     Egwald Economics: Microeconomics

Duality and the Translog Production / Cost Functions
Homothetic CES Technology

by

Elmer G. Wiens

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R. Translog (Transcendental Logarithmic) Duality and the CES Technology

I. Profit (Wealth) Maximizing Firm.

Production and cost functions (and profit functions) can be used to model how a profit (wealth) maximizing firm hires or purchases inputs (factors), such as labour, capital (structures and machinery), and materials and supplies, and combines these inputs through its production process to produce the products (outputs) that the firm sells (supplies) to its customers. The theory of duality links the production function models to the cost function models by way of a minimization or maximization framework. The cost function is derived from the production function by choosing the combination of factor quantities that minimize the cost of producing levels of output at given factor prices. Conversely, the production function is derived from the cost function by calculating the maximum level of output that can be obtained from specified combinations of inputs.

II. The Production and Cost Functions.

The Translog production function:

ln(q) = ln(A) + aL*ln(L) + aK*ln(K) + aM*ln(M) + bLL*ln(L)*ln(L) + bKK*ln(K)*ln(K) + bMM*ln(M)*ln(M)
+ bLK*ln(L)*ln(K) + bLM*ln(L)*ln(M) + bKM*ln(K)*ln(M)   =   f(L, K, M).

q = exp(f(L, K, M)) = F(L, K, M).

an equation in 10 parameters, A, aL, aK, aM, bLL, bKK, bMM, bLK, bLM, bKM, where L = labour, K = capital, M = materials and supplies, and q = product.

The Translog (total) cost function:

 ln(C(q;wL,wK,wM)) = c + cq * ln(q) + cL * ln(wL) + cK * ln(wK) + cM * log(wM)                                 + .5 * [dqq * ln(q)^2 + dLL * ln(wL)^2 + dKK * ln(wK)^2 + dMM * ln(wM)^2]                   + .5 * [(dLK + dKL) * ln(wL)*ln(wK) + (dLM + dML) * ln(wL)*ln(wM) + (dKM + dMK) * ln(wK)*log(wM)]                   + dLq * ln(wL)*ln(q) + dKq * ln(wK)*ln(q) + dMq * ln(wM)*ln(q)

an equation in 18 parameters, c, cq, cL, cK, cM, dqq, dLL, dKK, dMM, dLK, dKL, dLM, dML, dKM, dMK, dLq, dKq, and dMq, where wL, wK, and wM are the factor prices of L, K, and M respectively.

III. Duality between Production and Cost Functions.

Mathematically, the duality between a production function, q = F(L, K, M), and a cost function, C(q; wL, wK, wM), is expressed:

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,   L > 0, K > 0, M > 0}     (*)

F*(L, K, M) = maxq {q   :   C(q; wL, wK, wM)   <=   wL * L + wK * K + wM * M,   L > 0, K > 0, M > 0,   for all wL >= 0 , wK >= 0, wM >= 0}     (**),

with the questions, F == F*, and C == C*?

The dual functions C*, and F* will be derived from the estimated functions F, and C.

If the functions F(L, K, M), and C(q; wL, wK, wM) obey sufficient conditions, the above minimization and maximization problems can be solved by nonlinear optimization techniques, such as Newton's Method. Furthermore, the implicit function theorem can be exploited to facilitate such calculations.

IV. Constrained Optimization.

The (*) minimization problems are solved for the Generalized CES production function, the Translog production function, and for the Diewert (Generalized Leontief) production function.

The maximization problem (**) is somewhat more difficult than the minimization problem (*), since as specified, the requirements   —   for all wL >= 0 , wK >= 0, wM >= 0   —   imply an infinite number of constraints.

This difficulty is overcome with cost functions that factor:

C(q; wL, wK, wM) = q^1/nu * c(wL, wK, wM)

by solving the reduced dimension problem:

F*(L, K, M)^1/nu = 1 / maxwL,wK,wM {c(wL, wK, wM)   :   wL * L + wK * K + wM * M = 1,   wL >= 0, wK >= 0, wM >= 0)

However, with cost functions that do not factor, such as the Translog cost function, the following method can be used to reduce the dimensionality of the constraints of (**), if the cost function is linear homogeneous in factor prices.

Cost Function Linear Homogeneous in Factor Prices.

When (L, K, M) is the least cost combination of inputs at the specific combination of factor prices (wL, wK, wM):

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

With the cost function, C(q; wL, wK, wM), linear homogeneous in factor prices:

C(q; wL / wM, wK / wM, 1)   =   wL / wM * L + wK / wM * K + M,

assuming that wM > 0. Writing vL = wL / wM, and vK = wK / wM, the maximization problem (**) becomes the minimization problem:

F*(L, K, M) = minq, vL, vK {q   :   C(q; vL, vK, 1)   >=   vL * L + vK * K + M,   L > 0, K > 0, M > 0,   for all vL >= 0 , vK >= 0}     (***),

an optimization problem with one constraint.

V. Cost Function to Production Function: The Method of Langrange.

Using the Method of Lagrange, define the Langrangian function, H, of the minimization problem (***):

H(L, K, M, q, vL, vK, λ) = q + λ * (vL * L + vK * K + M - C(q; vL, vK, 1)),

where the new variable, λ, is called the Lagrange multiplier.

a.   First Order Necessary Conditions:

 0.   Hλ(L, K, M, q, vL, vK, λ) = vL * L + vK * K + M - C(q; vL, vK, 1) = 0 1.   Hq(L, K, M, q, vL, vK, λ) = 1 - λ * ∂C(q; vL, vK, 1)/∂q = 0 2.   HvL(L, K, M, q, vL, vK, λ) = λ * (L - ∂C(q; vL, vK, 1)/∂vL) = 0 3.   HvK(L, K, M, q, vL, vK, λ) = λ * (K - ∂C(q; vL, vK, 1)/∂vK) = 0

b.   Solution Functions:

We want to solve, simultaneously, these four equations for the variables λ, q, vL, and vK as functions of the variables L, K, and M, and the parameters of the cost function.

