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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

Cost Functions:   Cobb-Douglas Cost | Normalized Quadratic Cost | Translog Cost | Diewert Cost | Generalized CES-Translog Cost | Generalized CES-Diewert Cost | References and Links

Production Functions:   Cobb-Douglas | CES | Generalized CES | Translog | Diewert | Translog vs Diewert | Diewert vs Translog | Estimate Translog | Estimate Diewert | 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   =  

HλλHλqHλvLHλvK
HHqqHqvLHqvK
HvLλHvLqHvLvLHvLvK
HvKλHvKqHvKvLHvKvK
=
  0   -CqL - CvLK - 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   -CqL - 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,KHλ,M
Hq,LHq,KHq,M
HvL,LHvL,KHvL,M
HwK,LHwK,KHvK,M
=
vLvK1
000
λ00
0λ0

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

JΦ   =  

λLλKλM
qLqKqM
vLLvLKvLM
vKLvKKvKM

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
lnA0.0067230.00111.473
aL0.35792601444.192
aK0.2693260705.187
aM0.35870829.31
bLL-0.0193240-182.604
bKK-0.009840-60.732
bMM-0.0276090-206.138
bLK0.019438079.722
bLM0.0145310108.034
bKM0.01491066.89
R2 = 1 R2b = 1 # obs = 28
Observation Matrix Rank: 10

aL + aK + aM = 0.986
-2*bLL = 0.038648 =~ 0.033969 = bLK + bLM
-2*bKK = 0.01968 =~ 0.034348 = bLK + bKLM
-2*bMM = 0.055218 =~ 0.02944 = bLM + bKM

The estimated Translog production function:

ln(q) = 0.006723 + 0.357926 * ln(L) + 0.269326 * ln(K) + 0.3587 * ln(M) + -0.019324 * ln(L)*ln(L) + -0.00984 * ln(K)*ln(K) + -0.027609 * ln(M)*ln(M)
+ 0.019438 * ln(L)*ln(K) + 0.014531 * ln(L)*ln(M) + 0.01491 * 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
cL0.3609180.0001861940.538652
dLq9.8E-55.4E-51.815749
dLL0.034366.9E-5498.578515
dLK-0.0137216.7E-5-205.781539
dLM-0.0206395.5E-5-375.121318
cK0.2727050.0001931412.709886
dKq0.020655.4E-5382.776086
dKL-0.0137216.7E-5-205.781539
dKK0.030550.000104293.098109
dKM-0.0168297.5E-5-225.12634
cM0.3663770.0001921905.986887
dMq-0.0207485.4E-5-385.252656
dML-0.0206395.5E-5-375.121318
dMK-0.0168297.5E-5-225.12634
dMM0.0374677.7E-5484.119163
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
c1.1148520.0001447765.155903
cq101000
dqq0.0210222.5E-5853.163339
R2 = 1 R2b = 1 # obs = 28

1 = cq

The estimated Translog cost function:

ln(C(q;wL,wK,wM)) = 1.114852 + 1 * ln(q) + 0.360918 * ln(wL) + 0.272705 * ln(wK) + 0.366377 * log(wM)
+ .5 * [0.021022 * ln(q)^2 + 0.03436 * ln(wL)^2 + 0.03055 * ln(wK)^2 + 0.037467 * ln(wM)^2]
+ .5 * [-0.027442 * ln(wL)*ln(wK) + -0.041277 * ln(wL)*ln(wM) + -0.033657 * ln(wK)*log(wM)]
+ 9.8E-5 * ln(wL)*ln(q) + 0.02065 * ln(wK)*ln(q) + -0.020748 * 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 26.791.0602 1.8586
20.1955 26.991.1519 2.1164
30.1944 271.1661 2.1653
40.1944 271.1666 2.1666
50.1944 271.1666 2.1666

c. Solution Vector:

W = (L, K, M, q, vL, vK, λ) = (39.05, 21.4, 36.07, 27, 1.16656, 2.16665, 0.19439)

With λ = 0.19439, q = 27, vL = 1.16656, and vK = 2.16665, the first order conditions are:

