## Abstract

I establish the uniqueness of a stationary equilibrium in a Bewley–Aiyagari model when the agents’ utility function exhibits constant relative risk aversion bounded above by 1 and the production function exhibits a certain gross substitute property.

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## Notes

- 1.
- 2.
- 3.
For a textbook treatment of Bewley models, see Ljungqvist and Sargent (2004). Bewley models are used to study many macroeconomic phenomena. These include asset pricing (Huggett 1993), fiscal policy (Heathcote 2005), defaults (Chatterjee et al. 2007), capital taxation (Conesa et al. 2009), labor productivity (Heathcote et al. 2010), wealth distribution (Benhabib et al. 2015), borrowing constraints (Guerrieri and Lorenzoni 2017), monetary transmission mechanisms (Kaplan et al. 2018), and many more. For an excellent survey of quantitative results in Bewley models, see Heathcote et al. (2009).

- 4.
- 5.
I consider here the standard discrete time formulation. For a continuous time formulation of the BA model, see Achdou et al. (2018).

- 6.
- 7.
The assumptions that

*P*is irreducible and aperiodic guarantee that the Markov chain that governs the states’ dynamics is ergodic (see Lemma 2 for more details). - 8.
All the results in this paper can be easily extended to the case where the agents can borrow, as long as the borrowing limit is tighter than the natural borrowing limit \( -\frac{ws_{1}}{R -1}\) (see Aiyagari (1994)).

- 9.
The RRA measure is defined as \(RRA (c) = -\frac{c u^{ \prime \prime } \left( c\right) }{u^{ \prime } (c)}\). For CRRA utility functions we have \( -\frac{cu^{ \prime \prime }(c)}{u^{ \prime }(c)} =\gamma \).

- 10.
Let \(X^{t}:=\underbrace{X \times \cdots \times X}_{t~ \mathrm {t} \mathrm {i} \mathrm {m} \mathrm {e} \mathrm {s}}\) be the space of all finite savings-labor productivities histories of length

*t*. A*consumption plan*\(\pi \) is a function that assigns to every finite history a feasible consumption. That is, \(\pi (a(1) ,s(1) ,\cdots ,a(t) ,s(t)) \in C(a(t) ,s(t))\) for each history \((a(1) ,s(1) ,\ldots ,a(t) ,s(t)) \in X^{t}\) and for all \(t =1 ,2 ,\cdots \). - 11.
The probability measure on the space of all infinite histories \(X^{\mathbb {N}}\) is uniquely defined (see for example Bertsekas and Shreve (1978)).

- 12.
- 13.
Clearly, the savings policy function and the value function depend also on the wage

*w*. For notational convenience I omit*w*as an argument in those functions. - 14.
That is, for any \((a ,s) \in Z\), \(Q((a ,s) , \centerdot ;R)\) is a measure; and for any \(D \times B \in \mathcal {B}(A) \times 2^{S}\), \(Q( \cdot ,D \times B;R)\) is a measurable function (see Theorem 9.13 in Stokey and Lucas (1989)).

- 15.
A similar proof of ergodicity in a more general framework (but under stronger assumptions) can be found in Benhabib et al. (2015) and in Zhu (2017). These proofs all rely on a characterization of ergodicity that is given in Meyn et al. (2012). A different approach to proving Lemma 2 relies on the monotonicity of the Markov kernel

*Q*and uses techniques developed in Hopenhayn and Prescott (1992). This last approach is used in Huggett (1993) and Acemoglu and Jensen (2015). - 16.
I thank an anonymous referee for providing the proof of Lemma 3.

- 17.
When \(\gamma >1\) savings might be decreasing in the interest rate [for more details, see Toda (2018)].

- 18.
- 19.
Note that \(\gamma \le 1\) implies that \(cu^{ \prime }(c)\) is an increasing function.

- 20.

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### Publisher's Note

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The author wishes to thank Ehud Lehrer, Gabriel Carroll, two anonymous referees, and seminar participants at Stanford for their valuable comments.

## Appendix

### Appendix

I first introduce the notations and preliminary results that are needed to prove Theorem 1.

Let \(A =[0 ,{\overline{a}}]\), \(Z =A \times S\).

Denote by *b* the agent’s savings in the next period. For every \((a ,s_{i} ,R) \in Z \times I\) and \(b \in C (a ,s_{i} ,R)\), define the following function:

From Lemma 1, the value function is bounded on \(Z \times I\). Thus, I can use the standard Bellman’s equation approach to solve the agent’s problem.