λ = λ(L, K, M)
q = q(L, K, M)
vL = vL(L, K, M)
vK = vK(L, K, M)

c.   Jacobian Matrices:

The Jacobian matrix (bordered Hessian of H) of the four functions, Hλ, Hq, HvL, HvK, with respect to the choice variables, λ, q, vL, vK :

J3   =

 Hλλ Hλq HλvL HλvK Hqλ Hqq HqvL HqvK HvLλ HvLq HvLvL HvLvK HvKλ HvKq HvKvL HvKvK
=
 0 -Cq L - CvL K - CvK -Cq -λ*Cqq -λ*CqvL -λ*CqvK L - CvL C-λ*CqvL -λ*CvLvL -λ*CvLvK K - CvK -λ*CqvK -λ*CvKvL -λ*CvKvK

=   Jλ, q, vL, vK

The bordered principal minor of the bordered Hessian of the Langrangian function, H:

J2   =

 0 -Cq L - CvL -Cq -λ*Cqq -λ*CqvL L - CvL -λ*CqvL -λ*CvLvL

d.   Second Order Necessary Conditions:

The second order necessary conditions require that the Jacobian matrix (bordered Hessian of H) be positive definite at the solution vector W = (L, K, M, q, vL, vK, λ). The Jacobian matrix, J3, is a positive definite matrix if the determinant of J2 is negative, and the determinant of J3 is negative.

e.   Sufficient Conditions:

If the second order necessary conditions are satisfied, then the first order necessary conditions are sufficient for a minimum at W.

f. The Solution Functions' Comparative Statics.

The Jacobian matrix of the four functions, Hλ, Hq, HvL, HvK, with respect to the variables, L, K, M:

JL, K, M   =

 Hλ,L; Hλ,K Hλ,M Hq,L Hq,K Hq,M HvL,L HvL,K HvL,M HwK,L HwK,K HvK,M
=
 vL vK 1 0 0 0 λ 0 0 0 λ 0

The Jacobian matrix of the four solution functions, Φ = {λ, q, vL, vK}, with respect to the variables, L, K, and M:

JΦ   =

 λL λK λM qL qK qM vLL vLK vLM vKL vKK vKM

From the Implicit Function Theorem:

JL, K, M;   +   Jλ, q, vL, vM   * JΦ   =   0 (zero matrix)   →

JΦ   =   - (Jλ, q, vL, vK)-1   *   JL, K, M

for points (L, K, M) in a neighborhood of the point (L, K, M), and wL = wL(L, K, M), wK = wK(L, K, M), wM = wM(L, K, M), and λ = λ(L, K, M),

with:

 F*(L, K, M) = q(L, K, M), F*L(L, K, M) = qL(L, K, M), F*K(L, K, M) = qK(L, K, M), and F*M(L, K, M) = qM(L, K, M),

VI. Question: F == F*?

The obtained values for F*(L, K, M), F*L(L, K, M), F*K(L, K, M), and F*M(L, K, M) can be compared with the values of F(L, K, M), FL(L, K, M), FK(L, K, M), and FM(L, K, M), and with the corresponding values from the CES production function.

VII. Duality.  The Plan:

1. Specify the parameters of a CES production function, and thereby the corresponding parameters of its dual CES cost function.

2. Generate the CES data displayed in the table below, and estimate the parameters of a Translog production function, and the parameters of a Translog cost function.

3. Using Newton's Method with the implicit function theorem, obtain the production function that is dual to the estimated Translog cost function. Check that the derived production function corresponds with the estimated Translog production function, and the underlying CES production function.

4. Using Newton's Method with the implicit function theorem, obtain the cost function that is dual to the estimated Translog production function. Check that the derived cost function corresponds with the estimated Translog cost function, and the underlying CES cost function.

VIII. Generate CES Data.

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).

The CES cost function:

C(q;wL,wK,wM) = h(q) * c(wL,wK,wM) = (q/A)^1/nu * [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 cost function's factor prices, wL, wK, and wM, are positive real numbers.

The estimated coefficients of the estimated Translog production and cost functions will vary with the parameters sigma, nu, alpha, beta and gamma of the CES production function.

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
15 < q < 45;
4 <= wL* <= 11,   7<= wK* <= 16,   4 <= wM* <= 10

CES Production Function Parameters
elasticity of scale parameter: nu
elasticity of substitution: sigma
alpha
beta
gamma
Base Factor Prices
 wL* wK* wM*
Distribution to Randomize Factor Prices
 Use [-2, 2] Uniform distribution     Use .25 * Normal (μ = 0, σ2 = 1)

 The CES production function as specified: f(L, K, M) = 1 * [0.35 * (L^- 0.17647) + 0.4 * (K^- 0.17647) + 0.25 *(M^- 0.17647)]^(-1/0.17647) The CES cost function as specified: C(q;wL,wK,wM) = h(q) * c(wL,wK,wM) = (q/1)^1/1 * [0.35^(1/(1+0.17647))) * wL^(0.17647)/(1+0.17647))) + 0.4^(1/(1+0.17647))) * wK^(0.17647)/(1+0.17647))) + 0.25^(1/(1+0.17647))) * wM^(0.17647)/(1+0.17647)))]^((1+0.17647))/0.17647))

For these coefficients of the CES production / cost functions, I generated a sequence of factor prices, outputs, and the corresponding cost minimizing inputs. The factor prices are distributed about the base factor prices by adding a random number distributed uniformly in the [-2, 2] domain.

