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Egwald Economics: Microeconomics

Duality and the Translog Production / Cost Functions
Homothetic 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

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   =  

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 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-0.0001140-0.584
aL0.35015301984.589
aK0.39985103435.043
aM0.25004702785.563
bLL-0.0196990-314.693
bKK-0.0214330-553.198
bMM-0.0163620-497.491
bLK0.0247320269.601
bLM0.0146010301.888
bKM0.0181550372.999
R2 = 1 R2b = 1 # obs = 28
Observation Matrix Rank: 10

aL + aK + aM = 1
-2*bLL = 0.039398 =~ 0.039333 = bLK + bLM
-2*bKK = 0.042865 =~ 0.042888 = bLK + bKLM
-2*bMM = 0.032724 =~ 0.032756 = bLM + bKM

CES
Translog Cost Function

SVD Restricted Least Squares
Parameter Estimates
Parameter Coefficient std error t-ratio
c1.0832880.0002883759.665029
cq0.9999480.0001486750.99703
cL0.3487670.0001871862.983168
cK0.3897760.0001742242.078844
cM0.2614570.0001162259.502348
dqq1.3E-54.0E-50.320775
dLL0.0334520.000109305.816038
dKK0.0364538.6E-5422.746446
dMM0.027836.2E-5446.932702
2*dLK-0.0420750.000193-218.122444
2*dLM-0.0248299.3E-5-267.231754
2*dKM-0.030830.000103-299.162016
dLq-0.0001228.5E-5-1.438873
dKq5.4E-58.2E-50.654381
dMq6.9E-55.5E-51.24741
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) = -0.000114 + 0.350153 * ln(L) + 0.399851 * ln(K) + 0.250047 * ln(M) + -0.019699 * ln(L)*ln(L) + -0.021433 * ln(K)*ln(K) + -0.016362 * ln(M)*ln(M)
+ 0.024732 * ln(L)*ln(K) + 0.014601 * ln(L)*ln(M) + 0.018155 * ln(K)*ln(M)   =   f(L,K,M).
       

The estimated Translog cost function:

ln(C(q;wL,wK,wM)) = 1.083288 + 0.999948 * ln(q) + 0.348767 * ln(wL) + 0.389776 * ln(wK) + 0.261457 * log(wM)
+ .5 * [1.3E-5 * ln(q)^2 + 0.033452 * ln(wL)^2 + 0.036453 * ln(wK)^2 + 0.02783 * ln(wM)^2]
+ .5 * [-0.042075 * ln(wL)*ln(wK) + -0.024829 * ln(wL)*ln(wM) + -0.03083 * ln(wK)*log(wM)]
+ -0.000122 * ln(wL)*ln(q) + 5.4E-5 * ln(wK)*ln(q) + 6.9E-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 = 0.2713.

Newton's Method:

Using these estimates, Newton's Method provides:

Nonlinear Optimization: Newton's Method
Parameter Estimates
Iter #λqvLvK
  0   0.2713 331.0714 1.6429
10.2596 26.741.0478 1.8467
20.2367 26.991.149 2.1134
30.2354 271.1647 2.1645
40.2354 271.1651 2.166
50.2354 271.1651 2.166

c. Solution Vector:

W = (L, K, M, q, vL, vK, λ) = (33.2, 21.97, 28.43, 27, 1.16514, 2.16595, 0.23537)

With λ = 0.23537, q = 27, vL = 1.16514, and vK = 2.16595, the first order conditions are:

0.   Hλ(33.2, 21.97, 28.43, q, vL, vK, λ) = 1.16514 * 33.2 + 2.16595 * 21.97 + 28.43 - C(q; 1.16514, 2.16595, 1) = 0
1.   Hq(33.2, 21.97, 28.43, q, vL, vK, λ) = 1 - 0.23537 * ∂C(q; 1.16514, 2.16595, 1)/∂q = 0
2.   HvL(33.2, 21.97, 28.43, q, vL, vK, λ) = 0.23537 * (33.2 - ∂C(q; 1.16514, 2.16595, 1)/∂vL) = 0
3.   HvK(33.2, 21.97, 28.43, q, vL, vK, λ) = 0.23537 * (21.97 - ∂C(q; 1.16514, 2.16595, 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
=
0-4.2490-0
-4.249-0-0.289-0.192
0-0.2893.78-1.272
-0-0.192-1.2721.187

  Determinant(J3) = -51.80986

 

J2   =  

0-4.2490
-4.249-0-0.289
0-0.2893.78

  Determinant(J2) = -68.22831

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   =  

vLvK1
000
λ00
0λ0
  =  
1.165142.165951
000
0.235400
00.23540

From the Implicit Function Theorem:

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

JΦ   =  

λLλKλM
qLqKqM
vLLvLKvLM
vKLvKKvKM
  =  
0.0028070.00523-0.007317
0.274230.509790.23537
-0.04126-00.04815
5.0E-5-0.115980.08956

