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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

M. Generalized CES-Translog Cost Function

This web page replicates the procedures used to estimate the parameters of a Translog cost function to approximate a CES cost function. To determine the Translog's efficacy in estimating varying elasticities of substitution among pairs of inputs, on this web page we approximate a Generalized CES cost function with the Translog cost function.

Unlike the CES cost function, the Generalized CES cost function is not necessarily homothetic. Consequently, when we estimate the Translog cost function directly, we do not impose the homothetic conditions as we did when we approximated the CES cost function with the Translog cost function.

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 use the three factor Generalized CES production function to generate the required cost data, yielding a data set relating output, q, and factor prices, wL, wK, and wM, [randomized about the base prices (wL*, wK*, wM*)]   to the Generalized CES cost minimizing factor inputs, L, K, and M. The base factor prices and the randomizing distribution are specified in the form below.

The three factor Generalized CES production function is:

q = A * [alpha * L^-rhoL + beta * K^-rhoK + gamma *M^-rhoM]^(-nu/rho) = f(L,K,M).

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

The parameter nu permits one to adjust the returns to scale, while the parameter rho is the geometric mean of rhoL, rhoK, and rhoM:

rho = (rhoL * rhoK * rhoM)^1/3.

If rho = rhoL = rhoK = rhoM, we get the standard CES production function. If also, rho = 0, ie sigma = 1/(1+rho) = 1, we get the Cobb-Douglas production function.

Generalized CES Elasticity of Scale of Production:

εLKM = ε(L,K,M) = (nu / rho) * (alpha * rhoL * L^-rhoL + beta * rhoK * K^-rhoK + gamma * rhoM * M^-rhoM) / (alpha * L^-rhoL + beta * K^-rhoK + gamma * M^-rhoM).

See the Generalized CES production function.



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

   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.

II. The estimated coefficients of the Translog cost function will vary with the parameters nu, rho, rhoL, rhoK, rhoM, alpha, beta and gamma of the Generalized CES production function.

Set the parameters below to re-run with your own Generalized CES parameters.

The restrictions ensure that the least-cost problems can be solved to obtain the underlying Generalized CES cost function, using the parameters as specified.
Intermediate (and other) values of the parameters also work.

Restrictions:
.8 < nu < 1.1;
-.6 < rhoL = rho < .6;
-.6 < rho < -.2 → rhoK = .95 * rho, rhoM = rho/.95;
-.2 <= rho < -.1 → rhoK = .9 * rho, rhoM = rho / .9;
-.1 <= rho < 0 → rhoK = .87 * rho, rhoM = rho / .87;
0 <= rho < .1 → rhoK = .7 * rho, rhoM = rho / .7;
.1 <= rho < .2 → rhoK = .67 * rho, rhoM = rho / .67;
.2 <= rho < .6 → rhoK = .6 * rho, rhoM = rho / .6;
rho = 0 → nu = alpha + beta + gamma (Cobb-Douglas)
4 <= wL* <= 11,   7<= wK* <= 16,   4 <= wM* <= 10

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

The Generalized CES production function as specified:

q = 1 * [0.35 * L^- 0.17647 + 0.4 * K^- 0.11823 + 0.25 *M^- 0.26339]^(-1/0.17647) = f(L,K,M).

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

III. For these coefficients of the CES Generalized production function, I generated a sequence (displayed in the "Generalized CES-Translog Cost Function" table) of factor prices, outputs, and the corresponding cost minimizing inputs. 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
cL0.3604410.001433.965
dLq4.8E-500.535
dLL0.0343280239.928
dLK-0.0135270-46.082
dLM-0.0205390-198.035
R2 = 0.9998 R2b = 0.9997 # obs = 31

QR Least Squares
Parameter Estimates
Parameter Coefficient std error t-ratio
cK0.2729890.001447.087
dKq0.0207990318.611
dKL-0.0137770-130.979
dKK0.0302740140.287
dKM-0.0168260-220.678
R2 = 0.9999 R2b = 0.9998 # obs = 31

