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History versus expectations: Increasing returns or social influence?
The Journal of Socio-Economics 35 (2006) 877–888
History versus expectations: Increasing returns or social influence? Franz Wirl a,∗,1 , Gustav Feichtinger b,1,2 a
b
University of Vienna, Chair of Industry, Energy and Environment, Bruennerstrasse 72, Room 123, 1210 Vienna, Austria Technical University of Vienna, Chair of Operations Research and Nonlinear Dynamical Systems, Argentinierstr. 8/105-4, Austria Received 15 December 2003; accepted 21 November 2005
Abstract Krugman [Krugman, P., 1991a. History versus expectations. Quarterly Journal of Economics 106, 651–667.] links multiple equilibria (the domains of attractions depend on history and possibly also on expectations) to the technological property of increasing returns to scale. It is shown that the recently introduced characterization of (non-moderate) social influence [Glaeser, E.L., Sacerdote, B.I., Scheinkman J.A., 2003. The social multiplier. Journal of the European Economic Association 1, 345–353.] coupled with complementarity provides the proper criterion for thresholds and multiple equilibria instead of technological characterizations (increasing returns). © 2006 Elsevier Inc. All rights reserved. JEL classification: D90; C62; D62 Keywords: History dependence; Increasing returns; Social influence; Instability; Focus versus node
1. Introduction The widely quoted work of Krugman (1991a) stresses two conditions why the long run outcome of a competitive economy can depend on history and possibly also on expectations: increasing returns and an externality. The first condition is of course no surprise given Krugman’s extensive use of increasing returns in his work on strategic trade policy (e.g. Krugman, 1980) and geography ∗ 1 2
Corresponding author. Tel.: +43 14277 38101; fax: +43 14277 38104. E-mail addresses: [email protected] (F. Wirl), [email protected] (G. Feichtinger). Both authors acknowledge helpful comments from an anonymous referee and suggestions by the editor (Prof. Altman). Tel.: +43 1 58801 11927; fax: +43 1 58801 11999.
1053-5357/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.socec.2005.11.058
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(e.g. Krugman, 1991b). This linkage between increasing returns and history dependent economic evolutions (even for planning problems, i.e., absent any externality) is a significant research program in economics with obvious implications to neighboring fields like sociology, see, e.g. Altman (2000); this point has been popularized by Arthur (1989, 1994). The second reason given in Krugman (1991a), that of an externality, relates to the concept of (moderate) social influence, which has been recently introduced within a static context in Glaeser and Scheinkman (2003) by generalizing earlier works in particular that of Cooper and John (1988); Durlauf and Young (2001), Mailath and Postlewaite (2003) and DeMarzo et al. (2003) are related recent approaches to social interactions and Glaeser et al. (2003) applies the concept to quantify social multipliers. Krugman (1991a) tells his story in terms of a two-sectors economy in which “manufacturing” exhibits increasing and “agriculture” constant returns to scale. In order to avoid trivial solutions, “manufacturing” pays lower wages than agriculture at low scales (more precisely, if only few work in manufacturing) but higher wages if many (and in particular if all) work in manufacturing. This dependence of own pay on others’ actions is the (only) externality in this economy. In addition, the competitive agents face adjustment costs for any expansion of the time working in one of the sectors, which introduces dynamics (i.e., the competitive economy is described by two differential equations plus a transversality condition). Our paper extends the seminal work of Krugman (1991a) by allowing for a general objective instead of a specific one. An alternative interpretation is that our framework introduces dynamics into the so far static models of social interactions. The objective is to investigate within a general dynamic adjustment-cost framework whether ‘increasing returns to scale’, as claimed in Krugman (1991a) and in many follow-ups and related papers, or sufficiently strong social influences (also present but ignored in Krugman, 1991a) lead to multiple equilibria with domains of attraction depending on history and/or possibly also on expectations. Or put differently: do the results obtained in static models of social interactions survive the extension to adjustment dynamics? It turns out that the properties of the static theory – moderate social influence is sufficient for a unique competitive equilibrium – are robust and extend to dynamic adjustment cost models while Krugman’s (1991a) claim does not extend since increasing returns are in general neither necessary nor sufficient for history dependent evolutions. In other words, strong social interactions (plus complementarity) are a trigger for different long run equilibria and thresholds but not increasing returns. Of course, there are other reasons for multiple competitive equilibria: frictions in overlapping generations models (Kehoe and Levine, 1985), ‘sunspots’ (e.g. in Grandmont, 1985), and indeterminacy in endogenous growth models (e.g. in Benhabib and Perli, 1994). The paper is organized as follows. The framework is introduced in Section 2, the social optimum is characterized in Section 3, and the competitive equilibrium is derived and analyzed in Section 4. A summary concludes. 2. Framework The economy consists of identical competitive agents (with the aggregate normalized to one) solving the following intertemporal (using the constant discount rate r > 0) optimization problem ∞ c max (1) e−rt p(x(t), y(t)) − u(t)2 dt, {u(t)} 0 2 y˙ (t) = u(t),
y(0) = y0 .
(2)
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The individual control u(t) describes the adjustment (‘investment’) of the private stock, y(t), that induces costs,1 which are assumed to be quadratic (for reasons of simplicity only). p describes the agent’s net instantaneous benefit that depends on the private stock, e.g. capital, and on an economywide externality x(t). The externality, a given datum for each private agent, is determined by the aggregate over the private stocks and this aggregate (or average) equals the individual private stock x(t) = y(t) ∀t ≥ 0,
(3)
due to the above normalization and the assumption of symmetric equilibria. A static game with the payoffs p(x, y) is the limiting case (number of identical agents →∞) of the model in Cooper and John (1988) so that the above embedding within an adjustment-cost framework provides a simple dynamic version. An example for a positive externality, px > 0, is that productivity of private capital is enhanced with respect to the capital intensity of the entire economy. Pollution and/or a reduction of natural areas associated with an expansion of capital (or associated production) are examples of negative external effects, px < 0. These two examples highlight that externalities need not be ‘social’ and are used, e.g., in the endogenous growth literature (starting with Romer (1986), see also the survey and the classification in Benhabib and Rustichini (1994)). Externalities of the type (3) are dynamic in the sense that they are linked to a stock, while flow externalities, e.g. due to conspicuous consumption (Rauscher, 1997; Corneo and Jeanne, 1997, 2001; Futagami and Shibata, 1998 but see also Hirsch, 1976), are static and can thus not induce history dependence (Fisher and Hof, 2000); Wirl (2002) considers truly dynamic externalities, x˙ = g(x, y), in an otherwise related adjustment-cost framework but focuses on limit cycles (thereby in particular refuting Krugman (1991a) conjecture that limit cycles may be associated in a higher dimensional case with the unstable state). The economy-wide objective (internalizing the externality) P(y) ≡ p(y, y), is relevant for a social planner who will ∞ c max exp(−rt) P(y(t)) − u(t)2 . {u(t)} 0 2
(4)
(5)
2.1. Assumptions Existence of competitive equilibria requires non-increasing returns in the private stock, i.e., pyy < 0,
(6)
and the assumed strict concavity ensures that an interior solution satisfying py (x, y) = 0 determines the static choice, the maximum of p(x, y) with respect to y; otherwise, pyy = 0, the agent’s static choice is at the boundary of the feasible set of y. Instead of (joint) concavity of p(x, y), the weaker condition P (y) ≡ pxx (x, y) + 2pxy (x, y) + pyy (x, y)|x=y < 0
(7)
1 This adjustment cost framework (but absent the externality) dates back to Eisner and Strotz (1963) and Lucas (1967); the survey of Hamermesh and Pfann (1996) emphasizes empirical issues.
