Egwald Economics: Microeconomics

Cost Functions

by

Elmer G. Wiens

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Suppose we have a data set relating output quantities, q, to (cost minimizing) factor inputs, L, K, M, and input prices, wL, wK, and wM, and consequently data on the total cost of producing specific levels of outputs.

The three factor Normalized Quadratic (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,

and the unit cost function is:

 c(wL,wK,wM) = cL * wL + cK * wK + cM * wM + (1/2) * [dLL * (wL*wL) + dLK*(wL*wK) + dLM * (wL*wM)                  + dKL * (wK*wL) + dKK*(wK*wK) + dKM * (wK*wM)              + dML * (wM*wL) + dMK*(wM*wK) + dMM * (wM*wM)] * (wL + wK + wM)^-1

linear in its twelve parameters, cL, cK, cM, dLL, dLK, dLM, dKL, dKK, dKM, dML, dMK, and dMM.  Dividing the non-linear in variables, wL, wK, and wM, portion of the quadratic unit cost function by (wL + wK + wM) ensures that the unit cost function is homogeneous of degree one in prices, since c(t*wL, t*wK, t*wM) = t * c(wL,wK,wM) (Diewert and Wales, (1988), 327-342).  Multiplying the unit cost function by the returns to scale function to obtain the total cost function presumes that the production technology is homothetic.

It is convenient to express the normalized unit cost function in terms of vectors and matrices.

Define the vectors:

c = [cL, cK, cM]T, w = [wL, wK, wM]T, 1 = [1, 1, 1]T, and 0 = [0, 0, 0]T.

Define the matrix D by :

 D = dLL dLK dLM dKL dKK dKM dML dMK dMM

The unit cost function becomes:

c(w) = cT * w + (1/2) * wT * D * w / (1T * w),

a linear function in its parameters, c, and D.

Restrictions:

D = DT, and D * w* = 0,

where the reference vector w* = [wL*, wK*, wM*]T for the base prices, wL*, wK*, and wM*. The symmetry restriction D = DT ensures that dLK = dKL, dLM = dML, and dKM = dMK (Young's Theorem).

Consequently, as specified the Normalized Quadratic cost function apparently has 9 free parameters (actually only six free parameters — see below).

Using conventional matrix calculus (Intriligator, (1971), 497-500), the first and second order partial derivatives (Diewert and Fox, (2009), 158-164) of the unit cost function, c(w), are:

c(w) = c + D * w / ( 1T * w) - (1/2) * wT * D * w * 1 / (1T * w)2, and

2c(w) = D / ( 1T * w) - D * w * 1T / (1T * w)2 - 1 * wT * D / (1T * w)2 + wT * D * w * (1 * 1T) / (1T * w)3.

At the reference vector w*:

c(w*) = c,

2c(w*) = D / ( 1T * w*).

Shephard's lemma provides that the gradient vector of the unit cost function is the vector of unit input demand functions:

c(w) = [l(w), k(w), m(w)]T.

Re-write the factor prices as:

v = [vL, vK, vM] = [wL, wM, wK] / (wL + wM + wK).

The factor demands as functions of q, vL, vK, and vM are:

 L(q; vL, vK, vM) / q^(1/nu1) = l(vL, vK, vM) = cL + dLL * vL + dLK * vK + dLM * vM - (1/2) * [dLL * vL * vL + dLK * vL * vK + dLM * vL * vM           + dKL * vK * vL + dKK * vK * vK + dKM * vK * vM           + dML * vM * vL + dMK * vM * vK + dMM * vM * vM] K(q; vL, vK, vM) / q^(1/nu1) = k(vL, vK, vM) = cK + dKL * vL + dKK * vK + dKM * vM - (1/2) * [dLL * vL * vL + dLK * vL * vK + dLM * vL * vM           + dKL * vK * vL + dKK * vK * vK + dKM * vK * vM           + dML * vM * vL + dMK * vM * vK + dMM * vM * vM] M(q; vL, vK, vM) / q^(1/nu1) = m(vL, vK, vM) = cM + dML * vL + dMK * vK + dMM * vM - (1/2) * [dLL * vL * vL + dLK * vL * vK + dLM * vL * vM           + dKL * vK * vL + dKK * vK * vK + dKM * vK * vM           + dML * vM * vL + dMK * vM * vK + dMM * vM * vM]

À1:   Estimate the factor demand equations using linear least squares.

