Egwald Economics: Microeconomics

Cost Functions

by

Elmer G. Wiens

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N. Generalized CES-Diewert (Generalized Leontief) Cost Function

This web page replicates the procedures used to estimate the parameters of a Diewert cost function to approximate a CES cost function. To determine the Diewert cost function'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 Diewert cost function.

Unlike the CES cost function, the Generalized CES cost function is not necessarily homothetic, while the Diewert cost function is homothetic by construction. Moreover, the Generalized CES's elasticity of scale is a function of its factor inputs, i.e. its elasticity of scale, εLKM, varies with factor prices and with output.

Consequently, when we estimate the Diewert cost function, we need to estimate the parameter nu1 of its returns to scale function separately, as we did when we approximated the CES production function by the Diewert production function.

The three factor Diewert (Generalized Leontief) (total) cost function is:

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

where the returns to scale function is:

h(q) = q^(1/nu1)

a continuous, increasing function of q (q >= 1), with h(0) = 0 and h(1) = 1, where nu1 is a measure of the returns to scale,

and the unit cost function is:

c(wL,wK,wM) = cLL * wL + cKK * wK + cMM * wM + (dLK+dKL) * (wL*wK)^(1/2) + (dLM +dML) * (wL*wM)^(1/2) + (dKM+dMK) * (wK*wM)^(1/2)

linear in its parameters cLL, cKK, cMM, dLK, dKL, dLM, dML, dKM, and dMK.

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.

For methods À1, and À2, we shall set nu1 = εLKM where q = 30 units of output, i.e. at the elasticity of scale of the Generalized CES cost function for q = 30.

To illustrate what can happen when nu1 is set incorrectly, we shall repeat methods À1, and À2 as methods À3, and À4, with nu1 = 1, i.e. under the assumption of constant returns to scale along the domain of q.

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 demand equations separately.

Assuming dLK = dKL, dLM = dML, and dKM = dMK, the Diewert unit cost function becomes:

c(wL,wK,wM) = cLL * wL + cKK * wK + cMM * wM + 2 * dLK * (wL*wK)^(1/2) + 2 * dLM * (wL*wM)^(1/2) + 2 * dKM * (wK*wM)^(1/2)

Taking the partial derivative of the cost function with respect to an input price, we get the factor demand function for that input:

 ∂C/∂wL = L(q; wL, wK, wM) = q^(1/nu1) * [cLL + dLK * (wK / wL)^(1/2) + dLM * (wM / wL)^(1/2)] ∂C/∂wK = K(q; wL, wK, wM) = q^(1/nu1) * [cKK + dKL * (wL / wK)^(1/2) + dKM * (wM / wK)^(1/2)] ∂C/∂wM = M(q; wL, wK, wM) = q^(1/nu1) * [cMM + dML * (wL / wM)^(1/2) + dMK * (wK / wM)^(1/2)]

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

After dividing each factor demand equation by q(1/nu1), we can estimate the parameters of these three linear in parameters equations using linear multiple regression.

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

The substituted values of the estimated parameters determine the Diewert cost function.

I. Stage 1. Generate cost data with the Generalized CES production function, 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.

Stage 2. Obtain the Diewert cost function by substituting the estimates (obtained via linear multiple regression) of the parameters of the three factor demand equations into the cost function.

II. The estimated coefficients of the Diewert 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.

Generalized CES elasticity of scale at q = 30:     εLKM = nu1 = 0.92

III. For these coefficients of the CES Generalized production function, I generated a sequence (displayed in the "Generalized CES-Diewert 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
cLL-0.102520.005-22.778
dLK0.5208960.003149.482
dLM0.5174160.005114.055
 R2 = 0.9996 R2b = 0.9996 # obs = 31

QR Least Squares
Parameter Estimates
Parameter Coefficient std error t-ratio
cKK-0.1151680.02-5.752
dKL0.600330.02623.2
dKM0.405390.02515.92
 R2 = 0.9788 R2b = 0.9772 # obs = 31

QR Least Squares
Parameter Estimates
Parameter Coefficient std error t-ratio
cMM-0.1724330.037-4.681
dML0.4152990.03811.048
dMK0.4866390.02916.611
 R2 = 0.9742 R2b = 0.9723 # obs = 31

The three estimated factor demand functions are:

 ∂C/∂wL = L(q; wL, wK, wM) = q^(1/0.92) * [-0.1025 + 0.5209 * (wK / wL)^(1/2) + 0.5174 * (wM / wL)^(1/2)] ∂C/∂wK = K(q; wL, wK, wM) = q^(1/0.92) * [-0.1152 + 0.6003 * (wL / wK)^(1/2) + 0.4054 * (wM / wK)^(1/2)] ∂C/∂wM = M(q; wL, wK, wM) = q^(1/0.92) * [-0.1724 + 0.4153 * (wL / wM)^(1/2) + 0.4866 * (wK / wM)^(1/2)]

As estimated, generally dLK != dKL, dLM != dML, and dKM != dMK: Young's Theorem doesn't hold without constraints across equations. Consequently, I use the sum of the cross parameters in the Diewert cost function.

The estimated factor demand elasticities are obtained by:

 εL,wL = ∂ln(L(q;wL,wK,wM))/∂ln(wL) = -.5 * (q^1/nu1) / L(q;wL,wK,wM)) * (dLK * (wK/wL)^1/2 + dLM * (wM/wL)^1/2), εL,wK = ∂ln(L(q;wL,wK,wM))/∂ln(wK) = .5 * (q^1/nu1) / L(q;wL,wK,wM)) * dLK * (wK/wL)^1/2, εL,wM = ∂ln(L(q;wL,wK,wM))/∂ln(wM) = .5 * (q^1/nu1) / L(q;wL,wK,wM)) * dLM * (wM/wL)^1/2, εL,q = ∂ln(L(q;wL,wK,wM))/∂ln(q) = 1/nu1, etc.

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

 εL,wL εL,wK εL,wM εL,q εK,wL εK,wK εK,wM εK,q εM,wL εM,wK εM,wM εM,q
 -0.5472 0.3267 0.2205 1.087 0.3666 -0.5959 0.2292 1.087 0.226 0.3609 -0.5869 1.087

IV. The Diewert cost function as obtained from the unrestricted factor demand equations is:

C(q;wL,wK,wM) = q^(1/nu1) * [cLL * wL + cKK * wK + cMM * wM + (dLK+dKL) * (wL*wK)^(1/2) + (dLM +dML) * (wL*wM)^(1/2) + (dKM+dMK) * (wK*wM)^(1/2)]

= q^(1/0.92) * [-0.10252 * wL + -0.11517 * wK + -0.17243 * wM + 1.12123 * (wL*wK)^(1/2) + 0.93272 * (wL*wM)^(1/2) + 0.89203 * (wK*wM)^(1/2)]
(***)

V. The Restricted Factor Demand Equations.

Having obtained the unrestricted estimates of the coefficients of the factor demand 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
cLL-0.0924630.024549-3.766542
dLK0.5500940.01705932.246574
dLM0.4628310.01766726.197874
cKK-0.1126920.017039-6.613939
dKL0.5500940.01705932.246574
dKM0.4569110.01561129.269376
cMM-0.1803480.022944-7.860518
dML0.4628310.01766726.197874
dMK0.4569110.01561129.269376
 R2 = 0.9971 R2b = 0.9968 # obs = 93

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

The three estimated, restricted factor demand functions are:

