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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
cL0.3594830.001470.484
dLq0.00021903.199
dLL0.034620278.319
dLK-0.013580-52.011
dLM-0.0205730-200.275
R2 = 0.9998 R2b = 0.9998 # obs = 31

QR Least Squares
Parameter Estimates
Parameter Coefficient std error t-ratio
cK0.2708440.001476.061
dKq0.0208960410.426
dKL-0.0136350-147.211
dKK0.0310580159.747
dKM-0.0170880-223.412
R2 = 0.9999 R2b = 0.9999 # obs = 31

QR Least Squares
Parameter Estimates
Parameter Coefficient std error t-ratio
cM0.3696720.001383.098
dMq-0.0211140-244.517
dML-0.0209850-133.584
dMK-0.0174780-53.002
dMM0.0376610290.303
R2 = 0.9999 R2b = 0.9999 # obs = 31

The three estimated factor share functions are:

sL(q;wL,wK,wM) = 0.359483 + 0.000219 * ln(q) + 0.03462 * ln(wL) + -0.01358 * ln(wK) + -0.020573 * ln(wM),

sK(q;wL,wK,wM) = 0.270844 + 0.020896 * ln(q) + -0.013635 * ln(wL) + 0.031058 * ln(wK) + -0.017088 * ln(wM),

sM(q;wL,wK,wM) = 0.369672 + -0.021114 * ln(q) + -0.020985 * ln(wL) + -0.017478 * ln(wK) + 0.037661 * 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   =   0.359483 + 0.270844 + 0.369672   =   1
0   =?   dLL + dLK + dLM   =   0.03462 + -0.01358 + -0.020573  =   0.000467
0   =?   dKL + dKK + dKM   =   -0.013635 + 0.031058 + -0.017088  =   0.000335
0   =?   dML + dMK + dMM   =   -0.020985 + -0.017478 + 0.037661  =   -0.000803
0   =?   dLq + dKq + dMq   =   0.000219 + 0.020896 + -0.021114  =   -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
cL0.3599920.000464776.558059
dLq0.0002036.9E-52.963326
dLL0.0346130.000125276.666599
dLK-0.0136320.000113-120.474266
dLM-0.0207458.0E-5-260.30987
cK0.2709840.000759356.802348
dKq0.020896.9E-5303.652939
dKL-0.0136320.000113-120.474266
dKK0.0310490.000263118.063118
dKM-0.0171459.6E-5-178.013859
cM0.3683740.000431855.308244
dMq-0.0211276.9E-5-307.196027
dML-0.0207458.0E-5-260.30987
dMK-0.0171459.6E-5-178.013859
dMM0.0376770.000103364.722663
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) = 0.359992 + 0.000203 * ln(q) + 0.034613 * ln(wL) + -0.013632 * ln(wK) + -0.020745 * ln(wM),

sK(q;wL,wK,wM) = 0.270984 + 0.02089 * ln(q) + -0.013632 * ln(wL) + 0.031049 * ln(wK) + -0.017145 * ln(wM),

sM(q;wL,wK,wM) = 0.368374 + -0.021127 * ln(q) + -0.020745 * ln(wL) + -0.017145 * ln(wK) + 0.037677 * 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   =   0.359992 + 0.270984 + 0.368374   =   0.99935
0   =?   dLL + dLK + dLM   =   0.034613 + -0.013632 + -0.020745  =   0.000236
0   =?   dKL + dKK + dKM   =   -0.013632 + 0.031049 + -0.017145  =   0.000272
0   =?   dML + dMK + dMM   =   -0.020745 + -0.017145 + 0.037677  =   -0.000212
0   =?   dLq + dKq + dMq   =   0.000203 + 0.02089 + -0.021127  =   -3.4E-5

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
cL0.3605030.0002061750.40721
dLq0.0002125.7E-53.703402
dLL0.0344929.2E-5374.200343
dLK-0.0137668.7E-5-157.852319
dLM-0.0207266.3E-5-328.309083
cK0.2715940.0002091299.429144
dKq0.0209055.7E-5365.592743
dKL-0.0137668.7E-5-157.852319
dKK0.0308430.000116264.934076
dKM-0.0170767.3E-5-233.627785
cM0.3679030.0002011832.646516
dMq-0.0211175.7E-5-368.228363
dML-0.0207266.3E-5-328.309083
dMK-0.0170767.3E-5-233.627785
dMM0.0378027.7E-5491.634067
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) = 0.360503 + 0.000212 * ln(q) + 0.034492 * ln(wL) + -0.013766 * ln(wK) + -0.020726 * ln(wM),

sK(q;wL,wK,wM) = 0.271594 + 0.020905 * ln(q) + -0.013766 * ln(wL) + 0.030843 * ln(wK) + -0.017076 * ln(wM),

sM(q;wL,wK,wM) = 0.367903 + -0.021117 * ln(q) + -0.020726 * ln(wL) + -0.017076 * ln(wK) + 0.037802 * 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
c1.1158190.0001417921.477245
cq101000
dqq0.0208722.3E-5903.197409
R2 = 1 R2b = 1 # obs = 31

