Eco 601E: Advanced Microeconomics II (Fall, 2nd, 2013)
Midterm Exam: January 7 1. True or False (9 points)
Answer whether each of the following statements is true (T) or false (F). You do NOT need to explain the reason.
(a) If an agent is risk neutral, her certainty equivalence of a lottery must be equal to its expected value.
(b) The first-degree price discrimination is called the “perfect price discrimination” because the price charged by the monopoly firm is the same as the price determined in the perfectly competitive market.
(c) Any finite game has at least one Nash equilibrium in pure strategies. 2. Expected Utility (16 points)
Suppose that an individual can either exert effort or not. Her initial wealth is $100 and the cost of exerting effort is c. Her probability of facing a loss $75 (that is, her wealth becomes $25) is 1
3 if she exerts effort and 2
3 if she does not. Her wealth will not change (that is, it remains $100) with the rest of probability in each scenario. Let u(x) be her vNM utility function.
(a) Express her expected utilities in each scenario, i.e., exerting effort or not, by using u(x). You can assume that her expected utility is additively separable between effort cost and (probabilistic) monetary outcome: E[u(x)] − c. (b) Assume u(x) =√x. For what values of c is she willing to exert effort? 3. Monopoly (16 points)
Suppose a monopoly firm operates in two different markets, A and B. The inverse demand function for each market is given as follows.
p(qA) = 200 − qA
p(qB) = 120 − qB
1
The cost function is given by
C(qA, qB) = 1
2(qA+ qB)
2
(a) Derive the profit function of this monopoly firm, π(qA, qB).
(b) What are the optimal (i.e., profit maximizing) quantities qA∗ and qB∗? 4. Nash equilibrium (15 points)
Find all pure strategy Nash equilibria in each of the following games.
(a)
1 2 L R
U 1, 1 0, 1
D 1 , 0 0, 0
(b)
1 2 L M R
U 200, 0 50, 1 2, 2 D 80, 100 100, 50 1, 1
(c)
1 2 L M R
U 0, 0 2, 1 6, 2 M 1, 2 4, 4 5, 3 D 2, 6 3, 5 3, 4 5. Mixed Strategy (24 points)
Three firms (1, 2 and 3) put three items on the market and can advertise these products either on morning (= M ) or evening TV (= E). A firm advertises exactly once per day. If more than one firm advertises at the same time, their profits become 0. If exactly one firm advertises in the morning, its profit is 1; if exactly one firm advertises in the evening, its profit is 2. Firms must make their daily advertising decisions simultaneously.
(a) Derive all pure strategy Nash equilibria.
(b) Show that the following type of asymmetric Nash equilibria does NOT exist: One firm chooses pure strategy M , and other two firms use mixed strategies. (c) Derive a symmetric mixed strategy Nash equilibria. You may assume that
each firm chooses M with probability p and E with probability 1 − p, then calculate an equilibrium probability p.
2
