     Egwald Economics: Microeconomics

Production Functions

by

Elmer G. Wiens

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Please show your support by joining Egwald Web Services as a Facebook Fan: Follow Elmer Wiens on Twitter: Duality: Production / Cost Functions:   Cobb-Douglas Duality | CES Duality | Theory of Duality | Translog Duality - CES | Translog Duality - Generalized CES

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).

Separating the elasticity of scale, nu, from the other parameters is facilitated when:

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

The CES production function has the form:

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:
1. 35 = 1. * [.3 * (51.45^-.17647) + .4 * (38.82^-.17647) + .3 * (58.86 ^-.17647)]^(-.92/.17647)
2. c(q) = wL * L + wK * K + wM * M   -->   1216.78 = 7 * 51.45 + 13 * 38.82 + 6 * 58.86
3. Average cost = c(q)/q   -->   1216.78 / 35 = 34.77

Graph of the Average Cost and Marginal Cost
for a CES Production Function Average cost function Marginal cost function
Decreasing returns to scale

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.

CES Production Function Data
Decreasing Returns to Scale
qLKMcostave.costmarg.costsLKsLMsKM
5 6.21 4.68 7.07 146.77 29.3531.910.850.850.85
10 13.18 9.95 15.03 311.77 31.1833.890.850.850.85
15 20.48 15.46 23.35 484.44 32.335.10.850.850.85
20 28 21.13 31.93 662.28 33.1135.990.850.850.85
25 35.69 26.93 40.69 844.07 33.7636.70.850.850.85
30 43.51 32.83 49.61 1029.07 34.337.290.850.850.85
35 51.45 38.82 58.66 1216.78 34.7737.790.850.850.85
40 59.49 44.89 67.82 1406.85 35.1738.230.850.850.85
45 67.61 51.02 77.08 1599 35.5338.620.850.850.85
50 75.82 57.21 86.43 1793.02 35.8638.980.850.850.85

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)

Graph of the Average Cost and Marginal Cost
for a CES Production Function Average cost function Marginal cost function
Constant returns to scale

The average cost and marginal cost curves coincide, a consequence of constant returns to scale.

CES Production Function Data
Constant Returns to Scale
qLKMcostave.costmarg.costsLKsLMsKM
5 5.4 4.07 6.15 127.6 25.5225.520.850.850.85
10 10.79 8.14 12.3 255.2 25.5225.520.850.850.85
15 16.19 12.21 18.45 382.8 25.5225.520.850.850.85
20 21.58 16.28 24.6 510.4 25.5225.520.850.850.85
25 26.98 20.36 30.75 638 25.5225.520.850.850.85
30 32.37 24.43 36.91 765.59 25.5225.520.850.850.85
35 37.77 28.5 43.06 893.19 25.5225.520.850.850.85
40 43.16 32.57 49.21 1020.79 25.5225.520.850.850.85
45 48.56 36.64 55.36 1148.39 25.5225.520.850.850.85
50 53.96 40.71 61.51 1275.99 25.5225.520.850.850.85

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)

Graph of the Average Cost and Marginal Cost
for a CES Production Function Average cost function Marginal cost function
Increasing returns to scale

Both average cost and marginal cost are decreasing, with marginal cost below average cost, a consequence of increasing returns to scale.

CES Production Function Data
Increasing Returns to Scale
qLKMcostave.costmarg.costsLKsLMsKM
5 4.79 3.61 5.46 113.26 22.6520.970.850.850.85
10 9.1 6.87 10.37 215.18 21.5219.920.850.850.85
15 13.24 9.99 15.1 313.22 20.8819.330.850.850.85
20 17.29 13.04 19.71 408.82 20.4418.930.850.850.85
25 21.25 16.04 24.23 502.65 20.1118.620.850.850.85
30 25.16 18.99 28.69 595.09 19.8418.370.850.850.85
35 29.02 21.9 33.09 686.39 19.6118.160.850.850.85
40 32.84 24.78 37.44 776.72 19.4217.980.850.850.85
45 36.63 27.64 41.76 866.22 19.2517.820.850.850.85
50 40.38 30.47 46.04 954.99 19.117.680.850.850.85

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.

Graph of Average Cost and Marginal Cost
CES Production Function - Capital Fixed Average cost function Marginal cost function L.R. Average cost function
Decreasing economies of scale

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).

CES Production Function Data
Decreasing economies of scale
Fixed Capital
qLKMcostave.costmarg. cost3 factor
sLM
2 factor
sLM
5 2.44 32.82 2.09 456.3 91.269.140.850.85
10 7.34 32.82 6.29 515.74 51.5714.590.850.85
15 14.45 32.82 12.38 602.09 40.1419.960.850.85
20 23.81 32.82 20.39 715.64 35.7825.490.850.85
25 35.48 32.82 30.39 857.36 34.2931.240.850.85
30 49.58 32.82 42.46 1028.5 34.2837.260.850.85
35 66.22 32.82 56.72 1230.5 35.1643.590.850.85
40 85.54 32.82 73.26 1464.92 36.6250.230.850.85
45 107.65 32.82 92.2 1733.41 38.5257.220.850.85
50 132.72 32.82 113.67 2037.71 40.7564.560.850.85

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.

