115年 - 115 國立臺北大學_日間學士班轉學生招生考試試題_財政學系日間學士班2年級：經濟學#147740

科目：研究所、轉學考(插大)-經濟學 | 年份：115年 | 選擇題數：0 | 申論題數：12

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所屬科目：研究所、轉學考(插大)-經濟學

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申論題 (12)

1. Consider a monopolist facing the following market demand function:

$$Q = 120 - 2P$$

where $Q$ denotes quantity demanded and $P$ denotes price.

The firm's total cost function is given by:

$$TC = 20Q + 100$$

Answer the following questions:

(a) Derive the marginal revenue (MR) function and the marginal cost (MC) function. \((5\%)\)
(b) Determine the profit-maximizing output level \((Q^{*})\) and price \((P^{*})\). Clearly show all intermediate steps. \((10\%)\)
(c) Suppose the government imposes a per-unit tax of \( t = 5 \) on the monopolist's output. Determine the new profit-maximizing price and quantity, and compare the new price and quantity with the pre-tax equilibrium. (10%)
(d) Suppose now that a second firm enters the market and the market structure changes from monopoly to Cournot duopoly. Assume that no tax is imposed in this market.

Firm 1 is the incumbent firm (the former monopolist) with cost function:

$$TC_1 = 20q_1 + 100$$

Firm 2 is a new entrant with a different cost structure:

$$TC_2 = 30q_2 + 50$$

The market demand remains:

$$Q = 120 - 2P, \quad Q = q_1 + q_2$$

where $q_1$ and $q_2$ denote the output of firm 1 and firm 2 respectively.

Answer the following questions:

(d.1) Derive the profit function for each firm and obtain the reaction function of each firm. \((5\%)\)
(d.2) Solve for the Cournot equilibrium outputs \( q_{1}^{*} \) and \( q_{2}^{*} \). (5%)
(d.3) Determine the total market output \( Q^{*} \) and the equilibrium market price \( P^{*} \). (5%)
(d.4) Compare the Cournot equilibrium with the monopoly outcome in terms of output, price, and profit. Explain the economic intuition for why moving from monopoly to duopoly affects these outcomes. (10%)
2. Consider a worker whose utility depends on consumption and leisure. The worker's income comes from labor income and non-labor income. The worker's utility maximization problem is given by:

$$\underset{C,N}{Max} \quad U = \ln C + \beta \ln(H - N)$$

$$s.t. \quad C = wN + v$$

where $C$ denotes consumption, $N$ denotes hours of labor supplied, $H$ is the total time endowment, $w$ is the real wage, $v > 0$ is non-labor income, and $\beta > 0$ measures the relative importance of leisure in utility. The worker takes $w$, $v$, $H$, and $\beta$ as given.


Complete the following tasks:

(a) Explain the income effect and substitution effect of an increase in the real wage on labor supply. (Please provide detailed economic intuition; no mathematical derivation is required.) (10%)
(b) Solve for the worker's optimal consumption \( C^* \) and optimal labor supply \( N^* \). (5%)
(c) Analyze how the optimal labor supply \( N^* \) responds to: (i) an increase in the real wage \( w \); and (ii) an increase in the preference parameter \( \beta \). Clearly explain both the direction of change and the underlying economic intuition. (10%).
3. Consider a closed economy described by the Solow growth framework in continuous time. Output is produced using a Cobb-Douglas technology:

$$Y = K^{\alpha} N^{1-\alpha}, \quad 0 < \alpha < 1$$

where $Y$ represents total output, $K$ denotes the capital stock, and $N$ is the labor force (number of workers). The parameter $\alpha$ measures capital's share in output. The population grows at a constant rate $n > 0$, i.e., $\dot{N}/N = n$.

Capital accumulates according to:

$$\dot{K} = I - \delta K$$

where $I$ is total investment and $\delta \in (0,1)$ is the constant rate of capital depreciation. National saving is defined as a constant fraction of output:

$$S = sY$$

where $s \in (0,1)$ is the savings rate. We assume the goods market clears such that $S = I$.

Please answer the following questions:

(a) Derive the steady-state level of capital per worker \( k \) and the steady-state output per worker \( y \), where \( k = K / N \) and \( y = Y / N \). (15%)
(b) Determine the long-run growth rates of output per worker \( y \) and total output \( Y \). Based on your results, explain why the Solow model is classified as an exogenous growth model. (10%)