Existence and uniqueness of equilibrium in Lucas’ asset pricing model when utility is unbounded

Abstract

This note presents a proof of the existence of a unique equilibrium in a Lucas (Econometrica 46(6):1429–1445, 1978) economy when the utility function displays constant relative risk aversion, and the logarithm of dividends follow a normally distributed autoregressive process of order one with positive autocorrelation. We provide restrictions on the coefficient of relative risk aversion, the discount factor and the conditional variance of the consumption process that ensure the existence of a unique equilibrium.

Appendices

Appendix

A Proof of Part 2 of Lemma 2

Proof

(Part 2 of Lemma 2) As in the main text, denote by $S_{\varphi }$ the set of continuous and $\varphi$-bounded functions. The set $S'_{\varphi }$ is the set of continuous, $\varphi$-bounded, non-decreasing and concave functions, and $S''_{\varphi } \subset S'_{\varphi }$ imposes additionally strict monotonicity and concavity. We want to show that the contraction operator T maps any function $\tilde{f} \in S'_{\varphi }$ into the subset $S''_{\varphi }$. As the solution to the functional equation is characterized by $Tf = f$ and $S'_{\varphi }$ is a closed set, if the operator T transforms any non-decreasing and concave function into a strictly increasing and concave function, then the fixed point f is strictly increasing and concave (Corollary 1 of the Contraction Mapping Theorem in Stokey and Lucas 1989, p. 52). To show this result, we suppose first that h is strictly increasing and concave and take any $\tilde{f} \in S'_{\varphi }$. To begin, let us study whether $T\tilde{f}$ is strictly increasing. For any pair $\hat{y},y \in Y$ with $\hat{y} > y$, the function $T\tilde{f}$ satisfies:

$$\begin{aligned} T\tilde{f}(\hat{y})&= h(\hat{y}) + \beta \int _Z \tilde{f}( G(\hat{y},z')) Q(\mathrm{d}z')\\&> h(y) + \beta \int _Z \tilde{f}( G(y,z')) Q(\mathrm{d}z')\\&= T\tilde{f}(y). \end{aligned}$$

The inequality holds because G and h are strictly increasing and $\tilde{f}$ is non-decreasing. Hence, $T\tilde{f}$ is strictly increasing. To analyze concavity, define $y_{\omega } = \omega y + (1-\omega ) y'$, for any $y,y' \in Y$, $y \ne y'$, and $0< \omega < 1$. Strict concavity of h and G, together with $\tilde{f}$ being concave, imply that:

$$\begin{aligned} T\tilde{f}(y_\omega )&= h(y_\omega ) + \beta \int _Z \tilde{f}( G(y_\omega ,z')) Q(\mathrm{d}z') \\&> \omega \left[ h(y) + \beta \int _Z \tilde{f}( G(y,z')) Q(\mathrm{d}z') \right] + (1 - \omega ) \\&\quad \times \left[ h(y') + \beta \int _Z \tilde{f}( G(y',z')) Q(\mathrm{d}z') \right] \\&= \omega T\tilde{f}(y) + (1-\omega ) T \tilde{f}(y'). \end{aligned}$$

The function $T\tilde{f}$ is strictly concave. Taken together, we know that for any $\tilde{f} \in S'_{\varphi }$, $T\tilde{f} \in S''_{\varphi }$. Hence, f (such that $Tf=f$) must be an element of the set $S''_{\varphi }$, guaranteeing that f has the same functional form as h. Now, suppose h is convex and falling. We could again define the operator T as $Tf(y) = h(y) + \beta \int _Z f(G(y,z')) Q(\mathrm{d}z')$ and study into which subset a candidate solution is mapped into. To make the analysis brief though, we take a different route. We consider the modified operator $Tf_{-} = h_{-} + \beta \int _Z f_{-} (G(y,z')) Q(z')$, with $h_{-} = -h$ and $f_{-} = -f$. Under the same assumptions guaranteeing that there is a unique fixed point of the original contraction mapping, there exists a unique fixed point of the modified contraction mapping. As $h_{-}$ is strictly increasing and concave, the proof above applies to the modified contraction mapping. As $f_{-}$ is strictly increasing and concave, f is strictly decreasing and convex and inherits the properties of h. $\square$

B Limit condition on v

Let us take $v\in S_\phi$ such that $Hv=v$, with the operator H as defined in Sect. 4 of the main text. Our initial aim is to characterize lower and upper bounds on v in the functional space $S_\phi$. We show that if $v\ge 0$ (respectively, $\le$0), v can be bounded below (resp. above) using the zero function and above (resp. below) using function $\phi$ (resp. $-\phi$). To this end, we consider $\gamma \in (0,1)$. Define the set $S'_\phi =\left\{ f\in S_\phi : f\ge 0 \right\}$. This is a closed subset of $S_\phi$ (its complement in $S_\phi$ is open). We pick any $f\in S'_\phi$. Since the utility function u takes on positive values, $Hf \ge 0$ Thus, since f was arbitrary, $H:S'_\phi \rightarrow S'_\phi$. Then by Corollary 1 of the CMT (p. 52) in Stokey and Lucas (1989) $v\in S'_\phi$, i.e., $v\ge 0$. A similar argument shows that if $\gamma >1$, $v\le 0$. The remainder of this section shows that the discounted expected value of the upper bound converges to zero, implying that $\lim _{t\rightarrow \infty }\mathbb {E}_0[\beta ^tv(x_t,y_t)]=0$. We take $\gamma \in (0,1)$ (If $\gamma > 1$, the condition is equivalent, as the constant in the bound changes the sign of the bound.) Then for any t, $x_t\in X$ and $y_t\in Y$:

$$\begin{aligned} \mathbb {E}_0[\beta ^tv(x_t,y_t)]&\le \mathbb {E}_0[\beta ^t \phi ] \\&= \kappa \beta ^{t} \ \mathbb {E}_0 [\mathbb {E}_{t-1} [ \mathrm{max} \lbrace 1, y_{t}^{1-\gamma } \rbrace ] ]\\&\le \kappa \beta ^{t} \ \mathbb {E}_0 \left[ 0.5 + y_{t-1}^{\alpha (1-\gamma )} \int _0^{\infty } Q(\mathrm{d}z') (z')^{1-\gamma } \right] \\&\le \kappa \beta ^{t} [ 0.5 + y_{0}^{1-\gamma } \exp ( t (1-\gamma )^{2} \sigma ^{2}/2 ) ] \\&= \kappa [ \beta ^{t} 0.5 + y_{0}^{1-\gamma } \exp ( t [ \log (\beta ) + (1-\gamma )^{2} \sigma ^{2}/2 ] ). ] \end{aligned}$$

The first line follows from the fact that $v\in S_{\phi }$. The second line uses the definition of $\phi$ and the law of iterated expectations. The third line bounds the term in brackets. The strategy is identical to the one used in the proof of Proposition 1. The fourth line iterates until time zero and uses the fact that $\alpha \in (0,1)$. The fifth line factors $\beta ^{t}$ in. The entire sums converges to zero with $t\rightarrow \infty$ if $\log (\beta ) + (1-\gamma )^{2} \sigma ^{2}/2 < 0$. The proof for the cases $\gamma >1$ is analogous.