Egwald Economics: Microeconomics

Duality and the Translog Production / Cost Functions
Non-Homothetic Generalized CES Technology

by

Elmer G. Wiens

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Duality: Production / Cost Functions:   Cobb-Douglas Duality | CES Duality | Theory of Duality | Translog Duality - CES | Translog Duality - Generalized CES | References and Links

S. Translog (Transcendental Logarithmic) Duality and the Generalized 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 Generalized CES production function.

VII. Duality.  The Plan:

1. Specify the parameters of a Generalized CES production function, and obtain the derived Generalized CES cost function using Newton's Method.

2. Generate the Generalized 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 Translog production function corresponds with the estimated Translog production function, and the underlying Generalized 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 derived Generalized CES cost function.

VIII. Generate Generalized CES production / cost data.

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.

The estimated coefficients of the Translog production and cost functions and will vary with the parameters nu, rho, rhoL, rhoK, rhoM, alpha, beta and gamma of the Generalized CES 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 function, using the parameters as specified.
Intermediate (and other) values of the parameters also work. The program might "time-out" for values of rho >> 0 and rho << 0, yielding "NAN" values.

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)
4 <= wL* <= 11,   7<= wK* <= 16,   4 <= wM* <= 10

Generalized CES Production Function Parameters
nu:
rho:
Base Factor Prices
 wL* wK* wM*
Distribution to Randomize Factor Prices
 Use [-2, 2] Uniform distribution     Use .25 * Normal (μ = 0, σ2 = 1)

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).

 The factor prices are distributed about the base factor prices by adding a random number distributed uniformly in the [-2, 2] domain.

The Generalized CES cost function as derived from the Generalized CES production function:

Unlike the homothetic CES technology, the non-homothetic Generalized CES technology does not admit a closed form cost function. Consequently, the Generalized CES cost function is derived from the Generalized CES production function as demonstrated on the Generalized CES Production Function web page.

IX. Estimate the Translog Production and Cost Functions.

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

Generalized CES
Translog Production Function

SVD Least Squares
Parameter Estimates
Parameter Coefficient std error t-ratio
lnA0.0054550.0017.886
aL0.3570540850.189
aK0.2697580.001381.51
aM0.3599050.001580.428
bLL-0.019550-103.201
bKK-0.0095940-24.218
bMM-0.0279240-134.633
bLK0.019382050.442
bLM0.01526054.028
bKM0.0144230.00126.841
 R2 = 1 R2b = 1 # obs = 28 Observation Matrix Rank: 10
 aL + aK + aM = 0.987-2*bLL = 0.039099 =~ 0.034642 = bLK + bLM-2*bKK = 0.019188 =~ 0.033805 = bLK + bKLM-2*bMM = 0.055848 =~ 0.029683 = bLM + bKM

The estimated Translog production function:

ln(q) = 0.005455 + 0.357054 * ln(L) + 0.269758 * ln(K) + 0.359905 * ln(M) + -0.01955 * ln(L)*ln(L) + -0.009594 * ln(K)*ln(K) + -0.027924 * ln(M)*ln(M)
+ 0.019382 * ln(L)*ln(K) + 0.01526 * ln(L)*ln(M) + 0.014423 * ln(K)*ln(M)   =   f(L,K,M).

The Translog cost factor share functions are:

 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),

three linear equations in their 15 parameters.

Estimating the Translog factor share functions simultaneously with the specified constraints, from the Generalized CES data using QR least squares, yields the coefficient estimates:

QR Restricted Least Squares
Parameter Estimates
Parameter Coefficient std error t-ratio
cL0.3601780.0002161667.264402
dLq0.0002886.4E-54.503274
dLL0.0344739.1E-5378.965508
dLK-0.0136678.7E-5-156.903041
dLM-0.0208067.3E-5-285.720975
cK0.2722390.0002141270.631423
dKq0.0207616.6E-5314.94953
dKL-0.0136678.7E-5-156.903041
dKK0.0306390.000135226.483418
dKM-0.0169729.8E-5-173.15413
cM0.3675830.0002131722.462221
dMq-0.0210496.7E-5-316.467618
dML-0.0208067.3E-5-285.720975
dMK-0.0169729.8E-5-173.15413
dMM0.0377790.000103368.133588
 R2 = 1 R2b = 1 # obs = 84
 dLK = dKL, dLM = dML, dKM = dMK 1 = cL + cK + cM 0 = dLL + dLK + dLM 0 = dKL + dKK + dKM 0 = dML + dMK + dMM 0 = dLq + dKq + dMq

The first three constraints are the usual across equation constraints on the factor share functions. The last five constraints are necessary conditions for the estimated Translog cost function to be linear homogeneous in factor prices. See the Generalized CES-Translog Cost Function for more details.

