Egwald Economics: Microeconomics

Cost Functions

by

Elmer G. Wiens

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L. Diewert (Generalized Leontief) Cost Function

The three factor Diewert production function is:

q^1/nu = aLL * L + aKK * K + aMM * M + bLK * L^1/2 * K^1/2 + bLM * L^1/2 * M^1/2 + bKM * K^1/2 * M^1/2         (*)

=     f(L,K,M)

where L = labour, K = capital, M = materials and supplies, q = product output, and nu = elasticity of scale parameter.

If we have a data set relating these inputs to output for varying levels of inputs and output, we can estimate the parameters of the Diewert production function directly.

Suppose, however, we have a data set relating relating output quantities to total costs and input (factor) prices, wL, wK, and wM. Then, we can work with the Diewert (Generalized Leontief) cost function to estimate the parameters of the production technology.

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/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.

We shall use two methods to obtain estimates of the parameters:

À1:   Estimate the parameters the cost function by way of its factor demand 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 also:

À3:   Re-estimate the Diewert Production Function — to investigate properties of the underlying technology.

Assuming dLK = dKL, dLM = dML, and dKM = dMK, the 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)

À1:   Estimate the factor demand equations separately.

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/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)]

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/nu), 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. Duality.  The Plan:

1. Generate CES data and estimate the parameters of a Diewert production function.

2. Generate cost data with the estimated Diewert 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 Diewert cost minimizing factor inputs, L, K, and M. The base factor prices and the randomizing distribution are specified in the form below.

3. 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.

The estimated coefficients of the Diewert production and cost functions will vary with the parameters sigma, nu, alpha, beta and gamma of the CES production function.

CES Production Function:

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

where L = labour, K = capital, M = materials and supplies, and q = product. The parameter nu is a measure of the economies of scale, while the parameter rho yields the elasticity of substitution:

sigma = 1/(1 + rho).

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

Restrictions: .7 < nu < 1.3; .5 < sigma < 1.5;
.25 < alpha < .45, .3 < beta < .5, .2 < gamma < .35
sigma = 1 → nu = alpha + beta + gamma (Cobb-Douglas)
sigma < 1 → inputs complements; sigma > 1 → inputs substitutes
4 <= wL* <= 11,   7<= wK* <= 16,   4 <= wM* <= 10

CES Production Function Parameters
elasticity of scale parameter: nu
elasticity of substitution: sigma
alpha
beta
gamma
Base Factor Prices
 wL* wK* wM*
Distribution to Randomize Factor Prices
 Use [-2, 2] Uniform distribution     Use .25 * Normal (μ = 0, σ2 = 1)

The CES production function as specified:

q = 1 * [0.35 * (L^- 0.17647) + 0.4 * (K^- 0.17647) + 0.25 *(M^- 0.17647)]^(-1/0.17647)

II. For these coefficients of the CES production function, I generated a sequence of factor prices, outputs, and the corresponding cost minimizing inputs. Then I estimated the Diewert production function using multiple regression yielding the following coefficient estimates:

QR Least Squares
Parameter Estimates
Parameter Coefficient std error t-ratio
cLL-0.1480530.001-183.569
cKK-0.181890-424.452
cMM-0.179970-380.444
cLK0.6443770.001555.523
cLM0.3466190.001322.232
cKM0.519060.001465.723
 R2 = 1 R2b = 1 # obs = 182

nu = 1
aLL = -0.148053, aKK = -0.18189, aMM = -0.17997
bLK = 0.644377, bLM = 0.346619 bKM, = 0.51906

aLL + aKK + aMM + bLK + bLM + bKM = 1

The estimated Diewert production function:

q^1/1 = -0.148053 * L + -0.18189 * K + -0.17997 * M + 0.644377 * L^1/2 * K^1/2 + 0.346619 * L^1/2 * M^1/2 + 0.51906 * K^1/2 * M^1/2     =     f(L,K,M)

III. For these coefficients of the Diewert production function, I generated a new sequence (displayed in the "Diewert Cost Function" table) of factor prices, outputs, and the corresponding cost minimizing inputs.