0.   Hλ(39.05, 21.4, 36.07, q, vL, vK, λ) = 1.16656 * 39.05 + 2.16665 * 21.4 + 36.07 - C(q; 1.16656, 2.16665, 1) = 0
1.   Hq(39.05, 21.4, 36.07, q, vL, vK, λ) = 1 - 0.19439 * ∂C(q; 1.16656, 2.16665, 1)/∂q = 0
2.   HvL(39.05, 21.4, 36.07, q, vL, vK, λ) = 0.19439 * (39.05 - ∂C(q; 1.16656, 2.16665, 1)/∂vL) = 0
3.   HvK(39.05, 21.4, 36.07, q, vL, vK, λ) = 0.19439 * (21.4 - ∂C(q; 1.16656, 2.16665, 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   -CqL - CvLK - 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
=
0-5.1440-0
-5.144-0.004-0.305-0.176
0-0.3053.563-1.134
-0-0.176-1.1341.063

  Determinant(J3) = -66.15585

 

J2   =  

0-5.1440
-5.144-0.004-0.305
0-0.3053.563

  Determinant(J2) = -94.29841

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   =  

vLvK1
000
λ00
0λ0
  =  
1.166562.166651
000
0.194400
00.19440

From the Implicit Function Theorem:

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

JΦ   =  

λLλKλM
qLqKqM
vLLvLKvLM
vKLvKKvKM
  =  
0.0019090.003548-0.005152
0.226770.421170.19439
-0.035117.0E-50.04074
8.0E-5-0.113110.07568

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
1178.64 11.184.96 19.2914.07 25.68451.3819.2914.0725.71451.5516.9940.3020.3910.174170.3020.390.173
2185.86 14.766.3 29.2412.02 23.03493.8329.2512.0323.02493.9517.9960.1980.4990.213180.1980.4980.212
3195.24 12.94.48 29.212.25 27.57434.6129.2112.2527.58434.6718.9970.2110.520.181190.2110.520.18
4206.8 11.224.2 25.2415.03 31.14471.0525.2415.0331.15471.0819.9990.2660.440.165200.2660.4390.165
5218.56 12.126.48 26.5318.16 27.76627.0926.5318.1627.77627.1120.9990.2660.3770.202210.2660.3760.201
6226.02 12.086.54 33.8517.18 26.18582.5833.8617.1926.15582.5422.0010.2110.4230.229220.2110.4230.229
7237.56 12.146.38 31.2219.26 29.64658.9431.2219.2629.64658.923.0020.2450.3930.206230.2440.3920.206
8247.84 12.844.48 30.1918.26 39.12646.4130.1718.2639.12646.3324.0030.2670.4380.153240.2670.4380.153
9258.36 13.487.12 34.2821.17 32.11800.6534.2821.1732.1800.5425.0030.2420.3890.206250.2410.3890.206
10268.04 11.987.96 36.123.99 29.91815.8336.1223.9929.89815.726.0040.2380.3550.235260.2380.3550.235
11275.94 12.885.28 41.4119.82 37.01696.6741.419.8137.01696.5627.0040.2120.4590.188270.2120.4590.188
12285.28 11.464.78 43.2720.74 38647.843.2620.7338647.728.0040.210.4550.19280.210.4550.19
13295.82 14.34.72 45.7319.67 43.65753.4645.7219.6643.66753.3329.0050.2050.5030.166290.2050.5030.166
14307 136 43.7924.14 40.13861.1743.7824.1440.13861.0530.0040.2240.4160.192300.2240.4160.192
15318.74 11.544.66 36.6627.17 49.52864.6136.6427.1749.51864.4731.0050.2870.3790.153310.2870.3790.153
16325.4 14.266.48 56.8623.2 39.21891.9356.8423.1939.22891.8332.0030.1780.470.214320.1780.470.214
17335.86 13.167.04 55.7526.26 38.47943.1255.7426.2738.46943.0233.0030.1890.4240.227330.1890.4240.227
18347.26 11.965.1 46.2128.44 49.39927.4546.228.4349.38927.3634.0030.2440.4020.171340.2440.4020.171
19358.68 12.824.02 42.1728.48 63.07984.742.1828.4963984.6735.0010.2810.4150.13350.2810.4150.13
20366.74 12.827.22 56.5530.93 42.681085.8356.5530.9342.681085.836.0010.2060.3910.22360.2060.3910.22
21377.74 13.227.82 55.5233.4 43.941214.955.5233.443.941214.89370.2170.3710.219370.2170.3710.219
22387.1 12.184.92 52.7231.51 56.441035.7852.7331.5156.441035.837.9990.2380.4080.165380.2380.4080.165
23396.68 14.747.32 65.1231.42 47.811248.0465.1231.4147.821248.1138.9980.1910.4220.21390.1910.4220.21
24407.82 11.444.4 50.6434.82 64.111076.5150.6534.8464.081076.6239.9960.2650.3870.149400.2650.3880.149
25416.42 14.766.6 68.7532.06 52.891263.6568.7632.0552.921263.840.9960.190.4370.196410.1910.4380.196
26427.28 13.265.82 61.8535.33 58.491259.1361.8735.3358.51259.3341.9940.2220.4040.177420.2220.4040.177
27436.02 13.285.28 69.0233.4 60.191176.9469.0533.460.221177.1542.9930.20.4420.176430.2010.4420.176
28445.52 14.167.3 81.4534.74 50.741311.9281.4434.7550.781312.2543.990.1690.4340.224440.170.4350.224

