Practice Problems - Advanced Microeconomics Alessandro Di Nola A.Y. 2011-2012, Spring Term
Exercise 1 - Financial Contracts Consider the following asymmetric information problem in the …nancial market. The principal is a lender who provides a loan of size k to a borrow borrower. er. Capita Capitall costs costs Rk to the lender since it could be invested elsewhere in the economy to earn the risk-free interest rate R. The lende lenderr has thus a utility function V = t Rk. The borrower makes a pro…t U = f (k) t where f (k) is the production with k units of capital and t is the borrower’s repayment to the lender. We assume that f 0 > 0 and f 00 < 0. The parameter is a productivity shock drawn from = ; with respective probabilities 1 and .
1. Write down incentive incentive and participation participation constraints constraints directly in terms of the borrower’s borrower’s information rents U = f (k)t and U = f (k) t. 2. Write down and solve for the optimal contract assuming the lender perfectly observes the productivity of the borrower. 3. Write down the principal’s principal’s maximization maximization problem. problem. Clearly Clearly denote the constraint constraints. s. 4. Which Which constraints constraints are binding binding at the optimum? optimum? Explain. Explain. Is there a capital distortion distortion with with respect to the …rst-best outcome? Exercise 1 - Solution
1. The Incentive Incentive Compatibility Compatibility constraint constraintss for this problem (IC’s) are: f (k) t f (k) t; f (k) t f (k) t:
The Participat Participation ion Constraints Constraints (also called Individual Individual rationality rationality constraints, constraints, IR) are: f (k) t 0; f (k) t 0:
1
Using the de…nitions for information rents (and letting > 0), they can be rewritten as: U U f (k);
(ICL )
U U + f (k);
(ICH )
U 0;
(PCL )
U 0:
(PCH )
2. Assuming that the lender can observe the borrower’s productivity we can can …nd the …rst-best menu of contracts by solving: max V = t Rk t;k
s.t. f (k ) t 0:
(1)
Clearly the constraint is binding at the solution, hence the above problem is equivalent to: max V = f (k ) Rk ft;kg
The …rst-order condition with respect to k delivers: f 0 (k) = R:
n Therefore the menu of contracts o¤ered by the principal under full information is (
t ; k ) ; (t ; k
such that f 0 (k ) = R and f 0 (k ) = R and the optimal transfers t and t can be derived from (1) with equality. Note furthermore that k < k since f 0 (k ) =f 0 (k ) and f is concave. It is easy to see that the FB contract we just derived is not incentive compatible , since it violates the IC constraint for -agent. In fact
f (k ) t = f (k ) f (k ) = f (k ) > 0 = f (k ) t
meaning that the high-productivity borrower will not choose (t ; k ) but (t ; k ) if o¤ered the FB contract. 3. Under second-best (assuming asymmetric information), the principal is maximizing
f (k) Rk + (1 ) (f (k) Rk) U + (1 )U max f(U ;k);(U ;k)g s.t. constraints (ICL ) to (PCH ).
(2)
Note that the optimization variables are now (U ; k) ; U ; k (we substituted out transfers by using the de…nitions of information rents). A …rst approach to solve (2) could be to apply the Lagrangian techniques but there is a more practical route, that calls for the simpli…cation of the number of relevant constraints (see Prof. Pavoni’s slides). First we claim that the 2
o)
-agent participation constraint (PCH ) is always strictly satis…ed (hence it is redundant). In
fact, considering (ICH ) we have: U U + f (k ) f (k) > 0
where the second inequality holds since U 0 by (PCL ) and the last inequality holds since f (k ) > 0 as long as k > 0. Hence U > 0: Then we state and prove the following: . Claim: Both (ICH ) and (PCL ) must be binding at the solution. . Proof : Suppose not. Then either (PCL ) or (ICH ) is not binding (or both). Consider the case (PCL ) is not binding, i.e. U > 0. Then we can …nd another contract, call it B = (U "; k ) ; U "; k , " small enogh, that still satis…es the constraints. But under contract B the principal’s pro…t increases by " > 0. This contradicts the assumption that (U ; k) ; U ; k was the optimal contract. Hence U = 0. Consider instead case (ICH ) is not binding. Then U > U + f (k ) = f (k )
since we showed U = 0. But if U > f (k), then the Principal can decrease U by " > 0, for " small enough, and gain v" > 0. This again contradicts the assumption that (U ; k) ; U ; k was the optimal contract.
