A Production Economy¶
The economy in the previous section already showcases all the key ingredients of household heterogeneity. However, there is no production and hence no way to talk about returns. The \(r\) in the previous model is better understood as an attribute of a financial contract, but not a rate of return on a productive investment. This ultimately prevents us from drawing a clear parallel between the heterogeneous agent economy above and the representative agent economy in the neoclassical growth model, where prices reflect the marginal productivity of factors used in generating the aggregate output of the economy.
The economy described in Aiyagari (1994) fills the next portion of the gap between heterogeneous agent models and the representative agent real business cycle models. The key is the move from an endowment to a production economy. Production is (on purpose) just as in the neoclassical growth model: there is a representative firm operating a constant returns to scale technology, \(Y=f\left(z,K,L\right)\). As it turns out, doing this greatly simplifies market clearing. The reason is in the discussion in Profit Maximization: A note. The scale of operations of a firm with a constant-returns-to-scale technology is undefined, and so prices have to be such that the firm is indifferent between any scale. For this to happen, prices must reflect the marginal product of inputs.
Aiyagari also interprets the value of \(\epsilon\) as the value of "labor efficiency" instead of a direct income endowment. Under this interpretation, income depends on labor efficiency \(\left(\epsilon\right)\) and the market wage \(\left(w\right)\). We assume for simplicity that individuals do not care about leisure and thus supply one unit of labor inelastically. As before, \(\epsilon\) is stochastic and follows a Markov process \(Q\). This is the only source of variation in the economy and the realizations of \(\epsilon\) are independent across households.
The household's problem is then
The borrowing limit must satisfy
The Euler equation is, as before,
As above, the households take the (constant) prices as given. Their behavior induces a Markov process over the state \(\left(\epsilon,a\right)\) that is constructed in the same way as in the previous section and results in a stationary distribution of households \(\Gamma\).
The firm's cost minimization problem implies that the demand for inputs is perfectly elastic at equilibrium prices. Therefore, the only way to clear the market is to make producers indifferent between any scale of production by setting
Stationary recursive competitive equilibrium
A S-RCE is a set of a value function \(\left(V\right)\), a policy function \(\left(a^{\prime}\right)\), a distribution \(\left(\Gamma\right)\), and prices \(\left(r,w\right)\) such that:
- Given \(\left(r,w\right)\) the value and policy functions solve the agent's problem (the dynamic programming problem above).
- Given the policy function, \(\Gamma\) is a fixed point of the adjoint Markov operator of the endogenous Markov process defined by the (exogenous) Markov process for \(\epsilon\) with Transition function \(Q\) and the (endogenous) policy function for \(a^{\prime}\).
-
The capital and labor markets clear. That is, aggregate capital reflects the aggregate assets of the households and aggregate labor demand equals the supply of "efficiency units of labor" from the household
\[ K=\int\int a\cdot d\Gamma\left(\epsilon,a\right)\qquad L=\int\int\epsilon\cdot d\Gamma\left(\epsilon,a\right). \] -
Prices are consistent with firm optimization (in this case cost-minimization)
\[ r=f_{k}\left(z,K,L\right)-\delta\qquad w=f_{L}\left(z,K,L\right). \]
Algorithm: S-RCE Algorithm
Input: Guess for price \((r)\)
Output: \(V,a^{\prime},\Gamma,r\)
-
Solve the DP problem of the agent given \((r,w)\):
\((V,a^{\prime})=T(V;r)\) (a fixed point problem) -
Find stationary distribution method (or update dist. N times)
-
Update prices to ensure market clearing:
\(K=\sum_{i}\sum_j a\cdot\Gamma(i,j) \quad \longrightarrow \quad r = f_K(z,K,1)-\delta \quad w=f_L(z,K,1)\)
Dampen updating of prices if necessary
- Repeat (1)-(3) until prices converge
Note that the labor supply is exogenous! So, (i) we can simplify the integral because it does not depend on the level of assets, and (ii) we can normalize the level of \(\epsilon\) so that the market clearing condition becomes: \(L=\int\epsilon\Gamma_{\epsilon}\left(\epsilon\right)d\epsilon=1\), where \(\Gamma_{\epsilon}\) is the stationary distribution of the Markov process \(Q\) having normalized the levels so that the expected value of \(\epsilon\) is 1.
Finally, the definition of equilibrium also gives us an algorithm to find it. The key is that once we normalize the level of labor supply we realize that, just as in the neoclassical growth model, all the economy depends only on the aggregate level of capital, \(K\). Once \(K\) is known it implies prices \(r=f_{k}\left(z,K,1\right)-\delta\) and \(w=f_{L}\left(z,K,1\right)\). With this prices we can solve the household's problem and then arrive at at new level of \(K\). This means that the problem of finding an equilibrium is a fixed point problem on a single value, the value of aggregate capital. Incidentally, a good initial condition is the steady state value of capital in the corresponding complete-markets/representative-agent economy.