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
L. Diewert (Generalized Leontief) Cost Function
The three factor Diewert production function is:
q^1/nu = aLL * L + aKK * K + aMM * M + bLK * L^1/2 * K^1/2 + bLM * L^1/2 * M^1/2 + bKM * K^1/2 * M^1/2 (*)
where L = labour, K = capital, M = materials and supplies, q = product output, and nu = elasticity of scale parameter.
If we have a data set relating these inputs to output for varying levels of inputs and output, we can estimate the parameters of the Diewert production function directly.
Suppose, however, we have a data set relating relating output quantities to total costs and input (factor) prices, wL, wK, and wM. Then, we can work with the Diewert (Generalized Leontief) cost function to estimate the parameters of the production technology.
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/nu)
a continuous, increasing function of q (q >= 1), with h(0) = 0 and h(1) = 1,
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 demand 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 also:
À3: Re-estimate the Diewert Production Function — to investigate properties of the underlying technology.
Assuming dLK = dKL, dLM = dML, and dKM = dMK, the 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)
À1: Estimate the factor demand equations separately.
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/nu) * [cLL + dLK * (wK / wL)^(1/2) + dLM * (wM / wL)^(1/2)]
∂C/∂wK = K(q; wL, wK, wM) = q^(1/nu) * [cKK + dKL * (wL / wK)^(1/2) + dKM * (wM / wK)^(1/2)]
∂C/∂wM = M(q; wL, wK, wM) = q^(1/nu) * [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/nu), 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. Duality. The Plan:
1. Generate CES data and estimate the parameters of a Diewert production function.
2. Generate cost data with the estimated Diewert 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 Diewert cost minimizing factor inputs, L, K, and M. The base factor prices and the randomizing distribution are specified in the form below.
3. 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.
The estimated coefficients of the Diewert production and cost functions will vary with the parameters sigma, nu, alpha, beta and gamma of the CES production function.
CES Production Function:
q = A * [alpha * (L^-rho) + beta * (K^-rho) + gamma *(M^-rho)]^(-nu/rho) = f(L,K,M).
where L = labour, K = capital, M = materials and supplies, and q = product. The parameter nu is a measure of the economies of scale, while the parameter rho yields the elasticity of substitution:
sigma = 1/(1 + rho).
Set the parameters below to re-run with your own CES parameters.
Restrictions: .7 < nu < 1.3; .5 < sigma < 1.5;
.25 < alpha < .45, .3 < beta < .5, .2 < gamma < .35
sigma = 1 → nu = alpha + beta + gamma (Cobb-Douglas)
sigma < 1 → inputs complements; sigma > 1 → inputs substitutes
4 <= wL* <= 11, 7<= wK* <= 16, 4 <= wM* <= 10