111 年 國立臺灣大學經濟學系碩士班《個體經濟學》

Penny consumes three goods, beverages ($x_1$), dresses ($x_2$) and shoes ($x_3$). Her preference can be represented by the following utility function:
$U(x_1, x_2, x_3) = 4x_1 + \min[x_2, 2x_3]$.
The prices of these goods are $P_1, P_2$, and $P_3$, respectively. Penny has an income $Y$.
Which of the following statements regarding her optimal choice is (are) TRUE?
(A) She will spend all her income in buying beverages when $P_1 = 3$ and $P_2 = 1$.
(B) She will buy all the three goods if she has a sufficiently high income $Y$.
(C) There can be many optimal solutions.
(D) Other things being the same, if beverages become cheaper, she will buy more beverages.
(E) Other things being the same, if dresses become more expensive, she will buy fewer shoes.

Sheldon (player 1) and Raj (player 2) are deciding whether or not to enter a contest. The prize for the winner is $w$ and 0 for the loser. Player $i$'s utility function is:
$u_i = p_i w - k e_i - \lambda$.
$p_i$ is the probability that player $i$ wins the contest, which depends on the effort levels, $e_1, e_2 \in [0,\infty)$, made by each player. Specifically, let $p_i = \frac{e_i}{e_1+e_2}$ if both players enter the contest. It means that it is more likely that player $i$ wins if he makes more efforts or his opponent makes less efforts. On the other hand, if there is only one player entering the contest, he wins the prize with probability 1, i.e., $p_i = 1$, and the other player who stays out of the contest obtains 0. If neither player enters the contest, they each obtain 0. $k > 0$ is the marginal effort cost, and $\lambda$ is the fee that each player needs to pay to enter the contest. We assume that $w > \lambda > 0$. The game contains two stages: in the first stage, players simultaneously decide whether or not to enter the contest. If one does, he needs to pay the fee $\lambda$. In second stage, players simultaneously choose the effort levels, $e_1$ and $e_2$. The equilibrium effort levels are denoted by $(e_1^*, e_2^*)$.
Which of the following statements regarding the pure-strategy subgame perfect Nash equilibrium in this game is (are) TRUE?
(A) Both players always make efforts, i.e., $e_1^* > 0$ and $e_2^* > 0$.
(B) There is only one equilibrium.
(C) Each player wins the contest with equal probability, i.e., $p_1 = p_2 = 1/2$.
(D) When $k$ is smaller, other things being the same, the equilibrium payoff for the winner is larger.
(E) Even when $\lambda$ is very high, the case where neither one enters the contest never happens.

The Cheesecake factory is a monopoly in a small town which sells cakes in two periods. In the first period, the market demand is
$P_1 = 120-Q_1$,
and in the second period, the market demand is
$P_2 = 120 + \phi - Q_2$,
where $\phi$ is a shock on the demand side. The firm's total cost function is
$TC(Q) = Q^2$,
where $Q = Q_1 + Q_2$. In the beginning of the first period, the firm has to choose $Q_1$ and $Q_2$ to
maximize the total profit in two periods:
$\pi = P_1Q_1 + P_2Q_2 - (Q_1+Q_2)^2$.
Let the optimal prices and outputs in the two periods be $(P_1, Q_1)$ and $(P_2, Q_2)$, respectively.
Consider the case where they are all positive.
Which of the following statements regarding the monopolist's optimal decisions is (are) TRUE?
(A) $Q_1$ is not affected by the shock because it happens in the second period.
(B) Both $P_1$ and $P_2$ are greater than 100.
(C) If the shock is negative, i.e., $\phi < 0$, then the price is increasing over time, i.e., $P_2 > P_1$.
(D) If the shock is positive, i.e., $\phi > 0$, then $Q_1$ is greater than 20.
(E) The profit is increasing in $\phi$ if it is a positive shock.

There are two comic book stores operating in a duopolistic market, Stuart's (Store 1) and Jesse's (Store 2). They sell heterogeneous products and engage in price competition. Stuart's store is more popular than Jesse's in that it has a larger demand. Namely, the demand function for each store is the following:
$Q_1 = 60 - P_1 + \delta P_2$.
$Q_2 = 40 - P_2 + \delta P_1$,
where $0 < \delta < 1$, which captures the level of "differentiation" of these two stores. Assume that the firms have the same marginal cost, $MC = 20$. They simultaneously choose their prices. The equilibrium prices and outputs are denoted by $(P_1^*, Q_1^*)$ and $(P_2^*, Q_2^*)$.
Which of the following statements regarding the Nash equilibrium outcomes is (are) TRUE?
(A) Since this is a Bertrand-style competition, the equilibrium prices are equal to the marginal cost.
(B) Because Stuart faces a larger market demand, the equilibrium quantity sold by Stuart is always larger than that sold by Jesse.
(C) The reaction functions are positively sloping in that when a store increases its price, it is a best response for its opponent to increase its price as well in order to gain more profit.
(D) When their products are more differentiated, i.e., $\delta$ is higher, then other things being the same, each firm can charge a higher price.
(E) It is profitable for them to collude, and then they will do so in equilibrium.