 ∂C/∂wL = L(q; wL, wK, wM) = q^(1/0.92) * [-0.0925 + 0.5501 * (wK / wL)^(1/2) + 0.4628 * (wM / wL)^(1/2)] ∂C/∂wK = K(q; wL, wK, wM) = q^(1/0.92) * [-0.1127 + 0.5501 * (wL / wK)^(1/2) + 0.4569 * (wM / wK)^(1/2)] ∂C/∂wM = M(q; wL, wK, wM) = q^(1/0.92) * [-0.1803 + 0.4628 * (wL / wM)^(1/2) + 0.4569 * (wK / wM)^(1/2)]

The estimated factor demand elasticities are obtained by:

 εL,wL = ∂ln(L(q;wL,wK,wM))/∂ln(wL) = -.5 * (q^1/nu1) / L(q;wL,wK,wM)) * (dLK * (wK/wL)^1/2 + dLM * (wM/wL)^1/2), εL,wK = ∂ln(L(q;wL,wK,wM))/∂ln(wK) = .5 * (q^1/nu1) / L(q;wL,wK,wM)) * dLK * (wK/wL)^1/2, εL,wM = ∂ln(L(q;wL,wK,wM))/∂ln(wM) = .5 * (q^1/nu1) / L(q;wL,wK,wM)) * dLM * (wM/wL)^1/2, εL,wq = ∂ln(L(q;wL,wK,wM))/∂ln(q) = 1/nu, etc.

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

 εL,wL εL,wK εL,wM εL,q εK,wL εK,wK εK,wM εK,q εM,wL εM,wK εM,wM εM,q
 -0.5426 0.3452 0.1973 1.087 0.3356 -0.5937 0.2581 1.087 0.2519 0.3389 -0.5909 1.087

VI. The Diewert cost function as obtained from the restricted factor demand equations is:

C(q;wL,wK,wM) = q^(1/nu) * [cLL * wL + cKK * wK + cMM * wM + (dLK+dKL) * (wL*wK)^(1/2) + (dLM +dML) * (wL*wM)^(1/2) + (dKM+dMK) * (wK*wM)^(1/2)]

= q^(1/0.92) * [-0.09246 * wL + -0.11269 * wK + -0.18035 * wM + 1.10019 * (wL*wK)^(1/2) + 0.92566 * (wL*wM)^(1/2) + 0.91382 * (wK*wM)^(1/2)]
(***)

VII. Note:
1. C(q;wL,wK,wM) linear homogeneous in factor prices requires for any t > 0 that the unit cost function c(wL,wK,wM) obey:

c(t*wL,t*wK,t*wM) = t * c(wL,wK,wM).

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

2 * c(7, 13, 6) = 42.74 = 42.74 = c(14, 26,12).

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

 ∂2C/∂q∂wL = h'(q) * ∂c/∂wL = ∂2C/∂wL∂q, ∂2C/∂q∂wK = h'(q) * ∂c/∂wK = ∂2C/∂wK∂q, ∂2C/∂q∂wM = h'(q) * ∂c/∂wM = ∂2C/∂wM∂q, ∂2c/∂wL∂wK = .25 * (dLK+dKL)/(wL*wK)^(1/2) = ∂2c/∂wK∂wL, ∂2c/∂wL∂wM = .25 * (dLM+dML)/(wL*wM)^(1/2) = ∂2c/∂wM∂wL, ∂2c/∂wK∂wM = .25 * (dKM+dMK)/(wL*wK)^(1/2) = ∂2c/∂wM∂wK, and ∂2c/∂wL∂wL = -.25((dLK+dKL)*wK^(1/2) + (dLM+dML)*wM^(1/2))/wL^(3/2), ∂2c/∂wK∂wK = -.25((dKL+dLK)*wL^(1/2) + (dKM+dMK)*wM^(1/2))/wK^(3/2), ∂2c/∂wM∂wM = -.25((dML+dLM)*wL^(1/2) + (dMK+dKM)*wK^(1/2))/wM^(3/2).

3. The cost function C(q;wL,wK,wM) is a concave in factor prices if the Hessian matrix H = ∇2wwc(wL,wK,wM) of second order partial derivatives with respect to factor prices is negative semidefinite.   The values of the second order partial derivatives depend on factor prices. As an example, consider the case where wL = 7, wK = 13, and wM = 6:

H = ∇2wwc =
 -0.08415 0.02883 0.03571 0.02883 -0.02746 0.02587 0.03571 0.02587 -0.09771

The principal minors of H are H1 = -0.084154, H2 = 0.00148, 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 wL = 7, wK = 13, and wM = 6.

The eigenvalues of H are e1 = -0.1273, e2 = -0.082, and e3 = -0.
H3 = e1 * e2 * e3 = -0.

VIII. The factor share functions are:

 sL(q;wL,wK,wM) = wL * L(q; wL, wK, wM) / C(q;wL,wK,wM) = wL * ∂C/∂wL / C(q;wL,wK,wM) , sK(q;wL,wK,wM)= wK * K(q; wL, wK, wM) / C(q;wL,wK,wM) = wK * ∂C/∂wK / C(q;wL,wK,wM), sM(q;wL,wK,wM)= wM * M(q; wL, wK, wM) / C(q;wL,wK,wM) = wM * ∂C/∂wM / C(q;wL,wK,wM).
IX. 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) * M(q;wL,wK,wM)),

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

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

as seen in the following table.

