320 HN F MUTH We solve for the weights V, in terms of the weights W, in the following manner. Substituting from (3.7)and(3.8), we obtai 1+-=v,w+=(vW)4 Since the equality must hold for all shocks, the coefficients must satisfy the equations ViWi-g (=1,2,3, This is a system of equations with a triangular structure, so that it may be If the disturbances are independently distributed, as we assumed before, then we0=-1/B and all the others are zero. Equations(3. 14) therefore 少=0 (315b) pt=十 These are the results obtained before Suppose, at the other extreme, that an exogenous shock affects all future onditions of supply, instead of only the one period. This assumption would be appropriate if it represented how far technological change differed from its trend. Because ut is the sum of all the past es, ue=1(i=0, 1, 2,.) From(3.11) (316a) 1/B, (316b) From 3. 14)it can be seen that the expected price is a geometrically weighted moving average of past prices (3.17) 8 go p This prediction formula has been used by Nerlove [26] to estimate the supply elasticity of certain agricultural commodities. The only difference is that our analysis states that the coefficient of adjustment"in the ex pectations formula should depend on the demand and the supply coeffi- cients. The geometrically weighted moving average forecast is, in fact optimal under slightly more general conditions (when the disturbance is composed of both permanent and transitory components). In that case the coefficient will depend on the relative variances of the two components as well as the supply and demand coefficients. (See [24]320 JOHN F. MUTH We solve for the weights V1 in terms of the weights Wj in the following manner. Substituting from (3.7) and (3.8), we obtain 00 00 00 00 t (3.13) WiVt- EV IWiet-i-i = V Wi 8t-ti. {=1 ?~=1 i=0 J5 =1 Since the equality must hold for all shocks, the coefficients must satisfy the equations (3.14) Wi VWiy (i = 1,2,3,...). 1=1 This is a system of equations with a triangular structure, so that it may be solved successively for the coefficients V1, V2, V3,.... If the disturbances are independently distributed, as we assumed before, then wO -1 /8 and all the others are zero. Equations (3.14) therefore imply (3.15a) t (3.15b) Pt = P+Wost - lete These are the results obtained before. Suppose, at the other extreme, that an exogenous shock affects all future conditions of supply, instead of only the one period. This assumption would be appropriate if it represented how far technological change differed from its trend. Because ut is the sum of all the past ej, wi 1 (i = 0,1,2,...). From (3.1 1), (3.16a) Wo -1/fl, (3.16b) Wi l/0 +y) From (3.14) it can be seen that the expected price is a geometrically weighted moving average of past prices: (3.17) ( ) . y pt y P t: + yJt￾This prediction formula has been used by Nerlove [26] to estimate the supply elasticity of certain agricultural commodities. The only difference is that our analysis states that the "coefficient of adjustment" in the ex￾pectations formula should depend on the demand and the supply coeffi￾cients. The geometrically weighted moving average forecast is, in fact, optimal under slightly more general conditions (when the disturbance is composed of both permanent and transitory components). In that case the coefficient will depend on the relative variances of the two components as well as the supply and demand coefficients. (See [24].)