Download Answers to Midterm Exam II

Answers to Midterm Exam II
1. Suppose that the demand and supply of gas are
Q D  150  50 p , Q S  60  40 p
(a)Find the market equilibrium in price and quantity of gas. (b)Suppose that the government
imposes quantity tax on per unit sold: t = $0.5. Find the market price and quantity. (c) Find the
consumers’ surplus, firm’s surplus and government benefit. What is the social loss of a tax?
Answer 1: (a) let Q D  Q S , then p   1; Q   100 .
11
800
(b) let p d  p s  t , then the market price p   p d  , Q  
.
9
9
(c) respectively compute CS , PS and the government benefit :
1700
2125
3600
CS 
, PS 
, and the government benefit =
.
81
81
81
3825
However, the total loss = CS + PS =
.
81
3825 3600 225


So the deadweight loss =
.
81
81
81
2. In a pure exchange economy, Ollie's utility function is U(x, y) = 3x+y and Fawn's utility
function is U(x, y) = xy. Ollie's initial allocation is 1 x and no y’s. Fawn's initial allocation is no
x’s and 2 y’s. Draw an Edgeworth box for Fawn and Ollie. Put x’s on the horizontal axis and y’s
on the vertical axis. Measure goods for Ollie from the lower left and goods for Fawn from the
upper right. Mark the initial allocation with the letter W. The locus of Pareto optimal points
consists of two line segments. Describe these line segments in words or formulas and show them
on your graph.
Answer: The Edgeworth box is 1 unit wide and 2 units high. Along the contract curve, Fawn
consumes 3 times as much y as x. The contract curve consists of a line running from the upper
right corner of the box to the point on the bottom of the box where Fawn consumes all of the y
and 2/3 units of x and a line from this point to the lower left of the box.
3. A local electric utility monopoly faces an inverse daily demand for electricity of p = 12 - Q,
where price is in cents per kilowatt-hour and Q is in millions of KwH. The utility's cost is C(Q) =
Q2 + 40. (Below, derive actual numbers and show in a graph.)
a. If the monopoly cannot price discriminate, what are its profit-maximizing price and
quantity? What is the deadweight loss?
b. If the firm can perfectly price discriminate, how much output does it sell, what is the
lowest price that it charges, and what is the deadweight loss?
Answer: 3. a.
As with all firms, the monopoly maximizes profit at MC = MR. Since C(Q) =Q2 +40, MC =
C’(Q) = 2Q. We can use the fact that R=P*Q. From D, P=12-Q => R= (12-Q)*Q= 12Q-Q2. MR
= R’ = 12-2Q.
Setting MR = MC, we have Qm => 2Qm=12-2Qm => Qm=3. The price the monopoly charges,
Pm, is based upon what the market will bear, described by D => Pm=12-Qm= 12-3=9.
The deadweight loss is the economic surplus society foregoes by not producing at the social
optimum. At the social optimum, 12-Q=P=MC=2Q. Solving, Qso = 4 and Pso = 8. Deadweight
loss is therefore the shaded triangle on the graph. At Qm=3, MC = 2Q = 2*3=6. So the area of
DWL = 1/2 (9-6)(4-3) =1.5.
3.b.
When the monopoly perfectly price discriminates, it charges a different price for each unit
sold, namely the consumers’ (marginal) willingness to pay for each additional unit, again as
represented by D. In perfect price discrimination, the producer doesn’t have to lower the price on
each previously sold unit, only on the extra unit being sold. Therefore, the firm’s MR is always
the price consumers are willing to pay for the next unit i.e. MR is D.
The firm again maximizes by producing until MC = MR i.e. where the MC curve intersects
the demand curve. We thus have 12-Q = P = MR = MC = 2Q. Therefore, Qpd = 4 and Ppd =
12-Qpd = 12-4 = 8, where Ppd is the lowest price at which the firm sells (remember that the firm
sells all units prior to the 4 millionth at the higher price each commands on the demand curve).
Since this is the same as the socially optimal quantity found in a there is no DWL.
4. Firms A and B engage in a one-period game. Each firm’s profit depends on how much output
both firms produce:
a. Suppose we observe many pairs of firms playing this one-period game and all firms
choose to produce a high level of output. What must be true about the values of X and Y? Be
sure to explain your answer using the concept of Nash equilibrium (and, if relevant, dominant
strategies).
b. Each firm would earn higher profits if they both agreed to produce at the low output level.
Why do the firms produce high levels of output in the equilibrium? (That is, explain the
economic concepts behind your answer to part A).
Answer: 4.a. In all of our observations of firms playing this game, we always observe the outcome
that each firm produces at the high level of output. This must mean that (High, High) is the only
Nash equilibrium in this game. Recall that a Nash equilibrium is when both firms are earning the
highest profits they can, given the strategy played by the other firm. In other words, neither firm
has an incentive to change its behavior when it learns the other firm’s action.
So what must be true about X and Y for (High, High) to be the only Nash equilibrium in this
game? First, if (High, High) is a Nash equilibrium, it must be the case that when firm B plays
High, that firm A earns a higher profit playing High (8) than playing Low (X): X < 8. Analyzing
the game from B’s perspective yields the same conclusion. Next, none of the other combinations
of strategies can be a Nash equilibrium. Let’s look at (Low, Low) first. Given that B has played
Low, if (Low, Low) isn’t Nash, then A must earn a higher profit from switching to High (Y) than
when it stays with Low (10): Y > 10. Again, looking at it from B’s perspective gives the same
result. For completeness, note that (High, Low) and (Low, High) can’t be Nash given these two
conditions – the player who gets X always wants to switch and get 8 instead.
These two conditions also imply that playing High is a dominant strategy for both players:
playing High always pays more than playing Low regardless of the other player’s action. For
instance, if B plays low, A should play High rather than Low since Y >10; similarly, High pays
more than Low when B plays High because X < 8.
4.b. Specifically, what economics underlies the idea that the firm always has an incentive to
renege on the cartel agreement in this setup? (i.e. why does Y >10 make economic sense?) If one
firm in a duopoly can get the other firm to reduce its output, the residual demand curve that the
original firm faces will be a larger share of the total demand in the market – essentially, the
firm’s MR curve shifts out. If the firm wants to maximize profits, its response would be to
increase output. Thus, sticking to the cartel agreement means the firm isn’t maximizing profits,
since MR > MC at the low output level. The firm will earn higher profits by reneging on the
cartel agreement and expanding output in response to the other firm cutting back. In the context
of the game, if one firm plays Low, the best response is to play High, implying Y > 10.
The other reason that (Low, Low) will likely not be an equilibrium is that this is a one-shot game
– there’s no punishment for cheating on the deal. In a multi-period game, the firm considering
defection will weigh two alternatives. On the one hand, it can receive the collusive profit of 10
each time. On the other hand, it can renege this time, get Y > 10 once, and then be stuck with
getting the non-cooperative profit of 8 for the rest of the times it plays (the “punishment”). There
are some circumstances where it still makes sense to defect (if the number of games is small
and/or Y is “high”), but there are others where collusion is the profit-maximizing play (lots of
games and/or Y isn’t “too big”).