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is used to characterize concave economies (i.e., diminishing returns), while P > 0 stands for increasing returns. However, the aggregate economy needs not satisfy this law of diminishing returns. 2 Furthermore, the set of states satisfying P = 0 is finite, i.e., the social objective P has a finite number of local extrema, and the maximum maximorum is unique. This assumption rules out that the maximum of P is assumed along an interval, which induces indeterminacy already for the static problem and hence a fortiori for the dynamic social choice problem. And of course, all functions are assumed to be at least twice differentiable. 2.2. Example Krugman (1991a) Agents move between an ‘agricultural’ (subscript a) and a ‘manufacturing’ sector (subscript m), which pays higher wages if sufficiently many agents work already in factories (due to the increasing returns in manufacturing). The corresponding objective (of the representative agent) is p(x, y) = ywm (x) + (¯y − y)wa ,
(8)
where wa and wm are the wages paid in agriculture (constant, wa ) and in manufacturing respectively, wm (0) < wa , wm (¯y) > wa , wm > 0, wm = 0; 0 ≤ y ≤ y¯ = 1 denotes the time (total individual labor supply is given an equal to 1) spent in the factory and 0 < x˜ < 1 denotes the critical participation level at which manufacturing pays the same wage as agriculture, wm (˜x) = wa . The objective (8) is not jointly concave with respect to (x, y) and in fact P (y) is globally convex, P = 2wm > 0. Fukao and Benabou (1993) correct some of Krugman’s formal errors and Liski (2001) is a recent variant that attributes increasing returns to lower transaction costs in ‘thick’ markets for trades in CO2 permits. 2.3. Social influence Many papers account for the social impact of a reference group, here the aggregate or average over all agents. Yet it is only recently that Glaeser and Scheinkman (2003) introduces explicitly the condition of moderate social influence (see also Glaeser et al., 2003), which boils down in our framework to pxy (9) p < 1. yy
This means that the marginal benefit from the private stock (py ) is more sensitive to adaptations of the own compared to an increase of the average stock; of course, this inequality (9) needs only to hold along x = y (for the assumed symmetric equilibrium). This property of moderate social influence according to (9) is sufficient for a unique equilibrium in the static framework of Glaeser and Scheinkman (2003). Therefore, an alternative interpretation of the model (1)–(3) is that it is a first, explicitly dynamic version of Cooper and John (1988) and Glaeser and Scheinkman (2003) for ‘many’ agents. The existence of such a measure of social influence, whether of ‘moderate’ size or not, requires the strict inequality, pyy < 0, as assumed in (6). The example in Krugman (1991a) implies pyy = 0 2 Indeed (almost) all related papers on thresholds in socially optimal developments assume increasing returns, at least locally, see, e.g. the survey in Deissenberg et al. (2004).
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and pxy = wm ≥ 0 and thus infinite instead of moderate social influence. It is important to note that concavity and non-moderate social influence are not exclusive, and conversely, increasing returns for the economy at large can be compatible with moderate social influence. Therefore, the familiar attribution of the interesting properties of multiple equilibria depending on history and possibly also on expectations to increasing returns is questionable. This will be investigated in detail below. 3. Social optimum The social optimum requires solving a dynamic optimization problem – (5) subject to the dynamic constraint (2) – due to the possibility to control the externality (or the social influence). Defining the corresponding current value Hamiltonian, H ≡ P(y)−c/2(u)2 + λu in which λ denotes the costate, the first order optimality conditions (see, e.g. Seierstad and Sydsaeter, 1986) are: Hu = −cu + λ = 0
⇒u=
λ c
λ˙ = rλ − Hy = rλ − P .
(10) (11)
Substituting the optimal control from (10) into (2) yields y˙ =
λ , c
(12)
˙ which together with (11) determines the canonical equations. Therefore, steady states (˙y = 0 = λ) of the intertemporal, socially optimal program require that P (y) = px (x, y) + py (x, y)|x=y = 0,
(13)
i.e., they are characterized by (possibly local) extrema of the (static) objective P. The Jacobian of the canonical equations and its determinant are ⎡ ⎤ 1 P 0 A=⎣ . (14) c ⎦ ⇒ det(A) = c −P r Therefore (saddlepoint) stable steady states (det(A) < 0) must be located in the concave domain of P. Conversely, an unstable steady state y∞ is only possible if P is convex at y∞ due to det(A) > 0. Hence: Proposition 1. A strictly concave social objective, P(y) ≡ p(y, y) and P < 0, rules out unstable steady states and thus history dependent multiple equilibria,3 so that (at least local) convexities, P > 0, are necessary (but not sufficient, unless at a steady state) for the existence of multiple equilibria and history dependence.
3 This follows also directly from a theorem in Feichtinger and Wirl (2000), which shows that either growth in the state differential equation or a control-state intercation are necessary for an unstable steady state in concave optimization problems.
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4. Competitive equilibrium Defining the Hamiltonian (using again λ to denote the costate, which, of course is different from the one of the previous section) for an individual agent’s intertemporal optimization problem, c H ≡ p(x, y) − u2 + λu, 2 the necessary optimality conditions are Hu = −cu + λ = 0,
(15)
λ˙ = rλ − py (x, y).
(16)
Therefore, the dynamic, symmetric competitive equilibrium is described by: y˙ =
λ , c
y(0) = y,
λ˙ = rλ − py (y, y),
(17) (18)
due to the identity (3) between the individual stock y and the economy-wide externality x (for agents with rational expectations). The y˙ = 0 isocline is given by λ = 0, and λ˙ = 0 ⇔ λ = py /r. Hence, a steady state must satisfy py (x, y)|x=y = 0.
