Econ Theory Bull (2017) 5:65–80 DOI 10.1007/s40505-016-0102-3 RESEARCH ARTICLE

Edgeworth box economies with multiple equilibria Alexis Akira Toda1 · Kieran James Walsh2

Received: 11 June 2016 / Accepted: 23 June 2016 / Published online: 9 July 2016 © Society for the Advancement of Economic Theory 2016

Abstract We derive sufficient conditions for the existence of multiple equilibria in two-good, two-agent pure exchange economies with heterogeneous but symmetric preferences with identical Bernoulli utilities. When preferences are non-homothetic (for example, quadratic, quasi-linear, or HARA), multiple equilibria are possible even with identical endowments. Keywords Excess demand functions · Multiple equilibria · Edgeworth box JEL Classification C62 · D51 · D58

1 Introduction The Sonnenschein (1972, 1973)–Mantel (1974)–Debreu (1974) (SMD) theorems imply that strong assumptions on preferences, such as homogeneity across agents or homotheticity, are not sufficient for uniqueness or stability of equilibrium.1 Economies consisting solely of well-behaved agents satisfying the weak axiom of revealed pref1 Mantel (1976) shows the SMD result for homothetic preferences, and Kirman and Koch (1986) do so with identical preferences and collinear endowments. See Shafer and Sonnenschein (1982) for a survey of this literature.

B

Kieran James Walsh [email protected] Alexis Akira Toda [email protected]

1

Department of Economics, University of California San Diego, 9500 Gilman Dr., #0508, 92093-0508 La Jolla, CA, USA

2

Darden School of Business, University of Virginia, 100 Darden Blvd, Charlottesville, VA 22903, USA

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erences may aggregate into a badly behaved multi-humped demand curve violating the compensated law of demand. In such cases, macro-comparative statics can violate basic economic intuition (Kehoe 1985). While in some economies there are reasonable sufficient conditions for uniqueness and stability,2 in general the SMD theory tells us that “anything goes” for the market excess demand, and hence the existence of multiple equilibria is the rule rather than the exception. However, many of the examples established in the SMD papers are quite unintuitive, so it seems useful to provide examples of canonical economies with multiple equilibria. General sufficient conditions for the existence of multiple equilibria also highlight the necessary properties of preferences or endowments to obtain equilibrium uniqueness. In this paper, we provide such sufficient conditions in Edgeworth box economies with standard utility functions. In particular, we show how to construct economies with three equilibria when there are two goods, two agents, and identical Bernoulli utilities. Our strategy is based on the following simple observation. In an Edgeworth box economy with symmetric preferences and endowments, p1 = p2 = 1 is always an equilibrium price. If we can show that this equilibrium is unstable, since by the index theorem there are at least two more stable equilibria, it follows that there are in total at least three equilibria. Therefore, to construct an example with multiple equilibria, it suffices to show that the slope of the aggregate excess demand function at p1 = p2 = 1 has the “wrong” sign. For Proposition 1, agents have identical constant relative risk aversion (CRRA) Bernoulli utilities and heterogeneous but symmetric utility weights, which one can interpret as discount factors when the two goods are consumption across time or as subjective probabilities when the two goods are consumption across states. With symmetric endowments, when the risk aversion coefficient is greater than 2, we show how to choose endowments and utility weights to construct an economy with three equilibria, one of which is unstable. In Proposition 2, with quadratic utility and identical endowments we show how to choose utility weights to construct an economy with three equilibria. In Proposition 3, we provide an example with symmetric quasi-linear utility and identical endowments. In Proposition 4, we derive a sufficient condition for the existence of multiple equilibria with general additively separable preferences and show in Corollary 1 that at least one Bernoulli utility function must exhibit relative risk aversion greater than 2 at the symmetric equilibrium allocation to apply this argument. As applications, in Corollary 2 we show that any Bernoulli utility function with unbounded and increasing absolute risk aversion (IARA) can give rise to multiple equilibria, and in Proposition 5 we show how to choose the parameters of hyperbolic absolute risk aversion (HARA) preferences and utility weights to construct an economy with three equilibria. As these propositions show, even in an Edgeworth box economy with canonical utility functions, equilibrium may be non-unique and unstable. Intuitively, in endowment economies standard utility functions may still yield Giffen good behavior as rising prices may increase (decrease) the wealth of sellers (buyers). With sufficiently hetero-

