ dγ
(γ A , γ A )
.
ACCEPTED MANUSCRIPT Sufficient convexity in production with respect to the policy in question ensures that there are only asymmetric equilibria in the open economy, even though γ A is an interior autarky optimum. Note that here it is crucial that at a point of symmetry, the level of autarky and trade prices are the same, but the partial derivative of price w.r.t. to own policy is halved in trade compared to
T
autarky–trade lessens the effect of own policy on equilibrium price. This is crucial in allowing the
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signs of second order conditions to differ between trade and autarky, even at a point of symmetric policy choice. Steps 1, 2 and 3 combined complete the proof of proposition 1. Proof of Proposition 2.
Step 1: Since γ ∗ affects the welfare of the home country through
SC
terms-of-trade,
∂pT ∂U T (γ, γ ∗ ) T = f (p )(F (.) − C ) . 1 1 ∂γ ∗ ∂γ ∗
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Since an increase in γ confers comparative advantage in good 1,
MA
∂pT (γ, γ ∗ ) < 0. ∂γ ∗ Suppose γ ∗ > γ, which implies that the home country is an importer of good 1,
advantage in good 2.
∂U (γ, γ ∗ ) δγ ∗
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Step 2: Given γ ∗1 < γ ∗2 < γ 2 and
but
∂U (γ, γ ∗ ) > 0. δγ ∗
< 0. A similar proof works if an increase in γ confers comparative
PT
Similarly, γ ∗ < γ implies
ED
F1 < C1 , and
∂U T (γ,γ ∗ ) ∂γ ∗
< 0 for γ > γ ∗ ,
U T (γ 2 , γ ∗2 ) < U T (γ 2 , γ ∗1 ),
U T (γ 2 , γ ∗1 ) < U T (γ 1 , γ ∗1 )
since γ 1 is the best response to γ ∗1 . Similarly, given γ ∗2 < γ 2 < γ 1 and
∂U T (γ ∗ , γ) ∂γ
> 0 for γ ∗ < γ,
U T (γ ∗2 , γ 2 ) < U T (γ ∗2 , γ 1 ). But γ ∗1 is the best response to γ 1 , which implies that U T (γ ∗2 , γ 1 ) < U T (γ ∗1 , γ 1 ).
5.2
Derivation and Proofs for Section 3
The aggregate indirect utility is a function of the equilibrium price and the aggregate income, 31
ACCEPTED MANUSCRIPT
cpj
−µ
Y (p, γ).
Aggregate income, Y (p, γ), is given by wH (p)H e (γ) + wL (p)Le (γ), where , wH (p) = ch pj
−(1−η 2 )/(η 2 −η 1 )
.
T
η 2 /(η 2 −η 1 )
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wL (p) = cl pj
Here c, ch , cl , and cp denote constants that depend on the economic fundamentals. Let us first derive (15). A more able agent with initial ability h chooses he by maximizing
1−η 2 η 1 −η 2
wH (pj )he − β h (
he ǫ ), h
. From the FOC, one can derive
NU
where wH (pj ) = ch pj
−µ
SC
pj
1−a
ch
ǫβ h
1
hǫ ) ǫ−1 .
MA
he = (
η −η pj 1 2
Multiplying both sides by nH we arrive at (15). The second order condition of optimality requires ǫ > 1. The optimal skill-choice problem of the low-skilled agents can be solved similarly.
AC CE
L
e
1−a η −η 2
H ǫ pj 1 = ch′ ( nH β H
PT
He
ED
The optimal skill choice function is given by
1
) ǫ−1 , a = µη 1 + (1 − µ)η 2 ,
−a η −η 2
Lǫ pj 1 = cl ′ ( nl β L
(21)
1
) ǫ−1 , H = nH h, L = nl l.
where ch′ and cl′ are functions of η 1 , η 2 , µ and ǫ. Note that 1−a η −η 2
ch pj 1 ∂H e = −( ∂γ ǫnH
and
∂2H e ∂γ 2
1−a
ch p ǫnH
provided
− 1 −1 ∂β h 1 >0 )β h ǫ−1 ∂γ ǫ−1 −
is equal to
j η1 −η2
(
1
hǫ ) ǫ−1 (
1 1 1 )( + 1)β h hǫ ) ǫ−1 ( ǫ−1 ǫ−1
∂2βh ∂γ 2
1 −2 − ǫ−1
(
1−e η 1 −η 2
ch p ∂β h 2 ) −( ∂γ ǫ −
1
hǫ ) ǫ−1 (
1 )β ǫ−1 h
1 −1 − ǫ−1
2
∂ βh > 0, ∂γ 2
≤ 0. A similar result holds for Le .
In the competitive equilibrium of the closed economy, only domestic policy determines the equilibrium price, pA = cp (
H e η2 −η1 ) , Le 32
ACCEPTED MANUSCRIPT and in the open economy both domestic and foreign policy determine the equilibrium terms-oftrade, e
pT = (
H e + H ∗ η2 −η1 ) cp (µ, η 1 , η 2 ). Le + Le∗
pT (γ, γ ∗ ) = {
∗
1
β h1−ǫ + β h 1−ǫ 1 1−ǫ
βl
+
∗ 1 β l 1−ǫ
}
(η 1 −η 2)(1−ǫ) ǫ
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1
T
The equilibrium price in the open economy is given by,
(22)
c′p (η 1 , η 2 , µ, H, L, nH , nL ),
SC
where cp denotes a constant that does not depend on policy. Welfare in the open economy is given by e
e
He + H ∗ a e He H e + H ∗ −1 ) (L + ) ), ( Le + Le∗ qf (µ, η 1 , η 2 ) Le + Le∗
NU
U T = vq a (
MA
where
(23)
a = η 2 + µ(η 1 − η 2 ), (1 − µ)f1 + µf2 qf = , µf2 (η 1 /(1 − η 1 )) + (1 − µ)f1 (η 2 /(1 − η 2 ) = (((1 − η 1 )/(1 − η 2 ))(η 2 /(1 − η 2 ))−η2 (η 1 /(1 − η 1 ))η1 )1/(η2 −η1 ) ,
ED
f
f2 = (η 2 /(1 − η 2 ))η2 f η2 −1 /(η 1 /(1 − η 1 ) − η 2 /(1 − η 2 )),
PT
f1 = (η 1 /(1 − η 1 ))η1 f η1 −1 /(η 1 /(1 − η 1 ) − η 2 /(1 − η 2 )),
AC CE
v = (1 − η 1 )(η 1 /(1 − η 1 ))η1 f η1 . are constants depending on the production and demand parameters. Welfare in the closed economy is given by
1 1 He a e ) (L + ) e qf (µ, η 1 , η 2 ) Le L 1 )(H e )a (Le )1−a . = vq a (1 + qf
U A = vq a (
It is straightforward to show that both the autarky optimal policy γ A and open economy best response correspondence BR(γ ∗ ) are increasing in µ, η i . Proof of Proposition 3.
From proposition 1 U T (γ, γ ∗ ) is quasiconvex in γ, if
∂ 2 Y (.,γ) ∂γ 2
is
sufficiently large. Following proposition 1 we consider the case where ∂ 2 Y (pT , γ) ∂pT ∂p∂γ ∂γ
< (γ A ,γ A )
∂ 2 Y (pT , γ) ∂ 2 Y (pT , γ) dpA | < A A (γ ,γ ) ∂γ 2 ∂p∂γ dγ
. (γ A ,γ A )
In this case, given Proposition 1 there are only asymmetric PSNEs at the extremes, even though there is an interior γ A . Note that 33
ACCEPTED MANUSCRIPT
∂pT (γ, γ ∗ ) ∂γ
1
1
1
(η 1 − η 2)(1−ǫ) = c′p ǫ
∂{
T
∂β 1−ǫ ∂β 1−ǫ ′ ′ (p)sH (p) + wH (p)s′H (p)) H + (wL (p)sL (p) + wL (p)s′L (p)) L , = (wH ∂γ ∂γ 1
∗ 1
1
1 ∗ 1−ǫ
β h1−ǫ +β h 1−ǫ β l1−ǫ +β l
}
1
.{
∂γ
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∂ 2 Y (pT , γ) ∂p∂γ
1
1−ǫ ∂2βH ∂ 2 β L1−ǫ = wH (p)sH (p) + w (p)s (p) , L L ∂γ 2 ∂γ 2
∗
1
β h1−ǫ + β h 1−ǫ 1 1−ǫ
βl
+
∗ 1 β l 1−ǫ
}
(η 1 −η 2)(1−ǫ) ǫ
−1
.
SC
∂ 2 Y (., γ) ∂γ 2
∂ 2 Y (pT ,γ)
part A
is given by
2ǫ−1
2ǫ−1
wH (p)sH (p)β H1−ǫ + wL (p)sL (p)β L1−ǫ 2 (pT ,γ) | ∂ Y∂p∂γ |
1 ∗ 1 β 1−ǫ +β 1−ǫ h ∂{ h 1 ∗ 1 β 1−ǫ +β 1−ǫ l l
!
∂γ
1
{
∗ 1
β h1−ǫ +β h 1−ǫ 1 ∗ 1 β l1−ǫ +β l 1−ǫ
}
.
(η 1 −η 2)(1−ǫ) ǫ
"#
part B
−1
$
ED
We assume β H , β L , and
}
MA
ǫ2 − 1$ !ǫ "#
dpA dγ
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Given all the other parameters and
2 γ A , ∂ 2 Y (p∂γ T ,γ) ∂p∂γ
1
∗ 1
1
1 ∗ 1−ǫ
β h1−ǫ +β h 1−ǫ β l1−ǫ +β l
> 1 for any choice of γ by imposing parameter restrictions
PT
on bH , bL and c such that bH − c > 1 and bL − c > 1.This ensures that part B decreases in ǫ, and ǫ < 2 ensures that part A declines in ǫ. Hence, a fall in ǫ increases
∂ 2 Y (pT ,γ) 2
2 ∂γ ∂ Y (pT ,γ) ∂p∂γ
dpA dγ
if ǫ < 2 and
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bH − c > 1 and bL − c > 1. This means that for sufficiently small ǫ the objective function of the government in the open economy is quasiconvex in γ. From the first order condition for an interior optimum of autarky welfare, in order to keep γ A unchanged with a fall in ǫ , one needs to increase a(= η 2 + µ(η 1 − η 2 )). This is because, for given A
p,
∂( dU ) dγ ∂ǫ
A
< 0 and
∂( dU ) dγ ∂a
> 0. Thus, for sufficiently small ǫ, avoiding extreme preferences (µ close
to 1 or 0) and sufficiently different production processes (η 1 − η 2 sufficiently high) ensures that an interior autarky optimum exists and neither γ = 1 nor γ = 0 is a dominant strategy in the open economy. Hence, together a more progressive education system (sufficiently small ǫ), diversified preferences (an intermediate value of µ i.e. neither close to 0 or close to 1) and a large enough differences in skill intensities (η 1 − η 2 sufficiently high) ensure the existence of asymmetric PSNEs even though there is a unique interior autarky optimum. To illustrate that the ex-post gains from trade are not necessarily identical among the ex-ante identical countries, one needs to find parameter restrictions such that U T (γ = 1, γ ∗ = 0) > U T ∗ (γ ∗ = 0, γ = 1) > U A (γ A = 1).
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ACCEPTED MANUSCRIPT Let us denote H = H e (γ = 1, γ ∗ = 0) H = H e (γ = 0, γ ∗ = 1)
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L = Le (γ = 1, γ ∗ = 0).
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L = Le (γ = 0, γ ∗ = 1)
Thus, at NE (γ = 1, γ ∗ = 0) the Home country’s endowments are H and L, and the foreign country’s endowments are H and L. Evidently, H > H and L > L.
