     Egwald Economics: Microeconomics

Production Functions

by

Elmer G. Wiens

Egwald's popular web pages are provided without cost to users.
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

The standard CES production function permits one to obtain an elasticity of substitution between inputs different from 1, as dictated by the Cobb-Douglas production function. However, the CES elasticity of substitution must be constant for all pairs of inputs. The generalized CES production function permits varying elasticities of substitution among pairs of inputs.

B1. Generalized CES Production Function

I. 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.

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 functions, using the parameters as specified.
Intermediate (and other) values of the parameters also work.

Restrictions:
.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)

 Generalized CES Production Function Parameters nu: rho:

The Generalized CES production function as specified:

q = 1 * [0.35 * L^- 0.17647 + 0.4 * K^- 0.11823 + 0.25 *M^- 0.26339]^(-1/0.17647) = f(L,K,M).

Elasticity of Scale of Production:

Long Run: Capital Variable:

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).

Short Run: Captial Fixed: K = K:

εLKM = ε(L,K,M) = (nu / rho) * (alpha * rhoL * L^-rhoL + gamma * rhoM * M^-rhoM) / (alpha * L^-rhoL + beta * K^-rhoK + gamma * M^-rhoM).

See the Mathematical Notes below.

Curvature:

The Generalized CES production function, f(L,K,M), is concave to the origin of the 3-dimensional (L,K, M) space if its Hessian is negative (semi)definite. Define the matrices h1, h2, and h3:

 fLL fLK fLM fKL fKK fKM fML fMK fMM
=|h3|
 fLL fLK fKL fKK
=|h2|
 fLL
=|h1|

The Hessian of f(L,K,M) is negative definite if the determinants |h1|, |h2|, and |h3| alternate in sign, starting with negative. If one or more the determinants have a zero value, then the Hessian of f(L,K,M) is negative semidefinite, and f(L, K, M) is quasi-concave to the origin of the 3-dimensional space of (L,K,M).

II. The least-cost combination of inputs:

The entrepreneur, management, and employees of the profit maximizing firm choose the factor proportions and quantities, and output levels given the prices of factor inputs, and products. For any specified combination of positive factor prices, wL, wK, and wM, what combination of factor inputs, L, K, and M, will minimize the cost of producing any given level of positive output, q?

C(q; wL, wK, wM) = minL,K,M{ wL * L + wK * K + wM * M   :   q - f(L,K,M) = 0,   q > 0, wL > 0; wK > 0, and wM > 0 }

Equivalently: Find the values of L, K, M, and µ that minimize the Lagrangian when the factor prices are wL, wK, and wM:

G(q;L,K,M,µ) = wL * L + wK * K + wM * M + µ * [q - f(L,K,M)],

or equivalently, that maximize the Lagrangian G(q;L,K,M,µ) = - G(q;L,K,M,µ).

G(q;L,K,M,µ) = -(wL * L + wK * K + wM * M) - µ * [q - f(L,K,M)].

First Order Conditions:

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

To solve equations 0. to 3. numerically, for given q, wL, wK, and wM, I used the first order conditions 0. to 3. and the associated Jacobian.

Jacobian of Second Order Conditions (Bordered Hessian with F(L,K,M) = f(L,K,M)):

 Gµµ GµL GµK GµM GLµ GLL GLK GLM GKµ GKL GKK GKM GMµ GML GMK GMM
=
 0 FL FK FM FL GLL GLK GLM FK GKL GKK GKM FM GML GMK GMM
=|H3|
 0 FL FK FL GLL GLK FK GKL GKK
= |H2|

The Lagrangian G(q;L,K,M,µ) obtains a maximum at q, L, K, and M if the the bordered, principal minor determinant |H2| is positive, and the bordered determinant |H3| is negative. (See the table below, and the Mathematical Notes.)

III. Long Run: Capital Variable.

Suppose the firm buys its inputs at the prices: wL = 7   wK = 13   wM = 6 . Solve the least-cost problem (noting that µ = marginal cost) to obtain the data as displayed in the following table. The terms sLK, sLM, and sKM are the Allen partial elasticities of substitution.

