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The 1 CES production function The homogenous Constant Elasticity of Substitution (CES) production function takes the form Q = K + (1 )L n o 1/ exp(U), 1, 0 1, > 0, (1) where K is capital, L is labor, Q is output, and U is an error term satisfying E[U |K, L] = 0. The CES production function was introduced by Arrow, Chenery, Minhas and Solow1 in 1961 Formally, the elasticity of substitution measures the percentage change in factor proportions due to a percentage change in the marginal rate of technical substitution. In particular, for a canonical production function Q = f(K, L) with marginal products fK = f (K, L)/ K and fL = f (K, L)/ L, the marginal rate of technical substitution is fL /fK , i.e., minus the slope of the isoquant in point (L, K), hence the elasticity of substitution between capital and labor is given by: = d ln(L/K)/d ln(fL /fK ). In the case of the deterministic homogenous CES production function Q = { K + (1 )L } 1/ we have fL = K + (1 )L fK = K + (1 )L n n o (1+ )/ o (1+ )/ (1 )L 1 , K 1 , hence d ln (fL /fK ) = ( + 1)d ln (L/K) and thus = 1/( + 1). In order to estimate the parameters of the CES production function via EasyReg, rewrite (1) as ln(Q/L) = ln( ) 1 ln { [exp ( ln(L/K)) 1] + 1} + U ln { [exp ( ln(L/K)) 1] + 1} = ln( ) [exp ( ln(L/K)) 1] exp ( ln(L/K)) 1 ( ln(L/K)) + U. ln(L/K) (2) Arrow, K.J., H.B. Chenery, B.S. Minhas and R.M. Solow (1961), Capital-Labor Substitution and Economic E ciency , Review of Economics and Statistic. 1 1 It is an elementary calculus exercise to verify that lim ln { [exp ( ln(L/K)) 1] + 1} ln(1 + ) ln(1) = lim 0 [exp ( ln(L/K)) 1] d ln(x) = = 1, dx x=1 0 hence it follows that for 0 (2) becomes ln(Q/L) = ln( ) ln(L/K) + U. Thus, for = 0 the CES production function becomes a Cobb-Douglas production function: ln(Q) = ln( ) + ln(K) + (1 ) ln(L) + U. Y = g(x, b) + U, where Y = ln(Q/L), x = ln(L/K), b = (b(1), b(2), b(3))0 = (ln( ), , )0 , and g(x, b) = b(1) ln {b(3) [exp (b(2)x) 1] + 1} b(3) [exp (b(2)x) 1] exp (b(2)x) 1 (b(3)x) . b(2)x (4) (3) exp ( ln(L/K)) 1 exp( ) exp(0) d exp(x) lim = lim = = 1, 0 0 ln(L/K) dx x=0 The CES production function (2) is now a nonlinear regression model, 2 2.1 Specifying the CES production function in EasyReg Selection of the initial X variables In EasyReg a nonlinear regression function is build up recursively by augmenting the list of X variables with nonlinear transformations and linear and/or multiplicative combinations previous of X variables. In order to build up (4), we have to select two initial X variables, X(1) = ln(L/K) and X(2) = 1. The latter is necessary for two reasons: First, EasyReg does not allow to specify constants directly. Second the parameters b(2) and b(3) are common to di erent transformations. The constant X(2) = 1 enables us to associate them to new X variables. 2 2.2 Storing the parameters in X variables Thus, we need to specify three new X variables rst, each associated with a parameter, by selecting X(2) and choosing the Linear combination option three times: X(3) = b(1)X(2), X(4) = b(2)X(2), X(5) = b(3)X(2). Then (4) becomes: g(x, b) = X(3) ln {X(5) [exp (X(1)X(4)) X(2)] + 1} X(5) [exp (X(1)X(4)) X(2)] exp (X(1)X(4)) 1 X(1)X(5), X(1)X(4) Now the CES production function involved is build up recursively as follows. X(.) X(1) X(2) X(3) X(4) X(5) X(6) X(7) X(8) X(9) X(10) = = = = = = = = = = Transformation ln(L/K) 1 b(1)X(2) = b(1) b(2)X(2) = b(2) b(3)X(2) = b(3) X(1)X(4) exp(X6)) X(7) X(2) X(5)X(8) ln (X(9) + 1) /X(9) Actual transformation Transformation option ln( ) ln(L/K) exp ( ln(L/K)) exp ( ln(L/K)) 1 (exp ( ln(L/K)) 1) ln( (exp( ln(L/K)) 1)+1) (exp( ln(L/K)) 1) exp( ln(L/K)) 1 ln(L/K) Linear combination Linear combination Linear combination Multiply EXP(z) [z = X(6)] Subtract Multiply LOG(z+1)/z [z = X(9)] (EXP(z)-1)/z [z = X(6)] Multiply X(11) = (exp (X(6)) 1) /X(6) X(12) = X(1)X(5)X(10)X(11) X(13) = X(3) X(12) ln(L/K) ln( (exp( ln(L/K)) 1)+1) (exp( ln(L/K)) 1) exp( ln(L/K)) 1 ln(L/K) ln( ) ln(L/K) ln( (exp( ln(L/K)) 1)+1) (exp( ln(L/K)) 1) ln(L/K)) 1 exp( ln(L/K) = g(b, x) Subtract As you see, this is actually a simple computer program, like a macro in MS Word or Corel Wordperfect. 3 2.3 The non-homogenous CES production function n o / The general CES production function takes the form exp(U), 1, 0 1, > 0, > 0, (5) where is the degree of homogeneity: if K and L are both increased with a factor , then Q increases with a factor . Thus, if > 1 we have increasing return to scale, and if < 1 we have decreasing returns to scale. In this case (2) becomes ln(Q/L) = ln( ) ln { [exp ( ln(L/K)) 1] + 1} + U ln { [exp ( ln(L/K)) 1] + 1} = ln( ) [exp ( ln(L/K)) 1] exp ( ln(L/K)) 1 ( ln(L/K)) + U. ln(L/K) (6) Q = K + (1 )L The speci cation of (6) in EasyReg is the same as steps X(1) through X(12), with X(13) replaced with the following three steps: X Transformation X(13) = b(4)X(2) = b(4) (= ) X(14) = X(12)X(13) X(15) = X(3) X(14) Transformation option Linear combination Multiply Subtract 4

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