I. Linear least squares with restrictions on the parameter values.

For the purpose of estimating the factor demand equations, re-write these functions:

 L(q; vL, vK, vM) / q^(1/nu1) = l(vL, vK, vM) = cL + dLL * vL + dLK * vK + dLM * vM + [tdLL * vL * vL + tdLK * vL * vK + tdLM * vL * vM   + tdKL * vK * vL + tdKK * vK * vK + tdKM * vK * vM   + tdML * vM * vL + tdMK * vM * vK + tdMM * vM * vM] K(q; vL, vK, vM) / q^(1/nu1) = k(vL, vK, vM) = cK + dKL * vL + dKK * vK + dKM * vM + [tdLL * vL * vL + tdLK * vL * vK + tdLM * vL * vM   + tdKL * vK * vL + tdKK * vK * vK + tdKM * vK * vM   + tdML * vM * vL + tdMK * vM * vK + tdMM * vM * vM] M(q; vL, vK, vM) / q^(1/nu1) = m(vL, vK, vM) = cM + dML * vL + dMK * vK + dMM * vM + [tdLL * vL * vL + tdLK * vL * vK + tdLM * vL * vM   + tdKL * vK * vL + tdKK * vK * vK + tdKM * vK * vM   + tdML * vM * vL + tdMK * vM * vK + tdMM * vM * vM]

We can express the factor demands as functions of ten explanatory variables with their corresponding parameters:

12345678910
VariableconstantvLvKvMvL*vLvL*vKvL*vMvK*vKvK*vMvM*vM
LcLdLLdLKdLMtdLLtdLKtdLMtdKKtdKMtdMMβL
KcKdLKdKKdKMtdLLtdLKtdLMtdKKtdKMtdMMβK
LcLdLMdKMdMMtdLLtdLKtdLMtdKKtdKMtdMMβM

The construction of the variables along with their restrictions described below ensures the symmetry of the estimated matrix D, i.e. D = DT.

Suppose we have m observations on q, L , K, M, wL, wK, and wM.  Let V be the m x 10 matrix of observations of the explanatory variables constructed from v = [vL,vK,vM]T.  Let L, K, and M be the m-component vectors of observed factor demands.  Then we can estimate the factor demand equations as a group (Johnston, (1972), 238-241) along with the necessary constraints, D = DT, D * w* = 0, and the accounting constraints among the "d__" and "td__" parameters within and between the demand equations.

This group of equations can be represented as:

W*β=F
V*βL=L
V*βK=K
V*βM=M

Subject to the constraints on the thirty parameters, our objective is to find the values for the vector of parameters, β, that minimizes the distance between F and W * β, i.e. the norm || W * β - F ||.

Looking at the construction of the ten variables of the matrix V, one finds that the rank of V is six, i.e. V has six independent column vectors. Therefore, when one estimates the factor demand equations as a group, the 3*m by 30 matrix of observations, W, has 18 independent column vectors out of thirty.

Moreover, the matrix of restrictions, R, on the parameters has 24 rows, i.e. R is a 24 x 30 matrix, with a rank of 24. These restrictions can be expressed as:

R * β = r,

where r is a 24 component vector. The restrictions matrix, R, and restrictions vector, r:

cLdLLdLKdLMtdLLtdLKtdLMtdKKtdKMtdMMcKdLKdKKdKMtdLLtdLKtdLMtdKKtdKMtdMMcMdLMdKMdMMtdLLtdLKtdLMtdKKtdKMtdMMr
1010020000000000000000000000000 0
2001001000000000000000000000000 0
3000100100000000000000000000000 0
4000000000001000100000000000000 0
5000000000000100002000000000000 0
6000000000000010000100000000000 0
7000000000000000000000100001000 0
8000000000000000000000010000010 0
9000000000000000000000001000002 0
1000001000000000-1000000000000000 0
11000010000000000000000000-100000 0
12000001000000000-100000000000000 0
130000010000000000000000000-10000 0
140000001000000000-10000000000000 0
1500000010000000000000000000-1000 0
1600000001000000000-1000000000000 0
17000000010000000000000000000-100 0
18000000001000000000-100000000000 0
190000000010000000000000000000-10 0
200000000001000000000-10000000000 0
2100000000010000000000000000000-1 0
2200.2690.50.23100000000000000000000000000 0
23000000000000.2690.50.2310000000000000000 0
240000000000000000000000.2690.50.231000000 0

II. Objective: Solve the LSE problem:

Among all 30 component β vectors obeying:

R * β = r,

find the β vector that minimizes the norm:

|| W * β - F ||.

By construction the observation matrix W is rank deficient, making it infeasible to use the usual QR algorithm described on the web page, Linear and Restricted Multiple Regression, to solve the LS problem:

find the 30 component β vector that minimizes the norm:

|| W * β - F ||,

subject to the requirement:

R * β = r.

III. To determine the Normalized Quadratic cost function's efficacy in estimating the cost structure of a production technology, we shall use it to approximate cost data generated by a CES Production function. The estimated parameters of the Normalized Quadratic cost function 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
.5 < nu1 < 2;
4 <= wL* <= 11,   7<= wK* <= 16,   4 <= wM* <= 10

CES Production Function Parameters
CES elasticity of scale parameter: nu
elasticity of substitution: sigma
alpha
beta
gamma
Normalized Quadratic elasticity of scale: nu1
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) The factor prices are distributed about the base factor prices, wL*, wK*, wM*, by adding a random number distributed uniformly in the [-2, 2] domain.