Generalized CES Long Run Cost Data
Parameters: nu = 1, rho = 0.17647, rhoL = 0.17647, rhoK = 0.11823, rhoM = 0.26339
wL = 7, wK= 13, wM = 6LagrangianGen. CES Curvature
q f(L,K,M) LKMcostave. costmarg. costεLKMsLKsLMsKM|H2|*10|H3|*102|h1|*103|h2|*104|h3|*107
18 18 25.1713.48 23.96495.29 27.5229.680.927 0.8960.7930.834 1.228-5.325-7.391.3-10.39
20 20 28.2115.2 26.64554.94 27.7529.980.926 0.8950.7930.834 1.071-4.164-6.531.01-7.18
22 22 31.2816.94 29.33615.18 27.9630.250.924 0.8950.7920.834 0.946-3.332-5.840.8-5.13
24 24 34.3718.71 32.02675.95 28.1630.510.923 0.8950.7920.833 0.844-2.717-5.270.64-3.78
26 26 37.4920.5 34.72737.22 28.3530.760.922 0.8950.7920.833 0.76-2.251-4.80.53-2.84
28 28 40.6322.31 37.42798.97 28.5330.990.921 0.8950.7920.833 0.689-1.891-4.40.44-2.19
30 30 43.7924.14 40.13861.17 28.7131.210.92 0.8940.7920.833 0.629-1.607-4.050.37-1.71
32 32 46.9825.99 42.84923.81 28.8731.420.919 0.8940.7920.833 0.578-1.38-3.750.32-1.36
34 34 50.1827.86 45.56986.85 29.0231.620.918 0.8940.7910.833 0.533-1.195-3.490.27-1.1
36 36 53.4129.75 48.281050.28 29.1731.810.917 0.8940.7910.833 0.494-1.044-3.260.24-0.89
38 38 56.6531.65 511114.08 29.3231.990.916 0.8940.7910.832 0.46-0.918-3.060.21-0.74
40 40 59.9133.57 53.731178.25 29.4632.170.916 0.8940.7910.832 0.429-0.813-2.880.18-0.61
42 42 63.1935.51 56.461242.76 29.5932.340.915 0.8940.7910.832 0.402-0.724-2.720.16-0.51
44 44 66.4937.46 59.21307.61 29.7232.510.914 0.8940.7910.832 0.378-0.648-2.570.14-0.44
46 46 69.839.43 61.941372.78 29.8432.660.914 0.8940.7910.832 0.356-0.583-2.440.13-0.37
48 48 73.1341.4 64.691438.26 29.9632.820.913 0.8930.7910.832 0.336-0.526-2.320.12-0.32

The Lagrangian method for obtaining the least-cost combination of inputs, yields values for the solution variables µ, L, K, and M for specified values of q, wL, wK, and wM. That is, the solution variables are functions:

µ = µ(q; wL, wK, wM), L = L(q; wL, wK, wM), K = K(q; wL, wK, wM), M = M(q; wL, wK, wM).

The Lagrangian method also produces the Jacobian matrix (bordered Hessian of G = -G) of the four functions, Gµ, GL, GK, GM, with respect to the choice variables, μ, L, K, and M:

J3   =

=
 0 -FL -FK -FM -FL -µ * FLL -µ * FLK -µ * FLM -FK -µ * FKL -µ * FKK -µ * FKM -FM -µ * FML -µ * FMK -µ * FMM

The Jacobian matrix of the four solution functions, Φ = {µ, L, K, M}, with respect to the variables, q, wL, wK, and wM equals the negative of the matrix inverse of J3, yielding the comparative statics of the solution functions.

For example, at q = 30, wL = 7,   wK = 13, and   wM = 6 :

JΦ   =

 µq µwL µwK µwM Lq LwL LwK LwM Kq KwL KwK KwM Mq MwL MwK MwM
=
 0.107 1.587 0.921 1.355 1.587 -3.424 1.098 1.616 0.921 1.098 -1.024 0.937 1.355 1.616 0.937 -3.915

The Uzawa partial elasticities of substitution at these same values of q, wL, wK, and wM:

 uLK = C(q;wL,wK,wM) * LwK / (L * K)   =   0.894, uLM = C(q;wL,wK,wM) * LwM / (L * M)   =   0.792, uKL = C(q;wL,wK,wM) * KwL / (K * L)   =   0.894, uKM = C(q;wL,wK,wM) * KwM / (K * M)   =   0.833, uML = C(q;wL,wK,wM) * MwL / (M * L)   =   0.792, uMK = C(q;wL,wK,wM) * MwK / (M * K)   =   0.833,

The factor demand elasticities at these same values of q, wL, wK, and wM:

 εL,q εL,wL εL,wK εL,wM εK,q εK,wL εK,wK εK,wM εM,q εM,wL εM,wK εM,wM
=
 1.088 -0.547 0.326 0.221 1.144 0.318 -0.551 0.233 1.013 0.282 0.304 -0.585

where, for example, εL,wK = ∂ln(L(q; wL, wK, wM)) / ∂ln(wK) = (∂ln(L(q; wL, wK, wM)) / ∂wK) * (∂wK / ∂ln(wK)) = (1 / L(q; wL, wK, wM)) * (∂L(q; wL, wK, wM) / ∂wK) * wK = LwK * wK / L

The cost function C(q;wL,wK,wM) is a concave in factor prices if the Hessian matrix ∇2wwC(q;wL,wK,wM) of second order partial derivatives with respect to factor prices is negative semidefinite. The matrix ∇2wwC(q;wL,wK,wM) is negative semidefinite if its eigenvalues are nonpositive.

