Risk aversion and uniqueness of equilibrium: a polynomial approach

We study the connection between risk aversion, number of consumers and uniqueness of equilibrium. We consider an economy with two goods and $c$ impatience types, where each type has additive separable preferences with HARA Bernoulli utility function, $u_H(x):=\frac{\gamma}{1-\gamma}\left(b+\frac{a}{\gamma}x\right)^{1-\gamma}$. We show that if $\gamma\in \left(1, \frac{c}{c-1}\right]$, the equilibrium is unique. Moreover, the methods used, involving Newton's symmetric polynomials and Descartes' rule of signs, enable us to offer new sufficient conditions for uniqueness in a closed-form expression highlighting the role played by endowments, patience and specific HARA parameters. Finally, new necessary and sufficient conditions in ensuring uniqueness are derived for the particular case of CRRA Bernoulli utility functions with $\gamma =3$.