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For unequal welfare at the asymmetric Nash equilibrium under trade, namely U T (γ = 1, γ ∗ = 0) > U T ∗ (γ ∗ = 0, γ = 1), the parameter restrictions are that
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L+L H −H > qf. H +H L−L
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For the country with lower welfare under the asymmetric Nash equibirum to still gain from trade, namely U T ∗ (γ ∗ = 0, γ = 1) > U A (γ A = 1), one needs that
If
H+H L L+L H
H + H L aL 1 H L+L 1 ). ) (1 + ) > (1 + L qf qf L+L H L H +H
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(
> 1 and a(= η 2 +µ(η 1 −η 2 )) is sufficiently large, the above restriction can be satisfied.
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To briefly see a case of magnification of equilibrium welfare under trade compared to autarky when countries are initially different, consider the countries are otherwise identical except for their initial endowment of skilled labor H. Suppose that the home country is less skill abundant implying
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that the relative inherent ability (or productivity) of high-skill labour in the home country ( hl ) is ∗
lower than that in the foreign country ( hl∗ ) :
Note that
κ=
U A = vq a (1 + Hence, the solution to
dU A dγ
h l h∗ l∗
< 1.
1 )(H e )a (Le )1−a . qf ∗
= 0 does not depend on ( hl ) or ( hl∗ ). The interior autarky optimum is
the same across the two countries, but the optimal value of welfare in autarky is different. ∗
γA = γA aǫ U A (γ A ) = κ ǫ−1 < 1. ∗ A A U (γ ) However, in this case, in the open economy the choice of the countries’ policies depend on their initial
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ACCEPTED MANUSCRIPT natural comparative advantage. In particular, κ =
h l h∗ l∗
< 1 implies that in the Nash equilibrium each
country invests more in its area of natural comparative advantage, home country invests relatively more in basic education and Foreign country invests more in higher education. Let us consider the most extreme NE which aligns with their natural comparative advantage, in which the home ∗
aǫ U T (0, 1) < κ ǫ−1 , ∗ T U (1, 0)
>
bH −c bL .
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nH nL
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PT
ED
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if (η 1 − η 2 ) is relatively large and
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less welfare in the home country relative to the autarky implying
36
T
country chooses γ T = 0 and Foreign country chooses γ T = 1. Such a NE leads to proportionately
S0022-1996(17)30113-7 doi: 10.1016/j.jinteco.2017.08.009 INEC 3078
To appear in:
Journal of International Economics
Received date: Revised date: Accepted date:
1 December 2013 21 June 2017 29 August 2017
Please cite this article as: Chatterjee, Arpita, Endogenous Comparative Advantage, Gains From Trade and Symmetry-Breaking, Journal of International Economics (2017), doi: 10.1016/j.jinteco.2017.08.009
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ACCEPTED MANUSCRIPT
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Symmetry-Breaking
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Endogenous Comparative Advantage, Gains From Trade and
Arpita Chatterjee∗ University of New South Wales
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June 2017
Abstract
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Similar countries often choose very different policies and specialize in very distinct industries. This paper proposes a novel mechanism to explain policy diversity between similar countries from an open economy perspective. I study optimal policies in a two country model where policies affect the determinants of trade patterns. Symmetric countries choose asymmetric policies in equilibrium if aggregate income is sufficiently convex in own policy. Such high social increasing return to policy, despite constant returns to scale in production, generate enough gains from trade to sustain policy diversity in equilibrium. Any asymmetric equilibrium is welfare-improving compared to the common autarky optimum. Similarly, a more asymmetric Nash equilibrium Pareto dominates a less asymmetric one. As an application, I consider a model in which the government’s choice of education policy affects skill distribution and comparative advantage. In this application, the existence of an asymmetric Nash equilibrium in education policy requires that the elasticity of the education cost of agents with respect to skill is relatively low. (JEL Classification: F11, E62.) Keywords: Symmetry-breaking, Endogenous comparative advantage, Gains from trade, Education policy.
∗
School of Economics, UNSW Business School, University of New South Wales, Sydney, Australia. (Email : [email protected], Tel : +61293854314).
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Introduction
Why do similar countries choose very different domestic policies? 1 Existing theories (Benabou and Tirole, 2006; Alesina and Angeletos, 2005) explain this by modeling the optimal policy problem in a closed economy setting. They rely on coordination failure as the reason for policy diversity.
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However, studying each country separately, coordination failure cannot rule out the similarity of
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equilibrium policies as an equally plausible outcome. This is why Matsuyama (2002, page 241) notes that “coordination failure offers no compelling reason” for why we should expect to observe diversity across space, time, or groups.
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Matsuyama (2002, 2004) instead proposes symmetry-breaking – the existence of an asymmetric equilibrium in a symmetric setting– as an explanation for observed diversity. The logic of symmetrybreaking relies on modeling interdependence between the relevant spaces, times, or groups. In is the only stable equilibrium outcome.
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Matsuyama’s (2002, 2004) symmetric strategic setup, symmetry-breaking implies that asymmetry In this paper, I apply the logic of symmetry-breaking to explain why similar countries may
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choose different domestic policies in equilibrium. I model the policy problem in an open economy where two symmetric countries choose a domestic policy that affects their comparative advantage. I show that diversity in equilibrium policy arises because it allows countries to gain from international
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specialization and trade. 2 This provides an alternative answer for why similar countries choose different policies, without relying on coordination failure. In the optimal policy problem, however, any symmetric equilibrium is also stable. Hence, I
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cannot rely on Matsuyama’s (2002) argument of instability in symmetric equilibrium to ensure equilibrium asymmetry. Instead, I offer necessary and sufficient conditions for symmetry-breaking in equilibrium policy by relying on social increasing returns to policy. Under these conditions, the
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open economy equilibrium exhibits a stronger form of symmetry-breaking by guaranteeing that no symmetric Nash equilibrium exists, not just that any symmetric equilibrium is unstable. This sufficient condition is a close parallel to the sufficient condition in Amir, Garcia, and Knauff (2010) which similarly ruled out the existence of any symmetric equilibrium. Thus, in this symmetric setup, asymmetry is the only possible equilibrium outcome in the open economy. What kind of policies might affect international comparative advantage? International trade literature predicts comparative advantage on the basis of differences in domestic factor endowments (Heckscher-Ohlin), skill distribution (Grossman and Maggi, 2000; Bougheas and Riezman, 2007), sector-specific technologies (Ricardo) or institutions (Costinot, 2009; Levchenko, 2007). Typically, these differences are treated as entirely exogenous. 3 However, these determinants of trade pattern 1 Bosworth and Collins (2008), Panagariya (2006), and He and Kuijis (2007) highlight important policy differences between India and China. Arora and Gambardella (2005, Chapter 3) discuss differences in economic policies between Ireland and Greece. Important differences in economic policies between the US and Europe have received considerable attention in the literature (Alesina, Glaeser and Sacerdote, 2001; Krueger and Kumar, 2004). 2 Similar countries specialize in distinct industries in the open economy. This observation is the key motivation behind Grossman and Maggi (2000). Baumol and Gomory (2000) document that for the largest economies (Germany, Japan, and the US) the cross-industry pattern of specialization is remarkably stable. 3 Notable exceptions are Clarida and Findlay (1991, 1992) and Deardorff (1997).
2
ACCEPTED MANUSCRIPT are affected by the choice of domestic education policy, national policies on savings and capital accumulation, sector-specific R&D policies, and labor-market policies or credit market reforms. 4 In this paper, I endogenize the source of comparative advantage by allowing domestic policy to affect comparative advantage.5
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My general setup consists of a two-good, two-factor, two-country model in which both the goods
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and factor markets are perfectly competitive. The planner in each country chooses a single policy that affects productivity differently for different goods; for example, the policy in question affects the relative technological progress or skill composition of a country. Because the two countries are otherwise identical, a difference in government policy is the only potential source of comparative
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advantage. Both policies affect the equilibrium price under free trade. Hence to formulate an optimal policy the policy makers in each country need to take the relevant policies of their trade
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partners into consideration. Rather than each country choosing its policy in isolation, countries interact in the open economy to achieve the optimal design of the national policies that impact international comparative advantage.
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I model the optimal policy problem as a non-cooperative optimization in which each country chooses a policy to maximize its own aggregate welfare. The optimal policy problem in this setting is a symmetric game in which each player’s best response is decreasing in the action of the other player. Can these symmetric countries choose asymmetric policies in a pure strategy Nash equilibrium
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(PSNE)? I show that the countries face a clear trade-off. An asymmetric policy choice gives rise to endogenous comparative advantage and gains from trade, but also means that at least one of the identical countries is choosing a policy that would be suboptimal in the absence of trade
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opportunities. The gains from trade thus need to be weighed against the loss in welfare due to a suboptimal domestic policy.
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If an asymmetric equilibrium exists, it is associated with higher aggregate welfare for both countries compared to the common autarky optimum. In an asymmetric equilibrium, ex-ante identical countries are ex-post different with endogenous comparative advantage in different industries. This leads to welfare gains in the asymmetric equilibrium. More generally, a more asymmetric PSNE Pareto dominates a less asymmetric PSNE. When are the gains from trade due to asymmetric policy choice large enough to ensure symmetrybreaking in optimal policy? In the optimal policy problem, a symmetric equilibrium is always stable, thus Matsuyama’s (2002) symmetry-breaking condition does not apply in this setting. I rely on 4
Policymakers in many countries emphasize the international comparative advantage they derive from domestic policies in an increasingly integrated world. For example, international competitiveness within the knowledgeintensive service sector is an important factor in formulating education policies in service-exporting nations like India or Ireland. The Indian task force report on Human Resource Development in Information Technology (2000) makes 47 specific recommendations with a view ”to create a sustainable competitive advantage” in knowledge-led businesses. A very similar picture of the education policy in Ireland emerges from the Human Capital Priority program of Ireland’s National Development Plan (2007). 5 Note here that I am restricting attention to a single, national policy that affects the entire population of a country. If a government can choose different policies for different entities within a country (across different provinces or states, for example), a similar type of asymmetric equilibrium and endogenous internal trade may arise within the homogeneous population of a single country.
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ACCEPTED MANUSCRIPT quasiconvexity in the welfare function with respect to the country’s own policy, as used in Amir, Garcia and Knauff (2010) to rule out any symmetric equilibrium. 6 This quasiconvexity in welfare in own policy arises as a result of social increasing returns to policy. I show that sufficiently large social increasing return to policy, defined as convexity in aggregate income with respect to own
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policy, can ensure the existence of an asymmetric equilibrium. In fact, these conditions ensure Nash equilibrium in the open economy is asymmetric.
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a stronger form of symmetry-breaking compared to Matsuyama (2002) by guaranteeing that any I illustrate the existence of social increasing returns to policy in a specific application of education policy. In the application, the social planner allocates a fixed education budget between two
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categories of education, – higher and basic. 7 The agents have heterogeneous ability, and choose skill by incurring an education cost. The total and marginal costs of education increase in skill
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and decrease in ability. The education choice is intermediated by the government’s allocation of an education budget. Apart from the endogenous skill choice by agents and education policy choice by the government, the model is a standard Heckscher-Ohlin economy.