2 See Hens and Loeffler (1995), Kehoe (1998), and Geanakoplos and Walsh (2016) for recent treatments.

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geneous endowments or preferences, agents that are locally Giffen may dominate in aggregation, leading to upward sloping portions of the aggregate demand function. Sections 2, 3, 4, and 5 consider, respectively, CRRA, quadratic, quasi-linear, and general additively separable preferences. Section 6 provides further intuition and concludes.

2 CRRA preferences With homothetic preferences, it is well known that if either preferences are identical (Gorman 1953), endowments are collinear (Chipman 1974), or relative risk aversion is at most 1 (Hens and Loeffler 1995), then the equilibrium is unique. Therefore, to construct a CRRA example with multiple equilibria, agents must have different endowments and weights on goods with relative risk aversion greater than 1. Consider an Edgeworth box economy where the agents have symmetric CRRA preferences 1 1−γ 1 U2 (x1 , x2 ) = 1−γ U1 (x1 , x2 ) =

  1−γ 1−γ α γ x1 + (1 − α)γ x2 ,   1−γ 1−γ (1 − α)γ x1 + α γ x2 ,

where γ > 1 is the relative risk aversion and 0 < α < 1 determines the utility weights. The initial endowments are e1 = (e, 1 − e) and e2 = (1 − e, e), where 0 < e < 1. Let p1 = 1 and p2 = p be prices, xil ( p) be agent i’s demand for good l, and zl ( p) =

2  (xil ( p) − eil ) i=1

be the aggregate excess demand for good l. For notational simplicity, let ε = 1/γ < 1 be the elasticity of substitution. Proposition 1 Suppose that ε <1−

1 2



e 1−e + α 1−α

 .

(1)

Then the economy has at least three equilibria. In particular, for any γ > 2 we can construct an economy with three equilibria. Proof Let w1 = e + (1 − e) p be the wealth of agent 1. Since preferences are CRRA, his demand for good 1 is x11 =

αp1−ε w1 αp11−ε

+ (1 − α) p21−ε

=

α(e + (1 − e) p) . α + (1 − α) p 1−ε

(2)

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Since agents are symmetric, agent 2’s demand for good 1 can be obtained by changing α → 1 − α and e → 1 − e. Therefore, the aggregate excess demand for good 1 is z 1 ( p) = x11 + x21 − e − (1 − e) =

α(e + (1 − e) p) (1 − α)(1 − e + ep) + − 1. α + (1 − α) p 1−ε 1 − α + αp 1−ε

By symmetry, p = 1 is an equilibrium price. Since 0 < ε < 1, by direct substitution we have z 1 (1) = α + (1 − α) − 1 = 0 and z 1 (∞) = ∞. Therefore, if z 1 (1) < 0, we have z 1 ( p) < 0 when p > 1 is sufficiently small. Since z 1 (∞) = ∞, by the intermediate value theorem there exists p ∗ > 1 such that z 1 ( p ∗ ) = 0, so p ∗ is an equilibrium price. By symmetry, if p ∗ is an equilibrium price, so is 1/ p ∗ . Therefore, to show the existence of multiple equilibria, it suffices to show z 1 (1) < 0. Differentiating x11 ( p), we obtain α[(1 − e)(α + (1 − α) p 1−ε ) − (e + (1 − e) p)(1 − α)(1 − ε) p −ε ] (α + (1 − α) p 1−ε )2 = α(1 − e − (1 − α)(1 − ε))