IV. For these coefficients of the CES production function, I generated a sequence (displayed in the "Normalized Quadratic Cost Function" table) of factor prices, outputs, and the corresponding cost minimizing inputs. Then I used these data to estimate the coefficients of the factor demand equations as a group.

I used the following method to solve the LSE problem:

Among all 30 component β vectors obeying:

R * β = r,

find the β vector that minimizes the norm:

|| W * β - F ||.

1. Start by obtaining the Householder QR factorization of the transpose matrix, RT, where the RT matrix has 30 rows and 24 columns, with rank 24:

RT = Q * U,

where Q is a 30 by 30 orthogonal matrix, and U is a 30 by 24 matrix.

2. Multiplying the matrix R by Q will produce a 24 by 24 lower triangular matrix, T, and a 24 by 6 zero matrix, 0:

R * Q = [T | 0].

3. Multiplying the matrix W by Q will produce a 3*m by 24 matrix, H1, and a 3*m by 6 matrix, H2:

W * Q = [H1 | H2].

4. Solve by forward substitution for the 24 component vector, Γ1, the lower triangular problem:

T * Γ1 = r.

5. Compute the 30 component vector, f:

f = F - H1 * Γ1.

If the vector r = 0, f = F.

6. Solve for the 6 component vector, Γ2, using Householder QR factorization, the least squares problem:

H2 * Γ2 = f.

7. After creating the 30 component vector, Γ = [Γ1, Γ2]T, obtain the solution vector, β, by:

β = Q * Γ.

Note: The solution vector, β, is unique since the augemented matrix:

 M = | R  || W |

has full rank = 30, (Lawson and Hanson, (1974), 134-143).

V. The method described above obtains the unique solution vector, β.

I obtained the Covariance Matrix of the solution vector, β, as follows:

Form the matrix, tW, as:

tW = H2 * H2+ * W,

where H2+ is the pseudo-inverse matrix of H2.

The solution to the augmented least squares problem:

 | R   || tW | * β = | r  || F |

is also a solution to the original least squares problem:

|| W * β - F ||,

subject to the requirement:

R * β = r,

(Lawson and Hanson, (1974), 143).

I used the Covariance Matrix from the augmented system to obtain the t-statistics that are displayed below in the solution to the LSE problem. The reported "Observation Matrix Rank" is the rank of the augmented matrix of the augmented least squares problem.

LSE Restricted Least Squares
Parameter Estimates
Parameter Coefficient std error t-ratio
cL1.2335270.0187165.929266
dLL-2.7462860.075783-36.23867
dLK0.9236080.04476920.630326
dLM1.2028510.05501321.864712
tdLL1.3731430.03921335.017787
tdLK-0.9236080.045868-20.136319
tdLM-1.2028510.055228-21.779581
tdKK0.4201520.02848914.748107
tdKM-0.7431170.045444-16.352256
tdMM1.5067060.04216635.733071
cK0.805360.01577251.061087
dLK0.9236080.04901618.843009
dKK-0.8403040.044542-18.86527
dKM0.7431170.04883815.215867
tdLL1.3731430.04291231.999224
tdLK-0.9236080.047632-19.390301
tdLM-1.2028510.058116-20.697563
tdKK0.4201520.02389917.580574
tdKM-0.7431170.047386-15.682362
tdMM1.5067060.04573132.947025
cM1.0676960.01984953.7911
dLM1.2028510.05533121.739086
dKM0.7431170.04280817.35938
dMM-3.0134120.074127-40.652157
tdLL1.3731430.04344131.609593
tdLK-0.9236080.049321-18.726594
tdLM-1.2028510.055373-21.722545
tdKK0.4201520.03344712.561744
tdKM-0.7431170.04393-16.916004
tdMM1.5067060.03842939.207401
 R2 = 0.9956 R2b = 0.9933 # obs = 84 Observation Matrix Rank: 30

Check that the parameter estimates satisfy the restrictions matrix, R.

If the t-ratio = the parameter coefficient / std error seems large in absolute value
with nu = 1 & nu1 = 1 try setting nu = 1.3 & nu1 = 0.5.

