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Egwald Economics: Microeconomics

Cost Functions

by

Elmer G. Wiens

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

K. Translog (Transcendental Logarithmic) Cost 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.

If we have a data set relating these inputs to output for varying levels of inputs and output, we can estimate the parameters of the Translog production function directly.

Suppose, however, we have a data set relating output quantities to total costs and input (factor) prices, wL, wK, and wM. Then, we can work with the Translog cost function to estimate the parameters of the production technology.

The three factor Translog (total) cost function is:

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 linear in its 18 parameters, c, cq, cL, cK, cM, dqq, dLL, dKK, dMM, dLK, dKL, dLM, dML, dKM, dMK, dLq, dKq, and dMq.

We shall use two methods to obtain estimates of the parameters:

          À1: Estimate the parameters the cost function by way of its factor share functions — requires data relating output quantities to (cost minimizing) factor inputs, and input prices,

          À2: Estimate the parameters of the cost function directly — requires data relating output quantities to total (minimum) cost and input prices.

We shall also:

          À3:   Re-estimate the Translog Production Function — to investigate properties of the underlying technology.

À1:   Estimate the factor share equations separately.

Taking the partial derivative of the cost function, C(q;wL,wK,wM), with respect to an input price, we get the (nonlinear) factor demand function for that input:

∂C/∂wL = L(q; wL, wK, wM),

∂C/∂wK = K(q; wL, wK, wM),

∂C/∂wM = M(q; wL, wK, wM)

The (total) cost of producing q units of output equals the sum of the factor demand functions weighted by their input prices:

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

Write the factor share functions as:

sL(q;wL,wK,wM) = wL * L(q; wL, wK, wM) / C(q;wL,wK,wM) = wL * ∂C/∂wL / C(q;wL,wK,wM) = (∂ln(C)/∂wL )/ (∂ln(wL)/∂wL) = ∂ln(C)/∂ln(wL),

sK(q;wL,wK,wM)= wK * K(q; wL, wK, wM) / C(q;wL,wK,wM) = wK * ∂C/∂wK / C(q;wL,wK,wM) = (∂ln(C)/∂wK )/ (∂ln(wK)/∂wK) = ∂ln(C)/∂ln(wK),

sM(q;wL,wK,wM)= wM * M(q; wL, wK, wM) / C(q;wL,wK,wM) = wM * ∂C/∂wM / C(q;wL,wK,wM) = (∂ln(C)/∂wM )/ (∂ln(wL)/∂wM) = ∂ln(C)/∂ln(wM).

The Translog 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.

Since it is likely that, as measured numerically, dLK != dKL, dLM != dML, and dKM != dMK, these distinctions are maintained in the factor share equations.

Having obtained the unrestricted estimates of the coefficients of the factor share functions, we shall compute the restricted least squares estimates with dLK = dKL, dLM = dML, and dKM = dMK.

I. Duality.  The Plan:

      1. Generate CES data and estimate the parameters of a Translog production function.

      2. Generate cost data with the estimated Translog production function, yielding a data set relating output, q, and factor prices, wL, wK, and wM, [randomized about the base factor prices (wL*, wK*, wM*)]   to the Translog cost minimizing factor inputs, L, K, and M. The base factor prices and the randomizing distribution are specified in the form below.

      3. Stage 1. Obtain the estimates of the fifteen parameters of the three factor share equations using linear multiple regression.

      4. Stage 2. Substitute the values of these parameters into the Translog cost function, and estimate its remaining three parameters, c, cq, and dqq, using linear multiple regression.

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.

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).

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
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:

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

II. For these coefficients of the CES production function, I generated a sequence of factor prices, outputs, and the corresponding cost minimizing inputs. Then I estimated the Translog production function using multiple regression yielding the following coefficient estimates:

QR Least Squares
Parameter Estimates
Parameter Coefficient std error t-ratio
lnA6.0E-600.581
aL0.349891039932.852
aK0.399994063895.119
aM0.250116030768.879
bLL-0.0196660-1954.4
bKK-0.0213370-5734.291
bMM-0.0164370-3243.464
bLK0.02456502137.485
bLM0.01476601173.702
bKM0.01810801742.463
R2 = 1 R2b = 1 # obs = 182

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 = 0.039331 =~ 0.039331 = bLK + bLM
-2*bKK = 0.042673 =~ 0.042673 = bLK + bKLM
-2*bMM = 0.032875 =~ 0.032875 = bLM + bKM

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).
       

III. For these coefficients of the Translog production function, I generated a new sequence (displayed in the "Translog Cost Function" table) 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.

Then I used these data to estimate the coefficients of each factor share equation separately:

QR Least Squares
Parameter Estimates
Parameter Coefficient std error t-ratio
cL0.348464013403.997
dLq-6.0E-60-1.981
dLL0.03344606284.853
dLK-0.0208960-1904.037
dLM-0.0125360-2429.233
R2 = 1 R2b = 1 # obs = 31

QR Least Squares
Parameter Estimates
Parameter Coefficient std error t-ratio
cK0.38991609630.871
dKq3.0E-600.658
dKL-0.0209010-2521.941
dKK0.0363102124.504
dKM-0.0154010-1916.4
R2 = 1 R2b = 1 # obs = 31

QR Least Squares
Parameter Estimates
Parameter Coefficient std error t-ratio
cM0.2616207695.037
dMq3.0E-600.731
dML-0.0125450-1802.542
dMK-0.0154140-1073.974
dMM0.02793704139.594
R2 = 1 R2b = 1 # obs = 31

The three estimated factor share functions are:

sL(q;wL,wK,wM) = 0.348464 + -6.0E-6 * ln(q) + 0.033446 * ln(wL) + -0.020896 * ln(wK) + -0.012536 * ln(wM),

sK(q;wL,wK,wM) = 0.389916 + 3.0E-6 * ln(q) + -0.020901 * ln(wL) + 0.03631 * ln(wK) + -0.015401 * ln(wM),

sM(q;wL,wK,wM) = 0.26162 + 3.0E-6 * ln(q) + -0.012545 * ln(wL) + -0.015414 * ln(wK) + 0.027937 * ln(wM)

As estimated, generally dLK != dKL, dLM != dML, and dKM != dMK: Young's Theorem doesn't hold without constraints across equations.