1 = cq

VI. The Translog cost function as estimated is:

ln(C(q;wL,wK,wM)) = 1.115819 + 1 * ln(q) + 0.360503 * ln(wL) + 0.271594 * ln(wK) + 0.367903 * log(wM)
+ .5 * [0.020872 * ln(q)^2 + 0.034492 * ln(wL)^2 + 0.030843 * ln(wK)^2 + 0.037802 * ln(wM)^2]
+ .5 * [-0.027533 * ln(wL)*ln(wK) + -0.041452 * ln(wL)*ln(wM) + -0.034153 * ln(wK)*log(wM)]
+ 0.000212 * ln(wL)*ln(q) + 0.020905 * ln(wK)*ln(q) + -0.021117 * ln(wM)*ln(q)           (***)

VII. Check for linear homogeneity:

1   =?   cL + cK + cM   =   0.360503 + 0.271594 + 0.367903   =   1
0   =?   dLL + dLK + dLM   =   0.034492 + -0.013766 + -0.020726  =   0
0   =?   dKL + dKK + dKM   =   -0.013766 + 0.030843 + -0.017076  =   0
0   =?   dML + dMK + dMM   =   -0.020726 + -0.017076 + 0.037802  =   -0
0   =?   dLq + dKq + dMq   =   0.000212 + 0.020905 + -0.021117  =   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) = 1412.92 =? 1412.92 = C(25; 14, 26,12).

2 * C(30; 7, 13, 6) = 1722.08 =? 1722.08 = 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  =   0.020872,

0   =?   dLq  =   0.000212,     0   =?   dKq  =   0.020905,     0   =?   dMq  =   -0.021117.

      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 =? 722.22 = 30 * 24.07 = 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 =
-0.194740.115930.07881
0.11593-0.200780.08485
0.078810.08485-0.16366

The principal minors of H are H1 = -0.194742, H2 = 0.02566, 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 = -0.3141, e2 = -0.2451, 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
-0.54720.32570.22141.0878
0.3181-0.55090.23281.1446
0.28180.3034-0.58521.0117

