Figure 1-S05
U1
U2
P2
P
P1
1
2
x1
x2
Partial Answers to Micro Prelim of June 2005
1. Two farmers, 1 and 2, grow rice at opposite ends of an island. Every year, rain falls
on one end of the island, and not on the other (at different ends different years). At the
beginning of any given year, meteorologists say that the rain is equally likely to fall on
either end of the island that year. If rain falls at one end, the farm there harvests 3 units
of rice and the other farm harvests 1 unit. The farmers consume only rice and need rice
to survive. They are expected utility maximizers, obtaining utility ln x by consuming
x > 0 units in a year. At the beginning of one year, one of the farmers proposes that
they sign a contract according to which, whoever has the larger harvest will give some
to the other.
(a) What notation can be used to describe the contingent consumption resulting from
such a contract? In the rest of the problem, assume that the farmers are able to commit
themselves to such contracts.
We can let xsi be the rice consumption of farmer i if the rain falls on farmer s’s farm
(s = 1, 2). The side where the rain falls can be called the state of the world and
consumption in the two states can be treated as two different consumption goods. The
model then is mathematically identical to a pure exchange economy with two distinct
physical commodities and consumers with Cobb-Douglas utility functions.
(b) Find all the Pareto efficient allocations of contingent consumption for the two farmers
that can be attained through the contracts of part (a). Describe the Pareto efficient
contracts in English. Draw the allocations in an Edgeworth box, including indifference
curves of the farmers.
If there is no disposal, feasible allocations are strictly positive vectors ((x11 , x21 ), (x12 , x22 ))
with xs1 + xs2 = 4, s = 1, 2, since 4 is the total supply of rice in each state of the world.
If there is free disposal, xs1 + xs2 = 4 is replaced by xs1 + xs2 ≤ 4. The utility of farmer i
is the expected utility from rice consumption: ui (xi ) = (1/2) ln x1i + (1/2) ln x2i . Pareto
efficient allocations maximize the utility of farmer 1 subject to the feasibility conditions
and to a constraint on the utility of farmer 2, i.e., maximize u1 (x1 ) subject to u2 (x2 ) ≥ ū2
and the feasibility conditions. Since the objective function and constraint functions are
concave, the first order conditions are sufficient for a solution. The first order conditions
are equivalent to the condition that the farmers have equal marginal rates of substitution
(∂u(xi )/∂x1i )/(∂u(xi )/∂x2i ) = x2i /x1i , with u2 (x2 ) = ū2 . This along with the feasibility
conditions implies x1i = x2i , i = 1, 2, and every feasible allocation with this property
has some ū2 such that the optimization problem is solved. So the set of Pareto efficient
allocations is the set of feasible allocations in which a farmer gets the same rice consumption no matter where the rain falls. Both farmers are completely insured, though
one farmer may consume more than the other.
The set of feasible contingent consumption allocations can be represented by the
interior of an Edgeworth box with corners at (0, 0) and (4, 4). The Pareto efficient
allocations are represented by the diagonal, the segment joining (0, 0) and (4, 4) where
x11 = x21 , with the endpoints of the segment removed.
(c) Suppose that the farmers both believe that rain will fall with probability 1/2 on their
side of the island and with probability 1/2 on the other side. They sign a contract that
yields contingent consumption arising in a competitive equilibrium. What can be said
about the equilibrium prices and allocation? Be as specific as possible. Is the allocation
Pareto efficient? Justify your answers.
1
In competitive equilibrium, farmer i maximizes ui (xi ) subject to pxi ≤ pei , where
e1 = (3, 1) and e2 = (1, 3) are the endowment vectors of contingent consumption of
farmers 1 and 2. The Cobb-Douglas demand for “good 1” (i.e., for contingent consumption in state 1) by consumer 1 at price vector p = (p1 , p2 ) is x11 = pe1 /(2p1 )
and for consumer 2 is x12 = pe2 /(2p1 ), so the total demand is p(e1 + e2 )/(2p1 ) =
(4p1 + 4p2 )/(2p1 ) = 4, where the last equation is required by feasibility (demand equals
supply). Thus p1 = p2 in equilibrium. Since the farmers spend their entire budgets in
equilibrium, supply equaling demand for “good 1” implies the same for “good 2.” Only
relative prices are determined in equilibrium, not the price levels. Farmer 1 demands
[p · (3, 1)]/(2p1 ) = (3p1 + p1 )/(2p1 ) = 2 of “good 1,” so farmer 2 demands 2 units also.
