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

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.

(a) \( (3\%) \) only labor?

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

(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.

(c) \( (6\%) \) Explain the economic intuition behind the previous result.

(a) Show free-riding incentive: if some citizens provide human right protection, each citizen except the one providing the maximum level of human right protection has incentive to unilaterally deviate to provide zero human right protection, regardless of her value on human right protection.

(b) Show that if no one provides human right protection, one citizen among those who highly value human right has incentive to unilaterally deviate from providing zero human right protection.Please characterize the above Nash equilibrium. Specifically, how many citizens provide human right protection? Who is(are) the provider(s)? In equilibrium, what is the actual level of human right protection in the society?$^2$ For example, if Citizen 1 provides r1 = 0.1 and Citizen 2 provides r2 = 0.7, and no one else provides positive level of human right protection, the actual level of human right protection in the society is $r = \max\{r1, r2\} = 0.7$.

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