Let \(B (Z \times I)\) be the space of all bounded real-valued functions defined on \(Z \times I\). Define the operator \(T :B (Z \times I) \rightarrow B (Z \times I)\) by

Standard dynamic programming arguments (Blackwell 1965) show that the value function *V* is the unique fixed point of *T*, i.e., there is a unique function \(V \in B (Z \times I)\) such that \(T V =V\). The equation \(TV =V\) is called the Bellman equation.

From Theorem 9.8 in Stokey and Lucas (1989), the value function *V* is continuous and concave in the first argument, and thus *h*(*a*, *s*, *b*, *R*, *V*) is strictly concave in *b*. The unique savings level that attains the maximum of *h*(*a*, *s*, *b*, *R*, *V*) is the *savings policy function*:

Note that the savings policy function induces a consumption policy function \(\sigma (a ,s ,R)\) by the equation \(\sigma (a ,s ,R) =R a +ws -g (a ,s ,R)\).

Let \(g_{f}\left( a ,s ,R\right) ={\text {*}}{argmax}_{b \in C\left( a ,s ,R\right) }h\left( a ,s ,b ,R ,f\right) \) and \(\sigma _{f}\left( a ,s\right) =Ra +ws -g_{f}\left( a ,s ,R\right) \) for all \(\left( a ,s ,R\right) \in Z \times I\).

For \(f \in B\left( Z \times I\right) \) define

Denote by \(u^{ \prime }\) the derivative of *u* and by \(h^{ \prime }\left( a ,s ,b ,R ,f\right) \) the derivative of *h* with respect to *b*.

The *envelope theorem* (Benveniste and Scheinkman 1979) implies that *Tf* is differentiable.

In addition, if \(\sigma _{f}\left( a ,s ,R\right) >0\) (which will always be the case in this paper, since \(u^{ \prime }(0) =\infty \)), then

The last equation is called the *envelope condition*. Berge’s maximum theorem [see Theorem 17.31 in Aliprantis and Border (2006)] implies that \(\sigma _{f}\) is continuous. Thus, \((Tf)^{ \prime }\) is continuous as a composition of continuous functions. Since \(TV =V\), the value function is continuously differentiable and \(V^{ \prime }(a ,s ,R) =Ru^{ \prime }\left( \sigma (a ,s ,R)\right) \).

To prove Theorem 1 I need the following three Lemmas:

### Lemma 4

Let \(z :[0 ,\infty ) \rightarrow (0 ,\infty )\) be a concave function. Then \(\frac{\alpha }{z(\alpha x)}\) is increasing in \(\alpha \) on \((0 ,\infty )\) for all \(x >0\).

### Proof

Let \(\alpha _{2} \ge \alpha _{1} >0\) and \(x >0\). Since *z* is a concave function we have

which implies

Since \(z(0) >0\) the last inequality implies \(\alpha _{1}xz(\alpha _{2}x) \le \alpha _{2}xz(\alpha _{1}x)\) or

which proves the Lemma. \(\square \)

### Lemma 5

\(\sigma (a ,s ,R)\) is concave in *a* for all \((s ,R) \in S \times I\).

### Proof

See Jensen (2017). \(\square \)

The proof of the following simple lemma is given in Lehrer and Light (2018).

### Lemma 6

Let \(z :\mathbb {R} \rightarrow \mathbb {R}\) and \(f :\mathbb {R} \rightarrow \mathbb {R}\) be strictly concave, continuously differentiable functions. Let \(\phi >0\). Denote \(x_{f} ={\text {*}}{argmax}_{x \in [0 ,\phi ]}f \left( x\right) \) and \(x_{z} ={\text {*}}{argmax}_{x \in [0 ,\phi ]}z \left( x\right) \). If for all \(x \in [0 ,\phi ]\) we have \(f^{ \prime } \left( x\right) \ge z^{ \prime } (x)\), then \(x_{f} \ge x_{z}\). Furthermore, if \(x_{z} \in (0 ,\phi )\) and \(f^{ \prime } \left( x\right) >z^{ \prime } \left( x\right) \) for all \(x \in [0 ,\phi ]\), then \(x_{f} >x_{z}\).

### Theorem 1

Assume that \(\gamma \le 1\). Then \(R_{2} \ge R_{1}\) implies \(g(a ,s ,R_{2}) \ge g(a ,s ,R_{1})\) for all \((a ,s) \in A \times S\) .

