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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

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-0.1123680.004-27.521
dLK0.5268970.003173.559
dLM0.5192610.003160.52
R2 = 0.9997 R2b = 0.9997 # obs = 31

QR Least Squares
Parameter Estimates
Parameter Coefficient std error t-ratio
cKK-0.0696340.023-3.089
dKL0.5675730.03118.558
dKM0.3743350.02415.885
R2 = 0.9697 R2b = 0.9675 # obs = 31

QR Least Squares
Parameter Estimates
Parameter Coefficient std error t-ratio
cMM-0.1223930.033-3.664
dML0.4800220.04311.218
dMK0.4055330.03312.311
R2 = 0.9767 R2b = 0.9751 # obs = 31

The three estimated factor demand functions are:

∂C/∂wL = L(q; wL, wK, wM) = q^(1/0.92) * [-0.1124 + 0.5269 * (wK / wL)^(1/2) + 0.5193 * (wM / wL)^(1/2)]

∂C/∂wK = K(q; wL, wK, wM) = q^(1/0.92) * [-0.0696 + 0.5676 * (wL / wK)^(1/2) + 0.3743 * (wM / wK)^(1/2)]

∂C/∂wM = M(q; wL, wK, wM) = q^(1/0.92) * [-0.1224 + 0.48 * (wL / wM)^(1/2) + 0.4055 * (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,wLεL,wKεL,wMεL,q
εK,wLεK,wKεK,wMεK,q
εM,wLεM,wKεM,wMεM,q
-0.55170.33050.22131.087
0.3464-0.55790.21151.087
0.26110.3006-0.56161.087

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) * [-0.11237 * wL + -0.06963 * wK + -0.12239 * wM + 1.09447 * (wL*wK)^(1/2) + 0.99928 * (wL*wM)^(1/2) + 0.77987 * (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-0.1136840.027194-4.180474
dLK0.5389970.01869328.833846
dLM0.5026070.01682629.870042
cKK-0.0598160.017586-3.401296
dKL0.5389970.01869328.833846
dKM0.3904790.01489526.215732
cMM-0.124250.020046-6.198186
dML0.5026070.01682629.870042
dMK0.3904790.01489526.215732
R2 = 0.9972 R2b = 0.9969 # 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) * [-0.1137 + 0.539 * (wK / wL)^(1/2) + 0.5026 * (wM / wL)^(1/2)]

∂C/∂wK = K(q; wL, wK, wM) = q^(1/0.92) * [-0.0598 + 0.539 * (wL / wK)^(1/2) + 0.3905 * (wM / wK)^(1/2)]

∂C/∂wM = M(q; wL, wK, wM) = q^(1/0.92) * [-0.1243 + 0.5026 * (wL / wM)^(1/2) + 0.3905 * (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,wLεL,wKεL,wMεL,q
εK,wLεK,wKεK,wMεK,q
εM,wLεM,wKεM,wMεM,q
-0.55230.33810.21421.087
0.3291-0.54980.22071.087
0.27320.2893-0.56251.087

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) * [-0.11368 * wL + -0.05982 * wK + -0.12425 * wM + 1.07799 * (wL*wK)^(1/2) + 1.00521 * (wL*wM)^(1/2) + 0.78096 * (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) = 42.75 = 42.75 = 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 =
-0.08570.028250.03878
0.02825-0.025420.02211
0.038780.02211-0.09314

