UCSB ECON 100c – The used car supply in Metropolis

UCSB ECON 100c – The used car supply in Metropolis

Subject: Economics    / General Economics
Question
Economics 100C
Problem Set #2
Winter 2017 1. The used car supply in Metropolis consists of 10,000 cars. The value of these cars
ranges from $5,000 to $15,000 with exactly one car being worth each (integer)
dollar amount between these two figures. Used car owners are always willing to
sell their cars for what they are worth. Demanders of used cars in Metropolis
have no way of telling the value of a particular car. Their demand depends on
both the average value of cars in the market (P¯ ) and on the market price of the
cars themselves (P ) according to the demand function:
Q = 1.5P¯ P. a) If demanders base their estimate of P¯ on the entire used car market, what
will its value be and what will be the competitive equilibrium price of used
cars?
b) In the equilibrium described in part a), what will be the average value of
used cars actually traded in the market?
c) If demanders revise their estimate of P¯ on the basis of the average value of
cars actually traded, what will be the new competitive equilibrium price of
used cars? What is the average value of cars traded now?
2. Consider a “job action” involving a union and an employer as represented by
Figure 1. The union is either “tough” or “weak” which is privately known to
the union only. The prior belief of the employer is that the union is tough with
probability 0 < p < 1.
a) Provide an interpretation for the payo?s of this game.
b) List all the strategies for each player.
c) Suppose 0 < p < 2/3. Verify that the following strategies and beliefs consist
of a pooling equilibrium: s?u (Tough) = s?u (Weak) = NO Strike, s?e (Strike) = s?e (No Strike) = Hold Out,
1 1
µ? (Tough|No Strke) = p, µ? (Tough|Strike) = ,
2
where s?e and s?u denote the strategies for the employer and union, respectively.
d) Verify that the following strategies and beliefs consist of a separating equilibrium:
s?u (Tough) = Strike, s?u (Weak) = No Strike,
s?e (Strike) = Back Down, s?e (No Strike) = Hold Out,
µ? (Tough|Strke) = 1, µ? (Tough|No Strike) = 0.
3. A principal hires an agent to work on a project. The agent chooses e?ort e 2
{el , eh } with cost of e?ort c(e) = 0 if e = el and c(e) = 1 if e = eh . The project
succeeds with probability
p(eh ) = 1
and p(el ) = 0.
2 Denote by x0 the result in monetary terms if the project fails and x1 if the project
succeeds. The principal pays wage w0 if the project fails and w1 if the project
succeeds. The agent has utility function
u(w) c(e), where u0 > 0 and u00 < 0. The agent has reservation utility U = 0. The principal
has utility function B(m) = m.
a) Write down the maximization problem for the efficient contract when information is symmetric.
b) What wages will the principal o?er in the efficient contract in part a)?
c) Now suppose that the principal cannot observe the agent’s e?ort. Write down
the maximization problem for the efficient contract in this case.
d) What wages will the principal o?er in the efficient contract in part c)? 2 (6, 4) Back Down Back Down (5, 3) Union
(4, 3) No Strike
Hold Out (4, 6) Back Down Hold Out (3, 2) [ p ] Tough Employer
(6, 4) Strike 1 0 [1 p ] Employer Weak
No Strike
Strike
Union Hold Out Back Down Hold Out Figure 1 3 (3, 3) (1, 5)