# Closed form solution to consumption-saving model in discrete time

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 $$a'(a,y)$$.

• 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. Oct 15 at 18:03