VI. As estimated, the parameters of the Normalized Quadratic cost function are:

c = [cL, cK, cM]T = [1.233527, 0.80536, 1.067696], and

 D = dLL dLK dLM dKL dKK dKM dML dMK dMM
=
 -2.74629 0.923608 1.20285 0.923608 -0.840304 0.743117 1.20285 0.743117 -3.01341

The Normalized Quadratic cost function is:

C(q;vL,vK,vM) / q^(1/1) = c(vL,vK,vM) = 1.233527 * vL + 0.80536 * vK + 1.067696 * vM + (1/2) *
 [-2.746286 * (vL*vL) + 0.923608 * (vL*vK) + 1.202851 * (vL*vM) + 0.923608 * (vK*vL) + -0.840304 * (vK*vK) + 0.743117 * (vK*vM) + 1.202851 * (vM*vL) + 0.743117 * (vM*vK) + -3.013412 * (vM*vM)]

The factor demand functions are:

L(q;vL,vK,vM) / q^(1/1) = l(vL,vK,vM) = 1.233527 + -2.746286 * vL + 0.923608 * vK + 1.202851 * vM - (1/2) *
 [-2.746286 * (vL*vL) + 0.923608 * (vL*vK) + 1.202851 * (vL*vM) + 0.923608 * (vK*vL) + -0.840304 * (vK*vK) + 0.743117 * (vK*vM) + 1.202851 * (vM*vL) + 0.743117 * (vM*vK) + -3.013412 * (vM*vM)]

K(q;vL,vK,vM) / q^(1/1) = k(vL,vK,vM) = 0.80536 + 0.923608 * vL + -0.840304 * vK + 0.743117 * vM - (1/2) *
 [-2.746286 * (vL*vL) + 0.923608 * (vL*vK) + 1.202851 * (vL*vM) + 0.923608 * (vK*vL) + -0.840304 * (vK*vK) + 0.743117 * (vK*vM) + 1.202851 * (vM*vL) + 0.743117 * (vM*vK) + -3.013412 * (vM*vM)]

M(q;vL,vK,vM) / q^(1/1) = m(vL,vK,vM) = 1.067696 + 1.202851 * vL + 0.743117 * vK + -3.013412 * vM - (1/2) *
 [-2.746286 * (vL*vL) + 0.923608 * (vL*vK) + 1.202851 * (vL*vM) + 0.923608 * (vK*vL) + -0.840304 * (vK*vK) + 0.743117 * (vK*vM) + 1.202851 * (vM*vL) + 0.743117 * (vM*vK) + -3.013412 * (vM*vM)]

The Uzawa Partial Elasticities of Substitution in terms of the w factor price vector:

 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)),

VII. Table of Results: check that the costs, factor inputs, and Uzawa elasticities from the estimated Normalized Quadratic cost function at given levels of output agree with those obtained from the CES cost function.