X. Table of Results: check that the estimated Diewert 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-Diewert Cost Function
À1: Estimate the Translog cost function using restricted factor shares
Parameters: rho = 0.17647, rhoL = 0.17647, rhoK = 0.11823, rhoM = 0.26339
nu = 1
—   Generalized CES Cost Data   —
nu1 = 0.92
—   Diewert Cost Data   —
Factor SharesUzawa Elasticities2wwc(w)
obs #qwLwKwM LK MsLKsLMsKMεLKMcostest costest Lest Kest MsLsKsMuLKuLMuKLuKMuMLuMKe1e2e3
1185.3 14.286.46 30.7711.94 21.870.90.790.830.927474.88474.4430.5812.2621.250.3420.3690.2890.930.670.9310.671 -0.158-0.0830
2198.78 13.665.42 23.4314.65 29.060.890.790.830.924563.31563.7523.515.1227.840.3660.3660.2680.980.710.980.870.710.87 -0.138-0.07-0
3205.14 11.424.54 30.0913.86 27.630.90.790.830.923438.3438.5730.1714.1326.910.3540.3680.2790.960.670.960.950.670.95 -0.176-0.1090
4215.76 11.944.04 29.4614.42 32.520.890.790.830.919473.25474.2229.7514.6431.680.3610.3690.270.990.660.990.920.660.92 -0.193-0.103-0
5225.02 11.444.08 33.0214.94 32.180.890.790.830.92467.95468.6333.2515.131.610.3560.3690.2750.980.660.980.950.660.95 -0.194-0.114-0
6238.24 12.226.06 29.5119.49 31.370.90.790.830.925671.37671.0429.4319.9630.460.3610.3640.2750.950.740.950.890.740.89 -0.119-0.075-0
7246.3 12.467 37.2219.23 28.080.90.790.830.927670.64670.1236.9919.6527.470.3480.3650.2870.910.730.910.960.730.96 -0.126-0.0780
8255.18 14.645.04 42.416.03 35.230.890.790.830.917631.89632.4742.6315.9635.310.3490.3690.2810.970.630.970.990.630.99 -0.174-0.1030
9266.62 13.766.46 40.0319.86 33.310.890.790.830.922753.48753.8739.9620.0932.960.3510.3670.2820.930.70.930.950.70.95 -0.127-0.081-0
10276.64 11.667.4 40.7223.51 30.470.90.790.830.927769.97769.3940.4623.9729.910.3490.3630.2880.890.750.890.950.750.95 -0.117-0.0760
11287.8 12.685.68 37.5423.12 39.550.890.790.830.92810.62811.237.5623.3739.080.3610.3650.2740.960.720.960.890.720.89 -0.13-0.0780
12295.6 12.55.76 46.6821.91 36.870.890.790.830.92747.61748.1946.5921.9836.890.3490.3670.2840.930.690.930.970.690.97 -0.149-0.091-0
13307 136 43.7924.14 40.130.890.790.830.92861.17861.7543.7824.2540.010.3560.3660.2790.940.710.940.930.710.93 -0.127-0.082-0
14316.44 14.946.6 50.823.13 400.890.790.830.919936.67937.3550.7323.0740.290.3490.3680.2840.940.680.940.970.680.97 -0.131-0.080
15326.48 11.747.8 50.3128.56 34.90.90.790.830.925933.63933.9849.9928.9134.690.3470.3630.290.880.760.880.960.760.96 -0.119-0.073-0
16337.26 13.146.4 48.4727.49 43.140.890.790.830.919989.16989.7548.427.5143.260.3550.3650.280.940.720.940.930.720.93 -0.12-0.0790
17346 13.045.92 54.5126.41 44.020.890.790.830.918932.09932.5954.4426.2644.530.350.3670.2830.940.690.940.960.690.96 -0.141-0.0880
18358.06 14.94.1 46.3325.72 63.910.890.790.830.9091018.631018.2846.9925.3563.870.3720.3710.2571.020.651.020.880.650.88 -0.2-0.074-0
19366.22 11.85.06 53.1829.06 50.110.890.790.830.916927.2927.3253.2328.850.660.3570.3670.2760.950.70.950.920.70.92 -0.15-0.0930
20376.46 126.06 56.1331.34 47.080.890.790.830.9181024.011024.5555.9931.1847.630.3530.3650.2820.930.720.930.940.720.94 -0.13-0.086-0
21387.92 11.124.76 47.8934.07 57.770.890.790.830.9151033.121033.0147.8733.9258.140.3670.3650.2680.970.730.970.860.730.86 -0.155-0.079-0
22396.42 14.145.6 61.9229.84 54.640.890.790.830.9131125.451124.562.0129.2655.840.3540.3680.2780.960.670.960.950.670.95 -0.142-0.0870
23407.96 12.646.16 55.6635.74 54.390.890.790.830.9171229.831229.8855.5635.4455.140.360.3640.2760.940.730.940.90.730.9 -0.119-0.076-0
24418.96 14.187.9 58.7537.95 51.60.890.790.830.9191472.141472.8658.5637.6952.360.3560.3630.2810.920.750.920.910.750.91 -0.094-0.066-0
25426.92 13.34.08 59.132.09 71.330.890.790.830.9081126.821123.6859.6231.2372.510.3670.370.2631.010.661.010.890.660.89 -0.194-0.0860
26437.48 11.424.86 57.2938.16 64.170.890.790.830.9131176.131174.9757.2737.6165.260.3650.3650.270.970.720.970.880.720.88 -0.152-0.082-0
27446.78 11.94.16 60.8235.87 70.990.890.790.830.911134.521131.7761.0835.0272.350.3660.3680.2660.990.680.990.890.680.89 -0.185-0.088-0
28458.5 12.427.94 64.1544.71 53.50.890.790.830.921525.351526.6963.944.3454.510.3560.3610.2830.90.770.90.910.770.91 -0.094-0.070
29468.46 14.146.2 65.1840.31 65.790.890.790.830.9131529.391527.2665.1339.4967.40.3610.3660.2740.960.710.960.90.710.9 -0.119-0.071-0
30477.2 11.125.04 64.3842.7 67.440.890.790.830.9131278.321276.664.2741.8869.070.3620.3650.2730.960.730.960.880.730.88 -0.145-0.0850
31488.88 11.186.82 62.8349.96 61.260.890.790.830.9181534.31534.762.5349.3662.690.3620.360.2790.920.780.920.870.780.87 -0.103-0.075-0
AVE:336.96 12.735.79 47.8127.32 45.860.890.790.830.918943.91943.7947.8127.3245.860.3570.3660.2770.950.710.950.920.710.92-0.143-0.083-0

À2:   Estimate the Diewert cost function directly.

XI. 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 do not have data on the required levels of inputs, we can estimate the Diewert cost function directly. We will use the same sequence (displayed in the "Diewert 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.

QR Least Squares
Parameter Estimates
Parameter Coefficient std error t-ratio
cLL-0.0192640.074-0.261
cKK-0.1474760.037-4.016
cMM-0.1129270.053-2.128
2*dLK1.1385370.07315.566
2*dLM0.7157660.1126.39
2*dKM0.973350.07513.056
 R2 = 0.9999 R2b = 0.9999 # obs = 31

The Diewert cost function as obtained by direct estimation is:

C(q;wL,wK,wM) = q^(1/nu1) * [cLL * wL + cKK * wK + cMM * wM + 2 * dLK * (wL*wK)^(1/2) + 2 * dLM * (wL*wM)^(1/2) + 2 * dKM * (wK*wM)^(1/2)]

= q^(1/0.92) * [-0.01926 * wL + -0.14748 * wK + -0.11293 * wM + 1.13854 * (wL*wK)^(1/2) + 0.71577 * (wL*wM)^(1/2) + 0.97335 * (wK*wM)^(1/2)]
(***)

XII. Its three derived factor demand functions are:

 ∂C/∂wL = L(q; wL, wK, wM) = q^(1/0.92) * [-0.0193 + 0.5693 * (wK / wL)^(1/2) + 0.3579 * (wM / wL)^(1/2)] ∂C/∂wK = K(q; wL, wK, wM) = q^(1/0.92) * [-0.1475 + 0.5693 * (wL / wK)^(1/2) + 0.4867 * (wM / wK)^(1/2)] ∂C/∂wM = M(q; wL, wK, wM) = q^(1/0.92) * [-0.1129 + 0.3579 * (wL / wM)^(1/2) + 0.4867 * (wK / wM)^(1/2)]

The derived factor demand elasticities are obtained by:

 εL,wL = ∂ln(L(q;wL,wK,wM))/∂ln(wL) = -.5 * (q^1/nu1) / L(q;wL,wK,wM)) * (dLK * (wK/wL)^1/2 + dLM * (wM/wL)^1/2), εL,wK = ∂ln(L(q;wL,wK,wM))/∂ln(wK) = .5 * (q^1/nu1) / L(q;wL,wK,wM)) * dLK * (wK/wL)^1/2, εL,wM = ∂ln(L(q;wL,wK,wM))/∂ln(wM) = .5 * (q^1/nu1) / L(q;wL,wK,wM)) * dLM * (wM/wL)^1/2, εL,q = ∂ln(L(q;wL,wK,wM))/∂ln(q) = 1/nu, etc.

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

 εL,wL εL,wK εL,wM εL,q εK,wL εK,wK εK,wM εK,q εM,wL εM,wK εM,wM εM,q
 -0.5089 0.3566 0.1523 1.087 0.3476 -0.6227 0.2751 1.087 0.1952 0.3618 -0.557 1.087

XIII. 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 requires for any t > 0 that the unit cost function c(wL,wK,wM) obey:

c(t*wL,t*wK,t*wM) = t * c(wL,wK,wM).

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

2 * c(7, 13, 6) = 42.73 = 42.73 = c(14, 26,12).