 ( p) = x11

at p = 1. Changing α → 1 − α and e → 1 − e, the economy has at least three equilibria if   z 1 (1) = x11 (1) + x21 (1) = α(1 − e) + (1 − α)e − 2α(1 − α)(1 − ε) < 0   1−e 1 e + , ⇐⇒ ε < 1 − 2 α 1−α

which is (1). Finally, let us show that we can construct an economy with at least three equilibria when γ > 2. Set e = α 2 . Then (1) becomes 1 1 ε < 1 − (α + (1 + α)) = − α. 2 2 Hence, for any γ > 2 (ε < 1/2), by choosing α such that 0 < α < e = α 2 , we get an economy with at least three equilibria.

1 2

− ε and setting 

Remark 1 Since 0 < e < 1, 0 < α < 1, and the right-hand side of (1) is linear in e, we obtain   1 1 1 1 ε < 1 − min , < , 2 α 1−α 2 so we can construct an economy with three equilibria using the argument of Proposition 1 only if γ > 2. For CRRA preferences, it is well known that γ ≤ 1 is a sufficient condition for equilibrium uniqueness because it ensures goods are gross substitute (Hens and Loeffler 1995). To the best of our knowledge, whether multiple equilibria are possible or not when 1 < γ ≤ 2 is unknown.

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Excess demand of good 1

0.01

0.005

0

-0.005

-0.01 10 -1

10 0

10 1

Price of good 1 Fig. 1 Aggregate excess demand with symmetric CRRA preferences

For a concrete example, let γ = 3, α = 1/7, and e = 1/49. Using (2), we have 11 1 1 + 48 · 8 1 385 = , = 2 49 1 + 6 · 8 3 49 25 35 24 6 48 + 8 6 56 x21 (8) = = . = 2 49 6 + 8 3 49 10 35 x11 (8) =

Therefore, z 1 (8) =

11 24 + − 1 = 0, 35 35

so p = 8 is an equilibrium. By symmetry, so is p = 1/8. p = 1 is clearly an equilibrium. Figure 1 shows the graph of the excess demand of good 1 as a function of the price of good 1 (that is, we plot z 1 (1/ p)). We can see that the demand is increasing in the price around p = 1, exhibiting Giffen behavior.

3 Quadratic preferences Next, suppose that the agents have symmetric quadratic preferences U1 (x1 , x2 ) = αu(x1 ) + (1 − α)u(x2 ), U2 (x1 , x2 ) = (1 − α)u(x1 ) + αu(x2 ), 1 2 x is quadratic with risk tolerance parameter τ > 0 and utility where u(x) = x − 2τ weight 0 < α < 1. Suppose that agents have identical endowments e1 = e2 = (e, e).

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Proposition 2 Suppose that τ > e and either 0 < α < Then the economy has exactly three equilibria.

√ 2− 2 4

or

√ 2+ 2 4

< α < 1.

Proof First, let us compute agent 1’s demand. Let p1 = 1 and p2 = p be the prices, and w = (1 + p)e be the wealth. The Lagrangian is     1 2 1 2 L(x1 , x2 , λ) = α x1 − x1 + (1 − α) x2 − x2 + λ(w − x1 − px2 ). 2τ 2τ The first-order conditions are 

0 = α(1 − x1 /τ ) − λ ⇐⇒ 0 = (1 − α)(1 − x2 /τ ) − λp



x1 = τ − x2 = τ −

τλ α , τ λp 1−α .

Using the budget constraint, we obtain  w = x1 + px2 = (1 + p)τ − τ

p2 1 + α 1−α

 λ.

