STOCHASTIC

Consider first the effects of uncertainty regarding the consumer's future income. His
first-period budget constraint is Yt = d + Si where Yt is his (certain) income in period 1
and Si is savings. Future consumption is C2 = Y2 + Si (1 + r), where r is the rate of interest,
assumed to be known in the case of pure income risk, and Y2 is the uncertain income in
period 2.
(e) Show that expected utility is
EU = ju{Cu Y2 + (K, - dXl +/•)} dF(Y2).
(f) Show that the first- and second-order conditions for an optimal choice of Ct are
E[U1-(l+r)U2] = Q and D = E[Utl - 2(1+r) Ul2 + (l + r) 2 C/22] < 0,
respectively.
(g) Verify that D < 0.
(h) Show that the effect of an increase in present income is
dCJBYi = -{\+r)E[Ul2-(\+r)U22\ID.
(i) Assume that the expression in (h) is always positive. Show that this means that both
present and future consumption are not inferior goods.
Write future income as yY2 + 6, where y and 6 are multiplicative and additive shift factors,
respectively.
(j) Interpret the meaning of the two types of shift parameters.
Since y2 2: 0 a multiplicative shift around zero will increase the mean. Hence to maintain
a constant expected value, the additive shift must be negative. Taking the differential, the
requirement is that dE(yY2 + 6) = E(Y2 dy + d0) = 0 or that dO/dy =-E[Y2] = -ij>.
(k) Show that
(tL=_. = -(^) £c(f/i2 - (i+r)[/ " )(r2 ^ )] -
(1) Show that decreasing temporal risk aversion is sufficient for the expression in (k)
to be < 0. Hence increased uncertainty about future income increases savings.
Consider the case of capital risk. Suppose that in the first period the consumer can allocate
his resources (Ki) between present consumption (Ci) and capital investment K. Capital
investment is transformed into resources available for future consumption by the function
f(K, £), where £ is random. Consider the simple form C2 = K£, where £ g 0.
(m) Show that expected utility is £[/= J (/{C.^-CiK} dF(®.
(n) Show that necessary and sufficient conditions for a maximum of EU are
E[U1 -SUA = 0 and H = E[Uli-2£Ui2 + £ 2 U12'\ < 0.
Let the yield on capital be y(£—1)+0. For a multiplicative shift around zero to keep
the mean constant we must have dE[y(l;-1) + 0] = 0 or dd/dy = -£[{- 1] = -fi.
(o) Show that
(8CJ8y)lele,,= _, = -{\IH)KE{{Ui2-£U22)(£-n)} + (\IH)E[U2^-n)l
[Hint: Differentiate the first-order conditions with respect to y and evaluate the derivative
at (y,6) — (1,0).] We refer to the first term as the income effect [because of its similarity
to the expression in (k)] and the second term as the substitution effect.
(p) Show that the existence of risk aversion, i.e., U is concave, is a necessary and
sufficient condition for the substitution effect to be positive. Show that the additional
assumption of decreasing temporal risk aversion is sufficient for the income effect to be
negative. Hence the total effect is ambiguous without further assumptions.
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PART V DYNAMIC MODELS