Aiyagari Model

EC417_Macroeconomics

Aiyagari Model with Production

Consider a representative firm with Constant Return to Scale production technology:

$$Y=K^{\alpha }{L}^{1-\alpha }$$

Firm rents capital from households at rate $R$ and hires efficiency units of labor at wage rate $w$:

$$R=r+\delta =\alpha {\left(\frac{L}{K}\right)}^{1-\alpha }$$

and for labour demand, it would be:

$$w=\frac{\partial Y}{\partial L}=(1-\alpha ){\left(\frac{K}{L}\right)}^{\alpha }$$

Thus we could get:

$$w(r)=(1-\alpha )(\frac{\alpha }{r+\delta }{)}^{\frac{\alpha }{1-\alpha }}$$

For the households problem, he would face:

$$c+\underset{\text{Saving}}{\underbrace{a^{\prime }}}=\underset{\text{wealth I get if I save}}{\underbrace{(1+r)a}}+\underset{\text{Labour Income I get}}{\underbrace{w{y}_j}}$$

Market Clearing Conditions

Thus there are two markets to be cleared: Labour market and capital market.

• Labour Market:

$$L=\sum _j{\int }_a{y}_{j}g(a,y_j;r)da$$

We could also denote as: $L={\sum }_j{y}_j{\pi }_j$ where ${\pi }_j={\int }_{a}g(a,y_j;r)da$ is the stationary distribution of income process.

• Capital Market:

Same as the normal Aiyagari Model, we have:

$$A(r)=K(r)=L{\frac{\alpha }{r+\delta }}^{\frac{1}{1-\alpha }}$$

Recursive Competitive Equilibrium (RCE)

Given $r$ and $w$, the function $V(a,y)$ solves the household problem, we use Bellman Equation to solve:

$$V(a,y_j)=\underset{c}{\max }u(c)+\beta \sum _{j^{\prime }}{p}_{j{j}^{\prime }}V(a^{\prime },y_{j^{\prime }})\text{s.t.}{a}^{\prime }=(1+r)a+w{y}_j-c$$

Given $r$ and $w$, the firm side is able to solve their FOC.