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
1187.14 13.466.16 25.3313.39 23.980.90.790.830.927508.84509.0225.3413.39240.3550.3540.290.890.80.890.830.80.83 -0.308-0.2520
2195.2 11.27.96 32.0915.25 18.980.90.790.840.935488.77488.8732.1315.2618.960.3420.350.3090.880.80.880.840.80.84 -0.3-0.2630
3206.54 12.326.46 29.2815.61 24.660.90.790.830.928543.04543.1429.2915.624.670.3530.3540.2930.890.80.890.840.80.84 -0.307-0.2540
4215.6 13.467.16 35.4415.34 24.010.90.790.830.928576.89576.9535.4415.3524.010.3440.3580.2980.890.80.890.840.80.84 -0.305-0.2560
5227.68 13.346.94 30.9917.81 27.890.90.790.830.926669.12669.1630.9917.8127.90.3560.3550.2890.890.80.890.830.80.83 -0.309-0.2520
6235.32 13.387.12 40.116.78 25.980.90.790.830.926622.89622.940.0916.7925.980.3420.3610.2970.890.80.890.840.80.84 -0.306-0.2550
7246.34 13.525.74 36.3917.56 32.340.890.790.830.922653.67653.6436.3717.5532.350.3530.3630.2840.890.790.890.830.790.83 -0.312-0.2480
8255.42 12.227.14 42.2119.53 27.60.90.790.830.927664.45664.442.219.5327.590.3440.3590.2960.890.80.890.840.80.84 -0.306-0.2550
9267 12.44.62 34.8519.81 39.90.890.790.830.918673.89673.7934.8419.839.910.3620.3640.2740.90.790.90.830.790.83 -0.317-0.2410
10277.6 14.84.08 35.8618.75 48.250.890.790.830.912746.9746.7935.8618.7548.210.3650.3720.2630.90.780.90.830.780.83 -0.322-0.2340
11289 11.845.32 33.524.79 41.970.890.790.830.92818.37818.2233.4924.7941.980.3680.3590.2730.90.790.90.830.790.83 -0.317-0.2410
12297.78 13.36.52 40.8824.18 38.320.890.790.830.921889.56889.4340.8724.1838.330.3570.3620.2810.890.790.890.830.790.83 -0.313-0.2460
13307 136 43.7924.14 40.130.890.790.830.92861.17861.0443.7824.1440.140.3560.3640.280.890.790.890.830.790.83 -0.314-0.2450
14316.82 11.624.52 41.7224.78 47.020.890.790.830.916785784.8741.7124.7847.010.3620.3670.2710.90.790.90.830.790.83 -0.318-0.2390
15328.44 14.567.48 46.227.28 41.080.890.790.830.9211094.491094.3246.1927.2841.090.3560.3630.2810.890.790.890.830.790.83 -0.313-0.2460
16338.3 12.587.52 46.0430.52 40.240.890.790.830.9231068.671068.5346.0330.5140.250.3580.3590.2830.890.80.890.830.80.83 -0.312-0.2480
17346.26 14.985.34 54.5324.24 49.410.890.790.830.913968.27968.1454.5224.2349.410.3530.3750.2730.90.780.90.830.780.83 -0.318-0.240
18357.38 13.665.04 49.2727.41 53.60.890.790.830.9141008.161008.0449.2727.4153.580.3610.3710.2680.90.790.90.830.790.83 -0.32-0.2370
19367.18 11.47.08 52.3533.51 42.410.890.790.830.9221058.131058.0252.3433.542.410.3550.3610.2840.890.790.890.830.790.83 -0.312-0.2480
20377.32 13.35.56 53.2230.23 52.950.890.790.830.9151086.041085.9453.2230.2352.940.3590.370.2710.90.790.90.830.790.83 -0.318-0.2390
21386.14 12.45.46 58.7130.49 51.220.890.790.830.9161018.261018.1958.7130.4951.220.3540.3710.2750.90.790.90.830.790.83 -0.317-0.2410
22397.28 11.965.72 54.9334.19 53.090.890.790.830.9171112.481112.4354.9334.1953.080.3590.3680.2730.90.790.90.830.790.83 -0.317-0.2410
23407.26 13.946.84 61.8433.69 51.350.890.790.830.9171269.791269.7761.8333.6951.350.3540.370.2770.890.790.890.830.790.83 -0.316-0.2430
24415.6 13.984.06 65.6328.36 66.780.890.790.830.9071035.21035.2165.6928.3666.730.3550.3830.2620.90.780.90.830.780.83 -0.323-0.2320
25425.42 11.885.96 70.534.36 51.330.890.790.830.9161096.231096.2770.4934.3651.350.3490.3720.2790.890.790.890.840.790.84 -0.315-0.2440
26435.86 135.06 68.9233.19 60.860.890.790.830.9121143.271143.3468.9533.1860.860.3530.3770.2690.90.780.90.830.780.83 -0.32-0.2380
27445.98 13.546.76 75.4435.84 53.490.890.790.830.9161297.971298.175.4435.8453.510.3480.3740.2790.890.790.890.840.790.84 -0.315-0.2440
28457.1 11.74.54 61.5538.44 69.480.890.790.830.9111202.21202.4161.5938.4669.410.3640.3740.2620.90.780.90.830.780.83 -0.323-0.2330
29468.84 13.127.52 64.7144.49 58.070.890.790.830.9171592.361592.6164.7244.558.070.3590.3670.2740.90.790.90.830.790.83 -0.317-0.2410
30475.14 13.687.22 89.8237.26 52.990.890.790.830.9161353.991354.2989.7937.2753.030.3410.3760.2830.890.780.890.840.780.84 -0.314-0.2450
31487.04 12.947.44 76.2643.67 56.990.890.790.830.9171525.981526.3276.2743.68570.3520.370.2780.890.790.890.830.790.83 -0.315-0.2430
AVE:336.81 12.986.14 50.0826.93 44.070.890.790.830.919949.49949.4950.0826.9344.070.3540.3660.280.890.790.890.830.790.83-0.314-0.2450