Farmer 1’s equilibrium competitive demand for “good 2” with Cobb-Douglas utility is
[p · (3, 1)]/(2p2 ) = 2, so farmer 2 has the same demand. From the answer to (b), the
allocation is Pareto efficient.
(d) Suppose instead that both farmers are optimists. They both think that the rain
will fall on their side of the island with probability π ∈ (1/2, 1). Compare the possible
outcome(s) of competitive trading of contingent consumption to the outcome in part
(c). Be as specific as possible with the given information. What can be said about the
efficiency of a competitive equilibrium allocation in this case? Does it matter that the
farmers’ beliefs cannot both be correct?
The expected utility function of farmer 1 becomes π ln x11 + (1 − π) ln x21 and that of
farmer 2 is (1−π) ln x12 +π ln x22 . Farmer 1 now has stronger preference for “good 1” and
farmer 2 for “good 2,” that is, farmer 1, thinking that it is more likely for rain to fall on its
side of the island, has stronger preference for consumption contingent on that event, and
farmer 2 has the reverse preference. Farmer 2 has the utility function of farmer 1 with the
indices for the goods reversed, so by symmetry, we might expect the prices of the goods
to be equal in equilibrium. This turns out to be correct. The Cobb-Douglas demand for
“good 1” at price vector p by farmer 1 is x11 = πpe1 /p1 , and the demand by farmer 2 is
(1−π)pe2 /p1 , so total demand is [π(3p1 +p2 )+(1−π)(p1 +3p2 )]/p1 , which, in equilibrium,
equals the supply of 4. This implies p1 = p2 in equilibrium. In state 1, farmer 1 consumes
πpe1 /p1 = 4π units of rice and farmer 2 consumes (1−π)pe2 /p1 = 4(1−π). By symmetry,
the consumptions are reversed in state 2, so farmer 2 consumes 4π and farmer 1 consumes
4(1 − π).
By the first fundamental welfare theorem, the consumption allocation is Pareto efficient given the farmers’ preferences. This means that given their beliefs, it is not possible
to make one better off in expected utility without hurting the other. But at least one
farmer has incorrect beliefs. From the point of view of meteorologists, with the two
states equally likely, both farmers have higher expected utility at the allocation in part
(c) than in the one in (d) above. Each farmer has the same budget set in part (d) as in
part (c) since p1 = p2 in both cases. If the states are equally likely, the expected utility
of each farmer is maximized over its budget set at the consumption vector in part (c).
In that case consumption risk is removed. Thus, evaluated with beliefs that are more
reasonable than the farmers’ beliefs, the allocation in part (d) is Pareto inefficient.
(e) Suppose now that farmer 1 is an optimist, but farmer 2 is a pessimist. Both believe
the probability to be π ∈ (1/2, 1) that the rain will fall on farmer 1’s farm. Compare the
outcome(s) of competitive trading of contingent consumption to the outcomes in parts
2
(c) and (d) and explain the comparisons. Be as specific as possible with the given information. What can be said about the efficiency of a competitive equilibrium allocation in
this case? Can it be advantageous to have unrealistic beliefs in this situation? Explain.
Now each farmer i has expected utility ui (xi ) = π ln x1i +(1−π) ln x2i , given the farmer’s
belief. Total demand for rice in state 1 is πp(e1 + e2 )/p1 = πp(4, 4)/p1 = 4π(p1 + p2 )/p1 ,
which equals the total supply of 4 when p1 /p2 = π/(1 − π). Farmer 1’s demands are
x11 = π(3p1 + p2 )/p1 = 3π + π(p2 /p1 ) = 3π + (1 − π) = 1 + 2π and x21 = (1 − π)(3p1 +
p2 )/p2 = 3π + 1 − π = 1 + 2π. Thus, the optimistic farmer, farmer 1, gets more rice in
each state than in part (c). In state 1, farmer 1 gets less rice in this equilibrium than in
part (d) since 4π > 1 + 2π, whereas in state 2, farmer 1 gets more rice than in (d). Both
farmers are completely insured since they consume the same amount no matter where
the rain falls. For the farmers’ preferences, based on their own beliefs, the allocation
is Pareto efficient, by the first welfare theorem. By the answer to (b), the allocation
is also Pareto efficient for preferences determined by the meteorologists’ beliefs. The
optimism of farmer 1 and pessimism of farmer 2 increase their demands for rice in state
1 at any given prices. This raises equilibrium relative price of “good 1” and the value
of the endowment of farmer 1 relative to farmer 2. Farmer 1 gains and farmer 2 loses.