Note: C(q;wL,wK,wM) linear homogeneous in factor prices implies:

1   =?   cL + cK + cM   =   0.348464 + 0.389916 + 0.26162   =   1
0   =?   dLL + dLK + dLM   =   0.033446 + -0.020896 + -0.012536  =   1.4E-5
0   =?   dKL + dKK + dKM   =   -0.020901 + 0.03631 + -0.015401  =   8.0E-6
0   =?   dML + dMK + dMM   =   -0.012545 + -0.015414 + 0.027937  =   -2.2E-5
0   =?   dLq + dKq + dMq   =   -6.0E-6 + 3.0E-6 + 3.0E-6  =   -0

IV. The Restricted Factor Share Equations.

Having obtained the unrestricted estimates of the coefficients of the factor share functions, we compute the restricted least squares estimates with dLK = dKL, dLM = dML, and dKM = dMK.

QR Restricted Least Squares
Parameter Estimates
Parameter Coefficient std error t-ratio
cL0.3484772.2E-515772.546284
dLq-6.0E-64.0E-6-1.55786
dLL0.0334467.0E-65104.22607
dLK-0.0208996.0E-6-3567.030535
dLM-0.012544.0E-6-2870.001573
cK0.3899123.2E-512265.871142
dKq3.0E-64.0E-60.784664
dKL-0.0208996.0E-6-3567.030535
dKK0.0363121.3E-52726.994528
dKM-0.0154046.0E-6-2670.703087
cM0.261592.1E-512654.170383
dMq2.0E-64.0E-60.529051
dML-0.012544.0E-6-2870.001573
dMK-0.0154046.0E-6-2670.703087
dMM0.0279366.0E-64779.264063
R2 = 1 R2b = 1 # obs = 93

dLK = dKL, dLM = dML, and dKM = dMK

The three estimated, restricted factor share functions are:

sL(q;wL,wK,wM) = 0.348477 + -6.0E-6 * ln(q) + 0.033446 * ln(wL) + -0.020899 * ln(wK) + -0.01254 * ln(wM),

sK(q;wL,wK,wM) = 0.389912 + 3.0E-6 * ln(q) + -0.020899 * ln(wL) + 0.036312 * ln(wK) + -0.015404 * ln(wM),

sM(q;wL,wK,wM) = 0.26159 + 2.0E-6 * ln(q) + -0.01254 * ln(wL) + -0.015404 * ln(wK) + 0.027936 * ln(wM)

Imposing the dLK = dKL, dLM = dML, and dKM = dM restrictions may distort five necessary C(q;wL,wK,wM) linear homogeneous in factor prices restraints:

1   =?   cL + cK + cM   =   0.348477 + 0.389912 + 0.26159   =   0.99998
0   =?   dLL + dLK + dLM   =   0.033446 + -0.020899 + -0.01254  =   7.0E-6
0   =?   dKL + dKK + dKM   =   -0.020899 + 0.036312 + -0.015404  =   9.0E-6
0   =?   dML + dMK + dMM   =   -0.01254 + -0.015404 + 0.027936  =   -8.0E-6
0   =?   dLq + dKq + dMq   =   -6.0E-6 + 3.0E-6 + 2.0E-6  =   -1.0E-6

Let us impose these additional five restraints and re-estimate the Restricted Factor Share Equations:

QR Restricted Least Squares
Parameter Estimates
Parameter Coefficient std error t-ratio
cL0.3484921.0E-533343.0692
dLq-5.0E-63.0E-6-1.761565
dLL0.0334424.0E-67452.247241
dLK-0.0209045.0E-6-4540.876727
dLM-0.0125393.0E-6-3583.27177
cK0.3899331.1E-535538.199959
dKq4.0E-63.0E-61.231061
dKL-0.0209045.0E-6-4540.876727
dKK0.0363047.0E-64953.248688
dKM-0.0154015.0E-6-3137.982657
cM0.2615751.1E-524745.705209
dMq2.0E-63.0E-60.522921
dML-0.0125393.0E-6-3583.27177
dMK-0.0154015.0E-6-3137.982657
dMM0.0279395.0E-65842.236679
R2 = 1 R2b = 1 # obs = 93

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 three re-estimated, restricted factor share functions are:

sL(q;wL,wK,wM) = 0.348492 + -5.0E-6 * ln(q) + 0.033442 * ln(wL) + -0.020904 * ln(wK) + -0.012539 * ln(wM),

sK(q;wL,wK,wM) = 0.389933 + 4.0E-6 * ln(q) + -0.020904 * ln(wL) + 0.036304 * ln(wK) + -0.015401 * ln(wM),

sM(q;wL,wK,wM) = 0.261575 + 2.0E-6 * ln(q) + -0.012539 * ln(wL) + -0.015401 * ln(wK) + 0.027939 * ln(wM)

V.  To obtain estimates of the remaining three parameters, c, cq, and dqq, 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
c1.08316706461194.988367
cq10200709406235.71
dqq000
R2 = 1 R2b = 1 # obs = 31

1 = cq, 0 = dqq

VI. The Translog cost function as estimated is:

ln(C(q;wL,wK,wM)) = 1.083167 + 1 * ln(q) + 0.348492 * ln(wL) + 0.389933 * ln(wK) + 0.261575 * log(wM)
+ .5 * [0 * ln(q)^2 + 0.033442 * ln(wL)^2 + 0.036304 * ln(wK)^2 + 0.027939 * ln(wM)^2]
+ .5 * [-0.041807 * ln(wL)*ln(wK) + -0.025077 * ln(wL)*ln(wM) + -0.030801 * ln(wK)*log(wM)]
+ -5.0E-6 * ln(wL)*ln(q) + 4.0E-6 * ln(wK)*ln(q) + 2.0E-6 * ln(wM)*ln(q)           (***)

