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?