QR Least Squares
Parameter Estimates
Parameter Coefficient std error t-ratio
cM0.366570.001426.759
dMq-0.0208460-227.002
dML-0.0205520-138.893
dMK-0.0167470-55.164
dMM0.0373650348.36
R2 = 0.9999 R2b = 0.9999 # obs = 31

The three estimated factor share functions are:

sL(q;wL,wK,wM) = 0.360441 + 4.8E-5 * ln(q) + 0.034328 * ln(wL) + -0.013527 * ln(wK) + -0.020539 * ln(wM),

sK(q;wL,wK,wM) = 0.272989 + 0.020799 * ln(q) + -0.013777 * ln(wL) + 0.030274 * ln(wK) + -0.016826 * ln(wM),

sM(q;wL,wK,wM) = 0.36657 + -0.020846 * ln(q) + -0.020552 * ln(wL) + -0.016747 * ln(wK) + 0.037365 * 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   =   0.360441 + 0.272989 + 0.36657   =   1
0   =?   dLL + dLK + dLM   =   0.034328 + -0.013527 + -0.020539  =   0.000261
0   =?   dKL + dKK + dKM   =   -0.013777 + 0.030274 + -0.016826  =   -0.000328
0   =?   dML + dMK + dMM   =   -0.020552 + -0.016747 + 0.037365  =   6.7E-5
0   =?   dLq + dKq + dMq   =   4.8E-5 + 0.020799 + -0.020846  =   -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
cL0.3609430.000442816.436707
dLq5.8E-57.5E-50.777562
dLL0.0343170.000121283.74476
dLK-0.0137280.000109-125.718041
dLM-0.0205397.1E-5-289.582958
cK0.2728720.000695392.769266
dKq0.0207957.5E-5276.255967
dKL-0.0137280.000109-125.718041
dKK0.0302820.000249121.676068
dKM-0.0168168.3E-5-202.527354
cM0.3667110.000378969.035291
dMq-0.0208447.4E-5-281.671601
dML-0.0205397.1E-5-289.582958
dMK-0.0168168.3E-5-202.527354
dMM0.0373698.8E-5426.966196
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) = 0.360943 + 5.8E-5 * ln(q) + 0.034317 * ln(wL) + -0.013728 * ln(wK) + -0.020539 * ln(wM),

sK(q;wL,wK,wM) = 0.272872 + 0.020795 * ln(q) + -0.013728 * ln(wL) + 0.030282 * ln(wK) + -0.016816 * ln(wM),

sM(q;wL,wK,wM) = 0.366711 + -0.020844 * ln(q) + -0.020539 * ln(wL) + -0.016816 * ln(wK) + 0.037369 * 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   =   0.360943 + 0.272872 + 0.366711   =   1.000526
0   =?   dLL + dLK + dLM   =   0.034317 + -0.013728 + -0.020539  =   5.0E-5
0   =?   dKL + dKK + dKM   =   -0.013728 + 0.030282 + -0.016816  =   -0.000263
0   =?   dML + dMK + dMM   =   -0.020539 + -0.016816 + 0.037369  =   1.3E-5
0   =?   dLq + dKq + dMq   =   5.8E-5 + 0.020795 + -0.020844  =   9.0E-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
cL0.3610170.0002191647.990227
dLq6.1E-56.1E-50.986902
dLL0.034278.2E-5417.373198
dLK-0.0137117.8E-5-175.26615
dLM-0.0205595.7E-5-362.501588
cK0.2722590.0002241216.425533
dKq0.0207826.1E-5339.030346
dKL-0.0137117.8E-5-175.26615
dKK0.0305150.000106288.8897
dKM-0.0168046.6E-5-252.748695
cM0.3667240.0002161694.393199
dMq-0.0208436.1E-5-339.698514
dML-0.0205595.7E-5-362.501588
dMK-0.0168046.6E-5-252.748695
dMM0.0373637.0E-5537.393109
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) = 0.361017 + 6.1E-5 * ln(q) + 0.03427 * ln(wL) + -0.013711 * ln(wK) + -0.020559 * ln(wM),

sK(q;wL,wK,wM) = 0.272259 + 0.020782 * ln(q) + -0.013711 * ln(wL) + 0.030515 * ln(wK) + -0.016804 * ln(wM),

sM(q;wL,wK,wM) = 0.366724 + -0.020843 * ln(q) + -0.020559 * ln(wL) + -0.016804 * ln(wK) + 0.037363 * 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
c1.1155620.0001467653.875198
cq101000
dqq0.0209012.4E-5874.098763
R2 = 1 R2b = 1 # obs = 31

