Egwald Economics - Cost Functions: Generalized CES-Translog Cost Function

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Egwald Economics: Microeconomics

Cost Functions

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Elmer G. Wiens

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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:

ε_LKM = 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
cL	0.361633	0.001	493.55
dLq	-0.000101	0	-1.446
dLL	0.034539	0	319.385
dLK	-0.013943	0	-66.16
dLM	-0.020533	0	-230.12

R2 = 0.9998

R2b = 0.9998

# obs = 31

QR Least Squares
Parameter Estimates
Parameter	Coefficient	std error	t-ratio
cK	0.271546	0	712.946
dKq	0.020923	0	574.725
dKL	-0.013819	0	-245.835
dKK	0.030916	0	282.212
dKM	-0.01713	0	-369.333

R2 = 1

R2b = 0.9999

# obs = 31

QR Least Squares
Parameter Estimates
Parameter	Coefficient	std error	t-ratio
cM	0.36682	0.001	519.706
dMq	-0.020821	0	-308.634
dML	-0.02072	0	-198.896
dMK	-0.016973	0	-83.608
dMM	0.037663	0	438.189

R2 = 0.9999

R2b = 0.9999

# obs = 31

The three estimated factor share functions are:

sL(q;wL,wK,wM) = 0.361633 + -0.000101 * ln(q) + 0.034539 * ln(wL) + -0.013943 * ln(wK) + -0.020533 * ln(wM),

sK(q;wL,wK,wM) = 0.271546 + 0.020923 * ln(q) + -0.013819 * ln(wL) + 0.030916 * ln(wK) + -0.01713 * ln(wM),

sM(q;wL,wK,wM) = 0.36682 + -0.020821 * ln(q) + -0.02072 * ln(wL) + -0.016973 * ln(wK) + 0.037663 * 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 = 0.361633 + 0.271546 + 0.36682 = 1
0 =? dLL + dLK + dLM = 0.034539 + -0.013943 + -0.020533 = 6.4E-5
0 =? dKL + dKK + dKM = -0.013819 + 0.030916 + -0.01713 = -3.4E-5
0 =? dML + dMK + dMM = -0.02072 + -0.016973 + 0.037663 = -3.0E-5
0 =? dLq + dKq + dMq = -0.000101 + 0.020923 + -0.020821 = -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
cL	0.361414	0.000341	1058.363403
dLq	-7.8E-5	5.1E-5	-1.530045
dLL	0.034557	8.6E-5	399.68327
dLK	-0.013855	7.8E-5	-178.586645
dLM	-0.020603	5.5E-5	-371.50807
cK	0.271546	0.000589	461.232992
dKq	0.020921	5.7E-5	369.256117
dKL	-0.013855	7.8E-5	-178.586645
dKK	0.030926	0.00017	181.575591
dKM	-0.017103	6.6E-5	-258.432073
cM	0.36705	0.0003	1221.564648
dMq	-0.020841	5.1E-5	-411.237471
dML	-0.020603	5.5E-5	-371.50807
dMK	-0.017103	6.6E-5	-258.432073
dMM	0.037634	7.1E-5	533.39274

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) = 0.361414 + -7.8E-5 * ln(q) + 0.034557 * ln(wL) + -0.013855 * ln(wK) + -0.020603 * ln(wM),

sK(q;wL,wK,wM) = 0.271546 + 0.020921 * ln(q) + -0.013855 * ln(wL) + 0.030926 * ln(wK) + -0.017103 * ln(wM),