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

4. The cost function C(q;wL,wK,wM) is a concave in factor prices if the Hessian matrix H = ∇2wwc(wL,wK,wM) of second order partial derivatives with respect to factor prices is negative semidefinite.   The values of the second order partial derivatives depend on factor prices. As an example, consider the case where wL = 7, wK = 13, and wM = 6:

H = ∇2wwc =
 -0.07908 0.02984 0.02761 0.02984 -0.02878 0.02755 0.02761 0.02755 -0.09191

The principal minors of H are H1 = -0.07908, H2 = 0.001386, 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 wL = 7, wK = 13, and wM = 6.

The eigenvalues of H are e1 = -0.114, e2 = -0.0858, and e3 = 0.
H3 = e1 * e2 * e3 = 0.

XIV. Table of Results: check that the estimated Diewert 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-Diewert Cost Function
À2: Estimate the Diewert cost function directly
Parameters: rho = 0.17647, rhoL = 0.17647, rhoK = 0.11823, rhoM = 0.26339
nu = 1
—   Generalized CES Cost Data   —
nu1 = 0.92
—   Diewert Cost Data   —
Factor SharesUzawa Elasticities2wwc(w)
obs #qwLwKwM LK MsLKsLMsKMεLKMcostest costest Lest Kest MsLsKsMuLKuLMuKLuKMuMLuMKe1e2e3
1185.3 14.286.46 30.7711.94 21.870.90.790.830.927474.88474.5430.3212.1921.630.3390.3670.2950.970.510.971.060.511.06 -0.144-0.0860
2198.78 13.665.42 23.4314.65 29.060.890.790.830.924563.31564.1723.8615.1127.370.3710.3660.26310.5510.950.550.95 -0.127-0.0710
3205.14 11.424.54 30.0913.86 27.630.90.790.830.923438.3438.4130.2514.0526.980.3550.3660.2790.990.520.991.010.521.01 -0.159-0.114-0
4215.76 11.944.04 29.4614.42 32.520.890.790.830.919473.25474.1530.114.5331.50.3660.3660.2681.020.511.020.990.510.99 -0.176-0.106-0
5225.02 11.444.08 33.0214.94 32.180.890.790.830.92467.95468.433.4714.9831.630.3590.3660.2761.010.51.011.010.51.01 -0.176-0.119-0
6238.24 12.226.06 29.5119.49 31.370.90.790.830.925671.37671.0529.6320.0230.070.3640.3650.2720.970.580.970.950.580.95 -0.108-0.077-0
7246.3 12.467 37.2219.23 28.080.90.790.830.927670.64670.1736.6619.6827.710.3450.3660.2890.940.560.941.010.561.01 -0.114-0.0810
8255.18 14.645.04 42.416.03 35.230.890.790.830.917631.89631.942.6915.7735.70.350.3650.2851.010.481.011.050.481.05 -0.158-0.108-0
9266.62 13.766.46 40.0319.86 33.310.890.790.830.922753.48753.7439.8720.0533.120.350.3660.2840.970.540.971.010.541.01 -0.114-0.084-0
10276.64 11.667.4 40.7223.51 30.470.90.790.830.927769.97769.3340.0224.0930.10.3450.3650.290.930.590.9310.591 -0.106-0.079-0
11287.8 12.685.68 37.5423.12 39.550.890.790.830.92810.62811.2837.8623.3738.670.3640.3650.2710.980.560.980.960.560.96 -0.118-0.080
12295.6 12.55.76 46.6821.91 36.870.890.790.830.92747.61748.0946.4121.9237.190.3470.3660.2860.970.530.971.020.531.02 -0.134-0.096-0
13307 136 43.7924.14 40.130.890.790.830.92861.17861.5943.8724.2339.920.3560.3660.2780.980.550.980.990.550.99 -0.114-0.0860
14316.44 14.946.6 50.823.13 400.890.790.830.919936.67937.1650.5722.9740.650.3470.3660.2860.980.520.981.030.521.03 -0.118-0.0830
15326.48 11.747.8 50.3128.56 34.90.90.790.830.925933.63934.1449.2929.0735.050.3420.3650.2930.920.590.921.010.591.01 -0.109-0.0750
16337.26 13.146.4 48.4727.49 43.140.890.790.830.919989.16989.5548.4227.5243.190.3550.3650.2790.970.560.970.990.560.99 -0.107-0.0820
17346 13.045.92 54.5126.41 44.020.890.790.830.918932.09932.454.3126.1844.80.3490.3660.2840.980.530.981.020.531.02 -0.126-0.0920
18358.06 14.94.1 46.3325.72 63.910.890.790.830.9091018.631019.5748.1625.162.780.3810.3670.2521.040.51.040.960.50.96 -0.186-0.075-0
19366.22 11.85.06 53.1829.06 50.110.890.790.830.916927.2927.1753.4728.7450.490.3590.3660.2760.990.540.990.990.540.99 -0.135-0.0970
20376.46 126.06 56.1331.34 47.080.890.790.830.9181024.011024.3455.8831.247.680.3520.3660.2820.960.560.9610.561 -0.116-0.09-0
21387.92 11.124.76 47.8934.07 57.770.890.790.830.9151033.121033.9548.6333.9656.960.3730.3650.2620.990.570.990.930.570.93 -0.142-0.08-0
22396.42 14.145.6 61.9229.84 54.640.890.790.830.9131125.451124.0862.229.0955.970.3550.3660.27910.5211.010.521.01 -0.128-0.091-0
23407.96 12.646.16 55.6635.74 54.390.890.790.830.9171229.831229.7955.8435.554.630.3610.3650.2740.970.570.970.960.570.96 -0.107-0.079-0
24418.96 14.187.9 58.7537.95 51.60.890.790.830.9191472.141472.3958.4937.8452.110.3560.3640.280.950.580.950.970.580.97 -0.084-0.069-0
25426.92 13.34.08 59.132.09 71.330.890.790.830.9081126.821124.2660.7330.9771.610.3740.3660.261.030.511.030.970.510.97 -0.18-0.088-0
26437.48 11.424.86 57.2938.16 64.170.890.790.830.9131176.131175.595837.6164.230.3690.3650.2660.990.560.990.950.560.95 -0.139-0.0840
27446.78 11.94.16 60.8235.87 70.990.890.790.830.911134.521132.4762.0834.8571.360.3720.3660.2621.010.531.010.960.530.96 -0.17-0.09-0
28458.5 12.427.94 64.1544.71 53.50.890.790.830.921525.351525.8363.5844.6554.260.3540.3630.2820.930.60.930.970.60.97 -0.083-0.0740
29468.46 14.146.2 65.1840.31 65.790.890.790.830.9131529.391527.3965.6539.4766.740.3640.3650.2710.980.550.980.970.550.97 -0.108-0.073-0
30477.2 11.125.04 64.3842.7 67.440.890.790.830.9131278.321276.8964.8841.9368.170.3660.3650.2690.980.560.980.950.560.95 -0.132-0.087-0
31488.88 11.186.82 62.8349.96 61.260.890.790.830.9181534.31533.9962.7249.7361.740.3630.3620.2740.940.610.940.940.610.94 -0.093-0.0760
AVE:336.96 12.735.79 4827.3 45.610.890.790.830.918943.91943.84827.345.610.3580.3650.2760.980.550.980.990.550.99-0.129-0.086-0

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).

À3:   Estimate the factor demand equations separately, with nu1 = 1.