4. After simplifying the number of relevant constraints we can restate problem (2) as follows:
max f (k) Rk + (1 ) (f (k) Rk) U + (1 )U f(U ;k);(U ;k)g
(3)
s.t. U = f (k);
(3a)
U = 0;
(3b)
U U f (k):
(ICL )
Now we guess that (ICL ) is slack at the optimum (we will verify that the solution obtained by neglecting this constraint indeed satis…es ICL ). Thus we are left with only two remaining constraints, (3a) and (3b). Substituting (3a) and (3b) into (3) we obtain a reduced program with k; k as the only choice variables:
max f (k ) Rk + (1 ) (f (k) Rk) vf (k) fk;k g
The …rst-order condition with respect to k is: f 0 (k) = R:
which is equal to the FB condition: we therefore showed that there is no distortion at the top : SB k = k . The …rst-order condition with respect to k, however, reads as:
v
1v
f 0 (k SB ) = R: 3
Recalling that the FB output choice for the less productive borrower is k s.t. f 0 (k ) = R we have
v
1v
f 0 (kSB ) = f 0 (k )
(4)
from which1 it follows f 0 (kSB ) > f 0 (k ), hence kSB
> U f (k )
0 > U f (k ) But U = f (k) from (3a) hence 0> Inequality (5) is equivalent to kSB < k
h f k
SB
SB
f (k
SB
i)
(5)
since f is strictly increasing. But this is true since
k SB < k < k = k
SB
where the …rst inequality follows from (4), the second from our characterization of FB contract (see point 2), and the last equality comes from the fact that the SB contract calls for no distortion for the e¢cient type. 1. We can summarize our results for the second-best (SB) contract in the following proposition: Proposition. (a) At the optimum, only the incentive constraint for the -agent and the participation constraint for the -agent are binding. SB
(b) There is no capital distortion with respect to FB for the -agent, i.e. k = k , where f 0 (k ) = R (hence for the high-type the return on capital is equalized to the risk-free rate. (c) There is a downward distortion in loan given to the -agent: k SB < k ,where in particular
v
1v
f 0 (k SB ) = f 0 (k ):
Exercise 2 - My Little Red Corvette Everybody likes red Corvettes (testimony to this is the famous song by Prince "My Little Red Corvette"). Assume that there exist two types of potential Corvette purchasers. On one hand, ’snobs’ are willing to pay 25000 for a red Corvette but only 20000 for a Corvette of another colour. On the other hand, ’less snobbish’ purchasers are willing to pay 22000 for a red Corvette and 20000 if it is any other colour. The percentage of ’snob’ purchasers is , so that (1 )% of purchasers 1
Note that
v 1v
< for any v 2 (0; 1) :
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are ’less snobbish’. The production cost of a Corvette is independent of its colour. Assume the car seller does not know the type of the buyer (or, even if he knew it, the law would prevent him from price discrimination). Why are Corvettes of all colours seen on the streets? What parameters does your answer depend on? Exercise 2 - Solution It is clear that there are only two possibly optimal contracts the seller is willing to o¤er. The …rst is a separating contract: sell red corvettes at price 25000 and corvettes of other colour at price 20000. Snob consumers will then buy only red corvettes making zero net utility and less snobbish consumers will buy only other colour corvettes (in fact the price of a red corvett is higher than their willingness to pay for it). The expected pro…t for the seller under this separating contract is: = 25000 + (1 )20000: The other possible contract is a pooling contract asking a price of 22000 for a car of either colour. Clearly the snob consumers will purchase only red corvettes, earning a positive payo¤ equal to 25000 22000 = 3000. Less snobbish consumers will buy only red corvettes as well, earning a zero payo¤. Under this pooling contract the expected pro…t of the seller is 22000: Hence red corvettes are always sold and the car seller o¤ers the separating contract (charging a higher price for red corvettes) if and only if the parameter is such that 25000 + (1 )20000 22000;
i.e.
