Download Econ 210, Microeconomic Theory HW 8

Econ 210, Microeconomic Theory
HW 8 - Answers
Auctions and Price Discrimination
Professor Guse
November 12, 2015
1. First-price sealed bid auction. Assume there are n bidders. Each bidder i has a true
value, vi distributed uniformly on [0, 1]. Each bidder knows her own value but does
not know the value of the other bidders in the game.
(a) When there are only two bidders, show that there is an equilibrium in linear
strategies. In other words, show that if player 2 employs a strategy with the
form b2 (v2 ) = a2 v2 , that bidder 1’s best response is to employ a similar strategy.
ANSWER: Player 1’s problem is to maximize expected consumer surplus
max(v1 − b1 )P r(b1 > b2 )
b1
substituting in the assumed form for player 2’s strategy for choosing b2 we can
rewrite this as
max(v1 − b1 )P r(b1 > a2 v2 )
b1
⇔ max(v1 − b1 )
b1
b1
a2
Using the product rule to differentiate with respect to b1 , we get a first order
condition for an optimal choice of b1 :
(v1 − b1 )
⇔b1 =
1
v1
2
1
b1
−
=0
a2 a2
(b) Solve for the linear equilibrium in the n = 2 case and calculate the expected
revenue generated by the auction in that equilibrium. ANSWER. As we just
saw, there is a linear equilibrium in which both players bid half of their true
value for the object. To calculate the expected revenue we need to calculate
E(Rev) = E(max{
v1 v2
1
, }) = E(max{v1 , v2 })
2 2
2
In other words, since, in equlibrium, both player bid half their true value, the
highest bid will be half of the highest true value. So how do we calculate
the expected value of the maximum of a pair of random uniformly distributed
variables? To answer this question, we need to review what an expectation
is. By definition the expected value of a random variable x whose density is
described by f (x) is given by
E(x) =
Z
∞
xf (x)dx
x=−∞
We want
E(max{v1 , v2 }) =
=
Z
1
1
Z
max{v1 , v2 }dv1 dv2
v1 =0 v2 =0
Z 1 Z 1
v1 =0
v2 =v1
v2 dv1 dv2 +
Z
1
v1 =0
Z
v1
v1 dv1 dv2
v2 =0
Note that in the last step we just decomposed the integral into the lower and
upper triangles of the square. The first term represents all the cases where
v2 > v1 and the second case represents all the cases where v2 < v1 . Note also
that even though they look a little different they have to evaluate to the same
value since by symmetry the expected value of v1 when its bigger than v2 cannot
be different that the expected value of v2 when its bigger than v1 . Therefore we
really need only compute one of the terms and then multiply it by two for our
answer. Nevertheless I will explicitly evaluate both terms.
2
E(max{v1 , v2 }) =
Z
1
v1 =0
Z 1
Z
1
v2 dv1 dv2 +
v2 =v1
1
(1 − v12 )dv1 +
E(max{v1 , v2 }) =
2
v1 =0
1
1
1
= (1 − ) +
2
3
3
2
=
3
Z
Z
1
v1 =0
Z
v1
v1 dv1 dv2
v2 =0
1
v12 dv1 dv2
v1 =0
Since the bid will be half of the max value, the expected revenue will be
n = 2.
1
3
for
(c) Repeat the above two questions for the general case of n bidders. ANSWER
Again consider Player 1’s problem when facing off against linear strategies
b2 (v2 ) = av2 , b3 (v3 ) = av3 , ... (Note that we assume that Players 2, 3, ...
n following the same strategy.)
max(v1 − b1 )P r(b1 > b2 &b1 > b3 ...)
b1
Assuming independence and plugging in the other player’s strategic forms we
get
P r(b1 > b2 &b1 > b3 ) = P r(b1 > av2 ) × P r(b1 > av3 ) × ...
=
b1n−1
an−1
Plugging this into 1’s objective function, differentiating it w.r.t. b1 and setting
it equal to 0, we get the FOC:
−
Dividing thru by
b1n−2
an−1
b1n−1
(n − 1)b1n−2
+
(v
−
b
)
=0
1
1
an−1
an−1
we get
−b1 + (v1 − b1 )(n − 1) = 0
(n − 1)
v1
b1 =
n
3
So, 1’s best response when everyone is playing the av strategy is to play a linear
strategy and specifically one where she bids n−1
n times her true value. Therefore
everyone employing such a strategy consititutes an equilibrium. To calculate
n
the expected revenue, one needs to recognize that E(max{vi }ni=1 ) = n+1
. (This
follows by using the logic of integration we applied for the case of n = 2 above).
Therefore the expected revenue will be
n−1 n
n−1
=
n n+1
n+1
Note, therefore that expected revenue increases toward 1 as n increases for two
reasons. First, as the auction becomes more competetive, bidders bid closer to
their true value. Second, the max true value drawn will on average be higher
the more true values are drawn.
2. Second Degree Price Discrimination. Suppose there are two types of consumers ,L
and H, with demands for donuts given by the following inverse demand equations.
P L (y) = 100 − y
P H (y) = 150 − y
Assume that there is one firm that serves this market and that the marginal cost of
making donuts is zero.
(a) If the firm pursues a strategy of setting one price, what price should it set?
How much revenue will it earn? ANSWER Since cost is zero, this is a revenue
maximization problem. In order to solve it, we need to derive a revenue function.
But first we need an aggregate demand function. Horizontally adding the two
demand curves above (note they are presented in their vertical forms), we have
y Agg (p) = y L (p) + y H (p)
= max{100 − p, 0} + max{150 − p, 0}
= max{250 − 2p, 150 − p, 0}