The principal minors of H are H1 = -0.085704, H2 = 0.00138, 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 = -0.1285, e2 = -0.0758, 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
1185.72 11.124.4 24.9212.81 25.860.90.790.830.924398.74399.0124.9613.2524.750.3580.3690.2730.940.750.940.790.750.79 -0.172-0.096-0
2196.16 11.666.18 27.8514.74 23.240.90.790.830.929487.04486.0227.7715.1222.430.3520.3630.2850.90.780.90.810.780.81 -0.135-0.079-0
3208.42 13.347.78 27.316.92 24.350.90.790.830.93645.04643.4727.1917.323.610.3560.3590.2860.90.810.90.780.810.78 -0.099-0.0630
4216.24 12.545.38 30.5415.4 28.560.90.790.830.924537.33537.6330.5715.7727.730.3550.3680.2770.930.750.930.80.750.8 -0.146-0.084-0
5227.48 14.026.22 31.1516.73 29.930.90.790.830.924653.85654.1831.1617.129.160.3560.3660.2770.930.760.930.790.760.79 -0.123-0.072-0
6237.12 12.188 34.0319.93 25.610.90.790.830.93689.94688.8433.9620.225.130.3510.3570.2920.880.810.880.810.810.81 -0.112-0.065-0
7246.84 13.168 37.3319.78 27.050.90.790.830.928731.97731.4337.2620.0326.620.3480.360.2910.880.790.880.830.790.83 -0.118-0.063-0
8255.34 12.887.12 43.2618.87 27.970.90.790.830.926673.14673.2443.1219.127.660.3420.3650.2920.880.760.880.860.760.86 -0.156-0.069-0
9266.66 14.227.36 41.5420.16 31.250.90.790.830.924793.33793.7141.520.3630.950.3480.3650.2870.90.770.90.830.770.83 -0.125-0.066-0
10275.1 12.885.96 46.319.45 33.070.890.790.830.921683.7684.3646.2519.632.890.3450.3690.2860.90.740.90.850.740.85 -0.166-0.081-0
11285.1 14.525.82 49.8418.88 36.090.890.790.830.918738.37739.2749.8618.9636.030.3440.3720.2840.920.710.920.860.710.86 -0.171-0.083-0
12296.88 14.067.56 46.0423.36 34.540.90.790.830.923906.41907.164623.4634.50.3490.3640.2880.90.770.90.820.770.82 -0.12-0.0650
13307 136 43.7924.14 40.130.890.790.830.92861.17861.9843.824.2340.060.3560.3650.2790.930.770.930.790.770.79 -0.128-0.0760
14318.02 14.344.24 40.4222.96 54.60.890.790.830.912884.86886.5440.6423.2253.70.3680.3760.2570.990.730.990.740.730.74 -0.184-0.072-0
15325.32 12.166.12 53.724.85 38.440.890.790.830.92823.15824.0353.6524.8238.70.3460.3660.2870.90.760.90.840.760.84 -0.157-0.08-0
16335.48 13.744.38 52.8722.5 50.540.890.790.830.912820.26820.6453.1822.4150.520.3550.3750.270.960.70.960.820.70.82 -0.184-0.099-0
17346.38 11.84.22 47.426.33 53.030.890.790.830.914836.9837.3847.526.3452.980.3620.3710.2670.960.740.960.770.740.77 -0.178-0.089-0
18358.96 13.444.46 43.0228.68 60.680.890.790.830.9121041.561041.9842.9628.8660.360.3690.3720.2580.980.760.980.720.760.72 -0.172-0.066-0
19368.44 14.765.38 49.1428.76 56.480.890.790.830.9131143.061143.2149.1828.6956.640.3630.370.2670.960.750.960.760.750.76 -0.139-0.068-0
20376.54 14.584.8 56.5726.84 57.590.890.790.830.9111037.771037.2756.8126.657.890.3580.3740.2680.960.720.960.80.720.8 -0.162-0.085-0
21385.64 11.744.48 57.8829.24 55.270.890.790.830.913917.31917.0257.9728.9455.880.3570.370.2730.940.740.940.80.740.8 -0.173-0.0960
22398.42 11.446.9 52.1138.34 48.890.890.790.830.9211214.661216.445237.8650.080.360.3560.2840.910.830.910.760.830.76 -0.104-0.07-0
23407.08 13.784.66 57.5430.86 63.780.890.790.830.911129.81128.5557.6730.5264.320.3620.3730.2660.960.730.960.770.730.77 -0.163-0.08-0
24416.9 12.94.7 58.7532.69 63.280.890.790.830.9111124.491123.2458.7932.2864.080.3610.3710.2680.960.750.960.770.750.77 -0.16-0.0820
25428.08 12.967.96 62.0839.73 49.740.890.790.830.921412.51415.0162.0739.0551.180.3540.3580.2880.890.810.890.790.810.79 -0.101-0.064-0
26437.64 12.027.76 63.7241.55 49.750.890.790.830.921372.31375.2763.7540.7751.310.3540.3560.290.890.820.890.790.820.79 -0.105-0.067-0
27447.86 11.46.96 61.7243.21 53.840.890.790.830.9191352.51354.6961.6642.3655.620.3580.3570.2860.90.820.90.770.820.77 -0.107-0.072-0
28458.92 14.144.98 59.0938.12 74.470.890.790.830.9091436.931433.6658.9337.675.590.3670.3710.2630.970.760.970.740.760.74 -0.151-0.065-0
29465.66 13.424.48 74.5333.95 70.130.890.790.830.9081191.691187.7974.6633.171.650.3560.3740.270.950.710.950.810.710.81 -0.177-0.096-0
30477.24 14.685.86 72.7138.05 67.330.890.790.830.9111479.491476.3872.6537.1469.130.3560.3690.2740.940.740.940.80.740.8 -0.132-0.074-0
31487.1 14.687.02 78.1640.35 61.470.890.790.830.9141578.751577.678.0439.3263.570.3510.3660.2830.910.760.910.820.760.82 -0.12-0.068-0
AVE:336.89 13.155.97 49.2126.59 45.960.890.790.830.918954.77954.7449.2126.5945.960.3550.3670.2780.930.760.930.80.760.8-0.143-0.076-0