Solving for λ and substituting into the first-order condition, we obtain x11 = τ − x12 = τ −

(1 + p)τ − w τ −e =τ− (1 + p), α α 2 1 + 1−α p 1 + 1−α p2 (1 + p)τ − w 1−α α

+ p2

p=τ−

τ −e 1−α α

+ p2

p(1 + p),

(3a) (3b)

where we have used w = (1 + p)e. Note that so far we have ignored the possibility that the demand exceeds the satiation point τ , but since τ > e, the above expressions are less than τ , so it is the actual demand. Since agents have symmetric preferences but identical endowments, agent 2’s demand can be obtained by changing α → 1 − α. Therefore, the aggregate excess demand for good 1 is

1+ p 1+ p − z 1 ( p) = x11 +x21 −2e = (τ −e) 2− α 2 1 + 1−α p2 1 + 1−α α p

=: (τ − e) f ( p).

Since τ > e, the equilibrium condition is f ( p) = 0. Clearly f (1) = 0 and f (∞) > 0, so by the same argument as in the CRRA case, a sufficient condition for multiple equilibria is f  (1) < 0. By a straightforward calculation, we obtain 1+ d 1+ p = α 2 d p 1 + 1−α p

123

α 1−α

α p 2 − (1 + p)2 1−α p = (1 − α)(1 − 4α)  2 α 1 + 1−α p2

Edgeworth box economies with multiple equilibria

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Excess demand of good 1

0.2

0.1

0

-0.1

-0.2 10 -1

10 0

10 1

Price of good 1 Fig. 2 Aggregate excess demand with symmetric quadratic preferences

at p = 1. Changing α → 1 − α, it follows that f  (1) = −(1 − α)(1 − 4α) − α(1 − 4(1 − α)) = −1 + 8α − 8α 2 < 0 √ √ 2− 2 2+ 2 ⇐⇒ α < , α> . 4 4 To show that the economy has exactly three equilibria, note that the terms of f ( p) have denominators that are quadratic in p. By multiplying through these denominators, f ( p) = 0 is equivalent to a quartic equation and hence has at most 4 real roots. We can easily verify that f (0) = 0, but p = 0 is not an equilibrium, so there are exactly three equilibria.  For a concrete example, set α = 1/10 (τ, e do not matter). Figure 2 shows the graph of the excess demand of good 1 as a function of the price of good 1, ignoring the factor τ − e > 0 (that is, we plot f (1/ p)). We can see that the demand is increasing in the price around p = 1, exhibiting Giffen behavior. Remark 2 In the above discussion, we did not exclude the possibility that agents consume negative amounts. However, it is possible to construct examples in which consumption is always positive. To see this, note that in the demand formula (3a, 3b), p(1+ p) the expressions 1+1+α p p2 and 1−α are bounded as functions of p because the degree 2 1−α

α

+p

of the numerators does not exceed that of the denominators. Therefore, if we take τ > e sufficiently close to e, then the demands (3a, 3b) are positive, so all equilibria have positive consumption.

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4 Quasi-linear preferences Consider an Edgeworth box economy with symmetric quasi-linear preferences U1 (x1 , x2 ) = x1 + u(x2 ), U2 (x1 , x2 ) = u(x1 ) + x2 , where u  > 0, u  < 0, u  (0) = ∞, and u  (∞) = 0. Agents have identical endowments e1 = e2 = (e, e). Due to quasi-linearity, agent 1 (2) is allowed to consume good 1 (2) in negative amounts. Let v(y) = (u  )−1 (y) be the inverse function of the marginal utility, which is decreasing by assumption. Clearly v(0) = ∞ and v(∞) = 0. Proposition 3 Suppose that 0 < e < v(1) + 2v  (1).