### Proof

Assume \(\frac{1}{\beta }>R_{2} \ge R_{1} >0\) and \(w >0\). Let \(f :A \times S \times I \rightarrow \mathbb {R}\) be differentiable and concave in the first argument. Assume that \(f^{ \prime }(a ,s ,R_{2}) \ge f^{ \prime }(a ,s ,R_{1})\) for all \((a ,s) \in Z\). Let \((a ,s_{i}) \in Z\) and \(b \in C(a ,s_{i} ,R_{1})\). We have

Note that \(C(a ,s_{i} ,R_{1}) =[0 ,R_{1}a +s_{i}] =C(\frac{R_{1}}{R_{2}}a ,s_{i} ,R_{2})\). Thus, Lemma 6 implies that \(g_{f}(\frac{R_{1}}{R_{2}}a ,s_{i} ,R_{2}) \ge g_{f}(a ,s_{i} ,R_{1})\). That is, \(\sigma _{f}(a ,s_{i} ,R_{1}) \ge \sigma _{f}(\frac{R_{1}}{R_{2}}a ,s_{i} ,R_{2})\). We have

The first and last equalities follow from the envelope condition. The first inequality follows from the fact that \(u^{ \prime }\) is decreasing. To see why the second inequality holds, first note that by defining \(c_{2} =\sigma _{f}(\frac{R_{2}}{R_{2}}a ,s_{i} ,R_{2})\) and \(c_{1} =\sigma _{f}(\frac{R_{1}}{R_{2}}a ,s_{i} ,R_{2})\) and using^{Footnote 19}\(\gamma \le 1\) we have \(u^{ \prime }(\sigma _{f}(\frac{R_{1}}{R_{2}}a ,s_{i} ,R_{2}))\sigma _{f}(\frac{R_{1}}{R_{2}}a ,s_{i} ,R_{2}) \le u^{ \prime }(\sigma _{f}(a ,s_{i} ,R_{2}))\sigma _{f}(a ,s_{i} ,R_{2})\). Using Lemma 4 and the fact that \(\sigma _{f}\) is concave in *a* yields

Define \(f_{n} =T^{n}f : =T(T^{n -1}f)\) for \(n =1 ,2 , \ldots \) where \(T^{0}f : =f\). I conclude that \(f_{n}^{ \prime }(a ,s ,R)\) is increasing in *R* for every *n*. From the Banach fixed-point theorem \(f_{n} \rightarrow V\). The envelope condition implies that \(f_{n}^{ \prime } =u^{ \prime }\left( \sigma _{f_{n}}\right) \) for every *n*. Theorem 3.8 and Theorem 9.9 in Stokey and Lucas (1989) show that \(\sigma _{f_{n}} \rightarrow \sigma \). Thus, \(f_{n}^{ \prime }(a ,s ,R) =Ru^{ \prime }\left( \sigma _{f_{n}}\left( a ,s ,R\right) \right) \rightarrow Ru^{ \prime }\left( \sigma \left( a ,s ,R\right) \right) =V^{ \prime }(a ,s ,R)\) which implies that \(V^{ \prime }(a ,s ,R)\) is increasing in *R*. Inequality (3), Lemma 6, and the fact that *g* is increasing in *a* imply that \(g(a ,s ,R_{2}) \ge g(\frac{R_{1}}{R_{2}}a ,s ,R_{2}) \ge g(a ,s ,R_{1})\) - which proves the Theorem. \(\square \)

### Theorem 2

Assume that \(\gamma \le 1\). Then the aggregate savings increase with the interest rate for every fixed \(w >0\). That is, \(R_{2} \ge R_{1}\) implies \(AS(R_{2} ,w) \ge AS(R_{1} ,w)\).

### Proof

For two distributions \(\lambda _{2} ,\lambda _{1} \in \mathcal {P}(Z)\) write \(\lambda _{2} \succeq _{I_{a}}\lambda _{1}\) if for any bounded and continuous real-valued function *f*(*a*, *s*) that is increasing in *a* we have^{Footnote 20}

Let \(R_{2} \ge R_{1} >0\) and fix \(w >0\). I claim that if \(\lambda ( \cdot ;R_{2}) \succeq _{I_{a}}\lambda ( \cdot ;R_{1})\) then \(M\lambda ( \cdot ;R_{2}) \succeq _{I_{a}}M\lambda ( \cdot ;R_{1})\).

Assume that \(\lambda ( \cdot ;R_{1}) ,\lambda ( \cdot ;R_{2}) \in \mathcal {P}(Z)\) and \(\lambda ( \cdot ;R_{2}) \succeq _{I_{a}}\lambda ( \cdot ;R_{1})\). Let \(f :Z \rightarrow \mathbb {R}\) be a bounded and continuous function. First note that

To see that the last equality holds, first let \(f =1_{D \times B}\) for \(D \times B \in \mathcal {B}(A) \times 2^{S}\) where \(1_{D \times B}\) is the indicator function of \(D \times B\). We have

A standard argument shows that (4) holds for a general *f*. Now assume that *f* is increasing in *a*. Then

The first inequality follows from the facts that \(\sum \limits _{j =1}^{n}P_{sj}f(g(a ,s ,R_{2})) ,s_{j})\) is increasing in *a* and \(\lambda ( \cdot ;R_{2}) \succeq _{I_{a}}\lambda ( \cdot ;R_{1})\). The second inequality follows from the facts that \(g(a ,s ,R_{2}) \ge g(a ,s ,R_{1})\) for all \((a ,s) \in Z\) and *f* is increasing in the first argument.