VII. Check for linear homogeneity:

1   =?   cL + cK + cM   =   0.348492 + 0.389933 + 0.261575   =   1
0   =?   dLL + dLK + dLM   =   0.033442 + -0.020904 + -0.012539  =   -0
0   =?   dKL + dKK + dKM   =   -0.020904 + 0.036304 + -0.015401  =   0
0   =?   dML + dMK + dMM   =   -0.012539 + -0.015401 + 0.027939  =   0
0   =?   dLq + dKq + dMq   =   -5.0E-6 + 4.0E-6 + 2.0E-6  =   0

C(q;wL,wK,wM) linear homogeneous in factor prices requires for any t > 0, that C(q;wL,wK,wM) obey:

C(q;t*wL,t*wK,t*wM) = t * C(q;wL,wK,wM)

      For example, with wL = 7, wK = 13, wM = 6, and t = 2:

2 * C(25; 7, 13, 6) = 1275.29 =? 1275.29 = C(25; 14, 26,12).

2 * C(30; 7, 13, 6) = 1530.34 =? 1530.34 = C(30; 14, 26,12).

VIII. Check for homotheticity:

Can we write C(q;wL,wK,wM) as the product of the unit cost function C(1;wL,wK,wM) and a suitable increasing function h(q) with h(0) = 0, and h(1) = 1:

C(q;wL,wK,wM) =? h(q) * C(1;wL,wK,wM)

To do so it is necessary that:

1   =?   cq   =   1,     0   =?   dqq  =   0,

0   =?   dLq  =   -5.0E-6,     0   =?   dKq  =   4.0E-6,     0   =?   dMq  =   2.0E-6.

      For example, with wL = 7, wK = 13, wM = 6, and q = 30, try h(q) = q1/nu with nu = 1:

C(30; 7, 13, 6) = 765.17 =? 765.17 = 30 * 25.51 = 30^1/1 * C(1; 7, 13, 6).

Try it with sigma = 1!

IX. 1. The matrix ∇2C of second order partial derivatives of the cost function C(q;wL,wK,wM) is symmetric. To show this, write:

ln(C(q;wL,wK,wM) = C(q;wL,wK,wM), so C(q;wL,wK,wM) = exp(C(q;wL,wK,wM)).

2C/∂q∂wL = dLq/(q*wL) = ∂2C/∂wL∂q,

2C/∂q∂wK = dKq/(q*wK) = ∂2C/∂wK∂q,

2C/∂q∂wM = dMq/(q*wM) = ∂2C/∂wM∂q,

2C/∂wL∂wK = .25 * (dLK+dKL)/(wL*wK) = ∂2C/∂wK∂wL,

2C/∂wL∂wM = .25 * (dLM+dML)/(wL*wM) = ∂2C/∂wM∂wL,

2C/∂wK∂wM = .25 * (dKM+dMK)/(wL*wK) = ∂2C/∂wM∂wK.

        Since the matrix ∇2C(q;wL,wK,wM) is symmetric, the matrix ∇2C(q;wL,wK,wM) is also symmetric.

  2. The cost function C(q;wL,wK,wM) is a concave in factor prices if its Hessian matrix ∇2wwC of second order partial derivatives with respect to factor prices is negative semidefinite.   Following the procedure suggested by Diewert and Wales (1987), write sL = sL(q;wL,wK,wM), sK = sK(q;wL,wK,wM), and sM = sM(q;wL,wK,wM), and the matrices:

D =
dLLdLKdLM
dKLdKKdKM
dMLdMKdMM
S =
sL 0 0
 0sK 0
 0 0sM
SS =
sL*sLsL*sKsL*sM
sK*sLsK*sKsK*sM
sM*sLsM*sKsM*sM
W =
wL 0 0
 0wK 0
 0 0wM

Assuming C(q;wL,wK,wM) > 0, then ∇2wwC is negative semidefinite if and only if the matrix:

H = W * ∇2wwC * W   =   D   -   S   +   SS

is negative semidefinite. See the Mathematical Notes.

  The values of the second order partial derivatives depend on the amount of output, and on factor prices. As an example, consider the case where q = 30, wL = 7, wK = 13, and wM = 6:

W * ∇2wwC * W =
-0.190140.119080.07107
0.11908-0.206440.08736
0.071070.08736-0.15843

The principal minors of H are H1 = -0.190141, H2 = 0.025073, and H3 = -0.

If these principal minors alternate in sign, starting with negative, with H3 <= 0, the matrix H is negative (semi)definite,
and the cost function C(q;wL,wK,wM) is a concave function in factor prices at q = 30, wL = 7, wK = 13, and wM = 6.

The eigenvalues of H are e1 = -0.3198, e2 = -0.2352, and e3 = -0.
H3 = e1 * e2 * e3 = -0.

X. The three estimated factor demand functions are obtained by:

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.

    The estimated factor demand elasticities are obtained by:

εL,wL = ∂ln(L(q;wL,wK,wM))/∂ln(wL) = -1 + sL(q;wL,wK,wM) + dLL / sL(q;wL,wK,wM),

εL,wK = ∂ln(L(q;wL,wK,wM))/∂ln(wK) = sK(q;wL,wK,wM) + dLK / sL(q;wL,wK,wM),

εL,wM = ∂ln(L(q;wL,wK,wM))/∂ln(wM) = sM(q;wL,wK,wM) + dLM / sL(q;wL,wK,wM),

εL,q = ∂ln(L(q;wL,wK,wM))/∂ln(q) = q * ∂ln(C)/∂q + dLq / sL(q;wL,wK,wM), etc.