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I show that the endogenous skill choice of agents implies an increasing return to education policy. The degree of the social increasing return is governed by the elasticity of the education cost in skill. Previous literature on education policy (Benabou, 2002) identifies this elasticity as the determinant of progressiveness of the education system. This elasticity is the key institutional
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parameter for symmetry-breaking in this application, provided that the technologies of the two sectors are sufficiently different and consumer preference is reasonably diversified. To summarize, this paper makes three contributions. First, it provides a new answer, along
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the line of Matsuyama (2002), to the well-researched question (Benabou and Tirole, 2006; Alesina and Angeletos, 2005): ”Why do similar countries choose very different policies?”. This study shows
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that when comparative advantage is endogenous to domestic policy, gains from trade can be an explanation for equilibrium policy diversity between similar countries. 8 Second, I exploit the prop6
This approach to symmetry-breaking has many applications in the industrial organization literature. For example Amir (1996, 2000), Amir and Wooders (2000), Amir, Garcia and Knauff (2008), in the context of R&D investment and capacity choice under demand uncertainty by ex-ante identical firms. In this paper, I apply this approach to symmetry-breaking in a novel context: an optimal policy problem in which domestic policy affects international comparative advantage. It is also novel methodologically, since Amir, Garcia and Knauff’s (2010) condition ensures that the necessary condition of symmetric equilibrium is never satisfied. In this setting, however, the necessary condition for symmetric Nash equilibrium is satisfied if both countries choose their common autarky optimum policy. Instead, I rely on ruling out sufficient second order conditions of symmetric equilibrium. 7 Climent and Mukhopadhyay (2013) study the implications of a similar budget allocation problem for the Indian economy. 8 In a related work, Acemoglu, Robinson and Verdier (2015) argue that in a technologically interconnected world, the world equilibrium may be asymmetric, involving different economic institutions and technology levels for different countries. In their framework, in which the world technology frontier is affected by all countries, an asymmetric world equilibrium emerges provided the function aggregating the innovation decisions of all countries into the world technology frontier is sufficiently convex. This resonates with the sufficient condition for symmetry-breaking in my framework in terms of sufficient convexity of the aggregate income function with respect to own policy. However, the global linkages in their model are purely technological and in my model the global linkages are purely pecuniary and arise from world goods market equilibrium. Also, Acemoglu, Robinson and Verdier (2014) analyze properties of the world equilibrium, while I analyze both the non-cooperative Nash equilibrium and cooperative world equilibrium and show that the cooperative world equilibrium is more likely to be asymmetric. A related argument is developed in Aoki (2001), who also notes the possibility that international interconnections might be at the root of this type of
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ACCEPTED MANUSCRIPT erties of a standard general equilibrium model of international trade to characterize the conditions for, and welfare-implications of, an asymmetric Nash equilibrium in a symmetric non-cooperative policy game. I show that the conditions of symmetry-breaking that rely on payoff nonconcavities in a submodular game (Amir, Garcia and Knauff, 2010) are related to the concept of social increasing
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returns to policy, defined as convexity in aggregate income with respect to policy. I also estab-
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lish that greater equilibrium asymmetry gives rise to larger welfare gains. This naturally implies a Pareto ranking of asymmetric equilibria – a more asymmetric Nash equilibrium Pareto dominates a less asymmetric Nash equilibrium. Third, I contribute to the trade and education policy literature (Chang and Huang, 2012; Bougheas, Kneller and Riezman, 2009) by showing that endogenous skill
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choice by agents, in the presence of simple, linear education subsidies, can be a source of social increasing return to policy and symmetry-breaking in optimal policy. 9
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The rest of the paper is simply structured. In Section 2, I present the general framework and the optimal policy problem, and establish the conditions for symmetry-breaking in optimal policy and its consequent welfare implications. I analyze the application to education policy in Section 3.
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Section 4 concludes.
Optimal Policy in a General Framework
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I consider a two-good, two-factor, perfectly competitive world comprising two large, identical countries. I denote autarky variables using a superscript A and trade variables using a superscript T. All foreign country variables are designated by an asterisk. Let Fi denote the aggregate production
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of good i and Ci denote the aggregate demand of good i. In equilibrium, both production and consumption are functions of equilibrium price and the domestic policy, γ.
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Let u(C1 , C2 ) denote the direct utility function and I denote the aggregate income in a country. The equilibrium relative price of good 1 is pj , {j = A, T }, where j denotes the state of the economy (autarky A or free trade T ). Aggregate welfare, a function of equilibrium price and income, is denoted by V j , j = A, T. I assume homothetic preferences which imply that the aggregate demand for each good and indirect utility is linearly homogenous in income, institutional diversity (see also Rodrik, 2008). 9 Note that I derive conditions for symmetry-breaking and the consequent welfare implications in an optimal policy problem in a two country, two good economy. In this framework domestic policy is the only determinant of international comparative advantage and I do not impose any further structure on how the policy affects the economy. This flexibility implies that the framework can allow for many different types of policy problems. But without further structure on the economy, the sufficient conditions for symmetry-breaking rely on the convexity of the aggregate income function, an equilibrium object. In different policy problems, different restrictions on economic fundamentals can satisfy the requirement of sufficient convexity of the aggregate income function. The application to education policy highlights one such example. In this application, the sufficient condition relies on elasticity of the education cost function in a Hecksher-Ohlin economy with endogenous skill choice. Alternatively, in an economy with Grossman and Maggi (2000) production structure, submodularity in production can give rise to social increasing return to education policy and hence equilibrium policy diversity. This resonates with the key result of Chang and Huang (2012).
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Ci = fi (pj )I
(1)
Vj
(2)
= u(f1 (pj )I, f2 (pj )I) j
(3)
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T
= f (p )I where f (pj ) ≡ u(f1 (pj ), f2 (pj )).
Each government chooses a domestic policy γ and that is the only source of comparative advantage in the open economy. Without loss of generality, let an increase in γ confer a comparative
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advantage to good 1. I interpret γ as a policy that affects sector-specific technologies or the relative factor endowments. Government policy γ affects equilibrium quantities directly through its effects
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on economic fundamentals and indirectly through the equilibrium price. Profit maximization by competitive final good producers ensures that at any point (Q1 , Q2 ) on the production possibility frontier aggregate value of production (pj Q1 + Q2 ) is maximized (for any
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γ and any pj ) such that pj equals the marginal rate of transformation, dQ2 . p = dQ1 j
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In competitive equilibrium,
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Q1 = F1 (pj , γ) Q2 = F2 (pj , γ).
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Aggregate income in equilibrium is denoted by Y (pj , γ). From the equality of aggregate value of production and aggregate income, I = Y (pj , γ) = pj F1 (pj , γ) + F2 (pj , γ).
(4)
Thus, Y (pj , γ) is the same as the national income function in Copeland and Taylor (2003) or the revenue function in Dixit and Norman (1980). In the competitive equilibrium of the closed economy, Ci (pA , γ) = Fi (pA , γ) , i = 1, 2,
(5)
and hence the equilibrium price, pA , depends only on own policy γ. In the open economy, both countries’ policies determine the equilibrium terms-of-trade pT (.) from the goods market clearing of the world, Ci (pT , γ) + Ci∗ (pT , γ ∗ ) = Fi (pT , γ) + Fi∗ (pT , γ ∗ )
(6)
and the foreign policy γ ∗ affects domestic welfare through the terms-of-trade externality. The 6
ACCEPTED MANUSCRIPT economy, for any given γ, is a standard Walrasian economy. Below, I define the competitive equilibrium. Definition of Competitive Equilibrium For any given policy γ, the competitive equilibrium is defined by a sequence of equilibrium price and quantities {pj , Ci , Qi , Ci∗ ,
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Q∗i , Y (pj , γ), Y ∗ (pj , γ ∗ ), i = 1, 2, j = A, T } such that
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1. C1 , C2 solve the consumer’s utility maximization problem in the home country max u(C1 , C2 ) s.t. pj C1 + C2 = I.
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C1 ,C2
Similarly, Ci∗ solves the consumer optimization problem in the foreign country. In equilibrium
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Ci = fi (pj )I
2. Q1 , Q2 solve the producer’s profit maximization problem, which implies that aggregate income
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is maximized subject to the technology and factor endowment constraints of the economy. We denote the feasible set of (Q1 , Q2 ), given the technological and endowment constraints and a given policy γ, by P P F (γ).
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max I = pj Q1 + Q2 s.t. (Q1 , Q2 ) ∈ P P F (γ).
Q1 ,Q2
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Similarly, Q∗i solves the producer optimization problem in the foreign country. In equilibrium
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Q1 = F1 (pj , γ) Q2 = F2 (pj , γ), I = Y (pj , γ) = pj F1 (pj , γ) + F2 (pj , γ).
3. Markets clear.
Autarky: fi (pA )Y (pA , γ) = Fi (pA , γ), i = 1, 2.
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Open economy: fi (pT )Y (pT , γ)+fi (p T )Y (p T , γ ) = F i (p T , γ) + F ∗i (p T , γ ∗ ). I denote home aggregate welfare under autarky by U A (γ) and home aggregate welfare under ∗
trade as U T (γ, γ ∗ ). Similarly, Foreign aggregate welfare under autarky is U A (γ ∗ ) and Foreign aggregate welfare under trade is U T (γ ∗ , γ)− thus, for the welfare function for any country, the first variable in the argument refers to the country in question. Note that because the two countries are symmetric, welfare is the same function of policy in both countries. By definition of the indirect utility function,
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U A (γ) ≡ f (pA (γ))Y (pA (γ), γ),
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U T (γ, γ ∗ ) ≡ f (pT (γ, γ ∗ ))Y (pT (γ, γ ∗ ), γ). I assume that U A (γ) and U T (γ, γ ∗ ) are both differentiable up to the second order with respect to
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all the arguments. Similarly, one can define U A (γ ∗ ) and U T (γ ∗ , γ). The world welfare is denoted by W (.),
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W (γ, γ ∗ ) ≡ U T (γ, γ ∗ ) + U T (γ ∗ , γ).
Let us consider a bounded policy space γ ≤ γ ≤ γ. I also restrict attention to pure strategy Nash equilibrium (PSNE) in the non-cooperative game. 10 In addition, an assumption is that every
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individual residing within the national boundaries of a country is subject to the same policy, even though policies may differ across national boundaries. 11 Governments’ optimization problem is defined below:
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Definition of the Policy Problem In the closed economy, each government chooses γ to maximize aggregate welfare, U A (γ). The autarky optimal policy is therefore defined as γ A = arg max U A (γ).
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γ∈[γ,γ]
In the non-cooperative policy problem in the open economy, the home government chooses γ to
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maximize U T (γ, γ ∗ ) taking γ ∗ as given. I focus on pure strategy Nash equilibrium as the solution
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concept in this non-cooperative policy game. The best response functions are defined as BR(γ ∗ ) = arg maxU T (γ, γ ∗ ),
(7)
γ∈[γ,γ]
BR(γ) = arg maxU T (γ ∗ , γ). γ ∗ ∈[γ,γ]
In the cooperative policy problem in the open economy, the social planner chooses (γ, γ ∗ ) to maximize world welfare, W (γ, γ ∗ )
(γ P O , γ ∗P O ) = arg max {U T (γ, γ ∗ ) + U T (γ ∗ , γ)} (γ,γ ∗ )∈[γ,γ]2
The Nash equilibrium solution is denoted by (γ N E , γ ∗N E ) where the first argument refers to the 10
Note that in this setup a symmetric mixed strategy Nash equilibrium (MSNE) always exists (Dasgupta and Maskin 1986). In a game characterized by strategic substitutability, MSNE is usually unstable, and countries attain greater welfare in an asymmetric PSNE than in a symmetric MSNE. Moreover, it is standard in the policy literature to focus on the PSNE as the relevant solution concept. 11 Note that if the government of any country is allowed to impose different policies on different sections of its citizens, even within a country endogenous asymmetry, production specialization and internal trade may arise. Analysis of the policy problem of regional asymmetry within the domestic economy is similar to the cooperative policy problem in the world economy.
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ACCEPTED MANUSCRIPT equilibrium choice of the home country and the second argument refers to the equilibrium choice of the Foreign country. If there are more than one pairs of Nash equilibria, k-th Nash equilibrium is denoted as (γ N Ek , γ ∗N Ek ), k = 1, 2. Let us similarly define (γ P O , γ ∗P O ) as the Pareto optimum solution for the cooperative policy problem.