sM(q;wL,wK,wM) = 0.36705 + -0.020841 * ln(q) + -0.020603 * ln(wL) + -0.017103 * ln(wK) + 0.037634 * 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 = 0.361414 + 0.271546 + 0.36705 = 1.000011
0 =? dLL + dLK + dLM = 0.034557 + -0.013855 + -0.020603 = 9.9E-5
0 =? dKL + dKK + dKM = -0.013855 + 0.030926 + -0.017103 = -3.2E-5
0 =? dML + dMK + dMM = -0.020603 + -0.017103 + 0.037634 = -7.2E-5
0 =? dLq + dKq + dMq = -7.8E-5 + 0.020921 + -0.020841 = 3.0E-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
cL	0.36166	0.00016	2255.180284
dLq	-8.5E-5	4.2E-5	-2.033202
dLL	0.034506	6.8E-5	509.56275
dLK	-0.013901	5.6E-5	-246.20371
dLM	-0.020605	4.7E-5	-439.780785
cK	0.271459	0.000174	1560.280804
dKq	0.020923	4.3E-5	485.797392
dKL	-0.013901	5.6E-5	-246.20371
dKK	0.030964	6.9E-5	451.862152
dKM	-0.017063	4.7E-5	-361.292384
cM	0.366881	0.000161	2273.534912
dMq	-0.020838	4.2E-5	-494.290751
dML	-0.020605	4.7E-5	-439.780785
dMK	-0.017063	4.7E-5	-361.292384
dMM	0.037668	5.5E-5	683.020003

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) = 0.36166 + -8.5E-5 * ln(q) + 0.034506 * ln(wL) + -0.013901 * ln(wK) + -0.020605 * ln(wM),

sK(q;wL,wK,wM) = 0.271459 + 0.020923 * ln(q) + -0.013901 * ln(wL) + 0.030964 * ln(wK) + -0.017063 * ln(wM),

sM(q;wL,wK,wM) = 0.366881 + -0.020838 * ln(q) + -0.020605 * ln(wL) + -0.017063 * ln(wK) + 0.037668 * 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
c	1.115824	0.000132	8476.577347
cq	1	0	1000
dqq	0.020874	2.2E-5	966.593173

R2 = 1

R2b = 1

# obs = 31

1 = cq

VI. The Translog cost function as estimated is:

ln(C(q;wL,wK,wM)) = 1.115824 + 1 * ln(q) + 0.36166 * ln(wL) + 0.271459 * ln(wK) + 0.366881 * log(wM)
+ .5 * [0.020874 * ln(q)^² + 0.034506 * ln(wL)^² + 0.030964 * ln(wK)^² + 0.037668 * ln(wM)^²]
+ .5 * [-0.027802 * ln(wL)*ln(wK) + -0.04121 * ln(wL)*ln(wM) + -0.034126 * ln(wK)*log(wM)]
+ -8.5E-5 * ln(wL)*ln(q) + 0.020923 * ln(wK)*ln(q) + -0.020838 * ln(wM)*ln(q) (***)

VII. Check for linear homogeneity:

1 =? cL + cK + cM = 0.36166 + 0.271459 + 0.366881 = 1
0 =? dLL + dLK + dLM = 0.034506 + -0.013901 + -0.020605 = 0
0 =? dKL + dKK + dKM = -0.013901 + 0.030964 + -0.017063 = -0
0 =? dML + dMK + dMM = -0.020605 + -0.017063 + 0.037668 = 0
0 =? dLq + dKq + dMq = -8.5E-5 + 0.020923 + -0.020838 = 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) = 1722.08 =? 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 = 0.020874,

0 =? dLq = -8.5E-5, 0 =? dKq = 0.020923, 0 =? dMq = -0.020838.

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

C(30; 7, 13, 6) = 861.04 =? 722.29 = 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 ∇²_wwC 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 =

dLL	dLK	dLM
dKL	dKK	dKM
dML	dMK	dMM

S =

sL	0	0
0	sK	0
0	0	sM

SS =

sL*sL	sL*sK	sL*sM
sK*sL	sK*sK	sK*sM
sM*sL	sM*sK	sM*sM

W =

wL	0	0
0	wK	0
0	0	wM

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

H = W * ∇²_wwC * 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 * ∇²_wwC * W =

-0.19474	0.11581	0.07893
0.11581	-0.20065	0.08484
0.07893	0.08484	-0.16377

The principal minors of H are H1 = -0.194741, H2 = 0.025663, 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 = -0.3139, e2 = -0.2453, 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:

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

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

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

ε_L,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:

ε_L,wL	ε_L,wK	ε_L,wM	ε_L,q
ε_K,wL	ε_K,wK	ε_K,wM	ε_K,q
ε_M,wL	ε_M,wK	ε_M,wM	ε_M,q

-0.5471	0.3254	0.2217	1.0869
0.3178	-0.5506	0.2328	1.1446
0.2823	0.3034	-0.5857	1.0126

XI. Uzawa Partial Elasticities of Substitution:

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

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

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

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

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

u_MK = 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:

u_LK = u_KL, u_LM = u_ML, and u_KM = u_MK.

