STOCHASTIC

COMPUTATIONAL AND REVIEW EXERCISES
1. In the Dreze-Modligliani article it is convenient to assume that both present and future
consumption are not inferior goods or that the marginal propensity to consume is positive
but less than unity.
(a) Show that this condition requires that
(t/i/l/2) V22 -Ul2 0 is the borrowing-lending rate of interest, U is the utility function over
consumption C\, C2, and subscripts refer to partial differentiation.
(b) Show that the conditions are satisfied when £/(•) = log Ci + log C2 but not when
£/(•) = log(d + C2). What happens when £/(•) = log(aC\ +0C2) for aj > 0?
(c) Show that the conditions are satisfied when [/(•) = d'C 1 ^" for 0 < a < 1.
2. Consider an investor with $1000 to invest over two investment periods between a risky
and a risk-free asset. The risk-free asset returns 5% per period. In period 1 the risky asset
returns 0, 5, or 10% with equal probabilities. The returns in period 2 for the risky asset are
P,[»'2 = 3|r1=0] = i) pr[r2 = 5\rl=0] = i, p,\r2 = 3 |r, = 5] = ±,
/,r[r2 = 6|r1 = 5]=i> pr[r2 = 5|r1 = 10] = i, pr[r2 = S\rl = 10] = i,
/*[r2 = 12|r, = 10] = i.
Suppose that the goal is to maximize expected wealth at the end of period 2.
(a) Formulate the two-period decision problem.
(b) Show that the problem in (a) is equivalent to a linear program.
(c) Formulate and interpret the dual linear program. (Recall that {min b'y \ A'y g c, y S 0}
is the dual to {max c'x \ Ax g b, x =; 0} where c, x e £", b, ye E m , and A is an m x n
matrix.)
(d) Solve the linear program.
(e) For what risk-free return rates is it optimal to invest entirely in either the risky or
risk-free assets in both periods.
3. Consider an investor who wishes to allocate his funds between assets A and B. Asset A
has net returns of 0.01 and 0.04 over one- and two-month horizons, respectively, while
asset B returns 0.02 per month over either a one- or two-month horizon.
(a) Assume that there are no transaction costs, that A and B are perfectly liquid assets,
and that the investor wishes to maximize his wealth at the end of the second month.
Show that it is optimal to hold B in the first month and to sell B and buy A. What is the
two-month net rate of return?
(b) Suppose that the cost of switching from B to A at the end of the first month is 1.5%
of the value of the portfolio. Show that the investor will hold A during both months.
Suppose the investor is only concerned with his wealth at the end of the first month.
Show that he will hold B. When will the investor hold a mixture of A and 2??
Suppose asset A is illiquid. Over two months one dollar invested in A will increase in
value to (1 + ri) 2 . But A cannot be sold until the end of the second month. However, each
dollar invested in the liquid asset B can be realized at the end of the first month for 1 + r2
or at the end of the second month for (1 +r2) 2 . The individual expects to receive yi and y2
from exogenous sources at the end of each month. His problem is to decide on the fraction x
of his initial wealth that is invested in A to maximize his utility U(CU C2) over consumption
in the two months.
(c) What happens if r2 g ri ?
COMPUTATIONAL AND REVIEW EXERCISES 663