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Egwald Economics: Microeconomics

Production Functions

by

Elmer G. Wiens

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Cobb-Douglas | CES | Generalized CES | Translog | Diewert | Translog vs Diewert | Diewert vs Translog | Estimate Translog | Estimate Diewert | 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

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

With the Cobb-Douglas and CES production functions, I obtained an explicit cost function, total cost as a function of q, wL, wK, and wM, by minimizing the cost of producing a given level of output. Because the Translog production function is much more general (it has a flexible functional form permitting the partial elasticities of substitution between inputs to vary), I will use numerical analysis to obtain the cost functions associated with a given Translog production function.

C. Translog (Transcendental Logarithmic) Production Function

The three factor Translog production function is:

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).

where L = labour, K = capital, M = materials and supplies, and q = product.

I. To obtain estimates of the Translog production function, let us use the CES production function to generate a sequence of observations relating the CES least cost factor inputs to factor prices and levels of output.

The three factor CES production function is:

q = A * [alpha * (L^-rho) + beta * (K^-rho) + gamma *(M^-rho)]^(-nu/rho)

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 estimated coefficients of the 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 < 2;
.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

CES Production Function Parameters
elasticity of scale parameter: nu
elasticity of substitution: sigma
alpha
beta
gamma

The CES production function as specified:

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

II. Estimating the Translog production function using multiple regression yielded the following coefficient estimates:

QR Restricted Least Squares
Parameter Estimates
Parameter Coefficient std error t-ratio
lnA6.0E-61.0E-66.841089
lnL0.3498915.0E-665632.858402
lnK0.3999942.0E-6189076.722475
lnM0.2501165.0E-651826.630638
lnLlnL-0.0196669.0E-6-2117.574784
lnKlnK-0.0213373.0E-6-6313.512891
lnMlnM-0.0164375.0E-6-3521.69695
lnLlnK0.0245651.1E-52316.547107
lnLlnM0.0147661.2E-51270.359603
lnKlnM0.0181081.0E-51888.785962
R2 = 1 R2b = 1 # obs = 182

2*bLL + bLK + bLM = 0
2*bKK + bLK + bKM = 0
2*bMM + bLM + bKM = 0

lnA = 6.0E-6 aL = 0.349891 aK = 0.399994 aM = 0.250116
bLL = -0.019666 bKK = -0.021337 bMM = -0.016437
bLK = 0.024565 bLM = 0.014766 bKM = 0.018108

aL + aK + aM = 1
2 * bLL + bLK + bLM = 0
2 * bKK + bLK + bKM = 0
2 * bMM + bLM + bKM = 0

The estimated Translog production function:

ln(q) = 6.0E-6 + 0.349891 * ln(L) + 0.399994 * ln(K) + 0.250116 * ln(M) + -0.019666 * ln(L)*ln(L) + -0.021337 * ln(K)*ln(K) + -0.016437 * ln(M)*ln(M)
+ 0.024565 * ln(L)*ln(K) + 0.014766 * ln(L)*ln(M) + 0.018108 * ln(K)*ln(M)   =   f(L,K,M).
       

Elasticity of Scale of Production:

        Long Run: Capital Variable:

εLKM = ε(L,K,M) = aL + aK + aM + (2*bLL + bLK + bLM) * ln(L) + (2*bKK + bLK + bKM) * ln(K) + (2*bMM + bLM + bKM) * ln(M).

        Short Run: Captial Fixed: K = K:

εLKM = ε(L,K,M) = aL + aM + (2*bLL + bLM) * ln(L) + (bLK + bKM) * ln(K) + (2*bMM + bLM) * ln(M).

See the Mathematical Notes below.

Curvature:

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

FLLFLK FLM
FKLFKK FKM
FMLFMK FMM
=|h3|  
FLLFLK
FKLFKK
=|h2|  
FLL
=|h1|  

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

The curvature of the estimated Translog production function depends on the elasticity of substitution, sigma, of the CES production function.

sigma < 1 & nu = 1   →   F(L,K,M) is concave to the origin of the 3-dimensional space of (L,K,M),

sigma = 1   & nu = alpha + beta + gamma < 1 →   F(L,K,M) is concave to the origin of the 3-dimensional space of (L,K,M),
       sigma = 1   & nu = alpha + beta + gamma = 1 →   F(L,K,M) is quasi-concave to the origin of the 3-dimensional space of (L,K,M),
     sigma = 1   & nu = alpha + beta + gamma > 1 →   F(L,K,M) is NOT concave to the origin of the 3-dimensional space of (L,K,M),

sigma > 1 & nu = 1   →   F(L,K,M) is concave to the origin of the 3-dimensional space of (L,K,M),

Check the displayed curvature conditions for other values of sigma, nu, alpha, beta, and gamma.