CES Production Function:  Returns to Scale = 1, Elasticity of Substitution = 0.85
Normalized Quadratic:  Returns to Scale = 1;   Base Prices: wL* = 7, wK* = 13, wM* = 6
—   —   CES Cost Data   —   —     —   Normalized Quadratic Cost Data   —
Factor PricesFactor InputsFactor SharesUzawa Elasticities  Factor InputsFactor SharesUzawa Elasticities
obs #qwLwKwM LKMsLsKsM uLKuLMuKLuKMuMLuMKcostest costest Lest Kest MsLsKsMuLKuLMuKLuKMuMLuMK
1176.54 13.865.64 21.9312.98 18.690.3350.420.2460.850.850.850.850.850.85428.69428.4922.0612.818.940.3370.4140.2490.90.730.90.860.730.86
2185 14.86.8 28.8412.84 16.680.3220.4250.2530.850.850.850.850.850.85447.7448.132812.9317.170.3120.4270.2610.580.630.581.290.631.29
3197.2 14.646.04 24.0214.72 20.950.3360.4180.2460.850.850.850.850.850.85514.89514.7324.1514.5321.220.3380.4130.2490.930.760.930.840.760.84
4205.66 11.26.12 26.4216.57 18.570.3330.4140.2530.850.850.850.850.850.85448.8244926.6316.518.550.3360.4120.2530.720.990.721.10.991.1
5217.42 13.244.22 23.416.02 28.40.3430.420.2370.850.850.850.850.850.85505.53505.5523.5216.1627.740.3450.4230.2321.20.71.20.50.70.5
6227.22 13.345.3 26.1417.38 25.540.340.4170.2430.850.850.850.850.850.85555.96555.8126.2117.2425.760.340.4140.2461.030.81.030.70.80.7
7235.26 12.466.14 32.9217.72 21.680.3290.4190.2530.850.850.850.850.850.85527.06527.1632.8117.6621.90.3270.4180.2550.670.80.671.180.81.18
8246.54 13.287.52 32.4219.89 21.630.3320.4130.2550.850.850.850.850.850.85638.9639.0932.6919.8821.440.3350.4130.2520.6710.671.1811.18
9256.76 12.964.34 29.2818.87 32.060.340.420.2390.850.850.850.850.850.85581.61581.4129.5718.8231.720.3440.4190.2371.120.681.120.60.680.6
10267.76 11.025.6 28.0723.34 27.830.3450.4080.2470.850.850.850.850.850.85630.91630.9827.5523.3128.620.3390.4070.2541.051.151.050.621.150.62
11277.52 13.184.2 29.7820.7 36.710.3440.4190.2370.850.850.850.850.850.85650.99651.0329.8720.9435.820.3450.4240.2311.220.711.220.480.710.48
12285.48 12.646.38 39.6821.84 26.190.3290.4180.2530.850.850.850.850.850.85660.63660.7739.6321.7926.370.3290.4170.2550.670.830.671.180.831.18
13296.82 14.547.98 40.0123.55 26.30.3310.4150.2540.850.850.850.850.850.85825.12825.3440.223.5526.160.3320.4150.2530.660.940.661.190.941.19
14307 136 36.8924.42 31.590.3370.4150.2480.850.850.850.850.850.85765.17765.3237.0124.1632.030.3380.410.2510.910.90.910.850.90.85
15317.18 11.027.78 37.5129.19 26.320.3380.4040.2570.850.850.850.850.850.85795.84796.4238.4429.1425.620.3470.4030.250.661.390.661.081.391.08
16325.16 13.327.8 49.9524.99 26.410.3240.4180.2590.850.850.850.850.850.85796.63796.849.2625.6425.790.3190.4290.2520.480.890.481.480.891.48
17338.78 13.026.94 36.8729.55 33.830.3430.4080.2490.850.850.850.850.850.85943.21943.836.5629.3934.60.340.4050.2540.971.150.970.721.150.72
18345.1 12.567.64 52.0927.12 27.760.3250.4160.2590.850.850.850.850.850.85818.41818.2251.7327.8326.820.3220.4270.250.480.950.481.490.951.49
19357.34 11.387.6 42.0832.47 30.690.3390.4050.2560.850.850.850.850.850.85911.63912.542.8532.3530.250.3450.4030.2520.711.330.711.031.331.03
20368.1 11.124.3 36.0530.85 46.40.350.4110.2390.850.850.850.850.850.85834.64833.1734.631.6446.770.3360.4220.2411.310.981.310.380.980.38
21375.8 14.344.86 50.1526.03 43.790.3320.4260.2430.850.850.850.850.850.85876.95875.6350.6625.5744.260.3360.4190.2460.90.540.90.830.540.83
22388.6 13.065.22 40.5231.82 46.530.3460.4130.2410.850.850.850.850.850.851006.971006.2839.7232.1146.990.3390.4170.2441.20.931.20.50.930.5
23396.82 12.925.12 46.9730.57 45.030.3390.4180.2440.850.850.850.850.850.85945.88945.5947.1730.2945.430.340.4140.2461.010.781.010.730.780.73
24407.02 13.824.78 47.8730.15 49.860.3390.420.240.850.850.850.850.850.85991.09990.648.3529.9549.640.3430.4180.241.070.681.070.650.680.65
25416.46 11.945.76 50.7533.73 42.030.3370.4140.2490.850.850.850.850.850.85972.64972.9750.9533.3942.570.3380.410.2520.880.930.880.890.930.89
26426.62 13.44.16 49.9530.72 55.690.3390.4230.2380.850.850.850.850.850.85974.04973.7250.7930.6354.570.3450.4220.2331.110.611.110.590.610.59
27438.9 14.685.16 46.7734.24 55.840.3450.4160.2390.850.850.850.850.850.851206.961206.5446.4234.5955.360.3420.4210.2371.210.81.210.490.80.49
28447.74 14.47.34 55.3136.55 43.470.3360.4130.2510.850.850.850.850.850.851273.481274.0955.6136.2243.870.3380.4090.2530.830.970.830.950.970.95
AVE:30.56.85 13.045.96 37.9624.61 33.070.3370.4160.2470.850.850.850.850.850.85768.94768.937.9624.6133.070.3370.4160.2470.90.880.90.870.880.87

VIII. The Normalized Quadratic cost function as a flexible functional form.

Diewert (1976) defines a flexible function form as follows. Write c(w) as the unit Normalized Quadratic cost function, and c*(w) as the unit CES cost function. Then the unit Normalized Quadratic cost function, c(w), is a flexible functional form if it has enough free parameters to satisfy:

c(w) = c*(w),

c(w) = c*(w), and

2c(w) = 2c*(w),

at any factor price vector w = [wL, wK, wM]T in the domain of definition of c(w).

Since both c(w) and c*(w) are linear homogeneous functions (Diewert and Fox, (2009), 155), the 6 free parameters of c(w) classify the Normalized Quadratic cost function as a "parsimonious flexible functional form" within the present context.

However, when we estimate the unit Normalized Quadratic cost function, c(w), we choose these free parameters to minimize the distance between c(w) and c*(w) at the set of data points {q, wL, wK, wM, c*(w)}, subject to the augmented restrictions matrix, [R | r].

Consequently, 2c*(w) and 2c(w) can differ, as seen by comparing the Uzawa elasticities in the table above, and the second order partial derivatives of c*(w) and c(w) in the table below.

