Theory Field Exam
August
Consider a risk-neutral seller and a risk-neutral buyer. The seller owns an asset
(an invention, small firm, etc.) in which she can invest. Let I∈[0,∞) denote
her investment. After investing, an opportunity arises in which the seller can
sell the asset to a buyer. To motivate trade, assume the buyer, if he acquires
the asset, can take a subsequent action, b∈[0,¯
b], that affects the asset’s return
(to him at least). Finally, the asset yields a return, r, to its then owner, where
the return depends on investments made in it. Note the realization of roccurs
after the point at which exchange can occur. The payoffs to the buyer and
seller—ignoring transfers—are, respectively,
UB=0,if no exchange
r−b , if exchange and US=r−I , if no exchange
−I , if exchange ,
where “exchange” means ownership of the asset was passed to the buyer. Not
surprisingly, the buyer takes no action if there isn’t exchange.
Given investment Iand action b, the return rhas an expected value R(I, b).
Assume the expected return function, R:R+×[0,¯
b]→R, has the following
properties:
•For all b∈[0,¯
b], R(·, b) : R+→Ris a twice continuously differentiable,
strictly increasing, and strictly concave function;
•For any b∈[0,¯
b], there exists an ¯
I(b)<∞such that ∂R(I , b)/∂I < 1 if
I > ¯
I(b);
•∂R(0,0)/∂ I > 1; and
•For any I∈R+, argmaxb∈[0,¯
b]R(I, b)−bexists and, if I > 0, it is a subset
of (0,¯
b].
Define V(I) = maxb∈B R(I, b)−b. Finally, assume:
•There exists an I∗>0 such that I∗= argmaxIV(I)−I.
The expected return to the seller if she retains ownership is R(I , 0).
The timing of the game is first the seller decides her investment. At this time,
no contract exists between buyer and seller. Then there is possible exchange.
Then, if he took possession, the buyer decides on his investment. Finally ris
realized. Assume that the buyer never observes the seller’s investment nor any
signal of it prior to exchange. Assume the seller never observes the buyer’s
investment (if any). Assume only the owner of the asset at the time ris realized
observes r.
(a) Prove that, if I > 0, then efficiency dictates that exchange take place (i.e.,
the buyer end up with the asset).
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