IX. Estimate the Translog Production and Cost Functions.

Estimating the Translog production function and Translog cost function from the CES data using SVD least squares yields the coefficient estimates:

CES
Translog Production Function

SVD Least Squares
Parameter Estimates
Parameter Coefficient std error t-ratio
lnA-4.9E-50-0.286
aL0.34992602816.622
aK0.39983302778.33
aM0.25026302763.692
bLL-0.0196690-346.252
bKK-0.0212780-270.214
bMM-0.0164410-473.252
bLK0.0245470251.887
bLM0.0147770292.767
bKM0.0180640213.293
 R2 = 1 R2b = 1 # obs = 28 Observation Matrix Rank: 10
 aL + aK + aM = 1-2*bLL = 0.039339 =~ 0.039324 = bLK + bLM-2*bKK = 0.042556 =~ 0.04261 = bLK + bKLM-2*bMM = 0.032882 =~ 0.032841 = bLM + bKM

CES
Translog Cost Function

SVD Restricted Least Squares
Parameter Estimates
Parameter Coefficient std error t-ratio
c1.0832880.0002474381.481456
cq0.9999530.0001397216.774749
cL0.3484720.0001362567.385945
cK0.3898210.0001632395.895215
cM0.2617070.0001162247.575797
dqq3.0E-64.2E-50.067449
dLL0.033430.000108310.473282
dKK0.0362110.000138262.580678
dMM0.0279336.7E-5416.079217
2*dLK-0.0417080.0002-208.084935
2*dLM-0.0251510.00011-228.543017
2*dKM-0.0307140.000175-175.916992
dLq-2.9E-57.2E-5-0.401193
dKq0.0001110.0001021.08456
dMq-8.2E-56.2E-5-1.320887
 R2 = 1 R2b = 1 # obs = 28 Observation Matrix Rank: 15
 1 = cL + cK + cM 0 = dLL + dLK + dLM 0 = dKL + dKK + dKM 0 = dML + dMK + dMM 0 = dLq + dKq + dMq

The estimated Translog production function:

ln(q) = -4.9E-5 + 0.349926 * ln(L) + 0.399833 * ln(K) + 0.250263 * ln(M) + -0.019669 * ln(L)*ln(L) + -0.021278 * ln(K)*ln(K) + -0.016441 * ln(M)*ln(M)
+ 0.024547 * ln(L)*ln(K) + 0.014777 * ln(L)*ln(M) + 0.018064 * ln(K)*ln(M)   =   f(L,K,M).

The estimated Translog cost function:

ln(C(q;wL,wK,wM)) = 1.083288 + 0.999953 * ln(q) + 0.348472 * ln(wL) + 0.389821 * ln(wK) + 0.261707 * log(wM)
+ .5 * [3.0E-6 * ln(q)^2 + 0.03343 * ln(wL)^2 + 0.036211 * ln(wK)^2 + 0.027933 * ln(wM)^2]
+ .5 * [-0.041708 * ln(wL)*ln(wK) + -0.025151 * ln(wL)*ln(wM) + -0.030714 * ln(wK)*log(wM)]
+ -2.9E-5 * ln(wL)*ln(q) + 0.000111 * ln(wK)*ln(q) + -8.2E-5 * ln(wM)*ln(q)

X. Example: Cost Function to Production Function:

The dual Translog production function, F*, is obtained from the estimated Translog cost function, C, by:

F*(L, K, M) = minq, vL, vK {q   :   C(q; vL, vK, 1)   >=   vL * L + vK * K + M,   L > 0, K > 0, M > 0,   for all vL >= 0 , vK >= 0}     (***).

With L = 33.2, K = 21.97, and M = 28.43, (***) becomes:

F*(33.2, 21.97, 28.43) = minq, vL, vK {q   :   C(q; vL, vK, 1)   >=   vL * 33.2 + vK * 21.97 + 28.43,   for all vL >= 0 , vK >= 0}     (***).

XI. Constrained Optimization (Minimum): The Method of Lagrange:

H(33.2, 21.97, 28.43, q, vL, vK, λ) = q + λ * (vL * 33.2 + vK * 21.97 + 28.43 - C(q; vL, vK, 1)),

where λ is the Lagrange multiplier.

a. First Order Necessary Conditions:

 0.   Hλ(33.2, 21.97, 28.43, q, vL, vK, λ) = vL * 33.2 + vK * 21.97 + 28.43 - C(q; vL, vK, 1) = 0 1.   Hq(33.2, 21.97, 28.43, q, vL, vK, λ) = 1 - λ * ∂C(q; vL, vK, 1)/∂q = 0 2.   HvL(33.2, 21.97, 28.43, q, vL, vK, λ) = λ * (33.2 - ∂C(q; vL, vK, 1)/∂vL) = 0 3.   HvK(33.2, 21.97, 28.43, q, vL, vK, λ) = λ * (21.97 - ∂C(q; vL, vK, 1)/∂vK) = 0

b.   Solution Functions:

Solve these four equations simultaneously (using Newton's Method) for λ, q, vL, and vK as functions of the variables L, K, and M, and the parameters of the cost function, so that:

λ = λ(33.2, 21.97, 28.43)
q = q(33.2, 21.97, 28.43)
vL = vL(33.2, 21.97, 28.43)
vK = vK(33.2, 21.97, 28.43)

We know that the ranges of factor prices are: 4 <= wL <= 11, 7 <= wK <= 16, and 4 <= wM <= 10. Estimating wL = 7.5, wK = 11.5, and wM = 7 yields vL = wL / wM = 7.5 / 7, and vK = wK / wM = 11.5/7. With a range of 20 to 45 for output, estimate q = 33. From the first order condition 1., estimate λ = 1 / ∂C(33; 7.5/7, 11.5/7, 1)/∂q = 0.2712.

Newton's Method:

Using these estimates, Newton's Method provides:

Nonlinear Optimization: Newton's Method
Parameter Estimates
Iter #λqvLvK
0   0.2712 331.0714 1.6429
10.2595 26.741.0484 1.8471
20.2364 26.991.1506 2.1155
30.2352 271.1665 2.1672
40.2351 271.167 2.1687
50.2351 271.167 2.1687

c. Solution Vector:

W = (L, K, M, q, vL, vK, λ) = (33.2, 21.97, 28.43, 27, 1.16701, 2.16874, 0.23508)

With λ = 0.23508, q = 27, vL = 1.16701, and vK = 2.16874, the first order conditions are:

 0.   Hλ(33.2, 21.97, 28.43, q, vL, vK, λ) = 1.16701 * 33.2 + 2.16874 * 21.97 + 28.43 - C(q; 1.16701, 2.16874, 1) = 0 1.   Hq(33.2, 21.97, 28.43, q, vL, vK, λ) = 1 - 0.23508 * ∂C(q; 1.16701, 2.16874, 1)/∂q = 0 2.   HvL(33.2, 21.97, 28.43, q, vL, vK, λ) = 0.23508 * (33.2 - ∂C(q; 1.16701, 2.16874, 1)/∂vL) = 0 3.   HvK(33.2, 21.97, 28.43, q, vL, vK, λ) = 0.23508 * (21.97 - ∂C(q; 1.16701, 2.16874, 1)/∂vK) = -0

d. Second Order Necessary Conditions:

The second order necessary conditions require that the Jacobian matrix (bordered Hessian of H) be positive definite at the solution vector W = (L, K, M, q, vL, vK, λ). The Jacobian matrix, J3, is a positive definite matrix if the determinant of J2 is negative, and the determinant of J3 is negative.

