Path-dependent economic progress and regress: The negative role of subsidies in economic growth

Structural Change and Economic Dynamics 21 (2010) 197–205

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Path-dependent economic progress and regress: The negative role of subsidies in economic growth Tsutomu Harada ∗ Graduate School of Business Administration, Kobe University, 2-1 Rokkodai, Nada, Kobe, Japan

a r t i c l e

i n f o

Article history: Received October 2009 Received in revised form May 2010 Accepted June 2010 Available online 16 June 2010 JEL classification: O14 O33 O41 Keywords: Two-stage economic growth Economic progress and regress Deterministic AK stage Stochastic AK stage

a b s t r a c t This paper develops a two-stage economic growth model with real options and examines the effects of various subsidy policies. The economic stages are the deterministic and stochastic AK stages, and the economy may shift between the two, depending upon state variables and technological shocks. This model allows for path-dependent economic growth that accounts for both club convergence and divergence across countries. Moreover, it is shown that under certain conditions, a decrease in the subsidy rate facilitates the shift from the deterministic to stochastic AK stages, which is defined as “economic progress”, even in the face of an economic crisis, while more subsidies delay economic progress and promote the shift from the stochastic to deterministic AK stages, which is defined as “economic regress”. © 2010 Elsevier B.V. All rights reserved.

1. Introduction External shocks such as technological change, demand shifts, and policy shifts have played critical roles in the process of economic growth and development. For example, the second industrial revolution, especially in the US, was enabled by a series of innovations in machine-based manufacturing.1 World War I spurred economic growth in

∗ Tel.: +81 78 803 6957; fax: +81 78 803 6977. E-mail address: [email protected]. 1 During the 19th century, “the American system of manufactures” emerged first in the US, in which manufacturing goods were (1) produced by specialized machines, (2) highly standardized, and (3) made up of interchangeable component parts. Rosenberg (1994) describes: “While greater homogeneity of tastes was originally conducive to the introduction of goods produced according to the American system of manufactures, it is also true that, once this technology began to spread, it in turn shaped and influenced tastes in the direction of simplicity and functionality. Fur0954-349X/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.strueco.2010.06.001

the US and Japan during the 1910s,2 and Japanese economic growth during 1950s would have been impossible without the special procurement boom occasioned by the Korean War.3 Conversely, these external shocks sometimes result

thermore, although American factor endowment pushed in the direction of mechanization, the experience with mechanization in itself brought about an improvement in inventive activity and its more rapid diffusion (Rosenberg, 1994, p. 119). 2 Usually, high costs of war outweigh its positive economic effects. However, Goldstein (2003) states: “Countries that can fight wars beyond their borders avoid the most costly destruction (though not the other costs of war). For example, the Dutch toward the end of the Thirty Years’ War, the British during the Napoleonic Wars, the Japanese in World War I, and the Americans in both World Wars enjoyed this relative insulation from war’s destruction, which meanwhile weakened their economic rivals.” (p. 215). 3 After the adoption of the “Dodge Line”, an anti-inflationary policy in 1949, the Japanese economy encountered economic crisis due to the resulting deflation and money shortage. The recession caused by the Dodge Line was transformed into a boom overnight by the outbreak of

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in economic decline or slowdown, as in the cases of the bursting of the Japanese bubble economy in 1991, the Asian financial crisis in 1997, and the Lehman shock in 2008 (see for example, IMF, 2003, 2009). Although purposive R&D as an engine of economic growth has been extensively studied in endogenous growth literature, little attention has been paid to these external shocks and their relation to the process of economic growth. In contrast, the real business cycle literature focuses on external technological shocks as a cause of economic fluctuation, but in so doing, it assumes away the endogenous aspect of economic growth. This paper attempts to develop a simple model that integrates these two contrasting views of economic growth and fluctuations. That is, this paper models endogenous economic growth, while allowing for recurrent technological shocks that cause economic fluctuation and affect selection between two economic regimes: the deterministic and stochastic AK stages. In the former stage, no uncertainty exists and economic growth is driven by a standard AK model of endogenous growth. However, as financial wealth accumulates, the economy could switch to the latter stage, in which recurrent technological shocks always take place and cause economic fluctuations. Despite this, the average rate of economic growth in this stage is much higher than in the former. Thus, economic growth and development in this model are caused not only by increasing returns and structural change between the two stages, but also by external technological shocks. Indeed, technological shocks play a significant role in the switching between the two stages, which in turn affects economic slowdown as well as economic growth and development. The purpose of this paper is to examine the effects of various subsidy policies on economic growth in the two stages and the optimal timing for switching between the two stages. The role of public intervention and subsidies in economic growth has been extensively studied in both the deterministic and stochastic endogenous growth framework (see for example, Barro, 1990; Turnovsky, 1993; Corsetti, 1997). However, little attention has been paid to the role of taxation and subsidies in the process of switching between the deterministic and stochastic growth stages. We formulate this switching problem by taking a real options approach and evaluate how increases and decreases in the subsidy rates at each stage affect the optimal timing for switching. The real options approach has been extensively studied (Dixit and Pindyck, 1994), but this approach has been limited to partial equilibrium settings such as investment (Abel and Eberly, 1994), project management (McDonald and Siegel, 1985), and exchange rates (Krugman, 1991). One exception is Dixit and Rob (1994), who examined labor mobility between the two sectors in the face of technological shocks in the general equilibrium framework. But the model used in this paper differs from the latter in that the process of economic growth and development is explicitly incorporated and solved in three steps: (1) solving the deterministic AK stage, (2) solving the stochastic AK stage,