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

Then I used these data to estimate the coefficients of each factor demand equation separately:

QR Least Squares
Parameter Estimates
Parameter Coefficient std error t-ratio
cLL-0.1672910.003-53.555
dLK0.6488240.003209.746
dLM0.5535410.003171.961
 R2 = 0.9999 R2b = 0.9999 # obs = 31

QR Least Squares
Parameter Estimates
Parameter Coefficient std error t-ratio
cKK-0.0050440.002-2.036
dKL0.6542330.003260.225
dKM0.4992390.002202.936
 R2 = 0.9997 R2b = 0.9997 # obs = 31

QR Least Squares
Parameter Estimates
Parameter Coefficient std error t-ratio
cMM-0.2735020.005-53.895
dML0.5477370.005107.317
dMK0.4990960.00599.115
 R2 = 0.9997 R2b = 0.9997 # obs = 31

The three estimated factor demand functions are:

 ∂C/∂wL = L(q; wL, wK, wM) = q^(1/1) * [-0.1673 + 0.6488 * (wK / wL)^(1/2) + 0.5535 * (wM / wL)^(1/2)] ∂C/∂wK = K(q; wL, wK, wM) = q^(1/1) * [-0.005 + 0.6542 * (wL / wK)^(1/2) + 0.4992 * (wM / wK)^(1/2)] ∂C/∂wM = M(q; wL, wK, wM) = q^(1/1) * [-0.2735 + 0.5477 * (wL / wM)^(1/2) + 0.4991 * (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/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.

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
 -0.568 0.3596 0.2084 1 0.2948 -0.5031 0.2083 1 0.281 0.3489 -0.6299 1

IV. The Diewert cost function as obtained from the unrestricted 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) * [-0.16729 * wL + -0.00504 * wK + -0.2735 * wM + 1.30306 * (wL*wK)^(1/2) + 1.10128 * (wL*wM)^(1/2) + 0.99833 * (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-0.1680550.00332-50.620484
dLK0.6519270.002681243.168683
dLM0.5496970.002317237.203186
cKK-0.0024240.002922-0.829594
dKL0.6519270.002681243.168683
dKM0.4978810.002484200.437326
cMM-0.2738860.003131-87.48107
dML0.5496970.002317237.203186
dMK0.4978810.002484200.437326
 R2 = 0.9999 R2b = 0.9999 # 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) * [-0.1681 + 0.6519 * (wK / wL)^(1/2) + 0.5497 * (wM / wL)^(1/2)] ∂C/∂wK = K(q; wL, wK, wM) = q^(1/1) * [-0.0024 + 0.6519 * (wL / wK)^(1/2) + 0.4979 * (wM / wK)^(1/2)] ∂C/∂wM = M(q; wL, wK, wM) = q^(1/1) * [-0.2739 + 0.5497 * (wL / wM)^(1/2) + 0.4979 * (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/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,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
 -0.5684 0.3614 0.207 1 0.2938 -0.5015 0.2077 1 0.282 0.3481 -0.6301 1

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^(1/1) * [-0.16805 * wL + -0.00242 * wK + -0.27389 * wM + 1.30385 * (wL*wK)^(1/2) + 1.09939 * (wL*wM)^(1/2) + 0.99576 * (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) = 51.01 = 51.01 = 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 =
 -0.09981 0.03417 0.04241 0.03417 -0.03141 0.02819 0.04241 0.02819 -0.11055

The principal minors of H are H1 = -0.09981, H2 = 0.001967, 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 = -0.148, e2 = -0.0938, 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) * K(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 for a given level of output agrees with the minimized cost using the Diewert production function.