 

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   =  

GµµGµLGµKGµM
GGLLGLKGLM
GGKLGKKGKM
GGMLGMKGMM
=
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;   =  

GµqGµwLGµwKGµwM
GLqGLwLGLwKGLwM
GKqGKwLGKwKGKwM
GMqGMwLGMwKGMwM
=
1000
0100
0010
0001

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

JΦ   =  

µqµwLµwKµwM
LqLwLLwKLwM
KqKwLKwKKwM
MqMwLMwKMwM

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
131.7507 48.778.2939 45.3467
236.9889 48.2116.9041 43.3086
332.0783 45.2122.0133 41.3947
431.275 43.8523.9669 40.1888
531.2096 43.7924.1394 40.137
631.2093 43.7924.1404 40.1366
731.2093 43.7924.1404 40.1366

c. Solution Vector:

Z = (q, wL, wK, wM, L, K, M, µ) = (30, 7, 13, 6, 43.79, 24.14, 40.14, 31.21)

With µ = 31.21, L = 43.79, K = 24.14, and M = 40.14, the first order conditions are:

0.   Gµ(30; 7, 13, 6, L, K, M, μ) = 30 - F(43.79, 24.14, 40.14) = -0
1.   GL(30; 7, 13, 6, L, K, M, μ) = 7 - µ * FL(43.79, 24.14, 40.14) = -0
2.   GK(30; 7, 13, 6, L, K, M, μ) = 13 - µ * FK(43.79, 24.14, 40.14) = 0
3.   GM(30; 7, 13, 6, L, K, M, μ) = 13 - µ * FM(43.79, 24.14, 40.14) = -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
-0.2240.126-0.114-0.053
-0.417-0.1140.39-0.098
-0.192-0.053-0.0980.143

  Determinant(J3) = -0.01601

 

J2   =  

0-0.224-0.417
-0.2240.126-0.114
-0.417-0.1140.39

  Determinant(J2) = -0.06291

e. The Solution Functions' Comparative Statics:

JΦ   =  

C*q,qC*q,wLC*q,wKC*q,wM
C*wL,qC*wL,wLC*wL,wKC*wL,wM
C*wK,qC*wK,wLC*wK,wKC*wK,wM
C*wM,qC*wM,wLC*wM,wKC*wM,wM
=
µqµwLµwKµwM
LqLwLLwKLwM
KqKwLKwKKwM
MqMwLMwKMwM
=
0.1071.5870.921.355
1.587-3.4271.0971.621
0.921.097-1.0250.94
1.3551.6210.94-3.928

The estimated Translog cost function's comparative statics.

Cq,qCq,wLCq,wKCq,wM
CwL,qCwL,wLCwL,wKCwL,wM
CwK,qCwK,wLCwK,wKCwK,wM
CwM,qCwM,wLCwM,wKCwM,wM
=
0.1111.5880.9211.355
1.588-3.4251.0981.617
0.9211.098-1.0240.939
1.3551.6170.939-3.922

The underlying specified Generalized CES cost function's comparative statics:

Cq,qCq,wLCq,wKCq,wM
CwL,qCwL,wLCwL,wKCwL,wM
CwK,qCwK,wLCwK,wKCwK,wM
CwM,qCwM,wLCwM,wKCwM,wM
=
0.1071.5870.9211.355
1.587-3.4241.0981.616
0.9211.098-1.0240.937
1.3551.6160.937-3.915

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) = (43.79, 24.14, 40.14)