Jλ, q, vL, vK =

J3   =

 0 -Cq L - CvL K - CvK -Cq -λ*Cqq -λ*CqvL -λ*CqvK L - CvL C-λ*CqvL -λ*CvLvL -λ*CvLvK K - CvK -λ*CqvK -λ*CvKvL -λ*CvKvK
=
 0 -4.254 0 -0 -4.254 -0 -0.289 -0.191 0 -0.289 3.769 -1.271 -0 -0.191 -1.271 1.186

Determinant(J3) = -51.61622

J2   =

 0 -4.254 0 -4.254 -0 -0.289 0 -0.289 3.769

Determinant(J2) = -68.19397

e. Maximum Output:

With L = 33.2, K = 21.97, M = 28.43,

Dual Translog production function:

F*(33.2, 21.97, 28.43) = q(33.2, 21.97, 28.43) = 27,

Estimated Translog production function:

F(33.2, 21.97, 28.43) = 27.

Specified CES production function:

f(33.2, 21.97, 28.43) = 27.

f. The Solution Functions' Comparative Statics.

JL, K, M   =

 vL vK 1 0 0 0 λ 0 0 0 λ 0
=
 1.16701 2.16874 1 0 0 0 0.2351 0 0 0 0.2351 0

From the Implicit Function Theorem:

JΦ   =   - (Jλ, wL, wK, wM, q)-1   *   JL, K, M

JΦ   =

 λL λK λM qL qK qM vLL vLK vLM vKL vKK vKM
=
 0.002811 0.005219 -0.00732 0.27434 0.50983 0.23508 -0.04134 -1e-05 0.04829 -3e-05 -0.11599 0.08973

g. Comparing Partial Derivatives:

Partial Derivates of the dual Translog production function F*(L,K,M):

F*L(L,K,M) = F*(33.2, 21.97, 28.43) = qL(33.2, 21.97, 28.43) = 0.274,
F*K(L,K,M) = F*(33.2, 21.97, 28.43) = qK(33.2, 21.97, 28.43) = 0.51,
F*M(L,K,M) = F*(33.2, 21.97, 28.43) = qM(33.2, 21.97, 28.43) = 0.235.

Partial Derivatives of the estimated Translog production function F(L,K,M):

FL(33.2, 21.97, 28.43) = 0.274,
FK(33.2, 21.97, 28.43) = 0.51,
FM(33.2, 21.97, 28.43) = 0.235.

Partial Derivatives of the specified CES production function f(L,K,M):

fL(33.2, 21.97, 28.43) = 0.274,
fK(33.2, 21.97, 28.43) = 0.51,
fM(33.2, 21.97, 28.43) = 0.235.

XII. Table of Results.

Check that the derived dual production function corresponds with the estimated Translog production function by comparing the values for output, q, and the partial derivatives of the production function.

Translog Production / Cost Function Duality
Cost Function to Production Function
CES: Returns to Scale = 1, Elasticity of Substitution = 0.85
—       CES Data   —    —    Estimated Translog Cost   —   — Derived Dual Production — — Estimated Translog Production —
obs #qwLwKwM LK Mcost LKMcostqF*LF*KF*MqFLFKFM
1177.3 12.026.12 19.9514.62 17.41427.919.9414.6317.4427.9516.9980.290.4780.243170.290.4780.243
2185.26 12.566 25.7113.74 17.27411.4725.7113.7517.27411.5217.9980.230.550.262180.230.5490.263
3198.74 14.925.7 21.4315.23 23.15546.4921.4315.2423.14546.5618.9970.3040.5190.198190.3040.5190.198
4206.48 11.24.58 23.0316.2 23.24437.1423.0316.2123.23437.219.9970.2960.5130.209200.2960.5120.21
5218.8 14.16.74 23.9417.97 22.57616.1423.9417.9822.56616.2120.9980.30.4810.23210.30.4810.23
6227.34 12.224.14 23.8717.34 29.18507.9123.8717.3529.16507.9921.9960.3180.530.179220.3180.5290.179
7237.04 12.94 25.8617.31 31.42531.0925.8717.3331.39531.1922.9960.3050.5590.173230.3050.5590.173
8247.3 12.266.72 28.9220.85 23.31623.4428.9220.8623.3623.523.9970.2810.4720.259240.2810.4720.259
9258.38 11.967.6 28.3923.5 23.17695.0728.3823.5123.16695.1224.9980.3010.430.273250.3010.430.273
10267.8 14.227.04 32.1121.59 26.32742.7732.1121.626.31742.8625.9970.2730.4980.246260.2730.4980.246
11277.58 12.146.68 31.7123.8 26.52706.4731.7123.8126.51706.5426.9970.290.4640.255270.290.4640.255
12287.92 14.45.56 32.7822.09 33.27762.6832.7822.133.25762.7927.9960.2910.5290.204280.2910.5290.204
13295.68 11.727.82 40.9824.8 23.46706.9140.9824.8223.44706.9728.9980.2330.4810.32290.2330.4810.321
14307 136 36.8924.42 31.59765.1736.8924.4331.58765.2829.9960.2740.510.235300.2740.510.235
15316.1 11.585.24 38.4324.97 32.85695.738.4324.9832.84695.830.9960.2720.5160.233310.2720.5160.234
16328.06 11.866.86 36.2229.22 31.21852.636.2229.2431.2852.6931.9970.3020.4450.257320.3020.4450.257
17337.84 12.54.14 34.826.22 44.99786.8334.826.2544.94786.9832.9940.3290.5250.173330.3290.5240.174
18347.92 12.284.84 36.5728.22 41.76838.2636.5628.2441.73838.3933.9950.3210.4980.196340.3210.4980.196
19356.34 11.346.68 44.3830.33 31.9838.4144.3830.3531.88838.534.9960.2650.4740.279350.2650.4730.279
20365.28 14.185.94 53.4725.87 36.35865.0353.4725.8836.35865.1935.9930.220.590.247360.220.590.247
21375.24 12.624.22 49.2826.15 44.5775.9949.2926.1644.48776.1536.9930.250.6020.201370.250.6020.201
22386.32 13.167.06 51.4730.91 35.2980.5751.4730.9335.18980.737.9950.2450.510.273380.2450.510.274
23397.88 14.27.76 48.8733.18 37.21144.9648.8733.2137.181145.138.9950.2680.4840.264390.2680.4840.264
24407.34 11.245.16 44.134.39 44.7940.7944.0934.4144.68940.9239.9950.3120.4780.219400.3120.4780.219
25416.12 13.227.6 57.5633.51 35.971068.6357.5633.5335.951068.7740.9950.2350.5080.291410.2350.5070.292
26427.42 13.584.72 48.332.37 53.31049.548.332.3953.261049.7141.9920.2970.5440.189420.2970.5440.189
27438.72 12.764.68 44.1435.77 56.281104.7144.1335.8156.231104.942.9930.3390.4970.182430.3390.4970.182
28445.92 12.644.78 56.1533.01 50.59991.3956.1533.0350.57991.5843.9920.2630.5610.212440.2630.5610.212

