382 MERTON where the notation for partial derivatives is Jw =aJ/aW,Jt=a/at, Uc=aU/aC,J=aJ/aP,, JM=0/aPi aP,, and Jsw= 02J/aP, aw Because Lcc= cc= UcC <0, LCw,- pCw, =0, L MpmR=.wWw Lu,w,=0, k*j, a sufficient condition for a unique interior maximum is that Jww<O(i.e,, that J be strictly concave in W). That assumed, as in immediate consequence of differentiating (19) totally with respect to W, we have >0 (22) To solve explicitly for C* and w*, we solve the n +2 nondynamic implicit equations, ( 19H(21), for C*, and w*, and A as functions of Jw Jww, Jw, w, P, and t. Then, C* and w* are substituted in(18)which now becomes a sccond-ordcr 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 I. Define the inverse function G=[Uc]-. Then, from(19), (23) To solve for the wi * note that(20) is a linear system in wi* and hence can be solved explicitly, Define 22= [ou] n x n variance-covariance matrIx, Eliminating A from(20), the solution for wr* can be written as k*=h2(P,t)+m(P,W,t)8(P,t)1f(P,W,t),k=1,…,n,(25) where∑nh2=1,∑1gk=0,and∑1fk≡0.16 IS52- exists by the assumption on $2 in footnote 12. h (P,t)=eve/r; m(P, W, t)=-Jw/W/ww; ga(P,E ∑∑):P,W rP2-∑JmP∑n/rww382 MERTON where the notation for partial derivatives is Jw SE aJ/a W, Jt = aJ/at, UC = aUjaC, Ji = aJ/aPi, Jij = a2J/aPi aPj, and Jjw = azJ/aPj a W. Because Lee = +cc = UC, c 0, -&ok = &to, = 0, -&ok = ~a2W2Jww, L %*j = 0, k fj, 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 ac* aw > 0. To solve explicitly for C* and w*, we solve the n + 2 nondynamic implicit equations, (19)-(21), for C*, and w*, and X as functions of Jw , J ww > Jiw , 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, r) = 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 = [U&l. Then, from (19), C” = G(J, , t). (23) To solve for the wi*, note that (20) is a linear system in wi* and hence can be solved explicitly. Define 52 = [CT& the n x n variance-covariance matrix, [Vii] EE Q-l,15 (24) Eliminating X from (20), the solution for wk* can be written as wk* = h,(P, t) + m(P, K t> g,(P, t> +.ap, w, t), k = l,..., yt, (25) where C,” h, = 1, C: g, = 0, and C,“,fk E 0.16 I5 52-l exists by the assumption on 9 in footnote 12. n 16 h,(P, r> = c l+/r; m(P, w, t) = --Jw/WJwv ;