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

εL,wLεL,wKεL,wMεL,q
εK,wLεK,wKεK,wMεK,q
εM,wLεM,wKεM,wMεM,q
-0.56340.35290.21061
0.2871-0.49770.21061
0.28690.3526-0.63951

XI. Uzawa Partial Elasticities of Substitution:

uLK = C(q;wL,wK,wM) * ∂L(q;wL,wK,wM)/∂wK / (L(q;wL,wK,wM) * K(q;wL,wK,wM)),

uLM = C(q;wL,wK,wM) * ∂L(q;wL,wK,wM)/∂wM / (L(q;wL,wK,wM) * M(q;wL,wK,wM)),

uKL = C(q;wL,wK,wM) * ∂K(q;wL,wK,wM)/∂wL / (K(q;wL,wK,wM) * L(q;wL,wK,wM)),

uKM = C(q;wL,wK,wM) * ∂K(q;wL,wK,wM)/∂wM / (K(q;wL,wK,wM) * M(q;wL,wK,wM)),

uML = C(q;wL,wK,wM) * ∂M(q;wL,wK,wM)/∂wL / (M(q;wL,wK,wM) * L(q;wL,wK,wM)),

uMK = C(q;wL,wK,wM) * ∂M(q;wL,wK,wM)/∂wK / (M(q;wL,wK,wM) * K(q;wL,wK,wM)),

    where the partial derivatives of the factor demand functions are:

∂L(q;wL,wK,wM)/∂wK = (dLK * C / wK + sL * K) * C / (wL * L * K),

∂L(q;wL,wK,wM)/∂wM = (dLM * C / wM + sL * M) * C / (wL * L * M),

∂K(q;wL,wK,wM)/∂wL = (dKL * C / wL + sK * L) * C / (wK * K * L),

∂K(q;wL,wK,wM)/∂wM = (dKM * C / wM + sK * M) * C / (wK * K * M),

∂M(q;wL,wK,wM)/∂wL = (dML * C / wL + sM * L) * C / (wM * M * L),

∂M(q;wL,wK,wM)/∂wK = (dMK * C / wK + sM * K) * C / (wM * M * K)

With the dLK = dKL, dLM = dML, and dKM = dMK restrictions in the factor share functions, we get (approximate?) equality of the cross partial elasticities of substitution:

uLK = uKL,    uLM = uML,  and    uKM = uMK.

as seen in the following table.

XII. Table of Results: check that the estimated Translog cost function for a given level of output agrees with the minimized cost using the Translog production function.

Translog Cost Function
À1: Estimate the cost function using restricted factor shares
CES: Returns to Scale = 1, Elasticity of Substitution = 0.85
   —   Translog Production Data   —    —   Translog Cost Data   —   Factor Shares   —   Uzawa Elasticities   —   W * ∇2wwC * W
obs #qwLwKwM LK MsLKsLMsKMcostest costest Lest Kest MsLsKsMuLKuLMuKLuKMuMLuMKe1e2e3
1185.54 14.867.8 28.0413.58 15.750.850.850.86480.05480.0528.0413.5815.750.3240.420.2560.850.850.850.860.850.86 -0.317-0.2390
2195.64 12.55.6 25.6614.61 19.390.850.850.85435.96435.9625.6614.6119.390.3320.4190.2490.850.850.850.850.850.85 -0.319-0.235-0
3208.76 12.35.68 21.0317.66 22.840.850.850.85531.18531.1821.0317.6622.840.3470.4090.2440.850.850.850.850.850.85 -0.321-0.234-0
4216.48 14.45.76 27.7315.76 23.030.850.850.85539.26539.2627.7315.7623.030.3330.4210.2460.850.850.850.850.850.85 -0.321-0.233-0
5228.58 14.166.08 24.9418.25 25.110.850.850.85625.02625.0124.9418.2525.110.3420.4130.2440.850.850.850.850.850.85 -0.321-0.233-0
6235.08 12.544.04 30.8316.02 28.140.850.840.85471.14471.1430.8316.0228.140.3320.4260.2410.850.840.850.850.840.85 -0.322-0.229-0
7247.84 11.765.1 25.8420.52 27.990.850.850.85586.6586.625.8420.5227.990.3450.4110.2430.850.850.850.850.850.85 -0.322-0.233-0
8255.34 11.346.36 34.5720.42 22.380.850.850.85558.56558.5634.5720.4222.380.3310.4150.2550.850.850.850.850.850.85 -0.317-0.239-0
9267 11.84.1 28.5620.53 33.790.850.850.85580.63580.6428.5620.5333.790.3440.4170.2390.850.850.850.850.850.85 -0.324-0.229-0
10276.14 11.745.94 34.4122.22 26.590.850.850.85630.11630.1134.4122.2226.590.3350.4140.2510.850.850.850.850.850.85 -0.319-0.237-0
11286.8 11.945.1 32.8222.79 31.490.850.850.85655.88655.8832.8222.7931.490.340.4150.2450.850.850.850.850.850.85 -0.321-0.233-0
12297.94 12.386.44 33.1525.46 29.760.850.850.85770.03770.0333.1525.4629.760.3420.4090.2490.850.850.850.850.850.85 -0.319-0.237-0
13307 136 36.8924.41 31.590.850.850.85765.17765.1736.8924.4131.590.3370.4150.2480.850.850.850.850.850.85 -0.32-0.235-0
14317.38 11.64.46 33.4225.5 38.530.850.850.85714.3714.333.4225.538.520.3450.4140.2410.850.850.850.850.850.85 -0.323-0.231-0
15325.92 12.287.28 44.226.63 27.840.850.850.85791.39791.3944.226.6327.840.3310.4130.2560.850.850.850.850.850.85 -0.316-0.24-0
16336.72 11.184.22 36.5926.6 40.820.850.850.85715.47715.4736.5926.640.820.3440.4160.2410.850.850.850.850.850.85 -0.323-0.231-0
17346.78 11.324.22 37.6827.3 42.350.850.850.85743.24743.2437.6827.342.350.3440.4160.240.850.850.850.850.850.85 -0.323-0.23-0
18358.28 13.125.4 38.4429.12 41.530.850.850.85924.7924.738.4429.1341.530.3440.4130.2430.850.850.850.850.850.85 -0.322-0.232-0
19365.66 14.65.28 50.6725.36 40.390.850.850.85870.29870.2950.6725.3640.390.330.4250.2450.850.840.850.850.840.85 -0.321-0.232-0
20375.7 11.884.46 46.5127.9 43.040.850.850.85788.52788.5246.5127.943.040.3360.420.2430.850.850.850.850.850.85 -0.322-0.232-0
21388.24 144.64 41.5229.65 50.80.850.850.85992.89992.8941.5229.6550.790.3450.4180.2370.850.850.850.840.850.84 -0.325-0.228-0
22398.32 13.445.4 43.0832.11 46.740.850.850.851042.421042.4243.0832.1146.740.3440.4140.2420.850.850.850.850.850.85 -0.322-0.232-0
23408.36 13.17.46 46.7435.75 38.690.850.850.851147.671147.6746.7535.7438.690.3410.4080.2510.850.850.850.850.850.85 -0.318-0.239-0
24415.92 12.887.02 57.1333.06 37.130.850.850.851024.651024.6557.1333.0637.130.330.4160.2540.850.850.850.850.850.85 -0.317-0.239-0
25427.12 14.67.12 55.2633.62 41.510.850.850.851179.841179.8455.2633.6241.510.3330.4160.2510.850.850.850.850.850.85 -0.319-0.237-0
26435.4 13.286.56 62.932.8 40.060.850.850.851038.041038.0562.932.840.060.3270.420.2530.850.850.850.860.850.86 -0.318-0.238-0
27446.62 14.525.24 56.4632.44 51.740.850.850.851115.841115.8556.4632.4351.740.3350.4220.2430.850.850.850.850.850.85 -0.322-0.231-0
28456.42 13.865.82 59.0634.39 48.230.850.850.851136.521136.5259.0634.3948.230.3340.4190.2470.850.850.850.850.850.85 -0.32-0.234-0
29467.58 13.925.3 5436.08 54.980.850.850.851203.011203.015436.0854.980.340.4180.2420.850.850.850.850.850.85 -0.322-0.231-0
30475.82 13.125.64 63.5335.66 49.020.850.850.851114.051114.0563.5335.6649.020.3320.420.2480.850.850.850.850.850.85 -0.32-0.235-0
31486.78 14.76.16 63.1236.62 51.450.850.850.851283.271283.2763.1236.6251.450.3330.420.2470.850.850.850.850.850.85 -0.32-0.234-0
AVE:336.81 12.975.67 41.1226.22 36.220.850.850.85821.15821.1541.1226.2236.220.3370.4170.2460.850.850.850.850.850.85-0.32-0.234-0