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-0.2571510.103-2.505
cKK-0.1243850.061-2.029
cMM-0.1443120.101-1.43
2*dLK1.2633310.1369.297
2*dLM1.0367570.1099.506
2*dKM0.7808970.1276.129
R2 = 0.9998 R2b = 0.9998 # 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) * [-0.25715 * wL + -0.12438 * wK + -0.14431 * wM + 1.26333 * (wL*wK)^(1/2) + 1.03676 * (wL*wM)^(1/2) + 0.7809 * (wK*wM)^(1/2)]
        (***)

XII. Its three derived factor demand functions are:

∂C/∂wL = L(q; wL, wK, wM) = q^(1/0.92) * [-0.2572 + 0.6317 * (wK / wL)^(1/2) + 0.5184 * (wM / wL)^(1/2)]

∂C/∂wK = K(q; wL, wK, wM) = q^(1/0.92) * [-0.1244 + 0.6317 * (wL / wK)^(1/2) + 0.3904 * (wM / wK)^(1/2)]

∂C/∂wM = M(q; wL, wK, wM) = q^(1/0.92) * [-0.1443 + 0.5184 * (wL / wM)^(1/2) + 0.3904 * (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,wLεL,wKεL,wMεL,q
εK,wLεK,wKεK,wMεK,q
εM,wLεM,wKεM,wMεM,q
-0.61870.39720.22151.087
0.3835-0.60290.21941.087
0.28270.2902-0.57291.087

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) = 42.77 = 42.77 = 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 =
-0.095770.033110.03999
0.03311-0.028030.0221
0.039990.0221-0.09455