the Korean War, and “Japan settled on the road to long-term recovery” (Nakamura, 2003, p. 97).

and (3) solving the optimal timing for the switching. The real options approach is utilized in (3) alone, and (1) and (2) must be solved as optimal control and stochastic dynamic programming problems, respectively. Thus, the model used in this paper consists of a mixture of these three different techniques. It is shown that a decrease in subsidy rates in the deterministic AK stage always facilitates the shift from the deterministic to stochastic AK stages in the face of positive shocks and delays the reversion from the stochastic to deterministic AK stages in the face of negative shocks. In this paper, “economic progress” is defined as the switch from the deterministic to stochastic AK stages, since the latter stage allows for higher economic growth. Also “economic regress” refers to the reversion from the stochastic to deterministic AK stages. Although these two stages and their economic growth rates are exogenously fixed in this model, the switch between the two is endogenous. Thus, the subsidy policies do affect both “economic progress” and “economic regress” and the resulting economic growth rates. In the stochastic AK stage, the effect of an increase in tax rates on the stochastic part of net output also facilitates economic growth and delays economic decline. Moreover, when technological uncertainty reaches a sufficiently high level in this stage, the effect of a decrease in subsidy rates on net output plays a similar role in promoting economic growth, even in the face of economic crisis, since in this case the subsidies work as a form of insurance. In the standard endogenous growth models (see for example, Grossman and Helpman, 1991; Aghion and Howitt, 1998), taxes impede and subsidies promote economic growth. However, once staged economic growth and recurrent uncertainty are incorporated into the model, more taxes and a reduction in subsidies not only enable faster economic growth, but also facilitate economic recovery under certain conditions. This is one of the salient findings of this paper that shed new light on the negative role of subsidies in promoting economic growth and recovery. In related literature, Boucekkine et al. (2004) also studied the two-stage optimal control problem involving the two deterministic AK models, and Harada (2010) examined the switch from the Solow to AK economies using a similar technique. However, the model used in this paper differs significantly from the aforementioned in that uncertainty is introduced in the stochastic AK stages and bilateral, rather than unilateral, shifts between the two stages are considered. That is, the model used in this paper involves examination not only of the shift from the deterministic stage to the stochastic AK stage (“economic progress”), but also of the reversion from the latter to the former stage (“economic regress”). Consequently, the model in this paper examines the effects of various subsidy policies in the context of economic regress as well as economic progress. This also departs from the endogenous growth literature, which focuses primarily on economic growth and has tended to ignore economic decline. The rest of the paper is organized as follows. Section 2 presents a basic model of two-stage economic growth, and Section 3 evaluates the effects of subsidy policies on

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economic progress and regress. Finally, Section 4 presents our conclusions.

the stochastic AK stage. Then, the production function during this stage becomes

2. The model

dfS (k) = [S dt + dω]k.

2.1. Preferences and technology Consider a continuous time economy inhabited by a representative agent, the intertemporal utility function of which is



∞

U=

e−t

0

c(t)1−R dt, 1−R

0 < R < 1,

0 < ˇ < 1,

y = c + I = fi (k),

(2)

where AD , Y, K and J denote the productivity index, the output, capital and labor inputs, respectively, and the subscript D refers to the deterministic AK stage. For the sake of simplicity, the population is assumed to be stationary. Dividing this by J yields y = fD (k) = AD k1−ˇ ,

(3)

and AD = D kˇ ,

(4)

is assumed, where y and k are the per capita output and capital, respectively. Thus, the production function in the consumption sector during this stage becomes fD (k) = D k.

