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1.9 1.0 1.7 5 1.8 e 1.5 z 5 1.4 1.3 1.2 1.1 Figure 3 : 1900-57 t? 1 6s * + u 0 * 2 3 4 5 6 7 0 Short-term Interest Rate 147 I I I , A THEORETI CAL FRAME As an ai d i n i nterpreti ng the resul ts reported i n the l ast secti on and the addi ti onal resul ts to be reported i n Secti on I V, I i l l i ntroduce a si mpl e theoreti cal f ramework based on the model anal yzed i n Lucas and Stokey ( 1987) . The f r
  1.9 1.0 1.7 5 1.8 e 1.5 z 5 1.4 1.3 1.2 1.1 Figure 3 : 1900-57 t? 1 6s * + u 0 * 2 3 4 5 6 7 0 Short-term Interest Rate 147  III, A THEORETICAL FRAME As an aid in interpreting the results reported in the last section and the additional results to be reported in Section IV, I ill introduce a simple theoretical framework based on the model analyzed in Lucas and Stokey (1987). The framework has the advantage (relative to the framework eltzer used) of being explicit about the connection between the portfolio and transactions demands for money, and the disadvantage of being unrealistically stylized about the way trading occurs. It wll take some care to exploit the explicitness of this model wthout being led too far astray by its unrealistic features. We consider an economy in which the representative agent has the ultimate objective of maximzing the discounted expected utility fromconsumption of goods, B t E{ c t3 U(ct)} t=o This agent lives in a Markovian world, the state of which at t is arized by a vector St. The distribution of st+l, given St, is given by a fixed transition function F(s,A) = Pr(st+I E A(s = s} . In this setting, all equilibriumdate-t prices and quantities wll be fixed (no time subscript) functions of the current state, s+_. Agents are assumed to alternate between securities trading and goods trading in lockstep fashion. At the beginning of each period, all agents trade in securities, including money, in a single centralized market, all ith full knowedge of the current realization of St. hen securities trading is concluded, all agents disperse either to produce or to purchase consumption goods. Some of these goods can only be purchased wth money acquire4 during the course of securities trading: This transactions requirement is the sole reason for including cash in a portfolio, in Preference to interest bearing claims to future cash. Consider first the decision problemfacing an agent who is engaged in securities trading at a time in hich the state of the economy is s and his . (In a ten lized securities market is asset ote the value of this agent's expected, 148  discounted utility if he proceeds optimally fromthis point on. At this point, the agent is faced by a vector Q(s) of securities prices (in dollars, so the price of money is unity). He must choose money holdings and a vector of securities holdings z, subject to a portfolio constraint: + Q(s) z I: (2) Let be the indirect utility function he uses to make this choice. (Clearly G wll depend on s, since the current state variable includes all the information he has about the returns fromthese securities.) Then v(s,W must satisfy: v(s,W = max G(Mz,s) subject to (2). (3) Mz I call (3) the agent's portfolio problem Nowwhere does this indirect utility function G come from? Having completed securities trading, the agent is about to engage in purchasing a vector c of consumption goods. He wll also receive an endowment y s) of goods, but this he must sell for cash or future cash: He cannot consume his own endowment. The rules of trading in this goods market are summarized by a vector of constants a, where ai E [OJ] is the fraction of purchases of good i that must be covered by money. It wll be an expositional simplification in what follows to postulate a technology together wth a choice of units for measuring goods such that all goods sell for the same nomnal price P(s)- In this case, the agent's Clower- or cash-in-advance constraint is: P(s)a c 5 (4) The outcome (Mz) of the portfolio decision plus the outcome ( Y(S)) of his goods trades plus a given vector D(s') of nomnal returns (dividends, interest, principal) on securities ill determne this agent's nomnal wealth position 1 as of tomorroti, conditional o state s'. He begins next period wth his dollar holdings as plus the dividends and resale value of his securities, plus the dollar value of his endo of his goods purchasesI P(s)S c . That is: 149  W’ zz + [Q(s')+D(s')]*Z + f'(S)~i[YiO-ciI (5) These considerations determne what I call the transactions problem = max U(c) + B/ w(s', ')F(s,ds') subject to (4) , (6) where I is defined in (5). Elimnating the function G between (3) and (6) defines a functional equation in the value function v. See Lucas and Stokey (1987) for an analysis of this equation and its use in constructing an equilibriumfor this economy. My purpose here is not so much analysis as it is clarifying what we mean by a demand function for money, and hence in understanding what an empirical money demand function mght mean. Let me begin wth what eTtzer (1963) and certainly Hamburger (1977) meant by a demand function for money. Fromthe portfolio problem(3) one obtains the first order conditions: GM(Mz,s) = u , (7) G,_JJI~z,s) = Qju , j = l,...,m , 3 (8) here v is the multiplier associated wth the wealth constraint (2) and here j indexes the mavailable securities. These m+l equations ith (2) can be solved to obtain the demand functions for the hich have as arguments the prices Q and wealth W Singling out the demand function (in this sense) for money: (9) Note that the entire vector Q of securities prices enters Jn the right of (9). In practice, as in any empirical application of demand uld focus on the prices of securities thought to have strong substitution or complementary relationships wth money. In this spirit, eltzer used a long termbond yield in his econometric work. In the same experimented ith equities yields and other a respectable basis for an empirical study, 150
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