1 = cq

VI. The Translog cost function as estimated is:

ln(C(q;wL,wK,wM)) = 1.115562 + 1 * ln(q) + 0.361017 * ln(wL) + 0.272259 * ln(wK) + 0.366724 * log(wM)
+ .5 * [0.020901 * ln(q)^2 + 0.03427 * ln(wL)^2 + 0.030515 * ln(wK)^2 + 0.037363 * ln(wM)^2]
+ .5 * [-0.027422 * ln(wL)*ln(wK) + -0.041119 * ln(wL)*ln(wM) + -0.033607 * ln(wK)*log(wM)]
+ 6.1E-5 * ln(wL)*ln(q) + 0.020782 * ln(wK)*ln(q) + -0.020843 * ln(wM)*ln(q)           (***)

VII. Check for linear homogeneity:

1   =?   cL + cK + cM   =   0.361017 + 0.272259 + 0.366724   =   1
0   =?   dLL + dLK + dLM   =   0.03427 + -0.013711 + -0.020559  =   0
0   =?   dKL + dKK + dKM   =   -0.013711 + 0.030515 + -0.016804  =   0
0   =?   dML + dMK + dMM   =   -0.020559 + -0.016804 + 0.037363  =   0
0   =?   dLq + dKq + dMq   =   6.1E-5 + 0.020782 + -0.020843  =   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) = 1412.94 =? 1412.94 = C(25; 14, 26,12).

2 * C(30; 7, 13, 6) = 1722.09 =? 1722.09 = 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  =   0.020901,

0   =?   dLq  =   6.1E-5,     0   =?   dKq  =   0.020782,     0   =?   dMq  =   -0.020843.

      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) = 861.04 =? 722.39 = 30 * 24.08 = 30^1/1 * C(1; 7, 13, 6).

Try it with rho = 0!

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 =
-0.194970.115990.07898
0.11599-0.20110.08512
0.078980.08512-0.16409

The principal minors of H are H1 = -0.194966, H2 = 0.025755, 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 = -0.3144, e2 = -0.2457, 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
-0.54780.32590.22191.0873
0.3183-0.55180.23361.1442
0.28240.3043-0.58671.0126

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 * K),

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

With the dLK = dKL, dLM = dML, and dKM = dMK restrictions in the factor share functions, we get 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 and input amounts for a given level of output agree with the Generalized CES production function's minimized cost and inputs. Also, compare the Allen partial elasticities of substitution, sLK, sLM, and sMK, with the Uzawa partial elasticities of substitution, uLK, uLM, and uKM.