We assume S > D in what follows. If the opposite holds, there is no incentive to switch to the stochastic AK stage. Although the latter stage entails high risk, it remains attractive due to its higher returns. As S gets higher, the relative advantage of the stochastic AK stage increases. The consumption goods are used either for consumption or capital goods such that

(1)

where R is the inverse of elasticity of intertemporal substitution, and as we will see later, this must be less than unity.  refers to the time discount factor and c(t) denotes the per capita consumption level at time t. In the following notations, t is omitted unless it causes confusion. There are two stages of economic growth in this model: (1) the deterministic AK stage; (2) the stochastic AK stage.4 These stages represent different production functions and engines of growth. In the deterministic AK stage, the production function takes the form of Y = AD K 1−ˇ J ˇ ,

(6)

(5)

In the stochastic AK stage, the production function is given by dfS (k) = [S dt + dω]AS k1−ˇ , and AS = kˇ , is assumed, where df(k) is the instantaneous output flow, dω is the increment to a Wiener process with zero mean and unit variance, and S and  are positive constants that denote the instantaneous drift and standard deviation of productivity shocks, respectively. The subscript S refers to

i = D, S,

where I refers to the investment. The capital accumulation is given by k˙ = fi (k) − c, where the dot denotes the time derivative. In this specification, capital depreciation is assumed away in order to simplify the algebra and notation, but this does not change the qualitative results below. 2.2. The public sector Following Eaton (1981) and Corsetti (1997), we specify a general linear tax function as dT = i dy + ˛kdω,

i = D, S,

(7)

where T is cumulated tax revenue,  i is a time-invariant tax rate on net output in each stage, which can be called “deterministic tax rate”, and ˛ is a time-invariant tax rate on output exceeding or falling short of its expected level, which can be called “stochastic tax rate”. Note that the latter tax revenue accrues only in the stochastic AK stage, since no stochastic disturbances are assumed in the deterministic AK stage. Budget deficits are financed by issuing consols paying an instantaneous real coupon rate, u. Define B and qB as the number of consols and their prices in terms of consumption good. Then, the dynamics of public debt follow dBqB = −dT + BqB

u qB

+



dqB , qB

(8)

where dqB /qB represents capital gains on consols. Following Corsetti (1997), we make a simplifying assumption of a null government expenditure without loss of generality. To be consistent with this assumption, the time-invariant tax rate should be interpreted as subsidies such that  i < 0. However, ˛ is assumed to be positive. In the presence of a positive productivity shocks, it is equivalent to taxes, but in the presence of a negative shock, it becomes subsidies. 2.3. Financial assets

4

The qualitative results in this paper remain the same even if exogenous growth rates are assumed under constant returns to scale. However, we adopt the AK endogenous growth framework in this paper, since it seems more reasonable to allow for endogenous growth rather than exogenous growth.

We assume financial assets in this model include equity shares, consols and claim to labor income. These are freely traded in the competitive markets without transaction

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T. Harada / Structural Change and Economic Dynamics 21 (2010) 197–205

costs. Thus, the financial wealth of the representative agent is W = k + BqB + HqH ,

(9)

where H and qH are units and real price of claims to labor income.5 We assume policy parameters,  i (i = D, S) and ˛, are constant without policy changes. Hence, technology shocks (dω) are the only source of uncertainty in the stochastic AK stage. The after-tax equilibrium rate of return on each financial asset takes the form of rj dt + j dω,

j = k, B, H.

(10)

That is, the returns on financial assets consist of deterministic (rj dt) and stochastic parts ( j dω). The latter parts depend on technology shocks (dω) alone. Since technology shocks are the common stochastic element driving the returns of all financial assets, these returns are perfectly correlated.

the portfolio shares are immediately adjusted to the optimal ones in the stochastic AK stage, which are to be derived below. The objective function can be rewritten as



∞

maxUD = c

e−t

0

c(t)1−R dt, 1−R

(11)

subject to ˙ = Wrk − c = W (1 − ˇ)(1 − D )D − c. W

(12)

This is a standard optimal control problem, and solving this gives W = e−((−1(1−ˇ)(1−D )D )/R)t W0 , c=

(13)

 − (1 − R)(1 − ˇ)(1 − D )D −((−(1−ˇ)(1−D )D )/R)t e W0 . R (14)

2.4. Optimal switching In this model, the social planner is assumed to determine the timing of the switch between the deterministic and stochastic AK stages. “Economic progress” refers to the switch from the deterministic to stochastic AK stages, and “economic regress” to the one from the stochastic to deterministic AK stages. So the model in this paper differs from the related studies on the staged economic growth in that economic regress is allowed to take place. This economic regress has often been observed in the world, such as “the productivity slowdown” in the US during 1980s, “the lost decade” in Japan during 1990s, the Asian financial crisis in 1997, and “the Lehman shock” in 2008. Faced with these economic crises, some countries might have experienced structural changes, which could be regarded as the switch from high growing, but volatile to lower growing, but stable economic regimes. This switching problem can be formulated and solved using the real options approach, although the structure of the model in this paper is more complicated than the standard one. To take this approach, we need to solve the problem in three steps: (a) the optimal solutions in the deterministic AK stage; (b) the optimal solutions in the stochastic AK stage; (c) the optimal timing of the switch. Note that (a) and (b) are to be solved as if the economy stays there forever. Given these results, the optimal timing could be derived in (c). 2.4.1. Deterministic AK stage First, let us consider the optimal control problem in the deterministic AK stage. We will examine both competitive and socially optimal solutions, since the optimal tax rates internalize the AK externality in the market economy. Since no uncertainty exists and each return should be equal in this stage, the portfolio shares are indeterminate. Without loss of generality, we can assume the wealth consists of equity alone and W0 = k0 at time t = 0. However, at the time of the switch to the stochastic AK stage, we assume

