STOCHASTIC

CONSUMPTION UNDER UNCERTAINTY 329
is defined by the first- and second-order conditions
Ut - £/2(l + 0 = 0, Un - 2Un(l + 0 + 1/MU + 0 2 < 0. (A.2)
Let
S =def Ut(Un - 2l/12(l + 0 + f/22(l + O 2 )" 1 < 0. (A.3)
Through total differentiation of the first-order condition in (A.2), we find
dc1_dc1_ g(t/i/C/2)
1y~ ~ ~dy~l ~ ~ b(V + r) dc2 '
dc, ^SjUJU,) where 3(^/1/,) U1S - (UJUJ I/88 ,
rfy2 2c2 ' dc2 Us '
dcjdr = ((* - c0/(l + WdcJdyJ + S
In (A.5), ((^ — ^/(l + rj^dcjdyj) measures the income effect, and
S (< 0) the substitution effect, of a change in r on ^ . The absolute value
of 5 is also a measure of curvature of the indifference surfaces:
-1/S = (d'cjdc^a = (-l/t/2)(rf 2 t// where
*-*+(-a^r>*.
One then finds that (dcjdr), _j = $i — cx = (.^(.UJU^/dc^)- 1 and
rf 2 ^ I _ djdcjdr) dct _ . _ . 8Ke/i/C/J
rfr 2 \n,h 8c2 dr * Kyi Cl) dcf
This expression has the sign of d 2 (Cy C/2)/Sc2 2 , where
3c2 2 t/2 t/2 dcj " K '
The indirect utility function V(y, r) is defined by
V(y, r) = U(Cl{y, r), cJLy, 01 17, = t«l + 0), (A. 10)
480 PART V. DYNAMIC MODELS