The principal minors of H are H1 = -0.095767, H2 = 0.001588, 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 = -0.1359, e2 = -0.0825, 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
1185.72 11.124.4 24.9212.81 25.860.90.790.830.924398.74399.2224.9513.2924.70.3580.370.2721.10.771.10.790.770.79 -0.18-0.106-0
2196.16 11.666.18 27.8514.74 23.240.90.790.830.929487.04486.1327.7615.1922.320.3520.3640.2841.050.811.050.810.810.81 -0.146-0.0840
3208.42 13.347.78 27.316.92 24.350.90.790.830.93645.04643.326.8917.5323.520.3520.3640.2841.060.851.060.780.850.78 -0.106-0.068-0
4216.24 12.545.38 30.5415.4 28.560.90.790.830.924537.33537.8730.6415.7927.640.3550.3680.2761.090.781.090.80.780.8 -0.154-0.0910
5227.48 14.026.22 31.1516.73 29.930.90.790.830.924653.85654.4531.117.1929.080.3550.3680.2761.090.791.090.790.790.79 -0.13-0.078-0
6237.12 12.188 34.0319.93 25.610.90.790.830.93689.94688.6933.7920.3924.970.3490.3610.291.020.851.020.810.850.81 -0.122-0.068-0
7246.84 13.168 37.3319.78 27.050.90.790.830.928731.97731.4237.3220.126.440.3490.3620.2891.030.821.030.830.820.83 -0.129-0.066-0
8255.34 12.887.12 43.2618.87 27.970.90.790.830.926673.14672.9243.7418.9427.440.3470.3630.291.020.781.020.870.780.87 -0.171-0.072-0
9266.66 14.227.36 41.5420.16 31.250.90.790.830.924793.33793.841.7920.3230.770.3510.3640.2851.050.791.050.840.790.84 -0.136-0.070
10275.1 12.885.96 46.319.45 33.070.890.790.830.921683.7684.094719.3732.70.350.3650.2851.050.751.050.870.750.87 -0.182-0.0850
11285.1 14.525.82 49.8418.88 36.090.890.790.830.918738.37738.5350.9718.635.830.3520.3660.2821.070.721.070.880.720.88 -0.187-0.0880
12296.88 14.067.56 46.0423.36 34.540.90.790.830.923906.41907.2846.2223.4734.310.350.3640.2861.040.81.040.830.80.83 -0.13-0.0690
13307 136 43.7924.14 40.130.890.790.830.92861.17862.343.6924.3739.930.3550.3670.2781.080.81.080.790.80.79 -0.136-0.0820
14318.02 14.344.24 40.4222.96 54.60.890.790.830.912884.86886.9440.323.4153.770.3640.3790.2571.160.761.160.740.760.74 -0.188-0.0810
15325.32 12.166.12 53.724.85 38.440.890.790.830.92823.15823.9854.2324.6738.470.350.3640.2861.050.781.050.850.780.85 -0.171-0.084-0
16335.48 13.744.38 52.8722.5 50.540.890.790.830.912820.26820.6853.9622.1450.410.360.3710.2691.120.711.120.830.710.83 -0.193-0.108-0
17346.38 11.84.22 47.426.33 53.030.890.790.830.914836.9837.8147.2926.552.950.360.3730.2671.120.771.120.760.770.76 -0.183-0.1-0
18358.96 13.444.46 43.0228.68 60.680.890.790.830.9121041.561041.5142.0629.3860.470.3620.3790.2591.160.81.160.70.80.7 -0.175-0.075-0
19368.44 14.765.38 49.1428.76 56.480.890.790.830.9131143.061143.6548.7728.9656.620.360.3740.2661.130.781.130.750.780.75 -0.142-0.076-0
20376.54 14.584.8 56.5726.84 57.590.890.790.830.9111037.771037.7857.2426.4757.80.3610.3720.2671.120.731.120.80.730.8 -0.168-0.0940
21385.64 11.744.48 57.8829.24 55.270.890.790.830.913917.31917.558.228.9255.760.3580.370.2721.10.761.10.80.760.8 -0.181-0.105-0
22398.42 11.446.9 52.1138.34 48.890.890.790.830.9211214.661215.0450.8638.6649.930.3520.3640.2841.070.871.070.740.870.74 -0.109-0.0770
23407.08 13.784.66 57.5430.86 63.780.890.790.830.911129.81129.2357.5930.6264.280.3610.3740.2651.130.761.130.770.760.77 -0.168-0.09-0
24416.9 12.94.7 58.7532.69 63.280.890.790.830.9111124.491123.8458.5732.4664.030.360.3730.2681.120.771.120.770.770.77 -0.165-0.0910
25428.08 12.967.96 62.0839.73 49.740.890.790.830.921412.51414.6261.4639.5550.930.3510.3620.2871.040.851.040.780.850.78 -0.109-0.0680
26437.64 12.027.76 63.7241.55 49.750.890.790.830.921372.31374.763.0741.3251.050.3510.3610.2881.040.861.040.790.860.79 -0.114-0.07-0
27447.86 11.46.96 61.7243.21 53.840.890.790.830.9191352.51353.760.6243.1255.410.3520.3630.2851.060.861.060.760.860.76 -0.114-0.0770
28458.92 14.144.98 59.0938.12 74.470.890.790.830.9091436.931433.5757.9938.1675.650.3610.3760.2631.140.81.140.720.80.72 -0.154-0.074-0
29465.66 13.424.48 74.5333.95 70.130.890.790.830.9081191.691188.1375.5132.8271.50.360.3710.271.110.731.110.820.730.82 -0.186-0.105-0
30477.24 14.685.86 72.7138.05 67.330.890.790.830.9111479.491477.1372.8337.1768.970.3570.3690.2741.10.771.10.80.770.8 -0.138-0.0810
31487.1 14.687.02 78.1640.35 61.470.890.790.830.9141578.751578.0778.4139.3163.290.3530.3660.2821.060.781.060.820.780.82 -0.129-0.0720
AVE:336.89 13.155.97 49.1926.72 45.840.890.790.830.918954.77954.7749.1926.7245.840.3550.3680.2771.080.791.080.80.790.8-0.152-0.082-0



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
cLL-0.0368360.079-0.468
dLK0.7098290.05912.12
dLM0.5802730.0629.298
R2 = 0.9331 R2b = 0.9284 # obs = 31

