Exam, Microeconomics A, August 1993
Unocial translation
Consider an economy with two individuals and one future period, in which two possible states may occur.
The two individuals agree that the states have probabilities 1 and 2 = 1 , 1, respectively. Individual
i will get an exogenous income y^is in state s, but may in advance obtain the possibility of consuming
yis 6= y^is by buying or selling state contingent claims in a market. Only the two individuals participate
in the trade, but we assume a competitive outcome is obtained. The individuals' budgets for buying state
contingent claims consist only of the market values of the exogenous incomes y^is .
(a) Write down the optimization problem of individual i for the general case in which the individual
has some preference ordering over the two state contingent consumption magnitudes. Derive the
conditions which characterize the behavior, and illustrate these in a diagram.
(b) Consider in particular the case with preferences determined according to von Neumann and Morgenstern's expected utility theorem, with the preference function of individual i given by
vi (y) ,e,i y ;
where i is a positive constant. Show that the optimal yi1, in the case of an interior solution, may
be written as
w^i , p2 ln 21 pp12
yi1 =
;
p +p
i
1
2
where ps is the price of a claim to one unit of income in state s, and w^i = p1y^i1 + p2y^i2 :
(c) Assume both individuals have preference functions of the given type, and dene
1 :
= 1
1
1 + 2
Show that in equilibrium the price ratio is given as
p=
p1 1 (^y2 ,y^1 )
= e
;
p2 2
where y^s is dened as y^1s + y^2s.
(d) Give an interpretation of how the price ratio in equilibrium depends on 1 =2, on y^2 , y^1 , and on
1 and 2.
(e) Assume y^1 = y^2 . Give an economic interpretation of this case. Illustrate the individual's behavior
in a diagram as in part (a), and calculate the optimal yi1 and yi2 under the assumptions of part (b).
(f) Assume that in addition to the future period discussed above, there is a present period which we call
\today." Consider two investment projects, both requiring reduced consumption today. One gives a
certain additional income x in the future period. The other gives an additional income z in state 1
and nothing in state 2. What do the equilibrium prices from part (c) tell us about
(i) the individuals' ranking of the two projects, and
(ii) the individuals' willingness to pay (today) for the two projects?