(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.

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

6. Human Right Protection (24%) Human right protection in Taiwan is enforced by the judicial system (e.g. the Constitution and the Constitutional Court). This is due to the non-exclusive and non-rival properties of the public goods. Let's take a game theoretic approach to analyze the provision of human right protection There are $n \ge 2$ citizens in the society. Each of them, indexed by $i = 1, 2, \dots, n$, can freely decide to provide human right protection of level $ri$ as any real number within zero and one (that is, $0 \le ri \le 1$ from each citizen $i$). Human right protection of level $ri$ is at a linear cost of $c(ri) = c \cdot ri$ to the provider, with $c > 0$. If multiple people provide different levels of human right protection, the actual level everyone in the society enjoys is the maximum among them. Denote this actual level of human right protection in the society as $r$, and $r = \max\{ri\}$ (that is, $r$ equals to the highest level of human right protection among all providers).$^2$ Human right protection is a public good in the sense that if level $r$ is provided, every citizen enjoys a positive utility from it. Among the citizens, a proportion $m > 0$ of them highly value human right protection, and they earn utility $uh \cdot r$ from human right protection of level $r$. Among the citizens, a proportion $1 - m > 0$ of them slightly value human right protection, and they earn utility $ul \cdot r$ from human right protection of level $r$. Suppose that $uh > c > ul > 0$ and $n \cdot u_l > c$. The utility of human right and the cost of provision for each citizen are perfect information. There is no other externality. Please find the Nash equilibrium level of human right protection by explaining the following: (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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