(c) \((4\%)\) Suppose that one optimal solution \(x^{}\) is nonzero and independent of \(p_y\). Find Mary's price elasticity and income elasticity of demand for \(x\) under \(x^{}\).

2. A utility-maximizing consumer chooses between commodity $x$ and $y$ with the utility function $U(x, y) = \sqrt[x]{x + xy}$ 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.

(b) \((4\%)\) What is the substitution effect of this reduction in price?

(c) \((4\%)\) What is the compensating variation of this reduction in price?

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).

3-2. Two-Part Tariff (10%) Suppose in this part of the question that the Planet Gym adopts a two-part tariff: to set a membership fee and a price per session for customers who have paid the membership fee. The membership fee is required for entrance. Given this form of pricing strategy, the Planet Gym maximizes its profit. Please find the optimal membership fee, the optimal price per session, and the equilibrium gym sessions traded. Please accompany the above answers with a consistent figure (graph).

3-3. The Optimal Pricing Strategy (6%) From what you have found in 3-1 and 3-2, what is the optimal pricing strategy for the Planet Gym? Is the optimal pricing strategy efficient from the social perspective? Is the optimal pricing strategy also the best for the customers? Briefly explain. ¹ This assumption might not be realistic, but it allows us to focus on what we are really interested to analyze. 4. A cost-minimizing firm has the production function $Q(L, K) = KL + K + L$. Suppose that the price of labor services ($L$) is 1, and let $r$ denote the price of capital services ($K$). If the firm aims to produce 20 units of output, for what values of $r$ would the firm use (9%) (a) \((3\%)\) only labor?

(b) \((3\%)\) only capital?

(c) \((3\%)\) both labor and capital?

5. A price-taking firm in a perfectly competitive market uses three factors of production: labor ($L$), capital ($K$), and material ($K$). The firm's faces the production function $Q(L, K, M) = L^{\alpha} K^{\beta} M^{\gamma}$ with $\alpha, \beta, \gamma > 0$. (15%) (a) \((3\%)\) For what condition (in terms of \(\alpha, \beta, \gamma\)) would this production function exhibit constant returns to scale?

(b) \((6\%)\) Show that the total value created by the firm under constant returns to scale is the exact sum of the values created by each factor's marginal productivity.

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