To obtain estimates of the remaining three parameters, c, cq, and dqq, of the Translog cost function write:

ln(C(q;wL,wK,wM)) - {cL * ln(wL) + cK * ln(wK) + cM * log(wM) + .5 * [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)}
= R(q;wL,wK,wM) = c + cq * ln(q) + .5 * dqq * ln(q)^2

Estimate the linear equation:

R(q;wL,wK,wM) = c + cq * ln(q) + .5 * dqq * ln(q)^2

to obtain c, cq, and dqq.

QR Restricted Least Squares
Parameter Estimates
Parameter Coefficient std error t-ratio
c1.1150820.0001417894.657039
cq101000
dqq0.020992.4E-5865.906812
 R2 = 1 R2b = 1 # obs = 28

1 = cq

The estimated Translog cost function:

ln(C(q;wL,wK,wM)) = 1.115082 + 1 * ln(q) + 0.360178 * ln(wL) + 0.272239 * ln(wK) + 0.367583 * log(wM)
+ .5 * [0.02099 * ln(q)^2 + 0.034473 * ln(wL)^2 + 0.030639 * ln(wK)^2 + 0.037779 * ln(wM)^2]
+ .5 * [-0.027334 * ln(wL)*ln(wK) + -0.041613 * ln(wL)*ln(wM) + -0.033945 * ln(wK)*log(wM)]
+ 0.000288 * ln(wL)*ln(q) + 0.020761 * ln(wK)*ln(q) + -0.021049 * 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 = 39.05, K = 21.4, and M = 36.07, (***) becomes:

F*(39.05, 21.4, 36.07) = minq, vL, vK {q   :   C(q; vL, vK, 1)   >=   vL * 39.05 + vK * 21.4 + 36.07,   for all vL >= 0 , vK >= 0}     (***).

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

H(39.05, 21.4, 36.07, q, vL, vK, λ) = q + λ * (vL * 39.05 + vK * 21.4 + 36.07 - C(q; vL, vK, 1)),

where λ is the Lagrange multiplier.

a. First Order Necessary Conditions:

 0.   Hλ(39.05, 21.4, 36.07, q, vL, vK, λ) = vL * 39.05 + vK * 21.4 + 36.07 - C(q; vL, vK, 1) = 0 1.   Hq(39.05, 21.4, 36.07, q, vL, vK, λ) = 1 - λ * ∂C(q; vL, vK, 1)/∂q = 0 2.   HvL(39.05, 21.4, 36.07, q, vL, vK, λ) = λ * (39.05 - ∂C(q; vL, vK, 1)/∂vL) = 0 3.   HvK(39.05, 21.4, 36.07, q, vL, vK, λ) = λ * (21.4 - ∂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(39.05, 21.4, 36.07)
q = q(39.05, 21.4, 36.07)
vL = vL(39.05, 21.4, 36.07)
vK = vK(39.05, 21.4, 36.07)

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 = 0.218.

Newton's Method:

Using these estimates, Newton's Method provides:

Nonlinear Optimization: Newton's Method
Parameter Estimates
Iter #λqvLvK
0   0.218 331.0714 1.6429
10.2122 26.791.0599 1.8583
20.1956 26.991.1513 2.1156
30.1945 271.1655 2.1644
40.1945 271.1659 2.1657
50.1945 271.1659 2.1657

c. Solution Vector:

W = (L, K, M, q, vL, vK, λ) = (39.05, 21.4, 36.07, 27, 1.16593, 2.16573, 0.19446)

With λ = 0.19446, q = 27, vL = 1.16593, and vK = 2.16573, the first order conditions are:

 0.   Hλ(39.05, 21.4, 36.07, q, vL, vK, λ) = 1.16593 * 39.05 + 2.16573 * 21.4 + 36.07 - C(q; 1.16593, 2.16573, 1) = 0 1.   Hq(39.05, 21.4, 36.07, q, vL, vK, λ) = 1 - 0.19446 * ∂C(q; 1.16593, 2.16573, 1)/∂q = 0 2.   HvL(39.05, 21.4, 36.07, q, vL, vK, λ) = 0.19446 * (39.05 - ∂C(q; 1.16593, 2.16573, 1)/∂vL) = 0 3.   HvK(39.05, 21.4, 36.07, q, vL, vK, λ) = 0.19446 * (21.4 - ∂C(q; 1.16593, 2.16573, 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 -5.143 0 -0 -5.143 -0.004 -0.305 -0.176 0 -0.305 3.565 -1.136 -0 -0.176 -1.136 1.063

Determinant(J3) = -66.10881

J2   =

 0 -5.143 0 -5.143 -0.004 -0.305 0 -0.305 3.565

Determinant(J2) = -94.27043

e. Maximum Output:

With L = 39.05, K = 21.4, M = 36.07,

Dual Translog production function:

F*(39.05, 21.4, 36.07) = q(39.05, 21.4, 36.07) = 27,

Estimated Translog production function:

F(39.05, 21.4, 36.07) = 27.