5 Note that H differs from J, although both are closely related. H is the quantity of claims to labor income, while J is the labor input.

In addition, the transversality condition requires  > (1 − R)(1 − ˇ)(1 − D )D .

(15)

Regarding the socially optimal solution, the only difference lies in the fact that the social planner now internalizes the externality without imposing taxes such that the wealth accumulation proceeds as ˙ = WD − c. W

(16)

Then, the socially optimal solution becomes W = e−((−D )/R)t W0 , c=

(17)

 − (1 − R)D W0 e−((−D )/R)t . R

(18)

Thus, the socially optimal rate of growth is higher in this case, due to the internalization of externality. Then, the optimal tax rates in the market economy should satisfy D = 1 −

1 < 0. 1−ˇ

(19)

With this production subsidy, the distortion induced by the external effect of capital on labor productivity is completely offset. As a result, the competitive equilibrium achieves the socially optimal allocation. Given this result, the value function in this stage, as if the economy stays in this stage forever, is given by



VD (W0 ) = max 0

=

 1

∞

e−t

c(t)1−R dt 1−R

R 1 − R  − (1 − R)(1 − ˇ)(1 − D )D

R W0 . (20)

For these optimal solutions to exist, once again, we need (15) is satisfied. 2.4.2. Stochastic AK stage Next, let us examine solutions for the stochastic AK stage of economic growth. Consider the competitive equilibrium first. Given the initial condition, W1 , at the time of

T. Harada / Structural Change and Economic Dynamics 21 (2010) 197–205

the switch to this stage, t1 , the representative consumer solves



∞

maxE1 {c,n}

e−(t−t1 )

t1

c(t)1−R dt, 1−R

(21)

The optimal  S and ˛ are to be determined to satisfy this equality. Given these results, the expected value in this stage, as if the economy stays here forever, can be derived as



+W (1 − nk − nB )[rH dt + H dω] − cdt,

(22)

and W ≥ 0 where nk and nB denote the portfolio share of equity and consols, respectively. Since this stochastic dynamic programming problem has been extensively studied (see Eaton, 1981; Merton, 1990; Corsetti, 1997), and the derivation of the solution of the dynamic problem is long, it is reported in Appendix A. We simply state the main results as follows: nk k + (1 − nk − nB )H + nB B = , nk rk + nB rB + nH rH = c = W, nk (1 − R)  = nk − (1 − R)

S −  (n−1 k

 

(23)

− 1),

(24)



1−R

− S + 0.5R 2 ,

e−(t−t1 )

t1

dW = Wnk [rk dt + k dω] + WnB [rB dt + B dω]

nk =

∞

VS (W1 ) ≡ E1

subject to

(25) (26)

vS ≡

(1 − R)

1/(1−R)



c(t)1−R W11−R , dt = v1−R S 1−R 

(33)

1/(1−R) ,

 − (1 − R)(S − n−1 − 0.5R 2 ) k

(34)

where for the integral convergence, we need  − (1 − R)(S − n−1 − 0.5R 2 ) > 0. But it is easy to check this k inequality always holds when (28) is satisfied. However, this inequality further needs R < 1 be satisfied, since  > 0 holds. Otherwise, VS < 0 realizes, which implies that the economy never switches to the stochastic AK stage. Moreover, to make the model non-trivial, we need to have VS (W) > VD (W) at least for higher values of W. If this inequality does not hold, no switching takes place. In what follows, we assume this is always satisfied.

