Consider the simplest consumption-savings model of the following form: $$ \max_{\{c_t,a_{t+1}\}_t}\mathbb{E_0}\sum_{t\geq 0} \beta^tu(c_t) \\ a_{t+1} + c_t = (1+r)a_t + y_t \\ y_t \mid y_{t-1} \sim F $$ The Bellman equation associated is thus $$ v(a,y)= \max_{c,a'} u(c) + \beta \mathbb{E}[v(a',y') \mid y] \\ a' + c = (1+r)a + y \\ y' \mid y\sim F $$ I know how to derive the Euler equation and how to use algorythms to numerically find the optimal policies $c(a,y)$ and $a'(a,y)$. I wanted to know what are the known and most used closed form solutions for these policies, that is which utility $u$ and income distribution $F$ generates a tractable form for $c(a,y)$ and $a'(a,y)$.

  • $\begingroup$ Don't quote me on this, but I believe Brock and Mirman (1972) and Hercowitz Sampson (1991) both have closed form solutions to their optimal growth problems. $\endgroup$ Oct 15 at 18:03

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