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The countries are identical in terms of all economic fundamentals. When both countries choose
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the same policy (γ = γ ∗ ), they are endogenously in autarky. This gives us the first insightful property of the welfare function which I use throughout the paper:
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U T (γ, γ) = U A (γ).
(8)
A similar property applies to the equilibrium price under trade and autarky:
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pT (γ, γ) = pA (γ).
(9)
A very important difference between trade and autarky is that the equilibrium price is less sensitive
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to each country’s policy γ in symmetric trade equilibrium compared to the corresponding autarky: ∂pT (γ, γ) 1 dpA (γ) = . ∂γ 2 dγ
(10)
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This difference in the first order derivative of the equilibrium price helps us to derive an important difference between the second order derivative of the aggregate welfare function under trade and autarky.
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The aggregate welfare function in the open economy satisfies the gains from trade property. By this property, a country, for any given own policy, can gain in welfare terms by trading with a
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partner with a different policy,
U T (γ, γ ∗ ) U T (γ, γ) ∀γ ∗ , ∀γ,
(11)
with equality at γ = γ ∗ . Hence, the relative welfare of choosing γ i over γ j , γ i %= γ j , improves in open economy over the relative welfare in autarky, conditional on the trading partner choosing γ j .12 In this sense, trade favors asymmetry in policy. Below I summarize these properties of the welfare function. Comparing Welfare under Autarky and Trade at Symmetric Policies By symmetry 12
From (11), we know that U T (γ i , γ j ) U A (γ i ) . > U T (γ j , γ j ) U A (γ j )
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pT (γ, γ) = pA (γ), ∂pT (γ, γ) 1 dpA (γ) = . ∂γ 2 dγ
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U T (γ, γ) = U A (γ),
Gains from Trade Property of the Welfare Function under Trade By gains from trade property of the welfare function
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U T (γ, γ ∗ ) U T (γ, γ)∀γ ∗ , ∀γ, with equality at γ = γ ∗ .
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When the autarky optimum policy is not unique, it is straightforward to show that at least one asymmetric equilibrium exists in both the non-cooperative and cooperative policy problems of the
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open economy using (11). I therefore restrict attention to the case in which the autarky problem has a unique optimum, γ A . In this game, the unique autarky optimum policy γ A is also a unique candidate for a symmetric Nash equilibrium (in which γ N E =γ ∗N E = γ A ) . When do these symmetric countries choose asymmetric policies in the open economy if the
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autarky problem has a unique optimum? To answer this question, it is important to understand an ′
important property of asymmetric Nash equilibrium. Suppose that (γ ′N E , γ N∗ E ) is an asymmetric
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PSNE. By gains from trade (11),
′
U T (γ A , γ A ) ≤ U T (γ A , γ N∗ E ). ′
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Since γ ′N E is the best response to γ N∗ E , ′
′
U T (γ A , γ N∗ E ) ≤ U T (γ ′N E , γ N∗ E ).
(12)
Thus, at any asymmetric PSNE, both countries gain in aggregate welfare compared to the common autarky optimum. This result is a powerful one. It implies that in an open economy Nash equilibrium, symmetric countries will necessarily benefit from a strategic choice of different policies. This simple, intuitive result is used to identify necessary and sufficient conditions for the existence of equilibrium policy diversity. 13 Next, I show that the convexity of the aggregate income function with regard to own policy is crucial for equilibrium policy diversity for a bounded policy space γ ≤ γ ≤ γ. If the aggregate income function, Y (pT , γ), is concave in policy γ, an asymmetric NE doesn’t exist. If the aggregate income Y (pT , γ), is sufficiently convex in policy γ, all NE in the open economy are asymmetric. Next, I relate the conditions for equilibrium asymmetry relying on convexities in the aggregate income 13
Note that the welfare gain in the asymmetric equilibrium is therefore due to an increase in production specialization which allows for an expansion of the consumption possibility frontier through trade, and not necessarily because of price difference between autarky and free trade equilibrium.
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ACCEPTED MANUSCRIPT function to the conditions for symmetry-breaking in the literature (Matsuyama, 2002; Amir, Garcia and Knauff, 2010), and also to establish the welfare properties of asymmetric equilibria. In the non-cooperative policy problem, mild conditions ensure that the policy game is submodular which implies decreasing best responses for both players. This strategic substitutability
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implies that at least one PSNE exists in the non-cooperative policy game, according to Topkis
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(1978).14 The Pareto optimum is asymmetric whenever an asymmetric PSNE exists, given (12). It is straightforward to show that γ P O = γ ∗P O = γ A is the only candidate for a symmetric Pareto optimum. Alternatively, if γ P O = γ ∗P O = γ A is the unique Pareto optimum, a Pareto-improving asymmetric equilibrium cannot exist. If the world welfare W (., .) is strictly concave, the symmetric
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strategy profile at (γ A , γ A ) is the unique Pareto optimum, and an asymmetric PSNE does not exist. If the aggregate income function, Y (pT , γ), is concave in policy γ, the world welfare W (., .)
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is strictly concave. Thus, convexity in Y (pT , γ) in γ is necessary for the existence of an asymmetric PSNE.
To derive the properties of the welfare function and the best replies sufficient for the existence of
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an asymmetric PSNE, first note that the policy game does not satisfy the sufficient conditions for the existence of an asymmetric PSNE in a symmetric game in the literature. In Matsuyama (2002) an asymmetric PSNE exists if the symmetric PSNE is Cournot unstable. In this two-good model with terms-of-trade externality as the only source of strategic interaction, the symmetric equilibrium, if it
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exists, is stable. The stability of the symmetric equilibrium follows from the substitutability of the two goods in consumption. Amir, Garcia and Knauff (2010) rule out any symmetric equilibrium since in their game the payoff function does not satisfy the necessary condition for an interior
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optimum at any interior point of symmetry. In this case, the welfare function U T (γ, γ ∗ ) has slope 0 at (γ A , γ A ) as a result of the gains from trade property,
15
hence the exact sufficient condition
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outlined in Amir, Garcia and Knauff (2010) is not satisfied in my framework. Similar to Amir, Garcia and Knauff (2010), however, I rely on some form of payoff nonconcavity to rule out a symmetric PSNE in this case. The problem of ruling out any symmetric PSNE is technically equivalent to ensuring that ( γ A , γ A ) is a local minimum of U T (γ, γ A ), even though γ A is a local maximum of U A (γ). The sufficient condition for symmetry-breaking requires that the welfare function in the open economy, U T (γ, γ A ), is quasi-convex in own policy, γ, at (γ A , γ A ). Because (γ A , γ A ) is the only possible symmetric NE, this sufficient condition rules out symmetric equilibrium in this game by imposing a robust jump of the best replies across the diagonal of the strategy space, as illustrated in Figure 1a and similar to Amir, Garcia, Knauff (2010). 16 However, unlike Amir, Garcia, Knauff (2010), I rule out 2
T
∗
(γ,γ ) 0. The condition for guranteeing submodularity of the non-cooperative policy game is ∂ U∂γ∂γ ∗ 15 T A A The necessary first order condition of maximization of U (γ, γ ) is satisfied at γ since by gains from trade and endogenous autarky properties of the welfare function, U T (γ, γ A ) is an upper envelope of U A (γ) with equality γ = γ A , where γ A is the critical point of U A (γ). 16 Because the game is symmetric, any asymmetric PSNE appears in pairs. When there are only asymmetric PSNEs in this game, the total number of PSNEs is even. In a game with continuum action space, there are usually an odd number of PSNEs by Wilson’s Oddness Theorem (1971). This result is based on degree theory and requires continuity of the best response form. Ruling out symmetric equilibrium in this game involves a robust jump of the best replies 14
11
ACCEPTED MANUSCRIPT symmetric equilibrium by ensuring that the second order condition of a local maximum is never
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T
satisfied at a point of symmetry, even though the first order condition is satisfied.
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Figure 1a: Discontinuous Best Response Curves and Multiple Asymmetric Equilibria
across the diagonal of the strategy space as in Amir, Garcia, Knauff (2010), hence my results are consistent with Wilson’s Oddness Theorem (1971).
12
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Figure 1b: Discontinuous Best Response Curves and Extreme Asymmetric Equilibrium This quasiconvexity in the welfare function arises in the presence of social increasing return to policy, i.e. if aggregate income is convex in government policy. If
∂ 2 Y (pT ,γ) ∂γ 2
< 0, W (γ, γ ∗ ) is strictly
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concave and there is no asymmetric equilibrium in the open economy. On the other hand, U T (γ, γ ∗ ) is quasiconvex in γ at (γ A , γ A ), even though U A (γ) is quasiconcave at γ A , if
∂ 2 Y (pT ,γ) |(γ A ,γ A ) ∂γ 2
is
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positive and sufficiently large. The condition needed for the existence of a unique interior autarky optimum which is not a symmetric Nash Equilibrium is that 2 ∂ Y (pT , γ) ∂pT ∂p∂γ ∂γ
(γ A ,γ A )
∂ 2 Y (pT ,γ) |(γ A ,γ A ) ∂γ 2
2 ∂ Y (pT , γ) dpA ∂ 2 Y (pT , γ) < |(γ A ,γ A ) < . dγ (γ A ,γ A ) ∂γ 2 ∂p∂γ
2 T (p ,γ) dpA < ∂ Y∂p∂γ dγ
ensures that a unique interior optimum 2 T (p ,γ) ∂pT (γ A ) exists under autarky, and sufficient convexity in aggregate income ( ∂ Y∂p∂γ ∂γ A A < Here, the condition
∂ 2 Y (pT ,γ) |(γ A ,γ A ) ) ∂γ 2
(γ A ,γ A )
ensures that the symmetric policy choice at the autarky optimum,
(γ ,γ ) (γ A , γ A ),
is
not a Nash equilibrium in the open economy. It is possible to satisfy this condition because of the differences in slope of the price functions at a point of symmetry: 1 dpA (γ) ∂pT (γ, γ) = . ∂γ 2 dγ This implies that all PSNEs in the open economy are asymmetric, since the only candidate for a symmetric PSNE, (γ A , γ A ) is ruled out–γ A is a local minimum of U T (γ, γ ∗ ) if γ ∗ = γ A . This
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ACCEPTED MANUSCRIPT sufficient condition for symmetry-breaking, which relies on sufficient convexity of the aggregate income function is similar in spirit to Acemoglu, Robinson and Verdier’s (2015) sufficient condition for world asymmetric equilibrium relying on sufficiently convex world technology frontier. I
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summarize these conditions for the existence of an asymmetric equilibrium in the next proposition. Proposition 1 (Symmetry-Breaking and Social Increasing Return) If aggregate income is 2Y
(pT ,γ) ∂γ 2
< 0), no asymmetric PSNE exists in the optimal policy problem. If
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concave in policy ( ∂
2
T
(p ,γ) aggregate income is sufficiently convex in own policy (given an interior γ A , if ∂ Y∂γ |(γ A ,γ A ) is 2 ∂ 2 Y (pT ,γ) ∂pT ∂ 2 Y (pT ,γ) |(γ A ,γ A ) > ∂p∂γ ∂γ A A ), all PSNEs in the positive and sufficiently large, namely ∂γ 2 (γ ,γ )
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open economy are asymmetric. Whenever an asymmetric PSNE exists, the Pareto optimum is also asymmetric.
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Proof. See Appendix.
If U T (γ, γ ∗ ) is strictly quasiconvex in own policy over the entire policy space and γ A is an
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interior optimum in autarky, a unique pair of asymmetric NE is given by the extremes ( γ, γ).and (γ, γ). This extreme case of equilibrium policy diversity is demonstrated in Figure 1b, in which the
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best response of either country is to choose one of the extreme policies:
BR(γ ∗ ) = γ if γ A < γ ∗
(13)
γ
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BR(γ ∗ ) = γ if γ ≦ γ A
γ
This extreme case deserves a special mention since among all asymmetric PSNEs, both countries gain maximum aggregate welfare at the asymmetric PSNE that exhibits maximum policy difference.