(4)

Then the economy has at least three equilibria. Proof Let p1 = 1 and p2 = p be the prices and w = (1 + p)e be wealth. Using the budget constraint, agent 1 maximizes w − px2 + u(x2 ). The first-order condition is u  (x2 ) = p ⇐⇒ x2 = (u  )−1 ( p) = v( p). Therefore, the demand for good 1 is x11 = w − px12 = (1 + p)e − pv( p). By symmetry, agent 2’s demand for good 1 is x21 = v(1/ p). Therefore, the aggregate excess demand for good 1 is z 1 ( p) = x11 + x21 − 2e = ( p − 1)e − pv( p) + v(1/ p). Clearly z 1 (1) = 0, so p = 1 is an equilibrium price. Since z 1 ( p) = p(e − v( p)) − e + v(1/ p), v(0) = ∞, and v(∞) = 0, letting p → ∞ we obtain z 1 (∞) = ∞. By the same argument as in the CRRA case, the economy has at least three equilibria if z 1 (1) < 0. Since 1 z 1 ( p) = e − v( p) − pv  ( p) − v  (1/ p) 2 , p we have z 1 (1) = e−v(1)−2v  (1). Therefore, a sufficient condition for three equilibria is (4).  1−γ

As a concrete example, let u(x) = β x1−γ and set ε = 1/γ . Since u  (x) = βx −γ , we have v(y) = (y/β)−1/γ = β ε y −ε . Therefore, (4) is equivalent to e < β ε − 2β ε ε = β ε (1 − 2ε), so if γ > 2 (ε < 1/2), we can take e > 0 to satisfy this inequality.

123

(5)

Excess demand of good 1

Edgeworth box economies with multiple equilibria

73

0.2 0.1 0 -0.1 -0.2 -0.3 10 -2

10 -1

10 0

10 1

10 2

Price of good 1 Fig. 3 Aggregate excess demand with symmetric quasi-linear preferences

Figure 3 shows the graph of the excess demand of good 1 as a function of the price of good 1 (that is, we plot z 1 (1/ p)) when β = 1, γ = 3 (ε = 1/3), and e = 1/4. We can see that the demand is increasing in the price around p = 1, exhibiting Giffen behavior. Remark 3 In the above discussion, we did not exclude the possibility that agents consume negative amounts. However, it is possible to construct examples in which consumption is always positive. Since x11 = (1 + p)e − pv( p), for consumption to be positive in all equilibria, it suffices to show that (1 + p)e − pv( p) > 0 for all p. We now derive a sufficient condition for positive consumption when u(x) = 1−γ β x1−γ and hence v( p) = β ε p −ε . Let f ( p) = (1 + p)e − pv( p) = (1 + p)e − β ε p 1−ε . Since 0 < ε < 1, we have f  ( p) = e − β ε (1 − ε) p −ε , f  ( p) = β ε ε(1 − ε) p −ε−1 > 0, so f is strictly convex and attains a minimum at f  ( p) = 0 ⇐⇒ p = β



1−ε e

1/ε .

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At this p, we have

f ( p) = e + p(e − β ε p −ε ) = e − εβ ε p 1−ε

ε =e 1−β 1−ε



1−ε e

1/ε

.

Therefore, f ( p) > 0 if and only if 1−β

ε 1−ε



1−ε e

1/ε

> 0 ⇐⇒ e > β ε εε (1 − ε)1−ε .

Combining with (5), there exist three equilibria with positive consumption if εε (1 − ε)1−ε < β −ε e < 1 − 2ε.

(6)

Take the logarithm and define g(x) = x log x + (1 − x) log(1 − x) − log(1 − 2x) for 0 ≤ x ≤ 1/2. Then g is strictly convex, g(0) = 0, and g(1/2) = ∞. Hence, for 0 < ε < 1/2 with g(ε) < 0 to exist, it suffices to show g  (0) < 0. However, since g  (x) = log x − log(1 − x) +

2 → −∞ 1 − 2x

as x → 0, we are done. By numerically solving g(x) = 0, there exist three equilibria with positive consumption if 0 < ε < 0.1945 and e is chosen to satisfy (6).

