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Egwald Economics: Microeconomics Production Functions Cobb-Douglas | CES | Translog | Diewert | Translog vs Diewert | Diewert vs Translog | Estimate Translog | Estimate Diewert | References and Links While Cobb-Douglas production functions are great, because they are easy to estimate, the elasticity of substitution between factors is always equal to 1. CES production functions permit you to vary the elasticity of substitution. B. CES (Constant Elasticity of Substitution) Production Function The three factor CES production function is: 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). To separate the elasticity of scale, nu, from the other parameters requires that: alpha + beta + gamma = 1. I. Decreasing returns to scale: nu < 1 With decreasing returns to scale, a proportional increase in all inputs will increase output by less than the proportional constant. Let's assume the CES production function's parameters are: A = 1.0, alpha = .3, beta = .4, gamma = .3, rho = .17647, nu=.92. Note that rho = .17647 --> sigma = .85 Then, q = 1.0 * [.3 * (L^-.17647) + .4 * (K^-.17647) + .3 *(M^-.17647)]^(-.92/.17647) Suppose the firm can buy its factors at the prices: wL = 7, wK = 13, wM = 6. Its costs will be: c(q) = wL * L + wK * K + wM * M = 7 * L + 13 * K + 6 * L To produce 35 units of product at minimum cost, it should use: L = 51.45, K = 38.82, and M = 58.86 units of inputs.
Notes:
Since our CES production function has decreasing returns to scale, the average cost and marginal cost are increasing, and marginal cost is greater than average cost.
The terms sLK, sLM, and sKM are the Allen partial elasticities of substitution. Here they all equal .85, because this is the constant elasticity of substitution between inputs given the value of rho = .17647. These measures are important for such production functions as the translog and Diewert, where they are not necessarily constant. II. Constant returns to scale: nu = 1 With constant returns to scale, a proportional increase in all inputs will increase output by the proportional constant. A = 1.0, alpha = .3, beta = .4, gamma = .3, rho = .17647, nu=1.0, sigma=.85. factor prices: wL = 7, wK = 13, wM = 6. The CES production function has the form:
q = 1. * [.3 * (L^-.17647) + .4 * (K^-.17647) + .3 * (M ^-.17647)]^(-1/.17647)
The average cost and marginal cost curves coincide, a consequence of constant returns to scale.
III. Increasing returns to scale: nu = 1.08 > 1 With increasing returns to scale, a proportional increase in all inputs will increase output by more than the proportional constant.A = 1.0, alpha = .3, beta = .4, gamma = .3, rho = .17647, nu=1.08, sigma=.85. factor prices: wL = 7, wK = 13, wM = 6. The CES production function has the form:
q = 1. * [.3 * (L^-.17647) + .4 * (K^-.17647) + .3 * (M ^-.17647)]^(-1.08/.17647)
Both average cost and marginal cost are decreasing, with marginal cost below average cost, a consequence of increasing returns to scale.
IV. Short Run: Economists often assume that capital is fixed in the short-run. While the quantities of labour, and materials and supplies can be adjusted, changing the amount of capital services, quickly, is costly. To model the short-run production activities of a firm, capital will be set at the level that is associated with producing q = 30 units of product. Note that the values of the coefficients alpha, beta, and gamma have been changed below, to facilitate comparison with the translog and Diewert production functions. 1. Decreasing returns to scale: A = 1.0, alpha = .35, beta = .4, gamma = .25, rho = .17647, nu = .92, sigma = .85. factor prices: wL = 7, wK = 13, wM = 6. Set capital K = 32.82, the amount of capital associated with producing q = 30 units of product.
Now we get the traditional U-shaped average, short run cost curve, with a minimum to the left of q = 30. Because marginal cost is virtually a linear function of q, total cost (with capital fixed) is virtually a quadratic function of q (since its derivative, marginal cost, is linear).