2. Consider an economy with I rational agents, each endowed with one unit of time.
Agents contribute labor to the production of a single consumption good, y, which is
produced according to the technology y = f (L1 , ..., LI ), where Li denotes the amount
of labor time contributed by agent i. In addition to contributing time to the productive
activity, agents can contribute to a nonproductive, conflict activity. If each agent i devotes ci units of time to conflict, i = 1, . . . , I, then agentPi receives the share θi (c1 , . . . , cI )
of the total output of the consumption good, where i θi (c1 , ..., cI ) = 1. The agents
choose their pairs (Li , ci ) independently.
(a) Assume first that the agents care only about their consumption of the produced
good. Set up the maximization problem that determines the optimal time allocation for
agent i, given the choices made by the other agents.
Maximize θi (c)f (L) with respect to ci and Li subject to ci +Li = 1, where c = (c1 , . . . , cI )
and L = (L1 , . . . , LI ).
(b) State sufficient conditions on f and θ that ensure the existence of a solution to the
time allocation problem in part (a). Try to make the conditions as weak as possible.
It is enough for f and θ to be continuous since the constraint set is compact.
(c) Find what you expect to be all the agents’ time allocations for the case in which
X
ci
Li
and
θi (c1 , ..., cI ) = P
f (L1 , ..., LI ) =
, ∀i,
c
h
h
i
being as specific as possible with the given information. Justify your answer.
Since the agents choose independently, consider Nash equilibrium (NE). In NE, each
agent optimizes given the choices of the others. At an interior solution, the first order
condition for agent i is θii (c)f (L) = λi and θi (c) = λi , where
P θii is the partial
P derivative
2
of θi with respect to its ith argument. Here,
θ
(c)
=
[(
c
)
−
c
]/(
h h Pi
h ch ) , so the
P ii
first order condition can be rewritten as
P [( h ch ) − ci ]f (L) = ci h ch . Solving for ci ,
we see that it is the same for all i, so h ch = Ic1 and (I − 1)f (L) = Ic1 . Since each
agent i’s payoff is increasing in Li , in NE, Li + ci = 1, so every Li equals 1 − c1 , and
3
f (L) = I(1 − c1 ), Ic1 = (I − 1)f (L) = (I − 1)I(1 − c1 ) and c1 = (I − 1)/I = ch , and
Lh = 1 − c1 = 1/I, ∀h.
To answer the problem, it is enough to find one NE, but it can also be shown that
there are no others. The argument above shows that all agents with noninterior solutions
to their optimization problems choose the same (ci , Li ) in NE. If some agent j chooses
cj = 0, then Lj = 1 and the first order conditions for j are θjj (c)f (L) ≤ λj = θj (c),
which cannot hold since θjj (c)f (L) > 0 = θj (c). If cj = 1 and
P Lj = 0 then the
P first
order conditions are θjj (c)f (L) = λj ≥ θj (c), which implies ( h ch − 1)f (L) ≥ h ch .
In order for the solution to be different from the one in the paragraph above, the weak
inequality must be strict. But then every agent i prefers Li = 0, which implies f (L) = 0,
contradicting the first order condition. This shows that every agent chooses an interior
solution in NE, and the NE in the paragraph above is the only NE.
(d) How would the equilibrium allocation of time between the two activities change as
I increases? Comment on the efficiency properties of the outcome.
As I increases, in NE, each agent spends less time on production and more on conflict.
Time devoted to conflict by one agent is a negative externality for the others and the
outcome is inefficient. The payoffs would all be higher if conflict were banned and the
total output f (L) were evenly divided among the agents.
(e) Next, assume that the agents also care about leisure and, in particular, that they
have preferences represented by u(yi , `i ) = yi `i , where yi denotes i’s consumption of the
produced good and `i denotes i’s leisure. As above, the share of aggregate consumption
received by agent i is θi (c1 , . . . , cI ), where ci is the time i devotes to the conflict activity.
Set up the individual optimization problem that will determine the agents’ optimal
allocation of time.
Maximize θi (c)f (L)(1 − ci − Li ) with respect to ci ≥ 0 and Li ≥ 0, with 1 − ci − Li ≥ 0.