Howard is trying to find the Nash equilibrium in the following normal-form game. However, the number in the $\boxed{\phantom{x}}$ is missing. The only condition that he knows is that "M" is a dominated strategy.
\begin{tabular}{|c|c|c|c|}
\hline
\multicolumn{2}{|c|}{} & \multicolumn{2}{|c|}{2} \
\cline{3-4}
\multicolumn{2}{|c|}{} & L & R \
\hline
\multirow{2}{*}{1} & A & 2,6 & 1,3 \
\cline{2-4}
\cline{2-4}
\multicolumn{2}{|c|}{} & B & $\boxed{\phantom{x}}$ ,5 & 1,2 \
\hline
\multicolumn{2}{|c|}{} & C & 5,1 & 4,4 \
\hline
\end{tabular}
Considering only the numbers that satisfy the condition, which of the following statements regarding the Nash equilibrium in this game is (are) TRUE?
(A) "B" is never played in the equilibrium.
(B) It is required that the number in the $\boxed{\phantom{x}}$ be smaller than 2.
(C) There is a unique Nash equilibrium.
(D) There is a mixed-strategy Nash equilibrium for some numbers in the $\boxed{\phantom{x}}$
(E) (A, R) can never be played in a Nash equilibrium.

Penny consumes three goods, beverages ($x_1$), dresses ($x_2$) and shoes ($x_3$). Her preference can be represented by the following utility function:
$U(x_1, x_2, x_3) = 4x_1 + \min[x_2, 2x_3]$.
The prices of these goods are $P_1, P_2$, and $P_3$, respectively. Penny has an income $Y$.
Which of the following statements regarding her optimal choice is (are) TRUE?
(A) She will spend all her income in buying beverages when $P_1 = 3$ and $P_2 = 1$.
(B) She will buy all the three goods if she has a sufficiently high income $Y$.
(C) There can be many optimal solutions.
(D) Other things being the same, if beverages become cheaper, she will buy more beverages.
(E) Other things being the same, if dresses become more expensive, she will buy fewer shoes.

Sheldon (player 1) and Raj (player 2) are deciding whether or not to enter a contest. The prize for the winner is $w$ and 0 for the loser. Player $i$'s utility function is:
$u_i = p_i w - k e_i - \lambda$.
$p_i$ is the probability that player $i$ wins the contest, which depends on the effort levels, $e_1, e_2 \in [0,\infty)$, made by each player. Specifically, let $p_i = \frac{e_i}{e_1+e_2}$ if both players enter the contest. It means that it is more likely that player $i$ wins if he makes more efforts or his opponent makes less efforts. On the other hand, if there is only one player entering the contest, he wins the prize with probability 1, i.e., $p_i = 1$, and the other player who stays out of the contest obtains 0. If neither player enters the contest, they each obtain 0. $k > 0$ is the marginal effort cost, and $\lambda$ is the fee that each player needs to pay to enter the contest. We assume that $w > \lambda > 0$. The game contains two stages: in the first stage, players simultaneously decide whether or not to enter the contest. If one does, he needs to pay the fee $\lambda$. In second stage, players simultaneously choose the effort levels, $e_1$ and $e_2$. The equilibrium effort levels are denoted by $(e_1^*, e_2^*)$.
Which of the following statements regarding the pure-strategy subgame perfect Nash equilibrium in this game is (are) TRUE?
(A) Both players always make efforts, i.e., $e_1^* > 0$ and $e_2^* > 0$.
(B) There is only one equilibrium.
(C) Each player wins the contest with equal probability, i.e., $p_1 = p_2 = 1/2$.
(D) When $k$ is smaller, other things being the same, the equilibrium payoff for the winner is larger.
(E) Even when $\lambda$ is very high, the case where neither one enters the contest never happens.

The Cheesecake factory is a monopoly in a small town which sells cakes in two periods. In the first period, the market demand is
$P_1 = 120-Q_1$,
and in the second period, the market demand is
$P_2 = 120 + \phi - Q_2$,
where $\phi$ is a shock on the demand side. The firm's total cost function is
$TC(Q) = Q^2$,
where $Q = Q_1 + Q_2$. In the beginning of the first period, the firm has to choose $Q_1$ and $Q_2$ to
maximize the total profit in two periods:
$\pi = P_1Q_1 + P_2Q_2 - (Q_1+Q_2)^2$.
Let the optimal prices and outputs in the two periods be $(P_1, Q_1)$ and $(P_2, Q_2)$, respectively.
Consider the case where they are all positive.
Which of the following statements regarding the monopolist's optimal decisions is (are) TRUE?
(A) $Q_1$ is not affected by the shock because it happens in the second period.
(B) Both $P_1$ and $P_2$ are greater than 100.
(C) If the shock is negative, i.e., $\phi < 0$, then the price is increasing over time, i.e., $P_2 > P_1$.
(D) If the shock is positive, i.e., $\phi > 0$, then $Q_1$ is greater than 20.
(E) The profit is increasing in $\phi$ if it is a positive shock.