Diewert Cost Function
À1: Estimate the cost function using restricted factor demands
CES: Returns to Scale = 1, Elasticity of Substitution = 0.85
—   Diewert Production Data   —     —   Diewert Cost Data   —   Factor Shares   —   Uzawa Elasticities   —   2wwc(w)
obs #qwLwKwM LK MsLKsLMsKMcostest costest Lest Kest MsLsKsMuLKuLMuKLuKMuMLuMKe1e2e3
1187.44 14.027.42 22.9915.02 17.280.830.880.89509.86509.8422.9615.0217.30.3350.4130.2520.850.850.850.860.850.86 -0.129-0.081-0
2198.9 11.245.18 18.6417.4 22.510.930.910.73478.03478.0818.6917.422.420.3480.4090.2430.910.880.910.760.880.76 -0.161-0.0850
3205.08 13.427.5 31.1315.48 16.810.760.841.01492.01492.0231.1915.4216.890.3220.4210.2570.810.830.810.940.830.94 -0.193-0.081-0
4217.92 12.087.48 24.6519.26 19.370.830.950.86572.74572.7124.619.2619.410.340.4060.2540.850.90.850.840.90.84 -0.119-0.084-0
5228.42 12.645 23.1618.54 27.130.940.840.75565.04565.0823.1918.5427.080.3460.4150.240.910.840.910.780.840.78 -0.171-0.085-0
6236.44 14.247.38 31.9718.29 21.380.810.850.94624.11624.1131.9718.2721.420.330.4170.2530.840.840.840.890.840.89 -0.148-0.082-0
7248.86 14.525.66 26.5219.61 29.120.930.830.77684.61684.6426.5419.6229.070.3430.4160.240.910.830.910.790.830.79 -0.152-0.080
8258.62 11.924.56 24.8921.52 32.220.960.850.71618.02618.1124.9621.532.170.3480.4150.2370.930.840.930.750.840.75 -0.19-0.0850
9265.82 13.64.98 34.7718.82 29.810.890.750.87606.73606.834.7618.8629.720.3330.4230.2440.880.780.880.850.780.85 -0.187-0.111-0
10275.14 13.84.88 38.7518.62 30.580.870.720.91605.38605.4838.7718.6730.440.3290.4260.2450.870.760.870.870.760.87 -0.206-0.119-0
11287.72 12.967.24 33.8924.43 26.850.840.910.87772.7772.6633.8524.4426.880.3380.410.2520.850.870.850.850.870.85 -0.125-0.084-0
12297.96 14.784.62 33.0721.86 38.760.960.750.76765.5765.5733.0321.8838.810.3430.4220.2340.920.780.920.790.780.79 -0.195-0.087-0
13307 136 36.8924.4 31.620.870.850.86765.2765.1836.8824.4331.580.3370.4150.2480.870.840.870.840.840.84 -0.148-0.094-0
14318.94 13.265.48 32.6926.44 37.370.930.860.75847.62847.6632.7426.4437.280.3450.4140.2410.910.850.910.780.850.78 -0.155-0.0810
15327.78 11.326.88 36.3829.62 30.350.840.950.84827.21827.1336.3329.6430.380.3420.4060.2530.850.90.850.830.90.83 -0.124-0.09-0
16337.2 147.88 43.4927.69 30.110.810.890.92938.12938.1343.4327.6730.20.3330.4130.2540.840.860.840.880.860.88 -0.13-0.0780
17347.14 11.585.2 38.4528.64 37.930.90.860.8803.42803.4138.4628.6737.850.3420.4130.2450.890.850.890.810.850.81 -0.163-0.0980
18358.46 11.84.02 34.2429.48 48.130.980.820.68830.97831.0934.3329.4148.180.3490.4180.2330.940.830.940.740.830.74 -0.223-0.0860
19367.16 127.38 44.5232.11 32.370.810.930.89943.01943.0144.4232.132.490.3370.4080.2540.840.890.840.860.890.86 -0.13-0.085-0
20377.56 14.144.12 41.8627.49 51.40.970.730.74916.94916.9941.7927.4951.540.3440.4240.2320.930.780.930.780.780.78 -0.222-0.0920
21387.52 13.26.62 46.0631.98 38.590.860.880.861024.041023.9846.033238.570.3380.4130.2490.860.850.860.840.850.84 -0.134-0.087-0
22396.14 12.066.34 50.9132.13 37.150.830.870.91935.58935.5650.8632.1337.20.3340.4140.2520.850.850.850.870.850.87 -0.155-0.095-0
23408.06 13.584.32 43.2331.24 54.30.970.770.731007.321007.3843.2231.2354.390.3460.4210.2330.930.80.930.770.80.77 -0.208-0.087-0
24416.08 14.945.3 56.0729.04 47.350.890.740.881025.611025.7756.0529.1147.180.3320.4240.2440.880.770.880.860.770.86 -0.179-0.106-0
25426.5 12.287 54.6235.63 38.330.810.90.911060.811060.8254.5435.6138.440.3340.4120.2540.840.870.840.870.870.87 -0.144-0.0880
26438.7 14.864.84 47.0533.57 57.370.960.770.741185.861185.9547.0433.5657.430.3450.4210.2340.930.80.930.780.80.78 -0.185-0.081-0
27448.24 12.084.86 45.8337.48 54.090.940.850.741093.331093.4145.9137.4853.980.3460.4140.240.910.840.910.770.840.77 -0.176-0.088-0
28458.72 14.84.06 47.6234.22 66.2710.730.71190.741190.6647.5434.1466.70.3480.4240.2270.950.780.950.760.780.76 -0.232-0.08-0
29468.1 13.55.14 51.137.23 56.350.930.820.771206.111206.1851.1337.2556.260.3430.4170.240.910.820.910.790.820.79 -0.168-0.087-0
30475.04 14.646.24 72.8733.15 46.340.810.750.981141.781141.6973.0733.1446.190.3230.4250.2520.840.780.840.910.780.91 -0.199-0.097-0
31486.22 13.725.84 63.9836.5 50.810.860.80.891195.481195.563.9836.5550.710.3330.4190.2480.870.810.870.860.810.86 -0.163-0.099-0
AVE:337.38 13.235.79 39.7526.67 37.360.890.830.83846.25846.2839.7526.6737.360.3390.4160.2450.880.830.880.820.830.82-0.168-0.089-0