À2:   Estimating the Translog cost function directly.

XIII. If we have a data set relating the input (factor) prices, wL, wK, and wM, to the total (minimum) cost of producing output for varying levels output, but we do not have data on the required levels of inputs, we can estimate the Translog cost function directly. We will use the same sequence (displayed in the "Translog Cost Function" table) of factor prices, outputs, and total (minimum) cost as used in À1. The estimated coefficients of the cost function will vary with the parameters sigma, nu, alpha, beta and gamma of the CES production function used to generate these data with the Translog production function. We impose 9 restrictions on the estimates of the parameters as specified, including the 4 necessary homothetic conditions.

SVD Restricted Least Squares
Parameter Estimates
Parameter Coefficient std error t-ratio
c1.0831664.0E-6243223.571723
cq10318444348245.5
cL0.3484737.0E-646550.192513
cK0.3899481.2E-532341.516494
cM0.2615799.0E-630676.459525
dqq-00-0.517387
dLL0.0334331.1E-53175.630987
dKK0.0363051.6E-52290.697596
dMM0.0279341.0E-52787.395204
2*dLK-0.0418052.2E-5-1938.466718
2*dLM-0.0250621.0E-5-2539.657173
2*dKM-0.0308062.1E-5-1458.956469
dLq-00-0
dKq-00-1.0E-6
dMq-001000
R2 = 1 R2b = 1 # obs = 31
Observation Matrix Rank: 15

1 = cL + cK + cM
0 = dLL + dLK + dLM
0 = dKL + dKK + dKM
0 = dML + dMK + dMM
0 = dLq + dKq + dMq
0 = dLq, 0 = dKq, 0 = dMq, 1 = cq

XIV. The Translog cost function estimated directly is:

ln(C(q;wL,wK,wM)) = 1.083166 + 1 * ln(q) + 0.348473 * ln(wL) + 0.389948 * ln(wK) + 0.261579 * log(wM)
+ .5 * [-0 * ln(q)^2 + 0.033433 * ln(wL)^2 + 0.036305 * ln(wK)^2 + 0.027934 * ln(wM)^2]
+ .5 * [-0.041805 * ln(wL)*ln(wK) + -0.025062 * ln(wL)*ln(wM) + -0.030806 * ln(wK)*log(wM)]
+ -0 * ln(wL)*ln(q) + -0 * ln(wK)*ln(q) + -0 * ln(wM)*ln(q)           (***)

XV. Its three derived factor share functions are:

sL(q;wL,wK,wM) = 0.348473 + -0 * ln(q) + 0.033433 * ln(wL) + -0.020902 * ln(wK) + -0.012531 * ln(wM),

sK(q;wL,wK,wM) = 0.389948 + -0 * ln(q) + -0.020902 * ln(wL) + 0.036305 * ln(wK) + -0.015403 * ln(wM),

sM(q;wL,wK,wM) = 0.261579 + -0 * ln(q) + -125.31 * ln(wL) + -0.015403 * ln(wK) + 0.027934 * ln(wM)

XVI. Notes:
      1. As derived, dLK = dKL, dLM = dML, and dKM = dMK: Young's Theorem holds by construction.

      2. C(q;wL,wK,wM) linear homogeneous in factor prices implies:

1   =?   cL + cK + cM   =   0.348473 + 0.389948 + 0.261579   =   1
0   =?   dLL + dLK + dLM   =   0.033433 + -0.020902 + -0.012531  =   -0
0   =?   dKL + dKK + dKM   =   -0.020902 + 0.036305 + -0.015403  =   -0
0   =?   dML + dMK + dMM   =   -0.012531 + -0.015403 + 0.027934  =   0
0   =?   dLq + dKq + dMq   =   -0 + -0 + -0  =   -0