5 General additively separable preferences In both the CRRA and quasi-linear examples, we found that relative risk aversion must exceed 2 to construct an economy with multiple equilibria. We can show that this is not a coincidence. Consider an Edgeworth box economy with symmetric additively separable preferences and endowments. Utility functions are U1 (x1 , x2 ) = u(x1 ) + v(x2 ), U2 (x1 , x2 ) = v(x1 ) + u(x2 ), where u  > 0, u  < 0, v  > 0, v  < 0. Endowments are e1 = (a, b) and e2 = (b, a), where a, b > 0. Suppose that the demand of a good diverges to ∞ as its price approaches 0. Let x1∗ = (x ∗ , y ∗ ), x2∗ = (y ∗ , x ∗ ) be the symmetric equilibrium corresponding to the price p1 = p2 = 1.

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Proposition 4 Suppose that −

u  (x ∗ ) ∗ v  (y ∗ ) (x − a) −  ∗ (y ∗ − b) > 2.  ∗ u (x ) v (y )

(7)

Then the economy has at least three equilibria. Proof Let p1 = p and p2 = 1 be prices and x = x1 . Using the budget constraint, agent 1 maximizes u(x) + v( pa + b − px). The first-order condition is u  (x) − pv  ( pa + b − px) = 0. Let F(x, p) be the left-hand side. By the implicit function theorem, we have Fp v  (y) + pv  (y)(a − x) ∂x =− , = ∂p Fx u  (x) + p 2 v  (y)

(8)

where y = pa + b − px. The slope of agent 2’s demand can be obtained by changing u → v, a → b, and x → y. Therefore, letting z 1 ( p) be the aggregate excess demand, at the symmetric equilibrium price p = 1, we obtain v  (y ∗ ) + v  (y ∗ )(a − x ∗ ) u  (x ∗ ) + u  (x ∗ )(b − y ∗ ) + u  (x ∗ ) + v  (y ∗ ) v  (y ∗ ) + u  (x ∗ ) u  (x ∗ ) + u  (x ∗ )(x ∗ − a) + v  (y ∗ ) + v  (y ∗ )(y ∗ − b) = , u  (x ∗ ) + v  (y ∗ )

z 1 (1) =

where we have used the market clearing condition x ∗ + y ∗ = a + b ⇐⇒ a − x ∗ = y ∗ − b. By u  > 0, u  < 0, v  > 0, v  < 0, and the first-order condition u  (x ∗ ) = pv  (y ∗ ) = v  (y ∗ ), multiplying both sides by −u  (x ∗ ) − v  (y ∗ ) > 0 and dividing both sides by u  (x ∗ ) = v  (y ∗ ) > 0, it follows that z 1 (1) has the same sign as −2 −

v  (y ∗ ) ∗ u  (x ∗ ) ∗ (x (y − b). − a) − u  (x ∗ ) v  (y ∗ )

Since p is the price of the good 1, the economy has at least three equilibria if z 1 (1) > 0, which is equivalent to (7).  The following corollary shows that relative risk aversion must exceed 2 to apply the argument of Proposition 4. Corollary 1 Suppose that u  (0) = v  (0) = ∞, so x ∗ , y ∗ > 0. Let γu (x) = −xu  (x)/u  (x) be the relative risk aversion of u and define γv analogously. If (7) holds, then either γu (x ∗ ) > 2 or γv (y ∗ ) > 2. Proof Since by the market clearing condition x ∗ + y ∗ = a + b, it must be either b x ∗ ≥ a or y ∗ ≥ b. In the former case, since yb∗ = a+b−x ∗ ≥ 1, by (7) we have γu (x ∗ ) ≥ γu (x ∗ )(1 − a/x ∗ ) > 2 − γv (y ∗ )(1 − b/y ∗ ) ≥ 2.

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The latter case is similar.

