112 LUCAS If this is true, the function p satisfies(multiplying(4. 2) through by mx 8 ind substituting) q(x6)]6q(x/6) ∫p[)](x5)( Let us make the change of variable z= xB, and z'=x 8,and H(z, 0) be the joint density function of z and 0 and let H(z, e)be the density of A conditional on z. Then(5. 1)is equivalent to p(z)]p(z) j[05]0c.B,0)mhm(2) Equations(4.2)and (5. 2)are studied in the appendix. The result of nterest is THEOREM 1. Equation(5.2) has exactly one continuous solution p(z) on(0, oo) with z/ o(z) bounded. The function (z) is strictly positive and continuously differentiable. Further, mop(x/@)is the unique equilibrium price Proof. See the appendix, We turn next to the characteristics of the solution function It is convenient to begin this study by first examining two polar cases, one in which 8=1 with probahility one, and a second in which x= I with probability one The first of these two cases may be interpreted as applying to an economy in which all trading place in a single market, and no nonmonetary dis turbances are present. Then z is simply equal to x and, in view of Lemma I the current value of x is fully revealed to traders by the equilibrium price It should not be surprising that the following classical neutrality of money theorem holds THEOREM 2. Suppose 6=1 with probability one. Let y* be the un h(y)=v(y) Then P(m, x, 8=mx/y* is the unique solution to(4.2)112 LUCAS If this is true, the function qz satisfies (multiplying (4.2) through by mx/% and substituting): h [ %I&%) 1 e&e) ~- dG (E, x’, 8’ I$]. (5.1) Let us make the change of variable z = xl%, and z’ = x’/%‘, and let H(z, %) be the joint density function of z and % and let ii(z, %) be the density of % conditional on z. Then (5.1) is equivalent to: = s v’ iI ii(z, %) H(z’, %‘) d% dz’ d%‘. (5.2) Equations (4.2) and (5.2) are studied in the appendix. The result of interest is: THEOREM 1. Equation (5.2) has exactly one continuous solution y(z) on (0, 00) with z/v(z) bounded. The function y(z) is strictly positive and continuously dtzerentiable. Further, my(x/%) is the unique equilibrium price function. Proof See the appendix, We turn next to the characteristics of the solution function v. It is convenient to begin this study by first examining two polar cases, one in which % = 1 with probability one, and a second in which x = 1 with probability one. The first of these two cases may be interpreted as applying to an economy in which all trading place in a single market, and no nonmonetary dis￾turbances are present. Then z is simply equal to x and, in view of Lemma 1, the current value of x is fully revealed to traders by the equilibrium price. It should not be surprising that the following classical neutrality of money theorem holds. THEOREM 2. Suppose % = 1 with probability one. Let y* be the unique solution to h(y) = V’(Y). (5.3) Then p(m, x, %) = mx/y* is the unique solution to (4.2)