(f) Are the conditions described in part (b) still sufficient to ensure the existence of a
solution to the optimization problem in (e)? Explain.
Yes. The objective function is continuous if θi and f are, and the constraint set is
compact.
(g) Again for the functions f and θi specified in part (c), determine an equilibrium and
study its behavior as the population size varies.
Given the payoffs in (e) and letting `i = 1 − ci − Li , the first order conditions for i
at an interior solution are θii (c)f (L)`i − θi (c)f (L) = 0 and θi (c)[`i − f (L)] = 0, which
imply θii (c)`i = θi (c) and `i = f (L). By the same reasoning as in part (c), all agents
have interior solutions. Then `i = f (L) implies that all agents choose the same amount
of leisure. Since θii (c)f (L) = θi (c), as in (c), the agents all choose the same conflict
and labor times. Therefore the same computation as in (c) yields f (L)(I − 1) = Ic1 ,
so IL1 (I − 1) = Ic1 . Thus ci = c1 = (I − 1)Li and ILi = f (L) = `i = 1 − ci − Li =
1 − (I − 1)Li − Li = 1 − ILi , which implies Li = 1/(2I), ci = (I − 1)/(2I) and `i = 1/2.
The agents’ preference for leisure reduces the inefficiency by cutting their conflict activity
in half compared with the outcome in part (c). If I increases, agents reduce their labor
time and increase their conflict activity, holding their leisure fixed.
(h) Assume that the agents care about leisure, as in part (e). Describe an alternative
way of sharing the total output that yields a more efficient time allocation than the
4
allocation in (g). Try to make your alternative sharing rule as efficient as possible. Find
the resulting time allocation under P
your sharing rule and compare it to the outcome in
part (g), assuming f (L1 , . . . , LI ) = i Li .
There is no externality in production. The externality in preferences in (c) and (e)
arises only because of the sharing rule. The fundamental welfare theorems suggest that
efficiency might be achieved by allocating as in a competitive equilibrium. This can be
done by giving each agent its own output. If agent i spends Li units of time on labor
and ci on conflict, its consumption is yi = Li and its utility is Li (1 − ci − Li ). The
optimal choice is ci = 0 and Li = 1/2. Compared to part (g), the agents work more,
have the same amount of leisure and less conflict and have higher utility: 1/4 compared
to [f (L)/I]`i = 1/(4I).
3. A continuum of consumers of measure 1 is uniformly distributed along a street of
length 1. Two stores, 1 and 2, selling the same good are located at the two ends of the
street respectively. Each store can acquire one unit of the good at constant marginal
cost c. Consumers incur traveling cost t per unit distance of travel. Assume c > t > 0.
Each consumer needs to purchase one unit of the good and will buy the good from a
store where the consumer’s total cost (price plus transportation cost) is the lowest. The
stores simultaneously post nonnegative prices.
(a) Formulate the interaction described above as a normal form game played by the two
stores, i.e. specify strategy sets and payoff functions for the stores.
(b) When the store i sets the price pi , i = 1, 2, which consumers buy from store 1?
Strategy set of each firm: [0, +∞). Payoffs: firm i’s payoff is its profit, πi (p1 , p2 ).
Let f be a location such that the consumer there is indifferent between the stores:
1
p1 + f t = p2 + (1 − f )t, so f = p2 +t−p
. Consumers at locations less than f prefer store
2t
1 and consumers at locations above f prefer store 2. If f < 0, then all consumers prefer
store 2, and if f > 1, then all consumers prefer store 1. Thus the payoff functions are:

p1 − c : if p1 ≤ p2 − t

1 +t
(p1 − c)( p2 −p
π1 (p1 , p2 ) =
) : if p2 − t ≤ p1 ≤ p2 + t
2t

0 : if p1 ≥ p2 + t

p2 − c : if p2 ≤ p1 − t

p1 −p2 +t
π2 (p1 , p2 ) =
(p2 − c)( 2t ) : if p1 − t ≤ p2 ≤ p1 + t

0 : if p2 ≥ p1 + t
(c) Find the best response correspondences of the two stores.
(d) Is there a dominant strategy for either player? Justify your answer.
1 +t
Consider p̂1 that maximizes π1 = (p1 − c)( p2 −p
) with respect to p1 . We have: p̂1 =
2t
1
(p + t + c). In order for this to be a best response, we need
2 2
1
p2 − t ≤ p̂1 = (p2 + t + c) ≤ p2 + t
2
which yields p2 ∈ [c − t, c + 3t].