(1 − R)(ˇ(S − R 2 ) + (1 − ˇ) + (/(1 − R)) − S + 0.5R 2 ) , ˇ(S − R 2 ) + (1 − ˇ) + (1 − R)((/(1 − R) − S + 0.5R 2 )

 − (1 − R)(S − 0.5R 2 ) > 0,

201

(27)

(28)

− R 2 (

where  ≡  S S S + ˛) is a certainty equivalent tax rate. According to Corsetti (1997), this is a certainty equivalent tax rate since changes in  S and ˛ alter the equilibrium allocation only through changes in the values of . Even if  S and ˛ change, but  remains constant, the portfolio shares and consumption are not affected, as is obvious from (25), (26), and (27). Also, it should be noted that (28) is imposed in order to obtain positive consumption. From (22), (23), (24), and (25), the financial wealth follows )dt + Wdω, dW = W (S − n−1 k which implies E1 R)(S − n−1 k

∞

(29)

e−(t−t1 ) Wt1−R dt = W11−R /[ − (1 −

t1 − 0.5R 2 )],

where the expectation is taken conditional on the initial level of the wealth W1 at time t1 (see Dixit, 1993, p. 13). Next, consider the socially optimal solution. The social planner also maximizes (21), but the resource constraint now becomes

  dy − cdt c k˙ = = S − dt + dω. k k k

(30)

This problem can be solved using the method described in Appendix A. The first order condition is c = R−1 {(R − 1)(S − 0.5R 2 ) + }. k

(31)

The optimal tax policy equates this ratio to the competitive equilibrium consumption rate out of total capital, c/Wnk , which is given by (25)–(27). Solving this yields  =−

ˇ(S − R 2 ) . 1−ˇ

(32)

2.4.3. Optimal switching time Now, we are in a position to solve the optimal timing of the switch. While the firm is to decide the timing of the switch in a competitive equilibrium, the optimal timing of the switch is to be determined by the social planner who maximizes the social welfare. Thus, the social planner compares the social welfare between the two stages (20) and (33).6 It should also be noted that when tax rates are fixed at the optimal levels, the solution of this switching problem is indeed socially optimal. However, if tax rates are different from the optimal values, the solution of this switching problem becomes socially optimal, conditional on the competitive equilibria in the two regimes. In other words, timing is socially optimal, but the competitive equilibria in the two regimes are not Pareto optimal. Suppose that if the economy switches from one stage to another, the sunk cost h > 0 must be paid by the social planner. Denote the expected relative value in the stochastic AK stage by v(W ). According to the standard real options approach (e.g., Dixit and Pindyck, 1994), this value consists of two parts. The first, v1 (W ), is the added value of being in the stochastic AK stage rather than the deterministic AK stage forever. From (20) and (33), this is given by

v1 (W ) = VS (W ) − VD (W ),

(35)

6 If the firm is to determine the timing of the switch, it compares the expected profits between the two stages. In this case, the dynamics of k, instead of W, matters. Thus, the timing in the competitive equilibrium differs from the socially optimal one. However, since the dynamics of k and W are closely related due to the common stochastic element, the resulting economic progress and regress would show similar patterns.

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T. Harada / Structural Change and Economic Dynamics 21 (2010) 197–205

which is assumed to be positive, as we have described above. This implies the average growth rate in the stochastic AK stage is much higher than in the deterministic AK stage. The other part, v2 (W ), comprises giving up the option of staying in the deterministic AK stage, and acquiring in turn the option of staying in the stochastic AK stage, assuming rational expectations about future moves and optimal behavior in each stage, as calculated above. This is given by (see for example, Dixit and Pindyck, 1994) 2

v (W ) =

E[dv2 (W )] dt

.

(36)

The intuition for this equation is that the expected return on the asset (option value) is equal to the expected capital gain (the change in the option value). Since W follows the geometric Brownian motion (29), using Ito’s lemma, the RHS can be expanded as 0.5 2 W 2 v2WW (W ) + ( − n−1 )W v2W (W ) − v2 (W ) = 0, k (37) where the subscripts W and WW of v2 denote the first and second derivatives, respectively. The general solution to this differential equation is given by

v2 (W ) = G1 W 1 + G2 W 2 ,

(38)

where G1 and G2 are constants to be determined, and 1 and 2 are roots of the quadratic equation )x − 0.5 2 x(x − 1) = 0.

(x) ≡  − (S − n−1 k

(39)

Since  > 0, and (x) is negative when the absolute value of x is large of either sign, the roots are real and of opposite signs. Without loss of generality, assume 1 < 0 < 2 . It should also be noted that (1 − R) > 0 always holds by (28). This implies 1 < 1 − R < 2 . From (35) and (38), we have an expression for the expected relative value of staying in the stochastic AK stage as

v(W ) = v1 (W ) + G1 W 1 + G2 W 2 .