XVI. For these coefficients of the CES Generalized production function, I generated a sequence (displayed in the "Generalized CES-Diewert 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
cLL0.0925870.0651.422
dLK0.5569550.0511.05
dLM0.6636150.06610.114
 R2 = 0.9412 R2b = 0.937 # obs = 31

QR Least Squares
Parameter Estimates
Parameter Coefficient std error t-ratio
cKK-0.2660590.066-4.004
dKL0.9759670.08611.365
dKM0.5325010.0856.301
 R2 = 0.9055 R2b = 0.8988 # obs = 31

QR Least Squares
Parameter Estimates
Parameter Coefficient std error t-ratio
cMM-0.2608480.013-19.969
dML0.7413690.01355.62
dMK0.5431140.0152.281
 R2 = 0.9982 R2b = 0.9981 # obs = 31

The three estimated factor demand functions are:

 ∂C/∂wL = L(q; wL, wK, wM) = q^(1/1) * [0.0926 + 0.557 * (wK / wL)^(1/2) + 0.6636 * (wM / wL)^(1/2)] ∂C/∂wK = K(q; wL, wK, wM) = q^(1/1) * [-0.2661 + 0.976 * (wL / wK)^(1/2) + 0.5325 * (wM / wK)^(1/2)] ∂C/∂wM = M(q; wL, wK, wM) = q^(1/1) * [-0.2608 + 0.7414 * (wL / wM)^(1/2) + 0.5431 * (wK / wM)^(1/2)]

As estimated, generally dLK != dKL, dLM != dML, and dKM != dMK: Young's Theorem doesn't hold without constraints across equations. Consequently, I use the sum of the cross parameters in the Diewert cost function.

The estimated factor demand elasticities are obtained by:

 εL,wL = ∂ln(L(q;wL,wK,wM))/∂ln(wL) = -.5 * (q^1/nu1) / L(q;wL,wK,wM)) * (dLK * (wK/wL)^1/2 + dLM * (wM/wL)^1/2), εL,wK = ∂ln(L(q;wL,wK,wM))/∂ln(wK) = .5 * (q^1/nu1) / L(q;wL,wK,wM)) * dLK * (wK/wL)^1/2, εL,wM = ∂ln(L(q;wL,wK,wM))/∂ln(wM) = .5 * (q^1/nu1) / L(q;wL,wK,wM)) * dLM * (wM/wL)^1/2, εL,q = ∂ln(L(q;wL,wK,wM))/∂ln(q) = 1/nu1, etc.

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

 εL,wL εL,wK εL,wM εL,q εK,wL εK,wK εK,wM εK,q εM,wL εM,wK εM,wM εM,q
 -0.4684 0.2589 0.2095 1 0.4411 -0.6639 0.2228 1 0.2989 0.2984 -0.5974 1

XVII. The Diewert cost function as obtained from the unrestricted factor demand equations is:

C(q;wL,wK,wM) = q^(1/nu1) * [cLL * wL + cKK * wK + cMM * wM + (dLK+dKL) * (wL*wK)^(1/2) + (dLM +dML) * (wL*wM)^(1/2) + (dKM+dMK) * (wK*wM)^(1/2)]

= q^(1/1) * [0.09259 * wL + -0.26606 * wK + -0.26085 * wM + 1.53292 * (wL*wK)^(1/2) + 1.40498 * (wL*wM)^(1/2) + 1.07562 * (wK*wM)^(1/2)]
(***)

XVIII. The Restricted Factor Demand Equations.

Having obtained the unrestricted estimates of the coefficients of the factor demand 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
cLL-0.0244990.047618-0.514489
dLK0.6433440.0330919.442082
dLM0.6626820.03426919.337539
cKK-0.0624520.033051-1.8896
dKL0.6433440.0330919.442082
dKM0.5952190.03028119.656715
cMM-0.2520930.044505-5.664385
dML0.6626820.03426919.337539
dMK0.5952190.03028119.656715
 R2 = 0.9939 R2b = 0.9933 # obs = 93

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

The three estimated, restricted factor demand functions are:

 ∂C/∂wL = L(q; wL, wK, wM) = q^(1/1) * [-0.0245 + 0.6433 * (wK / wL)^(1/2) + 0.6627 * (wM / wL)^(1/2)] ∂C/∂wK = K(q; wL, wK, wM) = q^(1/1) * [-0.0625 + 0.6433 * (wL / wK)^(1/2) + 0.5952 * (wM / wK)^(1/2)] ∂C/∂wM = M(q; wL, wK, wM) = q^(1/1) * [-0.2521 + 0.6627 * (wL / wM)^(1/2) + 0.5952 * (wK / wM)^(1/2)]

The estimated factor demand elasticities are obtained by:

 εL,wL = ∂ln(L(q;wL,wK,wM))/∂ln(wL) = -.5 * (q^1/nu1) / L(q;wL,wK,wM)) * (dLK * (wK/wL)^1/2 + dLM * (wM/wL)^1/2), εL,wK = ∂ln(L(q;wL,wK,wM))/∂ln(wK) = .5 * (q^1/nu1) / L(q;wL,wK,wM)) * dLK * (wK/wL)^1/2, εL,wM = ∂ln(L(q;wL,wK,wM))/∂ln(wM) = .5 * (q^1/nu1) / L(q;wL,wK,wM)) * dLM * (wM/wL)^1/2, εL,wq = ∂ln(L(q;wL,wK,wM))/∂ln(q) = 1/nu, etc.

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

 εL,wL εL,wK εL,wM εL,q εK,wL εK,wK εK,wM εK,q εM,wL εM,wK εM,wM εM,q
 -0.5084 0.2991 0.2093 1 0.29 -0.5384 0.2484 1 0.2671 0.327 -0.5941 1

XIX. The Diewert cost function as obtained from the restricted factor demand equations is:

C(q;wL,wK,wM) = q^(1/nu) * [cLL * wL + cKK * wK + cMM * wM + (dLK+dKL) * (wL*wK)^(1/2) + (dLM +dML) * (wL*wM)^(1/2) + (dKM+dMK) * (wK*wM)^(1/2)]

= q^(1/1) * [-0.0245 * wL + -0.06245 * wK + -0.25209 * wM + 1.28669 * (wL*wK)^(1/2) + 1.32536 * (wL*wM)^(1/2) + 1.19044 * (wK*wM)^(1/2)]
(***)

XX. Note:
1. C(q;wL,wK,wM) linear homogeneous in factor prices requires for any t > 0 that the unit cost function c(wL,wK,wM) obey:

c(t*wL,t*wK,t*wM) = t * c(wL,wK,wM).

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

2 * c(7, 13, 6) = 57.76 = 57.76 = c(14, 26,12).

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

 ∂2C/∂q∂wL = h'(q) * ∂c/∂wL = ∂2C/∂wL∂q, ∂2C/∂q∂wK = h'(q) * ∂c/∂wK = ∂2C/∂wK∂q, ∂2C/∂q∂wM = h'(q) * ∂c/∂wM = ∂2C/∂wM∂q, ∂2c/∂wL∂wK = .25 * (dLK+dKL)/(wL*wK)^(1/2) = ∂2c/∂wK∂wL, ∂2c/∂wL∂wM = .25 * (dLM+dML)/(wL*wM)^(1/2) = ∂2c/∂wM∂wL, ∂2c/∂wK∂wM = .25 * (dKM+dMK)/(wL*wK)^(1/2) = ∂2c/∂wM∂wK, and ∂2c/∂wL∂wL = -.25((dLK+dKL)*wK^(1/2) + (dLM+dML)*wM^(1/2))/wL^(3/2), ∂2c/∂wK∂wK = -.25((dKL+dLK)*wL^(1/2) + (dKM+dMK)*wM^(1/2))/wK^(3/2), ∂2c/∂wM∂wM = -.25((dML+dLM)*wL^(1/2) + (dMK+dKM)*wK^(1/2))/wM^(3/2).