There are two comic book stores operating in a duopolistic market, Stuart's (Store 1) and Jesse's (Store 2). They sell heterogeneous products and engage in price competition. Stuart's store is more popular than Jesse's in that it has a larger demand. Namely, the demand function for each store is the following:
$Q_1 = 60 - P_1 + \delta P_2$.
$Q_2 = 40 - P_2 + \delta P_1$,
where $0 < \delta < 1$, which captures the level of "differentiation" of these two stores. Assume that the firms have the same marginal cost, $MC = 20$. They simultaneously choose their prices. The equilibrium prices and outputs are denoted by $(P_1^*, Q_1^*)$ and $(P_2^*, Q_2^*)$.
Which of the following statements regarding the Nash equilibrium outcomes is (are) TRUE?
(A) Since this is a Bertrand-style competition, the equilibrium prices are equal to the marginal cost.
(B) Because Stuart faces a larger market demand, the equilibrium quantity sold by Stuart is always larger than that sold by Jesse.
(C) The reaction functions are positively sloping in that when a store increases its price, it is a best response for its opponent to increase its price as well in order to gain more profit.
(D) When their products are more differentiated, i.e., $\delta$ is higher, then other things being the same, each firm can charge a higher price.
(E) It is profitable for them to collude, and then they will do so in equilibrium.

Howard is trying to find the Nash equilibrium in the following normal-form game. However, the number in the $\boxed{\phantom{x}}$ is missing. The only condition that he knows is that "M" is a dominated strategy.
\begin{tabular}{|c|c|c|c|}
\hline
\multicolumn{2}{|c|}{} & \multicolumn{2}{|c|}{2} \
\cline{3-4}
\multicolumn{2}{|c|}{} & L & R \
\hline
\multirow{2}{*}{1} & A & 2,6 & 1,3 \
\cline{2-4}
\cline{2-4}
\multicolumn{2}{|c|}{} & B & $\boxed{\phantom{x}}$ ,5 & 1,2 \
\hline
\multicolumn{2}{|c|}{} & C & 5,1 & 4,4 \
\hline
\end{tabular}
Considering only the numbers that satisfy the condition, which of the following statements regarding the Nash equilibrium in this game is (are) TRUE?
(A) "B" is never played in the equilibrium.
(B) It is required that the number in the $\boxed{\phantom{x}}$ be smaller than 2.
(C) There is a unique Nash equilibrium.
(D) There is a mixed-strategy Nash equilibrium for some numbers in the $\boxed{\phantom{x}}$
(E) (A, R) can never be played in a Nash equilibrium.

Suppose that the market for coffee is perfectly competitive around Pasadena. There are two types of coffee shops, domestic and foreign. Different types of coffee shops have different cost structures, but the ones of the same type are identical. There are currently 30 coffee shops of each type, and an individual firm of each type has the following total cost function in the short run:

Domestic shops: $STC_d(q)=1+4q+3q^2$.

Foreign shops: $STC_f(q)=2+2q+q^2$.

On the other hand, there are also two types of consumers, residents and tourists. An individual consumer of each type has the following demand function:

Residents: $q_w=1-0.1P$.

Tourists: $q_t=0.8-0.2P$.

That is, residents have a higher demand for coffee than tourists do. There are potentially 200 consumers of each type.

Which of the following statements regarding the short-run market equilibrium is (are) TRUE?

(A) Both domestic and foreign coffee shops keep staying in the market.

(B) The total output in equilibrium is $Q^*=105/2$.

(C) The equilibrium price is $P^*=41/8$.

(D) Both domestic and foreign coffee shops earn positive profit.

(E) Both residents and tourists buy coffee.

Amy (person A) and Bernadette (person B) work in a lab, both of whom need two kinds of equipment in doing experiments, funnels ($f$) and test tubes ($t$). They cannot produce the equipment by themselves and can only exchange what they have in hand. Due to the nature of their own experiments, they have different preferences over the equipment, which can be represented by the following utility functions:

Amy: $u_A(f_A,t_A)=\min[f_A,t_A]$.

Bernadette: $u_B(f_B,t_B)=100-(f_B-5)^2-(t_B-5)^2$.

Initially, the endowment that Amy has is $(e_f^A,e_t^A)=(4,8)$, and Bernadette has $(e_f^B,e_t^B)=(6,2)$.