(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 $r = \max\{r1, r2\} = 0.7$.

115年 - 115 國立臺北大學_碩士班一般入學考試試題_經濟學系：個體經濟學#146554

114年 - 114 國立臺北大學_碩士班一般入學考試試題_經濟學系：個體經濟學#146546

112年 - 112 國立臺北大學_碩士班一般入學考試試題_經濟學系：個體經濟學#146556

112年 - [無官方正解]112 國立臺北大學_碩士班一般入學考試試題_經濟學系：個體經濟學#146523

110年 - [非官方正解]110 國立臺灣大學_碩士班招生考試_經濟學研究所:個體經濟學#123412

110年 - 110 國立政治大學_碩士班暨碩士在職專班招生考試_ 經濟學系:個體經濟學#105665

110年 - 110 國立中央大學_碩士班招生考試_經濟學系/不分組(一般生):個體經濟學#105315

110年 - 110 國立清華大學碩士班考試入學試題_經濟學系:個體經濟學#105241

110年 - 110 國立中山大學_碩士暨碩士專班招生考試_經濟所:個體經濟學#104433

109年 - 109 國立臺北大學_碩士班一般入學考試試題_經濟學系：個體經濟學#146548