3. The cost function C(q;wL,wK,wM) is a concave in factor prices if the Hessian matrix H = ∇2wwc(wL,wK,wM) of second order partial derivatives with respect to factor prices is negative semidefinite.   The values of the second order partial derivatives depend on factor prices. As an example, consider the case where wL = 7, wK = 13, and wM = 6:

H = ∇2wwc =
 -0.10645 0.03372 0.05113 0.03372 -0.03371 0.0337 0.05113 0.0337 -0.13266

The principal minors of H are H1 = -0.106447, H2 = 0.002451, 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 wL = 7, wK = 13, and wM = 6.

The eigenvalues of H are e1 = -0.1726, e2 = -0.1002, and e3 = 0.
H3 = e1 * e2 * e3 = 0.

XXI. The factor share functions are:

 sL(q;wL,wK,wM) = wL * L(q; wL, wK, wM) / C(q;wL,wK,wM) = wL * ∂C/∂wL / C(q;wL,wK,wM) , sK(q;wL,wK,wM)= wK * K(q; wL, wK, wM) / C(q;wL,wK,wM) = wK * ∂C/∂wK / C(q;wL,wK,wM), sM(q;wL,wK,wM)= wM * M(q; wL, wK, wM) / C(q;wL,wK,wM) = wM * ∂C/∂wM / C(q;wL,wK,wM).
XXII. 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) * M(q;wL,wK,wM)),

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

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

as seen in the following table.

XXIII. Table of Results: check that the estimated Diewert 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-Diewert Cost Function
À3: Estimate the Translog cost function using restricted factor shares
Parameters: rho = 0.17647, rhoL = 0.17647, rhoK = 0.11823, rhoM = 0.26339
nu = 1
—   Generalized CES Cost Data   —
nu1 = 1
—   Diewert Cost Data   —
Factor SharesUzawa Elasticities2wwc(w)
obs #qwLwKwM LK MsLKsLMsKMεLKMcostest costest Lest Kest MsLsKsMuLKuLMuKLuKMuMLuMKe1e2e3
1185.3 14.286.46 30.7711.94 21.870.90.790.830.927474.88499.1831.7413.1422.20.3370.3760.2870.80.720.80.950.720.95 -0.206-0.106-0
2198.78 13.665.42 23.4314.65 29.060.890.790.830.924563.31589.8124.6715.7429.190.3670.3640.2680.850.750.850.840.750.84 -0.189-0.084-0
3205.14 11.424.54 30.0913.86 27.630.90.790.830.923438.3456.9731.1514.8927.940.350.3720.2780.830.720.830.910.720.91 -0.238-0.133-0
4215.76 11.944.04 29.4614.42 32.520.890.790.830.919473.25491.9630.5915.3432.810.3580.3720.2690.850.710.850.880.710.88 -0.262-0.124-0
5225.02 11.444.08 33.0214.94 32.180.890.790.830.92467.95484.3533.9715.8232.550.3520.3740.2740.840.710.840.90.710.9 -0.263-0.139-0
6238.24 12.226.06 29.5119.49 31.370.90.790.830.925671.37690.6530.5320.3531.420.3640.360.2760.820.780.820.860.780.86 -0.163-0.0910
7246.3 12.467 37.2219.23 28.080.90.790.830.927670.64686.9337.8920.1928.10.3480.3660.2860.780.770.780.930.770.93 -0.167-0.098-0
8255.18 14.645.04 42.416.03 35.230.890.790.830.917631.89647.2742.7716.7435.850.3420.3790.2790.840.680.840.930.680.93 -0.232-0.128-0
9266.62 13.766.46 40.0319.86 33.310.890.790.830.922753.48767.5540.520.5833.470.3490.3690.2820.810.750.810.910.750.91 -0.17-0.1-0
10276.64 11.667.4 40.7223.51 30.470.90.790.830.927769.97780.6641.2524.2230.320.3510.3620.2870.770.80.770.920.80.92 -0.155-0.096-0
11287.8 12.685.68 37.5423.12 39.550.890.790.830.92810.62820.5638.1223.5339.590.3620.3640.2740.830.760.830.860.760.86 -0.177-0.094-0
12295.6 12.55.76 46.6821.91 36.870.890.790.830.92747.61754.6946.6522.3937.070.3460.3710.2830.810.740.810.920.740.92 -0.198-0.114-0
13307 136 43.7924.14 40.130.890.790.830.92861.17866.4443.9724.4240.190.3550.3660.2780.820.750.820.890.750.89 -0.173-0.10
14316.44 14.946.6 50.823.13 400.890.790.830.919936.67940.1750.4123.4240.240.3450.3720.2820.810.730.810.930.730.93 -0.174-0.0990
15326.48 11.747.8 50.3128.56 34.90.90.790.830.925933.63933.7250.1928.8234.630.3480.3620.2890.760.80.760.930.80.93 -0.156-0.0930
16337.26 13.146.4 48.4727.49 43.140.890.790.830.919989.16986.9148.2927.4343.120.3550.3650.280.810.760.810.890.760.89 -0.162-0.0960
17346 13.045.92 54.5126.41 44.020.890.790.830.918932.09927.7253.7926.3544.150.3480.370.2820.810.740.810.920.740.92 -0.188-0.109-0
18358.06 14.94.1 46.3325.72 63.910.890.790.830.9091018.631010.1846.325.363.410.3690.3730.2570.890.690.890.840.690.84 -0.273-0.089-0
19366.22 11.85.06 53.1829.06 50.110.890.790.830.916927.2917.7352.5428.650.10.3560.3680.2760.830.740.830.890.740.89 -0.204-0.114-0
20376.46 126.06 56.1331.34 47.080.890.790.830.9181024.011011.4955.2830.846.980.3530.3650.2810.80.760.80.90.760.9 -0.175-0.106-0
21387.92 11.124.76 47.8934.07 57.770.890.790.830.9151033.121017.8447.5633.0657.470.370.3610.2690.840.760.840.830.760.83 -0.212-0.096-0
22396.42 14.145.6 61.9229.84 54.640.890.790.830.9131125.451105.5460.4229.0854.730.3510.3720.2770.830.720.830.910.720.91 -0.192-0.107-0
23407.96 12.646.16 55.6635.74 54.390.890.790.830.9171229.831206.1754.7734.5454.150.3610.3620.2770.820.770.820.870.770.87 -0.162-0.093-0
24418.96 14.187.9 58.7537.95 51.60.890.790.830.9191472.141441.4457.6936.6251.290.3590.360.2810.80.790.80.880.790.88 -0.128-0.081-0
25426.92 13.34.08 59.132.09 71.330.890.790.830.9081126.821097.3357.830.7170.80.3650.3720.2630.870.70.870.860.70.86 -0.265-0.104-0
26437.48 11.424.86 57.2938.16 64.170.890.790.830.9131176.131145.156.136.463.750.3660.3630.2710.840.760.840.850.760.85 -0.208-0.1-0
27446.78 11.94.16 60.8235.87 70.990.890.790.830.911134.521100.6259.2634.170.430.3650.3690.2660.860.720.860.850.720.85 -0.252-0.107-0
28458.5 12.427.94 64.1544.71 53.50.890.790.830.921525.351482.5162.7142.5653.010.360.3570.2840.780.810.780.890.810.89 -0.127-0.087-0
29468.46 14.146.2 65.1840.31 65.790.890.790.830.9131529.391479.5763.2338.1565.360.3620.3650.2740.830.750.830.870.750.87 -0.163-0.0860
30477.2 11.125.04 64.3842.7 67.440.890.790.830.9131278.321234.5862.4840.2366.930.3640.3620.2730.830.760.830.860.760.86 -0.198-0.103-0
31488.88 11.186.82 62.8349.96 61.260.890.790.830.9181534.31482.9361.3546.8460.780.3670.3530.280.80.810.80.850.810.85 -0.141-0.09-0
AVE:336.96 12.735.79 47.5527.11 45.550.890.790.830.918943.91937.3747.5527.1145.550.3560.3670.2770.820.750.820.890.750.89-0.193-0.102-0

À4:   Estimate the Diewert cost function directly, with nu1 = 1.