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, s_LK, s_LM, and s_MK, with the Uzawa partial elasticities of substitution, u_LK, u_LM, and u_KM.

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 Shares			Uzawa Elasticities						W * ∇²_wwC * W
obs #	q	wL	wK	wM	L	K	M	s_LK	s_LM	s_KM	ε_LKM	cost	est cost	est L	est K	est M	sL	sK	sM	u_LK	u_LM	u_KL	u_KM	u_ML	u_MK	e₁	e₂	e₃
1	18	5.02	13.6	4.16	28.21	10.85	27.37	0.89	0.79	0.83	0.92	402.99	403.13	28.22	10.85	27.38	0.351	0.366	0.283	0.89	0.79	0.89	0.84	0.79	0.84	-0.313	-0.247	-0
2	19	5.8	14.54	6.92	31.72	13.15	22.87	0.9	0.79	0.83	0.927	533.43	533.55	31.74	13.16	22.86	0.345	0.359	0.296	0.89	0.8	0.89	0.84	0.8	0.84	-0.306	-0.256	-0
3	20	8.22	14.32	4.28	24.75	14.03	35	0.89	0.79	0.83	0.918	554.13	554.21	24.75	14.03	35.01	0.367	0.362	0.27	0.9	0.79	0.9	0.83	0.79	0.83	-0.318	-0.239	-0
4	21	7.16	13.8	4.86	28.57	14.9	32.43	0.89	0.79	0.83	0.92	567.75	567.8	28.57	14.9	32.44	0.36	0.362	0.278	0.89	0.79	0.89	0.83	0.79	0.83	-0.315	-0.244	-0
5	22	8.94	14.42	4.9	27.08	16.52	36.54	0.89	0.79	0.83	0.919	659.29	659.31	27.08	16.51	36.55	0.367	0.361	0.272	0.9	0.79	0.9	0.83	0.79	0.83	-0.318	-0.24	-0
6	23	5.8	14.82	6.1	38.12	15.69	30	0.89	0.79	0.83	0.922	636.55	636.55	38.12	15.69	29.99	0.347	0.365	0.287	0.89	0.79	0.89	0.84	0.79	0.84	-0.31	-0.25	-0
7	24	6.54	13.24	6.32	36.31	18.35	30.65	0.9	0.79	0.83	0.924	674.17	674.14	36.31	18.35	30.65	0.352	0.36	0.287	0.89	0.8	0.89	0.84	0.8	0.84	-0.31	-0.25	-0
8	25	6.32	14.96	6.4	40.36	17.83	32.59	0.89	0.79	0.83	0.921	730.45	730.39	40.35	17.83	32.59	0.349	0.365	0.286	0.89	0.79	0.89	0.84	0.79	0.84	-0.311	-0.249	-0
9	26	6.1	13.76	5.58	40.52	18.7	35.46	0.89	0.79	0.83	0.92	702.31	702.24	40.51	18.7	35.46	0.352	0.366	0.282	0.89	0.79	0.89	0.83	0.79	0.83	-0.313	-0.246	-0
10	27	8.86	13.98	5.62	34.67	21.85	40.98	0.89	0.79	0.83	0.919	842.94	842.82	34.67	21.84	40.98	0.364	0.362	0.273	0.89	0.79	0.89	0.83	0.79	0.83	-0.317	-0.241	-0