I conclude that \(M^{t}\lambda ( \cdot ;R_{2}) \succeq _{I_{a}}M^{t}\lambda ( \cdot ;R_{1})\) for all \(t =1 ,2 ,3 ,\cdots \). Since \( \succeq _{I_{a}}\) is a closed order and \(M^{t}\lambda ( \cdot ;R)\) converges to \(\mu ( \cdot ;R)\) we have \(\mu ( \cdot ;R_{2}) \succeq _{I_{a}}\mu ( \cdot ;R_{1})\). Thus,

which proves that the aggregate savings increase in the interest rate. \(\square \)

### Lemma 3

Suppose that the elasticity of substitution *ES*(*K*, *N*) is bounded below by 1. Then the function *K*(*R*) / *w*(*R*) is strictly decreasing.

### Proof

**Step 1**. If \(F_{KN}(K ,1)K <F_{N}(K ,1)\) for all \(K \ge 0\), where \(F_{KN}\) is the mixed partial derivative, then *K*(*R*) / *w*(*R*) is strictly decreasing.

Let \(M(R) =F_{K}^{ -1}(R +\delta -1 ,1)\). We have

Since *F* is twice differentiable and strictly concave in *K*, *M*(*R*) is strictly decreasing and differentiable. Thus, \(M^{\prime }(R)<0\). I conclude that \(F_{N}(M(R) ,1) >F_{KN}(M(R) ,1)M(R)\) implies \(\frac{ \partial }{ \partial R}\frac{K(R)}{w(R)} <0\) which proves step 1.

**Step 2.** If \(ES(K ,1) \ge 1\) then \(F_{KN}(K ,1)K <F_{N}(K ,1)\). Note that

The last equality and the facts that \(F_{K}(K ,1) >0\), \(F_{KK}(K ,1) <0\), and \(ES(K ,1) \ge 1\) imply

That is, \(F_{KN}(K ,1)K <F_{N}(K ,1)\) which proves step 2. I conclude that the function *K*(*R*) / *w*(*R*) is strictly decreasing. \(\square \)

### Theorem 4

Assume that \(\gamma \le 1\) and \(ES(K ,N) \ge 1\). (i) If the discount factor increases then the unique equilibrium interest rate \(R^{ *}\) decreases. (ii) If the risk of the invariant distribution of labor productivities increases (in the sense of second-order stochastic dominance) then the unique equilibrium interest rate \(R^{ *}\) decreases. (iii) Suppose that the production function is given by the Cobb–Douglas production function \(F(K ,N) ={\overline{T}}K^{\alpha }N^{1 -\alpha }\) for some \(0<\alpha <1\) and \({\overline{T}} >0\). If the depreciation rate \(\delta \) increases or the output elasticity \(\alpha \) decreases then the unique equilibrium interest rate \(R^{ *}\) decreases.

### Proof

Consider the function \(v(R;e) =AS(R ,1;e) -\frac{K(R;e)}{w(R;e)}\) where *e* is some parameter (the discount factor, the invariant distribution of labor productivities, the depreciation rate, or the output elasticity). \(R^{ *}\) is an equilibrium price if \(v(R^{ *};e) =0\). Theorem 3 implies that *v* is strictly increasing in *R*. It is easy to see that if *v* is increasing in *e* then the unique equilibrium \(R^{ *}\) is decreasing in *e*. From Propositions 1 and 2 in Acemoglu and Jensen (2015), the aggregate supply of savings is increasing in the discount factor and in the risk of the invariant distribution of labor productivities, so parts (i) and (ii) follow. Furthermore, in the case of the Cobb–Douglas production function, the first-order conditions yield

It is easy to check that *k*(*R*) / *w*(*R*) is decreasing in the depreciation rate and increasing in the output elasticity, so part (iii) follows. \(\square \)

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### Cite this article

Light, B. Uniqueness of equilibrium in a Bewley–Aiyagari model.
*Econ Theory* **69, **435–450 (2020). https://doi.org/10.1007/s00199-018-1167-z

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### Keywords

- Bewley–Aiyagari model
- Uniqueness of equilibrium
- heterogeneous agents
- Aggregate savings
- Stationary equilibrium

### JEL Classification

- D52
- D90
- E21