(19)
i.e., the static sub-objective p is maximized with respect to the individual stock y, which corresponds also to the static equilibrium choice (i.e., for no adjustment costs, c = 0, and an interior solution, pyy < 0). As a consequence, the stationary solution of dynamic, intertemporal adjustments is identical to the static choice of y. The reason is that this ideal state can be reached at ‘no’ costs for infinitesimally small adjustments over time. An immediate consequence of this steady state condition (19) is that multiple interior equilibria require increasing and decreasing parts of py . The Jacobian of the competitive equilibrium (of course evaluated at a steady state) is
0 1/c J= . (20) −(pyy + pyx ) r Therefore, tr(J) = r
(21)
so that the eigenvalues tr(J) ± tr(J)2 − 4 det(J) εi = , (22) 2 are symmetric around r/2. Hence, indeterminacy (=both eigenvalues are negative or have negative real parts) and the related case of limit cycles due to Hopf bifurcations (existence of a pair or purely imaginary eigenvalues) are impossible. The determinant is det(J) =
pyy + pyx c
(23)
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so that the sign of (pyy + pyx ) determines not only the sign of the slope of the λ˙ = 0 isocline but also the sign of det(J) and thus whether a steady state is stable or not. Proposition 2. Any steady state y∞ of the competitive equilibrium, i.e., of the differential Eqs. (17) and (18), is unstable iff pyy (x, y∞ ) + pxy (x, y∞ )|x=y∞ > 0, This in turn implies non-moderate social influence (at least locally) pxy (x, y∞ ) p (x, y ) > 1 at x = y∞ , ∞ yy
(24)
(25)
and pyx (x, y∞ )|x=y∞ > 0,
(26)
i.e., marginal benefit of the private stock increases if the average x is increased (i.e., y and x are complementary or supermodular). Proof. An unstable steady state is necessary for multiple equilibria and an associated threshold. Since an unstable steady state y∞ is characterized by a positive determinant of the Jacobian, the first claim of the proposition follows directly from (23). This inequality (24), omitting the arguments, is equivalent to pxy > −pyy ⇔ −
pxy > 1, pyy
(27)
which implies (25) and thus non-moderate social influence. Inequality (26) follows directly from (27) and the assumption (6). QED. Therefore, any (saddlepoint) stable steady state must be located on a downward sloping part of the λ˙ = 0 isocline and an unstable one on an upward sloping part. The above characterizations (Proposition 2) of an unstable steady state and thereby of a threshold separating the domains of attracting stable steady states highlight the following: First, moderate social influence excludes unstable and thus also multiple steady states, but non-moderate social influence is not sufficient for an unstable steady state. Second, complementarity is necessary for an unstable steady state and thus for a threshold, a property already stressed in Cooper and John (1988). Third, both conditions – non-moderate social influence and complementarity – present at a steady state are sufficient for a history dependent development. Comparing the steady state condition of the social optimum, (13), with the one of the competitive equilibrium, (19), yields immediately the familiar implication (analogous in Cooper and John, 1988) that competition leads to under-investment in stocks with positive externalities, px > 0, and to over-accumulation of stocks with negative externalities, px < 0. Furthermore, decreasing returns of the aggregate economy, P < 0, ensure a unique and stable long run social optimum. Krugman’s special case – pyy = 0, pxy = wm > 0, pxx = 0 thus P = 2wm > 0 and det(J) = wm /c > 0 – assumes economy-wide increasing returns located within in the manufacturing sector. And this condition is indeed necessary in Krugman’s model to obtain an unstable steady state and an associated threshold that separates the domains of convergence to either of the two boundaries, y = 0, y = 1, due to the state constraints, 0 ≤ y ≤ 1. However, this specification implies not only increasing returns but simultaneously a very large (actually infinite) social influence and given
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that these two characteristics hold simultaneously it is difficult to see which one is responsible for history dependence. 4.1. Example (diminishing returns) and existence In order to stress the significance of social influence for the existence of multiple equilibria within the adjustment cost framework, we assume decreasing returns to scale at the level of the aggregate economy. This assumption ensures a unique steady state for the social optimum so that the existence of a threshold results then from (inefficient) competition, and, according to Proposition 2, from sufficient social interaction and complementarity. The reason for these marked differences in the evolution is that the condition for an unstable steady state of the competitive equilibrium, inequality (24) in Proposition 2, pyy + pxy > 0,
(28)
is weaker than the assumption of economy-wide (strict) concavity: P (y) ≡ pxx + 2pxy + pyy < 0.
(29)
Therefore, it is conceivable to have diminishing returns for the economy at large and thus a unique socially efficient long run outcome (this holds for the adjustment cost framework due to Proposition 1) but multiple steady states separated by thresholds for the competitive economy, e.g. if pxx is sufficiently negative. The existence of such outcomes can be easily shown, for example, by using a quadratic p that satisfies the above two inequalities (and is possibly only a local approximation that extends beyond the crucial domain in a way that allows for a positive marginal benefit globally). The example in Fig. 1 is based on a direct extension of Krugman’s model: p(x, y) = W(x, y) + wα (1 − y),
(30)
i.e., the payoff for working in manufacturing is given by a ‘production’ function W with the following specification of W(x, y) = xα y1−α + µxy − vx2 ,
0 < α < 1,
ν > 0.
(31)
The term {µxy − νx2 } accounts for positive (e.g. knowledge and skills increase with the number of workers in industry) and negative (e.g. due to crowding or pollution) spillovers. Since
Fig. 1. W = xα y1−α + µxy − νx2 , c = 0.5, r = 1/4, α = 0.5, µ = 0.1, wa = 0.55.
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P = 2(µ − ν), P is strictly concave for µ < ν. As a consequence, the welfare maximization has a unique interior and (saddlepoint) stable steady state (of course, a node) y∞ = 1 − wa /ν − µ. However, since Wxy µ pxy < −1, = = −1 − pyy Wyy α(1 − α)
(32)
the degree of social influence is non-moderate for any µ > 0. Conversely, assuming µ < 0 implies moderate social influence and yet a convex social objective for ν < µ < 0. This observation confirms that increasing returns to scale and social influence are two entirely different concepts. A proper choice of model parameters ensures that the wages in agriculture, wa , and in manufacturing, given by wm = ∂W(x, y)/∂y|x=y = (1 − α) + µy, intersect for 0 < y = x˜ < 1. This intersection determines a steady state, y∞ = x˜ , which is an unstable node such that the long run outcome, y → ±1 (due to the state constraints, 0 ≤ y ≤ 1) is completely determined by history, i.e., whether initially y0 x˜ . Fig. 1 shows a typical outcome. This figure is actually ‘drawn’ using numerical calculations (˜x = 1/2), which in turn verifies the existence claim in Proposition 3 below. Proposition 3. Thresholds separating multiple long run competitive outcomes are possible even if the law of diminishing returns holds (globally) at the aggregate level. Therefore, the law of diminishing returns allows for multiple equilibria and thus in particular for history dependence, but moderate social influence or substitutability, pxy < 0, rule this out. This raises the question why not only Krugman (1991a) but also all later following papers (and Liski (2001) is just a particular recent example) use and emphasize the property of increasing returns (plus an externality4 ) to obtain history dependent outcomes, although it is actually not necessary (but admittedly often helpful). 4.2. Focus versus node, expectations versus history In the following it is investigated whether an unstable steady state, which is now assumed to exist, is either a node or a focus (i.e., complex eigenvalues, thus a negative discriminant in (22)). These at first sight purely mathematical local stability properties have important economic consequences whether history (in case of a node) or in addition expectations (in case of a focus) determine the long run outcome. The reason for the latter is that the solution spirals (due to complex eigenvalues) to either of the (saddlepoint) stable steady states overlap for an interval of initial conditions y0 such that history is insufficient to determine the future evolution (see Fig. 2). This additional dependence on expectations (consistent, yet potentially self-fulfilling) may require coordination, which provides the core argument of Farmer’s (1999) approach to macroeconomics. A focus (complex eigenvalues) requires for the social optimum a local convexity (thus increasing returns) of a sufficient magnitude, P = pxx + 2pxy + pyy |x=y >
cr 2 4
at a steady state y = y∞ ,
because of the Jacobian (14) and its eigenvalues (replacing J by A in formula (22)).