11	28	6.56	11.48	7.9	43.08	25.02	30.19	0.9	0.79	0.83	0.928	808.35	808.23	43.09	25.02	30.17	0.35	0.355	0.295	0.89	0.8	0.89	0.84	0.8	0.84	-0.306	-0.255	-0
12	29	8.36	14.6	7.44	41.72	24.24	37.24	0.89	0.79	0.83	0.922	979.74	979.6	41.72	24.23	37.24	0.356	0.361	0.283	0.89	0.8	0.89	0.83	0.8	0.83	-0.312	-0.247	0
13	30	7	13	6	43.79	24.14	40.13	0.89	0.79	0.83	0.92	861.17	861.04	43.78	24.14	40.13	0.356	0.364	0.28	0.89	0.79	0.89	0.83	0.79	0.83	-0.314	-0.245	-0
14	31	7.84	12.86	7.86	45.15	27.86	36.47	0.9	0.79	0.83	0.924	998.94	998.79	45.15	27.85	36.47	0.354	0.359	0.287	0.89	0.8	0.89	0.83	0.8	0.83	-0.31	-0.25	0
15	32	5.3	12.34	5.32	52.4	23.82	41.86	0.89	0.79	0.83	0.918	794.4	794.29	52.38	23.82	41.86	0.35	0.37	0.28	0.89	0.79	0.89	0.84	0.79	0.84	-0.314	-0.245	-0
16	33	5.58	15	4.74	54.83	21.97	49.83	0.89	0.79	0.83	0.912	871.72	871.59	54.82	21.97	49.83	0.351	0.378	0.271	0.9	0.78	0.9	0.83	0.78	0.83	-0.319	-0.239	0
17	34	7.82	13.92	5.1	46.66	26.8	52.85	0.89	0.79	0.83	0.914	1007.53	1007.39	46.65	26.81	52.83	0.362	0.37	0.267	0.9	0.79	0.9	0.83	0.79	0.83	-0.32	-0.237	-0
18	35	6.78	14.56	7.56	57.59	28.28	42.05	0.89	0.79	0.83	0.92	1120.18	1120.06	57.57	28.28	42.06	0.348	0.368	0.284	0.89	0.79	0.89	0.84	0.79	0.84	-0.312	-0.247	-0
19	36	8.24	13.92	6.92	51.54	31.19	47.47	0.89	0.79	0.83	0.918	1187.37	1187.25	51.53	31.19	47.47	0.358	0.366	0.277	0.89	0.79	0.89	0.83	0.79	0.83	-0.315	-0.243	-0
20	37	5.6	13.42	7.34	65.79	29.5	41.89	0.89	0.79	0.83	0.92	1071.71	1071.65	65.76	29.5	41.89	0.344	0.369	0.287	0.89	0.79	0.89	0.84	0.79	0.84	-0.311	-0.249	-0
21	38	7.96	11.18	5.12	48.59	34.58	55.49	0.89	0.79	0.83	0.916	1057.49	1057.41	48.58	34.58	55.49	0.366	0.366	0.269	0.9	0.79	0.9	0.83	0.79	0.83	-0.319	-0.238	0
22	39	5.64	12.38	4.08	59.44	28.67	61	0.89	0.79	0.83	0.91	939.07	939.03	59.45	28.67	60.97	0.357	0.378	0.265	0.9	0.78	0.9	0.83	0.78	0.83	-0.321	-0.235	0
23	40	7.88	13.96	5.56	56.58	32.97	59.42	0.89	0.79	0.83	0.913	1236.54	1236.52	56.58	32.98	59.41	0.361	0.372	0.267	0.9	0.79	0.9	0.83	0.79	0.83	-0.32	-0.237	0