XXIV. 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 do not have data on the required levels of inputs, we can estimate the Diewert cost function directly. We will use the same sequence (displayed in the "Diewert 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.

Under these circumstances, with nu1 = 1, the Diewert cost function may fail to be concave in factor prices, depending on the generated sequence of random factor prices, as one or more of the eigenvalues of the Hessian matrix H = ∇2wwc(wL,wK,wM) turn positive.

SVD Least Squares
Parameter Estimates
Parameter Coefficient std error t-ratio
cLL-2.8531472.321-1.229
cKK0.5376231.1550.465
cMM-0.7559551.669-0.453
2*dLK2.3307382.3011.013
2*dLM6.5751973.5241.866
2*dKM-2.080962.345-0.887
 R2 = 0.9661 R2b = 0.9593 # obs = 31 Observation Matrix Rank: 6
The Diewert cost function as obtained by direct estimation is:

C(q;wL,wK,wM) = q^(1/nu1) * [cLL * wL + cKK * wK + cMM * wM + 2 * dLK * (wL*wK)^(1/2) + 2 * dLM * (wL*wM)^(1/2) + 2 * dKM * (wK*wM)^(1/2)]

= q^(1/1) * [-2.85315 * wL + 0.53762 * wK + -0.75596 * wM + 2.33074 * (wL*wK)^(1/2) + 6.5752 * (wL*wM)^(1/2) + -2.08096 * (wK*wM)^(1/2)]
(***)

XXVI. Its three derived factor demand functions are:

 ∂C/∂wL = L(q; wL, wK, wM) = q^(1/1) * [-2.8531 + 1.1654 * (wK / wL)^(1/2) + 3.2876 * (wM / wL)^(1/2)] ∂C/∂wK = K(q; wL, wK, wM) = q^(1/1) * [0.5376 + 1.1654 * (wL / wK)^(1/2) + -1.0405 * (wM / wK)^(1/2)] ∂C/∂wM = M(q; wL, wK, wM) = q^(1/1) * [-0.756 + 3.2876 * (wL / wM)^(1/2) + -1.0405 * (wK / wM)^(1/2)]

The derived factor demand elasticities are obtained by:

 εL,wL = ∂ln(L(q;wL,wK,wM))/∂ln(wL) = -.5 * (q^1/nu1) / L(q;wL,wK,wM)) * (dLK * (wK/wL)^1/2 + dLM * (wM/wL)^1/2), εL,wK = ∂ln(L(q;wL,wK,wM))/∂ln(wK) = .5 * (q^1/nu1) / L(q;wL,wK,wM)) * dLK * (wK/wL)^1/2, εL,wM = ∂ln(L(q;wL,wK,wM))/∂ln(wM) = .5 * (q^1/nu1) / L(q;wL,wK,wM)) * dLM * (wM/wL)^1/2, εL,q = ∂ln(L(q;wL,wK,wM))/∂ln(q) = 1/nu, etc.

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

 εL,wL εL,wK εL,wM εL,q εK,wL εK,wK εK,wM εK,q εM,wL εM,wK εM,wM εM,q
 -1.302 0.4464 0.8556 1 0.6234 -0.1081 -0.5153 1 1.4052 -0.6061 -0.7991 1

XXVII. 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 requires for any t > 0 that the unit cost function c(wL,wK,wM) obey:

c(t*wL,t*wK,t*wM) = t * c(wL,wK,wM).

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

2 * c(7, 13, 6) = 57.9 = 57.9 = c(14, 26,12).

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

4. The cost function C(q;wL,wK,wM) is a concave in factor prices if the Hessian matrix H = ∇2wwc(wL,wK,wM) of second order partial derivatives with respect to factor prices is negative semidefinite.   The values of the second order partial derivatives depend on factor prices. As an example, consider the case where wL = 7, wK = 13, and wM = 6:

H = ∇2wwc =
 -0.33085 0.06108 0.25364 0.06108 -0.0057 -0.05891 0.25364 -0.05891 -0.16829

The principal minors of H are H1 = -0.330847, H2 = -0.001844, 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 wL = 7, wK = 13, and wM = 6.

The eigenvalues of H are e1 = -0.5294, e2 = 0.0246, and e3 = 0.
H3 = e1 * e2 * e3 = -0.

