Let u: ${\mathbf R^n_+ \rightarrow \mathbf R}$ be a utility function of n goods which you buy in quantities $x_1,…,x_n$ to the prices $p_1,…,p_n$ under the budget K. So maximize $u(x_1,…,x_n)$ subject to the constraint $\sum_{i=1}^n p_i k_i=K$. Let L be the Lagrangian of this problem. Show that we have a solution to the optimization whenever $\nabla L = \mathbf 0$ (Use concavity of the utility function). For a given K let $V(K)$ be this maximum and let $\lambda (K)$ be the langrange multiplier. Show that $V'(K)=\lambda(K)$
I don't really get it. I mean you normally solve this by putting $\nabla L = \mathbf 0$ and solve the equations. Is the point of the exercise to prove the lagrange multiplier theorem? And the rest I can't figure out how to write $V(K)$. Help out a lost economist:)
So a few hints:
Firstly, let $x^*_i(K)$ be the optimal quantities of each good to consume at buget level $K$. Then $V(K)$ is simply:
$$V(K)= \max_x \; \mathcal{L}(x,K) = u(x_1^*(K), ... , x_n^*(K))$$
Second, keep in the back of your mind the Envelope Theorem.
Finally, bear in mind the intuition behind indirect utility functions.
If you're truly stuck, a good worked version of this problem and explanation of the full intuition can be found here, in particular section 1.3.
Hope this helps!