4
In fact, not even externalities are necessary in a more general framework, see Feichtinger and Wirl (2000).
(33)
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In contrast, a competitive equilibrium can give rise to a focus for much less stringent requirements about the returns to scale cr 2 (34) at a steady state y∞ 4 This follows from a negative discriminant in (22), from tr(J) = r and from det(J) given by (23). Inequality (34) demands sufficiently large and definitely non-moderate social influence: pxy cr 2 (35) p > 1 − 4p > 1. yy yy pyy + pxy |x=y >
Nevertheless it seems possible to obtain a focus for an economy with non-increasing returns, in particular, for small adjustment costs c that do not affect the steady state at all. Another possibility is to choose a small discount rate. Thus, small discount rates and small adjustment costs increase the complexity and in addition make the long run outcome dependent on expectations (at least locally within the overlap, see Fig. 2), while high discount rates and/or large adjustment costs reduce the determination of the long run outcome to history. This is surprising from a formal point of view (but well explained in Krugman, 1991a) since both, high discount rates and high adjustment costs, are factors fostering complex patterns, e.g. limit cycles, in Feichtinger et al. (1994) and Wirl (2002), and chaos, according in Sorger (1992). An interesting implication is that economies with agents characterized by low discount rates and small adjustment costs (such that a focus results) increase the set of initial conditions that allow to reach the more efficient high stock long run outcome. Proposition 4. A focus requires sufficiently large increasing returns for the social optimum, P > cr2 /4, but it can arise (and exists) in the competitive case even for assuming decreasing returns (for the aggregate economy) if −
pxy(x, y∞ ) cr 2 >1− pyy (x, y∞ ) 4pyy (x, y∞ )
at a steady state y∞ and x = y∞ .
That is, social influence is sufficiently non-moderate and private stock and externality are complements, pxy > 0. Fig. 2 shows a corresponding example of a focus based on (31) and the same parameters as in Fig. 1 except for smaller adjustment costs and thus of an overlap within the domain of decreasing returns to scale. Both long run outcomes y∞ = 0 or 1 can be reached within in this overlap, if all
Fig. 2. W = xα y1−α + µxy − νx2 , c = 0.2, r = l/4, α = 0.5,µ = 0.1, wa = 0.55.
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agents bet on the same (due to assumed symmetry), either y = 0 or 1, and choose the associated adjustment path u = λ/c, either the lower or the upper one in Fig. 2. These two possibilities (circle or square) are shown for one initial condition y0 within the overlap. This highlights that history is insufficient do determine the steady state outcome for any initial condition y0 within this overlap, and that the kind of expectation is ultimately decisive. 5. Summary This paper reconsidered Krugman (1991a) with respect to the economics and mechanics of thresholds and multiple equilibria in competitive economies. Krugman (1991a) emphasizes the importance of two characteristics – increasing returns to scale for the aggregate economy and an externality – for multiple long run outcomes depending either on history and/or on expectations. This paper in contrast shows that the assumption of increasing returns, although helpful in many cases, is not crucial. Sufficient degree of social interactions, non-moderate in the terminology introduced in Glaeser and Scheinkman (2003), and complementarity (raising the economy’s average increases an individual’s marginal utility, stressed already in Cooper and John (1988)) trigger thresholds and multiple equilibria. Furthermore, the property of non-moderate social influence (and thus the instability of a corresponding steady state) can arise for strict concave economies, i.e., economies satisfying even at the aggregate level the law of diminishing returns; and conversely, moderate social influence is compatible with increasing returns. Therefore, increasing returns are not, as often suggested, the source of history dependence, and are in fact neither necessary nor sufficient. References Altman, M., 2000. A behavioral model of path dependency: the economics of profitable inefficiency and market failure. Journal of Socio-Economics 29, 127–145. Arthur, W.B., 1989. Competing technologies, increasing returns, and lock-in by historical events. Economic Journal 99, 116–131. Arthur, W.B., 1994. Increasing Returns and Path Dependence in the Economy. University of Michigan Press, Ann Arbor. Benhabib, J., Perli, R., 1994. Uniqueness and indeterminacy: on the dynamics of endogenous growth. Journal of Economic Theory 63, 113–142. Benhabib, J., Rustichini, A., 1994. Introduction to the symposium on growth, fluctuations, and sunspots: confronting the data. Journal of Economic Theory 63, 1–18. Cooper, R., John, A., 1988. Coordinating coordination failures in Keynesian models. Quarterly Journal of Economics CII, 441–463. Corneo, G., Jeanne, O., 1997. On relative wealth effects and the optimality of growth. Economics Letters 54, 87–92. Corneo, G., Jeanne, O., 2001. On relative wealth effects and long-run growth. Research in Economics 55, 349–358. Deissenberg, C., Feichtinger, G., Semmler, W., Wirl, F., 2004. History dependency and global dynamics in intertemporal optimization models. In: William, A.B. (Ed.), Economic Complexity: Non-Linear Dynamics, Multi-Agents Economies, and Learning. North Holland, Amsterdam, pp. 91–122. DeMarzo, P.M., Vayanos, D., Zwiebel, J., 2003. Persuasion bias, social influence and unidimensional opinions. Quarterly Journal of Economics 118, 651–667. Durlauf, S.N., Young, H.P. (Eds.), 2001. Social Dynamics. MIT Press, Cambridge, MA. Eisner, R., Strotz, R., 1963. Determinants of business investment. In: Impacts on Monetary Policy. Prentice Hall. Farmer, R., 1999. The Macroeconomics of Self-fulfilling Prophecies, second ed. MIT Press. Feichtinger, G., Andreas, N., Wirl, F., 1994. Limit cycles in intertemporal adjustment models—theory and applications. Journal of Economic Dynamics and Control 18, 353–380. Feichtinger, G., Wirl, F., 2000. Instabilities in concave, dynamic, economic optimization. Journal of Optimization Theory and Applications 107, 277–288. Fisher, W.H., Hof, F.X., 2000. Relative consumption, economic growth, and taxation. Journal of Economics 72, 241–262.
