Risk Aversion and Uniqueness of Equilibrium in Economies with Two Goods and Arbitrary Endowments

Andrea Loi; Stefano Matta

doi:10.1515/bejte-2021-0150

Article

Risk Aversion and Uniqueness of Equilibrium in Economies with Two Goods and Arbitrary Endowments

Andrea Loi and Stefano Matta

Published/Copyright: October 13, 2022

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From the journal The B.E. Journal of Theoretical Economics Volume 23 Issue 2

Abstract

We study the connection between risk aversion, the number of consumers, and the uniqueness of equilibrium. We consider an economy with two goods and I impatience types, where each type has additive separable preferences with HARA Bernoulli utility function, u H ( x ) ≔ γ 1 − γ b + a γ x 1 − γ . We show that if γ ∈ 1 , I I − 1 , the economy has a unique regular equilibrium. Moreover, the methods used, including Newton’s symmetric polynomials and Descartes’ rule of signs, enable us to offer new sufficient conditions for uniqueness in a closed-form expression that highlight the role played by endowments, patience, and specific HARA parameters. Finally, we derive new necessary and sufficient conditions that ensure uniqueness for the particular case of CRRA Bernoulli utility functions with γ = 3.

Keywords: uniqueness of equilibrium; excess demand function; risk aversion; polynomial approximation; Descartes’ rule of signs; Newton’s symmetric polynomials

JEL Classification: C62; D51; D58

Corresponding author: Stefano Matta, Dipartimento di Scienze Economiche e Aziendali, Università di Cagliari, Cagliari, Italy, E-mail: smatta@unica.it

Funding source: Fondazione Banco di Sardegna

Acknowledgement

We would like to acknowledge Alexis Akira Toda and an anonymous referee for their useful comments and critical remarks.

Research funding: The first author was supported by INdAM. GNSAGA – Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni. Both authors were supported by STAGE – Funded by Fondazione di Sardegna.
Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix A

A.1 Derivation of Eq. (14)

Consumer i maximizes (see (1))

u i ( x , y ) = u H ( x ) + β i u H ( y ) ,

under the constraint

p e i + f i ≤ p x + y ,

where

u H ( x ) ≔ γ 1 − γ b + a γ x 1 − γ , γ > 0 , γ ≠ 1 , a > 0 , b ≥ 0 .

By monotonicity of preferences, the budget constraint is fulfilled as an equality. By substituting y = pe_i + f_i − px into the objective function, we turn the constrained maximization problem into the unconstrained problem of maximizing the following function

γ 1 − γ b + a γ x 1 − γ + β i γ 1 − γ b + a γ ( p e i + f i − p x ) 1 − γ .

The necessary (and sufficient) condition is

a b + a x γ − γ − a p β b + a ( f + e p − p x ) γ − γ = 0 .

By setting ϵ ≔ 1 γ and σ i ≔ β i ϵ we obtain i’s demand for good x:

b − b p ϵ σ i + a ϵ p e i + f i a ϵ p + σ i p ϵ .

By summing over consumers i = 1, …, I and denoting ∑ i = 1 I e i = r x ( I ) , we obtain (14), the aggregate excess demand function for good x.

A.2 Proofs of the Intermediate Results

Proof of Proposition 7

At an equilibrium price p, the function (14)

− p r x ( I ) + ∑ i = 1 I p e i + f i + b a ϵ − b a ϵ σ i p ϵ 1 + σ i p ϵ − 1 ,

vanishes, or equivalently

− p r x ( I ) ∏ i = 1 I 1 + σ i p ϵ − 1 + ∑ i = 1 I p e i + f i + b a ϵ − b a ϵ σ i p ϵ ∏ j = 1 I 1 + σ j p ϵ − 1 1 + σ i p ϵ − 1 = 0 .

By Eq. (15), we can write these products as

∏ i = 1 I 1 + σ i p ϵ − 1 = 1 + ∑ t = 1 I − 1 S t ( σ ) p t ( ϵ − 1 ) + σ 1 … σ I p I ( ϵ − 1 )

∏ j = 1 I 1 + σ j p ϵ − 1 1 + σ i p ϵ − 1 = 1 + ∑ t = 1 I − 1 S t ( σ − i ) p t ( ϵ − 1 )

and rewrite the expression accordingly:

− p r x ( I ) 1 + ∑ t = 1 I − 1 S t ( σ ) p t ( ϵ − 1 ) + σ 1 … σ I p I ( ϵ − 1 ) + ∑ i = 1 I p e i + f i + b a ϵ − b a ϵ σ i p ϵ 1 + ∑ t = 1 I − 1 S t ( σ − i ) p t ( ϵ − 1 ) .