Generalized CES-Translog Cost Function
À1: Estimate the Translog cost function using restricted factor shares
Parameters: nu = 1, rho = 0.17647, rhoL = 0.17647, rhoK = 0.11823, rhoM = 0.26339
   —   Generalized CES Cost Data   —    —   Translog Cost Data   —   Factor SharesUzawa ElasticitiesW * ∇2wwC * W
obs #qwLwKwM LK MsLKsLMsKMεLKMcostest costest Lest Kest MsLsKsMuLKuLMuKLuKMuMLuMKe1e2e3
1185.02 11.884.02 26.7911.59 26.80.890.790.830.922379.92380.0626.811.5926.820.3540.3620.2840.890.80.890.840.80.84 -0.312-0.2480
2196.86 12.126.98 27.3415.38 22.590.90.790.830.93531.6531.7527.3715.3822.580.3530.3510.2960.890.80.890.840.80.84 -0.306-0.2570
3207.06 12.146.76 28.2416.3 24.430.90.790.830.929562.48562.5928.2616.324.430.3550.3520.2940.890.80.890.840.80.84 -0.307-0.2550
4218.62 137.78 28.1718.29 25.550.90.790.830.93679.35679.4528.1918.2825.550.3580.350.2930.890.80.890.840.80.84 -0.308-0.2540
5227.88 12.045.46 28.0117.97 31.330.90.790.830.924608.1608.1328.0117.9631.350.3630.3560.2810.890.80.890.830.80.83 -0.313-0.247-0
6238.68 14.344.06 27.716.55 42.310.890.790.830.915649.59649.5427.6816.5642.320.370.3660.2650.90.790.90.830.790.83 -0.322-0.2360
7245.44 14.687.76 43.6117.21 26.710.90.790.830.926697.17697.1443.6117.2226.690.340.3630.2970.890.80.890.840.80.84 -0.307-0.2560
8255 13.667.04 45.6317.85 28.150.90.790.830.925670.1670.0645.6217.8628.130.340.3640.2960.890.80.890.840.80.84 -0.308-0.2550
9268.96 14.844.2 31.7119.04 47.90.890.790.830.913767.92767.7931.6919.0447.90.370.3680.2620.90.790.90.830.790.83 -0.323-0.234-0
10276.4 14.36 42.3219.74 36.210.890.790.830.92770.36770.2742.3119.7336.220.3520.3660.2820.890.790.890.840.790.84 -0.313-0.247-0
11285.44 14.785.6 47.9518.9 37.780.890.790.830.917751.78751.6847.9418.8937.790.3470.3720.2820.890.790.890.840.790.84 -0.314-0.2460
12297.74 11.267.68 40.3427.55 33.110.90.790.830.927876.62876.4940.3427.5433.090.3560.3540.290.890.80.890.840.80.84 -0.309-0.2530
13307 136 43.7924.14 40.130.890.790.830.92861.17861.0443.7824.1440.140.3560.3640.280.890.790.890.840.790.84 -0.314-0.2460
14317.44 12.364.26 40.1124.32 50.890.890.790.830.914815.81815.6640.124.3250.890.3660.3680.2660.90.790.90.830.790.83 -0.321-0.2360
15327.08 11.664.3 41.9325.68 50.610.890.790.830.915813.87813.7341.9125.6750.620.3650.3680.2670.90.790.90.830.790.83 -0.32-0.2380
16337.78 12.124.16 41.4526.66 55.390.890.790.830.913876.1875.9641.4426.6755.380.3680.3690.2630.90.790.90.830.790.83 -0.323-0.2350
17346.98 12.325.92 49.2428.58 45.130.890.790.830.919963.04962.9149.2328.5845.140.3570.3660.2780.890.790.890.830.790.83 -0.316-0.2440
18356.44 13.784.04 50.824.87 58.990.890.790.830.91908.14908.0150.8124.8558.980.360.3770.2620.90.780.90.830.780.83 -0.323-0.234-0
19367.7 13.166.08 51.0530.56 49.40.890.790.830.9171095.551095.4351.0430.5549.40.3590.3670.2740.90.790.90.830.790.83 -0.317-0.242-0
20378.9 11.786.9 48.2136.16 47.510.890.790.830.9211182.841182.7348.1936.1647.520.3630.360.2770.90.80.90.830.80.83 -0.315-0.2440
21388.92 13.327.98 53.2536.04 46.540.890.790.830.9211326.511326.4353.2436.0446.540.3580.3620.280.890.790.890.830.790.83 -0.314-0.2460
22396.24 14.427.44 67.3831.25 46.160.890.790.830.9181214.511214.4867.3731.2546.170.3460.3710.2830.890.790.890.840.790.84 -0.313-0.247-0
23408.88 11.965.4 50.0937.06 59.680.890.790.830.9151210.371210.3250.0737.0859.680.3670.3660.2660.90.790.90.830.790.83 -0.321-0.2370
24418.72 14.966.8 58.7535.3 56.860.890.790.830.9151427.061427.0858.7435.356.870.3590.370.2710.90.790.90.830.790.83 -0.319-0.240
25428.64 14.264.16 53.8833.37 76.840.890.790.830.9071261.091261.2253.9233.3976.730.3690.3780.2530.90.780.90.820.780.82 -0.328-0.2270
26437.48 12.96.26 62.9437.78 57.330.890.790.830.9151317.061317.1562.9437.7857.340.3570.370.2730.90.790.90.830.790.83 -0.318-0.2410
27447.08 14.267.6 71.7337.74 53.170.890.790.830.9171450.121450.2871.7337.7453.190.350.3710.2790.890.790.890.840.790.84 -0.315-0.2450
28457.38 14.16.46 69.138.03 60.350.890.790.830.9141436.11436.2969.1138.0360.370.3550.3730.2720.90.790.90.830.790.83 -0.319-0.240
29465.24 13.927.04 86.8136.01 53.180.890.790.830.9151330.531330.7886.836.0153.220.3420.3770.2820.890.790.890.840.790.84 -0.315-0.245-0
30475.98 11.324.46 69.8338.78 69.190.890.790.830.9111165.141165.3769.8738.7869.190.3590.3770.2650.90.780.90.830.780.83 -0.322-0.2350
31486.36 13.484.02 71.7736.08 80.910.890.790.830.9061268.031268.3471.8736.0780.860.360.3830.2560.90.780.90.830.780.83 -0.326-0.2290
AVE:337.2 13.175.89 48.3926.93 46.490.890.790.830.918963.49963.4948.3926.9346.490.3570.3660.2760.90.790.90.830.790.83-0.316-0.2430