          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 = 861.18; 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 8.64 11.184.96 19.2914.07 25.68451.38 1719.2914.0725.68 19.2814.07 25.7451.38 19.2914.0725.71451.55
218 5.86 14.766.3 29.2412.02 23.03493.83 1829.2412.0223.03 29.2512.03 23493.83 29.2512.0323.02493.95
319 5.24 12.94.48 29.212.25 27.57434.61 1929.212.2527.57 29.212.25 27.58434.61 29.2112.2527.58434.67
420 6.8 11.224.2 25.2415.03 31.14471.05 2025.2415.0331.14 25.2315.03 31.16471.05 25.2415.0331.15471.08
521 8.56 12.126.48 26.5318.16 27.76627.09 2126.5318.1627.76 26.5318.16 27.76627.1 26.5318.1627.77627.11
622 6.02 12.086.54 33.8517.18 26.18582.58 2233.8517.1826.18 33.8717.19 26.14582.56 33.8617.1926.15582.54
723 7.56 12.146.38 31.2219.26 29.64658.94 2331.2219.2629.64 31.2219.26 29.64658.95 31.2219.2629.64658.9
824 7.84 12.844.48 30.1918.26 39.12646.41 2430.1918.2639.12 30.1718.26 39.14646.4 30.1718.2639.12646.33
925 8.36 13.487.12 34.2821.17 32.11800.65 2534.2821.1732.11 34.2921.17 32.11800.66 34.2821.1732.1800.54
1026 8.04 11.987.96 36.123.99 29.91815.83 2636.123.9929.91 36.1224 29.88815.82 36.1223.9929.89815.7
1127 5.94 12.885.28 41.4119.82 37.01696.67 2741.4119.8237.01 41.4119.82 37.02696.67 41.419.8137.01696.56
1228 5.28 11.464.78 43.2720.74 38647.8 2843.2720.7438 43.2720.74 38.01647.81 43.2620.7338647.7
1329 5.82 14.34.72 45.7319.67 43.65753.46 2945.7319.6743.65 45.7219.67 43.67753.45 45.7219.6643.66753.33
1430 7 136 43.7924.14 40.13861.17 3043.7924.1440.13 43.7924.14 40.14861.18 43.7824.1440.13861.05
1531 8.74 11.544.66 36.6627.17 49.52864.61 3136.6627.1749.52 36.6427.17 49.53864.59 36.6427.1749.51864.47
1632 5.4 14.266.48 56.8623.2 39.21891.93 3256.8623.239.21 56.8623.2 39.21891.94 56.8423.1939.22891.83
1733 5.86 13.167.04 55.7526.26 38.47943.12 3355.7526.2638.47 55.7626.27 38.45943.12 55.7426.2738.46943.02
1834 7.26 11.965.1 46.2128.44 49.39927.45 3446.2128.4449.39 46.2128.43 49.4927.44 46.228.4349.38927.36
1935 8.68 12.824.02 42.1728.48 63.07984.7 3542.1728.4863.07 42.1828.5 63984.73 42.1828.4963984.67
2036 6.74 12.827.22 56.5530.93 42.681085.83 3656.5530.9342.68 56.5630.93 42.671085.84 56.5530.9342.681085.8
2137 7.74 13.227.82 55.5233.4 43.941214.9 3755.5233.443.94 55.5333.4 43.941214.91 55.5233.443.941214.89
2238 7.1 12.184.92 52.7231.51 56.441035.78 3852.7231.5156.44 52.7331.51 56.441035.78 52.7331.5156.441035.8
2339 6.68 14.747.32 65.1231.42 47.811248.04 3965.1231.4247.81 65.1231.41 47.811248.04 65.1231.4147.821248.11
2440 7.82 11.444.4 50.6434.82 64.111076.51 4050.6434.8264.11 50.6534.83 64.081076.52 50.6534.8464.081076.62
2541 6.42 14.766.6 68.7532.06 52.891263.65 4168.7532.0652.89 68.7532.05 52.91263.65 68.7632.0552.921263.8
2642 7.28 13.265.82 61.8535.33 58.491259.13 4261.8535.3358.49 61.8635.32 58.491259.13 61.8735.3358.51259.33
2743 6.02 13.285.28 69.0233.4 60.191176.94 4369.0233.460.19 69.0333.4 60.21176.93 69.0533.460.221177.15
2844 5.52 14.167.3 81.4534.74 50.741311.92 4481.4534.7450.74 81.4334.74 50.751311.93 81.4434.7550.781312.25




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)),

 

 
   

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