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 sigma, nu, alpha, beta and gamma of the CES production function used to generate these data with the Diewert production function.

QR Least Squares
Parameter Estimates
Parameter Coefficient std error t-ratio
cLL-0.1725640.003-50.609
cKK-0.0023350.003-0.696
cMM-0.2677250.004-65.696
2*dLK1.3097390.005239.142
2*dLM1.1001620.005233.284
2*dKM0.9880430.006167.825
 R2 = 1 R2b = 1 # obs = 31

The Diewert cost function as obtained by direct estimation is:

C(q;wL,wK,wM) = q^(1/nu) * [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) * [-0.17256 * wL + -0.00234 * wK + -0.26772 * wM + 1.30974 * (wL*wK)^(1/2) + 1.10016 * (wL*wM)^(1/2) + 0.98804 * (wK*wM)^(1/2)]
(***)

XII. Its three derived factor demand functions are:

 ∂C/∂wL = L(q; wL, wK, wM) = q^(1/1) * [-0.1726 + 0.6549 * (wK / wL)^(1/2) + 0.5501 * (wM / wL)^(1/2)] ∂C/∂wK = K(q; wL, wK, wM) = q^(1/1) * [-0.0023 + 0.6549 * (wL / wK)^(1/2) + 0.494 * (wM / wK)^(1/2)] ∂C/∂wM = M(q; wL, wK, wM) = q^(1/1) * [-0.2677 + 0.5501 * (wL / wM)^(1/2) + 0.494 * (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/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.

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
 -0.5702 0.363 0.2072 1 0.2952 -0.5014 0.2062 1 0.282 0.3451 -0.6271 1

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) = 51.01 = 51.01 = 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 =
 -0.10012 0.03432 0.04244 0.03432 -0.03139 0.02797 0.04244 0.02797 -0.11011

The principal minors of H are H1 = -0.100122, H2 = 0.001965, 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 = -0.1479, e2 = -0.0937, and e3 = 0.
H3 = e1 * e2 * e3 = 0.

XIV. Table of Results: check that the estimated Diewert cost function for a given level of output agrees with the minimized cost using the Diewert production function.

