Utility Max Problem

Question

I have a utility function $U(x,y)=\frac{xy}{x+y}$ and a budget of $200=2x+2y, P_x=P_y=2$.

But for the first 50 units of product 1 sell for 2 dollars but for "$x>50$" the price of product 1 falls to 1 dollar per unit. Assume you can buy as much of product 2 as you like for $2.

I understand to set up the problem as

$\mathcal L=\frac{xy}{x+y}+\lambda(200-2x-2y)$ and take the partial derivatives without the $x>50$ condition.

How would I go about adding the price change condition? Since this has an inequality do I need to use the Kuhn Tucker conditions??

Answer 1

As pointed out in a comment, you can just optimize the two cases separately, i.e. one case with $x \leq 50$ and one case with $x \geq 50$. However, note that your problem is symmetric with respect to $x$ and $y$. Yet $x$ is cheaper if you take at least half of your units to be $x$. So you can reason that the global optimal solution must happen for the case $x \geq 50$ and then just solve that case.

Answer 2

Set up two inequalities: $$ 200 \le 2x + 2y $$ $$ 200 \le x + 2y $$ and do not use any equality constraints.

Solve for the critical points using the complementary slackness conditions, and then plug them into the objective function to determine which is the most preferred point; i.e. $$ \mathcal{L} = \dfrac{xy}{x+y} - \mu_1 (2x+2y-200) - \mu_2 (x+2y-200). $$

If one of the solutions to the FONCs is not feasible because of the discontinuous price, point that out and discard it as a possibility.