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Final

ECO370Y5 Study Guide - Final Guide: Nash Equilibrium, The Seller, Risk Neutral


Department
Economics
Course Code
ECO370Y5
Professor
babar
Study Guide
Final

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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 specied 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
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
2A2
2
Therefore the …rst-best problem is:
max
p;A pv2+ (1 p)v+Ap2
2A2
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 pay 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
2A2
2
where the term v1+vv1
2comes from the fact that if an in 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
2A2
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
2A2
2
Therefore the second-best problem is:
max
p;A pv2
2+ (1 p)v
2p2
2A2
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
sellers 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 sellers 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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