Diewert Cost Function
À2: Estimate the Diewert cost function directly
CES: Returns to Scale = 1, Elasticity of Substitution = 0.85
—   Diewert Production Data   —     —   Diewert Cost Data   —   Factor Shares   —   Uzawa Elasticities   —   2wwc(w)
obs #qwLwKwM LK MsLKsLMsKMcostest costest Lest Kest MsLsKsMuLKuLMuKLuKMuMLuMKe1e2e3
1187.44 14.027.42 22.9915.02 17.280.830.880.89509.86509.8522.9615.0117.320.3350.4130.2520.850.850.850.850.850.85 -0.129-0.0810
2198.9 11.245.18 18.6417.4 22.510.930.910.73478.03478.0418.6817.422.440.3480.4090.2430.910.880.910.750.880.75 -0.16-0.0850
3205.08 13.427.5 31.1315.48 16.810.760.841.01492.01492.0331.215.416.920.3220.420.2580.810.830.810.930.830.93 -0.193-0.08-0
4217.92 12.087.48 24.6519.26 19.370.830.950.86572.74572.7424.5919.2519.450.340.4060.2540.850.90.850.830.90.83 -0.119-0.084-0
5228.42 12.645 23.1618.54 27.130.940.840.75565.04565.0423.1818.5427.090.3450.4150.240.920.840.920.770.840.77 -0.171-0.086-0
6236.44 14.247.38 31.9718.29 21.380.810.850.94624.11624.1131.9718.2621.440.330.4170.2540.840.840.840.880.840.88 -0.149-0.0820
7248.86 14.525.66 26.5219.61 29.120.930.830.77684.61684.626.5319.6229.080.3430.4160.240.910.830.910.780.830.78 -0.151-0.08-0
8258.62 11.924.56 24.8921.52 32.220.960.850.71618.02618.0524.9421.532.180.3480.4150.2370.930.840.930.750.840.75 -0.19-0.085-0
9265.82 13.64.98 34.7718.82 29.810.890.750.87606.73606.7734.7718.8529.730.3340.4230.2440.890.780.890.850.780.85 -0.187-0.111-0
10275.14 13.84.88 38.7518.62 30.580.870.720.91605.38605.4338.7818.6630.440.3290.4250.2450.880.760.880.870.760.87 -0.206-0.119-0
11287.72 12.967.24 33.8924.43 26.850.840.910.87772.7772.6833.8424.4326.920.3380.410.2520.860.870.860.840.870.84 -0.125-0.083-0
12297.96 14.784.62 33.0721.86 38.760.960.750.76765.5765.5333.0321.8838.80.3430.4220.2340.930.790.930.780.790.78 -0.194-0.088-0
13307 136 36.8924.4 31.620.870.850.86765.2765.1636.8724.4131.610.3370.4150.2480.880.840.880.830.840.83 -0.148-0.0940
14318.94 13.265.48 32.6926.44 37.370.930.860.75847.62847.6132.7326.4437.30.3450.4140.2410.910.850.910.770.850.77 -0.154-0.081-0
15327.78 11.326.88 36.3829.62 30.350.840.950.84827.21827.1636.3129.6230.430.3420.4050.2530.860.90.860.820.90.82 -0.124-0.0890
16337.2 147.88 43.4927.69 30.110.810.890.92938.12938.1743.4327.6530.250.3330.4130.2540.840.860.840.870.860.87 -0.131-0.078-0
17347.14 11.585.2 38.4528.64 37.930.90.860.8803.42803.3838.4528.6637.880.3420.4130.2450.890.850.890.80.850.8 -0.162-0.0980
18358.46 11.84.02 34.2429.48 48.130.980.820.68830.97831.0234.329.4248.180.3490.4180.2330.940.830.940.740.830.74 -0.223-0.086-0
19367.16 127.38 44.5232.11 32.370.810.930.89943.01943.0744.4132.0732.550.3370.4080.2550.840.890.840.850.890.85 -0.13-0.085-0
20377.56 14.144.12 41.8627.49 51.40.970.730.74916.94916.9441.7827.551.530.3440.4240.2320.940.780.940.770.780.77 -0.222-0.092-0
21387.52 13.26.62 46.0631.98 38.590.860.880.861024.041023.9846.0231.9938.610.3380.4120.250.870.850.870.830.850.83 -0.134-0.087-0
22396.14 12.066.34 50.9132.13 37.150.830.870.91935.58935.5850.8632.137.240.3340.4140.2520.850.850.850.860.850.86 -0.156-0.095-0
23408.06 13.584.32 43.2331.24 54.30.970.770.731007.321007.3243.2131.2354.380.3460.4210.2330.930.80.930.770.80.77 -0.208-0.087-0
24416.08 14.945.3 56.0729.04 47.350.890.740.881025.611025.756.0729.147.190.3320.4240.2440.890.770.890.850.770.85 -0.179-0.1050
25426.5 12.287 54.6235.63 38.330.810.90.911060.811060.8654.5335.5838.50.3340.4120.2540.840.870.840.870.870.87 -0.144-0.088-0
26438.7 14.864.84 47.0533.57 57.370.960.770.741185.861185.8847.0233.5757.420.3450.4210.2340.930.80.930.770.80.77 -0.184-0.081-0
27448.24 12.084.86 45.8337.48 54.090.940.850.741093.331093.3345.8837.4854.010.3460.4140.240.920.840.920.770.840.77 -0.175-0.088-0
28458.72 14.84.06 47.6234.22 66.2710.730.71190.741190.647.5234.1666.670.3480.4250.2270.950.780.950.750.780.75 -0.231-0.08-0
29468.1 13.55.14 51.137.23 56.350.930.820.771206.111206.1151.1137.2556.280.3430.4170.240.910.820.910.780.820.78 -0.167-0.087-0
30475.04 14.646.24 72.8733.15 46.340.810.750.981141.781141.673.1133.1146.220.3230.4250.2530.840.780.840.910.780.91 -0.199-0.0960
31486.22 13.725.84 63.9836.5 50.810.860.80.891195.481195.4563.9936.5250.740.3330.4190.2480.870.810.870.850.810.85 -0.164-0.099-0
AVE:337.38 13.235.79 39.7426.67 37.380.890.830.83846.25846.2539.7426.6737.380.3390.4160.2450.890.830.890.820.830.82-0.168-0.089-0