24	41	6.72	12.06	5.42	60.06	34.71	56.51	0.89	0.79	0.83	0.915	1128.53	1128.54	60.06	34.71	56.51	0.358	0.371	0.271	0.9	0.79	0.9	0.83	0.79	0.83	-0.318	-0.24	-0
25	42	7.06	12.46	5.54	60.97	35.79	58.56	0.89	0.79	0.83	0.914	1200.79	1200.83	60.97	35.79	58.56	0.358	0.371	0.27	0.9	0.79	0.9	0.83	0.79	0.83	-0.319	-0.239	0
26	43	6.42	11.24	6.48	65.89	39.12	51.59	0.89	0.79	0.83	0.918	1197.01	1197.09	65.89	39.12	51.6	0.353	0.367	0.279	0.89	0.79	0.89	0.83	0.79	0.83	-0.314	-0.245	-0
27	44	8.46	11	7.68	59.9	46.17	51.34	0.89	0.79	0.83	0.921	1408.95	1409.08	59.91	46.17	51.36	0.36	0.36	0.28	0.89	0.8	0.89	0.83	0.8	0.83	-0.314	-0.246	-0
28	45	5.92	14.54	4.04	71.42	31.46	75.79	0.89	0.79	0.83	0.905	1186.36	1186.53	71.5	31.45	75.72	0.357	0.385	0.258	0.9	0.78	0.9	0.83	0.78	0.83	-0.325	-0.23	0
29	46	7.16	14.16	7.02	73.37	39.28	58.34	0.89	0.79	0.83	0.915	1491.16	1491.4	73.37	39.29	58.36	0.352	0.373	0.275	0.89	0.79	0.89	0.83	0.79	0.83	-0.317	-0.242	-0
30	47	5.16	11.98	5.1	79.42	36.99	62.4	0.89	0.79	0.83	0.912	1171.14	1171.37	79.42	36.99	62.44	0.35	0.378	0.272	0.89	0.78	0.89	0.83	0.78	0.83	-0.318	-0.239	0
31	48	6.66	11.18	7.66	75.42	46.81	52.75	0.89	0.79	0.83	0.92	1429.7	1430.04	75.42	46.83	52.77	0.351	0.366	0.283	0.89	0.79	0.89	0.84	0.79	0.84	-0.313	-0.247	-0
AVE:	33	6.87	13.31	5.97	49.8	26.81	44.74	0.89	0.79	0.83	0.918	950.06	950.06	49.8	26.81	44.74	0.355	0.367	0.278	0.89	0.79	0.89	0.83	0.79	0.83	-0.315	-0.244	-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 "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
c	1.09523	0.000529	2068.824416
cq	1.012184	0.000265	3812.53557
cL	0.361583	0.000271	1336.658217
cK	0.2714	0.000265	1025.814106
cM	0.367017	0.000129	2849.402317
dqq	0.017297	6.8E-5	254.975747
dLL	0.034584	0.000136	253.843387
dKK	0.030946	0.000116	266.459945
dMM	0.037642	9.7E-5	387.511213
2*dLK	-0.027888	0.000255	-109.276292
2*dLM	-0.04128	0.000161	-257.173401
2*dKM	-0.034004	0.000122	-277.683974
dLq	-9.3E-5	0.000118	-0.795466
dKq	0.041879	0.000121	346.714074
dMq	-0.041785	6.5E-5	-643.862815