Graph of Average Cost and Marginal Cost
CES Production Function - Capital Fixed Average cost function Marginal cost function L.R. Average cost function
Constant economies of scale

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).

CES Production Function Data
Constant economies of scale
Fixed Capital
qLKMcostave.costmarg. cost3 factor
sLM
2 factor
sLM
5 2.27 24.42 1.94 344.96 68.997.90.850.85
10 6.31 24.42 5.4 394 39.411.640.850.85
15 11.82 24.42 10.13 460.93 30.7315.120.850.85
20 18.76 24.42 16.07 545.1 27.2518.550.850.85
25 27.11 24.42 23.22 646.44 25.86220.850.85
30 36.89 24.42 31.59 765.17 25.5125.510.850.85
35 48.13 24.42 41.22 901.63 25.7629.090.850.85
40 60.87 24.42 52.13 1056.26 26.4132.770.850.85
45 75.14 24.42 64.36 1229.55 27.3236.560.850.85
50 91 24.42 77.94 1422.05 28.4440.460.850.85

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.

Graph of Average Cost and Marginal Cost
CES Production Function - Capital Fixed Average cost function Marginal cost function L.R. Average cost function
Increasing economies of scale

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).

CES Production Function Data
Constant economies of scale
Fixed Capital
qLKMcostave.costmarg. cost3 factor
sLM
2 factor
sLM
5 2.14 18.98 1.83 272.65 54.536.960.850.85
10 5.55 18.98 4.76 314.12 31.419.550.850.85
15 9.97 18.98 8.54 367.73 24.5211.870.850.85
20 15.31 18.98 13.11 432.59 21.6314.070.850.85
25 21.55 18.98 18.46 508.32 20.3316.220.850.85
30 28.67 18.98 24.56 594.76 19.8318.360.850.85
35 36.67 18.98 31.41 691.89 19.7720.50.850.85
40 45.56 18.98 39.02 799.74 19.9922.650.850.85
45 55.34 18.98 47.39 918.41 20.4124.820.850.85
50 66.01 18.98 56.54 1048.02 20.9627.030.850.85

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. Isoquants

A given level of output, q = q, can be produced by different combinations of factor inputs, L, K, and M. Fixing the level of product output at q = q, we obtain an equation from the CES production function:

q = A * [alpha * (L^-rho) + beta * (K^-rho) + gamma *(M^-rho)]^(-nu/rho) = f(L,K,M).

for the 3-dimensional isoquant surface, when q = q.

The isoquant surface is tangent to the isocost plane:

C(q) = wL * L + wK * K + wM * M

at the cost minimizing combination of factor inputs, (L, K, M ) = (L, K, M).

Consider again the specific CES production function:

q = 1 * [0.3 * (L^-0.17647) + 0.4 * (K^-0.17647) + 0.3 *(M^-0.17647)]^(-1.08/-0.17647) = f(L,K,M).

When q = 30 and (wL, wK, wM) = (7, 13, 6), the cost minimizing inputs are:

(L, K, M) = (25.16, 18.99, 28.69), and

C(30) = 7 * 25.16 + 13 * 18.99 + 6 * 28.69 = 595.09.

Solving the CES equation for L, K, and M in turn, we get:

 1. L = [(q / A)^(-rho/nu) - beta * K^-rho - gamma * M^-rho) / alpha]^(-1/rho), 2. K = [(q / A)^(-rho/nu) - alpha * L^-rho - gamma * M^-rho) / beta]^(-1/rho), 3. M = [(q / A)^(-rho/nu) - alpha * L^-rho - beta * K^-rho) / gamma]^(-1/rho),

three equations for the 3-dimensional isoquant surface.

By fixing the amount of input for one factor, we obtain a 2-dimensional isoquant curve. As examples, fixing M = M in equation 2, and K = K in equation 3, we get:

 2. → L-K Isoquant:   K = [(q / A)^(-rho/nu) - alpha * L^-rho - gamma * M^-rho) / beta]^(-1/rho), 3. → L-M Isoquant:   M = [(q / A)^(-rho/nu) - alpha * L^-rho - beta * K^-rho) / gamma]^(-1/rho),

with K and M as functions of one variable, L. The following diagrams graph, in blue, the L-K and L-M isoquants for q = 24, 30, and 36.

The yellow lines represent the isocost lines, combinations of L, K, and M that can be purchased at a constant total cost at the prices wL = 7, wK = 13, and wM = 6.

The slope of an L-K isocost line is mK = -wL / wK = -7 / 13; the slope of an L-M isocost line is mM = -wL / wM = -7 / 6.