Specified Generalized CES production function:

f(39.05, 21.4, 36.07) = 27.

f. The Solution Functions' Comparative Statics.

JL, K, M   =

 vL vK 1 0 0 0 λ 0 0 0 λ 0
=
 1.16593 2.16573 1 0 0 0 0.1945 0 0 0 0.1945 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
=
 0.001912 0.003553 -0.005167 0.22672 0.42114 0.19446 -0.03511 5e-05 0.04081 6e-05 -0.11311 0.07581

g. Comparing Partial Derivatives:

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

F*L(L,K,M) = F*(39.05, 21.4, 36.07) = qL(39.05, 21.4, 36.07) = 0.227,
F*K(L,K,M) = F*(39.05, 21.4, 36.07) = qK(39.05, 21.4, 36.07) = 0.421,
F*M(L,K,M) = F*(39.05, 21.4, 36.07) = qM(39.05, 21.4, 36.07) = 0.194.

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

FL(39.05, 21.4, 36.07) = 0.227,
FK(39.05, 21.4, 36.07) = 0.421,
FM(39.05, 21.4, 36.07) = 0.194.

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

fL(39.05, 21.4, 36.07) = 0.227,
fK(39.05, 21.4, 36.07) = 0.421,
fM(39.05, 21.4, 36.07) = 0.194.

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
Generalized CES: Returns to Scale = 1, rho = 0.17647, rhoL = 0.17647, rhoK = 0.11823, rhoM = 0.26339
—       Generalized CES Data   —    —    Estimated Translog Cost   —   — Derived Dual Production — — Estimated Translog Production —
obs #qwLwKwM LK Mcost LKMcostqF*LF*KF*MqFLFKFM
1175.04 11.486.78 28.0812.6 18.57412.1628.112.6118.56412.2816.9950.1940.4430.261170.1940.4420.26
2188.4 13.287.46 24.1314.89 22.4567.5224.1414.8922.41567.6717.9960.2480.3920.221180.2480.3920.22
3198.42 11.746.44 23.7516.39 24.85552.4723.7616.3824.87552.5618.9970.270.3750.206190.2690.3750.206
4207.58 157.04 29.3214.96 25.92629.129.3214.9625.93629.1519.9990.2230.4420.208200.2230.4420.207
5215.84 13.027.22 34.3215.87 23.93579.8334.3215.8723.92579.84210.1970.4380.243210.1960.4380.243
6226.94 11.386.96 31.1718.87 25.82610.7531.1718.8725.82610.74220.2330.3810.233220.2320.3810.233
7237.52 14.846.84 33.9217.48 30.18720.8733.9117.4830.19720.8223.0020.2220.4380.202230.2220.4370.202
8245.24 11.127.24 40.0619.51 25.31610.240.0619.5225.29610.1324.0030.1920.4070.265240.1910.4070.264
9258.9 13.165.06 30.520.23 39.65738.330.4820.2339.66738.1725.0040.2770.410.158250.2770.410.158
10265.66 13.625.78 42.4218.52 33.92688.442.418.5233.93688.3126.0030.1970.4740.201260.1970.4730.201
11277.16 13.14.92 37.0220.5 40.88734.8137.0120.540.88734.6827.0040.2410.4420.166270.2410.4420.166
12287.18 14.57.6 43.7822.39 33.95897.0343.7622.3933.95896.928.0040.2070.4180.219280.2070.4180.219
13298.62 13.67.08 39.6625.17 37.85952.1139.6525.1637.85951.9629.0040.2420.3820.199290.2420.3820.199
14307 136 43.7924.14 40.13861.1743.7824.1440.13861.0530.0040.2240.4160.192300.2240.4160.192
15318.1 12.247.24 42.8528.38 38.04969.8542.8428.3738.05969.7231.0040.2390.3610.214310.2390.3610.214
16326.74 14.95.3 48.8323.17 47.55926.3748.8323.1647.55926.2432.0040.2130.470.167320.2130.470.168
17337.18 14.686.3 50.425.68 44.91021.7150.3925.6844.91021.633.0030.2130.4350.187330.2130.4350.187
18348.8 13.085.52 43.6129.33 51.18105043.629.3451.171049.934.0030.2610.3880.164340.2610.3880.164
19358.38 14.284.36 45.2927.01 61.471033.2445.3127.0261.381033.23350.2580.440.134350.2590.440.134
20367.5 14.226.88 54.5829.88 46.691155.4154.5729.8846.691155.3736.0010.2140.4060.197360.2140.4070.197
21378.7 11.965.54 46.7733.86 53.981110.8946.7733.8753.961110.87370.2650.3650.169370.2660.3650.169
22385.78 14.426.76 66.9128.96 46.561119.0766.8928.9646.591119.137.9990.180.4480.21380.180.4490.211
23397.06 14.025.82 59.0931.17 54.711172.5259.1131.1654.711172.5838.9980.2140.4260.177390.2150.4260.177
24408.68 11.225.66 50.1638.51 56.541187.4950.1638.5256.531187.5939.9970.2680.3460.175400.2680.3470.175
25417.48 14.127.8 64.635.81 49.351373.8164.635.8249.361373.9840.9950.2050.3860.214410.2050.3870.214
26427.2 12.745.48 60.6235.49 59.681215.7360.6535.559.671215.9341.9940.2270.4020.173420.2270.4020.173
27436.48 11.325.5 63.437.64 57.081150.8463.4237.6457.081151.0642.9920.2210.3870.188430.2220.3870.188
28445.72 14.445.92 76.7633.11 58.291262.3176.7933.158.331262.5643.9920.1820.4580.188440.1820.4590.188