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Fukao, K., Benabou, R., 1993. History versus expectations: a comment. Quarterly Journal of Economics 108, 535–542. Futagami, K., Shibata, A., 1998. Keeping one step ahead of the joneses: status, the distribution of wealth, and long run growth. Journal of Economic Behavior and Organization 36, 93–111. Glaeser, E.L., Scheinkman, J.A., 2003. Non-market interaction. In: De-watripoint, M., Hansen, L.P., Turnovsky, S. (Eds.), Advances in Economics and Econometrics: Theory and Applications, Eighth World Congress, vol. 1. Cambridge University Press, Cambridge, pp. 339–369. Glaeser, E.L., Sacerdote, B.I., Scheinkman, J.A., 2003. The social multiplier. Journal of the European Economic Association 1, 345–353. Grandmont, J.M., 1985. On endogenous competitive business cycles. Econometrica 53, 995–1045. Hamermesh, D.S., Pfann, G.A., 1996. Adjustment costs in factor demand. Journal of Economic Literature XXXIV, 1264–1292. Hirsch, F., 1976. Social Limits to Growth. Harvard University Press, Cambridge, MA. Kehoe, T.J., Levine, D.K., 1985. Comparative statics and perfect foresight in infinite horizon economies. Econometrica 53, 433–453. Krugman, P., 1980. Scale economics, product differentiation, and the pattern of trade. American Economic Review 70, 950–959. Krugman, P., 1991a. History versus expectations. Quarterly Journal of Economics 106, 651–667. Krugman, P., 1991b. Increasing returns and economic geography. Journal of Political Economy 99, 483–499. Liski, M., 2001. Thin versus thick CO2 market. Journal of Environmental Economics and Management 41, 295–311. Lucas, R.E., 1967. Adjustment costs and the theory of supply. Journal of Political Economy 75, 321–334. Mailath, G.J., Postlewaite, A., 2003. The social context of economic decisions. Journal of the European Economic Association 1, 354–362. Rauscher, M., 1997. Conspicuous consumption, economic growth, and taxation. Journal of Economics 66, 35–42. Romer, P., 1986. Increasing returns and long run growth. Journal of Political Economy 94, 1002–1037. Seierstad, A., Sydsaeter, K., 1986. Optimal Control Theory with Economic Applications. North-Holland, Amsterdam. Sorger, G., 1992. On the minimal rate of impatience for complicated economic growth paths. Journal of Economic Theory 56, 160–179. Wirl, F., 2002. Stability and limit cycles in competitive equilibria subject to adjustment costs and dynamic spillovers. Journal of Economic Dynamics and Control 26, 375–398.
History versus expectations: Increasing returns or social influence? Franz Wirl a,∗,1 , Gustav Feichtinger b,1,2 a
b
University of Vienna, Chair of Industry, Energy and Environment, Bruennerstrasse 72, Room 123, 1210 Vienna, Austria Technical University of Vienna, Chair of Operations Research and Nonlinear Dynamical Systems, Argentinierstr. 8/105-4, Austria Received 15 December 2003; accepted 21 November 2005
Abstract Krugman [Krugman, P., 1991a. History versus expectations. Quarterly Journal of Economics 106, 651–667.] links multiple equilibria (the domains of attractions depend on history and possibly also on expectations) to the technological property of increasing returns to scale. It is shown that the recently introduced characterization of (non-moderate) social influence [Glaeser, E.L., Sacerdote, B.I., Scheinkman J.A., 2003. The social multiplier. Journal of the European Economic Association 1, 345–353.] coupled with complementarity provides the proper criterion for thresholds and multiple equilibria instead of technological characterizations (increasing returns). © 2006 Elsevier Inc. All rights reserved. JEL classification: D90; C62; D62 Keywords: History dependence; Increasing returns; Social influence; Instability; Focus versus node
1. Introduction The widely quoted work of Krugman (1991a) stresses two conditions why the long run outcome of a competitive economy can depend on history and possibly also on expectations: increasing returns and an externality. The first condition is of course no surprise given Krugman’s extensive use of increasing returns in his work on strategic trade policy (e.g. Krugman, 1980) and geography ∗ 1 2
Corresponding author. Tel.: +43 14277 38101; fax: +43 14277 38104. E-mail addresses: [email protected] (F. Wirl), [email protected] (G. Feichtinger). Both authors acknowledge helpful comments from an anonymous referee and suggestions by the editor (Prof. Altman). Tel.: +43 1 58801 11927; fax: +43 1 58801 11999.
1053-5357/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.socec.2005.11.058
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(e.g. Krugman, 1991b). This linkage between increasing returns and history dependent economic evolutions (even for planning problems, i.e., absent any externality) is a significant research program in economics with obvious implications to neighboring fields like sociology, see, e.g. Altman (2000); this point has been popularized by Arthur (1989, 1994). The second reason given in Krugman (1991a), that of an externality, relates to the concept of (moderate) social influence, which has been recently introduced within a static context in Glaeser and Scheinkman (2003) by generalizing earlier works in particular that of Cooper and John (1988); Durlauf and Young (2001), Mailath and Postlewaite (2003) and DeMarzo et al. (2003) are related recent approaches to social interactions and Glaeser et al. (2003) applies the concept to quantify social multipliers. Krugman (1991a) tells his story in terms of a two-sectors economy in which “manufacturing” exhibits increasing and “agriculture” constant returns to scale. In order to avoid trivial solutions, “manufacturing” pays lower wages than agriculture at low scales (more precisely, if only few work in manufacturing) but higher wages if many (and in particular if all) work in manufacturing. This dependence of own pay on others’ actions is the (only) externality in this economy. In addition, the competitive agents face adjustment costs for any expansion of the time working in one of the sectors, which introduces dynamics (i.e., the competitive economy is described by two differential equations plus a transversality condition). Our paper extends the seminal work of Krugman (1991a) by allowing for a general objective instead of a specific one. An alternative interpretation is that our framework introduces dynamics into the so far static models of social interactions. The objective is to investigate within a general dynamic adjustment-cost framework whether ‘increasing returns to scale’, as claimed in Krugman (1991a) and in many follow-ups and related papers, or sufficiently strong social influences (also present but ignored in Krugman, 1991a) lead to multiple equilibria with domains of attraction depending on history and/or possibly also on expectations. Or put differently: do the results obtained in static models of social interactions survive the extension to adjustment dynamics? It turns out that the properties of the static theory – moderate social influence is sufficient for a unique competitive equilibrium – are robust and extend to dynamic adjustment cost models while Krugman’s (1991a) claim does not extend since increasing returns are in general neither necessary nor sufficient for history dependent evolutions. In other words, strong social interactions (plus complementarity) are a trigger for different long run equilibria and thresholds but not increasing returns. Of course, there are other reasons for multiple competitive equilibria: frictions in overlapping generations models (Kehoe and Levine, 1985), ‘sunspots’ (e.g. in Grandmont, 1985), and indeterminacy in endogenous growth models (e.g. in Benhabib and Perli, 1994). The paper is organized as follows. The framework is introduced in Section 2, the social optimum is characterized in Section 3, and the competitive equilibrium is derived and analyzed in Section 4. A summary concludes. 2. Framework The economy consists of identical competitive agents (with the aggregate normalized to one) solving the following intertemporal (using the constant discount rate r > 0) optimization problem ∞ c max (1) e−rt p(x(t), y(t)) − u(t)2 dt, {u(t)} 0 2 y˙ (t) = u(t),
y(0) = y0 .