IX. Table of Results: check that second order partial derivatives of the estimated Normalized Quadratic cost function for a given level of output agree with the second order partial derivatives of the CES cost function. If all of the eigenvalues of 2c*(w) and 2c(w) are nonpositive, the cost functions c*(w*) and c(w*) are concave in factor prices.

(Write:   ∂(∂c(wL,wK,wM)/∂wL)/∂wL = cLL,   ∂(∂c(wL,wK,wM)/∂wL)/∂wK = cLK,   etc.)

CES Production Function:  Returns to Scale = 1, Elasticity of Substitution = 0.85
Normalized Quadratic:  Returns to Scale = 1;   Base Prices: wL* = 7, wK* = 13, wM* = 6
—     —   CES Cost Data   —     —     —     —   Normalized Quadratic Cost Data   —     —
Factor Prices2c*(w)Eigenvalues 2c(w)Eigenvalues
obs #qwLwKwM cLLcLKcLMcKKcKMcMMe1e2e3 cLLcLKcLMcKKcKMcMMe1e2e3
1176.5413.865.64 -0.1120.0330.048-0.0270.028-0.125-0.167-0.097-0 -0.110.0350.042-0.0280.029-0.119-0.157-0.1010
218514.86.8 -0.1850.0390.051-0.0240.023-0.086-0.21-0.0840 -0.1270.0260.037-0.0250.035-0.105-0.155-0.1020
3197.214.646.04 -0.0990.0310.044-0.0260.027-0.117-0.153-0.09-0 -0.1010.0330.04-0.0270.026-0.112-0.147-0.0930
4205.6611.26.12 -0.1320.0410.046-0.0370.029-0.096-0.167-0.098-0 -0.1280.0350.054-0.0380.038-0.119-0.178-0.1070
5217.4213.244.22 -0.0840.030.053-0.0280.036-0.208-0.23-0.09-0 -0.1010.0430.043-0.0310.021-0.141-0.169-0.1050
6227.2213.345.3 -0.0920.0320.046-0.0290.031-0.141-0.17-0.0930 -0.1030.0380.044-0.0310.026-0.125-0.159-0.099-0
7235.2612.466.14 -0.1550.0410.05-0.0310.027-0.098-0.188-0.096-0 -0.1310.0320.047-0.0320.038-0.117-0.172-0.108-0
8246.5413.287.52 -0.1170.0360.039-0.0310.024-0.076-0.144-0.08-0 -0.110.0280.046-0.0320.033-0.097-0.15-0.09-0
9256.7612.964.34 -0.0970.0320.055-0.0290.035-0.191-0.218-0.0990 -0.110.0430.044-0.0310.025-0.142-0.173-0.110
10267.7611.025.6 -0.0770.0340.04-0.0410.034-0.122-0.148-0.093-0 -0.0980.0410.055-0.0420.025-0.126-0.169-0.097-0
11277.5213.184.2 -0.0820.030.053-0.0290.037-0.21-0.232-0.088-0 -0.10.0430.043-0.0310.021-0.142-0.169-0.1040
12285.4812.646.38 -0.1470.040.048-0.0310.026-0.093-0.179-0.092-0 -0.1270.0310.047-0.0320.037-0.113-0.167-0.105-0
13296.8214.547.98 -0.1150.0330.037-0.0280.022-0.072-0.14-0.0750 -0.1040.0260.041-0.0290.031-0.091-0.14-0.085-0
14307136 -0.0990.0330.043-0.0310.029-0.112-0.149-0.093-0 -0.1060.0360.046-0.0320.029-0.116-0.157-0.096-0
15317.1811.027.78 -0.0950.0380.034-0.0430.026-0.069-0.123-0.0840 -0.1060.030.055-0.0430.033-0.097-0.157-0.0890
16325.1613.327.8 -0.1740.0420.044-0.0290.022-0.067-0.196-0.0740 -0.1280.0240.044-0.0320.038-0.095-0.158-0.0960
17338.7813.026.94 -0.0710.030.034-0.0350.027-0.094-0.119-0.0810 -0.0870.0340.047-0.0350.024-0.104-0.143-0.0830
18345.112.567.64 -0.1720.0430.044-0.0320.023-0.067-0.195-0.0760 -0.1310.0250.047-0.0340.04-0.097-0.164-0.0980
19357.3411.387.6 -0.0920.0360.034-0.0410.027-0.073-0.121-0.085-0 -0.1040.0310.054-0.0410.031-0.099-0.155-0.088-0
20368.111.124.3 -0.0680.0310.047-0.0390.04-0.194-0.215-0.0860 -0.0940.0480.053-0.0420.019-0.148-0.181-0.103-0
21375.814.344.86 -0.1330.0340.058-0.0240.03-0.157-0.204-0.11-0 -0.1210.0360.038-0.0240.029-0.131-0.164-0.1120
22388.613.065.22 -0.0690.0290.042-0.0320.033-0.151-0.172-0.08-0 -0.0880.040.045-0.0340.02-0.125-0.156-0.091-0
23396.8212.925.12 -0.0990.0330.049-0.030.032-0.145-0.177-0.098-0 -0.1090.0390.045-0.0310.027-0.129-0.165-0.1040
24407.0213.824.78 -0.0960.0310.051-0.0270.032-0.168-0.196-0.095-0 -0.1050.0390.041-0.0280.024-0.131-0.161-0.1030
25416.4611.945.76 -0.1080.0360.045-0.0340.03-0.114-0.156-0.0990 -0.1140.0370.051-0.0360.032-0.122-0.169-0.103-0
26426.6213.44.16 -0.1010.0320.058-0.0270.036-0.206-0.234-0.10 -0.1120.0420.041-0.0280.024-0.144-0.172-0.112-0
27438.914.685.16 -0.0680.0260.043-0.0270.031-0.163-0.182-0.0760 -0.0850.0380.04-0.0290.018-0.12-0.146-0.088-0
28447.7414.47.34 -0.0920.0310.036-0.0290.024-0.086-0.126-0.08-0 -0.0950.030.042-0.030.027-0.097-0.139-0.0840
AVE:30.56.85 13.045.96 -0.1080.0340.045-0.0310.029-0.125-0.175-0.089-0 -0.1080.0350.045-0.0330.029-0.118-0.16-0.098-0