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The welfare properties of different asymmetric PSNEs follow from a simple generalization of the gains from trade property (11). A country not only gains in welfare by trading with a partner who has a different policy, but it gains even more from trade as the difference between its own policy and the policy of its trading partner grows. 17 Thus, for any given own policy, the welfare of a country increases with an increase in foreign policy if the foreign policy is greater than the own policy:
∂U T (γ, γ ∗ ) ∂γ ∗ T ∂U (γ, γ ∗ ) ∂γ ∗
> 0 f or γ ∗ > γ, <
(14)
0 f or γ ∗ < γ.
This property of comparative-advantage driven trade and the definition of the Nash equilibrium allow us to rank various asymmetric PSNEs in terms of associated welfare. Consider two asymmetric PSNEs (γ N E1 , γ ∗N E1 ) and (γ N E2 , γ ∗N E2 ), as illustrated in Figure 1a, such that the home country 17 Ethier (2008) highlights a similar result, ”the greater the differences, the greater the gains” in comparative advantage-driven trade.
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ACCEPTED MANUSCRIPT is an exporter of good 1 in both these equilibria (γ N EK > γ ∗N EK , k = 1, 2). The countries are more different in the first PSNE than in the second, γ ∗N E1 < γ ∗N E2 < γ N E2 < γ N E1 . Both countries attain a higher welfare in (γ N E1 , γ ∗N E1 ) than in (γ N E2 , γ ∗N E2 ). Hence, an equilibrium with greater asymmetry increases the welfare of both countries. 18 I describe this property in the
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next proposition. Proposition 2 provides a welfare-ranking of multiple asymmetric PSNEs on any
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given side of the diagonal of the strategy space.
Proposition 2 (Welfare Ranking of Asymmetric PSNEs) Consider two asymmetric PSNEs – (γ N E1 , γ ∗N E1 ) and (γ N E2 , γ ∗N E2 ) such that γ ∗N E1 < γ ∗N E2 < γ N E2 < γ N E1 . Both countries have
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a higher welfare at (γ N E1 , γ ∗N E1 ) than at (γ N E2 , γ ∗N E2 ). Proof. See Appendix.
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The analysis so far provides a characterization of the existence conditions for, and welfare implications of, symmetry-breaking in a non-cooperative policy game in the open economy. However, this analysis has an important shortcoming. Here the necessary and sufficient conditions for the
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existence of an asymmetric PSNE crucially depend on convexity of the aggregate income function with respect to policy, and the aggregate income function is an equilibrium object. In the absence of any more structure on the economy, it is difficult to pinpoint the restrictions on economic fundamentals that ensure convexity of the aggregate income function.
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In order to address this, I next consider an application to education policy in a standard Heckscher-Ohlin model, in which the agents optimally choose their skill. I show that social increas-
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ing return in aggregate production arises from the optimality of agents’ skill choice, even though the technologies are constant returns to scale. This ensures that the necessary condition for the existence of an asymmetric PSNE is always satisfied. I further characterize the parameter restrictions
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sufficient to guarantee the existence of an asymmetric PSNE. In this application, I also show that if the countries are initially asymmetric, such initial asymmetry in autarkic welfare is amplified under free trade, given simple parameter restrictions.
3
An Application to Education Policy
I consider an application of the abstract general policy problem to education policy. Each Government allocates a fixed education budget, financed by lumpsum taxation, between higher and 18
While any asymmetric NE is always better for both countries compared to the autarky optimum, the two countries are not necessarily equally well-off. To see this, consider a common equilibrium price under free trade. Note that at (γ N E , γ ∗N E ) the welfare of the two countries is given by: f (pT )Y (pT , γ N E ) and f (pT )Y (pT , γ ∗N E ). Thus, asymmetry of welfare would require that Y (pT , γ N E ) = Y (pT , γ ∗N E ). While the two countries face the same equilibrium price in free trade, their production is subject to different production possibility constraints under the asymmetric NE. Hence, it is possible that the maximum value of income differs between the two countries.
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ACCEPTED MANUSCRIPT primary education. I consider a simple model of endogenous skill choice by heterogenous agents. There are two types of labor in each country. High types are born with the ability h and low types are born with the ability l, h > l. There are nL low type workers and nH high type workers. A positively skewed skill distribution implies nL > nH .
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Each type of worker chooses a skill level that incurs a cost of education (measured in welfare
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terms). Both the total and marginal cost of education rise in the level of skill and decrease in ability. The cost of education is affected by the allocation of education budget by the government. Let he denote the level of skill chosen by high type workers and le denote the level of skill chosen by low type workers. The total effective endowment of high-skilled and low-skilled labor is given
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by
H e = he nH and Le = le nL .
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As before, u denotes the direct utility function,
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u(C1 , C2 ) = C1µ C21−µ , where µ is the expenditure share of good 1 and Ci is the aggregate consumption of good i. Here, Ci is given by
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L nH cH i + nL ci
L where cH i and ci denote individual consumption of high and low type agents. Good 2 is the
numeraire and pj is the relative price of good 1 in state j = A, T . I assume that factor price
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equalization holds over the entire strategy space at the equilibrium price for the specified set of parameters and functional forms.
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H High types choose skill, he , and consumption, cH 1 and c2 , to maximize utility
he ǫ ) , β h > 0, 1 < ǫ < 2, h H s.t. pj cH ≤ wH he , 1 + c2
H u(cH 1 , c2 ) − β H (
given equilibrium wage, price and educational institutional parameters. Here, wH (wL ) refers to the wage of high and low type workers. Low type workers solve a similar optimization problem. e
The welfare cost of higher education is captured by β H ( hh )ǫ . The total and marginal costs of higher education rise in the skill that the agent acquires, he , and falls in the ability of the agent, h. Note that a fall in β t , t = H, L reduces the cost of education. I define β t as the quality of educational institutions, thus ability of the agents is complementary to the quality of educational institutions in the acquisition of skill. Here, ǫ is the elasticity of the cost of education with respect ′′
to skill. Note that given an ability h, the relative cost of acquiring higher skill
h′′
>
β ( h )ǫ h′ , H hh′ , β H ( h )ǫ
increases in ǫ. I define ǫ as the progressivity of the education system, following the interpretation
in Benabou (2002). The condition ǫ > 1 ensures that the second order condition of optimality of the agents’ skill choice is satisfied.
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ACCEPTED MANUSCRIPT Producers maximize profit given the equilibrium wage, price and effective endowments of skills . Producers of the two goods employ high and low skilled labor with different intensities eη
e1−η i
Fi (Hie , Lei ) = Hi i Li
T
where η i denotes the share of high-skilled labor per unit cost of good i. Good 1 is relatively more
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intensive in high-skilled labor, η 1 > η 2.
The optimal skill choice function, incorporating the equilibrium wage rates obtained from the
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zero profit conditions, is given by
1−a
L
e
1 H ǫ p η1 −η2 ǫ−1 = c h′ ( ) , ch′ = f (η 1 , η 2 , µ, ǫ), a = µη 1 + (1 − µ)η 2 , nH β H
(15)
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H
e
−a
1 Lǫ p η1 −η2 ǫ−1 = cl ′ ( ) , cl′ = f (η 1 , η 2 , µ, ǫ). nL β L
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where H (= hnH ) and L (= lnL ) denote the original endowments of the skills in the economy. In competitive equilibrium all the optimization conditions hold and both labor and goods markets clear as in Section 2.
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Government policy choice, γ, affects the cost of education and hence optimal skill choice (15), as specified below, thus, domestic education policy affects aggregate welfare both directly, via its effect on factor distributions of the economy, and indirectly via its effect on the equilibrium price. As in
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Section 2, aggregate indirect utility in the competitive equilibrium of the economy is a function of
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the equilibrium price and the aggregate income, cpj
−µ
Y (pj , γ).
Aggregate income, Y (pj , γ), is given by wH (pj )H e (γ) + wL (pj )Le (γ), where wL (pj ) = cl pj
η 2 /(η 2 −η 1 )
, wH (p) = ch pj
−(1−η 2 )/(η 2 −η 1 )
.
Here c, ch , cl , and cp denote constants that depend on the economic fundamentals. The equilibrium price, pj , depends only on domestic policy in autarky (pA (γ)) and on both countries’ policy in the open economy (pT (γ, γ ∗ )). This economy is thus an example of the general framework outlined in Section 2. The government has a total education budget T = 1, and chooses a fraction γ ∈ [0, 1] to spend on higher education. The remainder goes to primary education. The government expenditure on higher education, γ, improves the quality of higher education institutions, β H = g(γ), g ′ < 0.
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ACCEPTED MANUSCRIPT Similarly, β L = g(1−γ).19 Higher education and basic education institutions have a baseline quality level (bH and bL ,respectively), and that government policy affects the institutions in a simple linear fashion, = bH − cγ, bH > c > 0,
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β L = bL − c(1 − γ), bL > c > 0.
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βH
Note that the government always has the option of improving both types of institution equally (γ = .5), but may choose to attach different priorities to different institutions. From the optimal
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skill choice function, (15), the condition ǫ > 1 ensures that H e is a convex function of γ, given pj . Thus when agents choose their skill levels optimally, optimal skill function in the economy is convex in government policy, even though government policy affects education costs in a simple
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linear fashion. This ensures that the necessary condition for symmetry-breaking is satisfied in this framework.
The government takes the optimal response of the agents and market clearing conditions as
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given and allocates the education budget to maximize the aggregate indirect utility by choosing γ. As before, each government in the closed economy chooses γ to maximize aggregate welfare, U A (γ). In the non-cooperative policy problem in the open economy, each government maximizes U T (γ, γ ∗ )
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taking the other country’s policy as given. In the cooperative policy problem in the open economy, the social planner maximizes the world welfare, W (γ, γ ∗ ).
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Note that in this model,
1
1
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1−ǫ ∂β H ∂β L1−ǫ ∂ 2 Y (., γ) = wH (p)sH (p) + wL (p)sL (p) , ∂γ 2 ∂γ 2 ∂γ 2
(16)
where st (p), t = H, L, are given from (15) as: 1−α
sH (p) = p (η1 −η2 )(ǫ−1) , −α
st (p) = p (η1 −η2 )(ǫ−1) .