XIII. Production Function to Cost Function: The Method of Langrange

The dual cost function, C*(q; wL, wK, wM), is obtained from a production function, q = F(L, K, M), by the constrained optimization:

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,   L > 0, K > 0, M > 0 }     (*)

Define the Langrangian function, G, of the least-cost problem (*):

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

where the new variable, μ, is called the Lagrange multiplier.

a.   First Order Necessary Conditions:

 0.   Gµ(q; wL, wK, wM, L, K, M, μ) = q - F(L, K, M) = 0 1.   GL(q; wL, wK, wM, L, K, M, μ) = wL - µ * FL(L, K, M) = 0 2.   GK(q; wL, wK, wM, L, K, M, μ) = wK - µ * FK(L, K, M) = 0 3.   GM(q; wL, wK, wM, L, K, M, μ) = wM - µ * FM(L, K, M) = 0

b.   Solution Functions:

We want to solve, simultaneously, these four equations for the variables µ, L, K, and M as functions of the variables q, wL, wK, and wM, and the parameters of the production function.

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

c.   Jacobian Matrices:

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

J3   =

 Gµµ GµL GµK GµM GLµ GLL GLK GLM GKµ GKL GKK GKM GMµ GML GMK GMM
=
 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

=   Jµ, L, K, M

The bordered principal minor of the bordered Hessian of the Langrangian function, G:

J2   =

 0 -FL -FK -fL -µ * FLL -µ * FLK -fK -µ * FKL -µ * FKK

d.   Second Order Necessary Conditions:

The second order necessary conditions require that the Jacobian matrix (bordered Hessian of G) be positive definite at the solution vector Z = (q, wL, wK, wM, L, K, M, µ). The Jacobian matrix, J3, is a positive definite matrix if the determinants of J2 and J3 are both negative.

e.   Sufficient Conditions:

If the second order necessary conditions are satisfied, then the first order necessary conditions are sufficient for a minimum at Z.

f. The Solution Functions' Comparative Statics.

The Jacobian matrix of the four functions, Gµ, GL, GK, GM, with respect to the variables, q, wL, wK, and wM:

Jq, wL, wK, wM;   =

 Gµq GµwL GµwK GµwM GLq GLwL GLwK GLwM GKq GKwL GKwK GKwM GMq GMwL GMwK GMwM
=
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1

The Jacobian matrix of the four solution functions, Φ = {µ, L, K, M}, with respect to the variables, q, wL, wK, and wM:

JΦ   =

 µq µwL µwK µwM Lq LwL LwK LwM Kq KwL KwK KwM Mq MwL MwK MwM

From the Implicit Function Theorem:

Jq, wL, wK, wM;   +   Jµ, L, K, M   * JΦ   =   0 (zero matrix)   →

JΦ   =   - (Jµ, L, K, M)-1

for points (q; wL, wK, wM) in a neighborhood of the point (q; wL, wK, wM), and L = L(q; wL, wK, wM), K = K(q; wL, wK, wM), M = M(q; wL, wK, wM), and µ = µ(q; wL, wK, wM).

XIV. Question C == C*?

The obtained values of the solution functions at q, wL, wK, and wM:

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

and the calculated value of the dual Translog cost function:

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

can be compared with the corresponding values of the estimated Translog cost function, and the underlying CES cost function. Moreover, these functions' corresponding comparative static values can also be compared.

XV. Example: Production Function to Cost Function:

The dual Translog cost function, C*, is obtained from the estimated Translog production function, F, by:

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,   L > 0, K > 0, M > 0 }     (*)

With q = 30, wL = 7, wK = 13, and wM = 6, (*) becomes:

C*(30; 7, 13, 6) = minL,K,M{ 7 * L + 13 * K + 6 * M   :   30 - F(L,K,M) = 0,     L > 0, K > 0, M > 0 }     (*)

The Langrangian function, G, of the least-cost problem (*):

G(30; 7, 13, 6, L, K, M, μ) = 7 * L + 13 * K + 6 * M + μ * (30 - F(L,K,M))

where the new variable, μ, is called the Lagrange multiplier.

a.   First Order Necessary Conditions:

 0.   Gµ(30; 7, 13, 6, L, K, M, μ) = 30 - F(L, K, M) = 0 1.   GL(30; 7, 13, 6, L, K, M, μ) = 7 - µ * FL(L, K, M) = 0 2.   GK(30; 7, 13, 6, L, K, M, μ) = 13 - µ * FK(L, K, M) = 0 3.   GM(30; 7, 13, 6, L, K, M, μ) = 13 - µ * FM(L, K, M) = 0

b.   Solution Functions:

We want to solve, simultaneously, these four equations for the variables µ, L, K, and M as functions of the variables q, wL, wK, and wM, and the parameters of the production function, so that.