2wwC(q;wL,wK,wM)   =

 LwL LwK LwM KwL KwK KwM MwL MwK MwM
=
 -3.424 1.098 1.616 1.098 -1.024 0.937 1.616 0.937 -3.915

The eigenvalues of ∇2wwC  are e1 = -5.3039, e2 = -3.0594, and e3 = 0.

IV. Short Run: Capital Fixed: K = K = 24.14.

Let the firm buy its inputs at the same prices: wL = 7   wK = 13   wM = 6 . Set the level of capital at the least cost level for q = 30.   Solve the least-cost problem holding capital fixed to obtain the data in the following table. (See the discussion of the short-run elasticity of substitution, sLM, on the translog production function page.)

The Lagrangian G(q;L,K,M,µ) obtains a maximum at q, L, K, and M if the the bordered determinant |H2| is positive. The Generalized CES production function, f(L, K, M), is concave to the origin of the 2-dimensional (L, M) space if |h1| < 0, and |h2| > 0. (See the table below, and the Mathematical Notes.)

Generalized CES Short Run Cost Data
Parameters: nu = 1, rho = 0.17647, rhoL = 0.17647, rhoK = 0.11823, rhoM = 0.26339
wL = 7, wK= 13, wM = 6; K = 24.14
q f(L,K,M) LKMcostave. costmarg. costεLKMεLKM3 factor
sLM
2 factor
sLM
|H2|*102|h1|*103|h2|*104
1818 18.4924.14 17.98551.2 30.62 20.640.9450.639 0.790.828.599-14.051.68
2020 21.9624.14 21.1594.19 29.71 22.320.940.628 0.790.826.237-11.011.06
2222 25.7224.14 24.45640.56 29.12 24.030.9350.618 0.790.824.625-8.790.69
2424 29.7724.14 28.02690.37 28.77 25.770.9310.609 0.790.823.494-7.120.46
2626 34.1324.14 31.82743.69 28.6 27.540.9270.6 0.790.822.682-5.840.32
2828 38.824.14 35.85800.59 28.59 29.360.9230.592 0.790.822.088-4.840.22
3030 43.7824.14 40.12861.03 28.7 31.210.920.585 0.790.821.646-4.050.16
3232 49.0924.14 44.64925.35 28.92 33.10.9160.577 0.790.821.311-3.420.11
3434 54.7424.14 49.41993.48 29.22 35.040.9130.571 0.790.821.054-2.910.08
3636 60.7224.14 54.441065.53 29.6 37.020.910.564 0.790.820.855-2.50.06
3838 67.0524.14 59.731141.58 30.04 39.040.9080.558 0.790.820.698-2.150.05
4040 73.7424.14 65.281221.73 30.54 41.120.9050.552 0.790.820.575-1.860.04
4242 80.824.14 71.11306.07 31.1 43.240.9020.547 0.790.820.476-1.620.03
4444 88.2324.14 77.21394.69 31.7 45.420.90.541 0.790.820.397-1.420.02
4646 96.0624.14 83.581487.7 32.34 47.640.8970.536 0.790.820.332-1.250.02
4848 104.2724.14 90.241585.19 33.02 49.920.8950.531 0.790.820.28-1.10.01

The Lagrangian method for obtaining the least-cost combination of inputs, yields values for the solution variables µ, L, and M for specified values of q, wL, wK, and wM. That is, the solution variables are functions:

µ = µ(q; wL, wK, wM), L = L(q; wL, wK, wM), M = M(q; wL, wK, wM).

The Lagrangian method also produces the Jacobian matrix (bordered Hessian of G = -G) of the three functions, Gµ, GL, GM, with respect to the choice variables, μ, L, and M:

J2   =

=
 0 -FL -FM -FL -µ * FLL -µ * FLM -FM -µ * FML -µ * FMM

The Jacobian matrix of the three solution functions, Φ = {µ, L, M}, with respect to the variables, q, wL, wK, and wM equals the negative of the matrix inverse of J2, yielding the comparative statics of the solution functions.