By expanding and rearranging, we immediately get

− r x ( I ) σ 1 … σ I p I ( ϵ − 1 ) + 1 − ∑ t = 1 I − 1 r x ( I ) S t ( σ ) − ∑ i = 1 I e i S t ( σ − i ) p t ( ϵ − 1 ) + 1 + ∑ t = 1 I − 1 ∑ i = 1 I f i + b a ϵ S t ( σ − i ) p t ( ϵ − 1 ) − b a ϵ ∑ i = 1 I σ i p ϵ − b a ϵ ∑ t = 1 I − 1 ∑ i = 1 I σ i S t ( σ − i ) p t ( ϵ − 1 ) + ϵ + r y ( I ) + I b a ϵ .

Note now that by the change of index u ≔ t + 1, one gets

− b a ϵ ∑ t = 1 I − 1 ∑ i = 1 I σ i S t ( σ − i ) p t ( ϵ − 1 ) + ϵ = − b a ϵ ∑ t = 1 I − 1 ∑ i = 1 I σ i S t ( σ − i ) p ( t + 1 ) ( ϵ − 1 ) + 1 = − b a ϵ ∑ u = 2 I ∑ i = 1 I σ i S u − 1 ( σ − i ) p u ( ϵ − 1 ) + 1 = b a ϵ ∑ i = 1 I σ i S t ( σ − i ) p ( ϵ − 1 ) + 1 − b a ϵ ∑ t = 1 I − 1 ∑ i = 1 I σ i S t − 1 ( σ − i ) p t ( ϵ − 1 ) + 1 − b a ϵ ∑ i = 1 I σ i S I − 1 ( σ − i ) p I ( ϵ − 1 ) + 1 = b a ϵ ∑ i = 1 I σ i p ϵ − b a ϵ ∑ t = 1 I − 1 ∑ i = 1 I σ i S t − 1 ( σ − i ) p t ( ϵ − 1 ) + 1 − I b a ϵ σ 1 … σ I p I ( ϵ − 1 ) + 1 ,

where in the last equality we use ∑ i = 1 I σ i S I − 1 ( σ − i ) = I σ 1 ⋯ σ I .

By inserting this last equality into the previous expression, one gets

− σ 1 … σ I r x ( I ) + I b a ϵ p I ( ϵ − 1 ) + 1 − ∑ t = 1 I − 1 r x ( I ) S t ( σ ) − ∑ i = 1 I e i S t ( σ − i ) + b a ϵ ∑ i = 1 I σ i S t − 1 ( σ − i ) p t ( ϵ − 1 ) + 1 + ∑ t = 1 I − 1 ∑ i = 1 I f i + b a ϵ S t ( σ − i ) p t ( ϵ − 1 ) + r y ( I ) + I b a ϵ ,

and the proposition follows. □

Proof of Lemma 8

We work by induction on I ≥ 2 for all t such that 1 ≤ t ≤ I − 1. The base on the induction is immediate ( σ = σ 1 , σ 2 ) :

F ( 1,2 ) = ( e 1 + e 2 ) S 1 ( σ ) = − e 1 S 1 ( σ − 1 ) − e 2 S 1 ( σ − 2 ) = ( e 1 + e 2 ) ( σ 1 + σ 2 ) − e 1 σ 2 − e 2 σ 1 = e 1 σ 1 + e 2 σ 2 > 0 F ( 2,2 ) = ( e 1 + e 2 ) S 2 ( σ ) − e 1 S 2 ( σ − 1 ) − e 2 S 2 ( σ − 2 ) = ( e 1 + e 2 ) σ 1 σ 2 > 0 .

Assume now, by the induction hypothesis, that

F ( t , I − 1 ) > 0

for each integer 1 ≤ t ≤ I − 2. By (16) and (17), Eq. (22) reads as

F ( t , I ) = r x ( I − 1 ) S t ( σ − I ) + σ I r x ( I ) S t − 1 ( σ − I ) − ∑ j = 1 I − 1 e j S t ( σ − ( I , j ) ) + σ I S t ( σ − ( I , j ) ) ,

that we can rewrite as

F ( t , I ) = F ( t , I − 1 ) + σ I F ( t − 1 , I − 1 ) + σ I e I S t − 1 ( σ − I ) ,

which is strictly positive by the induction assumption. □

Proof of Lemma 9

For fixed (e, f, σ, I), the aggregate excess demand function (18) depends on the price p and the parameter ϵ. Let p₀ be a regular equilibrium of the function of one variable z e , f , p , ϵ 0 , σ , I . By the implicit function theorem, the regularity property holds true after a small perturbation of ϵ₀. Hence, the result follows by the denseness of Q in R . □

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Received: 2021-11-13

Revised: 2022-07-04

Accepted: 2022-09-20

Published Online: 2022-10-13

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Articles in the same Issue

https://doi.org/10.1515/bejte-2021-0150

Keywords for this article

uniqueness of equilibrium; excess demand function; risk aversion; polynomial approximation; Descartes’ rule of signs; Newton’s symmetric polynomials