For q = 30, the L-K isocost line has a K-intercept at (C(30) - wM * M) / wK = (595.09 - 6 * 28.69)/13 = 32.53,
while the L-M isocost line has a M-intercept at (C(30) - wK * K) / wM = (595.09 - 13 * 18.99)/6 = 58.04.

For q = 30, the L-K isoquant is tangent to the L-K isocost line at (L, K) = (25.16, 18.99), while the L-M isoquant is tangent to the L-M isocost line at (L, M) = (25.16, 28.69).

L-K Isoquants, M = ML-M Isoquants, K = K  The red rays emanating from the origin in the diagrams intersect each isoquant at the same angle. Consequently, any isoquant is a radial projection of each other isoquant. In particular, any isoquant is a radial projection of the unit isoquant, i.e. the isoquant for q = 1. Production functions with this property are called homothetic production functions.

VI. Formulae

The three factor CES production function is:

q = f(L,K,M) = A * [alpha * (L^-rho) + beta * (K^-rho) + gamma *(M^-rho)]^(-nu/rho).

a. Marginal Product of labour:

∂f(L, K, M)/∂L = fL = alpha*nu*L ^-(1+rho)*A*[alpha * (L^-rho) + beta * (K^-rho) + gamma *(M^-rho)]^(-nu/rho) - 1

= alpha*nu*L ^-(1+rho) * A^-(rho/nu) * A^(1+rho/nu)*[alpha*(L^-rho) + beta*(K^-rho) + gamma*(M^-rho)]^(-nu/rho)(1+rho/nu)

= alpha*nu*L ^-(1+rho) * A^-(rho/nu) * q^(1+rho/nu)

b. Marginal cost function: if (L,K,M) is the cost minimizing combination of inputs at prices (wL,wK,wM) for output q, then
C'(q) = ∂C/∂q = wL / (∂f(L,K,M)/∂L) = µ

VII. 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)]

1. GL = wL - µ * fL = 0
2. GK = wK - µ * fK = 0
3. GM = wM - µ * fM = 0
4. Gµ = q - f(L,K,M) = 0

From equations a., b., and c. we get:

1. wL / wK = fL / fK = (alpha/beta) * (K/L)^(1+rho) --> K = L * (wL*beta/(wK*alpha))^(1/(1+rho))
2. wL / wM = fL / fM = (alpha/gamma) * (M/L)^(1+rho) --> M = L * (wL*gamma/(wM*alpha))^(1/(1+rho))
3. wK / wM = fK / fM = (beta/gamma) * (M/K)^(1+rho)

Substituting equations e. and f. into the CES production function and solving for L yields;

1. L = (q/A)^1/nu * [alpha + beta*(wL*beta/(wK*alpha))^(-rho/(1+rho)) + gamma(wL*gamma/(wM*alpha))^(-rho/(1+rho))]^(1/rho)

= (q/A)^1/nu * (alpha / wL)^(1/(1+rho)) * [alpha^(1/(1+rho)) * wL^(rho/(1+rho)) + beta^(1/(1+rho)) * wK^(rho/(1+rho)) + gamma^(1/(1+rho)) * wM^(rho/(1+rho))]^(1/rho)

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.

VIII. CES Cost 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/A)^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) = [alpha^(1/(1+rho)) * wL^(rho/(1+rho)) + beta^(1/(1+rho)) * wK^(rho/(1+rho)) + gamma^(1/(1+rho)) * wM^(rho/(1+rho))]^((1+rho)/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.

IX. 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:

∂C/∂wL = h(q) * [(alpha / wL) * c(wL,wK,wM)]^(1/(1+rho)) = L

∂C/∂wK = h(q) * [(beta / wK) * c(wL,wK,wM)]^(1/(1+rho)) = K

∂C/∂wM = h(q) * [(gamma / wM) * c(wL,wK,wM)]^(1/(1+rho)) = M

X. 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.

XI. Elasticity of substitution between inputs (sigma).

From equation e. of Part VI 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

XII. Allen partial elasticity of substitution

Writing the production function as q = F(L,K,M), let the bordered Hessian be:

 F = 0 FL FK FM FL FLL FLK FLM FK FKL FKK FKM FM FML FMK FMM

If |F| is the determinant of the bordered Hessian and |FLK| is the cofactor associated with FLK, then the Allen elasticity of substitution is defined as:

sLK = ((FL * L + FK * K + FM *M) / (L * K)) * (|FLK|/|F|)

XIII. 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:

 F = 0 FL FM FL FLL FLM FM FML FMM

The 2-factor elasticity of substitution between L and M is:

sLM = - (FL * L + FM * M) / (L * M ) * (FL * FM) / |F|

XIII. Further examples:

The web page, "The Duality of Production and Cost Functions," permits one to specify the parameters of the CES (or Cobb-Douglas) production function, and to ascertain the curvature of the production function and corresponding cost function. Copyright © Elmer G. Wiens:   Egwald Web Services All Rights Reserved.    Inquiries 