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Let u: ${\bf R}_+^n \to {\bf R}$ be a utility function of $n$ goods which you buy in quantities $x_1, \dots, x_n$ to the prices $p_1, \dots, p_n$ under the budget $K$:

  1. maximize $u(x_1, \dots, x_n)$ subject to the constraint $\sum_{i=1}^n p_i k_i = K$.
  2. Let $L$ be the Lagrangian of this problem. Show that we have a solution to the optimization whenever $\nabla L = {\bf 0}$ (use concavity of the utility function).
  3. For a given $K$, let $V(K)$ be this maximum and let $\lambda (K)$ be the Lagrange multiplier. Show that $V'(K)=\lambda(K)$.

I don't really get it. I mean you normally solve this by putting $\nabla L = {\bf 0}$ 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 $V(K)$.

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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!

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