Egwald Economics: Microeconomics
Elmer G. Wiens
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| Normalized Quadratic Cost
| Translog Cost
| Diewert Cost
| Generalized CES-Translog Cost
| Generalized CES-Diewert Cost
| References and Links
N. Generalized CES-Diewert (Generalized Leontief) Cost Function
This web page replicates the procedures used to estimate the parameters of a Diewert cost function to approximate a CES cost function. To determine the Diewert cost function's efficacy in estimating varying elasticities of substitution among pairs of inputs, on this web page we approximate a Generalized CES cost function with the Diewert cost function.
Unlike the CES cost function, the Generalized CES cost function is not necessarily homothetic, while the Diewert cost function is homothetic by construction. Moreover, the Generalized CES's elasticity of scale is a function of its factor inputs, i.e. its elasticity of scale, εLKM, varies with factor prices and with output.
Consequently, when we estimate the Diewert cost function, we need to estimate the parameter nu1 of its returns to scale function separately, as we did when we approximated the CES production function by the Diewert production function.
The three factor Diewert (Generalized Leontief) (total) cost function is:
C(q;wL,wK,wM) = h(q) * c(wL,wK,wM) (**)
where the returns to scale function is:
h(q) = q^(1/nu1)
a continuous, increasing function of q (q >= 1), with h(0) = 0 and h(1) = 1, where nu1 is a measure of the returns to scale,
and the unit cost function is:
c(wL,wK,wM) = cLL * wL + cKK * wK + cMM * wM + (dLK+dKL) * (wL*wK)^(1/2) + (dLM +dML) * (wL*wM)^(1/2) + (dKM+dMK) * (wK*wM)^(1/2)
linear in its parameters cLL, cKK, cMM, dLK, dKL, dLM, dML, dKM, and dMK.
We shall use two methods to obtain estimates of the parameters:
À1: Estimate the parameters the cost function by way of its factor share functions — requires data relating output quantities to (cost minimizing) factor inputs, and input prices,
À2: Estimate the parameters of the cost function directly — requires data relating output quantities to total (minimum) cost and input prices.
We shall use the three factor Generalized CES production function to generate the required cost data, yielding a data set relating output, q, and factor prices, wL, wK, and wM, [randomized about the base prices (wL*, wK*, wM*)] to the Generalized CES cost minimizing factor inputs, L, K, and M.
For methods À1, and À2, we shall set nu1 = εLKM where q = 30 units of output, i.e. at the elasticity of scale of the Generalized CES cost function for q = 30.
To illustrate what can happen when nu1 is set incorrectly, we shall repeat methods À1, and À2 as methods À3, and À4, with nu1 = 1, i.e. under the assumption of constant returns to scale along the domain of q.
The three factor Generalized CES production function is:
q = A * [alpha * L^-rhoL + beta * K^-rhoK + gamma *M^-rhoM]^(-nu/rho) = f(L,K,M).
where L = labour, K = capital, M = materials and supplies, and q = product.
The parameter nu permits one to adjust the returns to scale, while the parameter rho is the geometric mean of rhoL, rhoK, and rhoM:
rho = (rhoL * rhoK * rhoM)^1/3.
If rho = rhoL = rhoK = rhoM, we get the standard CES production function. If also, rho = 0, ie sigma = 1/(1+rho) = 1, we get the Cobb-Douglas production function.
Generalized CES Elasticity of Scale of Production:
εLKM = ε(L,K,M) = (nu / rho) * (alpha * rhoL * L^-rhoL + beta * rhoK * K^-rhoK + gamma * rhoM * M^-rhoM) / (alpha * L^-rhoL + beta * K^-rhoK + gamma * M^-rhoM).
See the Generalized CES production function.
À1: Estimate the factor demand equations separately.
Assuming dLK = dKL, dLM = dML, and dKM = dMK, the Diewert unit cost function becomes:
c(wL,wK,wM) = cLL * wL + cKK * wK + cMM * wM + 2 * dLK * (wL*wK)^(1/2) + 2 * dLM * (wL*wM)^(1/2) + 2 * dKM * (wK*wM)^(1/2)
Taking the partial derivative of the cost function with respect to an input price, we get the factor demand function for that input:
∂C/∂wL = L(q; wL, wK, wM) = q^(1/nu1) * [cLL + dLK * (wK / wL)^(1/2) + dLM * (wM / wL)^(1/2)]
∂C/∂wK = K(q; wL, wK, wM) = q^(1/nu1) * [cKK + dKL * (wL / wK)^(1/2) + dKM * (wM / wK)^(1/2)]
∂C/∂wM = M(q; wL, wK, wM) = q^(1/nu1) * [cMM + dML * (wL / wM)^(1/2) + dMK * (wK / wM)^(1/2)]
Since it is likely that, as measured numerically, dLK != dKL, dLM != dML, and dKM != dMK, these distinctions are maintained in the factor demand equations.
After dividing each factor demand equation by q(1/nu1), we can estimate the parameters of these three linear in parameters equations using linear multiple regression.
Having obtained the unrestricted estimates of the coefficients of the factor demand functions, we shall compute the restricted least squares estimates with dLK = dKL, dLM = dML, and dKM = dMK.
The substituted values of the estimated parameters determine the Diewert cost function.
I. Stage 1. Generate cost data with the Generalized CES production function, yielding a data set relating output, q, and factor prices, wL, wK, and wM, [randomized about the base prices (wL*, wK*, wM*)] to the Generalized CES cost minimizing factor inputs, L, K, and M. The base factor prices and the randomizing distribution are specified in the form below.
Stage 2. Obtain the Diewert cost function by substituting the estimates (obtained via linear multiple regression) of the parameters of the three factor demand equations into the cost function.
II. The estimated coefficients of the Diewert cost function will vary with the parameters nu, rho, rhoL, rhoK, rhoM, alpha, beta and gamma of the Generalized CES production function.
Set the parameters below to re-run with your own Generalized CES parameters.
The restrictions ensure that the least-cost problems can be solved to obtain the underlying Generalized CES cost function, using the parameters as specified.
Intermediate (and other) values of the parameters also work.
.8 < nu < 1.1;
-.6 < rhoL = rho < .6;
-.6 < rho < -.2 → rhoK = .95 * rho, rhoM = rho/.95;
-.2 <= rho < -.1 → rhoK = .9 * rho, rhoM = rho / .9;
-.1 <= rho < 0 → rhoK = .87 * rho, rhoM = rho / .87;
0 <= rho < .1 → rhoK = .7 * rho, rhoM = rho / .7;
.1 <= rho < .2 → rhoK = .67 * rho, rhoM = rho / .67;
.2 <= rho < .6 → rhoK = .6 * rho, rhoM = rho / .6;
rho = 0 → nu = alpha + beta + gamma (Cobb-Douglas)
4 <= wL* <= 11, 7<= wK* <= 16, 4 <= wM* <= 10