2 5
(i.e. if and only if the percentage of snob consumers is high enough ).
Exercise 3 - Price Discrimination I A company o¤ers the only ‡ight service between two cities. The cost per passenger is 400. Assume that the airline’s potential clients can be divided into two groups: executives who travel for business reasons, and tourists who travel for vacations. For this particular service, if the meeting that an executive wants to attend takes place, then he is willing to pay a price of 1000. On the other hand, tourists are willing to pay 600 for the same trip. 1. What price will the airline charge if it can perfectly observe whether an individual is an executive or a tourist? What are the pro…ts of the airline in this case? Now assume that the airline cannot distinguish between tourists and executives. 2. What price (or prices) will the company charge for a ticket? What are the pro…ts of the airline now? Assume that tourists are perfectly informed as to when they have their vacations, and the probability that they must cancel their trip is zero. On the other hand executives believe that there is a 50% chance that their meeting will be cancelled or will be changed to another date. Assume now that the company can …x not only the price of the ticket but also the refund conditions when a ticket is cancelled. 3. What contracts (or tickets) will the airline o¤er? Discuss.
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Exercise 3 - Solution 1. If the company can observe the buyer’s type (full information case) it will set the price of a ticket for an executive at 1000 and the price of a ticket for a tourist at 600, thus extracting all the surplus. Let denote the percentage of travelers who are executives. Then the pro…ts of the airline are 1000 + 600(1 ) 400 > 200. 2. Under asymmetric information the company cannot observe the traveller’s type. If she proposes the …rst best contract, the executives clearly will not choose the ticket designed for them but will buy the cheaper ticket for tourists, earning a net utility equal to 1000 600 = 400. In other words, the menu of contracts we derived in (1) is not incentive-compatible here. The only feasible contract is a pooling contract asking a price of 600 to both types of travellers. The pro…ts of the airline are now equal to 600 400 = 200 (strictly less than the …rst best). 3. Now the airline is able to o¤er a contract that leads to self-selection. Consider for example the following menu of contracts: (a) tourist-tari¤ at price 600 with no refund and (b) businesstari¤ at price 1000 with a full refund clause in case the ‡ight is cancelled by the traveller. Clearly such contract is incentive compatible: if the executive buys (b) he gets zero in expected utility but if he pretends to be a tourist and buys (a) his expected utility is 12 (1000 600) + 1 2 (0 600) = 100. Remark - This exercise illustrates a typical feature of adverse selection problems: under asymmetric information, if the principal can set only the price then only pooling contracts are incentive-compatible. Instead, if he is allowed to choose also other variables (such as the refund clause in the above exercise) he is able to discriminate among di¤erent types of customers. Exercise 4 - Price Discrimination II Air Shangri-la is the only airline allowed to ‡y between the islands of Shangri-la and Nirvana. There are two types of passengers, tourist and business. Business travelers are willing to pay more than tourists. The airline, however, cannot tell directly whether a ticket purchaser is a tourist or a business traveler. The two types do di¤er, however, in how much they are willing to pay to avoid having to purchase their tickets in advance. More speci…cally, the utility levels of each of the two types net of the price of the ticket, P , for any given amount of time W prior to the ‡ight that the ticket is purchased are given by Business :
v B P W;
Tourist :
v T P W;
where 0 < B < T . (Note that for any given level of W ), the business traveler is willing to pay more for his ticket. Also, the business traveler is willing to pay more for any given reduction in W:). The proportion of travelers who are tourists is . Assume that the cost of transporting a passenger is c. Assume in (1) to (4) that Air Shangri-la wants to carry both types of passengers. 6
