Uncertainty and Risk - DARP

Microeconomics CHAPTER 8. UNCERTAINTY AND RISK
Exercise 8.14 Every year Alf sells apples from his orchard. Although the market
price of apples remains constant (and equal to 1), the output of Alf’s orchard
is variable yielding an amount R 1 ; R 2 in good and poor years respectively; the
probability of good and poor years is known to be 1 and respectively. A
buyer, Bill, o¤ers Alf a contract for his apple crop which stipulates a down payment
(irrespective of whether the year is good or poor) and a bonus if the year
turns out to be good.
1. Assuming Alf is risk averse, use an Edgeworth box diagram to sketch the
set of such contracts which he would be prepared to accept. Assuming that
Bill is also risk averse, sketch his indi¤erence curves in the same diagram.
2. Assuming that Bill knows the shape of Alf’s acceptance set, illustrate the
optimum contract on the diagram. Write down the …rst-order conditions
for this in terms of Alf’s and Bill’s utility functions.
b
x RED
a
x BLUE
0 b
R 2
•
D
0 a a
R 1
x RED
b
x BLUE
Figure 8.2: Acceptable contracts
Outline Answer:
1. In Figure 8.2 the contours represent Alf’s indi¤erence curves: note that
they are convex to the point 0 a (risk aversion) and that they have the same
slope [1 ] = where they cross the 45 ray through 0 a (consequence of
von-Neumann utility function). Point D represents the initial endowment;
Alf’s endowment is (R 1 ; R 2 ). Alf’s indi¤erence curve through point D
represents the boundary of the set of consumptions that Alf would regard
as being at least as good as the initial endowment: the shaded area is his
acceptance set. The buyer (Bill) has an endowment K that is independent
of the state of the world –see Figure 8.3. Note that the indi¤erence curves
for Bill also have the slope [1 ] = where they cross the 45 ray through
0 b :
2. Point E in Figure ?? represents the optimum contract (from Bill’s point
of view) since it is a point of common tangency of two indi¤erence curves.
cFrank Cowell 2006 134