(40)

Then, two conditions must be satisfied for optimality of each switch. That is, value matching and smooth pasting conditions. The value matching conditions are ¯ ) = h, v(W

v(W - ) = −h,

(41)

¯ where W - < W denote the threshold levels for the optimal ¯ , it is optimal to switch to switching. That is, when W > W the stochastic AK stage, and when W < W - , the economy should regress to the deterministic AK stage, if it stays in ¯ the stochastic AK stage. When W - < W < W , no switching takes place. The smooth pasting conditions are given by ¯ ) = vW (W ) = 0. vW (W -

(42)

These two conditions and the limiting argument deter¯ and W . Consider the switch from the mine G1 , G2 , W deterministic to stochastic AK stages, first. In this case, ¯ , when the asset value hits the upper threshold level, W the switching takes place. As W goes to zero, the option to switch is very far from being exercised, implying that the term in the negative power of W should be absent. Other-

wise, this term goes to infinity, as W goes to zero. Thus, we have

v(W ) = v1 (W ) + G2 W 2 . From the value matching and smooth pasting conditions, we obtain ¯ 1−R− 2 , G2 = −(1 − R) 2−1 v1−R W S



¯ = W

2 2 − 1 + R

(43)

1/(1−R)

1/(1−R) v−1 (h + VD ) . S

(44)

Following a similar argument, the switch from the stochastic to deterministic AK stages requires the value function as

v(W ) = v1 (W ) + G1 W 1 . Applying the value matching and smooth pasting conditions yields 1−R− 1 G1 = −(1 − R) 1−1 v1−R W , S -



W - =

1 1 − 1 + R

(45)

1/(1−R)

1/(1−R) v−1 (−h + VD ) . S

(46)

Note that although 1 < 0, since 1 < 1 − R holds, as described above, the first term on the RHS of (46) is posi¯ tive. Since W - < W always holds from (44) and (46), if the ¯ current state variable is in the region of W - < W < W , the economy could be either in the deterministic or stochastic AK stages, depending upon its history. That is, if the economy has shifted from the deterministic to stochastic AK stages, the current stage should be the latter, while if it has remained in the deterministic AK stage or reverted from the stochastic AK stage, the current stage should be the deterministic AK stage. Thus, even if the state variable W is the same among economies, their stages could differ. This result is summarized in the following proposition: Proposition 1. lows:

The economic stage is determined as fol-

¯ < W , then the economy is in the stochastic AK stage. • If W • If W < W , then the economy is in the deterministic AK stage. ¯ , then the economy is in either the deter• If W < W < W ministic or stochastic AK stages, depending upon its historical path. In other words, economic growth in this model is pathdependent such that both convergence and divergence could co-exist among several countries in terms of growth rates and stages. Howitt (2000) and Howitt and Mayer-Foulkes (2002) also attempt to account for not only convergence but also divergence in growth rates among multiple countries. In their models, countries that either conduct or implement modern R&D converge to the same growth rate in the long run, but those that do not make R&D investments diverge from those that do. In contrast, in this paper, the growth rates of countries could differ in the steady state, even if their state variables are the same. This is because history

T. Harada / Structural Change and Economic Dynamics 21 (2010) 197–205

203

matters in this model. Thus, countries with the same state variables could belong to different growth stages or “convergence clubs”.

i.e., deterministic ( S ) and stochastic (˛) subsidies. The former is provided on the net output, while the latter on the stochastic part of net output. The effects of these subsidies on economic progress and regress are as follows:

3. Subsidy policies

Proposition 3.

In this section, various subsidy policies are examined based upon the above model. In particular, we are interested in how they influence the process of economic progress and regress. Note that, as described above, the tax in this model should be interpreted as subsidies, since the null government expenditure is assumed. To avoid unnecessary confusion, we will use “subsidy” instead of “tax”, except for stochastic taxes, in what follows. A change in the subsidy rate primarily work through the effects on the household’s portfolio allocation, which in turn affects income, consumption, and production. Since the optimal timing of the switch is to be determined by the household in this model, the change in the subsidy rate does shift the optimal timing. In general, more subsidies (lower ) will be associated with lower debt-to-capital ratios (nB /nk ), but this does not necessarily hold in some cases. For example, if R < 1, a decrease in subsidies may reduce the quantity of capital in portfolio, even though the debt-to-capital ratio is declining.

• If S < R 2 , then a decrease in deterministic subsidy rates in the stochastic AK stage facilitates economic ¯ /∂S < 0, progress and delays economic regress (i.e., ∂W ∂W /∂ < 0). S • If S > R 2 , then an increase in deterministic subsidy rates in the stochastic AK stage facilitates economic ¯ /∂S > 0, progress and delays economic regress (i.e., ∂W > 0). ∂W /∂ S • An increase in stochastic tax rates in the stochastic AK stage always facilitates economic progress and delays ¯ /∂˛ < 0, ∂W /∂˛ < 0). economic regress (i.e., ∂W -