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
c1.094260.0007481463.836196
cq1.0128370.0004122455.383093
cL0.3607980.0003551016.627599
cK0.2710940.000482562.133599
cM0.3681090.0002771328.837102
dqq0.0170870.00012142.893529
dLL0.0344620.000248139.170523
dKK0.030860.00033791.566508
dMM0.0381080.00015253.343806
2*dLK-0.0272130.000546-49.843549
2*dLM-0.041710.000269-155.344903
2*dKM-0.0345060.000325-106.251947
dLq0.0002070.0001961.056117
dKq0.0420520.00026161.736305
dMq-0.0422590.000158-267.001714
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)) = 1.09426 + 1.012837 * ln(q) + 0.360798 * ln(wL) + 0.271094 * ln(wK) + 0.368109 * log(wM)
+ .5 * [0.017087 * ln(q)^2 + 0.034462 * ln(wL)^2 + 0.03086 * ln(wK)^2 + 0.038108 * ln(wM)^2]
+ .5 * [-0.027213 * ln(wL)*ln(wK) + -0.04171 * ln(wL)*ln(wM) + -0.034506 * ln(wK)*log(wM)]
+ 0.000207 * ln(wL)*ln(q) + 0.042052 * ln(wK)*ln(q) + -0.042259 * ln(wM)*ln(q)           (***)

XV. Its three derived factor share functions are:

sL(q;wL,wK,wM) = 0.360798 + 0.000207 * ln(q) + 0.034462 * ln(wL) + -0.013607 * ln(wK) + -0.020855 * ln(wM),

sK(q;wL,wK,wM) = 0.271094 + 0.042052 * ln(q) + -0.013607 * ln(wL) + 0.03086 * ln(wK) + -0.017253 * ln(wM),

sM(q;wL,wK,wM) = 0.368109 + -0.042259 * ln(q) + -208.55 * ln(wL) + -0.017253 * ln(wK) + 0.038108 * 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   =   0.360798 + 0.271094 + 0.368109   =   1
0   =?   dLL + dLK + dLM   =   0.034462 + -0.013607 + -0.020855  =   -0
0   =?   dKL + dKK + dKM   =   -0.013607 + 0.03086 + -0.017253  =   -0
0   =?   dML + dMK + dMM   =   -0.020855 + -0.017253 + 0.038108  =   0
0   =?   dLq + dKq + dMq   =   0.000207 + 0.042052 + -0.042259  =   -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 =
-0.194890.14170.05319
0.1417-0.215030.07333
0.053190.07333-0.12652

The principal minors of H are H1 = -0.194887, H2 = 0.021829, 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 = -0.3486, e2 = -0.1879, 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) = 1489.03 =? 1489.03 = C(25; 14, 26,12).

2 * C(30; 7, 13, 6) = 1820.36 =? 1820.36 = 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   =   1.012837,     0   =?   dqq  =   0.017087,

0   =?   dLq  =   0.000207,     0   =?   dKq  =   0.042052,     0   =?   dMq  =   -0.042259.

      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) = 744.52 =? 588.83 = 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) = 910.18 =? 706.59 = 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
-0.5470.39770.14931.1041
0.3251-0.49330.16821.2
0.25590.3529-0.60880.9002