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 "Generalized CES-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 nu, rho, rhoL, rhoK, rhoM, alpha, beta and gamma of the Generalized CES production function used to generate these data. We impose 5 restrictions on the estimates of the parameters as specified:

SVD Restricted Least Squares
Parameter Estimates
Parameter Coefficient std error t-ratio
c1.092740.0006641646.376388
cq1.013340.000382663.884099
cL0.3610860.000382945.446375
cK0.2727860.000398685.647054
cM0.3661280.0002251628.751518
dqq0.0170260.000113150.731655
dLL0.0342780.000221155.114069
dKK0.030490.000282108.257359
dMM0.0374230.000157238.779646
2*dLK-0.0273450.000481-56.805781
2*dLM-0.0412120.0002-206.24876
2*dKM-0.0336340.000319-105.378508
dLq8.0E-50.0001660.479132
dKq0.0412710.00019217.736783
dMq-0.0413510.000116-355.895493
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

XIV. The Translog cost function estimated directly is:

ln(C(q;wL,wK,wM)) = 1.09274 + 1.01334 * ln(q) + 0.361086 * ln(wL) + 0.272786 * ln(wK) + 0.366128 * log(wM)
+ .5 * [0.017026 * ln(q)^2 + 0.034278 * ln(wL)^2 + 0.03049 * ln(wK)^2 + 0.037423 * ln(wM)^2]
+ .5 * [-0.027345 * ln(wL)*ln(wK) + -0.041212 * ln(wL)*ln(wM) + -0.033634 * ln(wK)*log(wM)]
+ 8.0E-5 * ln(wL)*ln(q) + 0.041271 * ln(wK)*ln(q) + -0.041351 * ln(wM)*ln(q)           (***)

XV. Its three derived factor share functions are:

sL(q;wL,wK,wM) = 0.361086 + 8.0E-5 * ln(q) + 0.034278 * ln(wL) + -0.013672 * ln(wK) + -0.020606 * ln(wM),

sK(q;wL,wK,wM) = 0.272786 + 0.041271 * ln(q) + -0.013672 * ln(wL) + 0.03049 * ln(wK) + -0.016817 * ln(wM),

sM(q;wL,wK,wM) = 0.366128 + -0.041351 * ln(q) + -206.06 * ln(wL) + -0.016817 * ln(wK) + 0.037423 * 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   =   0.361086 + 0.272786 + 0.366128   =   1
0   =?   dLL + dLK + dLM   =   0.034278 + -0.013672 + -0.020606  =   0
0   =?   dKL + dKK + dKM   =   -0.013672 + 0.03049 + -0.016817  =   0
0   =?   dML + dMK + dMM   =   -0.020606 + -0.016817 + 0.037423  =   -0
0   =?   dLq + dKq + dMq   =   8.0E-5 + 0.041271 + -0.041351  =   -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 =
-0.195010.141080.05392
0.14108-0.215240.07415
0.053920.07415-0.12807

The principal minors of H are H1 = -0.195006, H2 = 0.022068, 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 = -0.3482, e2 = -0.1901, 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) = 1487.56 =? 1487.56 = C(25; 14, 26,12).