µ = µ(30; 7, 13, 6)
L = L(30; 7, 13, 6)
K = K(30; 7, 13, 6)
M = M(30; 7, 13, 6)

Suppose we think that the factor inputs for q = 30 are about 20 units. Estimate L = 20, K = 20, and M = 20, and guess µ = 10 to start Newton's Method.

Newton's Method:

Using these estimates, Newton's Method provides:

Nonlinear Optimization: Newton's Method
Parameter Estimates
Iter #µLKM
0   10 2020 20
125.9999 40.2118.9549 33.3851
225.8906 36.8923.605 31.8193
325.5108 36.924.3929 31.6044
425.5058 36.8924.4146 31.5953
525.5058 36.8924.4146 31.5953
625.5058 36.8924.4146 31.5953

c. Solution Vector:

Z = (q, wL, wK, wM, L, K, M, µ) = (30, 7, 13, 6, 36.89, 24.41, 31.6, 25.51)

With µ = 25.51, L = 36.89, K = 24.41, and M = 31.6, the first order conditions are:

 0.   Gµ(30; 7, 13, 6, L, K, M, μ) = 30 - F(36.89, 24.41, 31.6) = -0 1.   GL(30; 7, 13, 6, L, K, M, μ) = 7 - µ * FL(36.89, 24.41, 31.6) = -0 2.   GK(30; 7, 13, 6, L, K, M, μ) = 13 - µ * FK(36.89, 24.41, 31.6) = 0 3.   GM(30; 7, 13, 6, L, K, M, μ) = 13 - µ * FM(36.89, 24.41, 31.6) = -0

d. Second Order Necessary Conditions:

The second order necessary conditions require that the Jacobian matrix (bordered Hessian of G) be positive definite at the solution vector Z = (q, wL, wK, wM, L, K, M, µ). The Jacobian matrix, J3, is a positive definite matrix if the determinants of J2 and J3 are both negative.

Jµ, L, K, 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
=
 0 -0.274 -0.51 -0.235 -0.274 0.148 -0.14 -0.065 -0.51 -0.14 0.366 -0.12 -0.235 -0.065 -0.12 0.168

Determinant(J3) = -0.03121

J2   =

 0 -0.274 -0.51 -0.274 0.148 -0.14 -0.51 -0.14 0.366

Determinant(J2) = -0.1051

e. The Solution Functions' Comparative Statics:

JΦ   =

 C*q,q C*q,wL C*q,wK C*q,wM C*wL,q C*wL,wL C*wL,wK C*wL,wM C*wK,q C*wK,wL C*wK,wK C*wK,wM C*wM,q C*wM,wL C*wM,wK C*wM,wM
=
 µq µwL µwK µwM Lq LwL LwK LwM Kq KwL KwK KwM Mq MwL MwK MwM
=
 0 1.23 0.814 1.053 1.23 -2.969 1.001 1.294 0.814 1.001 -0.935 0.857 1.053 1.294 0.857 -3.367

The estimated Translog cost function's comparative statics.

 Cq,q Cq,wL Cq,wK Cq,wM CwL,q CwL,wL CwL,wK CwL,wM CwK,q CwK,wL CwK,wK CwK,wM CwM,q CwM,wL CwM,wK CwM,wM
=
 0 1.23 0.815 1.052 1.23 -2.969 1.002 1.293 0.815 1.002 -0.935 0.857 1.052 1.293 0.857 -3.367

The underlying CES cost function's comparative statics:

 Cq,q Cq,wL Cq,wK Cq,wM CwL,q CwL,wL CwL,wK CwL,wM CwK,q CwK,wL CwK,wK CwK,wM CwM,q CwM,wL CwM,wK CwM,wM
=
 0 1.23 0.814 1.053 1.23 -2.968 1 1.295 0.814 1 -0.934 0.857 1.053 1.295 0.857 -3.367

f. Comparing Least-Cost Combinations of Inputs:

With q = 30, wL = 7, wK = 13, and wM = 6,

Derived Dual Translog cost function factor demands: (L, K, M) = (36.89, 24.41, 31.6)

Estimated Translog cost function factor demands: (L, K, M) = (36.89, 24.43, 31.58)

Specified CES cost function factor demands: (L, K, M) = (36.89, 24.42, 31.59)

g. Comparing total costs and marginal costs:

Derived Dual Translog cost function: total cost = 765.17; marginal cost = µ = 25.51.

Estimated Dual Translog cost function: total cost = 765.28; marginal cost = 25.51.

Specified CES cost function: total cost = 765.17; marginal cost = 25.51.

XVI. Table of Results.

Check that the derived dual cost function corresponds with the estimated Translog cost function by comparing the values of the inputs, L, K, and M.