(2)
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The individual control u(t) describes the adjustment (‘investment’) of the private stock, y(t), that induces costs,1 which are assumed to be quadratic (for reasons of simplicity only). p describes the agent’s net instantaneous benefit that depends on the private stock, e.g. capital, and on an economywide externality x(t). The externality, a given datum for each private agent, is determined by the aggregate over the private stocks and this aggregate (or average) equals the individual private stock x(t) = y(t) ∀t ≥ 0,
(3)
due to the above normalization and the assumption of symmetric equilibria. A static game with the payoffs p(x, y) is the limiting case (number of identical agents →∞) of the model in Cooper and John (1988) so that the above embedding within an adjustment-cost framework provides a simple dynamic version. An example for a positive externality, px > 0, is that productivity of private capital is enhanced with respect to the capital intensity of the entire economy. Pollution and/or a reduction of natural areas associated with an expansion of capital (or associated production) are examples of negative external effects, px < 0. These two examples highlight that externalities need not be ‘social’ and are used, e.g., in the endogenous growth literature (starting with Romer (1986), see also the survey and the classification in Benhabib and Rustichini (1994)). Externalities of the type (3) are dynamic in the sense that they are linked to a stock, while flow externalities, e.g. due to conspicuous consumption (Rauscher, 1997; Corneo and Jeanne, 1997, 2001; Futagami and Shibata, 1998 but see also Hirsch, 1976), are static and can thus not induce history dependence (Fisher and Hof, 2000); Wirl (2002) considers truly dynamic externalities, x˙ = g(x, y), in an otherwise related adjustment-cost framework but focuses on limit cycles (thereby in particular refuting Krugman (1991a) conjecture that limit cycles may be associated in a higher dimensional case with the unstable state). The economy-wide objective (internalizing the externality) P(y) ≡ p(y, y), is relevant for a social planner who will ∞ c max exp(−rt) P(y(t)) − u(t)2 . {u(t)} 0 2
(4)
(5)
2.1. Assumptions Existence of competitive equilibria requires non-increasing returns in the private stock, i.e., pyy < 0,
(6)
and the assumed strict concavity ensures that an interior solution satisfying py (x, y) = 0 determines the static choice, the maximum of p(x, y) with respect to y; otherwise, pyy = 0, the agent’s static choice is at the boundary of the feasible set of y. Instead of (joint) concavity of p(x, y), the weaker condition P (y) ≡ pxx (x, y) + 2pxy (x, y) + pyy (x, y)|x=y < 0
(7)
1 This adjustment cost framework (but absent the externality) dates back to Eisner and Strotz (1963) and Lucas (1967); the survey of Hamermesh and Pfann (1996) emphasizes empirical issues.
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is used to characterize concave economies (i.e., diminishing returns), while P > 0 stands for increasing returns. However, the aggregate economy needs not satisfy this law of diminishing returns. 2 Furthermore, the set of states satisfying P = 0 is finite, i.e., the social objective P has a finite number of local extrema, and the maximum maximorum is unique. This assumption rules out that the maximum of P is assumed along an interval, which induces indeterminacy already for the static problem and hence a fortiori for the dynamic social choice problem. And of course, all functions are assumed to be at least twice differentiable. 2.2. Example Krugman (1991a) Agents move between an ‘agricultural’ (subscript a) and a ‘manufacturing’ sector (subscript m), which pays higher wages if sufficiently many agents work already in factories (due to the increasing returns in manufacturing). The corresponding objective (of the representative agent) is p(x, y) = ywm (x) + (¯y − y)wa ,
(8)
where wa and wm are the wages paid in agriculture (constant, wa ) and in manufacturing respectively, wm (0) < wa , wm (¯y) > wa , wm > 0, wm = 0; 0 ≤ y ≤ y¯ = 1 denotes the time (total individual labor supply is given an equal to 1) spent in the factory and 0 < x˜ < 1 denotes the critical participation level at which manufacturing pays the same wage as agriculture, wm (˜x) = wa . The objective (8) is not jointly concave with respect to (x, y) and in fact P (y) is globally convex, P = 2wm > 0. Fukao and Benabou (1993) correct some of Krugman’s formal errors and Liski (2001) is a recent variant that attributes increasing returns to lower transaction costs in ‘thick’ markets for trades in CO2 permits. 2.3. Social influence Many papers account for the social impact of a reference group, here the aggregate or average over all agents. Yet it is only recently that Glaeser and Scheinkman (2003) introduces explicitly the condition of moderate social influence (see also Glaeser et al., 2003), which boils down in our framework to pxy (9) p < 1. yy
This means that the marginal benefit from the private stock (py ) is more sensitive to adaptations of the own compared to an increase of the average stock; of course, this inequality (9) needs only to hold along x = y (for the assumed symmetric equilibrium). This property of moderate social influence according to (9) is sufficient for a unique equilibrium in the static framework of Glaeser and Scheinkman (2003). Therefore, an alternative interpretation of the model (1)–(3) is that it is a first, explicitly dynamic version of Cooper and John (1988) and Glaeser and Scheinkman (2003) for ‘many’ agents. The existence of such a measure of social influence, whether of ‘moderate’ size or not, requires the strict inequality, pyy < 0, as assumed in (6). The example in Krugman (1991a) implies pyy = 0 2 Indeed (almost) all related papers on thresholds in socially optimal developments assume increasing returns, at least locally, see, e.g. the survey in Deissenberg et al. (2004).
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and pxy = wm ≥ 0 and thus infinite instead of moderate social influence. It is important to note that concavity and non-moderate social influence are not exclusive, and conversely, increasing returns for the economy at large can be compatible with moderate social influence. Therefore, the familiar attribution of the interesting properties of multiple equilibria depending on history and possibly also on expectations to increasing returns is questionable. This will be investigated in detail below. 3. Social optimum The social optimum requires solving a dynamic optimization problem – (5) subject to the dynamic constraint (2) – due to the possibility to control the externality (or the social influence). Defining the corresponding current value Hamiltonian, H ≡ P(y)−c/2(u)2 + λu in which λ denotes the costate, the first order optimality conditions (see, e.g. Seierstad and Sydsaeter, 1986) are: Hu = −cu + λ = 0
⇒u=
λ c
λ˙ = rλ − Hy = rλ − P .
(10) (11)
Substituting the optimal control from (10) into (2) yields y˙ =
λ , c
(12)
˙ which together with (11) determines the canonical equations. Therefore, steady states (˙y = 0 = λ) of the intertemporal, socially optimal program require that P (y) = px (x, y) + py (x, y)|x=y = 0,
(13)
i.e., they are characterized by (possibly local) extrema of the (static) objective P. The Jacobian of the canonical equations and its determinant are ⎡ ⎤ 1 P 0 A=⎣ . (14) c ⎦ ⇒ det(A) = c −P r Therefore (saddlepoint) stable steady states (det(A) < 0) must be located in the concave domain of P. Conversely, an unstable steady state y∞ is only possible if P is convex at y∞ due to det(A) > 0. Hence: Proposition 1. A strictly concave social objective, P(y) ≡ p(y, y) and P < 0, rules out unstable steady states and thus history dependent multiple equilibria,3 so that (at least local) convexities, P > 0, are necessary (but not sufficient, unless at a steady state) for the existence of multiple equilibria and history dependence.
3 This follows also directly from a theorem in Feichtinger and Wirl (2000), which shows that either growth in the state differential equation or a control-state intercation are necessary for an unstable steady state in concave optimization problems.
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4. Competitive equilibrium Defining the Hamiltonian (using again λ to denote the costate, which, of course is different from the one of the previous section) for an individual agent’s intertemporal optimization problem, c H ≡ p(x, y) − u2 + λu, 2 the necessary optimality conditions are Hu = −cu + λ = 0,
(15)
λ˙ = rλ − py (x, y).
(16)
Therefore, the dynamic, symmetric competitive equilibrium is described by: y˙ =
λ , c
y(0) = y,
λ˙ = rλ − py (y, y),
(17) (18)
due to the identity (3) between the individual stock y and the economy-wide externality x (for agents with rational expectations). The y˙ = 0 isocline is given by λ = 0, and λ˙ = 0 ⇔ λ = py /r. Hence, a steady state must satisfy py (x, y)|x=y = 0.