III. Least-cost combination of inputs

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

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

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

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

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

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

First Order Conditions:

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

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

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

GµµGµLGµKGµM
GGLLGLKGLM
GGKLGKKGKM
GGMLGMKGMM
=
0FLFKFM
FLGLLGLKGLM
FKGKLGKKGKM
FMGMLGMKGMM
=|H3|  
0FLFK
FLGLLGLK
FKGKLGKK
= |H2|

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

IV. Long Run: Capital Variable.

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

Translog Long Run Cost Data
Returns to Scale = 1, Elasticity of Substitution = 0.85
wL = 7, wK= 13, wM = 6
—   CES Data   ——   —   Estimated Translog Data   —   —AESLagrangianTranslog Curvature
qLKMcost q LK Mcost ave. costmarg. costεLKMsLKsLMsKM|H2|*10|H3|*102|h1|*102|h2|*104|h3|*107
16 19.6713.02 16.85408.09 16 19.6713.02 16.85408.09 25.5125.511 0.8510.850.85 1.971-10.975-1.091.87-0
18 22.1314.65 18.96459.1 18 22.1314.65 18.96459.1 25.5125.511 0.8510.850.85 1.752-8.672-0.971.48-0
20 24.5916.28 21.06510.11 20 24.5916.28 21.06510.11 25.5125.511 0.8510.850.85 1.577-7.024-0.871.2-0
22 27.0517.9 23.17561.13 22 27.0517.9 23.17561.13 25.5125.511 0.8510.850.85 1.433-5.805-0.790.99-0
24 29.5119.53 25.27612.14 24 29.5119.53 25.27612.14 25.5125.511 0.8510.850.85 1.314-4.878-0.720.83-0
26 31.9721.16 27.38663.15 26 31.9721.16 27.38663.15 25.5125.511 0.8510.850.85 1.213-4.156-0.670.71-0
28 34.4322.79 29.49714.16 28 34.4322.79 29.49714.16 25.5125.511 0.8510.850.85 1.126-3.584-0.620.61-0
30 36.8924.42 31.59765.17 30 36.8924.41 31.59765.17 25.5125.511 0.8510.850.85 1.051-3.122-0.580.53-0
32 39.3526.04 33.7816.18 32 39.3526.04 33.7816.18 25.5125.511 0.8510.850.85 0.985-2.744-0.540.47-0
34 41.8127.67 35.8867.19 34 41.8127.67 35.81867.2 25.5125.511 0.8510.850.85 0.927-2.43-0.510.41-0
36 44.2729.3 37.91918.21 36 44.2729.3 37.91918.21 25.5125.511 0.8510.850.85 0.876-2.168-0.480.37-0
38 46.7330.93 40.02969.22 38 46.7330.93 40.02969.22 25.5125.511 0.8510.850.85 0.83-1.946-0.460.33-0
40 49.1832.55 42.121020.23 40 49.1832.55 42.121020.23 25.5125.511 0.8510.850.85 0.788-1.756-0.430.3-0
42 51.6434.18 44.231071.24 42 51.6434.18 44.231071.24 25.5125.511 0.8510.850.85 0.751-1.593-0.410.27-0
44 54.135.81 46.341122.25 44 54.135.81 46.341122.25 25.5125.511 0.8510.850.85 0.717-1.451-0.40.25-0
46 56.5637.44 48.441173.26 46 56.5637.44 48.441173.26 25.5125.511 0.8510.850.85 0.686-1.328-0.380.23-0
48 59.0239.06 50.551224.27 48 59.0239.06 50.551224.28 25.5125.511 0.8510.850.85 0.657-1.219-0.360.21-0