If p2 ≤ c − t, then choosing any p1 < p2 + t would entail negative payoff, so best choice
is any price not below p2 + t, which generates zero payoff.
If p2 > c + 3t, then the best choice is p̂1 = p2 − t.
Thus, the best response correspondence is
5


Bi (pj ) =
[pj + t, +∞), : if pj ≤ c − t
{ 12 (pj + t + c)} : if c − t < pj ≤ c + 3t

{pj − t} : c + 3t < pj
The best response varies with the rival’s price, so there is no dominant strategy.
(e) Show that the strategies in set [0, t) are strictly dominated for both players.
A strategy p1 ∈ [0, t) is strictly dominated by p01 = c because for any p2 ≥ 0: π1 (p1 , p2 ) <
0 = π1 (c, p2 ).
(f) What is the result of Iterated Deletion of Strictly Dominated Strategies in the limit
as the rounds of deletion go to infinity? Justify your answer.
[c + t, +∞) survives Iterated Deletion of Strictly Dominated Strategies.
After first round, a player is left with choices [p1 , +∞) with p1 = t. If p1 < c − t, then
p1 + t < 21 (p1 + t + c). So in the second round, what is left is [p2 , +∞) with p2 = p1 + t.
Evidently, it will happen for large n that pn > c − t. Then pn + t > 12 (pn + t + c). The
n
. As n −→ ∞, we have pn −→ c+t.
iteration then leaves [pn+1 , +∞) with pn+1 = p2 + t+c
2
(g) Find all pure-strategy Nash equilibria (NE). Justify your answer.
The unique pure-strategy NE: p∗1 = p∗2 = c + t, with πi (p∗1 , p∗2 ) = 2t . Since each store
plays a best response, it is NE. To show there is no other NE, plot the best response
correspondences on [c + t, +∞).
4. Firm B has patented a process for making a new drug and wants to sell its patent
to a single drug manufacturer. Every drug manufacturer knows that if it is given the
patent for free, it could produce the drug and make a net profit of either P1 or P2
(0 < P1 < P2 ) depending on the complexity of the production process (P1 if the process
is more complex). Firm B knows the complexity of the production process, but the drug
manufacturers initially do not. Each manufacturer initially believes that the process
would be less complex (and the net profit would be P2 ) with probability λ ∈ (0, 1).
(Firm B would want to tell them if the process is less complex, but would not necessarily
be believed.)
By spending x on related research and legal fees, firm B can get more patents related
to its new drug and can gain net profit of either π1 (x) or π2 (x) (possibly negative). The
net profit will be π1 (x) if the production process for the new drug is more complex. Each
πi is strictly concave, with πi (0) = 0, πi0 (0) > 0, i = 1, 2, π20 (x) > π10 (x), ∀x > 0, and
with π10 (1) = 0 and π20 (2) = 0.
Firm B chooses its expenditure x and offers its patent for sale. Drug manufacturers, knowing the information above and knowing x chosen by B, bid independently for
the patent, and B sells it to a highest bidder. The firms are rational expected profit
maximizers, and all the above information is common knowledge.
(a) Explain briefly why firm B might want to sell its patent to only one firm instead of
allowing several firms to pay to use the patent.
If the patent is sold or licensed to n firms, which compete in the markets that use it, their
total resulting profit could be less than the profit of one firm that is given a monopoly
right to use the patent.
(b) Interpret the restrictions on the functions πi above.
6
Profit from related research is 0 if expenditure on it is 0. Starting from any related
research expenditure level x, additional expenditure yields less profit if x is higher or if
the production process is more complex. Maximum profit is attained at expenditure 1
if the production process is more complex and at expenditure 2 if it is less complex.
(c) Define a perfect Bayesian equilibrium (PBE) for the game played by the firms.
The definition of PBE is the same as in signaling model in MW 13.C, with variables
reinterpreted as in answer (d) below. A PBE is a choice of expenditure by B for each
type it could be (with more or less complex production process), and for each drug
manufacturer (firm) a function assigning a bid to each expenditure x ≥ 0 by B, along
with a function assigning a belief for each x choice by B. The belief is a joint probability
distribution over possible types of B and bids by firms that bid earlier in the sequence
of moves, with the property that the firms use Bayes rule and equilibrium strategies
whenever possible to assign probabilities, assuming that behavior strategies of different
players are statistically independent, and they agree on the probability that B is a given
type, given its choice of x.