Note that q=30 is the point at which the short run (capital fixed), average cost curve is tangent to the long run, CES average cost curve. Here I have listed two measures of the short-run elasticity of substitution between L and M. The 3-factor measure of sLM uses the Allen partial elasticity of substitution formula. The 2-factor measure of sLM uses the standard short-run formula, which assumes that only L and M can vary with output, with a fixed amount of capital present. Here again, these short-run elasticities all equal .85, because the production function is CES. 2. Constant returns to scale: A = 1.0, alpha = .35, beta = .4, gamma = .25, rho = .17647, nu = 1.0, sigma = .85. factor prices: wL = 7, wK = 13, wM = 6. Set capital K = 24.42, the amount of capital associated with producing q = 30 units of product.
The traditional U-shaped average, short run cost curve has a minimum at q = 30. Because marginal cost is virtually a linear function of q, total cost (with capital fixed) is virtually a quadratic function of q (since its derivative, marginal cost, is linear).
Note that q=30 is the point at which the short run (capital fixed), average cost curve is tangent to the long run, CES average cost curve. 3. Increasing returns to scale: A = 1.0, alpha = .35, beta = .4, gamma = .25, rho = .17647, nu = 1.08, sigma = .85. factor prices: wL = 7, wK = 13, wM = 6. Set capital K = 18.98, the amount of capital associated with producing q = 30 units of product.
The traditional U-shaped average, short run cost curve has a minimum to the right of q = 30. Because marginal cost is virtually a linear function of q, total cost (with capital fixed) is virtually a quadratic function of q (since its derivative, marginal cost, is linear).
Note that q=30 is the point at which the short run (capital fixed), average cost curve is tangent to the long run, CES average cost curve. V. Formulae
a. Marginal Product of labour:
b. Marginal cost function: if (L,K,M) is the cost minimizing combination of inputs at prices (wL,wK,wM) for output q, then VI. Least-cost combination of inputs Find the values of L, K, M, and µ that minimize the Lagrangian: G(q;L,K,M,µ) = wL * L + wK * K + wM* M + µ * [q - f(L,K,M)]
From equations a., b., and c. we get:
Substituting equations e. and f. into the CES production function and solving for L yields;
Finally, substituting e., f. and h. into the cost function: C(q) = wL * L + wk * K + wM * M yields the cost function, as a function of output, depending on the input prices and the parameters of the CES production function. If we actually solve explicitly for C(q): C(q;wL,wK,wM) = h(q) * c(wL,wK,wM) where the returns to scale function is:
h(q) = q^1/nu and the unit cost function is:
c(wL,wK,wM) = (1/A)^1/nu * [alpha^(1/(1+rho) * wL^(rho/(1+rho) + beta^(1/(1+rho) * wK^(rho/(1+rho) + The unit cost function c(wL, wK, wM) looks, interestingly, like its parent - the CES production function. The CES production function is called homothetic, because the CES cost function can be separated (factored) into a function of output, q, times a function of input prices, wL, wK, and wM. VIII. Factor demand functions:
If we take the derivative of the cost function with respect to an input price, we get the factor demand function for that input: IX. Properties of the unit CES Cost Function, c(wL,wK,wM). a. c is linear homogeneous in factor prices. b. c is concave in factor prices. X. Elasticity of substitution between inputs (sigma).
From equation e. of Part V we get: K/L = [(beta / alpha)* (wl / wK)]^1/(1+rho) --> ln(K/L) = (1/(1+rho))*ln(beta/alpha) + (1/(1+rho))*ln(wL/wK) sigma = d(ln(K/L))/d(ln(wL/wK)) = 1/(1+rho)
a. K and L substitutes: -1 < rho < 0, then 1 < sigma < infinity
b. K and L complements: 0 < rho < infinity, then 0 < sigma < 1 XI. Allen partial elasticity of substitution Writing the production function as q = F(L,K,M), let the bordered Hessian be:
sLK = ((FL * L + FK * K + FM *M) / (L * K)) * (|FLK|/|F|) XII. Two factor elasticity of substitution Let the production function be q = F(L,K,M), where K is underlined to indicate it is constant in the short-run. Then the two factor (L and M) bordered Hessian is:
The 2-factor elasticity of substitution between L and M is: sLM = - (FL * L + FM * M) / (L * M ) * (FL * FM) / |F| |
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