Given 1 < ǫ < 2, a fall in ǫ increases the convexity of (16) provided bH − c > 1 and bL − c > 1. Hence, if ǫ is sufficiently low, (η 1 − η 2 ) is sufficiently high and consumers do not strongly prefer either of the goods, all PSNE in the open economy are asymmetric. 20 The following proposition summarizes the condition for equilibrium policy diversity in the open economy Proposition 3 If 1 < ǫ ≤ǫ < 2 and bH − c > 1 and bL − c > 1, (η 1 − η 2 ) is sufficiently high and consumers do not strongly prefer either of the goods, only asymmetric PSNEs exist in the open 19
The education system assumptions essentially mean that education is publicly funded and that government decides the type of institution to emphasize. 20 I have formulated the application in terms of a subsidy to education policy, with the government choosing to allocate fixed resources between different categories of education. Equivalently, I can formulate the education problem as the allocation of one unit of lumpsum tax between different skill groups in the form of a wage subsidy, and this alterative formulation leaves the analysis essentially unchanged. In this formulation, the optimization problem of the
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ACCEPTED MANUSCRIPT economy optimal policy problem. Proof. See Appendix. In this asymmetric Nash equilibrium, these symmetric countries may attain different welfare levels under trade, both of which are higher than their autarky optimum welfare. Thus, while both
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countries are better off in an asymmetric Nash equilibrium compared to their common autarky
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optimum, the ex-post gains from trade are not necessarily identical among the ex-ante identical countries. I illustrate this in an example in the context of the education budget allocation problem. Suppose that γ A = 1 implying that both countries find it optimal to invest all their resources
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into higher education in autarky, and suppose in an asymmetric Nash equilibrium γ N E1 = 1 and γ ∗N E1 = 0. Thus, in the open economy in a Nash equilibrium one country invests all its resources into higher education and the other country invests all its resources into basic education. To illustrate
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that the ex-post gains from trade are not necessarily identical among the ex-ante identical countries, one needs to find parameter restrictions such that
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U T (γ = 1, γ ∗ = 0) > U T ∗ (γ ∗ = 0, γ = 1) > U A (γ A = 1). I show in the appendix that this is more likely if (η 1 − η 2 ) is sufficiently high and consumers do not strongly prefer either of the goods, similar conditions under which existence of an asymmetric
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Nash equilibrium is more likely. Thus, parameter restrictions that make an asymmetric Nash equilibrium more likely, also increases the possibility of asymmetric welfare outcomes at such a
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Nash equilibrium. If gains from trade are large, a country may prefer to choose the ”bad” policy– here, investing only in basic education–to benefit from such gains, even though it would have preferred to be the country with the ”good” policy–here, investing only in higher education. 21
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A related question to ask in this context is: if these countries are initially asymmetric can they attain proportionally more (than their initial asymmetry) different welfare under asymmetric Nash equilibrium than under autarky? To answer this question, I model the initial asymmetry of countries in the traditional Heckscher-Ohlin way: the countries are otherwise identical except for their initial endowment of skilled labor H. Suppose that the home country is less skill abundant high skill type is reduced to
he ǫ ) }, β h h
>
0, ǫ > 1,
H s.t. pcH 1 + c2
≤
wH g(γ)he ,
H max {u(cH 1 , c2 ) − β H (
H e cH 1 ,c2 ,h
given equilibrium wage, price and educational institutional parameters. Low type workers solve a similar optimization problem. I formulate the wage subsidy schedules as follows: g(γ)
=
b + cγ, b > c > 0,
g(1 − γ)
=
b + c(1 − γ).
In this case, the optimality of skill choice also ensures that the necessary condition for the existence of an asymmetric PSNE is satisfied. The endogenous skill functions (15) and the sufficient conditions for the existence of an asymmetric PSNE, as illustrated in proposition 3, essentially remain unchanged in this alternative formulation of the policy problem. 21 I sincerely thank an anonymous referee for pointing out this possibility to me.
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ACCEPTED MANUSCRIPT implying that the relative inherent ability (or productivity) of high-skill labour in the home country ∗
( hl ) is lower than that in the foreign country ( hl∗ ) : < 1.
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κ=
h l h∗ l∗
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This is the typical nature of factor endowment differences assumed in a Heckscher-Ohlin economy with pre-existing comparative advantage. The skill distributions in the two countries are assumed to be the same (nH = n∗H < nL = n∗L ). It is straightforward to show that the optimal choice of policy in autarky, γ A , does not depend on this factor endowment difference. Hence, γ A = γ A∗ , ∗
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though welfare at the autarky optimum is different, U A (γ A ) %= U A (γ A∗ ). Suppose initial welfare asymmetry (at the autarky equilibrium) is captured by
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aǫ U A (γ A ) = κ ǫ−1 < 1. ∗ A A∗ U (γ )
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Let us consider the case where sufficient conditions for asymmetric NE are satisfied and ( γ N E = 0, γ ∗N E = 1) is the NE. If (η 1 − η 2 ) is sufficiently large and
nH nL
>
bH −c bL ,
the relative welfare at the
asymmetric NE is magnified compared to the autarky welfare asymmetry,
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U A (γ A ) U T (γ = 0, γ ∗ = 1) < . U T ∗ (γ ∗ = 1, γ = 0) U A∗ (γ A )
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The application to education policy in this section provides a setting in which aggregate income is convex in government policy. This leads to an asymmetric equilibrium policy choice for symmetric countries to encourage trade. In the presence of standard constant returns to scale technology in a
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Heckscher-Ohlin set-up, this social increasing return to policy arises as a result of the endogenous skill choice of agents. Other examples of frameworks and policy problems that may generate sufficient convexity in the aggregate income function and equilibrium policy diversity include an economy with Grossman-Maggi (2000) production structure.
4
22
Conclusion
In this paper, I show that similar countries may choose different policies because these policies allow them to specialize in different industries, and therefore gain from international trade. I find that even identical countries may optimally choose different policies in an open economy, and b oth 22
In Chatterjee (2012) I consider an alternative specification which shares the production structure of Grossman and Maggi (2000) but has a similar specification of consumer preferences and government policy as before. For simplicity, I assume that allocation of education budget by the government directly affects the effective endowment of skill. The submodular production technology implies that even when government policy affects skills in a linear fashion, aggregate production and income can be sufficiently convex in policy, and this gives rise to symmetrybreaking in optimal education policy in the open economy. Thus, in this case an additional convexity introduced by the endogenous skill choice problem of agents, is not necessary for convexity of aggregate income with respect to policy.
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ACCEPTED MANUSCRIPT of these symmetric countries may gain in aggregate welfare in any asymmetric Nash equilibrium compared to the autarky optimum. In an application in the competitive economy, an asymmetric equilibrium arises if education policy affects the determinants of trade, namely effective endowments of skill, in a strong convex fashion. I construct a model in which agents optimally choose their skill
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levels given government policy, and show that the optimal skill is a convex function of government
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policy giving rise to social increasing returns to the policy.
A natural extension of the pure welfare maximizing optimal policy problem is to allow governments to also care about redistributive equity. In the presence of redistributive concerns, it is no longer true that both countries attain higher social welfare at an asymmetric PSNE compared to
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any symmetric PSNE. Also, allowing for endogenous comparative advantage in a political economy framework, the political preferences of the government can be a source of comparative advantage
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in an open economy. Exploring the implications for an asymmetric policy choice among nearly identical economies in a more complete political economy setting is an important part of my future research agenda.
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This paper outlines a general mechanism that applies to many different policies which potentially affect comparative advantage in open economies. For any particular application, it is important to model the domestic economic environment carefully. For example, education policy is an important policy for encouraging trade, but there are also several reasons why education policy is important
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for the domestic economy. Since future human capital is typically not accepted as collateral, the availability of credit for financing educational expenditure is limited. This aspect of the education policy has received attention in both trade and macroeconomic policy literature (Benabou, 2002;
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Ranjan, 2001; Chesnokova and Krishna, 2009). Also, skills learned in the earlier stages of academic development are complementary to the acquisition of advanced skills. In future work, I intend
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to incorporate these aspects of education in a more complete application, to study the interplay between the domestic and international motives of optimal education policy.
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ACCEPTED MANUSCRIPT Acknowledgement This paper has benefitted immensely from the very helpful comments of the Editor, Giovanni Maggi, and two anonymous referees. This paper is based on the first chapter of my Ph.D. thesis, ”Why Do Similar Countries Choose Different Policies? Endogenous Comparative Advantage and Welfare Gains.” I am indebted to Gene Grossman and Esteban Rossi-Hansberg for
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invaluable guidance and encouragement. I would like to thank Rabah Amir, Roland Benabou, Saroj
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Bhattarai, Varanya Chaubey, Arnaud Costinot, Avinash Dixit, Arghya Ghosh, Richard Holden, Kala Krishna, Devashish Mitra, Vinayak Tripathi, Robert Staiger, Alan Woodland, and participants of various workshops and seminars for their comments and suggestions. I thank the International Economics Section (Princeton University), Social Policy Program (Woodrow Wilson School), and
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UNSW Business School (University of New South Wales) for their financial support, and Indian Statistical Institute (Calcutta) for their hospitality during my research visits. All remaining errors
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are mine.
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ACCEPTED MANUSCRIPT References [1] Acemoglu, D., J. Robinson and T. Verdier, ”Asymmetric growth and institutions in an interdependent world (2015), forthcoming Journal of Political Economy.
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[2] Alesina, A., and G. M., Angeletos, ”Fairness and redistribution”, The American Economic
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Review, Vol. 95, No. 4, ( 2005), pp. 960-980. doi:10.1257/0002828054825655 [3] Alesina, A., E., Glaeser, and B., Sacerdote, ”Why doesn’t the United States have a EuropeanStyle welfare state?”, Brookings Papers on Economic Activity, Vol. 2001, No. 2, (2001), pp.
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187-254. doi:10.1353/eca.2001.0014
[4] Amir, R., ”Cournot oligopoly and the theory of supermodular games”, Games and Economic
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Behavior, Vol. 15, Issue 2, (1996), pp. 132-148. doi:10.1006/game.1996.0062 [5] Amir, R., ”R&D returns, market structure, and research joint ventures”, Journal of Institu-
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tional and Theoretical Economics, Vol. 156, No. 4, (2000), pp. 583-598. [6] Amir, R., F., Garcia and M., Knauff, ”Symmetry-breaking in two-player games via strategic substitutes and diagonal nonconcavity: A synthesis”, Journal of Economic Theory 145 (2010)
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1968-1986. doi:10.1016/j.jet.2010.01.013
[7] Amir, R., and J. Wooders, ”One-way spillovers, endogenous innovator/imitator roles, and research joint ventures”,
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31,
1-25 (2000).
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doi:10.1006/game.1999.0734
[8] Arora, A., and A. Gambardella, (2005). From Underdogs to Tigers. Oxford University Press.
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No. 2, (1978), pp. 305-321. doi:10.1287/opre.26.2.305
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ACCEPTED MANUSCRIPT 5
Appendix
5.1
Derivation and Proofs for Section 2
T
From homothetic demand, aggregate demand for good i, Ci , is linearly homogenous in income,
Here, I is aggregate income,
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I = pQ1 + Q2 ,
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Ci = fi (pj ) ∗ I.
where Qi stands for aggregate production of good i. The aggregate indirect utility is given by
NU
V = u(f1 (pj ) ∗ I, f2 (pj ) ∗ I) = If (pj ),
MA
where f (p) = u(f1 (p), f2 (p)). From Roy’s identity
f ′ (pj )I = −C1 = −f1 (p)I. f (pj )
(17)
ED
The marginal effect of p on indirect utility is given by
PT
f ′ (pj )I + f (pj )Q1 + f (pj )(pj
From (17),
−dQ2 dQ1
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The condition p = M RT =
f (pj )(Q1 +
dQ1 dQ2 + ). dp dp
implies that the marginal effect of p on indirect utility is f (pj )(Q1 +
f ′ (pj ) )I). f (pj )
f ′ (pj ) I) = f (pj ) ∗ (Q1 − f1 (pj )I) = f (pj ) ∗ (Q1 − C1 ). f (pj )
Hence from (5), the marginal effect of p on indirect utility is zero in autarky. Assume that the policy instrument γ provides absolute advantage in good 1 and absolute disadvantage in good 2. Since the total value of consumption equals income, f1 (pj ) ∗ pj + f2 (pj ) = 1.
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Proof of Proposition 1. This proof is completed in three steps. Step 1: Uniqueness and first order optimality of (possible) symmetric Pareto optimum and Nash equilibrium: The symmetric strategy profile in which both countries choose the autarky optimum, (γ A , γ A ), is the only candidate for a symmetric Pareto optimum and for a symmetric PSNE. If γ A is an interior autarky optimum, (γ A , γ A ) satisfies the first order neces27
ACCEPTED MANUSCRIPT sary condition of local optimality for W (γ, γ ∗ ), and γ A satisfies the first order necessary condition ∗
of local optimality for U T (γ, γ A ) w.r.t. γ and for U T (γ A , γ ∗ ) w.r.t. γ ∗ . There are three results to prove in this step. First, (γ A , γ A ) is the only possible symmetric Pareto optimum.The result that (γ A , γ A ) is the
T
only candidate for a symmetric Pareto optimum directly follows from the symmetry of the setup
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which ensures that W (γ, γ) = 2U A (γ), and the optimality of γ A in autarky. Second, we need to prove that (γ A , γ A ), is the only candidate for a symmetric PSNE. Suppose ! ) by deviating to γ A , that (! γ, γ !), γ ! %= γ A is a PSNE. The home country attains a payoff of U T (γ A , γ given that the foreign country chooses γ !. The resulting change in payoff (U T (γ A , γ ! ) − U T (! γ, γ !))