1. Draw the indi¤erence curves of the two types in (P; W )-space. Draw the airline’s isopro…t curves. Now formulate the optimal (pro…t-maximizing) price discrimination problem mathematically that Air Shangri-la would want to solve. (Hint : Impose nonnegativity of prices as a constraint since, if it charged a negative price, it would sell an in…nite number of tickets at this price). 2. Show that in the optimal solution, tourists are indi¤erent between buying a ticket and not going at all. 3. Show that in the optimal solution, business travelers never buy their ticket prior to the ‡ight and are just indi¤erent between doing this and buying when tourists buy. 4. Describe fully the optimal price discrimination scheme under the assumption that they sell to both types. How does it depend on the underlying parameters ; B ; T ; and c? 5. Under what circumstances will Air Shangri-la choose to serve only business travelers? Exercise 4 - Solution (1) We represent the indi¤erence curves for the two types of travelers and the isopro…t curve for the airline in the picture below:
From …gure above we note that the indi¤erence curve for business travelers is ‡atter since business travelers are willing to pay more for a given reduction in W . Furthermore, the pro…t function of the airline depends only on price P and production cost c, as stated in the text, and hence it is a vertical line in the (P; W ) space. 7
The optimal (second-best) price discrimination problem that Air Shangri-la would want to solve is max
fP T ;W T ;P B ;W B g
P T + (1 )P B c
(1)
s.t. T P T + W T
T P B + W B ;
(IC1)
B P B + W B
B P T + W T ;
(IC2)
T P T + W T
v;
(P1)
B P B + W B
v;
(P2)
P T ; W T ; P B ; W B
0:
(2)
(2) We are asked to show that in the optimal solution tourists are indi¤erent between buying a ticket or not. This is equivalent to say that the participation constraint (P1) for the T -agent (here, the low type) is binding at the optimum. First, we show that (P2) is redundant. In fact, (IC2) together with (P1) imply: B P B + W B B P T + W T < T P T + W T v;
hence B P B + W B < v . Since (P2) is slack, it can be safely neglected. Now we are ready to show that (P1) is binding. If it is not the case, i.e. if T P T + W T < v
then the Principal (hereafter, P) could increase P T and P B by " > 0 and all the relevant constraints would still be satis…ed, provided " is small enough. But this modi…cation would earn the P higher pro…t, a contradiction. (3) We are asked to show that W B = 0 and that (IC2) is binding. Assume that f(P T ; W T ); (P B ; W B )g is an optimal, incentive compatible contract, and assume in negation that W B > 0. Now we can reduce W B by " > 0 and increase P B by " so that the B type utility does not change, and the …rm earns higher pro…ts from the B type. We need to check that the T type will not choose this new compensation package. Indeed , B
T P T + W T T P B + W B = T
" P B + T
+ (W B ") < T
" P B + B
+ (W B "):
It follows that f(P T ; W T ); (P B ; W B )g cannot be an optimal, incentive compatible contract. Therefore, we must have W B = 0. It remains to show that the (IC2) constraint is binding. If, in an optimal contract, the business travelers were not indi¤erent between (P T ; W T ) and (P B ; W B ); i.e. if B P B + W B < B P T + W T ; we could slightly raise P B and all the constraints would remain satis…ed (recall that (P2) is slack), and the …rm would earn higher pro…ts from the B types. Therefore in an optimal contract we must have the business types indi¤erent between (P T ; W T ) and (P B ; W B ). 8
(4) Using our results from the previous points, we are allowed to restate the …rm’s price discrimination program as: max (3) P T + (1 )P B c fP T ;W T ;P B g
s.t. T P T + W T B P B P T ; W T ; P B
= v;
(3a)
= B P T + W T ;
(3b)
0:
By substituting (3a) and (3b) into (3) we can leave W T as the only choice variable: max
fW T g
v W T T
+ (1 )
v W T W T + T B
c
s.t. 0 W T v2 : Note that the objective function is linear in W T so we have to deal with corner solutions. We need to distinguish two cases: (a) If 1 1 > 0 then the …rm’s pro…t is strictly increasing in W T over the interval [0; v ] so that B
T
W T = v . It follows that the optimal scheme will be f(P T ; W T ); (P B ; W B )g = only B types will be served.
