STOCHASTIC

382 MERTON
where the notation for partial derivatives is Jw = dJjdW, Jt HS dj/dt,
Uc = dU/8C, Jt = 87/ePj, /„• = a 2 J/8Pi dP,, and Jiw = 8*J/dPt 8W.
Because Lcc = $cc = Ucc < °. L cwk = 4>cwk = 0, L„k„t = ak i W i Jww,
Lw „. = 0, k =£ j, a sufficient condition for a unique interior maximum
is that Jww < 0 (i.e., that J be strictly concave in W). That assumed, as
an immediate consequence of differentiating (19) totally with respect to
W, we have
dC*
w > 0. (22)
To solve explicitly for C* and w*, we solve the n + 2 nondynamic
implicit equations, (19)—(21), for C*, and w*, and A as functions of Jw ,
Jww, Jjw, W, P, and t. Then, C* and w* are substituted in (18) which
now becomes a second-order partial differential equation for J, subject
to the boundary condition J(W, P, T) = B{W, T). Having (in principle
at least) solved this equation for J, we then substitute back into (19)-(21)
to derive the optimal rules as functions of W, P, and t. Define the inverse
function G = [Uc]- 1 . Then, from (19),
C* = G(JW , t). (23)
To solve for the w{*, note that (20) is a linear system in wt* and hence
can be solved explicitly. Define
Q ~ [oru], the n X n variance-covariance matrix,
[»„] = A" 1 , 15 (24)
1 1
Eliminating A from (20), the solution for wk* can be written as
wk* = hk(P, t) + m(P, W, t) gk{P, t) + fk(P, W, t), k = \,..., n, (25)
where T.i K = 1, •£igk = 0, and tlf* = 0. 16
16 Q- 1 exists by the assumption on Q in footnote 12.
18 hk(P, r) = £ vkijr; m(P, Wt t) = -JwjWJww ;
i
**(/•, 0 = -^ E v*, (re, - £ £ v0oJ ;/t(P, *r, o
i V i i /
= ^V*»P* - E • / "» /, < E v *< jrwJww .
630 PART V. DYNAMIC MODELS