2 * C(30; 7, 13, 6) = 1818.45 =? 1818.45 = 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   =   1.01334,     0   =?   dqq  =   0.017026,

0   =?   dLq  =   8.0E-5,     0   =?   dKq  =   0.041271,     0   =?   dMq  =   -0.041351.

      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) = 743.78 =? 588.66 = 25 * 23.55 = 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) = 909.23 =? 706.39 = 30 * 23.55 = 30^1/1 * C(1; 7, 13, 6).

Try it with rho = 0!

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
-0.54770.39620.15141.1034
0.3246-0.49520.17061.1981
0.25760.3543-0.61190.9056

XX. Table of Results: check that the estimated Translog cost function and input amounts for a given level of output agree with the Generalized CES production function's minimized cost and inputs. Also, compare the Allen partial elasticities of substitution, sLK, sLM, and sMK, with the Uzawa partial elasticities of substitution, uLK, uLM, and uKM. Comparing the estimated cost and input amounts obtained from the Translog cost function with the Generalized CES data, one can conclude that method À1 obtains more accurate results than method À2 for rho > 0.

Generalized CES-Translog Cost Function
À2: Estimate the Translog cost function directly
Parameters: nu = 1, rho = 0.17647, rhoL = 0.17647, rhoK = 0.11823, rhoM = 0.26339
   —   Generalized CES Cost Data   —    —   Translog Cost Data   —   Factor SharesUzawa ElasticitiesW * ∇2wwC * W
obs #qwLwKwM LK MsLKsLMsKMεLKMcostest costest Lest Kest MsLsKsMuLKuLMuKLuKMuMLuMKe1e2e3
1185.02 11.884.02 26.7911.59 26.80.890.790.830.922379.92405.328.5914.422.560.3540.4220.2240.910.740.910.820.740.82 -0.342-0.2020
2196.86 12.126.98 27.3415.38 22.590.90.790.830.93531.6549.7228.318.6618.540.3530.4110.2350.910.750.910.830.750.83 -0.336-0.2120
3207.06 12.146.76 28.2416.3 24.430.90.790.830.929562.48583.2229.3119.8719.970.3550.4140.2320.910.750.910.820.750.82 -0.338-0.2090
4218.62 137.78 28.1718.29 25.550.90.790.830.93679.35701.6429.1322.2720.70.3580.4130.2290.910.750.910.820.750.82 -0.339-0.2080
5227.88 12.045.46 28.0117.97 31.330.90.790.830.924608.1639.629.4722.2825.470.3630.4190.2170.910.740.910.820.740.82 -0.345-0.198-0
6238.68 14.344.06 27.716.55 42.310.890.790.830.915649.59704.9130.0621.1534.650.370.430.20.910.720.910.80.720.8 -0.353-0.1820
7245.44 14.687.76 43.6117.21 26.710.90.790.830.926697.17726.8945.4921.221.670.340.4280.2310.910.740.910.830.740.83 -0.338-0.2070
8255 13.667.04 45.6317.85 28.150.90.790.830.925670.1700.2347.6922.0722.770.3410.4310.2290.910.740.910.830.740.83 -0.339-0.2050
9268.96 14.844.2 31.7119.04 47.90.890.790.830.913767.92836.0134.5324.5338.730.370.4350.1950.920.710.920.80.710.8 -0.356-0.1780
10276.4 14.36 42.3219.74 36.210.890.790.830.92770.36817.2644.9124.8329.140.3520.4340.2140.910.730.910.820.730.82 -0.346-0.1940
11285.44 14.785.6 47.9518.9 37.780.890.790.830.917751.78803.6651.2823.9430.510.3470.440.2130.910.720.910.820.720.82 -0.346-0.1920
