Department

EconomicsCourse Code

ECO370Y5Professor

babarStudy Guide

FinalThis

**preview**shows pages 1-2. to view the full**8 pages of the document.**Harvard University

Economics 2060 –Contract Theory

Problem Set 3 –Solutions

Please email gegorov@fas.harvard.edu with any questions, comments, or suggestions.

1. Multi-Task and Incomplete Contracts

Consider the relationship between a shipper Sand a trucker T.Shires Tto carry a

load from Boston to Chicago. When Tarrives in Chicago, Swill have another load (the

“back-haul”) whose characteristics cannot be speci…ed in a contract in advance. The value

of this load is v. While driving from Boston to Chicago, Tcan engage in two activities:

she can search for an alternative back-haul, and she can look after the truck. Alternative

back-hauls have value v2with probability pand v1with probability 1p, where v2> v >

v1, and v2v1<1. The probability pis a choice variable for the trucker and the e¤ort

cost of pis p2=2. The value of the truck, A, is also a choice variable for the trucker and the

e¤ort cost of Ais A2=2. There is symmetric information throughout, but Ais not veri…able

and contracts on the back-haul cannot be written until the trucker reaches Chicago. Both

parties are risk neutral and there is no discounting.

1. Solve for the …rst-best levels of pand A.

The aggregate surplus is:

S=pv2+ (1 p)v+Ap2

2A2

2

Therefore the …rst-best problem is:

max

p;A pv2+ (1 p)v+Ap2

2A2

2

This yields the following …rst-order conditions:

p=v2v

A= 1=

2. Assume that the trucker owns the truck. Take this to mean that: (i) In the absence

of renegotiation, the trucker has the right to take the alternative back-haul, and the

shipper’s payo¤ is zero (since there are no other truckers to carry her load); (ii) the

trucker receives the full value of the truck, A(she is the residual claimant). Suppose,

however, that renegotiation takes place if the alternative back-haul is ine¢ cient, and

that this proceeds according to Nash bargaining with a 50 : 50 split. Solve for the

second-best values of pand A.

1

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The Trucker’s payo¤ under Trucker ownership is:

uT

T=pv2+ (1 p)v1+vv1

2+Ap2

2A2

2

where the term v1+vv1

2comes from the fact that if an ine¢ cient backhaul is found the

parties renegotiate and Treceives her threat point (v1) plus half the gains from trade (vv1).

Therefore the second-best problem is:

max

p;A pv2+ (1 p)v1+vv1

2+Ap2

2A2

2

which yields the following …rst-order conditions:

v2v1vv1

2p= 0 (1)

1 = A (2)

These imply:

ASB

T= 1= A

and:

pSB

T=1

2(2v2v1v)> p

3. Assume now that the shipper owns the truck. Take this to mean that: (i) In the

absence of renegotiation, the trucker does not have the right to take the alternative

back-haul, and so receives zero, and the shipper also receives zero (since she needs

the trucker to carry his load); (ii) the shipper receives the full value A(she is the

residual claimant). Suppose again that renegotiation takes place as above. Solve for

the second-best values of pand A.

The Trucker’s payo¤ under Shipper ownership is:

uT

S=pv2

2+ (1 p)v

2p2

2A2

2

Therefore the second-best problem is:

max

p;A pv2

2+ (1 p)v

2p2

2A2

2

which yields the following …rst-order conditions:

v2

2v

2p= 0 (3)

and:

A = 0 (4)

These imply:

ASB

S= 0 < A

and:

pSB

S=1

2(v2v)< p

2

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4. Provide conditions under which trucker ownership is optimal, and conditions under

which that shipper ownership is optimal. Discuss the trade-o¤ between shipper and

trucker ownership.

Under Trucker ownership total surplus is:

ST=pSB

Tv2+ (1 pSB

T)v+ASB

TpSB

T2

2ASB

T2

2

=1

2(2v2v1v)v2+11

2(2v2v1v)v

+1= 1

2(2v2v1v)2

2(1=)2

2

=v(8 + 3v) + 4

+ 2vv1v2

1+ 4v2(v22v)

8(5)

Under Shipper ownership total surplus is:

SS=pSB

Sv2+ (1 pSB

S)v+ASB

SpSB

S2

2ASB

S2

2

=1

2(v2v1)v2+11

2(v2v1)v1

2(v2v1)2

2(6)

=v+3(v2v)2

8(7)

Trucker ownership is optimal if and only if STSS)

v(8 + 3v) + 4

+ 2vv1v2

1+ 4v2(v22v)

8v+3(v2v)2

8

which simpli…es to: 1

21

8v2

1v2

2+1

4v(v2v1)

This implies that Trucker ownership will be optimal for su¢ ciently small. Trucker owner-

ship provides optimal incentives for taking care of the truck, but poor incentives for search.

Shipper ownership provides poor incentives for care of the truck, and too much incentive for

search.

2. Holdup Problem

There are two contracting parties, a prospective buyer (B) and prospective seller (S). The

seller’s cost of production of the good is c2 fcL; cHg,P(c=cL) = i, and the buyer’s

valuation of the good is v2 fvL; vHg,P(v=vH) = j, where iand jare seller’s and

buyer’s investments; their costs are (i)and (j), respectively (convex, satisfying (0) = 0;

0(0) = 0,(1) = 1, same for ). Assume vH> cH> vL> cL.

The timing is as follows: …rst, the parties sign a contract, then they undertake investments

iand j, then cand vare realized, and then the contract is ful…lled.

3

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