3.1. Subsidies in the deterministic AK stage First, let us examine the effect of subsidies in the deterministic AK stage in the context of economic progress and regress. Although a decrease in subsidies obviously reduces the value of this stage, this in turn increases the relative advantage of the stochastic AK stage. As a result, reducing subsidies in this stage always facilitates the shift to the stochastic AK stage. Even in the face of negative shocks, a decrease in subsidies enhances the value of remaining in the stochastic AK stage, which delays reversion to the deterministic AK stage if the economy has already shifted to the former. This result is summarized in the next proposition. Proposition 2. In the deterministic AK stage, a decrease in ¯ /∂D < subsidies always facilitates economic progress (∂W 0) and delays economic regress (∂W /∂ < 0). D Proof. See Appendix B. Note that an announcement of subsidy decreases is effective even if the economy stays in the stochastic AK stage, since this announcement alters the threshold levels, ¯ and W . Therefore, this subsidy policy could be carried W out, regardless of the current economic stage. In addition, although the divergence from the optimal subsidy rate reduces the social welfare in the deterministic AK stage, if the economy remains in the stochastic AK stage, this announcement does not directly affect social welfare, while it delaying economic regress. 3.2. Subsidies in the stochastic AK stage Next, consider the subsidy effects in the stochastic AK stage. In this stage, two kinds of subsidies are available,

In the stochastic AK stage,

Proof. See Appendix C. Surprisingly, the results indicate that imposing lower subsidies in this stage under certain conditions also facilitates economic progress, but the economic reasoning for these results is quite intuitive. An increase in stochastic tax rates implies that when the economy encounters negative shocks, tax reduction takes place, while more taxes are levied in the presence of positive shocks. Thus, the stochastic tax works as insurance that smoothes consumption over time. Since the representative agent is risk averse, an increase in stochastic tax rates always enhances the value of the stochastic AK stage. Second, a decrease in deterministic subsidy rates usually reduces the value of the stochastic AK stage. However, if uncertainty is high enough, a deterministic subsidy also works as insurance since it is provided on the stochastic part of net income as well. Indeed, a certainty equivalent tax rate decreases with a reduction in deterministic subsidy rates under high uncertainty, since ∂/∂ S = S − R 2 . Thus, a reduction in deterministic subsidies in this situation implies a decrease in taxes in the face of negative shocks (note that subsidies in this case become taxes if dω < 0). Due to risk aversion, this reduction in deterministic subsidies increases the value of the stochastic AK stage. If subsidies are introduced in the standard endogenous growth models (see for example, Grossman and Helpman, 1991; Aghion and Howitt, 1998), the growth rate must increase, since entrepreneurs with subsidies could make more R&D investment in these models. In contrast, this paper predicts that reducing subsidies facilitates economic progress, and hence, economic growth. This counterintuitive result is due to the “insurance” function of the subsidies in volatile environment. However, it should be noted once again that a change in the subsidy rates implies a diversion from the Pareto optimal allocation in the market economy. If the economy is in the deterministic AK stage, these changes do not directly affect the economy until it switches to the stochastic AK stage. The goal of these subsidy policies should be facilitating economic progress, or avoiding the reversion from the stochastic to deterministic AK stages, rather than maximizing the expected utility of the representative agent.

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T. Harada / Structural Change and Economic Dynamics 21 (2010) 197–205

rk − rH = R(k − H )[nk k + (1 − nk − nB )H + nB B ], (A4)

4. Concluding remarks This paper developed a model of path-dependent economic progress and regress and examined the effect of subsidies in two stages. This model differs significantly from existing growth models in that recurrent technological uncertainty and staged growth are introduced. Three main results were obtained. First, the model allows for path-dependent economic growth that accounts for both club convergence and divergence. In this model, even if the state variables are the same across the countries, since optimal switching depends upon not only the state variables, but also on history, the resulting stages might differ. Club convergence is enabled by similar histories as well as the similar state variable in this model. Second, it is shown that a decrease in the subsidy rate in the deterministic AK stage always facilitates economic progress and delays economic regress. Thus, regardless of whether the economy is in the process of economic growth or decline, an announcement of a reduction in subsidies would appear to be an effective measure in order to facilitate economic progress. Third, an increase in the stochastic tax rates also facilitates economic progress and regress, and a decrease in the deterministic subsidy rates in the stochastic AK stage has a similar effect when technological uncertainty is sufficiently high. This is because stochastic taxes and deterministic subsidies work as insurance, allowing realization of higher social welfare with risk aversion. This negative aspect of subsidies under certain conditions has not received much attention in previous studies. However, in situations in which recurrent uncertainty is significant, this negative role of subsidies should be taken into account in designing growth policies. Acknowledgements

rB − rH = R(B − H )[nk k + (1 − nk − nB )H + nB B ]. (A5) Regarding rk and  k , since rk dt + k dω =

dY − dT = (1 − ˇ)(1 − S )S dt dk +(1 − ˇ)[1 − (S + ˛)]dω,

it is immediate to obtain. rk = (1 − ˇ)(1 − S )S ,

(A6)

k = (1 − ˇ)[1 − (S + ˛)].