 250 − 2p if p < 100
150 − p if 100 < p < 150
=

0
if p > 150
Therefore revenue as a funciton of price is
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R(p) = y Agg (p) × p

 250p − 2p2 if p < 100
150p − p2 if 100 < p < 150
=

0
if p > 150
Marginal revenue is
M R(p) = y Agg (p) × p

 250 − 4p if p < 100
150 − 2p if 100 < p < 150
=

0
if p > 150
Normally the first order condition for maximizing revenue would be M R = 0.
However we must be cautious when working with a discontinuous function. One
thing we can say is that M R = 0 in the lower segment of the price ranges at
p = 250
4 = 62.5, so we can say that this is the best price to set of all the prices
between 0 and 100. The firm’s revenue at that price would be R(62.5) = 7812.5
However we need to check whether we can achieve a higher level of revenue in
the upper range - between 100 and 150. Marginal revenue is negative at 100
and just gets more negative at higher prices. Therefore in that price range the
solution is a corner; it’s 100 and at that price, revenue would be 5000. Therefore
the revenue maximizing price is 62.5.
(b) If the firm pursues a strategy of marketing two different sized donut packs,
what would be the optimal sized donuts packs and the prices for them? Would
this strategy make more revenue than the single-price approach?ANSWER
Following the example in Varian, the optimal size of the the large package would
be 150 donuts (where the MWTP is zero for the high demand group), while the
optimal size for the smaller package is the quantity at which the MWTP of the
High demand group is exactly twice the MWTP of the Low demand group:
P H (y) = 2P L (y)
⇔150 − y = 2(100 − y)
⇔y = 50
Again following the example in Varian, the menu would look as follows
5
size
50
150
price
3750
8750
where 3750 is the area under the L-type’s MWTP curve beteen quantities of
0 and 50 and 8750 is that same area plus the area under the H-type’s MWTP
curve between quantities of 50 and 150.
3. Suppose that demand for water by each industrial users (I) is
M W T PI (QI ) = .5 − .00005QI
While demand for water by each residential user (R) is
M W T PR (QR ) = 1 − .01QR
There are 1 industrial user and 100 residential users.
The fixed cost to build a water supply system to supply both groups is 5000, the
marginal cost of supplying the water, once the system is built is $.01 per gallon. If
each case below, say what water consumption levels for each group would be as well
as profit for the water company.
(a) If the water company were allowed to charge a single per-unit price. What would
it charge to maximize profit? ANSWER. First find aggregate demand curve
by re-writing each demand function according to its horizontal interpretation.
(i.e. in terms of quantity)...
.5 − P
= 10000 − 20000P
.00005
1−P
= 100 − 100P
QR (P ) =
.01
QI (P ) =
Now add these together multiplying the residential by 100 (since there are 100
of them) to get aggregate demand.
Q(P ) = QI (P ) + 100QR (P ) = 20000 − 30000P
Invert this to get aggregate marginal willingness to pay.
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∀P ≤ .5
(1)
P (Q) =
20000 − Q
30000
(2)
Total revenue is price × quantity and marginal revenue is the first derivative of
TR w.r.t Q
Q(20000 − Q)
30000
20000 − 2Q
⇒M R(Q) =
30000
T R(Q) =
Setting M R equal to M C we get the profit maximizing quantity to provide for
a sigle price monopoly which is about 10,000 gallons.