For example, at q = 30,   K = K = 24.14,   wL = 7,   wK = 13,  and   wM = 6 :

JΦ   =

 µq µwL µwM Lq LwL LwM Mq MwL MwM
=
 0.935 2.575 2.197 2.575 -2.245 2.62 2.197 2.62 -3.056

The factor demand elasticities at q = 30, wL = 7,   wK = 13, and   wM = 6 :

 εL,q εL,wL εL,wM εM,q εM,wL εM,wM
=
 1.764 -0.359 0.359 1.643 0.457 -0.457

where, for example, εL,wM = ∂ln(L(q; wL, wK, wM)) / ∂ln(wM) = (∂ln(L(q; wL, wK, wM)) / ∂wM) * (∂wM / ∂ln(wM)) = (1 / L(q; wL, wK, wM)) * (∂L(q; wL, wK, wM) / ∂wM) * wM = LwM * wM / L

The short run cost function C(q;wL,wK,wM) is a concave in factor prices if the Hessian matrix ∇2wwC(q;wL,wK,wM) of second order partial derivatives with respect to factor prices is negative semidefinite. The matrix ∇2wwC(q;wL,wK,wM) is negative semidefinite if its eigenvalues are nonpositive.

2wwC(q;wL,wK,wM)   =

 LwL LwM MwL MwM
=
 -2.245 2.62 2.62 -3.056

The eigenvalues of ∇2wwC  are e1 = -5.3017, and e2 = -0 .

Graph of Average Cost and Marginal Cost
Generalized CES Production / Cost Functions - Capital Fixed

wL = 7, wK= 13, wM = 6. Average cost function Marginal cost function L.R. Average cost function

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^-rhoL) + beta * (K^-rhoK) + gamma *(M^-rhoM)]^(-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 the Generalized CES production function as specified above:

q = 1 * [0.35 * L^- 0.17647 + 0.4 * K^- 0.11823 + 0.25 *M^- 0.26339]^(-1/0.17647)

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

(L, K, M) = (43.79, 24.14, 40.13), and

C(30) = 7 * 43.79 + 13 * 24.14 + 6 * 40.13 = 861.17.

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

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

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

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 = (861.17 - 6 * 40.13)/13 = 47.72,
while the L-M isocost line has a M-intercept at (C(30) - wK * K) / wM = (861.17 - 13 * 24.14)/6 = 91.22.

For q = 30, the L-K isoquant is tangent to the L-K isocost line at (L, K) = (43.79, 24.14), while the L-M isoquant is tangent to the L-M isocost line at (L, M) = (43.79, 40.13).

L-K Isoquants, M = ML-M Isoquants, K = K  Mathematical Notes

I. The Generalized CES Production Function

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.

Elasticity of Scale of Production:

Long Run: Capital Variable:

Using the partial derivatives of f(L,K,M) below we get:
dq/q = (nu / rho) * (alpha * rhoL * L^-rhoL * dL / L + beta * rhoK * K^-rhoK * dK / K + gamma * rhoM * M^-rhoM * dM / M) / (alpha * L^-rhoL + beta * K^-rhoK + gamma * M^-rhoM)

Substitute du/u = dL/L = dK/K = dM/M in the equation above, and divide by du/u.

Then,   (dq/q)/(du/u) = ε(L,K,M),   and

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).

If rho = 0, εLKM = ε(L,K,M) = alpha + beta + gamma.

Short Run: Captial Fixed: K = K:

Set dK = 0 in the equation for dq/q with du/u = dL/L = dM/M, and divide by du/u.

εLKM = ε(L,K,M) = (nu / rho) * (alpha * rhoL * L^-rhoL + gamma * rhoM * M^-rhoM) / (alpha * L^-rhoL + beta * K^-rhoK + gamma * M^-rhoM).

If rho = 0, εLKM = ε(L,K,M) = alpha + gamma.

Curvature:

The Generalized CES production function, f(L,K,M), is concave to the origin of the 3-dimensional (L,K, M) space if its Hessian is negative (semi)definite. Define the matrices h1, h2, and h3:

 fLL fLK fLM fKL fKK fKM fML fMK fMM
=|h3|
 fLL fLK fKL fKK
=|h2|
 fLL
=|h1|

The Hessian of f(L,K,M) is negative definite if the determinants |h1|, |h2|, and |h3| alternate in sign, starting with negative. If one or more the determinants have a zero value, then the Hessian of f(L,K,M) is negative semidefinite, and f(L, K, M) is quasi-concave to the origin of the 3-dimensional space of (L,K,M).