(19)
i.e., the static sub-objective p is maximized with respect to the individual stock y, which corresponds also to the static equilibrium choice (i.e., for no adjustment costs, c = 0, and an interior solution, pyy < 0). As a consequence, the stationary solution of dynamic, intertemporal adjustments is identical to the static choice of y. The reason is that this ideal state can be reached at ‘no’ costs for infinitesimally small adjustments over time. An immediate consequence of this steady state condition (19) is that multiple interior equilibria require increasing and decreasing parts of py . The Jacobian of the competitive equilibrium (of course evaluated at a steady state) is
0 1/c J= . (20) −(pyy + pyx ) r Therefore, tr(J) = r
(21)
so that the eigenvalues tr(J) ± tr(J)2 − 4 det(J) εi = , (22) 2 are symmetric around r/2. Hence, indeterminacy (=both eigenvalues are negative or have negative real parts) and the related case of limit cycles due to Hopf bifurcations (existence of a pair or purely imaginary eigenvalues) are impossible. The determinant is det(J) =
pyy + pyx c
(23)
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so that the sign of (pyy + pyx ) determines not only the sign of the slope of the λ˙ = 0 isocline but also the sign of det(J) and thus whether a steady state is stable or not. Proposition 2. Any steady state y∞ of the competitive equilibrium, i.e., of the differential Eqs. (17) and (18), is unstable iff pyy (x, y∞ ) + pxy (x, y∞ )|x=y∞ > 0, This in turn implies non-moderate social influence (at least locally) pxy (x, y∞ ) p (x, y ) > 1 at x = y∞ , ∞ yy
(24)
(25)
and pyx (x, y∞ )|x=y∞ > 0,
(26)
i.e., marginal benefit of the private stock increases if the average x is increased (i.e., y and x are complementary or supermodular). Proof. An unstable steady state is necessary for multiple equilibria and an associated threshold. Since an unstable steady state y∞ is characterized by a positive determinant of the Jacobian, the first claim of the proposition follows directly from (23). This inequality (24), omitting the arguments, is equivalent to pxy > −pyy ⇔ −
pxy > 1, pyy
(27)
which implies (25) and thus non-moderate social influence. Inequality (26) follows directly from (27) and the assumption (6). QED. Therefore, any (saddlepoint) stable steady state must be located on a downward sloping part of the λ˙ = 0 isocline and an unstable one on an upward sloping part. The above characterizations (Proposition 2) of an unstable steady state and thereby of a threshold separating the domains of attracting stable steady states highlight the following: First, moderate social influence excludes unstable and thus also multiple steady states, but non-moderate social influence is not sufficient for an unstable steady state. Second, complementarity is necessary for an unstable steady state and thus for a threshold, a property already stressed in Cooper and John (1988). Third, both conditions – non-moderate social influence and complementarity – present at a steady state are sufficient for a history dependent development. Comparing the steady state condition of the social optimum, (13), with the one of the competitive equilibrium, (19), yields immediately the familiar implication (analogous in Cooper and John, 1988) that competition leads to under-investment in stocks with positive externalities, px > 0, and to over-accumulation of stocks with negative externalities, px < 0. Furthermore, decreasing returns of the aggregate economy, P < 0, ensure a unique and stable long run social optimum. Krugman’s special case – pyy = 0, pxy = wm > 0, pxx = 0 thus P = 2wm > 0 and det(J) = wm /c > 0 – assumes economy-wide increasing returns located within in the manufacturing sector. And this condition is indeed necessary in Krugman’s model to obtain an unstable steady state and an associated threshold that separates the domains of convergence to either of the two boundaries, y = 0, y = 1, due to the state constraints, 0 ≤ y ≤ 1. However, this specification implies not only increasing returns but simultaneously a very large (actually infinite) social influence and given
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that these two characteristics hold simultaneously it is difficult to see which one is responsible for history dependence. 4.1. Example (diminishing returns) and existence In order to stress the significance of social influence for the existence of multiple equilibria within the adjustment cost framework, we assume decreasing returns to scale at the level of the aggregate economy. This assumption ensures a unique steady state for the social optimum so that the existence of a threshold results then from (inefficient) competition, and, according to Proposition 2, from sufficient social interaction and complementarity. The reason for these marked differences in the evolution is that the condition for an unstable steady state of the competitive equilibrium, inequality (24) in Proposition 2, pyy + pxy > 0,
(28)
is weaker than the assumption of economy-wide (strict) concavity: P (y) ≡ pxx + 2pxy + pyy < 0.
(29)
Therefore, it is conceivable to have diminishing returns for the economy at large and thus a unique socially efficient long run outcome (this holds for the adjustment cost framework due to Proposition 1) but multiple steady states separated by thresholds for the competitive economy, e.g. if pxx is sufficiently negative. The existence of such outcomes can be easily shown, for example, by using a quadratic p that satisfies the above two inequalities (and is possibly only a local approximation that extends beyond the crucial domain in a way that allows for a positive marginal benefit globally). The example in Fig. 1 is based on a direct extension of Krugman’s model: p(x, y) = W(x, y) + wα (1 − y),
(30)
i.e., the payoff for working in manufacturing is given by a ‘production’ function W with the following specification of W(x, y) = xα y1−α + µxy − vx2 ,
0 < α < 1,
ν > 0.
(31)
The term {µxy − νx2 } accounts for positive (e.g. knowledge and skills increase with the number of workers in industry) and negative (e.g. due to crowding or pollution) spillovers. Since
Fig. 1. W = xα y1−α + µxy − νx2 , c = 0.5, r = 1/4, α = 0.5, µ = 0.1, wa = 0.55.
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P = 2(µ − ν), P is strictly concave for µ < ν. As a consequence, the welfare maximization has a unique interior and (saddlepoint) stable steady state (of course, a node) y∞ = 1 − wa /ν − µ. However, since Wxy µ pxy < −1, = = −1 − pyy Wyy α(1 − α)
(32)
the degree of social influence is non-moderate for any µ > 0. Conversely, assuming µ < 0 implies moderate social influence and yet a convex social objective for ν < µ < 0. This observation confirms that increasing returns to scale and social influence are two entirely different concepts. A proper choice of model parameters ensures that the wages in agriculture, wa , and in manufacturing, given by wm = ∂W(x, y)/∂y|x=y = (1 − α) + µy, intersect for 0 < y = x˜ < 1. This intersection determines a steady state, y∞ = x˜ , which is an unstable node such that the long run outcome, y → ±1 (due to the state constraints, 0 ≤ y ≤ 1) is completely determined by history, i.e., whether initially y0 x˜ . Fig. 1 shows a typical outcome. This figure is actually ‘drawn’ using numerical calculations (˜x = 1/2), which in turn verifies the existence claim in Proposition 3 below. Proposition 3. Thresholds separating multiple long run competitive outcomes are possible even if the law of diminishing returns holds (globally) at the aggregate level. Therefore, the law of diminishing returns allows for multiple equilibria and thus in particular for history dependence, but moderate social influence or substitutability, pxy < 0, rule this out. This raises the question why not only Krugman (1991a) but also all later following papers (and Liski (2001) is just a particular recent example) use and emphasize the property of increasing returns (plus an externality4 ) to obtain history dependent outcomes, although it is actually not necessary (but admittedly often helpful). 4.2. Focus versus node, expectations versus history In the following it is investigated whether an unstable steady state, which is now assumed to exist, is either a node or a focus (i.e., complex eigenvalues, thus a negative discriminant in (22)). These at first sight purely mathematical local stability properties have important economic consequences whether history (in case of a node) or in addition expectations (in case of a focus) determine the long run outcome. The reason for the latter is that the solution spirals (due to complex eigenvalues) to either of the (saddlepoint) stable steady states overlap for an interval of initial conditions y0 such that history is insufficient to determine the future evolution (see Fig. 2). This additional dependence on expectations (consistent, yet potentially self-fulfilling) may require coordination, which provides the core argument of Farmer’s (1999) approach to macroeconomics. A focus (complex eigenvalues) requires for the social optimum a local convexity (thus increasing returns) of a sufficient magnitude, P = pxx + 2pxy + pyy |x=y >
cr 2 4
at a steady state y = y∞ ,
because of the Jacobian (14) and its eigenvalues (replacing J by A in formula (22)).
