Do policy functions exist for Finite Horizon Dynamic programming problems? I've been looking at the cake eating problem over a finite horizion and have been trying to figure out if we can derive a policy function for such a problem. My work is written below.





sequence form of such a problem is:

$$\max _{c_t,x_{t+1}}U(c_t)=\sum_{i=0}^T\beta^tu(c_t),\ \ \ \ \ \ 1&gt;\beta&gt;0$$
$$s.t. x_{t+1}=x_{t}-c_t$$

supposing that the instantaneous utility function for such preferences are $u(c_t)=\ln(c_t)$ it follows that the function equation for such a problem is:

$$v(x_{t+1})=\max_{x_{t+1}}\{\ln(x_t-x_{t+1})+\beta v(x_{t+1})\}$$

In this case im supposing $T=2$ (we have three periods starting from period zero) and suppose $x_0&gt;0$. Solving the problem from backwards induction we know.
$$v(x_2)=\ln(x_2)$$
it follows from this result that for the previous periods value function we have:
$$v(x_1)=\max_{x_2}\{\ln(x_1-x_2)+\beta v(x_2)\}$$
$$v(x_1)=\max_{x_2}\{\ln(x_1-x_2)+\beta \ln(x_2)\}$$

after taking the first order condition for this result and solving for $x_2$ we get:
$$x^* _2=\frac{\beta x_1}{1+\beta}$$

we know now our value function is properly defined as:

$$v(x_1)=\ln\left(x_1-\frac{\beta x_1}{1+\beta}\right)+\beta \ln\left(\frac{\beta x_1}{1+\beta} \right)$$

Simplifying:
$$v(x_1)=\ln\left(\frac{ x_1}{1+\beta}\right)+\beta \ln\left(\frac{\beta x_1}{1+\beta}\right) $$

For our $v(x_0)$ we follow the same procedure before

$$v(x_0)=\max_{x_1}\{\ln(x_0-x_1)+\beta v(x_1)\}$$
$$v(x_0)=\max_{x_1} \left \{\ln(x_0-x_1)+\beta \left[\ln\left(\frac{ x_1}{1+\beta}\right)+\beta \ln\left(\frac{\beta x_1}{1+\beta}\right) \right]\right\}$$

upon solving such an optimization problem we find:

$$x_1^*=\frac{(\beta+\beta^2)x_0}{1+\beta+\beta^2}$$

From this work that I've done above we can see that the policy functions change each period.





Given the solving above do time invariant policy functions exist?
No, they cannot by definition.
To derive a time-invariant policy function, you need to have an infinite horizon problem. This is because the structure of the solution remains the same, no matter when you look at it. Intuitively, this is because any sequence of periods from any time period to infinity looks the same. The technical conditions for this can be found in a number of textbooks (Adda and Cooper, Sargent, as well as Stokey/Lucas).

In your example, the number of periods left determine the solution structure.