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

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

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

J3   =  

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

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

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

JΦ   =  

µqµwLµwKµwM
LqLwLLwKLwM
KqKwLKwKKwM
MqMwLMwKMwM
=
01.230.8141.053
1.23-2.9691.0011.294
0.8141.001-0.9350.857
1.0531.2940.857-3.367

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

uLK = C(q;wL,wK,wM) * LwK / (L * K)   =   0.851,
uLM = C(q;wL,wK,wM) * LwM / (L * M)   =   0.85,
uKL = C(q;wL,wK,wM) * KwL / (K * L)   =   0.851,
uKM = C(q;wL,wK,wM) * KwM / (K * M)   =   0.85,
uML = C(q;wL,wK,wM) * MwL / (M * L)   =   0.85,
uMK = C(q;wL,wK,wM) * MwK / (M * K)   =   0.85,

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

εL,qεL,wLεL,wKεL,wM
εK,qεK,wLεK,wKεK,wM
εM,qεM,wLεM,wKεM,wM
=
1-0.5630.3530.211
10.287-0.4980.211
10.2870.353-0.639

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

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

2wwC(q;wL,wK,wM)   =  

LwLLwKLwM
KwLKwKKwM
MwLMwKMwM
=
-2.9691.0011.294
1.001-0.9350.857
1.2940.857-3.367

The eigenvalues of ∇2wwC  are e1 = -4.4778, e2 = -2.7932, and e3 = 0.

V. Short Run: Capital Fixed: K = K = 24.41.

Let the firm buy its inputs at the same prices: wL = 7   wK = 13   wM = 6 . Set the level of capital at the least cost level for q = 30.   Solving the least-cost problem holding capital fixed, we can compare the short-run CES cost data with the estimated Translog short-run cost data as displayed in the following table. (See the discussion of the short-run elasticity of substitution, sLM, below.)

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

Translog Short Run Cost Data
Returns to Scale = 1, Elasticity of Substitution = 0.85
wL = 7, wK= 13, wM = 6; K = 24.41
—   CES Data   ——   —   Estimated Translog Data   —   —AES   
qLKMcostq LK Mcost ave. costmarg. costεLKMεLKM3 factor
sLM
2 factor
sLM
|H2|*102|h1|*103|h2|*104
1613.124.41 11.22476.3916 13.124.41 11.22476.43 29.78 15.810.629 0.870.8621.216-25.225.84
1815.8224.41 13.55509.3718 15.8124.41 13.55509.36 28.3 17.1610.621 0.860.8614.888-19.383.47
2018.7624.41 16.07545.0920 18.7624.41 16.07545.08 27.25 18.5410.614 0.860.8610.78-15.242.16
2221.9324.41 18.78583.5622 21.9224.41 18.78583.55 26.52 19.9110.607 0.860.858.003-12.211.39
2425.3224.41 21.69624.7924 25.3224.41 21.69624.78 26.03 21.310.601 0.860.856.068-9.950.93
2628.9524.41 24.79668.7926 28.9524.41 24.79668.77 25.72 22.6910.596 0.850.854.684-8.220.63
2832.824.41 28.09715.5728 32.7924.41 28.09715.47 25.55 24.0910.59 0.850.853.675-6.870.44
3036.8924.41 31.59765.1730 36.8824.41 31.59765.08 25.5 25.5110.585 0.850.852.919-5.80.32
3241.2124.41 35.29817.6132 41.224.41 35.29817.52 25.55 26.9310.58 0.850.852.347-4.940.23
3445.7624.41 39.19872.934 45.7624.41 39.19872.82 25.67 28.3710.576 0.850.851.907-4.240.17
3650.5624.41 43.3931.0936 50.5524.41 43.29931.01 25.86 29.8210.572 0.840.851.564-3.670.13
3855.5924.41 47.61992.238 55.5824.41 47.6992.11 26.11 31.2810.568 0.840.851.294-3.190.1
4060.8724.41 52.131056.2740 60.8624.41 52.121056.15 26.4 32.7610.564 0.840.851.078-2.790.07
4266.3924.41 56.861123.3142 66.3824.41 56.851123.16 26.74 34.2510.56 0.840.840.905-2.460.06
4472.1724.41 61.811193.3844 72.1524.41 61.791193.18 27.12 35.7610.557 0.840.840.765-2.170.05
4678.1924.41 66.961266.5146 78.1724.41 66.941266.23 27.53 37.2910.553 0.830.840.65-1.930.04
4884.4724.41 72.341342.7248 84.4424.41 72.311342.36 27.97 38.8410.55 0.830.840.555-1.720.03

We get a U-shaped, short run average cost curve, with capital fixed. The short run average cost curve is (approx.) tangent to the long run average cost curve, at q = 30. The elasticity of scale, εLKM, is constant along the short run average cost curve, and equals the long run elasticity of scale, εLKM. The short run elasticity of scale with capital fixed at K = 24.41 is a decreasing function along the short run average cost curve, since sigma is less than 1.