4
In fact, not even externalities are necessary in a more general framework, see Feichtinger and Wirl (2000).
(33)
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In contrast, a competitive equilibrium can give rise to a focus for much less stringent requirements about the returns to scale cr 2 (34) at a steady state y∞ 4 This follows from a negative discriminant in (22), from tr(J) = r and from det(J) given by (23). Inequality (34) demands sufficiently large and definitely non-moderate social influence: pxy cr 2 (35) p > 1 − 4p > 1. yy yy pyy + pxy |x=y >
Nevertheless it seems possible to obtain a focus for an economy with non-increasing returns, in particular, for small adjustment costs c that do not affect the steady state at all. Another possibility is to choose a small discount rate. Thus, small discount rates and small adjustment costs increase the complexity and in addition make the long run outcome dependent on expectations (at least locally within the overlap, see Fig. 2), while high discount rates and/or large adjustment costs reduce the determination of the long run outcome to history. This is surprising from a formal point of view (but well explained in Krugman, 1991a) since both, high discount rates and high adjustment costs, are factors fostering complex patterns, e.g. limit cycles, in Feichtinger et al. (1994) and Wirl (2002), and chaos, according in Sorger (1992). An interesting implication is that economies with agents characterized by low discount rates and small adjustment costs (such that a focus results) increase the set of initial conditions that allow to reach the more efficient high stock long run outcome. Proposition 4. A focus requires sufficiently large increasing returns for the social optimum, P > cr2 /4, but it can arise (and exists) in the competitive case even for assuming decreasing returns (for the aggregate economy) if −
pxy(x, y∞ ) cr 2 >1− pyy (x, y∞ ) 4pyy (x, y∞ )
at a steady state y∞ and x = y∞ .
That is, social influence is sufficiently non-moderate and private stock and externality are complements, pxy > 0. Fig. 2 shows a corresponding example of a focus based on (31) and the same parameters as in Fig. 1 except for smaller adjustment costs and thus of an overlap within the domain of decreasing returns to scale. Both long run outcomes y∞ = 0 or 1 can be reached within in this overlap, if all
Fig. 2. W = xα y1−α + µxy − νx2 , c = 0.2, r = l/4, α = 0.5,µ = 0.1, wa = 0.55.
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agents bet on the same (due to assumed symmetry), either y = 0 or 1, and choose the associated adjustment path u = λ/c, either the lower or the upper one in Fig. 2. These two possibilities (circle or square) are shown for one initial condition y0 within the overlap. This highlights that history is insufficient do determine the steady state outcome for any initial condition y0 within this overlap, and that the kind of expectation is ultimately decisive. 5. Summary This paper reconsidered Krugman (1991a) with respect to the economics and mechanics of thresholds and multiple equilibria in competitive economies. Krugman (1991a) emphasizes the importance of two characteristics – increasing returns to scale for the aggregate economy and an externality – for multiple long run outcomes depending either on history and/or on expectations. This paper in contrast shows that the assumption of increasing returns, although helpful in many cases, is not crucial. Sufficient degree of social interactions, non-moderate in the terminology introduced in Glaeser and Scheinkman (2003), and complementarity (raising the economy’s average increases an individual’s marginal utility, stressed already in Cooper and John (1988)) trigger thresholds and multiple equilibria. Furthermore, the property of non-moderate social influence (and thus the instability of a corresponding steady state) can arise for strict concave economies, i.e., economies satisfying even at the aggregate level the law of diminishing returns; and conversely, moderate social influence is compatible with increasing returns. Therefore, increasing returns are not, as often suggested, the source of history dependence, and are in fact neither necessary nor sufficient. References Altman, M., 2000. A behavioral model of path dependency: the economics of profitable inefficiency and market failure. Journal of Socio-Economics 29, 127–145. Arthur, W.B., 1989. Competing technologies, increasing returns, and lock-in by historical events. Economic Journal 99, 116–131. Arthur, W.B., 1994. Increasing Returns and Path Dependence in the Economy. University of Michigan Press, Ann Arbor. Benhabib, J., Perli, R., 1994. Uniqueness and indeterminacy: on the dynamics of endogenous growth. Journal of Economic Theory 63, 113–142. Benhabib, J., Rustichini, A., 1994. Introduction to the symposium on growth, fluctuations, and sunspots: confronting the data. Journal of Economic Theory 63, 1–18. Cooper, R., John, A., 1988. Coordinating coordination failures in Keynesian models. Quarterly Journal of Economics CII, 441–463. Corneo, G., Jeanne, O., 1997. On relative wealth effects and the optimality of growth. Economics Letters 54, 87–92. Corneo, G., Jeanne, O., 2001. On relative wealth effects and long-run growth. Research in Economics 55, 349–358. Deissenberg, C., Feichtinger, G., Semmler, W., Wirl, F., 2004. History dependency and global dynamics in intertemporal optimization models. In: William, A.B. (Ed.), Economic Complexity: Non-Linear Dynamics, Multi-Agents Economies, and Learning. North Holland, Amsterdam, pp. 91–122. DeMarzo, P.M., Vayanos, D., Zwiebel, J., 2003. Persuasion bias, social influence and unidimensional opinions. Quarterly Journal of Economics 118, 651–667. Durlauf, S.N., Young, H.P. (Eds.), 2001. Social Dynamics. MIT Press, Cambridge, MA. Eisner, R., Strotz, R., 1963. Determinants of business investment. In: Impacts on Monetary Policy. Prentice Hall. Farmer, R., 1999. The Macroeconomics of Self-fulfilling Prophecies, second ed. MIT Press. Feichtinger, G., Andreas, N., Wirl, F., 1994. Limit cycles in intertemporal adjustment models—theory and applications. Journal of Economic Dynamics and Control 18, 353–380. Feichtinger, G., Wirl, F., 2000. Instabilities in concave, dynamic, economic optimization. Journal of Optimization Theory and Applications 107, 277–288. Fisher, W.H., Hof, F.X., 2000. Relative consumption, economic growth, and taxation. Journal of Economics 72, 241–262.
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