(d) The model above is most similar to which one of the asymmetric information models
in Microeconomic Theory, by Mas-Colell, et. al. (the competitive labor market model
with hidden productivities; the signaling model with hidden productivities; the screening
model with hidden productivities; the principal-agent model with hidden effort; the
principal-agent model with hidden preferences)? Explain briefly which variables in the
model above correspond to which variables in the model in Mas-Colell, et. al.
The model has the structure of Spences labor market with signaling (MW 13.C). Firm
B corresponds to the worker in Spences model and the other firms corrrespond to the
potential employers. High and low complexity of development correspond to high and
low worker productivity. Expenditure on related research corresponds to investment in
education.
(e) Explain which pooling and/or separating outcomes can arise in pure strategy PBE
in the game above. Justify the claim that these are PBE outcomes. Illustrate them
in one or more graphs, with x on the horizontal axis. Label your graph(s) clearly and
explain what the important points in the graph(s) represent.
Independent bidding by the drug manufacturers (call them D-firms) with common knowledge of the potential profits and common beliefs about B’s type yield bids equal to the
expected profit given B’s choice of x (by the same argument as in a model of Bertrand
competition with constant marginal costs that are identical across firms). Therefore, B
is assured at least a price of P1 and payoff u1 ≡ π1 (1) + P1 if it is type 1 (with the more
complex production process), and u2 ≡ π2 (2) + P1 if it is type 2. PBE outcomes can be
illustrated in a graph with P1 and P2 on the vertical axis and levels of x on the horizontal. The important indifference curves of B are sets of (x, p) such that π1 (x) + p = u1
(for type 1) and π2 (x) + p = u2 (for type 2). Their graphs are labeled U 1 and U 2 in
Figure 1-S05. The vertical intercept of the U1 curve is below that of the U2 curve. The
U1 curve is tangent to the horizontal line of height P1 at x = 1. The U2 curve is tangent
to that line at x = 2. The U1 curve reaches height P2 at x = x1 and the U2 curve
reaches that height at x = x2 . It follows that π1 (x1 ) + P2 = u1 and π2 (x2 ) + P2 = u2 .
In pooling equilibria, the bid is P = λP2 + (1 − λ)P1 between P1 and P2 . The
expenditure must be such that types 1 and 2 get payoffs u1 and u2 or more, so the x
7
must be on or above both curves U1 and U2. The set of possible contracts in Figure
1-S05 is the broken line segment at the height P between the curves U1 and U2. Every
contract (x̄, P ) of this form can be a PBE outcome since it arises if the firms believe that
B is of type 2 with probability λ if it chooses x = x̄ and with probability 0 otherwise.
In separating equilibrium, type 1 gets P1 , and since its payoff must be at least u1 , it
must choose x = 1. Type 2 gets P2 , and since it gets payoff u2 or more, it must choose
related research level x̄ ≤ x2 . If it chooses x̄ < x1 , then type 1 prefers that contract
to its own equilibrium contract, so type 2 must choose x̄ ∈ [x1 , x2 ], getting a contract
in the thick segment at the height P2 between the U1 and U2 curves in Figure 1-S05.
These outcomes, with type 1 getting (1, P1 ) and type 2 getting (x̄, P2 ) arise in PBE if
for example the D-firms believe that B is of type 2 for sure if it chooses x̄ and believe
that it is of type 1 for sure otherwise (so that they bid P1 otherwise).
(f) Under what conditions is there a Pareto efficient pure strategy PBE outcome? Explain.
In a Pareto efficient outcome, type 1 chooses x = 1 and type 2 chooses x = 2 since
these choices maximize total surplus no matter how it is divided among the agents. In
a pooling equilibrium, the types choose the same x, so that cannot be Pareto efficient.
The only way for a separating PBE to have a Pareto efficient outcome is if x1 ≤ 2. In
that case, there is a Pareto efficient PBE in which type 2 gets (2, P2 ). In Figure 1-S05,
since 2 < x1 , there is no Pareto efficient PBE outcome.
(g) Which pure strategy PBE outcome is most plausible? Explain.
The intuitive criterion argument suggests that all pooling equilibria are implausible.
Consider a pooling PBE contract (x̄, P ) and let the corresponding indifference curve of
type 1 reach height P2 at x = x0 . Type 1 would not choose x > x0 even if it would
receive P2 , but the D-firms must believe that there is positive probability that it would.
