Download Problem Set 5 Due 4/25

Problem Set 5 Due 4/25
Q1. A retailer faces downstream demand of P=240-2Q and has one supplier who has 0
marginal cost. The retailer’s only cost is what it pays the supplier. What will happen in
this market in terms of price, quantity and profits? What would happen if there were two
retailers who competed according to the Cournot model? What would happen if one of
the supplier became vertical integrated and only sold its output through its own retailer?
Initially we have one retailer and one supplier. The retailer has MC=Ps denoting the
price of the supplier. The MR of the retailer is 240-4Q, so setting these two equal gives
Ps=240-4Q, the demand for the supplier. The suppliers MR =240-8Q and its MC =0. So
the quantity will be where 240-8Q=0 or Q=30. The supplier will charge 240-4Q=120 and
the retailer will charge 240-2Q=180. The profits of the retailer are (180-120)*30=1800.
The supplier will make (120-0)*30 - any fixed costs it might have.
If there are two retailers then MR for retailer 1 equals 240-4q1-2q2. Setting this equal to
the retailer’s MC=Ps we have the best response function: 240-4q1-2q2 = Ps. For the
second retailer we get the similar expression: 240-2q1-4q2 = Ps. Given the symmetry, we
know that in the end q1 = q2 so we have q1 = (240-Ps)/6. For a given price set by the
supplier the total quantity it will sell is q1+q2 = (240-Ps)/3. That is Q=(240-Ps)/3 or
Ps=240-3Q, So MRs=240-6Q. As its MC=0 the firm want to sell where 240-6Q=0 or
Q=40. Each firm will thus buy 20 units and the supplier’s price will be 120. The retail
price will be 240-2(20+20)=160. The retailers will each make (160-120)*20=800. The
supplier will make (120-0)*40- any fixed costs. Note that this is better for the supplier
but worse for the retailers.
If the firm vertically integrates and shuts down the other retailer, then MC=0 and
MR=240-4Q so the optimal choice is 0=240-4Q or Q=60. The price will be 120 and the
profits will be 120*60 – any fixed costs.
Q2. A firm has 5 potential markets in which to enter: A, B, C, D, and E. Before
entering in any market, the owner-workers agree that they should invest in some
background research on the different markets. To be fair, they decide to allocate funds
based on the group ranking of the five projects. The following table shows how the
individuals rank the projects:
# voters
49
48
3
1st
A
B
C
2nd
B
E
B
3rd
C
D
E
4th
D
C
D
5th
E
A
A
Rank these five alternatives using the Borda-Count method as we discussed in class.
A first place vote gets 1 point, a second place vote gets 2 points, etc.
A: 49*1+48*5+3*5=304
B: 49*2+48*1+3*2=152
C: 49*3+48*4+3*1=294
D: 49*4+48*3+3*4=352
E: 49*5+48*2+3*3=350
Hence the rankings would be B>C>A>E>D
An alternative approach is the method of pairwise comparisons. Using this method, you
compare each pair of alternatives using majority voting with the winner getting 1 point.
In the event of a tie both alternatives get a ½ point. The rankings are then based on the
number of points each alternative receives. How would the five alternatives be ranked
under this method?
Comparing A&B: A receives 49 votes and B receives 51 votes: winner is B (+1 for B)
A&C: +1 for C,
A&D: +1 for D
A&E: +1 for E
B&C: +1 for B
B&D: +1 for B
B&E: +1 for B
C&D: +1 for C
C&E: +1 for C
D&E: +1 for E
Totals: A-0, B-4, C-3, D-1, E-2
Rankings: B>C>E>D>A
A third approach is recursive plurality. With this method each person votes for their
favorite alternative. The alternative receiving the most votes is ranked first. After
removing the winner of the first round from each person’s preferences, another round of
voting is held to determine the second place alternative and so on until all the alternatives
are ranked. So in this problem A would be ranked first with 49 votes in the first round.
With A removed, B would get 97 votes in the second round and be ranked second. How
would the other three alternatives be ranked?
Initial Voting: A-49, B-48, C-3, so A wins (remove it from set)
2nd Round: B-97, C-3, so B wins (remove it from set)
3rd Round: C-52, E-48 so C wins
4th Round: D-49, E-51 so D wins
5th Round: E-100 so E wins
Rankings: A>B>C>D>E
Notice that everyone prefers B to E and that in all 3 systems rank B above E. This is
because all three systems have the Pareto property so they must either be a dictatorship or
not have independence of irrelevant alternatives.
Q3. A worker has preferences given by u(w,e) = w - .5e2 where w denotes wealth and e
denotes effort. This worker has an outside offer that will earn her a utility of u*=10.
Sales revenue is increasing in the worker’s effort, TR= 20e, so the marginal benefit of
effort is 20. The only cost to this firm is the wage it must pay to the worker. To get the
worker to be willing to accept the job and exert a given level of effort, what does her
wage offer have to be (that is what is the individual rationality constraint)? What is the
expression for the firm’s profit? The marginal cost of effort is just e (the cost of effort is
.5e2 so the derivative is just e). Assuming that effort is observable, what level of effort
should the firm specify in its contract, what wage should it pay and what profit will the
firm earn?
To accept the job it must be that w - .5e2 > 10. So w > .5e2 + 10 and the firm will push
this down to equality. The firm’s profit is 20e-wage or 20e-.5e2 + 10. The optimal effort
is the one that maximizes this express, so we have 20=e. The optimal wage is .5(20)2 +
10= 210. The firm will earn 20*20-210=190. The firm could not observe effort then the
worker would pick e=0. The value of this to the firm is 0 and thus the firm would not
hire the worker. A way around this problem is to sell the worker the right to her output
for 190. This way the firm makes the same profit it would if effort is observable and the
worker will end up selecting an effort of 20 and just cover their opportunity cost.