X. The Factor Demand Price Elasticities of the Normalized Quadratic cost function

The estimated factor demand elasticities are obtained by:

 εL,wL = ∂ln(l(wL,wK,wM))/∂ln(wL) = wL * cLL / l(wL,wK,wM), εL,wK = ∂ln(l(wL,wK,wM))/∂ln(wK) = wK * cLK / l(wL,wK,wM), εL,wM = ∂ln(l(wL,wK,wM))/∂ln(wM) = wM * cLM / l(wL,wK,wM), εK,wL = ∂ln(k(wL,wK,wM))/∂ln(wL) = wL * cKL / k(wL,wK,wM), etc.

XI. Table of Results: check that factor demand price elasticities of the estimated Normalized Quadratic cost function for a given level of output and factor prices agree with the factor demand price elasticities of the CES cost function.

CES Production Function:  Returns to Scale = 1, Elasticity of Substitution = 0.85
Normalized Quadratic:  Returns to Scale = 1;   Base Prices: wL* = 7, wK* = 13, wM* = 6
—     —   CES Cost Data   —     —       —     —   Normalized Quadratic Cost Data   —     —
Factor PricesFactor Demand Price ElasticitiesFactor Demand Price Elasticities
obs #qwLwKwM εL,wLεL,wKεL,wMεK,wKεK,wMεM,wM εL,wLεL,wKεL,wMεK,wKεK,wMεM,wM
1176.5413.865.64 -0.5660.3570.209-0.4930.209-0.641 -0.5560.3740.182-0.3020.125-0.519
218514.86.8 -0.5760.3610.215-0.4890.215-0.635 -0.410.2460.163-0.2380.155-0.458
3197.214.646.04 -0.5650.3560.209-0.4940.209-0.641 -0.5730.3830.19-0.3140.125-0.53
4205.6611.26.12 -0.5670.3510.215-0.4990.215-0.635 -0.5440.2950.249-0.3220.173-0.547
5217.4213.244.22 -0.5580.3570.201-0.4930.201-0.649 -0.6710.510.161-0.3650.08-0.533
6227.2213.345.3 -0.5610.3540.207-0.4960.207-0.643 -0.6240.4270.196-0.3450.114-0.554
7235.2612.466.14 -0.5710.3560.215-0.4940.215-0.635 -0.4840.2810.203-0.280.162-0.502
8246.5413.287.52 -0.5680.3510.216-0.4990.216-0.634 -0.5270.2750.252-0.3160.181-0.538
9256.7612.964.34 -0.5610.3570.203-0.4930.203-0.647 -0.6280.4680.161-0.3350.091-0.521
10267.7611.025.6 -0.5570.3470.21-0.5030.21-0.64 -0.720.4290.291-0.4360.134-0.668
11277.5213.184.2 -0.5580.3560.201-0.4940.201-0.649 -0.680.5170.163-0.3730.078-0.538
12285.4812.646.38 -0.570.3550.215-0.4950.215-0.635 -0.4910.280.211-0.2860.165-0.508
13296.8214.547.98 -0.5690.3530.216-0.4970.216-0.634 -0.5120.2730.239-0.3050.177-0.526
14307136 -0.5630.3530.211-0.4970.211-0.639 -0.5990.3740.225-0.3410.139-0.564
15317.1811.027.78 -0.5620.3440.219-0.5060.219-0.631 -0.6150.2680.347-0.380.205-0.611
16325.1613.327.8 -0.5750.3550.22-0.4950.22-0.63 -0.4280.2050.223-0.2740.195-0.48
17338.7813.026.94 -0.5580.3470.212-0.5030.212-0.638 -0.6880.3940.294-0.4130.148-0.648
18345.112.567.64 -0.5740.3540.22-0.4960.22-0.63 -0.440.2030.237-0.2830.2-0.487
19357.3411.387.6 -0.5620.3450.217-0.5050.217-0.633 -0.6210.2870.334-0.3810.195-0.615
20368.111.124.3 -0.5530.3490.203-0.5010.203-0.647 -0.7890.5520.237-0.4860.083-0.661
21375.814.344.86 -0.5680.3620.206-0.4880.206-0.644 -0.5110.3780.133-0.2560.103-0.464
22388.613.065.22 -0.5560.3510.205-0.4990.205-0.645 -0.7250.4980.227-0.4270.099-0.623
23396.8212.925.12 -0.5620.3550.207-0.4950.207-0.643 -0.6120.420.192-0.3360.115-0.546
24407.0213.824.78 -0.5620.3570.204-0.4930.204-0.646 -0.6110.4490.163-0.3250.097-0.519
25416.4611.945.76 -0.5630.3520.212-0.4980.212-0.638 -0.5950.360.235-0.3410.146-0.567
26426.6213.44.16 -0.5610.3590.202-0.4910.202-0.648 -0.6110.4690.142-0.3150.084-0.495
27438.914.685.16 -0.5570.3540.203-0.4960.203-0.647 -0.6990.510.189-0.3970.087-0.574
28447.7414.47.34 -0.5640.3510.213-0.4990.213-0.637 -0.5830.3380.245-0.3390.157-0.566
AVE:30.56.85 13.045.96 -0.5640.3540.21-0.4960.21-0.64 -0.5910.3740.217-0.340.136-0.549

