2. A utility-maximizing consumer chooses between commodity $x$ and $y$ with the utility function U(x, y) = [x]x + xy while receiving a total income of 19. Let p_x denote the price of $x$, and suppose that the price of $y$ is 1. Answer the following questions. (14%) (a) (6%) Find the optimal solutions for x and y when p_x = 0.4 and 0.2. Now suppose that p_x decreases from 0.4 to 0.2.

3. Pricing Strategy of a Monopoly (26%) Planet Gym is the only gym in town. Providing a session of workout to a customer is at a marginal cost of $c$. This marginal cost is perfect information. The gym is large enough so that even in peak hours there is no externality between the customers.¹ Planet Gym has two forms of feasible pricing strategies. It can set a uniform price per session for all customers with free entrance, or it can set a two-part tariff with a membership fee and a price per session conditional on paying the membership fee. The market demand of gym session at price $p$ per session is $q = \Delta - \delta \cdot p$. The market demand is perfect information. To guarantee existence of an equilibrium, assume that $\Delta > \delta \cdot c$. 3-1. Uniform Price per Session (10%) Suppose in this part of the question that the Planet Gym takes a simple pricing strategy: to set a uniform price per session. Given this form of pricing strategy, it maximizes its profit. Please find Planet Gym's supply of gym sessions. Please accompany your answer with a consistent figure (graph).

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