University of Maryland Professor Rachel Kranton

Econ 682

Suggested Answers for Theory of the Firm Problem Set

(1) See Eswaran and Kotwal’s paper.

(2) (i) The first best level of surplus:

maxIb,Is

TS = 4Ib + 3Is − 12(Ib)

2 − 12(Is)

2 − C − Ib − Is

∂TS

∂Ib= 4− Ib − 1 = 0 ⇒ I∗b = 3

∂TS

∂Is= 3− Is − 1 = 0 ⇒ I∗s = 2

TS∗ = (4) · (3) + 3 · (2)− 12(9)− 1

2(4)− c− 3− 2

= 6.5− c

(ii) p splits the surplus between the two parties. Ex post payoffs are

πE.P.b =1

2[V (Ib, Is)− c]

πE.P.s =1

2[V (Ib, Is)− c]

Anticipating this, ex ante the buyer chooses Ib to

max1

2[V (Ib, Is)− c]− Ib.

The seller chooses Is to

max1

2[V (Ib, Is)− c]− Is.

πE.A.b =1

2[V (Ib, Is)− c]− Ib

∂πE.A.b

∂Ib=

1

2[4− Ib]− 1 = 0

⇒ I∗∗b = 2

πE.A.s =1

2[V (Ib, Is)− c]− Ib

∂πE.A.b

∂Is=

1

2[3− Is]− 1 = 0

⇒ I∗∗s = 2

TS in this case is 5− c. The levels are not efficient because neither partly receives the full benefit

for their investment.

(iii) Buyer chooses p. Ex post buyer will choose p = c; seller gets zero surplus. In this case,

seller will choose ex ante Is = 0 and buyer sets Ib = I∗b = 3. Buyer gets all the benefit from its

investment.

TS = 4 · 3 + 12(9)− 3− c = 4.5− c

Seller sets price. Will set p = V ; buyer gets 0. Buyer will then invest nothing and seller sets

Is = I∗s = 2

TS = 4− c

(iv) Bargaining power (a la Grossman and Hart) can bias amount of investment. The party

with the bargaining power extracts all the surplus and thus has an incentive to invest efficiently.

(3) (i) Note that the total surplus is v + s− (c+ s) = v − c which is the same as if quality were

not an issue. By setting quality equal to s and price equal to c+ s, we allow the buyer to capture

all the surplus. Hence, she will make the efficient amount of investment. If, however, quality

is not contractible, then the commitment to a price is meaningless: the seller can always adjust

quality to determine the real price, p− s. Ex post the seller can threaten to produce a very low

quality if they do not contract on quality and give him some of the surplus in the process. If you

cannot contract on quality, and lowering quality and increasing price are good substitutes, then

it is as if price itself were not contractible. Hence the buyer knows she will only obtain 12of the

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surplus (through Nash bargaining), and so she chooses investment to maximize (v−c)x2 − x2

2 . Thus

x = (v−c)2

(ii) The efficient level of investment maximizes x(v+ s− c− s)− x2

2 . This implies x∗ = v− c.

Unconstrained bargaining (non-integration) and seller control result in the price (v+c)2 +s. Hence,

x will be chosen to maximize x(v−c)2 − x2

2 . This implies xub = xsc = v−c

2 (sc: seller control).

Buyer control allows the buyer to order the seller to make the product which results in an x that

maximizes x(v+s)+(1−x) (c+ s) /2−x2/2. This implies xbc = v− c+ 32(s+c) (For elaboration

of this answer see next page)

If c and s are small, x∗ ∼= xbc ∼= v > v2∼= xsc. Hence, buyer control is socially preferred to

seller control. If v is very close to c, then x∗ = xsc ∼= 0 < 32(s + c) = xbc. Hence, seller control

yields an outcome that is closer to the efficient investment level.

(iii) Since the sellers produce an identical good, one might assume they will set the Bertrand

price: p = c + s. In that case, the buyers max problem is the same as the social welfare max.

problem. Hence, x = x∗ = v − c. This suggests that competition can alleviate the under-

investment problem. Hence a single seller might want to license his product to another firm in

order to assure the buyer that competition will prevent “hold-ups’ by the sellers.

If c and s are small, x∗ ∼= xbc ∼= v > v2∼= xsc. Hence, buyer control is preferred to seller

control. If v is very close to c, x∗ ∼= xsc ∼= 0 < 12(s+ c) ∼= xbc. In this case, seller control yields

an outcome closer to the efficient investment level.

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