À3:   Re-estimate the Diewert Production Function.

XV. Assuming we had 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 that we did not have data on the cost minimizing levels of inputs, we estimated the Diewert cost function directly. With the derived factor demand functions to determine cost minimizing inputs, we can estimate a Diewert production function using the data in the Diewert Cost Function table.

QR Least Squares
Parameter Estimates
Parameter Coefficient std error t-ratio
cLL-0.1461820.001-130.458
cKK-0.1799170.004-51.383
cMM-0.1787020.001-150.182
cLK0.6419090.003194.208
cLM0.3448570.002184.367
cKM0.5182110.003155.447
 R2 = 1 R2b = 1 # obs = 31

The estimated Diewert production function using the cost data is:

q^1/1 = -0.146182 * L + -0.179917 * K + -0.178702 * M + 0.641909 * L^1/2 * K^1/2 + 0.344857 * L^1/2 * M^1/2 + 0.518211 * K^1/2 * M^1/2     =     f(L,K,M)

The coefficients of this production function are approximately equal to the coefficients of the original Diewert production function.

XVI. Knowing the Diewert production function permits us to investigate further properties of the production / cost technology. For example, we can determine the short-run average and marginal cost functions in relation to the long-run average and marginal cost functions obtained from the Diewert cost function, and the short-run Allen elasticities of substitution.

For example, if we set capital at the least cost level for q = 30, then K = 24.4, and we can solve for the least cost inputs of labour, L, and materials and supplies, M, for various levels of output, q.   (See the Diewert Production Function web page.)

Diewert Short Run Cost Data
CES: Returns to Scale = 1, Elasticity of Substitution = 0.85
wL = 7, wK= 13, wM = 6
qest q LK Mtotal cost ave. costmarg. cost3 factor
sLM
2 factor
sLM
1414 10.7224.4 9.27447.94 32 14.05 1.391.15
1616 13.1624.4 11.36477.46 29.84 15.52 1.311.1
1818 15.8324.4 13.65509.95 28.33 16.97 1.241.06
2020 18.7324.4 16.16545.3 27.26 18.43 1.161.02
2222 21.8924.4 18.85583.59 26.53 19.86 1.090.98
2424 25.2824.4 21.75624.74 26.03 21.29 1.030.95
2626 28.9224.4 24.86668.83 25.72 22.7 0.970.91
2828 32.7624.4 28.16715.52 25.55 24.12 0.910.88
3030 36.8824.4 31.65765.28 25.51 25.5 0.850.86
3232 41.2124.4 35.34817.7 25.55 26.89 0.80.83
3434 45.7724.4 39.21872.86 25.67 28.26 0.750.81
3636 50.5624.4 43.27930.78 25.85 29.63 0.70.78
3838 55.5624.4 47.5991.21 26.08 31 0.660.76
4040 60.8124.4 51.961054.65 26.37 32.35 0.620.74
4242 66.2724.4 56.591120.67 26.68 33.69 0.580.72
4444 71.9524.4 61.421189.36 27.03 35.02 0.540.71
4646 77.8524.4 66.421260.71 27.41 36.34 0.510.69
4848 83.9724.4 71.621334.69 27.81 37.65 0.480.67
5050 90.2924.4 76.991411.24 28.22 38.96 0.440.66
5252 96.8424.4 82.531490.34 28.66 40.25 0.410.64

We get a U-shaped, short run average cost curve, with capital fixed.

As seen in the diagram below, the short run average cost curve is (approx.) tangent to the long run average cost curve, at q = 30.

Graph of Average Cost and Marginal Cost
Diewert Cost / Production Functions - Capital Fixed

 Average cost function Marginal cost function L.R. Average cost function

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).