QR Least Squares
Parameter Estimates
Parameter Coefficient std error t-ratio
cKK-0.1141030.075-1.515
dKL0.8674440.1028.488
dKM0.4277040.0795.431
R2 = 0.842 R2b = 0.8307 # obs = 31

QR Least Squares
Parameter Estimates
Parameter Coefficient std error t-ratio
cMM-0.2614130.009-30.04
dML0.7341520.01165.868
dMK0.550350.00964.139
R2 = 0.9992 R2b = 0.9992 # obs = 31

The three estimated factor demand functions are:

∂C/∂wL = L(q; wL, wK, wM) = q^(1/1) * [-0.0368 + 0.7098 * (wK / wL)^(1/2) + 0.5803 * (wM / wL)^(1/2)]

∂C/∂wK = K(q; wL, wK, wM) = q^(1/1) * [-0.1141 + 0.8674 * (wL / wK)^(1/2) + 0.4277 * (wM / wK)^(1/2)]

∂C/∂wM = M(q; wL, wK, wM) = q^(1/1) * [-0.2614 + 0.7342 * (wL / wM)^(1/2) + 0.5504 * (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,wLεL,wKεL,wMεL,q
εK,wLεK,wKεK,wMεK,q
εM,wLεM,wKεM,wMεM,q
-0.51250.32950.1831
0.3915-0.57020.17871
0.29550.3019-0.59741

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) * [-0.03684 * wL + -0.1141 * wK + -0.26141 * wM + 1.57727 * (wL*wK)^(1/2) + 1.31443 * (wL*wM)^(1/2) + 0.97805 * (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-0.1047440.051685-2.0266
dLK0.7055680.03552819.859517
dLM0.6594180.0319820.619696
cKK-0.0949340.033424-2.840308
dKL0.7055680.03552819.859517
dKM0.5733870.02830920.254721
cMM-0.2148470.038099-5.639132
dML0.6594180.0319820.619696
dMK0.5733870.02830920.254721
R2 = 0.9945 R2b = 0.9939 # 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) * [-0.1047 + 0.7056 * (wK / wL)^(1/2) + 0.6594 * (wM / wL)^(1/2)]

∂C/∂wK = K(q; wL, wK, wM) = q^(1/1) * [-0.0949 + 0.7056 * (wL / wK)^(1/2) + 0.5734 * (wM / wK)^(1/2)]

∂C/∂wM = M(q; wL, wK, wM) = q^(1/1) * [-0.2148 + 0.6594 * (wL / wM)^(1/2) + 0.5734 * (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,wLεL,wKεL,wMεL,q
εK,wLεK,wKεK,wMεK,q
εM,wLεM,wKεM,wMεM,q
-0.53570.32770.2081
0.3187-0.55840.23981
0.26550.3146-0.58011

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) * [-0.10474 * wL + -0.09493 * wK + -0.21485 * wM + 1.41114 * (wL*wK)^(1/2) + 1.31884 * (wL*wM)^(1/2) + 1.14677 * (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 =
-0.112290.036980.05088
0.03698-0.03490.03246
0.050880.03246-0.12969