Translog Production / Cost Function Duality
Production Function to Cost Function
CES: Returns to Scale = 1, Elasticity of Substitution = 0.85
—   CES Data   — — Est. Translog Production — — Derived Dual Translog Cost —   — Estimated Translog Cost —
obs #qwLwKwMLKMcost q LK ML KMcostLKMcost
117 7.3 12.026.12 19.9514.62 17.41427.9 1719.9514.6217.41 19.9414.62 17.41427.91 19.9414.6317.4427.95
218 5.26 12.566 25.7113.74 17.27411.47 1825.7113.7417.27 25.7113.74 17.27411.47 25.7113.7517.27411.52
319 8.74 14.925.7 21.4315.23 23.15546.49 1921.4315.2323.15 21.4215.23 23.15546.48 21.4315.2423.14546.56
420 6.48 11.24.58 23.0316.2 23.24437.14 2023.0316.223.24 23.0316.2 23.24437.14 23.0316.2123.23437.2
521 8.8 14.16.74 23.9417.97 22.57616.14 2123.9417.9722.57 23.9417.97 22.57616.14 23.9417.9822.56616.21
622 7.34 12.224.14 23.8717.34 29.18507.91 2223.8717.3429.18 23.8717.34 29.17507.91 23.8717.3529.16507.99
723 7.04 12.94 25.8617.31 31.42531.09 2325.8617.3131.42 25.8717.32 31.4531.09 25.8717.3331.39531.19
824 7.3 12.266.72 28.9220.85 23.31623.44 2428.9220.8523.31 28.9220.85 23.31623.44 28.9220.8623.3623.5
925 8.38 11.967.6 28.3923.5 23.17695.07 2528.3923.523.17 28.3923.5 23.17695.07 28.3823.5123.16695.12
1026 7.8 14.227.04 32.1121.59 26.32742.77 2632.1121.5926.32 32.1121.59 26.32742.77 32.1121.626.31742.86
1127 7.58 12.146.68 31.7123.8 26.52706.47 2731.7123.826.52 31.7123.8 26.52706.47 31.7123.8126.51706.54
1228 7.92 14.45.56 32.7822.09 33.27762.68 2832.7822.0933.27 32.7822.09 33.27762.67 32.7822.133.25762.79
1329 5.68 11.727.82 40.9824.8 23.46706.91 2940.9824.823.46 40.9824.81 23.45706.91 40.9824.8223.44706.97
1430 7 136 36.8924.42 31.59765.17 3036.8924.4231.59 36.8924.41 31.6765.17 36.8924.4331.58765.28
1531 6.1 11.585.24 38.4324.97 32.85695.7 3138.4324.9732.85 38.4324.97 32.86695.7 38.4324.9832.84695.8
1632 8.06 11.866.86 36.2229.22 31.21852.6 3236.2229.2231.21 36.2229.22 31.21852.6 36.2229.2431.2852.69
1733 7.84 12.54.14 34.826.22 44.99786.83 3334.826.2244.99 34.826.23 44.96786.83 34.826.2544.94786.98
1834 7.92 12.284.84 36.5728.22 41.76838.26 3436.5728.2241.76 36.5728.22 41.75838.25 36.5628.2441.73838.39
1935 6.34 11.346.68 44.3830.33 31.9838.41 3544.3830.3331.9 44.3830.33 31.89838.41 44.3830.3531.88838.5
2036 5.28 14.185.94 53.4725.87 36.35865.03 3653.4725.8736.35 53.4725.87 36.36865.04 53.4725.8836.35865.19
2137 5.24 12.624.22 49.2826.15 44.5775.99 3749.2826.1544.5 49.2826.14 44.5775.99 49.2926.1644.48776.15
2238 6.32 13.167.06 51.4730.91 35.2980.57 3851.4730.9135.2 51.4730.91 35.19980.57 51.4730.9335.18980.7
2339 7.88 14.27.76 48.8733.18 37.21144.96 3948.8733.1837.2 48.8733.19 37.191144.96 48.8733.2137.181145.1
2440 7.34 11.245.16 44.134.39 44.7940.79 4044.134.3944.7 44.0934.39 44.7940.79 44.0934.4144.68940.92
2541 6.12 13.227.6 57.5633.51 35.971068.63 4157.5633.5135.97 57.5633.51 35.971068.64 57.5633.5335.951068.77
2642 7.42 13.584.72 48.332.37 53.31049.5 4248.332.3753.3 48.332.37 53.291049.51 48.332.3953.261049.71
2743 8.72 12.764.68 44.1435.77 56.281104.71 4344.1435.7756.28 44.1335.79 56.261104.71 44.1335.8156.231104.9
2844 5.92 12.644.78 56.1533.01 50.59991.39 4456.1533.0150.59 56.1533 50.59991.39 56.1533.0350.57991.58

Mathematical Notes

1. The Translog (Transcendental Logarithmic) Production Function:

 ln(q) = ln(A) + aL * ln(L) + aK * ln(K) + aM * ln(M)               + bLL * ln(L) * ln(L) + bKK * ln(K) * ln(K) + bMM * ln(M) * ln(M)                   + bLK * ln(L) * ln(K) + bLM * ln(L) * ln(M) + bKM * ln(K) * ln(M)   =   f(L,K,M).

an equation in 10 parameters, A, aL, aK, aM, bLL, bKK, bMM, bLK, bLM, bKM, where L = labour, K = capital, M = materials and supplies, and q = product.

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

 fL(L,K,M) = (1/L) * [aL + 2 * bLL * ln(L) + bLK * ln(K) + bLM * ln(M)] = (1/L) * vL , fK(L,K,M) = (1/K) * [aK + 2 * bKK * ln(K) + bLK * ln(L) + bKM * ln(M)] = (1/K) * vK, fM(L,K,M) = (1/M) * [aM + 2 * bMM * ln(M) + bLM * ln(L) + bKM * ln(K)] = (1/M)* vM, fLL = (1/L^2) * [2 * bLL - vL],   fLK = bLK / (L*K),   fLM = bLM / (L*M),   fKK = (1/K^2) * [2 * bKK - vK],   fKL = bLK / (L*K),   fKM = bKM / (K*M), fMM = (1/M^2) * [2 * bMM - vM],   fML = bLM / (L*M),   fMK = bKM / (K*M).

3. The partial derivatives of F(L,K,M) = exp(f(L,K,M)):

 FL = fL * exp(f(L,K,M)),   FK = fK * exp(f(L,K,M)),   FM = fM * exp(f(L,K,M)), FLL = [fLL + fL * fL] * exp(f(L,K,M)),   FLK = [fLK + fL * fK] * exp(f(L,K,M)),   FLM = [fLM + fL * fM] * exp(f(L,K,M)),   FKL = FLK,   FKK = [fKK + fK * fK] * exp(f(L,K,M)),   FKM = [fKM + fK * fM] * exp(f(L,K,M)),   FML = FLM,   FMK = FKM,   FMM = [fMM + fM * fM] * exp(f(L,K,M)),

4. The Translog Cost Function:

 ln(C(q;wL,wK,wM)) = c + cq * ln(q) + cL * ln(wL) + cK * ln(wK) + cM * log(wM)                                 + .5 * [dqq * ln(q)^2 + dLL * ln(wL)^2 + dKK * ln(wK)^2 + dMM * ln(wM)^2]                   + .5 * [(dLK + dKL) * ln(wL)*ln(wK) + (dLM + dML) * ln(wL)*ln(wM) + (dKM + dMK) * ln(wK)*log(wM)]                   + dLq * ln(wL)*ln(q) + dKq * ln(wK)*ln(q) + dMq * ln(wM)*ln(q)   =   C(q;wL,wK,wM)

an equation in 18 parameters, c, cq, cL, cK, cM, dqq, dLL, dKK, dMM, dLK, dKL, dLM, dML, dKM, dMK, dLq, dKq, and dMq, where wL, wK, and wM are the factor prices of L, K, and M respectively.