Here I listed two measures of the short-run elasticity of substitution between L and M. The 3-factor measure of sLM uses the Allen partial elasticity of substitution formula. The 2-factor measure of sLM uses the standard short-run formula, which assumes that only L and M can vary with output, with a fixed amount of capital present. Another option would be to estimate a two factor production function q = F(L,M), and then to compute sLM. But then capital, K, is a missing variable from the estimation, skewing the estimates of the coefficients of the production function F

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

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

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

J2   =  

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

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

For example, at q = 30,   K = K = 24.41,   wL = 7,   wK = 13,  and   wM = 6 :

JΦ   =  

µqµwLµwM
LqLwLLwM
MqMwLMwM
=
0.7092.1011.799
2.101-1.8962.212
1.7992.212-2.581

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

εL,qεL,wLεL,wM
εM,qεM,wLεM,wM
=
1.709-0.360.36
1.7090.49-0.49

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

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

2wwC(q;wL,wK,wM)   =  

LwLLwM
MwLMwM
=
-1.8962.212
2.212-2.581

The eigenvalues of ∇2wwC  are e1 = -4.4764, and e2 = 0 .




Graph of Average Cost and Marginal Cost
Translog Cost / Production Functions - Capital Fixed
 
  Average cost function
  Marginal cost function
  L.R. Average cost function

VI. Long Run: Capital Variable.

Let us increase the price of M, materials and supplies. Now the firm buys its inputs at the prices: wL = 7   wK = 13   wM = 7 . Solving the least-cost problem, (and noting that µ = marginal cost,) we can compare the CES cost data with the estimated Translog cost data as displayed in the following table. The terms sLK, sLM, and sKM are the Allen partial elasticities of substitution.

Translog Long Run Cost Data
Returns to Scale = 1, Elasticity of Substitution = 0.85
wL = 7, wK= 13, wM = 7
—   CES Data   ——   —   Estimated Translog Data   —   —AESLagrangianTranslog Curvature
qLKMcost q LK Mcost ave. costmarg. costεLKMsLKsLMsKM|H2|*10|H3|*10^2|h1|*102|h2|*104|h3|*107
16 20.3313.45 15.27424.12 16 20.3313.46 15.27424.12 26.5126.511 0.850.850.85 1.767-12.72-1.021.65-0
18 22.8715.14 17.18477.13 18 22.8715.14 17.18477.13 26.5126.511 0.850.850.85 1.571-10.051-0.91.3-0
20 25.4116.82 19.09530.15 20 25.4116.82 19.09530.15 26.5126.511 0.850.850.85 1.414-8.141-0.811.06-0
22 27.9518.5 21583.16 22 27.9518.5 21583.16 26.5126.511 0.850.850.85 1.285-6.728-0.740.87-0
24 30.4920.18 22.91636.18 24 30.4920.18 22.91636.18 26.5126.511 0.850.850.85 1.178-5.654-0.680.73-0
26 33.0321.86 24.82689.19 26 33.0321.86 24.82689.19 26.5126.511 0.850.850.85 1.088-4.817-0.620.62-0
28 35.5723.55 26.73742.21 28 35.5823.55 26.72742.21 26.5126.511 0.850.850.85 1.01-4.154-0.580.54-0
30 38.1225.23 28.64795.22 30 38.1225.23 28.63795.22 26.5126.511 0.850.850.85 0.943-3.618-0.540.47-0
32 40.6626.91 30.54848.24 32 40.6626.91 30.54848.24 26.5126.511 0.850.850.85 0.884-3.18-0.510.41-0
34 43.228.59 32.45901.25 34 43.228.59 32.45901.25 26.5126.511 0.850.850.85 0.832-2.817-0.480.37-0
36 45.7430.27 34.36954.27 36 45.7430.27 34.36954.27 26.5126.511 0.850.850.85 0.786-2.513-0.450.33-0
38 48.2831.96 36.271007.28 38 48.2831.96 36.271007.28 26.5126.511 0.850.850.85 0.744-2.255-0.430.29-0
40 50.8233.64 38.181060.29 40 50.8233.64 38.181060.3 26.5126.511 0.850.850.85 0.707-2.035-0.410.26-0
42 53.3635.32 40.091113.31 42 53.3635.32 40.091113.31 26.5126.511 0.850.850.85 0.673-1.846-0.390.24-0
44 55.937 421166.32 44 55.9137 421166.33 26.5126.511 0.850.850.85 0.643-1.682-0.370.22-0
46 58.4438.68 43.911219.34 46 58.4538.68 43.91219.34 26.5126.511 0.850.850.85 0.615-1.539-0.350.2-0
48 60.9940.36 45.821272.35 48 60.9940.37 45.811272.36 26.5126.511 0.850.850.85 0.589-1.413-0.340.18-0