n(0
o
; v); ( v ; 0) : B
(b) If instead 1 1 < 0 then the …rm’s pro…t is strictly decreasing in W T over the interval [0; v ] so that W T = 0. It follows that the optimal scheme will be f(P T ; W T ); (P B ; W B )g = B
T
n 0 0o: both types will be served for the same price ( v ; T
;
v ; T
pooling contract ).
Since the direction of the inequality in the condition above determines the type of scheme, it is easy to see how changes in ; T and B will a¤ect the optimal scheme. In particular, if the proportion of B types is large enough ( small enough) the …rm will choose to serve only the B types (i.e. case (a) applies). If the B types su¤er less from prices (B is smaller) then the …rm is more likely to serve only them (i.e. case (a) applies again). If the T types su¤er less from prices (T is smaller) then the …rm is more likely to serve them as well as the B types (i.e. case (b) applies). Changes in the cost parameter c are discussed in point (5) below. (5) As long as c < v , and we are in case (a) described in point (4) above, the …rm will decide to serve only the business types. If, however, we are in case (b) above, and v < c < v , then the scheme described above cannot be optimal because the …rm is losing money. In such a case the …rm will choose the scheme described in case (a) above, and serve only the business types. If c > v the …rm will choose not to operate at all. B
T
B
2
The second constraint on W T comes from the fect that P T = (v W T ) =T and P T 0.
9
B
Exercise 5 - Regulation revisited The regulator is concerned with costumer welfare, therefore he wants to force a natural monopoly to charge competitive prices. The problem is that the regulator does not know as much about the …rm’s cost structure as he would like to. More precisely, we will consider a natural monopoly with an exogenous cost parameter which can take on two values: L and H , where L < H . The …rm’s cost of producing the good is c = e; where e stands for e¤ort. Exerting e¤ort has cost (e) =
e2
2
;
which is increasing and convex in e. We will assume that the regulator wants the good to be produced for the lowest possible payment P = s + c, which consists of two parts: the subsidy s and the accounting cost c. The payo¤ of the …rm is P c (e):
1. Assume that regulator has perfect information: he can observe and c. Write down the optimization problem for the regulator and solve it.
From now on we consider the incomplete information case. The regulator can observe c and can not observe , but he believes it is H with probability (1 ) and L with probability . 2. Write down the regulator’s minimization problem. 3. Show that the IR (participation) constraint for type L is redundant. 4. Argue that the IR constraint for type H is binding at the optimum. 5. Argue that the IC constraint for type L is binding at the optimum. 6. Show that IC constraints imply cH cL . 7. Argue that the IC constraint for the high type is redundant at the optimum. 8. Now we get to one of our favorite after-class activities. Compute the regulator’s minimization problem.