Corollary 2 Suppose that u(x) exhibits increasing absolute risk aversion (IARA) and that −u  (x)/u  (x) is unbounded. Then for any symmetric initial endowment e1 = (a, b) and e2 = (b, a) in the domain of u, there exists 0 < α < 1 such that the symmetric Edgeworth box economy with utility functions U1 (x1 , x2 ) = αu(x1 ) + (1 − α)u(x2 ), U2 (x1 , x2 ) = (1 − α)u(x1 ) + αu(x2 ) has at least three equilibria. Proof Take any x > a in the domain of u and define y = a + b − x < b. Since u  (y)  (y)  (b) is IARA, − uu  (y) is positive and decreasing in x. Therefore, − uu  (y) < − uu  (b) , and hence    u  (x) u  (y) u (x) u  (y) −  (x − a) −  (y − b) = −  +  (x − a) u (x) u (y) u (x) u (y)    u (x) u  (b) > −  +  (x − a) → ∞ u (x) u (b) as we make x larger because by assumption u is IARA and −u  (x)/u  (x) is unbounded. Therefore, the left-hand side will exceed 2 eventually, so there exists x ∗ such that (7) holds. Define y ∗ = a + b − x ∗ and 0<α=

u  (y ∗ ) < 1. u  (x ∗ ) + u  (y ∗ )

Then λ := αu  (x ∗ ) = (1−α)u  (y ∗ ) > 0, so the first-order condition holds with prices p1 = p2 = 1 and Lagrange multiplier λ > 0. Therefore, x1∗ = (x ∗ , y ∗ ), x2∗ = (y ∗ , x ∗ ) is a symmetric equilibrium. Since (7) holds by construction, by Proposition 4 there are at least three equilibria.  Using Proposition 4, we can derive a sufficient condition for the existence of multiple equilibria with hyperbolic absolute risk aversion (HARA) preferences. A Bernoulli utility function u(x) is called HARA if the absolute risk aversion is a hyperbolic function: −

1 u  (x) = , u  (x) ε(x − c)

where ε = 0 and c ∈ R. If ε > 0, then c is the subsistence consumption; if ε < 0, it is a bliss point. One can easily show that  u(x) =

123

1 ε−1 (ε(x

− c))1−1/ε , (ε = 1) log(x − c) (ε = 1)

(9)

Edgeworth box economies with multiple equilibria

77

up to an affine transformation. Consider an Edgeworth box economy with utility functions U1 (x1 , x2 ) = α 1/ε u(x1 ) + (1 − α)1/ε u(x2 ), U2 (x1 , x2 ) = (1 − α)1/ε u(x1 ) + α 1/ε u(x2 ) and symmetric endowments e1 = (a, b), e2 = (b, a). Note that the CRRA economy in Sect. 2 is a special case by setting ε = 1/γ , a = e, b = 1 − e, and c = 0. The quadratic economy in Sect. 3 is a special case by setting ε = −1, a = b = e, c = τ , and flipping α, 1 − α. Proposition 5 Suppose that e := a = b < c and 1 1 < 0. <1− ε (1 − 2α)2

(10)

Then the economy has at least three equilibria. Proof We apply Proposition 4. Let x1∗ = (x ∗ , y ∗ ) and x2∗ = (y ∗ , x ∗ ) be the symmetric equilibrium corresponding to p1 = p2 = 1. Using the property of HARA utility functions, u  (x ∗ ) ∗ v  (y ∗ ) ∗ (x (y − b) > 2 − a) − u  (x ∗ ) v  (y ∗ ) y∗ − b x∗ − a + > 2. ⇐⇒ ∗ ε(x − c) ε(y ∗ − c)

(7) ⇐⇒ −

(11)

By market clearing, we have x ∗ + y ∗ = a + b. By the first-order condition, we have α 1/ε (ε(x ∗ − c))−1/ε = (1 − α)1/ε (ε(y ∗ − c))−1/ε ⇐⇒

y∗ − c x∗ − c = . α 1−α

Solving these two equations, we obtain x ∗ = α(a + b) + (1 − 2α)c, y ∗ = (1 − α)(a + b) − (1 − 2α)c. Setting e = a = b < c, we obtain c−e 1 x∗ − a =1+ =1− , x∗ − c 2α(e − c) 2α c−e 1 y∗ − b =1+ =1− . ∗ y −c 2(1 − α)(e − c) 2(1 − α)