20000 − 2Q
= .01
30000
⇒20000 − 2Q = 300
19700
∼ 10000
⇒Q =
2
Plus back into equation ??, the inverse of the aggregate demand curve, to find
the price that would be set so that this quantity is demanded
Psp (Q = 10000) =
⇒Psp =
20000 − 10000
30000
1
3
(b) If the water company could charge different per-unit price to industrial users
and residential users, what would these prices be? ANSWER. Since marginal
cost is constant we can simply set each individual marginal revenue equal to the
marginal cost. Notice that since we have linear demand, the MR curves are just
the individual MWTP curve with twice the vertical slope.
First do industrial customers
M W T PI (QI ) = .5 − .00005QI
⇒M RI = .5 − .0001QI
Setting M RI equal to M C, we get
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.5 − .0001QI = .01
.5 − .01
⇒QI =
= 4900
.0001
⇒PI = .5 − .00005(4900) = .255
while for residental
M W T PR (QR ) = 1 − .01QR
⇒M R = 1 − .02QR
.99
= 49.5
.02
⇒ PR = 1 − .01(49.5) ≃ .50
⇒ QR =
So under a single price monopoly, both industrial and residential customers
would pay about 33 cents per gallong, but if the monopoly can discriminate
between residential and industrial then it would charge the industrial about .25
per gallon and residential customers about .50.
(c) Suppose the local community will not agree to the new water system unless,
the firm charges only the marginal cost for each unit of water. What does the
company tell the community? ANSWER At .01 per gallong total demand
according to equation ?? woudl be
Q(P = .01) = 20000 − 30000(.01) = 19700
At a price of .01, total revenue would be $197. However costs would be 5000 +
.01(19700) = 5197, hence the firm would be losing 5000 - its entire fixed cost.
Therefore at that price, this firm would exit in the long run (or never enter to
begin with).
(d) Suppose that the community does not like what it hears. Instead it decides to
let the water company charge a single price which is just high enough that it will
not require a subsidy. What price do they dictate? ANSWER. First calculate
average cost , AC...
AC =
Now set AC = MWTP.
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5000 + .01Q
Q
(3)
⇒
⇒
⇒
⇒
5000 + .01Q
20000 − Q
=
Q
30000
150, 000, 000 + 300Q = 20000Q − Q2
Q2 − 19700Q + 150, 000, 000 = 0
p
19700 ± 197002 − 600, 000, 000
Q=
2
p
19700 ± 197002 − 600, 000, 000
Q=
2
It turns out that that there is no real-valued solution since the expression inside
the root sign is negative. This can be interpretted as, “ the demand curve never
intersects the average cost curve”. In fact, it turns out that this company makes
zero profit in parts (a) and (b) as well. The only way for this water company to
make a positive profit would be to charge fixed “service” charges either instead
of or in addition to a per-unit charge of .01. For example charging the industrial
customer a $1000 “hook-up” fee and each residential customer a $40 “hook-up”
fee in addition to charging everyone $0.01 per gallon. Doing this, the company
would break even and each customer would willingly hook-up and still derive
some consumer surplus.
(e) Come up with a pricing plan that lowers prices to consumers and still allows the
water company to break even.
ANSWER. For example, one could increase the hook-up fee charge to the
industrial customer.
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