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)) = 1.09523 + 1.012184 * ln(q) + 0.361583 * ln(wL) + 0.2714 * ln(wK) + 0.367017 * log(wM)
+ .5 * [0.017297 * ln(q)^² + 0.034584 * ln(wL)^² + 0.030946 * ln(wK)^² + 0.037642 * ln(wM)^²]
+ .5 * [-0.027888 * ln(wL)*ln(wK) + -0.04128 * ln(wL)*ln(wM) + -0.034004 * ln(wK)*log(wM)]
+ -9.3E-5 * ln(wL)*ln(q) + 0.041879 * ln(wK)*ln(q) + -0.041785 * ln(wM)*ln(q) (***)

XV. Its three derived factor share functions are:

sL(q;wL,wK,wM) = 0.361583 + -9.3E-5 * ln(q) + 0.034584 * ln(wL) + -0.013944 * ln(wK) + -0.02064 * ln(wM),

sK(q;wL,wK,wM) = 0.2714 + 0.041879 * ln(q) + -0.013944 * ln(wL) + 0.030946 * ln(wK) + -0.017002 * ln(wM),

sM(q;wL,wK,wM) = 0.367017 + -0.041785 * ln(q) + -206.4 * ln(wL) + -0.017002 * ln(wK) + 0.037642 * 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 = 0.361583 + 0.2714 + 0.367017 = 1
0 =? dLL + dLK + dLM = 0.034584 + -0.013944 + -0.02064 = -0
0 =? dKL + dKK + dKM = -0.013944 + 0.030946 + -0.017002 = -0
0 =? dML + dMK + dMM = -0.02064 + -0.017002 + 0.037642 = -0
0 =? dLq + dKq + dMq = -9.3E-5 + 0.041879 + -0.041785 = -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 ∇²_wwC 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 =

dLL	dLK	dLM
dKL	dKK	dKM
dML	dMK	dMM

S =

sL	0	0
0	sK	0
0	0	sM

SS =

sL*sL	sL*sK	sL*sM
sK*sL	sK*sK	sK*sM
sM*sL	sM*sK	sM*sM

W =

wL	0	0
0	wK	0
0	0	wM

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

H = W * ∇²_wwC * 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 * ∇²_wwC * W =

-0.19463	0.14105	0.05357
0.14105	-0.21491	0.07385
0.05357	0.07385	-0.12743

The principal minors of H are H1 = -0.194627, H2 = 0.021931, 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 = -0.3478, e2 = -0.1892, 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) = 1488.62 =? 1488.62 = C(25; 14, 26,12).

2 * C(30; 7, 13, 6) = 1819.81 =? 1819.81 = 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 = 1.012184, 0 =? dqq = 0.017297,

0 =? dLq = -9.3E-5, 0 =? dKq = 0.041879, 0 =? dMq = -0.041785.

As examples:

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

C(25; 7, 13, 6) = 744.31 =? 589.6 = 25 * 23.58 = 25^^1/1 * C(1; 7, 13, 6).

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

C(30; 7, 13, 6) = 909.9 =? 707.52 = 30 * 23.58 = 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:

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

ε_L,wL	ε_L,wK	ε_L,wM	ε_L,q
ε_K,wL	ε_K,wK	ε_K,wM	ε_K,q
ε_M,wL	ε_M,wK	ε_M,wM	ε_M,q

-0.547	0.3964	0.1506	1.1031
0.3238	-0.4933	0.1695	1.1995
0.2569	0.3541	-0.611	0.903