The principal minors of H are H1 = -0.112288, H2 = 0.002551, 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 = -0.1043, 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
1185.72 11.124.4 24.9212.81 25.860.90.790.830.924398.74419.2626.2313.8926.070.3580.3680.2740.920.730.920.850.730.85 -0.234-0.131-0
2196.16 11.666.18 27.8514.74 23.240.90.790.830.929487.04508.282915.8723.390.3510.3640.2840.870.760.870.880.760.88 -0.179-0.110
3208.42 13.347.78 27.316.92 24.350.90.790.830.93645.04669.8528.3418.0724.440.3560.360.2840.870.790.870.850.790.85 -0.132-0.088-0
4216.24 12.545.38 30.5415.4 28.560.90.790.830.924537.33557.4131.6616.3528.790.3540.3680.2780.90.730.90.870.730.87 -0.196-0.115-0
5227.48 14.026.22 31.1516.73 29.930.90.790.830.924653.85675.532.1817.6530.120.3560.3660.2770.90.740.90.860.740.86 -0.166-0.0980
6237.12 12.188 34.0319.93 25.610.90.790.830.93689.94708.2634.8920.9125.640.3510.360.290.850.80.850.880.80.88 -0.147-0.091-0
7246.84 13.168 37.3319.78 27.050.90.790.830.928731.97749.4338.0920.6627.130.3480.3630.290.850.780.850.90.780.9 -0.155-0.089-0
8255.34 12.887.12 43.2618.87 27.970.90.790.830.926673.14687.6343.8119.6428.190.340.3680.2920.850.740.850.930.740.93 -0.205-0.0980
9266.66 14.227.36 41.5420.16 31.250.90.790.830.924793.33807.842.1120.8131.450.3470.3660.2870.870.750.870.90.750.9 -0.164-0.093-0
10275.1 12.885.96 46.319.45 33.070.890.790.830.921683.7694.3946.6919.9633.430.3430.370.2870.880.720.880.920.720.92 -0.22-0.115-0
11285.1 14.525.82 49.8418.88 36.090.890.790.830.918738.37747.8150.1319.2136.630.3420.3730.2850.890.690.890.930.690.93 -0.226-0.1170
12296.88 14.067.56 46.0423.36 34.540.90.790.830.923906.41914.4846.2623.7534.690.3480.3650.2870.870.760.870.90.760.9 -0.158-0.091-0
13307 136 43.7924.14 40.130.890.790.830.92861.17866.444.0224.3740.240.3560.3660.2790.90.750.90.860.750.86 -0.173-0.104-0
14318.02 14.344.24 40.4222.96 54.60.890.790.830.912884.86888.2640.8623.0854.140.3690.3730.2580.960.70.960.810.70.81 -0.254-0.096-0
15325.32 12.166.12 53.724.85 38.440.890.790.830.92823.15823.7353.4224.9138.660.3450.3680.2870.870.740.870.910.740.91 -0.207-0.112-0
16335.48 13.744.38 52.8722.5 50.540.890.790.830.912820.26817.8452.8722.2550.760.3540.3740.2720.930.680.930.880.680.88 -0.249-0.1350
17346.38 11.84.22 47.426.33 53.030.890.790.830.914836.9832.4747.326.0752.860.3620.370.2680.930.720.930.830.720.83 -0.244-0.120
18358.96 13.444.46 43.0228.68 60.680.890.790.830.9121041.561033.4942.8628.460.030.3720.3690.2590.960.730.960.790.730.79 -0.237-0.088-0
19368.44 14.765.38 49.1428.76 56.480.890.790.830.9131143.061130.9448.7728.2556.190.3640.3690.2670.930.730.930.830.730.83 -0.19-0.0910
20376.54 14.584.8 56.5726.84 57.590.890.790.830.9111037.771023.4956.0126.1457.50.3580.3720.270.930.690.930.860.690.86 -0.221-0.1150
21385.64 11.744.48 57.8829.24 55.270.890.790.830.913917.31902.9157.0428.4455.220.3560.370.2740.920.710.920.860.710.86 -0.234-0.131-0
22398.42 11.446.9 52.1138.34 48.890.890.790.830.9211214.661194.9751.2737.2748.820.3610.3570.2820.880.810.880.830.810.83 -0.14-0.096-0
23407.08 13.784.66 57.5430.86 63.780.890.790.830.911129.81106.0956.5829.7763.360.3620.3710.2670.940.710.940.840.710.84 -0.223-0.108-0
24416.9 12.94.7 58.7532.69 63.280.890.790.830.9111124.491098.6457.5731.4562.90.3620.3690.2690.930.720.930.840.720.84 -0.218-0.110
25428.08 12.967.96 62.0839.73 49.740.890.790.830.921412.51380.8760.6238.2849.610.3550.3590.2860.860.790.860.860.790.86 -0.134-0.089-0
26437.64 12.027.76 63.7241.55 49.750.890.790.830.921372.31339.2162.1339.9249.580.3540.3580.2870.850.80.850.860.80.86 -0.139-0.0930
27447.86 11.46.96 61.7243.21 53.840.890.790.830.9191352.51316.7660.0841.3153.670.3590.3580.2840.870.80.870.840.80.84 -0.143-0.0990
28458.92 14.144.98 59.0938.12 74.470.890.790.830.9091436.931391.1557.4336.2673.520.3680.3690.2630.940.730.940.80.730.8 -0.207-0.0870
29465.66 13.424.48 74.5333.95 70.130.890.790.830.9081191.691150.172.1431.9569.860.3550.3730.2720.930.690.930.880.690.88 -0.24-0.130
30477.24 14.685.86 72.7138.05 67.330.890.790.830.9111479.491427.0870.1835.8567.010.3560.3690.2750.910.720.910.860.720.86 -0.178-0.1020
31487.1 14.687.02 78.1640.35 61.470.890.790.830.9141578.751522.2475.1438.0361.320.350.3670.2830.880.740.880.880.740.88 -0.159-0.095-0
AVE:336.89 13.155.97 48.8926.41 45.650.890.790.830.918954.77947.9648.8926.4145.650.3550.3670.2780.90.740.90.860.740.86-0.193-0.104-0