XXVIII. Table of Results: check that the estimated Diewert 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-Diewert Cost Function
À4: Estimate the Diewert cost function directly
Parameters: rho = 0.17647, rhoL = 0.17647, rhoK = 0.11823, rhoM = 0.26339
nu = 1
—   Generalized CES Cost Data   —
nu1 = 1
—   Diewert Cost Data   —
Factor SharesUzawa Elasticities2wwc(w)
obs #qwLwKwM LK MsLKsLMsKMεLKMcostest costest Lest Kest MsLsKsMuLKuLMuKLuKMuMLuMKe1e2e3
1185.3 14.286.46 30.7711.94 21.870.90.790.830.927474.88475.8448.419.8612.150.5390.2960.1651.24.091.2-3.874.09-3.87 -0.6760.0420
2198.78 13.665.42 23.4314.65 29.060.890.790.830.924563.31592.322.4915.5133.750.3330.3580.3091.723.531.72-1.33.53-1.3 -0.4840.01-0
3205.14 11.424.54 30.0913.86 27.630.90.790.830.923438.3453.5739.4713.2721.840.4470.3340.2191.323.581.32-2.263.58-2.26 -0.7170.0380
4215.76 11.944.04 29.4614.42 32.520.890.790.830.919473.25494.0333.1415.58290.3860.3760.2371.413.681.41-1.723.68-1.72 -0.6860.0280
5225.02 11.444.08 33.0214.94 32.180.890.790.830.92467.95482.8241.1415.1425.270.4280.3590.2141.313.711.31-2.113.71-2.11 -0.7490.039-0
6238.24 12.226.06 29.5119.49 31.370.90.790.830.925671.37699.7131.8617.5236.80.3750.3060.3191.673.191.67-1.513.19-1.51 -0.4750.015-0
7246.3 12.467 37.2219.23 28.080.90.790.830.927670.64679.554.0314.0723.390.5010.2580.2411.413.191.41-2.763.19-2.76 -0.5620.031-0
8255.18 14.645.04 42.416.03 35.230.890.790.830.917631.89632.4858.7215.5120.090.4810.3590.161.164.311.16-3.074.31-3.07 -0.7050.046-0
9266.62 13.766.46 40.0319.86 33.310.890.790.830.922753.48760.5153.9416.4627.390.470.2980.2331.363.361.36-2.423.36-2.42 -0.5450.0290
10276.64 11.667.4 40.7223.51 30.470.90.790.830.927769.97782.9558.3715.8828.410.4950.2360.2691.512.991.51-2.622.99-2.62 -0.5310.0290
11287.8 12.685.68 37.5423.12 39.550.890.790.830.92810.62827.4940.2721.1543.180.380.3240.2961.593.291.59-1.563.29-1.56 -0.5030.017-0
12295.6 12.55.76 46.6821.91 36.870.890.790.830.92747.61741.6664.4417.7327.630.4870.2990.2151.313.51.31-2.693.5-2.69 -0.6410.0360
13307 136 43.7924.14 40.130.890.790.830.92861.17868.4653.3620.5837.910.430.3080.2621.453.271.45-1.973.27-1.97 -0.5290.0250
14316.44 14.946.6 50.823.13 400.890.790.830.919936.67921.7469.7518.9528.710.4870.3070.2061.283.61.28-2.753.6-2.75 -0.5580.032-0
15326.48 11.747.8 50.3128.56 34.90.90.790.830.925933.63930.8374.3217.7730.850.5170.2240.2591.5131.51-2.953-2.95 -0.5410.030
16337.26 13.146.4 48.4727.49 43.140.890.790.830.919989.16990.4459.4522.3641.40.4360.2970.2681.473.21.47-23.2-2 -0.5070.0240
17346 13.045.92 54.5126.41 44.020.890.790.830.918932.09915.8372.4421.3234.320.4750.3040.2221.333.451.33-2.523.45-2.52 -0.6010.0330
18358.06 14.94.1 46.3325.72 63.910.890.790.830.9091018.631014.6537.6629.7165.450.2990.4360.2641.694.121.69-1.224.12-1.22 -0.5830.0090
19366.22 11.85.06 53.1829.06 50.110.890.790.830.916927.2919.7261.8225.2946.810.4180.3240.2581.443.351.44-1.883.35-1.88 -0.6050.0270
20376.46 126.06 56.1331.34 47.080.890.790.830.9181024.011012.1171.0224.1743.450.4530.2870.261.443.191.44-2.183.19-2.18 -0.5620.028-0
21387.92 11.124.76 47.8934.07 57.770.890.790.830.9151033.121021.7440.931.9371.990.3170.3480.3351.853.531.85-1.213.53-1.21 -0.5480.0080
22396.42 14.145.6 61.9229.84 54.640.890.790.830.9131125.451098.4675.9326.0643.320.4440.3350.2211.323.571.32-2.223.57-2.22 -0.5760.03-0
23407.96 12.646.16 55.6635.74 54.390.890.790.830.9171229.831219.4660.329.4459.630.3940.3050.3011.63.181.6-1.643.18-1.64 -0.4820.0170
24418.96 14.187.9 58.7537.95 51.60.890.790.830.9191472.141461.7869.728.1855.40.4270.2730.2991.583.031.58-1.893.03-1.89 -0.4110.0180
25426.92 13.34.08 59.132.09 71.330.890.790.830.9081126.821104.2154.0533.6869.170.3390.4060.2561.553.841.55-1.413.84-1.41 -0.620.0170
26437.48 11.424.86 57.2938.16 64.170.890.790.830.9131176.131152.6953.1834.4974.290.3450.3420.3131.73.421.7-1.353.42-1.35 -0.5530.0130
27446.78 11.94.16 60.8235.87 70.990.890.790.830.911134.521105.3855.735.2973.980.3420.380.2781.63.651.6-1.383.65-1.38 -0.6240.016-0
28458.5 12.427.94 64.1544.71 53.50.890.790.830.921525.351517.5277.9830.1460.490.4370.2470.3171.652.91.65-1.962.9-1.96 -0.4270.0190
29468.46 14.146.2 65.1840.31 65.790.890.790.830.9131529.391490.6467.5234.569.60.3830.3270.2891.573.311.57-1.593.31-1.59 -0.4620.0160
30477.2 11.125.04 64.3842.7 67.440.890.790.830.9131278.321246.0163.2536.4276.510.3650.3250.3091.663.31.66-1.463.3-1.46 -0.5550.0160
31488.88 11.186.82 62.8349.96 61.260.890.790.830.9181534.31523.5464.1136.6579.840.3740.2690.3571.823.021.82-1.493.02-1.49 -0.4360.013-0
AVE:336.96 12.735.79 55.7723.37 44.90.890.790.830.918943.91939.9455.7723.3744.90.4190.3180.2631.53.431.5-2.033.43-2.03-0.5630.0240

Mathematical Notes

1. The Diewert (Generalized Leontief) cost function

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

 where the returns to scale function is: h(q) = q^(1/nu), a continuous, increasing function of q (q >= 1), with h(0) = 0 and h(1) = 1, and the unit cost function is: c(wL,wK,wM) = cLL * wL + cKK * wK + cMM * wM + (dLK+dKL) * (wL*wK)^(1/2) + (dLM +dML) * (wL*wM)^(1/2) + (dKM+dMK) * (wK*wM)^(1/2),

linear in its parameters cLL, cKK, cMM, dLK, dKL, dLM, dML, dKM, and dMK.

2. The Factor Demand Functions:

 ∂C/∂wL = L(q; wL, wK, wM) = q^(1/nu) * [cLL + dLK * (wK / wL)^(1/2) + dLM * (wM / wL)^(1/2)] ∂C/∂wK = K(q; wL, wK, wM) = q^(1/nu) * [cKK + dKL * (wL / wK)^(1/2) + dKM * (wM / wK)^(1/2)] ∂C/∂wM = M(q; wL, wK, wM) = q^(1/nu) * [cMM + dML * (wL / wM)^(1/2) + dMK * (wK / wM)^(1/2)]

3. The Factor Share Functions:

 sL(q;wL,wK,wM) = wL * L(q; wL, wK, wM) / C(q;wL,wK,wM) = wL * ∂C/∂wL / C(q;wL,wK,wM) , sK(q;wL,wK,wM)= wK * K(q; wL, wK, wM) / C(q;wL,wK,wM) = wK * ∂C/∂wK / C(q;wL,wK,wM), sM(q;wL,wK,wM)= wM * M(q; wL, wK, wM) / C(q;wL,wK,wM) = wM * ∂C/∂wM / C(q;wL,wK,wM).

4. The Factor Demand Elasticities:

 εL,wL = ∂ln(L(q;wL,wK,wM))/∂ln(wL) = -.5 * (q^1/nu) / L(q;wL,wK,wM)) * (dLK * (wK/wL)^1/2 + dLM * (wM/wL)^1/2), εL,wK = ∂ln(L(q;wL,wK,wM))/∂ln(wK) = .5 * (q^1/nu) / L(q;wL,wK,wM)) * dLK * (wK/wL)^1/2, εL,wM = ∂ln(L(q;wL,wK,wM))/∂ln(wM) = .5 * (q^1/nu) / L(q;wL,wK,wM)) * dLM * (wM/wL)^1/2, εL,q = ∂ln(L(q;wL,wK,wM))/∂ln(q) = 1/nu, etc.

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

 ∂2c/∂wL∂wL = -.25((dLK+dKL)*wK^(1/2) + (dLM+dML)*wM^(1/2))/wL^(3/2), ∂2c/∂wL∂wK = .25 * (dLK+dKL)/(wL*wK)^(1/2) = ∂2c/∂wK∂wL, ∂2c/∂wL∂wM = .25 * (dLM+dML)/(wL*wM)^(1/2) = ∂2c/∂wM∂wL, ∂2c/∂wK∂wK = -.25((dKL+dLK)*wL^(1/2) + (dKM+dMK)*wM^(1/2))/wK^(3/2), ∂2c/∂wK∂wM = .25 * (dKM+dMK)/(wL*wK)^(1/2) = ∂2c/∂wM∂wK, and ∂2c/∂wM∂wM = -.25((dML+dLM)*wL^(1/2) + (dMK+dKM)*wK^(1/2))/wM^(3/2).