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, s_LK, s_LM, and s_MK, with the Uzawa partial elasticities of substitution, u_LK, u_LM, and u_KM. 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 Shares			Uzawa Elasticities						W * ∇²_wwC * W
obs #	q	wL	wK	wM	L	K	M	s_LK	s_LM	s_KM	ε_LKM	cost	est cost	est L	est K	est M	sL	sK	sM	u_LK	u_LM	u_KL	u_KM	u_ML	u_MK	e₁	e₂	e₃
1	18	5.02	13.6	4.16	28.21	10.85	27.37	0.89	0.79	0.83	0.92	402.99	432.93	30.3	13.58	23.13	0.351	0.426	0.222	0.91	0.74	0.91	0.82	0.74	0.82	-0.342	-0.201	-0
2	19	5.8	14.54	6.92	31.72	13.15	22.87	0.9	0.79	0.83	0.927	533.43	558.42	33.2	16.14	18.96	0.345	0.42	0.235	0.9	0.75	0.9	0.83	0.75	0.83	-0.336	-0.21	-0
3	20	8.22	14.32	4.28	24.75	14.03	35	0.89	0.79	0.83	0.918	554.13	597.68	26.69	17.74	29.02	0.367	0.425	0.208	0.91	0.73	0.91	0.81	0.73	0.81	-0.349	-0.189	-0
4	21	7.16	13.8	4.86	28.57	14.9	32.43	0.89	0.79	0.83	0.92	567.75	606.77	30.52	18.72	26.73	0.36	0.426	0.214	0.91	0.73	0.91	0.81	0.73	0.81	-0.346	-0.194	-0
5	22	8.94	14.42	4.9	27.08	16.52	36.54	0.89	0.79	0.83	0.919	659.29	706.94	29.02	20.88	29.88	0.367	0.426	0.207	0.91	0.73	0.91	0.81	0.73	0.81	-0.349	-0.189	-0
6	23	5.8	14.82	6.1	38.12	15.69	30	0.89	0.79	0.83	0.922	636.55	674.77	40.39	19.62	24.55	0.347	0.431	0.222	0.91	0.73	0.91	0.82	0.73	0.82	-0.341	-0.2	-0
7	24	6.54	13.24	6.32	36.31	18.35	30.65	0.9	0.79	0.83	0.924	674.17	708.18	38.13	22.83	24.76	0.352	0.427	0.221	0.91	0.73	0.91	0.82	0.73	0.82	-0.342	-0.2	-0
8	25	6.32	14.96	6.4	40.36	17.83	32.59	0.89	0.79	0.83	0.921	730.45	773.47	42.71	22.37	26.39	0.349	0.433	0.218	0.91	0.73	0.91	0.82	0.73	0.82	-0.343	-0.197	-0
9	26	6.1	13.76	5.58	40.52	18.7	35.46	0.89	0.79	0.83	0.92	702.31	746.91	43.07	23.59	28.6	0.352	0.435	0.214	0.91	0.73	0.91	0.82	0.73	0.82	-0.345	-0.193	-0
10	27	8.86	13.98	5.62	34.67	21.85	40.98	0.89	0.79	0.83	0.919	842.94	897.59	36.91	27.69	32.65	0.364	0.431	0.204	0.91	0.72	0.91	0.81	0.72	0.81	-0.35	-0.186	-0
11	28	6.56	11.48	7.9	43.08	25.02	30.19	0.9	0.79	0.83	0.928	808.35	829.72	44.22	30.72	23.66	0.35	0.425	0.225	0.91	0.74	0.91	0.82	0.74	0.82	-0.34	-0.203	-0
12	29	8.36	14.6	7.44	41.72	24.24	37.24	0.89	0.79	0.83	0.922	979.74	1027.42	43.74	30.38	29.34	0.356	0.432	0.212	0.91	0.73	0.91	0.81	0.73	0.81	-0.346	-0.193	-0
13	30	7	13	6	43.79	24.14	40.13	0.89	0.79	0.83	0.92	861.17	909.9	46.25	30.49	31.63	0.356	0.436	0.209	0.91	0.72	0.91	0.81	0.72	0.81	-0.348	-0.189	-0
14	31	7.84	12.86	7.86	45.15	27.86	36.47	0.9	0.79	0.83	0.924	998.94	1034.94	46.77	34.64	28.34	0.354	0.43	0.215	0.91	0.73	0.91	0.82	0.73	0.82	-0.345	-0.195	-0
15	32	5.3	12.34	5.32	52.4	23.82	41.86	0.89	0.79	0.83	0.918	794.4	844.42	55.67	30.29	33.01	0.349	0.443	0.208	0.91	0.72	0.91	0.82	0.72	0.82	-0.347	-0.188	-0
16	33	5.58	15	4.74	54.83	21.97	49.83	0.89	0.79	0.83	0.912	871.72	948.39	59.63	28.53	39.61	0.351	0.451	0.198	0.91	0.7	0.91	0.81	0.7	0.81	-0.352	-0.179	-0