Otherwise, on observing the choice x slightly above x0 they would believe that B is type 2
for sure and would bid P2 , offering a contract that type 2 prefers to the pooling contract.
This shows that the D-firms’ beliefs are unreasonable in a pooling eq.
The same argument implies that with reasonable beliefs, the equilibrium bid is P2
whenever x ≥ x1 . Then the best choice of x for type 2 is x = 2 if 2 ≥ x1 and is x1
otherwise. Therefore, the most plausible outcome is separating equilibrium with (1, P1 )
for type 1 and (x, P2 ) for type 2, where x = max{2, x1 }.
(h) Under what conditions is there a constrained efficient pure strategy PBE outcome in
this model? (In such an outcome it is impossible for a planner, using only the information
available to the drug manufacturers, to obtain a Pareto improvement by having some
firm(s) change strategies.) Explain.
No pooling PBE outcome is constrained efficient. To see this, consider a pooling PBE
with both types getting (x̄, P ). The planner can offer two contracts (1, P ) and (2, P ),
which make at least one type better off and the other no worse off and are feasible
since the expected payment is still P . No separating PBE outcome other than the most
plausible one in (g) can be constrained efficient. The planner can improve on all the
other outcomes by changing the contract with payment P2 to the contract taken by type
2 in the most plausible PBE.
We have shown so far that the only possible constrained efficient outcome is the most
plausible outcome described in (g). If x1 ≤ 2, then as described above, that outcome
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is Pareto efficient, hence constrained efficient. The last possible case is when x1 > 2,
as in Figure 1-S05. Whether the outcome in (g) is constrained efficient in this case
depends on the slopes of the indifference curves of the two types at the contract (x1 , P2 ).
Note that the slope of type i’s indifference curve at this contract is −πi0 (x1 ). If type 2’s
indifference curve is steep enough, then this outcome is constrained inefficient because
the planner can offer a sufficiently lower payment P 0 for x slightly below x1 . (Note that
if this contract is taken by type 1, the expected payment to B is larger than in the
original PBE, so the D-firms lose. Thus type 1 must be offered a higher payment than
P1 .) As shown in Eco 700, if 1 − λ < π20 (x1 )/π10 (x1 ), then the planner can make expected
profit by offering contracts (1, p1 ) and (x, p2 ), with p1 > P1 , x < x1 and p2 < P2 such
that type 1 prefers the first contract, type 2 prefers the second to the outcome in (g).
In that case, the PBE outcome in (g) is constrained inefficient. The outcome in (g) is
constrained efficient if and only if the reverse inequality 1 − λ ≥ π20 (x1 )/π10 (x1 ) holds.
In that case, the probability that B is of type 1 is too high, so that raising its payment
makes forces the planner, operating the D-firms, to have negative expected profit.
(i) Could firm B do better by setting the price of its patent instead of letting the drug
manufacturers bid for it? Explain.
The interaction in this case can be formalized as a game in which B, knowing its type,
chooses a price and level of x, which it offers to a D-firm. (Since D-firms are identical,
it doesn’t matter which one is offered the contract.) A pure PBE of this new game is
a contract offer for each type of firm B, a probability belief of the D-firm about the
type of firm B given any offered contract and, for each possible contract, the D-firm’s
choice whether to accept or reject it. The D-firm must accept any contract that gives
it positive expected profit and reject any contract that gives negative expected profit.
The contract offers must maximize B’s expected payoff, given the strategy of the D-firm.
The set of PBE outcomes is exactly the same as the set of PBE outcomes in the game
in the previous parts of the problem (by the same reasoning). There are multiple PBE
outcomes in both games, so it is possible that B’s expected payoff is higher in some
equilibrium with price-setting than in some equilibrium with bidding. But even when
B sets its price it cannot choose the equilibrium. If B chooses a contract that is not
optimal given the D-firm’s expectation and equilibrium strategy, B risks having its offer
rejected. B sets its price based on its expectation about what the D-firm will accept in
equilibrium. The intuitive criterion argument in part (g) also applies to the game in (i).
It implies that the only equilibrium outcome with reasonable beliefs is the separating
outcome that is most favorable to B. Since this is true in both games, when the players
have beliefs that are reasonable according to the intuitive criterion, B does no better by
setting the price than by letting the D-firms bid for the patent.
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