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
cLL0.9220392.2940.402
cKK1.6369141.371.195
cMM-0.2988372.255-0.133
2*dLK-1.5135883.037-0.498
2*dLM3.8313682.4371.572
2*dKM-0.8511612.847-0.299
R2 = 0.9537 R2b = 0.9444 # 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) * [0.92204 * wL + 1.63691 * wK + -0.29884 * wM + -1.51359 * (wL*wK)^(1/2) + 3.83137 * (wL*wM)^(1/2) + -0.85116 * (wK*wM)^(1/2)]
        (***)

XXVI. Its three derived factor demand functions are:

∂C/∂wL = L(q; wL, wK, wM) = q^(1/1) * [0.922 + -0.7568 * (wK / wL)^(1/2) + 1.9157 * (wM / wL)^(1/2)]

∂C/∂wK = K(q; wL, wK, wM) = q^(1/1) * [1.6369 + -0.7568 * (wL / wK)^(1/2) + -0.4256 * (wM / wK)^(1/2)]

∂C/∂wM = M(q; wL, wK, wM) = q^(1/1) * [-0.2988 + 1.9157 * (wL / wM)^(1/2) + -0.4256 * (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,wLεL,wKεL,wMεL,q
εK,wLεK,wKεK,wMεK,q
εM,wLεM,wKεM,wMεM,q
-0.223-0.30980.53281
-0.35040.5328-0.18241
0.9044-0.2738-0.63061

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) = 57.63 = 57.63 = 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 =
-0.05302-0.039670.1478
-0.039670.03248-0.02409
0.1478-0.02409-0.12023

