380 MERTON (by convention, the n-th asset), then on=0, the instantaneous rate of rcturn, an, will be called r, and(14)is rewritten as dW=∑w(4-r)W+(rW-C+d+∑Wod;,(14 where m=n-l and the wi, . Wm are unconstrained by virtue of the fact that the relation wn=1-2i w will ensure that the identity constraint in(14)is satisfied. 4. OPTIMAL PORTFOLIO AND CONSUMPTION RULES THE EQUATIONS OF OPTIMALITY The problem of choosing optimal portfolio and consumption rules for an individual who lives T years is formulated as follows max Eo U(C(), t)dt+B(W(T), T) subject to: w(O)= Wo: the budget constraint(14), which in the case of "risk-free"asset becomes(14); and where the utility function(during life)U is assumed to be strictly concave in C and the bequest "function B is assumed also to be concave in w.2 programming is used. Define J(W,P, t)=mxE U(C, s)ds+ B(W(T),T) where as before, "Et "is the conditional expectation operator, conditional nd Pa)=Pi. Define (w, C; W, P,t)=U(C, t)+9[] 12 Where there is no"risk-free" asset, it is assumed that no asse as a linear combination of the other assets, implying that the nn matrix of returns, s=oil, where ou E Pu; ; is nonsingular there is a""asset, the same assumption is made about the variance-covariance matrix380 MERTON (by convention, the n-th asset), then (T, = 0, the instantaneous rate of return, E, , will be called r, and (14) is rewritten as dW = 2 wi(ai - r) W dt + (r W - C) dt + dy + f Wiai dzi , (14’) 1 1 where m = n - 1 and the wr ,..., w, are unconstrained by virtue of the fact that the relation w, = 1 - Cy wi will ensure that the identity constraint in (14) is satisfied. 4. OPTIMAL PORTFOLIO AND CONSUMPTION RULES: THE EQUATIONS OF OPTIMALITY The problem of choosing optimal portfolio and consumption rules for an individual who lives T years is formulated as follows: max E,, [ ,: u(C@>, t> dt + WV”), T,] (15) subject to: W(0) = W, ; the budget constraint (14), which in the case of a “risk-free” asset becomes (14’); and where the utility function (during life) U is assumed to be strictly concave in C and the “bequest” function B is assumed also to be concave in W.12 To derive the optimal rules, the technique of stochastic dynamic programming is used. Define J( W, P, t) = E Et [jr U(C, s> ds + WV), U] > (16) ,w where as before, “E,” is the conditional expectation operator, conditional on W(t) = Wand P,(t) = Pi . Define rb(w, c; w, P, t> = WC, 0 + aa (17) I2 Where there is no “risk-free” asset, it is assumed that no asset can be expressed as a linear combination of the other assets, implying that the n x it variance-covariance matrix of returns, 8 = [ud, where oij = pij~ioj, is nonsingular. In the case when there is a “risk-free” asset, the same assumption is made about the “reduced” m x m variance-covariance matrix