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
1187.14 13.466.16 25.3313.39 23.980.90.790.830.927508.84533.6226.5916.4419.870.3560.4150.2290.910.740.910.820.740.82 -0.339-0.206-0
2195.2 11.27.96 32.0915.25 18.980.90.790.840.935488.77499.1432.8418.3315.470.3420.4110.2470.90.750.90.830.750.83 -0.331-0.219-0
3206.54 12.326.46 29.2815.61 24.660.90.790.830.928543.04565.5930.5319.1420.150.3530.4170.230.910.740.910.820.740.82 -0.339-0.207-0
4215.6 13.467.16 35.4415.34 24.010.90.790.830.928576.89600.6336.9418.8319.60.3440.4220.2340.910.740.910.830.740.83 -0.337-0.209-0
5227.68 13.346.94 30.9917.81 27.890.90.790.830.926669.12698.1932.3721.9722.540.3560.420.2240.910.740.910.820.740.82 -0.341-0.202-0
6235.32 13.387.12 40.116.78 25.980.90.790.830.926622.89649.2941.8420.6921.050.3430.4260.2310.910.740.910.820.740.82 -0.338-0.206-0
7246.34 13.525.74 36.3917.56 32.340.890.790.830.922653.67692.238.5622.0126.160.3530.430.2170.910.730.910.810.730.81 -0.344-0.196-0
8255.42 12.227.14 42.2119.53 27.60.90.790.830.927664.45689.0143.8124.0622.060.3450.4270.2290.910.740.910.820.740.82 -0.339-0.204-0
9267 12.44.62 34.8519.81 39.90.890.790.830.918673.89721.1337.3325.1831.960.3620.4330.2050.910.720.910.810.720.81 -0.35-0.186-0
10277.6 14.84.08 35.8618.75 48.250.890.790.830.912746.9816.8439.2724.3438.770.3650.4410.1940.920.710.920.80.710.8 -0.355-0.176-0
11289 11.845.32 33.524.79 41.970.890.790.830.92818.37865.6935.4731.3532.960.3690.4290.2030.910.720.910.80.720.8 -0.352-0.184-0
12297.78 13.36.52 40.8824.18 38.320.890.790.830.921889.56935.6843.0430.4130.110.3580.4320.210.910.720.910.810.720.81 -0.348-0.19-0
13307 136 43.7924.14 40.130.890.790.830.92861.17910.1846.3330.5231.530.3560.4360.2080.910.720.910.810.720.81 -0.349-0.188-0
14316.82 11.624.52 41.7224.78 47.020.890.790.830.916785840.544.7231.7636.840.3630.4390.1980.910.710.910.80.710.8 -0.353-0.18-0
15328.44 14.567.48 46.227.28 41.080.890.790.830.9211094.491148.9748.5534.3931.890.3570.4360.2080.910.720.910.810.720.81 -0.349-0.188-0
16338.3 12.587.52 46.0430.52 40.240.890.790.830.9231068.671109.9347.8638.1730.910.3580.4330.2090.910.720.910.810.720.81 -0.348-0.189-0
17346.26 14.985.34 54.5324.24 49.410.890.790.830.913968.271045.2758.9431.3338.750.3530.4490.1980.910.70.910.810.70.81 -0.352-0.179-0
18357.38 13.665.04 49.2727.41 53.60.890.790.830.9141008.161086.3453.1635.4841.540.3610.4460.1930.920.70.920.80.70.8 -0.355-0.175-0
19367.18 11.47.08 52.3533.51 42.410.890.790.830.9221058.131096.8154.3141.9832.250.3560.4360.2080.910.720.910.810.720.81 -0.348-0.188-0
20377.32 13.35.56 53.2230.23 52.950.890.790.830.9151086.041160.556.9438.9340.650.3590.4460.1950.920.70.920.80.70.8 -0.354-0.176-0
21386.14 12.45.46 58.7130.49 51.220.890.790.830.9161018.261084.2262.5939.1539.280.3540.4480.1980.910.70.910.810.70.81 -0.353-0.179-0
22397.28 11.965.72 54.9334.19 53.090.890.790.830.9171112.481177.6258.2143.7840.260.360.4450.1960.910.70.910.80.70.8 -0.354-0.177-0
23407.26 13.946.84 61.8433.69 51.350.890.790.830.9171269.791341.9165.4243.0738.980.3540.4470.1990.910.70.910.810.70.81 -0.352-0.179-0
24415.6 13.984.06 65.6328.36 66.780.890.790.830.9071035.21140.2772.4537.651.440.3560.4610.1830.920.680.920.80.680.8 -0.359-0.165-0
25425.42 11.885.96 70.534.36 51.330.890.790.830.9161096.231157.2474.543.9238.880.3490.4510.20.910.70.910.810.70.81 -0.351-0.18-0
26435.86 135.06 68.9233.19 60.860.890.790.830.9121143.271231.974.3843.2546.210.3540.4560.190.920.690.920.80.690.8 -0.356-0.171-0
27445.98 13.546.76 75.4435.84 53.490.890.790.830.9161297.971371.6579.845.9240.330.3480.4530.1990.910.70.910.810.70.81 -0.351-0.179-0
28457.1 11.74.54 61.5538.44 69.480.890.790.830.9111202.21297.1166.5250.3751.880.3640.4540.1820.920.680.920.790.680.79 -0.36-0.164-0
29468.84 13.127.52 64.7144.49 58.070.890.790.830.9171592.361665.4167.7456.7542.820.360.4470.1930.920.70.920.80.70.8 -0.355-0.175-0
30475.14 13.687.22 89.8237.26 52.990.890.790.830.9161353.991425.7294.6447.6639.780.3410.4570.2010.910.70.910.810.70.81 -0.35-0.181-0
31487.04 12.947.44 76.2643.67 56.990.890.790.830.9171525.981596.2479.8555.7242.080.3520.4520.1960.910.70.910.810.70.81 -0.353-0.177-0
AVE:336.81 12.986.14 53.0834.27 34.10.890.790.830.919949.491004.9953.0834.2734.10.3550.4390.2070.910.710.910.810.710.81-0.349-0.186-0




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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