12297.74 11.267.68 40.3427.55 33.110.90.790.830.927876.62900.2441.4533.8425.820.3560.4230.220.910.740.910.820.740.82 -0.343-0.2-0
13307 136 43.7924.14 40.130.890.790.830.92861.17909.2346.2530.431.720.3560.4350.2090.910.720.910.820.720.82 -0.348-0.190
14317.44 12.364.26 40.1124.32 50.890.890.790.830.914815.81879.8343.2731.2840.220.3660.4390.1950.910.710.910.80.710.8 -0.355-0.1780
15327.08 11.664.3 41.9325.68 50.610.890.790.830.915813.87874.1145.0432.9439.80.3650.4390.1960.910.710.910.80.710.8 -0.355-0.1790
16337.78 12.124.16 41.4526.66 55.390.890.790.830.913876.1946.4544.7934.4543.380.3680.4410.1910.920.710.920.80.710.8 -0.357-0.1740
17346.98 12.325.92 49.2428.58 45.130.890.790.830.919963.041015.8251.9536.1535.110.3570.4380.2050.910.720.910.810.720.81 -0.35-0.1860
18356.44 13.784.04 50.824.87 58.990.890.790.830.91908.14993.7355.6432.4946.460.3610.4510.1890.920.70.920.80.70.8 -0.357-0.1720
19367.7 13.166.08 51.0530.56 49.40.890.790.830.9171095.551159.9754.0738.8738.170.3590.4410.20.910.710.910.810.710.81 -0.352-0.1820
20378.9 11.786.9 48.2136.16 47.510.890.790.830.9211182.841230.9750.1845.4236.140.3630.4350.2030.910.720.910.810.720.81 -0.352-0.1850
21388.92 13.327.98 53.2536.04 46.540.890.790.830.9211326.511378.5455.3645.2335.380.3580.4370.2050.910.720.910.810.720.81 -0.35-0.1860
22396.24 14.427.44 67.3831.25 46.160.890.790.830.9181214.511276.870.8639.5435.550.3460.4470.2070.910.710.910.820.710.82 -0.348-0.1870
23408.88 11.965.4 50.0937.06 59.680.890.790.830.9151210.371285.9753.2347.5845.240.3680.4420.190.920.70.920.80.70.8 -0.357-0.1730
24418.72 14.966.8 58.7535.3 56.860.890.790.830.9151427.06151662.4445.2743.290.3590.4470.1940.910.70.910.810.70.81 -0.355-0.1770
25428.64 14.264.16 53.8833.37 76.840.890.790.830.9071261.091387.0359.3344.2258.610.370.4550.1760.920.680.920.790.680.79 -0.364-0.160
26437.48 12.96.26 62.9437.78 57.330.890.790.830.9151317.061393.1266.648.3443.340.3580.4480.1950.910.70.910.810.70.81 -0.354-0.1770
27447.08 14.267.6 71.7337.74 53.170.890.790.830.9171450.121523.1675.3747.9740.190.350.4490.2010.910.710.910.810.710.81 -0.351-0.1820
28457.38 14.16.46 69.138.03 60.350.890.790.830.9141436.11526.9373.5148.9345.580.3550.4520.1930.910.70.910.810.70.81 -0.355-0.1750
29465.24 13.927.04 86.8136.01 53.180.890.790.830.9151330.531404.191.6345.9640.370.3420.4560.2020.910.70.910.820.70.82 -0.35-0.1830
30475.98 11.324.46 69.8338.78 69.190.890.790.830.9111165.141254.6775.2650.5552.10.3590.4560.1850.920.690.920.80.690.8 -0.358-0.1680
31486.36 13.484.02 71.7736.08 80.910.890.790.830.9061268.031396.7879.1947.9961.240.3610.4630.1760.920.680.920.790.680.79 -0.362-0.160
AVE:337.2 13.175.89 51.4334.28 36.210.890.790.830.918963.491023.2951.4334.2836.210.3570.4380.2050.910.720.910.810.720.81-0.35-0.1860




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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