The principal minors of H are H1 = -0.053017, H2 = -0.003295, 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 = -0.2383, e2 = 0.0976, 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
1185.72 11.124.4 24.9212.81 25.860.90.790.830.924398.74420.4427.8514.8821.760.3790.3930.228-0.872.38-0.87-0.712.38-0.71 -0.3130.1230
2196.16 11.666.18 27.8514.74 23.240.90.790.830.929487.04503.6134.1914.7619.550.4180.3420.24-0.852.22-0.85-0.832.22-0.83 -0.2460.1040
3208.42 13.347.78 27.316.92 24.350.90.790.830.93645.04671.4536.2214.2122.740.4540.2820.263-0.931.93-0.93-0.871.93-0.87 -0.1930.0820
4216.24 12.545.38 30.5415.4 28.560.90.790.830.924537.33556.3334.1917.3123.410.3830.390.226-0.842.41-0.84-0.752.41-0.75 -0.2640.1080
5227.48 14.026.22 31.1516.73 29.930.90.790.830.924653.85674.835.9217.6125.590.3980.3660.236-0.872.27-0.87-0.752.27-0.75 -0.2270.0920
6237.12 12.188 34.0319.93 25.610.90.790.830.93689.94702.2145.1516.4122.620.4580.2850.258-0.892.01-0.89-0.942.01-0.94 -0.2040.089-0
7246.84 13.168 37.3319.78 27.050.90.790.830.928731.97736.9446.6618.2322.240.4330.3260.241-0.832.21-0.83-0.92.21-0.9 -0.2060.0880
8255.34 12.887.12 43.2618.87 27.970.90.790.830.926673.14670.0148.9720.8319.690.390.40.209-0.752.7-0.75-0.912.7-0.91 -0.2440.1010
9266.66 14.227.36 41.5420.16 31.250.90.790.830.924793.33795.7447.5821.1324.230.3980.3780.224-0.82.46-0.8-0.842.46-0.84 -0.2130.0910
10275.1 12.885.96 46.319.45 33.070.890.790.830.921683.7685.8948.3423.5222.890.3590.4420.199-0.762.91-0.76-0.842.91-0.84 -0.2650.1140
11285.1 14.525.82 49.8418.88 36.090.890.790.830.918738.37749.1547.3625.7323.020.3220.4990.179-0.763.38-0.76-0.823.38-0.82 -0.2630.1160
12296.88 14.067.56 46.0423.36 34.540.90.790.830.923906.41901.0153.623.0727.50.4090.360.231-0.812.35-0.81-0.852.35-0.85 -0.2080.0890
13307 136 43.7924.14 40.130.890.790.830.92861.17864.4649.9323.7734.320.4040.3580.238-0.872.24-0.87-0.772.24-0.77 -0.2380.0980
14318.02 14.344.24 40.4222.96 54.60.890.790.830.912884.86901.340.3926.0348.150.3590.4140.227-0.942.36-0.94-0.612.36-0.61 -0.3120.0980
15325.32 12.166.12 53.724.85 38.440.890.790.830.92823.15810.4558.6426.728.40.3850.4010.214-0.782.61-0.78-0.842.61-0.84 -0.260.1110
16335.48 13.744.38 52.8722.5 50.540.890.790.830.912820.26833.8847.430.3235.980.3110.50.189-0.843.15-0.84-0.693.15-0.69 -0.3060.1270
17346.38 11.84.22 47.426.33 53.030.890.790.830.914836.9839.0649.3328.0845.730.3750.3950.23-0.92.33-0.9-0.672.33-0.67 -0.3180.116-0
18358.96 13.444.46 43.0228.68 60.680.890.790.830.9121041.561048.2247.1427.0858.720.4030.3470.25-0.992.01-0.99-0.632.01-0.63 -0.3040.0910
19368.44 14.765.38 49.1428.76 56.480.890.790.830.9131143.061140.2852.2329.0850.240.3870.3760.237-0.922.22-0.92-0.672.22-0.67 -0.250.0890
20376.54 14.584.8 56.5726.84 57.590.890.790.830.9111037.771037.0453.0332.7844.240.3340.4610.205-0.862.8-0.86-0.672.8-0.67 -0.2780.1090
21385.64 11.744.48 57.8829.24 55.270.890.790.830.913917.31906.2558.4332.2844.140.3640.4180.218-0.852.54-0.85-0.712.54-0.71 -0.3070.123-0
22398.42 11.446.9 52.1138.34 48.890.890.790.830.9211214.661217.3569.1925.6349.510.4790.2410.281-1.031.74-1.03-0.91.74-0.9 -0.2120.09-0
23407.08 13.784.66 57.5430.86 63.780.890.790.830.911129.81117.1356.8233.8853.220.360.4180.222-0.892.46-0.89-0.662.46-0.66 -0.2860.1040
24416.9 12.94.7 58.7532.69 63.280.890.790.830.9111124.491106.3860.233.8954.010.3750.3950.229-0.892.35-0.89-0.682.35-0.68 -0.2860.1060
25428.08 12.967.96 62.0839.73 49.740.890.790.830.921412.51380.978.3329.6445.70.4580.2780.263-0.921.93-0.92-0.91.93-0.9 -0.1930.0840
26437.64 12.027.76 63.7241.55 49.750.890.790.830.921372.31340.681.8529.7446.110.4660.2670.267-0.941.9-0.94-0.931.9-0.93 -0.2020.0880
27447.86 11.46.96 61.7243.21 53.840.890.790.830.9191352.51331.379.7829.7452.460.4710.2550.274-0.991.81-0.99-0.91.81-0.9 -0.2140.0920
28458.92 14.144.98 59.0938.12 74.470.890.790.830.9091436.931407.4763.0335.2569.650.3990.3540.246-0.962.07-0.96-0.652.07-0.65 -0.270.0880
29465.66 13.424.48 74.5333.95 70.130.890.790.830.9081191.691166.0667.2141.3851.420.3260.4760.198-0.842.95-0.84-0.692.95-0.69 -0.3010.123-0
30477.24 14.685.86 72.7138.05 67.330.890.790.830.9111479.491429.3473.6939.3254.370.3730.4040.223-0.852.47-0.85-0.722.47-0.72 -0.2370.0950
31487.1 14.687.02 78.1640.35 61.470.890.790.830.9141578.751508.8583.4639.1848.590.3930.3810.226-0.822.42-0.82-0.82.42-0.8 -0.2130.09-0
AVE:336.89 13.155.97 54.0726.5 38.390.890.790.830.918954.77950.1254.0726.538.390.3940.3740.231-0.872.37-0.87-0.782.37-0.78-0.2530.1010



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).

 

 
   

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