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
131.7463 48.758.3039 45.3779
236.9631 48.1916.9133 43.3362
332.0738 45.1922.0195 41.4001
431.274 43.8423.9672 40.1984
531.2092 43.7824.1383 40.1467
631.2088 43.7824.1393 40.1463
731.2088 43.7824.1393 40.1463

c. Solution Vector:

Z = (q, wL, wK, wM, L, K, M, µ) = (30, 7, 13, 6, 43.78, 24.14, 40.15, 31.21)

With µ = 31.21, L = 43.78, K = 24.14, and M = 40.15, the first order conditions are:

 0.   Gµ(30; 7, 13, 6, L, K, M, μ) = 30 - F(43.78, 24.14, 40.15) = -0 1.   GL(30; 7, 13, 6, L, K, M, μ) = 7 - µ * FL(43.78, 24.14, 40.15) = -0 2.   GK(30; 7, 13, 6, L, K, M, μ) = 13 - µ * FK(43.78, 24.14, 40.15) = 0 3.   GM(30; 7, 13, 6, L, K, M, μ) = 13 - µ * FM(43.78, 24.14, 40.15) = -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 -0.224 -0.417 -0.192 -0.224 0.127 -0.114 -0.053 -0.417 -0.114 0.389 -0.097 -0.192 -0.053 -0.097 0.143

Determinant(J3) = -0.01604

J2   =

 0 -0.224 -0.417 -0.224 0.127 -0.114 -0.417 -0.114 0.389

Determinant(J2) = -0.06291

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
=
 0.107 1.588 0.921 1.354 1.588 -3.419 1.098 1.61 0.921 1.098 -1.027 0.943 1.354 1.61 0.943 -3.922

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
=
 0.111 1.588 0.921 1.354 1.588 -3.422 1.098 1.614 0.921 1.098 -1.024 0.938 1.354 1.614 0.938 -3.915

The underlying specified Generalized 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
=
 0.107 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

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) = (43.78, 24.14, 40.15)

Estimated Translog cost function factor demands: (L, K, M) = (43.78, 24.14, 40.13)

Specified Generalized CES cost function factor demands: (L, K, M) = (43.79, 24.14, 40.13)

g. Comparing total costs and marginal costs:

Derived Dual Translog cost function: total cost = 861.17; marginal cost = µ = 31.21.

Estimated Dual Translog cost function: total cost = 861.05; marginal cost = 31.21.

Specified Generalized CES cost function: total cost = 861.17; marginal cost = 31.21.