      3. The matrix ∇2C of second order partial derivatives of the cost function C(q;wL,wK,wM) is symmetric.

     4. The cost function C(q;wL,wK,wM) is a concave in factor prices if its Hessian matrix ∇2wwC of second order partial derivatives with respect to factor prices is negative semidefinite.   Following the procedure suggested by Diewert and Wales (1987), write sL = sL(q;wL,wK,wM), sK = sK(q;wL,wK,wM), and sM = sM(q;wL,wK,wM), and the matrices:

D =
dLLdLKdLM
dKLdKKdKM
dMLdMKdMM
S =
sL 0 0
 0sK 0
 0 0sM
SS =
sL*sLsL*sKsL*sM
sK*sLsK*sKsK*sM
sM*sLsM*sKsM*sM
W =
wL 0 0
 0wK 0
 0 0wM

Assuming C(q;wL,wK,wM) > 0, then ∇2wwC is negative semidefinite if and only if the matrix:

H = W * ∇2wwC * W   =   D   -   S   +   SS

is negative semidefinite. See the Mathematical Notes.

  The values of the second order partial derivatives depend on the amount of output, and on factor prices. As an example, consider the case where q = 30, wL = 7, wK = 13, and wM = 6:

W * ∇2wwC * W =
-0.190150.119080.07107
0.11908-0.206430.08736
0.071070.08736-0.15843

The principal minors of H are H1 = -0.190149, H2 = 0.025074, and H3 = -0.

If these principal minors alternate in sign, starting with negative, with H3 <= 0, the matrix H is negative (semi)definite,
and the cost function C(q;wL,wK,wM) is a concave function in factor prices at q = 30, wL = 7, wK = 13, and wM = 6.

The eigenvalues of H are e1 = -0.3198, e2 = -0.2352, and e3 = -0.
H3 = e1 * e2 * e3 = -0.

XVII. Check for linear homogeneity:

C(q;wL,wK,wM) linear homogeneous in factor prices requires for any t > 0, that C(q;wL,wK,wM) obey:

C(q;t*wL,t*wK,t*wM) = t * C(q;wL,wK,wM)

      For example, with wL = 7, wK = 13, wM = 6, and t = 2:

2 * C(25; 7, 13, 6) = 1275.29 =? 1275.29 = C(25; 14, 26,12).

2 * C(30; 7, 13, 6) = 1530.34 =? 1530.34 = C(30; 14, 26,12).

XVIII. Check for homotheticity:

Can we write C(q;wL,wK,wM) as the product of the unit cost function C(1;wL,wK,wM) and a suitable increasing function h(q) with h(0) = 0, and h(1) = 1:

C(q;wL,wK,wM) =? h(q) * C(1;wL,wK,wM)

To do so it is necessary that:

1   =?   cq   =   1,     0   =?   dqq  =   -0,

0   =?   dLq  =   -0,     0   =?   dKq  =   -0,     0   =?   dMq  =   -0.

      As examples:

          a. with wL = 7, wK = 13, wM = 6, and q = 25, try h(q) = q1/nu with nu = 1:

C(25; 7, 13, 6) = 637.64 =? 637.64 = 25 * 25.51 = 25^1/1 * C(1; 7, 13, 6).

          b. with wL = 7, wK = 13, wM = 6, and q = 30, try h(q) = q1/nu with nu = 1:

C(30; 7, 13, 6) = 765.17 =? 765.17 = 30 * 25.51 = 30^1/1 * C(1; 7, 13, 6).

Try it with sigma = 1!

XIX. The three derived factor demand functions are obtained by:

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.

    The factor demand elasticities are obtained by:

εL,wL = ∂ln(L(q;wL,wK,wM))/∂ln(wL) = -1 + sL(q;wL,wK,wM) + dLL / sL(q;wL,wK,wM),

εL,wK = ∂ln(L(q;wL,wK,wM))/∂ln(wK) = sK(q;wL,wK,wM) + dLK / sL(q;wL,wK,wM),

εL,wM = ∂ln(L(q;wL,wK,wM))/∂ln(wM) = sM(q;wL,wK,wM) + dLM / sL(q;wL,wK,wM),

εL,q = ∂ln(L(q;wL,wK,wM))/∂ln(q) = q * ∂ln(C)/∂q + dLq / sL(q;wL,wK,wM), etc.

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

εL,wLεL,wKεL,wMεL,q
εK,wLεK,wKεK,wMεK,q
εM,wLεM,wKεM,wMεM,q
-0.56350.35290.21061
0.2871-0.49770.21061
0.28690.3526-0.63951

XX. Table of Results: check that the estimated Translog cost function for a given level of output agrees with the minimized cost using the Translog production function.