17	34	7.82	13.92	5.1	46.66	26.8	52.85	0.89	0.79	0.83	0.914	1007.53	1084.99	50.23	34.62	41.22	0.362	0.444	0.194	0.91	0.71	0.91	0.8	0.71	0.8	-0.355	-0.176	-0
18	35	6.78	14.56	7.56	57.59	28.28	42.05	0.89	0.79	0.83	0.92	1120.18	1176.22	60.43	35.71	32.6	0.348	0.442	0.21	0.91	0.72	0.91	0.82	0.72	0.82	-0.347	-0.189	-0
19	36	8.24	13.92	6.92	51.54	31.19	47.47	0.89	0.79	0.83	0.918	1187.37	1251.27	54.29	39.61	36.49	0.358	0.441	0.202	0.91	0.71	0.91	0.81	0.71	0.81	-0.351	-0.183	-0
20	37	5.6	13.42	7.34	65.79	29.5	41.89	0.89	0.79	0.83	0.92	1071.71	1121.8	68.81	37.2	32.32	0.343	0.445	0.211	0.91	0.72	0.91	0.82	0.72	0.82	-0.346	-0.191	-0
21	38	7.96	11.18	5.12	48.59	34.58	55.49	0.89	0.79	0.83	0.916	1057.49	1122.22	51.54	44.34	42.22	0.366	0.442	0.193	0.91	0.71	0.91	0.8	0.71	0.8	-0.355	-0.175	-0
22	39	5.64	12.38	4.08	59.44	28.67	61	0.89	0.79	0.83	0.91	939.07	1022.48	64.71	37.55	47.21	0.357	0.455	0.188	0.91	0.69	0.91	0.8	0.69	0.8	-0.356	-0.171	-0
23	40	7.88	13.96	5.56	56.58	32.97	59.42	0.89	0.79	0.83	0.913	1236.54	1327.59	60.73	42.75	45.37	0.36	0.45	0.19	0.91	0.7	0.91	0.8	0.7	0.8	-0.356	-0.173	-0
24	41	6.72	12.06	5.42	60.06	34.71	56.51	0.89	0.79	0.83	0.915	1128.53	1200.9	63.89	44.68	42.94	0.358	0.449	0.194	0.91	0.7	0.91	0.8	0.7	0.8	-0.354	-0.176	-0
25	42	7.06	12.46	5.54	60.97	35.79	58.56	0.89	0.79	0.83	0.914	1200.79	1279.37	64.93	46.17	44.35	0.358	0.45	0.192	0.91	0.7	0.91	0.8	0.7	0.8	-0.355	-0.174	-0
26	43	6.42	11.24	6.48	65.89	39.12	51.59	0.89	0.79	0.83	0.918	1197.01	1250.07	68.78	49.61	38.71	0.353	0.446	0.201	0.91	0.71	0.91	0.81	0.71	0.81	-0.351	-0.182	-0
27	44	8.46	11	7.68	59.9	46.17	51.34	0.89	0.79	0.83	0.921	1408.95	1449.61	61.61	57.94	37.9	0.36	0.44	0.201	0.91	0.71	0.91	0.81	0.71	0.81	-0.351	-0.183	-0
28	45	5.92	14.54	4.04	71.42	31.46	75.79	0.89	0.79	0.83	0.905	1186.36	1313.79	79.13	42.03	57.98	0.357	0.465	0.178	0.92	0.68	0.92	0.79	0.68	0.79	-0.36	-0.161	-0
29	46	7.16	14.16	7.02	73.37	39.28	58.34	0.89	0.79	0.83	0.915	1491.16	1577.43	77.58	50.48	43.75	0.352	0.453	0.195	0.91	0.7	0.91	0.81	0.7	0.81	-0.353	-0.176	-0
30	47	5.16	11.98	5.1	79.42	36.99	62.4	0.89	0.79	0.83	0.912	1171.14	1254.61	85.03	48.06	47.08	0.35	0.459	0.191	0.91	0.69	0.91	0.81	0.69	0.81	-0.354	-0.173	-0
31	48	6.66	11.18	7.66	75.42	46.81	52.75	0.89	0.79	0.83	0.92	1429.7	1474.23	77.72	58.97	38.82	0.351	0.447	0.202	0.91	0.71	0.91	0.81	0.71	0.81	-0.35	-0.183	-0
AVE:	33	6.87	13.31	5.97	52.8	34.13	34.81	0.89	0.79	0.83	0.918	950.06	1006.61	52.8	34.13	34.81	0.355	0.439	0.206	0.91	0.72	0.91	0.81	0.72	0.81	-0.349	-0.186	-0

Mathematical Notes

1. The Translog Cost Function:

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).

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
ε_L,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
ε_L,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
ε_L,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