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
Generalized CES: Returns to Scale = 1, rho = 0.17647, rhoL = 0.17647, rhoK = 0.11823, rhoM = 0.26339
—   Generalized CES Data   — — Est. Translog Production — — Derived Dual Translog Cost —   — Estimated Translog Cost —
obs #qwLwKwMLKMcost q LK ML KMcostLKMcost
117 5.04 11.486.78 28.0812.6 18.57412.16 1728.0812.618.57 28.0912.62 18.53412.14 28.112.6118.56412.28
218 8.4 13.287.46 24.1314.89 22.4567.52 1824.1314.8922.4 24.1314.9 22.38567.53 24.1414.8922.41567.67
319 8.42 11.746.44 23.7516.39 24.85552.47 1923.7516.3924.85 23.7516.4 24.84552.47 23.7616.3824.87552.56
420 7.58 157.04 29.3214.96 25.92629.1 2029.3214.9625.92 29.3214.96 25.93629.12 29.3214.9625.93629.15
521 5.84 13.027.22 34.3215.87 23.93579.83 2134.3215.8723.93 34.3215.88 23.92579.84 34.3215.8723.92579.84
622 6.94 11.386.96 31.1718.87 25.82610.75 2231.1718.8725.82 31.1718.88 25.8610.75 31.1718.8725.82610.74
723 7.52 14.846.84 33.9217.48 30.18720.87 2333.9217.4830.18 33.9117.48 30.19720.88 33.9117.4830.19720.82
824 5.24 11.127.24 40.0619.51 25.31610.2 2440.0619.5125.31 40.0619.54 25.28610.19 40.0619.5225.29610.13
925 8.9 13.165.06 30.520.23 39.65738.3 2530.520.2339.65 30.4920.23 39.67738.27 30.4820.2339.66738.17
1026 5.66 13.625.78 42.4218.52 33.92688.4 2642.4218.5233.92 42.418.52 33.95688.42 42.418.5233.93688.31
1127 7.16 13.14.92 37.0220.5 40.88734.81 2737.0220.540.88 37.0220.5 40.9734.79 37.0120.540.88734.68
1228 7.18 14.57.6 43.7822.39 33.95897.03 2843.7822.3933.95 43.7722.4 33.95897.05 43.7622.3933.95896.9
1329 8.62 13.67.08 39.6625.17 37.85952.11 2939.6625.1737.85 39.6525.17 37.85952.1 39.6525.1637.85951.96
1430 7 136 43.7924.14 40.13861.17 3043.7924.1440.13 43.7824.14 40.15861.17 43.7824.1440.13861.05
1531 8.1 12.247.24 42.8528.38 38.04969.85 3142.8528.3838.04 42.8428.38 38.04969.84 42.8428.3738.05969.72
1632 6.74 14.95.3 48.8323.17 47.55926.37 3248.8323.1747.55 48.8423.16 47.58926.34 48.8323.1647.55926.24
1733 7.18 14.686.3 50.425.68 44.91021.71 3350.425.6844.9 50.3925.68 44.921021.7 50.3925.6844.91021.6
1834 8.8 13.085.52 43.6129.33 51.181050 3443.6129.3351.18 43.6229.34 51.171049.99 43.629.3451.171049.9
1935 8.38 14.284.36 45.2927.01 61.471033.24 3545.2927.0161.47 45.3327.01 61.381033.28 45.3127.0261.381033.23
2036 7.5 14.226.88 54.5829.88 46.691155.41 3654.5829.8846.69 54.5729.88 46.71155.41 54.5729.8846.691155.37
2137 8.7 11.965.54 46.7733.86 53.981110.89 3746.7733.8653.98 46.7833.86 53.951110.89 46.7733.8753.961110.87
2238 5.78 14.426.76 66.9128.96 46.561119.07 3866.9128.9646.56 66.8728.96 46.61119.1 66.8928.9646.591119.1
2339 7.06 14.025.82 59.0931.17 54.711172.52 3959.0931.1754.71 59.131.16 54.711172.51 59.1131.1654.711172.58
2440 8.68 11.225.66 50.1638.51 56.541187.49 4050.1638.5156.54 50.1738.52 56.511187.5 50.1638.5256.531187.59
2541 7.48 14.127.8 64.635.81 49.351373.81 4164.635.8149.35 64.5835.82 49.361373.81 64.635.8249.361373.98
2642 7.2 12.745.48 60.6235.49 59.681215.73 4260.6235.4959.68 60.6435.49 59.661215.74 60.6535.559.671215.93
2743 6.48 11.325.5 63.437.64 57.081150.84 4363.437.6457.08 63.4137.64 57.071150.83 63.4237.6457.081151.06
2844 5.72 14.445.92 76.7633.11 58.291262.31 4476.7633.1158.29 76.7533.1 58.331262.29 76.7933.158.331262.56

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)),