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
1175.38 14.26.54 25.5112.52 16.24421.2925.512.5316.24421.3316.9980.2170.5730.264170.2170.5730.264
2186.98 13.965.84 22.6114.05 19.76469.3322.614.0519.77469.3517.9990.2680.5350.224180.2680.5350.224
3196.04 13.825.14 25.1313.93 21.65455.5125.1113.9321.66455.5418.9990.2520.5760.214190.2520.5760.214
4206.36 13.585.22 25.6115.05 22.76486.0725.615.0622.77486.0919.9990.2620.5590.215200.2620.5590.215
5218.12 13.27.94 25.3518.79 19.41607.9425.3418.7919.41607.9720.9990.280.4560.274210.2810.4560.274
6225.62 12.745.2 29.5116.49 23.68498.9929.4916.4923.69499.0221.9990.2480.5620.229220.2480.5620.229
7236.96 11.544.18 25.2418.4 29.25510.2425.2218.429.26510.23230.3140.520.188230.3140.520.188
8246.36 11.825.02 29.0119.19 26.65545.142919.226.66545.1623.9990.280.520.221240.280.520.221
9258.52 14.55.8 28.4120.25 29.6707.3728.3920.2629.61707.37250.3010.5130.205250.3010.5120.205
10265.08 14.124.34 36.9217.35 31.71570.1236.917.3531.73570.1625.9980.2320.6440.198260.2320.6440.198
11276.86 14.724.36 32.8719.24 36.29666.9132.8519.2536.3666.9226.9990.2780.5960.177270.2780.5960.176
12285.6 14.327.14 41.9721.17 25.65721.3441.9421.1825.66721.4327.9970.2170.5560.277280.2170.5560.277
13295.98 13.145.22 3821.8 32.04680.9437.9821.8132.06680.9728.9980.2540.560.222290.2550.560.222
14307 136 36.8924.42 31.59765.1736.8724.4231.61765.229.9990.2740.510.235300.2740.510.235
15316.72 11.045.74 36.4926.8 31.34720.9836.4626.8131.3672130.9990.2890.4750.247310.2890.4750.247
16326.64 12.584.26 37.3224.29 40.89727.537.324.2940.9727.5320.2920.5540.187320.2920.5530.187
17336.44 13.067.98 45.2927.82 28.36881.2845.2727.8328.36881.3532.9970.2410.4890.299330.2410.4890.299
18347.78 11.27.68 39.4432.41 29.96899.8839.4232.4229.96899.9133.9990.2940.4230.29340.2940.4230.29
19357.14 12.786.48 4329.37 35.08909.6142.9729.3735.1909.6634.9980.2740.4920.249350.2750.4920.249
20366.5 12.75.02 44.127.95 41.27848.8144.0727.9641.29848.8335.9990.2750.5390.213360.2760.5390.213
21376.28 12.386.78 48.830.7 34.35919.4748.7730.7234.36919.5436.9970.2520.4980.273370.2530.4980.273
22387.98 12.965.86 43.1532.01 42.151006.1843.1232.0242.181006.19380.3010.490.221380.3010.4890.221
23396.7 11.37.1 48.5234.85 34.7965.2348.534.8634.7965.2938.9980.2710.4570.287390.2710.4570.287
24406.84 11.984.44 45.4731.63 49.33909.0345.4431.6549.35909.03400.3010.5270.195400.3010.5270.195
25417.18 11.126.46 47.836.92 39.291007.5947.7736.9339.311007.6240.9990.2920.4530.263410.2920.4520.263
26428.2 14.56.32 49.6434.25 46.531197.8249.634.2646.561197.8441.9990.2870.5090.222420.2870.5080.222
27437.56 13.547.82 54.3437.09 39.671223.1854.3137.139.681223.2642.9970.2660.4760.275430.2660.4760.275
28447.16 14.226.08 55.2934.57 47.731177.6855.2534.5847.771177.7443.9980.2670.5310.227440.2670.5310.227

 

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
126.0001 40.2118.9556 33.3804
225.8909 36.8923.6056 31.8181
325.5108 36.924.3937 31.6029
425.5058 36.8924.4154 31.5938
525.5058 36.8924.4154 31.5938
625.5058 36.8924.4154 31.5938

c. Solution Vector:

Z = (q, wL, wK, wM, L, K, M, µ) = (30, 7, 13, 6, 36.89, 24.42, 31.59, 25.51)

With µ = 25.51, L = 36.89, K = 24.42, and M = 31.59, the first order conditions are:

0.   Gµ(30; 7, 13, 6, L, K, M, μ) = 30 - F(36.89, 24.42, 31.59) = -0
1.   GL(30; 7, 13, 6, L, K, M, μ) = 7 - µ * FL(36.89, 24.42, 31.59) = -0
2.   GK(30; 7, 13, 6, L, K, M, μ) = 13 - µ * FK(36.89, 24.42, 31.59) = 0
3.   GM(30; 7, 13, 6, L, K, M, μ) = 13 - µ * FM(36.89, 24.42, 31.59) = -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
-0.2740.148-0.14-0.064
-0.51-0.140.367-0.12
-0.235-0.064-0.120.168

  Determinant(J3) = -0.03122

 