SC
! ) − U A (γ A )) and an increase in autarky utility consists of a gain from trade component ((U T (γ A , γ
(U A (γ A ) − U A (! γ )). Hence, the autarky optimum, γ A , is a profitable unilateral deviation from γ !, if the other country chooses γ !. Thus (! γ, γ !), γ ! %= γ A cannot be a PSNE.
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Third, if γ A is an interior autarky optimum, (γ A , γ A ) satisfies the first order necessary con-
dition of local optimality for W (γ, γ ∗ ), and γ A satisfies the first order necessary condition of local ∗
MA
optimality for U T (γ, γ A ) w.r.t. γ and for U T (γ A , γ ∗ ) w.r.t. γ ∗ . To see this, note that if γ A is an interior autarky optimum, the first order condition (FOC) of maximization must be satisfied at γ A . Note that in equilibrium
From the definition of U A (.),
ED
I = Y (pj , γ) = pj F1 (pj , γ) + F2 (pj , γ).
AC CE
From Roy’s identity,
PT
dpA f ′ (pA ) ∂Y dU A (.) )Y ) + f (pA ) = f (pA )(F1 + . dγ dγ f (pA ) ∂γ
f (pA )(F1 +
f ′ (pA ) )Y ) = f (pA )(F1 (pA , γ) − C1 ), f (pA )
which equals zero from (5). Therefore, dU A (.) ∂Y = f (pA ) = 0 at γ A from the FOC. dγ ∂γ
Similarly, from the definition of W (.) and using (6), ∂Y ∂W (.) = f (pA ) . ∂γ ∂γ Therefore, the FOC of optimization of the world welfare is satisfied at (γ A , γ A ). If γ A is an interior autarky optimum, γ A satisfies the first order necessary condition of local
28
ACCEPTED MANUSCRIPT ∗
optimality for U T (γ, γ A ) w.r.t. γ and for U T (γ A , γ ∗ ) w.r.t. γ ∗ . The expression for f (pT )(F1 ((pT , γ)) − C1 )
∂U T (γ, γ A ) ∂γ
is
∂pT ∂Y + f (pT ) ∂γ ∂γ
T
At a symmetric strategy, excess demand of good 1, (F1 − C1 ), is zero and the equilibrium free trade economy is essentially autarky. Hence, at (γ A , γ A ) pT (γ, γ) = pA (γ)
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price is the same as the autarky equilibrium price, since a symmetric strategy choice in the open
NU
SC
F1 (pT (γ, γ), γ) − C1 = 0, and ∂U T (γ, γ A ) ∂Y ∂W (γ, γ A ) = = f (pj ) . ∂γ ∂γ ∂γ Step 2: Existence of an asymmetric Nash equilibrium and the second order properties of the payoff functions: If γ A is an interior autarky optimum and if W (γ, γ ∗ ) is globally
MA
strictly concave, (γ A , γ A ) is the unique maximum of W (γ, γ ∗ ) and an asymmetric PSNE does not exist. If γ A is an interior autarky optimum and U T (γ, γ ∗ ) is quasiconvex in its own strategy ( γ) at (γ A , γ A ), there is only asymmetric Nash equilibrium in the open economy . If U T (γ, γ ∗ ) is
ED
strictly quasiconvex in own strategy ( γ), a unique pair of asymmetric NE is given by the extremes (γ, γ).and (γ, γ).
There are two results to prove in this step.
PT
First, if γ A is an interior autarky optimum and if W (γ, γ ∗ ) is globally strictly concave , (γ A , γ A ) is the unique maximum of W (γ, γ ∗ ). In that case, there is a unique symmetric Pareto optimum in which both countries choose the autarky optimal policy, γ A . If an asymmetric PSNE exists, it
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is a Pareto improvement over the autarky optimum, thus if W (γ, γ ∗ ) is strictly concave, a Pareto improvement over (γ A , γ A ) is not possible. If W (γ, γ ∗ ) is strictly concave, an asymmetric PSNE does not exist.
Second, if U T (γ, γ ∗ ) is quasiconvex in own strategy ( γ) at (γ A , γ A ), the symmetric strategy (γ A , γ A ) is a local minimum of U T (γ, γ ∗ ). This implies that the home country government has a ′
unilateral profitable deviation to a γ %= γ A if the Foreign government chooses γ A . Hence, in this case, (γ A , γ A ) is not a PSNE and a symmetric PSNE does not exist. Any Nash equilibrium, in this case, must be asymmetric. If U T (γ, γ ∗ ) is strictly quasiconvex in own strategy ( γ), there is a unique pair of asymmetric NE given by the extremes (γ, γ).and (γ, γ). From the definition of strict quasiconvexity, U T (λγ 1 + (1 − λ)γ 2 ) < max{U T (γ 1 ), U T (γ 2 )}, ∀γ 1 , γ 2 ∈ [γ, γ]2 . where γ 1 = (γ, γ) and γ 2 = (γ, γ). Any γ ′ ∈ (γ, γ) yields a lower pay-off than γ ′ = γ or γ ′ = γ, for any given foreign policy γ ∈ [γ, γ]. Hence, ∀γ ∈ [γ, γ], the best response is either γ or γ. Neither γ nor γ is an autarky optimum. Therefore, γ (γ) is not the best response to γ (γ), and the best
29
ACCEPTED MANUSCRIPT response to γ is γ and vice versa. The only two PSNEs are (γ, γ) and (γ, γ). Step 3: Existence of asymmetric Nash equilibrium and social increasing return : 2 (pT ,γ) If ∂ Y∂γ < 0, W (γ, γ ∗ ) 2 2 T ∂ Y (p ,γ) |(γ A ,γ A ) is positive ∂γ 2
is strictly concave in (γ, γ ∗ ) and an asymmetric NE does not exist. if 2 T (p ,γ) ∂pT ∂ 2 Y (pT ,γ) and sufficiently large ( ∂ Y∂p∂γ |(γ A ,γ A ) ), U T (γ, ∂γ A A < ∂γ 2 (γ ,γ )
T
γ ∗ ) is quasiconvex in γ at (γ A , γ A ) even though U A (γ) is quasiconcave at γ A , and all Nash equilibria
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in the open economy are asymmetric.
The sufficient condition for equilibrium asymmetry in step 2 requires U T (γ, γ ∗ ) to be quasiconvex in γ at (γ A , γ A ), even though γ A is an interior autarky optimum. Let us consider the difference
SC
between the curvatures of U T (γ, γ ∗ ) at (γ A , γ A ) and U A (γ) at γ A . Note that the curvature of U A (γ) is given by,
2 2 A dpA ′ A ∂Y d2 U A (γ) A ∂ Y A ∂ Y (p , γ) (f = (p ) ) ) . + f (p ) + f (p dγ dγ 2 ∂γ ∂p∂γ ∂γ 2
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First, note that,
MA
∂ 2 Y (pA , γ) ∂F1 . = ∂γ ∂γ∂p
Since γ gives absolute advantage in good 1 and absolute disadvantage in good 2,
Hence,
∂ 2 Y (pA ,γ) ∂γ∂p
ED
δF2 /δγ < 0, δF1 /δγ > 0.
0.
d2 U A (γ) dγ 2
PT
At the interior autarky optimum γ A ,
∂Y ∂γ
∂ 2 Y (pA ,γ) |γ A < 0, ∂γ 2 2 Also, ∂∂γY2 < 0 implies
= 0. Hence, if
|γ A < 0 and γ A is the unique interior optimum.
it implies that that the world
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welfare function is strictly concave and that there is a unique Pareto optimum, and there is no asymmetric PSNE in the non-cooperative policy game. This proves the necessary condition for the existence of an asymmetric PSNE. In fact, even if
∂ 2 Y (pA ,γ) ∂γ 2
> 0, as long as
∂ 2 Y (pT ,γ) |(γ A ,γ A ) ∂γ 2
2 T (p ,γ) dpA < ∂ Y∂p∂γ dγ
(γ A ,γ A )
γ A is the
unique interior optimum. In the non-cooperative policy problem, at (γ A , γ A )
At (γ A , γ A ), if
∂ 2 Y (pT ,γ) |(γ A ,γ A ) ∂γ 2
∂2U T ∂ 2 Y ∂pT ∂2Y j + = f (p )( ). ∂p∂γ ∂γ ∂γ 2 ∂γ 2 2 T (p ,γ) ∂pT > ∂ Y∂p∂γ ∂γ
(γ A ,γ A )
ensures that
(20) ∂2U T ∂γ 2
is positive. This ensures
that (γ A , γ A ) is a local minimum. Hence, an asymmetric PSNE must exist. Note that at (γ A , γ A ), ∂pT ∂γ
A
T A = .5 dp dγ and p (γ, γ) = p (γ). It is possible that
2 ∂ Y (pT , γ) ∂pT ∂p∂γ ∂γ
(γ A , γ A )
∂ 2 Y (pT , γ) < |(γ A , ∂γ 2
30
γA)
2 ∂ Y (pT , γ) dpA < ∂p∂γ dγ
(γ A , γ A )
.
ACCEPTED MANUSCRIPT Sufficient convexity in production with respect to the policy in question ensures that there are only asymmetric equilibria in the open economy, even though γ A is an interior autarky optimum. Note that here it is crucial that at a point of symmetry, the level of autarky and trade prices are the same, but the partial derivative of price w.r.t. to own policy is halved in trade compared to
T
autarky–trade lessens the effect of own policy on equilibrium price. This is crucial in allowing the
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signs of second order conditions to differ between trade and autarky, even at a point of symmetric policy choice. Steps 1, 2 and 3 combined complete the proof of proposition 1. Proof of Proposition 2.
Step 1: Since γ ∗ affects the welfare of the home country through
SC
terms-of-trade,
∂pT ∂U T (γ, γ ∗ ) T = f (p )(F (.) − C ) . 1 1 ∂γ ∗ ∂γ ∗
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Since an increase in γ confers comparative advantage in good 1,
MA
∂pT (γ, γ ∗ ) < 0. ∂γ ∗ Suppose γ ∗ > γ, which implies that the home country is an importer of good 1,
advantage in good 2.
∂U (γ, γ ∗ ) δγ ∗
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Step 2: Given γ ∗1 < γ ∗2 < γ 2 and
but
∂U (γ, γ ∗ ) > 0. δγ ∗
< 0. A similar proof works if an increase in γ confers comparative
PT
Similarly, γ ∗ < γ implies
ED
F1 < C1 , and
∂U T (γ,γ ∗ ) ∂γ ∗
< 0 for γ > γ ∗ ,
U T (γ 2 , γ ∗2 ) < U T (γ 2 , γ ∗1 ),
U T (γ 2 , γ ∗1 ) < U T (γ 1 , γ ∗1 )
since γ 1 is the best response to γ ∗1 . Similarly, given γ ∗2 < γ 2 < γ 1 and
∂U T (γ ∗ , γ) ∂γ
> 0 for γ ∗ < γ,
U T (γ ∗2 , γ 2 ) < U T (γ ∗2 , γ 1 ). But γ ∗1 is the best response to γ 1 , which implies that U T (γ ∗2 , γ 1 ) < U T (γ ∗1 , γ 1 ).