VII. Short Run: Capital Fixed: K = K = 25.23.

Let the firm buy its inputs at the same (wM increased) prices: wL = 7   wK = 13   wM = 7 . Set the level of capital at the least cost level for q = 30: then .   Solving the least-cost problem holding capital fixed, we can compare the short-run CES cost data with the estimated Translog short-run cost data as displayed in the following table. (See the discussion of the short-run elasticity of substitution, sLM, above.)

Translog Short Run Cost Data
Returns to Scale = 1, Elasticity of Substitution = 0.85
wL = 7, wK= 13, wM = 7; K = 25.23
—   CES Data   ——   —   Estimated Translog Data   —   —AES   
qLKMcostq LK Mcost ave. costmarg. costεLKMεLKM3 factor
sLM
2 factor
sLM
|H2|*102|h1|*103|h2|*104
1613.5925.23 10.21494.5216 13.625.23 10.2494.62 30.91 16.510.632 0.870.8625.178-23.346.58
1816.3925.23 12.32528.9418 16.425.23 12.3528.9 29.38 17.9110.624 0.870.8617.731-17.983.93
2019.4325.23 14.6566.1920 19.4325.23 14.59566.13 28.31 19.3110.616 0.860.8612.853-14.162.45
2222.725.23 17.06606.2822 22.725.23 17.05606.23 27.56 20.7410.61 0.860.869.56-11.371.58
2426.225.23 19.69649.2124 26.225.23 19.69649.18 27.05 22.1710.604 0.860.857.26-9.271.06
2629.9425.23 22.4969526 29.9325.23 22.5694.99 26.73 23.6110.598 0.860.855.613-7.670.72
2833.9125.23 25.48743.6628 33.8925.23 25.5743.68 26.56 25.0510.593 0.850.854.406-6.420.51
3038.1225.23 28.63795.2230 38.125.23 28.64795.14 26.5 26.5210.588 0.850.853.51-5.420.36
3242.5625.23 31.97849.732 42.5325.23 31.99849.59 26.55 27.9810.583 0.850.852.826-4.620.27
3447.2425.23 35.49907.1234 47.225.23 35.51906.98 26.68 29.4610.578 0.850.852.3-3.980.2
3652.1725.23 39.19967.5135.99 52.1125.23 39.22967.33 26.87 30.9610.574 0.850.851.889-3.440.15
3857.3425.23 43.081030.9137.99 57.2625.23 43.121030.67 27.12 32.4710.57 0.840.851.565-30.11
4062.7625.23 47.151097.3339.99 62.6625.23 47.211097.03 27.43 33.9910.566 0.840.851.306-2.630.09
4268.4325.23 51.411166.8242 68.3525.23 51.491166.81 27.78 35.510.563 0.840.851.096-2.310.07
4474.3525.23 55.861239.4144 74.2425.23 55.961239.36 28.17 37.0510.559 0.840.840.927-2.050.05
4680.5325.23 60.51315.1346 80.3925.23 60.621315.03 28.59 38.6110.556 0.840.840.788-1.820.04
4886.9625.23 65.331394.0148 86.7925.23 65.471393.85 29.04 40.1910.552 0.830.840.674-1.630.03

We get a U-shaped, short run average cost curve, with capital fixed. The short run average cost curve is (approx.) tangent to the long run average cost curve, at q = 30. The elasticity of scale, εLKM, is constant along the short run average cost curve, and equals the long run elasticity of scale, εLKM. The short run elasticity of scale with capital fixed at K = 25.23 is a decreasing function along the short run average cost curve, since sigma is less than 1.