Translog Cost Function
À2: Estimate the Translog cost function directly
CES: Returns to Scale = 1, Elasticity of Substitution = 0.85
   —   Translog Production Data   —    —   Translog Cost Data   —   Factor Shares   —   Uzawa Elasticities   —   W * ∇2wwC * W
obs #qwLwKwM LK MsLKsLMsKMcostest costest Lest Kest MsLsKsMuLKuLMuKLuKMuMLuMKe1e2e3
1185.54 14.867.8 28.0413.58 15.750.850.850.86480.05480.0528.0413.5815.750.3240.420.2560.850.850.850.860.850.86 -0.317-0.239-0
2195.64 12.55.6 25.6614.61 19.390.850.850.85435.96435.9625.6614.6119.390.3320.4190.2490.850.850.850.850.850.85 -0.319-0.235-0
3208.76 12.35.68 21.0317.66 22.840.850.850.85531.18531.1821.0317.6622.840.3470.4090.2440.850.850.850.850.850.85 -0.321-0.234-0
4216.48 14.45.76 27.7315.76 23.030.850.850.85539.26539.2627.7315.7623.030.3330.4210.2460.850.850.850.850.850.85 -0.321-0.233-0
5228.58 14.166.08 24.9418.25 25.110.850.850.85625.02625.0224.9418.2525.110.3420.4130.2440.850.850.850.850.850.85 -0.321-0.233-0
6235.08 12.544.04 30.8316.02 28.140.850.840.85471.14471.1430.8316.0228.140.3320.4260.2410.850.840.850.850.840.85 -0.322-0.229-0
7247.84 11.765.1 25.8420.52 27.990.850.850.85586.6586.625.8420.5227.990.3450.4110.2430.850.850.850.850.850.85 -0.322-0.233-0
8255.34 11.346.36 34.5720.42 22.380.850.850.85558.56558.5634.5720.4222.380.3310.4150.2550.850.850.850.850.850.85 -0.317-0.239-0
9267 11.84.1 28.5620.53 33.790.850.850.85580.63580.6428.5620.5333.790.3440.4170.2390.850.850.850.850.850.85 -0.324-0.229-0
10276.14 11.745.94 34.4122.22 26.590.850.850.85630.11630.1134.4122.2226.590.3350.4140.2510.850.850.850.850.850.85 -0.319-0.237-0
11286.8 11.945.1 32.8222.79 31.490.850.850.85655.88655.8832.8222.7931.490.340.4150.2450.850.850.850.850.850.85 -0.321-0.233-0
12297.94 12.386.44 33.1525.46 29.760.850.850.85770.03770.0333.1525.4629.760.3420.4090.2490.850.850.850.850.850.85 -0.319-0.237-0
13307 136 36.8924.41 31.590.850.850.85765.17765.1736.8924.4131.590.3370.4150.2480.850.850.850.850.850.85 -0.32-0.235-0
14317.38 11.64.46 33.4225.5 38.530.850.850.85714.3714.333.4225.538.520.3450.4140.2410.850.850.850.850.850.85 -0.323-0.231-0
15325.92 12.287.28 44.226.63 27.840.850.850.85791.39791.3944.226.6327.840.3310.4130.2560.850.850.850.850.850.85 -0.316-0.24-0
16336.72 11.184.22 36.5926.6 40.820.850.850.85715.47715.4736.5926.640.820.3440.4160.2410.850.850.850.850.850.85 -0.323-0.231-0
17346.78 11.324.22 37.6827.3 42.350.850.850.85743.24743.2437.6827.342.350.3440.4160.240.850.850.850.850.850.85 -0.323-0.23-0
18358.28 13.125.4 38.4429.12 41.530.850.850.85924.7924.738.4429.1341.530.3440.4130.2430.850.850.850.850.850.85 -0.322-0.232-0
19365.66 14.65.28 50.6725.36 40.390.850.850.85870.29870.2950.6725.3640.390.330.4250.2450.850.840.850.850.840.85 -0.321-0.232-0
20375.7 11.884.46 46.5127.9 43.040.850.850.85788.52788.5246.5127.943.040.3360.420.2430.850.850.850.850.850.85 -0.322-0.232-0
21388.24 144.64 41.5229.65 50.80.850.850.85992.89992.8941.5229.6550.790.3450.4180.2370.850.850.850.840.850.84 -0.325-0.228-0
22398.32 13.445.4 43.0832.11 46.740.850.850.851042.421042.4243.0832.1146.740.3440.4140.2420.850.850.850.850.850.85 -0.322-0.232-0
23408.36 13.17.46 46.7435.75 38.690.850.850.851147.671147.6746.7535.7438.690.3410.4080.2510.850.850.850.850.850.85 -0.318-0.239-0
24415.92 12.887.02 57.1333.06 37.130.850.850.851024.651024.6557.1333.0637.130.330.4160.2540.850.850.850.850.850.85 -0.317-0.239-0
25427.12 14.67.12 55.2633.62 41.510.850.850.851179.841179.8455.2633.6241.510.3330.4160.2510.850.850.850.850.850.85 -0.319-0.237-0
26435.4 13.286.56 62.932.8 40.060.850.850.851038.041038.0562.932.840.060.3270.420.2530.850.850.850.860.850.86 -0.318-0.238-0
27446.62 14.525.24 56.4632.44 51.740.850.850.851115.841115.8456.4632.4451.740.3350.4220.2430.850.850.850.850.850.85 -0.322-0.231-0
28456.42 13.865.82 59.0634.39 48.230.850.850.851136.521136.5259.0634.3948.230.3340.4190.2470.850.850.850.850.850.85 -0.32-0.234-0
29467.58 13.925.3 5436.08 54.980.850.850.851203.011203.015436.0854.980.340.4180.2420.850.850.850.850.850.85 -0.322-0.231-0
30475.82 13.125.64 63.5335.66 49.020.850.850.851114.051114.0563.5335.6649.020.3320.420.2480.850.850.850.850.850.85 -0.32-0.235-0
31486.78 14.76.16 63.1236.62 51.450.850.850.851283.271283.2763.1236.6251.450.3330.420.2470.850.850.850.850.850.85 -0.32-0.234-0
AVE:336.81 12.975.67 41.1226.22 36.220.850.850.85821.15821.1541.1226.2236.220.3370.4170.2460.850.850.850.850.850.85-0.32-0.234-0




À3:   Re-estimate the Translog Production Function.

XXI. Assuming we had a data set relating the input (factor) prices, wL, wK, and wM, to the total (minimum) cost of producing output for varying levels output, but that we did not have data on the cost minimizing levels of inputs, we estimated the Translog cost function directly. With the derived factor demand functions to determine cost minimizing inputs, we can estimate a Translog production function using the data in the Translog Cost Function table.

SVD Least Squares
Parameter Estimates
Parameter Coefficient std error t-ratio
lnA-3.0E-60-0.134
aL0.349875032928.127
aK0.400015021720.811
aM0.250118016938.935
bLL-0.0196630-3321.139
bKK-0.0213390-2469.427
bMM-0.0164350-2493.473
bLK0.02456601951.878
bLM0.01476502364.035
bKM0.01810601349.804
R2 = 1 R2b = 1 # obs = 31
Observation Matrix Rank: 10

The estimated Translog production function using the cost data is:

ln(q) = -3.0E-6 + 0.349875 * ln(L) + 0.400015 * ln(K) + 0.250118 * ln(M) + -0.019663 * ln(L)*ln(L) + -0.021339 * ln(K)*ln(K) + -0.016435 * ln(M)*ln(M)
+ 0.024566 * ln(L)*ln(K) + 0.014765 * ln(L)*ln(M) + 0.018106 * ln(K)*ln(M)   =   f(L,K,M).
       