Note: C(q; wL, wK, wM) = exp(C(q; wL, wK, wM)), a change in the notation above where C = CES cost function.

5. The Factor Share Functions:

 ∂ln(C)/∂ln(wL) = (∂ln(C)/∂wL )/ (∂ln(wL)/∂wL) = (1 / C(q;wL,wK,wM)) * ∂ln(C)/∂wL * wL = wL * L(q; wL, wK, wM) / C(q;wL,wK,wM) = sL(q;wL,wK,wM), ∂ln(C)/∂ln(wK) = (∂ln(C)/∂wK )/ (∂ln(wK)/∂wK) = (1 / C(q;wL,wK,wM)) * ∂ln(C)/∂wK * wK = wK * K(q; wL, wK, wM) / C(q;wL,wK,wM) = sK(q;wL,wK,wM), ∂ln(C)/∂ln(wM) = (∂ln(C)/∂wM )/ (∂ln(wM)/∂wM) = (1 / C(q;wL,wK,wM)) * ∂ln(C)/∂wM * wM = wM * M(q; wL, wK, wM) / C(q;wL,wK,wM) = sM(q;wL,wK,wM).

 sL(q;wL,wK,wM) = cL + dLq * ln(q) + dLL * ln(wL) + dLK * ln(wK) + dLM * ln(wM), sK(q;wL,wK,wM) = cK + dKq * ln(q) + dKL * ln(wL) + dKK * ln(wK) + dKM * ln(wM), sM(q;wL,wK,wM) = cM + dMq * ln(q) + dML * ln(wL) + dMK * ln(wK) + dMM * ln(wM),

6. The Factor Demand Functions:

 L(q;wL,wK,wM) = sL(q;wL,wK,wM) * C(q;wL,wK,wM) / wL, K(q;wL,wK,wM) = sK(q;wL,wK,wM) * C(q;wL,wK,wM) / wK, M(q;wL,wK,wM) = sM(q;wL,wK,wM) * C(q;wL,wK,wM) / wM.

7. The partial derivatives of C(q; wL, wK, wM) = exp(C(q; wL, wK, wM)):

 CwL(q;wL,wK,wM) = L(q;wL,wK,wM) = sL(q;wL,wK,wM) * C(q;wL,wK,wM) / wL, CwK(q;wL,wK,wM) = K(q;wL,wK,wM) = sK(q;wL,wK,wM) * C(q;wL,wK,wM) / wK, CwM(q;wL,wK,wM) = M(q;wL,wK,wM) = sM(q;wL,wK,wM) * C(q;wL,wK,wM) / wM,

Writing:

 Cq(q;wL,wK,wM) = (ecq + edqq * log(q) + edLq * log(wL) + edKq * log(wK) + edMq * log(wM) ) / q, Cq,q(q;wL,wK,wM) = (edqq - (ecq + edqq * log(q) + edLq * log(wL) + edKq * log(wK) + edMq * log(wM) )) / (q * q), Cq(q;wL,wK,wM) = Cq(q;wL,wK,wM) * C(q;wL,wK,wM), Cq,q(q;wL,wK,wM) = ( Cq,q(q;wL,wK,wM) + Cq(q;wL,wK,wM) * Cq(q;wL,wK,wM) ) * C(q;wL,wK,wM);

then:

 CwL,q(q;wL,wK,wM) = ( edLq * C(q; wL, wK, wM) / q + sL(q,wL,wK,wM) * Cq(q;wL,wK,wM) ) / wL, CwL,wL(q;wL,wK,wM) = ( L(q;wL,wK,wM) / wL ) * (-1 + sL(q,wL,wK,wM) + edLL / sL(q,wL,wK,wM)), CwL,wK(q;wL,wK,wM) = ( L(q,wL,wK,wM) / wK ) * (sK(q,wL,wK,wM) + edLK / sL(q,wL,wK,wM)), CwL,wM(q;wL,wK,wM) = ( L(q,wL,wK,wM) / wM ) * (sM(q,wL,wK,wM) + edLM / sL(q,wL,wK,wM)), CwK,q(q;wL,wK,wM) = ( edKq * C(q; wL, wK, wM) / q + sK(q,wL,wK,wM) * Cq(q;wL,wK,wM) ) / wK, CwK,wL(q;wL,wK,wM) = ( K(q;wL,wK,wM) / wL ) * (sL(q,wL,wK,wM) + edKL / sK(q,wL,wK,wM)), CwK,wK(q;wL,wK,wM) = ( K(q,wL,wK,wM) / wK ) * (-1 + sK(q,wL,wK,wM) + edKK / sK(q,wL,wK,wM)), CwK,wM(q;wL,wK,wM) = ( K(q,wL,wK,wM) / wM ) * (sM(q,wL,wK,wM) + edKM / sK(q,wL,wK,wM)), CwM,q(q;wL,wK,wM) = ( edMq * C(q; wL, wK, wM) / q + sM(q,wL,wK,wM) * Cq(q;wL,wK,wM) ) / wM, CwM,wL(q;wL,wK,wM) = ( M(q;wL,wK,wM) / wL ) * (sL(q,wL,wK,wM) + edML / sM(q,wL,wK,wM)), CwM,wK(q;wL,wK,wM) = ( M(q,wL,wK,wM) / wK ) * (sK(q,wL,wK,wM) + edMK / sM(q,wL,wK,wM)), CwM,wM(q;wL,wK,wM) = ( M(q,wL,wK,wM) / wM ) * (-1 + sM(q,wL,wK,wM) + edMM / sM(q,wL,wK,wM)), Copyright © Elmer G. Wiens:   Egwald Web Services All Rights Reserved.    Inquiries 