Graph of Average Cost and Marginal Cost
Translog Cost / Production Functions - Capital Fixed
 
  Average cost function
  Marginal cost function
  L.R. Average cost function

VIII. Isoquants

A given level of output, q = q, can be produced by different combinations of factor inputs, L, K, and M. Fixing the level of product output at q = q, we obtain an equation from the 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).

for the 3-dimensional isoquant surface, when q = q.

The isoquant surface is tangent to the isocost plane:

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

at the cost minimizing combination of factor inputs, (L, K, M ) = (L, K, M).

Consider the Translog production function as estimated above:

ln(q) = 6.0E-6 + 0.349891 * ln(L) + 0.399994 * ln(K) + 0.250116 * ln(M) + -0.019666 * ln(L)*ln(L) + -0.021337 * ln(K)*ln(K) + -0.016437 * ln(M)*ln(M)
+ 0.024565 * ln(L)*ln(K) + 0.014766 * ln(L)*ln(M) + 0.018108 * ln(K)*ln(M)

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

(L, K, M) = (36.89, 24.41, 31.59), and

C(30) = 7 * 36.89 + 13 * 24.41 + 6 * 31.59 = 765.17.

Solving the Translog equation for L, K, and M in turn, we get three equations for the 3-dimensional isoquant surface. By fixing the amount of input for one factor in these equations, we obtain three 2-dimensional isoquant curves.

Firstly, consider the L-K Isoquant. Set M = M, and let:

a = bKK, b = aK + bLK*log(L) + bKM * log(M),
c = log(q) - log(A) - aL * log(L) - aM * log(M) - bLL * log(L) * log(L) - bMM * log(M) * log(M) - bLM * log(L) * log(M).

The L-K isoquant expressed as K as a function of L:

K = exp( (-b + sqrt(b*b+4*a*c)) / (2*a) ).

Secondly, consider the L-M Isoquant. Set K = K, and let:

a = bMM, b = aM + bLM*log(L) + bKM * log(K),
c = log(q) - log(A) - aL * log(L) - aK * log(K) - bLL * log(L) * log(L) - bKK * log(K) * log(K) - bLK * log(L) * log(K).

The L-M isoquant expressed as M as a function of L:

M = exp( (-b + sqrt(b*b+4*a*c)) / (2*a) ).

The following diagrams graph, in blue, the L-K and L-M isoquants for q = 24, 30, and 36.

The yellow lines represent the isocost lines, combinations of L, K, and M that can be purchased at a constant total cost at the prices wL = 7, wK = 13, and wM = 6.

The slope of an L-K isocost line is mK = -wL / wK = -7 / 13; the slope of an L-M isocost line is mM = -wL / wM = -7 / 6.

For q = 30, the L-K isocost line has a K-intercept at (C(30) - wM * M) / wK = (765.17 - 6 * 31.59)/13 = 44.28,
while the L-M isocost line has a M-intercept at (C(30) - wK * K) / wM = (765.17 - 13 * 24.41)/6 = 74.63.

For q = 30, the L-K isoquant is tangent to the L-K isocost line at (L, K) = (36.89, 24.41), while the L-M isoquant is tangent to the L-M isocost line at (L, M) = (36.89, 31.59).

Translog Production Function isoquants
L-K Isoquants, M = ML-M Isoquants, K = K

It can be profitable to compare the Translog isoquants with the isoquants of the CES production function, from which the Translog's isoquants were derived.

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

(L, K, M) = (36.89, 24.42, 31.59), and

C(30) = 7 * 36.89 + 13 * 24.42 + 6 * 31.59 = 765.17.

CES Production Function Isoquants
L-K Isoquants, M = ML-M Isoquants, K = K



Mathematical Notes

I. 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).

where L = labour, K = capital, M = materials and supplies, and q = product.