The coefficients of this production function are approximately? equal to the coefficients of the original Translog production function.

XXII. Knowing the Translog production function permits us to investigate further properties of the production / cost technology. For example, we can determine the short-run average and marginal cost functions in relation to the long-run average and marginal cost functions obtained from the Translog cost function, and the short-run Allen elasticities of substitution.

For example, if we set capital at the least cost level for q = 30, then K = 24.41, and we can solve for the least cost inputs of labour, L, and materials and supplies, M, for various levels of output, q.   (See the Translog Production Function web page.)

Translog Short Run Cost Data
CES: Returns to Scale = 1, Elasticity of Substitution = 0.85
wL = 7, wK= 13, wM = 6
qest q LK Mtotal cost ave. costmarg. cost3 factor
sLM
2 factor
sLM
1414 10.6124.41 9.09446.25 31.87 14.420.870.86
1616 13.124.41 11.22476.45 29.78 15.80.870.86
1818 15.8224.41 13.55509.43 28.3 17.180.860.86
2020 18.7624.41 16.07545.13 27.26 18.540.860.86
2222 21.9224.41 18.79583.6 26.53 19.910.860.85
2424 25.3124.41 21.71624.85 26.04 21.30.860.85
2626 28.9324.41 24.81668.72 25.72 22.70.850.85
2828 32.7824.41 28.11715.53 25.55 24.10.850.85
3030 36.8824.41 31.62765.22 25.51 25.50.850.85
3232 41.1924.41 35.32817.66 25.55 26.920.850.85
3434 45.7524.41 39.21872.92 25.67 28.360.850.85
3636 50.5524.41 43.3931.07 25.86 29.810.840.85
3838 55.624.41 47.6992.16 26.11 31.280.840.85
4040 60.8824.41 52.111056.19 26.4 32.760.840.85
4242 66.4124.41 56.821123.16 26.74 34.260.840.84
4444 72.1824.41 61.751193.17 27.12 35.770.840.84
4646 78.224.41 66.91266.22 27.53 37.290.830.84
4848 84.4724.41 72.271342.33 27.97 38.840.830.84
5050 9124.41 77.861421.54 28.43 40.40.830.84
5252 97.7824.41 83.691503.99 28.92 41.980.830.84

We get a U-shaped, short run average cost curve, with capital fixed.

As seen in the diagram below, the short run average cost curve is (approximately) tangent to the long run average cost curve, at q = 30.

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

 




Mathematical Notes

1. 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)
        (**)

2. 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),

3. 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.

4. The Factor Demand Elasticities:

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

∂L/∂wL = [(∂sL/∂wL * C + sL * ∂C/∂wL) * wL - (sL * C)(1)] / wL^2 = [(dLL/wL * wL*L/sL + sL * L) * wL - (sL * wL * L/sL)] / wL^2 = (L /wL) * (dLL/sL + sL - 1), so
εL,L = ∂ln(L)/∂ln(wL) = ∂ln(L)/∂wL * ∂wL/∂ln(wL) = ∂L/∂wL * wL/L = (- 1 + sL + dLL/sL).

∂L/∂wK = [(∂sL/∂wK * C + sL * ∂C/∂wK] / wL = [dLK/wK * (wL *L / sL) + ((wL * L) * sK / (wK * K))*K] / wL = [dLK*L/(wK*sL) + L*sK/wK], so
εL,K = ∂ln(L)/∂ln(wK) = ∂ln(L)/∂wK * ∂wK/∂ln(wK) = ∂L/∂wK * wK/L = (sK + dLK/sL).

∂L/∂q = [(∂sL/∂q * C + sL * ∂C/∂q] / wL = [(dLq/q)*(wL*L/sL) + (wL*L/C)*C*(∂ln(C)/∂q)]/wL, so
εL,q = ∂ln(L)/∂ln(q) = ∂ln(L)/∂q * ∂q/∂ln(q) = (∂L/∂q)*(q/L) = q * ∂ln(C)/∂q + dLq / sL,

5. Cost Function C(q;wL,wK,wM) Concave in Factor Prices:

∂ln(C)/∂ln(wL) = sL = wL * L / C = wL * ∂C/∂wL / C → ∂C/∂wL = sL * C / wL,

2ln(C)/∂ln(wL)∂ln(wL) = [(∂wL/∂ln(wL) * ∂C/∂wL + wL * ∂2C/∂wL∂ln(wL)) * C - wL * ∂C/∂wL * ∂C/∂ln(wL)] / C2
          = [(wL * ∂C/∂wL + wL * ∂2C/∂wL∂wL * wL) * C - wL * ∂C/∂wL * ∂C/∂wL * wL] / C2
          = wL * ∂C/∂wL / C - wL * ∂C/∂wL * wL * ∂C/∂wL / C2 + wL * wL * ∂2C/∂wL∂wL / C
          = sL - sL * sL + wL * wL * ∂2C/∂wL∂wL / C = dLL, so

wL * wL * ∂2C/∂wL∂wL / C = dLL - sL + sL * sL

2ln(C)/∂ln(wL)∂ln(wK) = [(∂wL/∂ln(wK) * L + wL * ∂L/∂ln(wK)) * C - wL * L * ∂C/∂ln(wK)] / C2
          = [(0 + wL * ∂2C/∂wL∂wK * wK) * C - wL * L * ∂C/∂wK * wK] / C2
          = - wL * L * wK * K / C2 + wL * wK * ∂2C/∂wL∂wK / C
          = - sL * sK + wL * wK * ∂2C/∂wL∂wK / C = dLK, so

wL * wK * ∂2C/∂wL∂wK / C = dLK + sL * sK

 

 
   

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