II. The partial derivatives of f(L,K,M)

 fL(L,K,M) = nu * alpha * (rhoL/rho) * A^(-rho/nu) * L ^-(1 + rhoL) * f(L,K,M)^(1 + rho/nu), fK(L,K,M) = nu * beta * (rhoK/rho) * A^(-rho/nu) * K ^-(1 + rhoK) * f(L,K,M)^(1 + rho/nu), fM(L,K,M) = nu * gamma *(rhoM/rho) * A^(-rho/nu) * M ^-(1 + rhoM) * f(L,K,M)^(1 + rho/nu), fLL =(fL(L,K,M))/f(L,K,M) *( (1 + rho/nu) * fL(L,K,M) - (1 + rhoL)*f(L,K,M)/L ),   fLK = (fL(L,K,M))/f(L,K,M))*( (1 + rho/nu) * fK(L,K,M) ),   fLM = fL(L,K,M))/f(L,K,M))*( (1 + rho/nu) * fM(L,K,M) ),   fKK = (fK(L,K,M))/f(L,K,M) *( (1 + rho/nu) * fK(L,K,M) - (1 + rhoK)*f(L,K,M)/K ),   fKL = fLK,   fKM = fK(L,K,M))/f(L,K,M))*( (1 + rho/nu) * fM(L,K,M) ), fMM = (fM(L,K,M))/f(L,K,M) *( (1 + rho/nu) * fM(L,K,M) - (1 + rhoM)*f(L,K,M)/M ),   fML = fLM,   fMK = fKM.

III. The least-cost combination of inputs:

Find the values of L, K, M, and µ that minimize the Lagrangian when the factor prices are wL, wK, and wM:

G(q;L,K,M,µ) = wL * L + wK * K + wM * M + µ * [q - f(L,K,M)],

or equivalently, that maximize the Lagrangian G(q;L,K,M,µ) = - G(q;L,K,M,µ).

G(q;L,K,M,µ) = -(wL * L + wK * K + wM * M) - µ * [q - f(L,K,M)].

First Order Conditions:

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

Jacobian of Second Order Conditions (Bordered Hessian with F(L,K,M) = f(L,K,M)):

 Gµµ GµL GµK GµM GLµ GLL GLK GLM GKµ GKL GKK GKM GMµ GML GMK GMM
=
 0 FL FK FM FL GLL GLK GLM FK GKL GKK GKM FM GML GMK GMM
=|H3|
 0 FL FK FL GLL GLK FK GKL GKK
= |H2|

where:

 (0,0) Gµµ = 0,   (0,1) GµL = fL,   (0,2) GµK = fK,   (0,3) GµM = fM (1,0) GLµ = GµL,   (1,1) GLL = µ * fLL,   (1,2) GLK = µ * fLK,   (1,3) GLM = µ * fLM,   (2,0) GKµ = GµK,   (2,1) GKL = GLK,   (2,2) GKK = µ * fKK,   (2,3) GKM = µ * fKM (3,0) GMµ = GµM,   (3,1) GML = GLM,   (3,2) GMK = GKM,   (3,3) GMM = µ * fMM,

IV. The least-cost combination of inputs: Capital fixed:

With the value of the capital value fixed at K = K, find the values of L, M, and µ that minimize the Lagrangian when the factor prices are wL, wK, and wM:

G(q;L,K,M,µ) = wL * L + wK * K + wM * M + µ * [q - f(L,K,M)],

or equivalently, that maximize the Lagrangian G(q;L,K,M,µ) = - G(q;L,K,M,µ).

G(q;L,K,M,µ) = -(wL * L + wK * K + wM * M) - µ * [q - f(L,K,M)].

First Order Conditions: K = K:

 0. Gµ = f(L,K,M) - q = 0 1. GL = -wL + µ * fL = 0 2. GM = -wM + µ * fM = 0

Jacobian of Second Order Conditions: K = K: (Bordered Hessian with F(L,K,M) = f(L,K,M)):

 Gµµ GµL GµM GLµ GLL GLM GMµ GML GMM
=
 0 FL FM FL GLL GLM FM GML GMM
=|H2|

where:

 (0,0) Gµµ = 0,   (0,1) GµL = fL,   (0,2) GµM = fM (1,0) GLµ = GµL,   (1,1) GLL = µ * fLL,   (1,2) GLM = µ * fLM,   (2,0) GMµ = GµM,   (2,1) GML = GLM,   (2,2) GMM = µ * fMM,

V. 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|)

VI. 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| Copyright © Elmer G. Wiens:   Egwald Web Services All Rights Reserved.    Inquiries 