(A7)

In the steady states, we have





dW dk c dqB B dqH H = dt + dω = = . = S − W qB B qH H k k Substitute this into (8) yields rB =  S −



B = 1 +

n c + k S S , nB k

(A8)



nk (S + ˛) . nB

(A9)

Then, substituting (A6), (A7), (A8), and (A9) into (22) gives rH =



S −

H =  +

c k



+

(1 − S )ˇnk S , 1 − nk − nB

(A10)

ˇ[1 − (S + ˛)]nk . 1 − nk − nB

(A11)

Note that while rk and rB were derived explicitly, rH was obtained by substituting rk and rB into the wealth accumulation equation (22). This is because the production function (6), is represented as a function of the per capita capital. From (A6) to (A11), we can calculate nk k + (1 − nk − nB )H + nB B = .

I would like to thank two anonymous referees for very useful comments and suggestions. I also benefited from the comments by seminar participants at Rokko seminar, Kobe University. This paper has been financially supported by the research grant at the Graduate School of Business Administration, Kobe University. All errors in this paper are mine.

nk rk + nB rB + nH rH =

S −  (n−1 k

(A12)

− 1).

(A13)

Thus, substituting these into (A4), (A5) and (22) yields rk − rH = R(k − H ),

(A14)

rB − rH = R(B − H ),

(A15)

dW =

W (S − n−1 )dt k

+ Wdω.

(A16)

Appendix A. Solutions for the stochastic AK stage

Using (A16), (A1) and (A2), we can derive the expression for  as

Given the stochastic dynamic programming in the text, the Bellman equation becomes

 =



V = maxE c,n



c 1−R + VW dW + 0.5VWW dW 2 , 1−R

(A1)

where the subscripts indicate the partial derivatives. Then, we can guess the value function as  −R W 1−R . V (W ) = 1−R

(A2)

The first order conditions with respect to c, nk , and nB give c = W,

(A3)

nk (1 − R) nk − (1 − R)

 

1−R



− S + 0.5R 2 .

(A17)

In order to exclude the negative consumption, we need to have  > 0. By inspection of (A18) below, we can see nk > 1 − R is satisfied. Given this result,  > 0 holds if  − (1 − R)(S − 0.5R 2 ) > 0.

(A18)

Finally, using (A17) and  ≡  S S − R 2 ( S + ˛), we can solve (A14) and (A15) for nk and nB as nk =

(1 − R)(ˇ(S − R 2 ) + (1 − ˇ) + (/1 − R) − S + 0.5R 2 ) , ˇ(S − R 2 ) + (1 − ˇ) + (1 − R)((/1 − R) − S + 0.5R 2 )

(A19)

nB =

R . ˇ(S − R 2 ) + (1 − ˇ) + (1 − R)((/1 − R) − S + 0.5R 2 )

(A20)

T. Harada / Structural Change and Economic Dynamics 21 (2010) 197–205

Appendix B. Proof of propositions 2 From (20) and R < 1, it is easy to derive ∂VD /∂ D < 0. Since  D appears only through VD in (44) and (46), this implies ¯ /∂D < 0 and ∂W /∂D < 0 always holds. Q.E.D. that ∂W Appendix C. Proof of propositions 3 Differentiating nk with respect to  yields ∂nk ∝ −R(1 − ˇ){ − (1 − R)(S − 0.5R 2 )} < 0. ∂ Substituting (26) into (34), through some algebra, we obtain

vS ∝

1 [(1 − R){nk − (1 − R)}−R nRk ]

1/1−R

.

Differentiating this with respect to nk yields ∂vs /∂nk > 0. Hence, we have ∂vs /∂ < 0. But according to (44) and ¯ /∂ > 0 and ∂W /∂ > 0. So (46), this implies that ∂W an increase in certainty equivalent taxes always raises the threshold levels. Then, since  ≡  S S − R 2 ( S + ˛), it is immediate to obtain ∂/∂˛ < 0 and ∂/∂S = S − R 2 . ¯ /∂˛ < 0 and ∂W /∂˛ < 0 always hold, Hence, we obtain ∂W ¯ /∂S < 0 and ∂W /∂S < 0 are also and if nS − R 2 < 0, ∂W satisfied Q.E.D. References Abel, A.B., Eberly, J.C., 1994. A unified model of investment under uncertainty. American Economic Review 84, 1369–1384. Aghion, P., Howitt, P., 1998. Endogenous Growth Theory. The MIT Press, Cambridge. Barro, R.J., 1990. Government spending in a simple model of endogenous growth. Journal of Political Economy 98, S103–S126.

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