5.2
Derivation and Proofs for Section 3
The aggregate indirect utility is a function of the equilibrium price and the aggregate income, 31
ACCEPTED MANUSCRIPT
cpj
−µ
Y (p, γ).
Aggregate income, Y (p, γ), is given by wH (p)H e (γ) + wL (p)Le (γ), where , wH (p) = ch pj
−(1−η 2 )/(η 2 −η 1 )
.
T
η 2 /(η 2 −η 1 )
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wL (p) = cl pj
Here c, ch , cl , and cp denote constants that depend on the economic fundamentals. Let us first derive (15). A more able agent with initial ability h chooses he by maximizing
1−η 2 η 1 −η 2
wH (pj )he − β h (
he ǫ ), h
. From the FOC, one can derive
NU
where wH (pj ) = ch pj
−µ
SC
pj
1−a
ch
ǫβ h
1
hǫ ) ǫ−1 .
MA
he = (
η −η pj 1 2
Multiplying both sides by nH we arrive at (15). The second order condition of optimality requires ǫ > 1. The optimal skill-choice problem of the low-skilled agents can be solved similarly.
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L
e
1−a η −η 2
H ǫ pj 1 = ch′ ( nH β H
PT
He
ED
The optimal skill choice function is given by
1
) ǫ−1 , a = µη 1 + (1 − µ)η 2 ,
−a η −η 2
Lǫ pj 1 = cl ′ ( nl β L
(21)
1
) ǫ−1 , H = nH h, L = nl l.
where ch′ and cl′ are functions of η 1 , η 2 , µ and ǫ. Note that 1−a η −η 2
ch pj 1 ∂H e = −( ∂γ ǫnH
and
∂2H e ∂γ 2
1−a
ch p ǫnH
provided
− 1 −1 ∂β h 1 >0 )β h ǫ−1 ∂γ ǫ−1 −
is equal to
j η1 −η2
(
1
hǫ ) ǫ−1 (
1 1 1 )( + 1)β h hǫ ) ǫ−1 ( ǫ−1 ǫ−1
∂2βh ∂γ 2
1 −2 − ǫ−1
(
1−e η 1 −η 2
ch p ∂β h 2 ) −( ∂γ ǫ −
1
hǫ ) ǫ−1 (
1 )β ǫ−1 h
1 −1 − ǫ−1
2
∂ βh > 0, ∂γ 2
≤ 0. A similar result holds for Le .
In the competitive equilibrium of the closed economy, only domestic policy determines the equilibrium price, pA = cp (
H e η2 −η1 ) , Le 32
ACCEPTED MANUSCRIPT and in the open economy both domestic and foreign policy determine the equilibrium terms-oftrade, e
pT = (
H e + H ∗ η2 −η1 ) cp (µ, η 1 , η 2 ). Le + Le∗
pT (γ, γ ∗ ) = {
∗
1
β h1−ǫ + β h 1−ǫ 1 1−ǫ
βl
+
∗ 1 β l 1−ǫ
}
(η 1 −η 2)(1−ǫ) ǫ
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1
T
The equilibrium price in the open economy is given by,
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c′p (η 1 , η 2 , µ, H, L, nH , nL ),
SC
where cp denotes a constant that does not depend on policy. Welfare in the open economy is given by e
e
He + H ∗ a e He H e + H ∗ −1 ) (L + ) ), ( Le + Le∗ qf (µ, η 1 , η 2 ) Le + Le∗
NU
U T = vq a (
MA
where
(23)
a = η 2 + µ(η 1 − η 2 ), (1 − µ)f1 + µf2 qf = , µf2 (η 1 /(1 − η 1 )) + (1 − µ)f1 (η 2 /(1 − η 2 ) = (((1 − η 1 )/(1 − η 2 ))(η 2 /(1 − η 2 ))−η2 (η 1 /(1 − η 1 ))η1 )1/(η2 −η1 ) ,
ED
f
f2 = (η 2 /(1 − η 2 ))η2 f η2 −1 /(η 1 /(1 − η 1 ) − η 2 /(1 − η 2 )),
PT
f1 = (η 1 /(1 − η 1 ))η1 f η1 −1 /(η 1 /(1 − η 1 ) − η 2 /(1 − η 2 )),
AC CE
v = (1 − η 1 )(η 1 /(1 − η 1 ))η1 f η1 . are constants depending on the production and demand parameters. Welfare in the closed economy is given by
1 1 He a e ) (L + ) e qf (µ, η 1 , η 2 ) Le L 1 )(H e )a (Le )1−a . = vq a (1 + qf
U A = vq a (
It is straightforward to show that both the autarky optimal policy γ A and open economy best response correspondence BR(γ ∗ ) are increasing in µ, η i . Proof of Proposition 3.
From proposition 1 U T (γ, γ ∗ ) is quasiconvex in γ, if
∂ 2 Y (.,γ) ∂γ 2
is
sufficiently large. Following proposition 1 we consider the case where ∂ 2 Y (pT , γ) ∂pT ∂p∂γ ∂γ
< (γ A ,γ A )
∂ 2 Y (pT , γ) ∂ 2 Y (pT , γ) dpA | < A A (γ ,γ ) ∂γ 2 ∂p∂γ dγ
. (γ A ,γ A )
In this case, given Proposition 1 there are only asymmetric PSNEs at the extremes, even though there is an interior γ A . Note that 33
ACCEPTED MANUSCRIPT
∂pT (γ, γ ∗ ) ∂γ
1
1
1
(η 1 − η 2)(1−ǫ) = c′p ǫ
∂{
T
∂β 1−ǫ ∂β 1−ǫ ′ ′ (p)sH (p) + wH (p)s′H (p)) H + (wL (p)sL (p) + wL (p)s′L (p)) L , = (wH ∂γ ∂γ 1
∗ 1
1
1 ∗ 1−ǫ
β h1−ǫ +β h 1−ǫ β l1−ǫ +β l
}
1
.{
∂γ
RI P
∂ 2 Y (pT , γ) ∂p∂γ
1
1−ǫ ∂2βH ∂ 2 β L1−ǫ = wH (p)sH (p) + w (p)s (p) , L L ∂γ 2 ∂γ 2
∗
1
β h1−ǫ + β h 1−ǫ 1 1−ǫ
βl
+
∗ 1 β l 1−ǫ
}
(η 1 −η 2)(1−ǫ) ǫ
−1
.
SC
∂ 2 Y (., γ) ∂γ 2
∂ 2 Y (pT ,γ)
part A
is given by
2ǫ−1
2ǫ−1
wH (p)sH (p)β H1−ǫ + wL (p)sL (p)β L1−ǫ 2 (pT ,γ) | ∂ Y∂p∂γ |
1 ∗ 1 β 1−ǫ +β 1−ǫ h ∂{ h 1 ∗ 1 β 1−ǫ +β 1−ǫ l l
!
∂γ
1
{
∗ 1
β h1−ǫ +β h 1−ǫ 1 ∗ 1 β l1−ǫ +β l 1−ǫ
}
.
(η 1 −η 2)(1−ǫ) ǫ
"#
part B
−1
$
ED
We assume β H , β L , and
}
MA
ǫ2 − 1$ !ǫ "#
dpA dγ
NU
Given all the other parameters and
2 γ A , ∂ 2 Y (p∂γ T ,γ) ∂p∂γ
1
∗ 1
1
1 ∗ 1−ǫ
β h1−ǫ +β h 1−ǫ β l1−ǫ +β l
> 1 for any choice of γ by imposing parameter restrictions
PT
on bH , bL and c such that bH − c > 1 and bL − c > 1.This ensures that part B decreases in ǫ, and ǫ < 2 ensures that part A declines in ǫ. Hence, a fall in ǫ increases
∂ 2 Y (pT ,γ) 2
2 ∂γ ∂ Y (pT ,γ) ∂p∂γ
dpA dγ
if ǫ < 2 and
AC CE
bH − c > 1 and bL − c > 1. This means that for sufficiently small ǫ the objective function of the government in the open economy is quasiconvex in γ. From the first order condition for an interior optimum of autarky welfare, in order to keep γ A unchanged with a fall in ǫ , one needs to increase a(= η 2 + µ(η 1 − η 2 )). This is because, for given A
p,
∂( dU ) dγ ∂ǫ
A
< 0 and
∂( dU ) dγ ∂a
> 0. Thus, for sufficiently small ǫ, avoiding extreme preferences (µ close
to 1 or 0) and sufficiently different production processes (η 1 − η 2 sufficiently high) ensures that an interior autarky optimum exists and neither γ = 1 nor γ = 0 is a dominant strategy in the open economy. Hence, together a more progressive education system (sufficiently small ǫ), diversified preferences (an intermediate value of µ i.e. neither close to 0 or close to 1) and a large enough differences in skill intensities (η 1 − η 2 sufficiently high) ensure the existence of asymmetric PSNEs even though there is a unique interior autarky optimum. To illustrate that the ex-post gains from trade are not necessarily identical among the ex-ante identical countries, one needs to find parameter restrictions such that U T (γ = 1, γ ∗ = 0) > U T ∗ (γ ∗ = 0, γ = 1) > U A (γ A = 1).
34
ACCEPTED MANUSCRIPT Let us denote H = H e (γ = 1, γ ∗ = 0) H = H e (γ = 0, γ ∗ = 1)
RI P
L = Le (γ = 1, γ ∗ = 0).
T
L = Le (γ = 0, γ ∗ = 1)
Thus, at NE (γ = 1, γ ∗ = 0) the Home country’s endowments are H and L, and the foreign country’s endowments are H and L. Evidently, H > H and L > L.
SC
For unequal welfare at the asymmetric Nash equilibrium under trade, namely U T (γ = 1, γ ∗ = 0) > U T ∗ (γ ∗ = 0, γ = 1), the parameter restrictions are that
NU
L+L H −H > qf. H +H L−L
MA
For the country with lower welfare under the asymmetric Nash equibirum to still gain from trade, namely U T ∗ (γ ∗ = 0, γ = 1) > U A (γ A = 1), one needs that
If
H+H L L+L H
H + H L aL 1 H L+L 1 ). ) (1 + ) > (1 + L qf qf L+L H L H +H
ED
(
> 1 and a(= η 2 +µ(η 1 −η 2 )) is sufficiently large, the above restriction can be satisfied.
PT
To briefly see a case of magnification of equilibrium welfare under trade compared to autarky when countries are initially different, consider the countries are otherwise identical except for their initial endowment of skilled labor H. Suppose that the home country is less skill abundant implying
AC CE
that the relative inherent ability (or productivity) of high-skill labour in the home country ( hl ) is ∗
lower than that in the foreign country ( hl∗ ) :
Note that
κ=
U A = vq a (1 + Hence, the solution to
dU A dγ
h l h∗ l∗
< 1.
1 )(H e )a (Le )1−a . qf ∗
= 0 does not depend on ( hl ) or ( hl∗ ). The interior autarky optimum is
the same across the two countries, but the optimal value of welfare in autarky is different. ∗
γA = γA aǫ U A (γ A ) = κ ǫ−1 < 1. ∗ A A U (γ ) However, in this case, in the open economy the choice of the countries’ policies depend on their initial
35
ACCEPTED MANUSCRIPT natural comparative advantage. In particular, κ =
h l h∗ l∗
< 1 implies that in the Nash equilibrium each
country invests more in its area of natural comparative advantage, home country invests relatively more in basic education and Foreign country invests more in higher education. Let us consider the most extreme NE which aligns with their natural comparative advantage, in which the home ∗
aǫ U T (0, 1) < κ ǫ−1 , ∗ T U (1, 0)
>
bH −c bL .
SC
nH nL
AC CE
PT
ED
MA
NU
if (η 1 − η 2 ) is relatively large and
RI P
less welfare in the home country relative to the autarky implying
36
T
country chooses γ T = 0 and Foreign country chooses γ T = 1. Such a NE leads to proportionately