Elasticity of Scale of Production:

        Long Run: Capital Variable:

dln(q) = dq/q,   dln(L) = dL/L,   dln(K) = dK/K,   dln(M) = dM/M

dq/q = aL * dL/L + aK * dK/K + aM * dM/M + 2 * bLL * ln(L) * dL/L + 2 * bKK * ln(K) * dK/K + 2 * bMM * ln(M) * dM/M
          + bLK * ln(K) * dL/L + blk * ln(L) * dK/K + bLM * ln(M) * dL/L + bLM * ln(L) * dM/M + bKM * ln(M) * dK/K + bKM * ln(K) * dM/M

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

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

εLKM = ε(L,K,M) = aL + aK + aM + (2*bLL + bLK + bLM) * ln(L) + (2*bKK + bLK + bKM) * ln(K) + (2*bMM + bLM + bKM) * ln(M).

        Short Run: Captial Fixed: K = K:

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

εLKM = ε(L,K,M) = aL + aM + (2*bLL + bLM) * ln(L) + (bLK + bKM) * ln(K) + (2*bMM + bLM) * ln(M).

Curvature:

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

FLLFLK FLM
FKLFKK FKM
FMLFMK FMM
=|h3|  
FLLFLK
FKLFKK
=|h2|  
FLL
=|h1|  

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

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

fL(L,K,M) = (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).

III. The least-cost combination of inputs:

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

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

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

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

First Order Conditions:

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

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

GµµGµLGµKGµM
GGLLGLKGLM
GGKLGKKGKM
GGMLGMKGMM
=
0FLFKFM
FLGLLGLKGLM
FKGKLGKKGKM
FMGMLGMKGMM
=|H3|  
0FLFK
FLGLLGLK
FKGKLGKK
= |H2|

 where:

(0,0) Gµµ = 0,   (0,1) GµL = fL * exp(f(L,K,M)),   (0,2) GµK = fK * exp(f(L,K,M)),   (0,3) GµM = fM * exp(f(L,K,M)),
(1,0) G = GµL,   (1,1) GLL = µ * [fLL + fL * fL] * exp(f(L,K,M)),   (1,2) GLK = µ * [fLK + fL * fK] * exp(f(L,K,M)),   (1,3) GLM = µ * [fLM + fL * fM] * exp(f(L,K,M)),  
(2,0) G = GµK,   (2,1) GKL = GLK,   (2,2) GKK = µ * [fKK + fK * fK] * exp(f(L,K,M)),   (2,3) GKM = µ * [fKM + fK * fM] * exp(f(L,K,M)),  
(3,0) G = GµM,   (3,1) GML = GLM,   (3,2) GMK = GKM,   (3,3) GMM = µ * [fMM + fM * fM] * exp(f(L,K,M)),  

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

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

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

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

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

First Order Conditions: K = K

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

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

GµµGµLGµM
GGLLGLM
GGMLGMM
=
0FLFM
FLGLLGLM
FMGMLGMM
=|H2|

 where:

(0,0) Gµµ = 0,   (0,1) GµL = fL * exp(f(L,K,M)),   (0,2) GµM = fM * exp(f(L,K,M)),
(1,0) G = GµL,   (1,1) GLL = µ * [fLL + fL * fL] * exp(f(L,K,M)),   (1,2) GLM = µ * [fLM + fL * fM] * exp(f(L,K,M)),  
(2,0) G = GµM,   (2,1) GML = GLM,   (2,2) GMM = µ * [fMM + fM * fM] * exp(f(L,K,M)),  

V. Allen partial elasticity of substitution

Writing the production function as q = F(L,K,M), let the bordered Hessian be:

F =
0FLFK FM
FLFLLFLK FLM
FKFKLFKK FKM
FMFMLFMK FMM

If |F| is the determinant of the bordered Hessian and |FLK| is the cofactor associated with FLK, then the Allen elasticity of substitution is defined as:

sLK = ((FL * L + FK * K + FM *M) / (L * K)) * (|FLK|/|F|)

VI. Two factor elasticity of substitution

Let the production function be q = F(L,K,M), where K is underlined to indicate it is constant in the short-run. Then the two factor (L and M) bordered Hessian is:

F =
0FLFM
FLFLLFLM
FMFMLFMM

The 2-factor elasticity of substitution between L and M is:

sLM = - (FL * L + FM * M) / (L * M ) * (FL * FM) / |F|

 

 
   

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