J2   =  

0-0.274-0.51
-0.2740.148-0.14
-0.51-0.140.367

  Determinant(J2) = -0.10518

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
=
01.2290.8141.053
1.229-2.96911.297
0.8141-0.9340.857
1.0531.2970.857-3.369

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
=
01.2280.8141.054
1.228-2.96811.297
0.8141-0.9340.858
1.0541.2970.858-3.371

The underlying 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
=
01.230.8141.053
1.23-2.96811.295
0.8141-0.9340.857
1.0531.2950.857-3.367

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, 24.42, 31.59)

          Estimated Translog cost function factor demands: (L, K, M) = (36.87, 24.42, 31.61)

          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 = 765.2; 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 5.38 14.26.54 25.5112.52 16.24421.29 1725.5112.5216.24 25.5112.53 16.23421.29 25.512.5316.24421.33
218 6.98 13.965.84 22.6114.05 19.76469.33 1822.6114.0519.76 22.6114.05 19.76469.33 22.614.0519.77469.35
319 6.04 13.825.14 25.1313.93 21.65455.51 1925.1313.9321.65 25.1313.93 21.65455.51 25.1113.9321.66455.54
420 6.36 13.585.22 25.6115.05 22.76486.07 2025.6115.0522.76 25.6115.05 22.76486.07 25.615.0622.77486.09
521 8.12 13.27.94 25.3518.79 19.41607.94 2125.3518.7919.41 25.3518.79 19.41607.94 25.3418.7919.41607.97
622 5.62 12.745.2 29.5116.49 23.68498.99 2229.5116.4923.68 29.5116.49 23.68498.99 29.4916.4923.69499.02
723 6.96 11.544.18 25.2418.4 29.25510.24 2325.2418.429.25 25.2418.4 29.25510.24 25.2218.429.26510.23
824 6.36 11.825.02 29.0119.19 26.65545.14 2429.0119.1926.65 29.0119.19 26.65545.14 2919.226.66545.16
925 8.52 14.55.8 28.4120.25 29.6707.37 2528.4120.2529.6 28.4120.25 29.6707.37 28.3920.2629.61707.37
1026 5.08 14.124.34 36.9217.35 31.71570.12 2636.9217.3531.71 36.9217.34 31.71570.12 36.917.3531.73570.16
1127 6.86 14.724.36 32.8719.24 36.29666.91 2732.8719.2436.29 32.8719.24 36.28666.91 32.8519.2536.3666.92
1228 5.6 14.327.14 41.9721.17 25.65721.34 2841.9721.1725.65 41.9721.17 25.65721.34 41.9421.1825.66721.43
1329 5.98 13.145.22 3821.8 32.04680.94 293821.832.04 3821.8 32.04680.94 37.9821.8132.06680.97
1430 7 136 36.8924.42 31.59765.17 3036.8924.4231.59 36.8924.42 31.59765.17 36.8724.4231.61765.2
1531 6.72 11.045.74 36.4926.8 31.34720.98 3136.4926.831.34 36.4926.8 31.34720.98 36.4626.8131.36721
1632 6.64 12.584.26 37.3224.29 40.89727.5 3237.3224.2940.89 37.3224.29 40.88727.5 37.324.2940.9727.5
1733 6.44 13.067.98 45.2927.82 28.36881.28 3345.2927.8228.36 45.327.82 28.34881.27 45.2727.8328.36881.35
1834 7.78 11.27.68 39.4432.41 29.96899.88 3439.4432.4129.96 39.4532.41 29.95899.88 39.4232.4229.96899.91
1935 7.14 12.786.48 4329.37 35.08909.61 354329.3735.08 4329.37 35.08909.62 42.9729.3735.1909.66
2036 6.5 12.75.02 44.127.95 41.27848.81 3644.127.9541.27 44.0927.96 41.27848.81 44.0727.9641.29848.83
2137 6.28 12.386.78 48.830.7 34.35919.47 3748.830.734.35 48.830.71 34.35919.47 48.7730.7234.36919.54
2238 7.98 12.965.86 43.1532.01 42.151006.18 3843.1532.0142.15 43.1532.01 42.151006.18 43.1232.0242.181006.19
2339 6.7 11.37.1 48.5234.85 34.7965.23 3948.5234.8534.7 48.5234.85 34.69965.23 48.534.8634.7965.29
2440 6.84 11.984.44 45.4731.63 49.33909.03 4045.4731.6349.33 45.4731.64 49.32909.03 45.4431.6549.35909.03
2541 7.18 11.126.46 47.836.92 39.291007.59 4147.836.9239.29 47.836.92 39.291007.59 47.7736.9339.311007.62
2642 8.2 14.56.32 49.6434.25 46.531197.82 4249.6434.2546.53 49.6334.25 46.541197.82 49.634.2646.561197.84
2743 7.56 13.547.82 54.3437.09 39.671223.18 4354.3437.0939.67 54.3437.09 39.661223.18 54.3137.139.681223.26
2844 7.16 14.226.08 55.2934.57 47.731177.68 4455.2934.5747.73 55.2934.57 47.741177.68 55.2534.5847.771177.74




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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