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Excess demand of good 1

0.25 0.2 0.15 0.1 0.05 0 -0.05 10 -1

10 0

10 1

Price of good 1 Fig. 4 Aggregate excess demand with symmetric HARA preferences

Therefore, (11) ⇐⇒

  1 1 1 (1 − 2α)2 1 − > 1. 2− > 2 ⇐⇒ − ε 2α 2(1 − α) ε 4α(1 − α)

Since 0 < α < 1, we obtain ε < 0 (and α = 1/2) necessarily and (10).



Remark 4 When preferences are quadratic (ε = −1), then (10) reduces to −1 < 1 −

1 ⇐⇒ 2(1 − 2α)2 > 1 ⇐⇒ 1 − 8α + 8α 2 > 0, (1 − 2α)2

which is the same inequality as in Sect. 3, as it should be. With some algebra, we can show that agent 1’s demand for good 1 when p1 = p is x11 =

(1 − α − αp −ε )c + αp −ε (1 + p)e . 1 − α + αp 1−ε

Agent 2’s demand can be obtained by changing α → 1 − α, and we can compute the aggregate excess demand. As a concrete example, set e = 1, c = 2, ε = −1/4, and α = 1/4. Figure 4 shows the graph of the excess demand of good 1 as a function of the price of good 1. We can see that the demand is increasing in the price around p = 1, exhibiting Giffen behavior.

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6 Discussion and conclusion Kehoe (1998) and Kubler and Schmedders (2010) present numerical examples of multiple equilibria with symmetric CRRA preferences, which are special cases of Proposition 1. Hens and Loeffler (1995) show the uniqueness and stability of equilibrium in a multi-state, multi-agent economy when either (1) agents have relative risk aversion at most 1, (2) agents have heterogeneous constant absolute risk aversion (CARA) preferences, or (3) agents have homogeneous beliefs and there is no aggregate risk (that is, the aggregate endowment is the same in all states). Kubler et al (2013) show that HARA preferences can exhibit Giffen good behavior, although they do not construct an example with multiple equilibria. Geanakoplos and Walsh (2016) show the uniqueness and stability of equilibrium in a two-good, multi-agent economy when agents have identical endowments and heterogeneous decreasing absolute risk aversion (DARA) preferences. The settings of Propositions 2, 5, and Corollary 2 violate the premises of their result since quadratic or HARA utilities with ε < 0 exhibit increasing absolute risk aversion. As they explain, increasing abolsute risk aversion means that sellers have the strongest income effects, which amplifies the aggregate income effect. More generally, it appears that non-uniqueness arises when either sellers have strong income effects or preferences/endowments are highly heterogeneous.3 Indeed, in this paper we have only considered economies with heterogeneous preferences and/or endowments. The following proposition confirms this intuition that some heterogeneity is in fact necessary for the multiplicity of equilibria. Although this result is well known, we present it for completeness. Proposition 6 Consider an economy with identical endowments and identical, weakly monotonic, strictly quasi-concave preferences. Then the unique equilibrium allocation is the initial endowment. Proof Let U be the utility function and e be the endowment of each agent. Suppose that there are I agents indexed by i = 1, . . . , I and that {xi } is an equilibrium I allocation xi ≤ e. different from the initial endowment. By market clearing, we have 1I i=1 Since the initial endowment is always affordable, we have U (xi ) ≥ U (e). By the weak monotonicity and strict quasi-concavity, it follows that U (e) ≥ U

I 1 xi I i=1

> min U (xi ) ≥ U (e), i



which is a contradiction. 3 See also the Engel curve arguments of Freixas and Mas-Colell (1987) and Mas-Colell (1991).

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