Exercise 5 - Solution (1) Recall the regulator’s problem: min s + e
fs;eg
10
s.t. s
e2
0: (IR) 2 Since the regulator’s payment to the …rm (which he is minimizing) is strictly increasing in the subsidy s, for each e he will decrease the subsidy as much as possible. Therefore at the optimum the …rm’s participation constraint (or IR, individual rationality) is binding and we have s = e2 . Plugging this equation back into the original minimization problem we obtain a new minimization problem: 2
min feg
e2
2
+e
Being this function strictly convex, the solution is obtained by computing the …rst-order condition: e = 1:
(2) The regulator is minimizing the expected payment subject to the …rm’s IC and IR constraints: (sL + cL ) + (1 )(sH + cH ) min fsL ;cL ;sH ;cH g
s.t. (H cH )2 sH 2 (L cL )2 sL 2
(H cL )2 sL ; 2 (L cH )2 sH ; 2
(H cH )2 sH 2 (L cL )2 sL 2
(ICH ) (ICL )
0;
(IRH )
0:
(IRL )
Notice that the e¤ort e in IC and IR constraints was replaced by c. (3) We are asked to show that the IR (participation) constraint for type L is redundant. Note that sL
(L cL )2 2
(L cH )2 ; 2 (H cH )2 > sH ; 2 0; sH
where the …rst inequality is IC for the type L, the second follows from the fact that H > L and the third is IR for type H . (4) We prove the claim by contradiction. Suppose at some optimal contract, which satis…es all four constraints, (IRH ) was not binding. Then one could decrease both sL and sH by the same amount, still have all the four constraints satis…ed, and the regulator’s payment would be lower. Therefore the starting contract is not optimal, a contradiction. 11
(5) Again by contradiction. Suppose we started with a contract for which all the constraints were satis…ed and the IC for L was not binding. Then we could decrease sL by a little bit and the IC for L would still not be binding. All the other constraints would be satis…ed after the decrease (make sure you understand why). Since sL is lower in the new contract the regulator pays less. Therefore the starting contract is not optimal, a contradiction. (6) Substracting the two IC constraints yields: (L cH )2 (H cH )2 (L cL )2 (H cL )2 ; which after some boring algebra yields cH (H L ) cL (H L ):
After canceling out some terms one obtains cH cL . (7) We are asked to argue that the IC constraint for the high type is redundant at the optimum. From (5) we know sL ( 2c ) = sH ( 2c ) . After rearranging it one obtains sH = sL L
(L cL )2 2
sH
+
(L cH )2 . 2
(H cH )2 2
L
2
L
H
2
Now, plugging this into sH
(H cH )2 2
one has
(L cL )2 (L cH )2 (H cH )2 + 2 2 2 2 2 (H cL ) (H cL ) (L cL )2 (L cH )2 (H cH )2 = sL + + 2 2 2 2 2 2 (H cL ) sL : 2 = sL
The inequality follows, after some rearranging, from the fact that cH cL . (8) Using the fact that high type’s IR constraint and low type’s IC constraints are binding and the other two irrelevant, we can rewrite the original problem as follows: min [(H cH ) (L cH ) + (L cL ) + cL ] + (1 ) [(H cH ) + cH ]
cL ;cH
Taking the …rst-order conditions with respect to cL and cH , yields: L csb L = 1;
1 (H csb (L csb H ) H ) = 1: 1 1 The two equations can be rewritten as: esb L
= L csb L = 1;
sb esb H = H cH = 1
(H L ): 1
Exercise 6: Car Insurance Recently, a friend of ours mentioned that he had gone to insure his car and he had been o¤ered several di¤erent policies. He could choose between an expensive contract with full insurance, or a contract with voluntary excess of $ 900 (a voluntary excess clause means that the company will 12
pay all losses from accidents over and above the …rst $900 but not below). The policy with excess was signi…cantly cheaper. Our friend argued the insurance companies o¤er contracts with excess clause since that way they make the drivers who su¤er many accidents pay each time $ 900. Is this a reasonable argument? Whether your answer is a¢rmative or not, you should argue in terms of an adverse selection situation (the insurance company cannot observe whether or not the driver is reckless). Exercise 6 - Solution The argument is not very resonable. If insurance companies could determine only the price per unit of coverage, the only incentive compatible contract would be a pooling one (assuming that the companies cannot distinguish "reckless" drivers from "safe" ones). But under a pooling contract it is possible that the insurance companies always make losses or that in equilibrium only the reckless drivers buy the insurance (remember Akerlof’s example with "lemons"), leading to a market failure. Therefore it is more e¢cient to design more complex contracts (like the one proposed in the text) that leave greater leeway to insurance companies: by o¤ering di¤erent price/coverage packages between which the clients can freely choose, there is hope that the bad-risk guys will self-select themselves. We then have a separating equilibrium in which safe drivers will buy the cheaper policy with voluntary excess and "reckless" drivers will buy the more expensive one without voluntary excess.
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