# Optimization: Dynamic Programming vs Kuhn-Tucker

Considering the standard utility maximization of representative household which lives forever, one may use dynamic programming and Kuhn-Tucker in case of discrete time. For instance, one would like to solve,

max $\Sigma^∞_tU(C(t),N(t))$ subject to $P(t)C(t)+Q(t)B(t)<B(t−1)+W(t)N(t)+D(t)$

where $C(t)$ is consumption, $B$ is the bond, $Q$ is the bond price, $D(t)$ is a dividend, and $N(t)$ is the amount of labor.

Does the interpretation differ when one use Dynamic programming or Kuhn-Tucker? Will it be something like this: In DP all the paths are optimized along t, but in Kuhn-Tucker only the path at time t is optimized.

If so, how you can make the above statement?

## 1 Answer

I would say that the main difference stems from the solution method, which results in your statement about all paths versus only the path at time t being true.

Dynamic programming (at least when done numerically) consists of backward induction. One tries to identify the optimal action for all possible values of the state variable in the final period, and then reasons backwards following the state equation. In this way one gets a solution not just for the path we're currently on, but also all other paths.

Similarly, if one uses the guess-and-verify method to solve the value function of the Bellman equation, the value function one guesses defines the optimal decision for all possible values of the state variable. Thus one gets a solution for all possible paths, including the current one.

Kuhn-Tucker basically works the other way around. One formulates the necessary and sufficient conditions, and solves the resulting difference equation(s) using the initial conditions as starting points.

• Thank you very much for your comment. So in most of the cases both methods could be used to find optimal behavior of the household, but the interpretation might be different. Is this right? – Roy_Oishi Nov 12 '17 at 13:36
• Indeed both methods can be used to find the optimal solution. I am not sure what you mean with "the interpretation might be different" The interpretation of what? The solution would be the same in both cases, the main difference is that DP would provide you also with host of "other solutions" in case you are not currently on the optimal path. – Maarten Punt Nov 13 '17 at 8:55
• I think one can say this consumer has no incentives to follow another path of the consumption in the future when DP is applied as all the path is optimized (sort of commitment). On the other hand, if the optimization of the consumption path is solved by Kuhn-Tucker, the household might get out of the consumption path in the future because only path at time t and time t+1 is optimized. – Roy_Oishi Nov 13 '17 at 9:34
• No, the consumer wouldn't have an incentive to leave the optimal path anyway, otherwise it wouldn't be optimal. It is more like: if the consumer would make a "mistake" or if we somehow take him/her off the optimal path, DP provides the way back. KT doesn't just optimize the path at t and t+1, it optimizes the full path. The reason we don't usually solve it for all t, is that the solution is usually a general one relating t and t+1 for all future t (provided the consumer stays on the optimal path) – Maarten Punt Nov 13 '17 at 15:26