XII. Curvature of the Normalized Quadratic cost function.

Approximating a CES cost function with a Normalized Quadratic cost function provides a "best of all possible worlds" test for the effectiveness of the Normalized Quadratic cost function. Diewert and Wales (1987) prove that a necessary and sufficient condition for the estimated cost function, c(w), to be concave in factor prices is that the estimated matrix:

 D = dLL dLK dLM dKL dKK dKM dML dMK dMM
=
 -2.74629 0.923608 1.20285 0.923608 -0.840304 0.743117 1.20285 0.743117 -3.01341

be a symmetric, negative semidefinite matrix.

The eigenvalues of the matrix D are:

e1 = -4.092, e2 = -2.508, e3 = 0.

The matrix D is negative semidefinite, since its three eigenvalues are nonpositive.

Furthermore, at the (constant) reference vector, v*, of base prices:

v* = w* / (wL* + wK* + wM*) = [vL*, vK*, vM*]T = [0.2692, 0.5, 0.2308]T = [7, 13, 6]T / 26, and

D * v* = 0,

making v* the eigenvector corresponding to the zero eigenvalue of the D matrix.

If D is a symmetric, negative semidefinite matrix, then -D is a symmetric, positive semidefinite matrix that can be decomposed into the product of a lower triangular matrix L and its transpose LT:

-D = L * LT, where

 L = bLL 0 0 bKL bKK 0 bML bMK bMM
 LT = bLL bKL bML 0 bKK bMK 0 0 bMM

Applying the reference vector, v*, to LT, we get the equations:

 bLL * vL* + bKL * vK* + bML * vM* = 0, bKK * vK* + bMK * vM* = 0, bMM * vM* = 0. bMM = 0, bMK = - bKK * vK* / vM*, bML = -(bLL * vL* + bKL * vK*) / vM*.

Consequently, the matrices L, LT, -D, and D can be expressed in terms of three parameters, bLL, bKL, and bKK.

 -D = B = bLL^2 bLL*bKL bLL*(-bLL*vL* - bKL*vK*)/vM* bLL*bKL bKL^2 + bKK^2 bKL*(-bLL*vL*-bKL*vK*)/vM* - bKK^2 * vK*/vM* bLL*(-bLL*vL* - bKL*vK*)/vM* bKL*(-bLL*vL* - bKL*vK*)/vM* - bKK^2 * vK*/vM* (bLL*vL* + bKL*vK*)^2 /vM*^2 + bKK^2 * vK*^2 /vM*^2
=
 bLL 0 0 bKL bKK 0 -(bLL*vL*+bKL*vK*)/vM* -bKK*vK*/vM* 0
*
 bLL bKL -(bLL*vL*+bKL*vK*)/vM* 0 bKK -bKK*vK*/vM* 0 0 0

The -B matrix is a reparameterization of the of the D matrix. By construction -B contains all the restrictions inherent in the restrictions matrix and vector, R, r, of the original parameters, and by construction the -B matrix is a symmetric, negative semidefinite matrix.

The reparameterized unit normalized quadratic cost function can be written as:

ç(w) = bT * w + (1/2) * wT * (-B) * w / (1T * w),

a nonlinear function in its parameters, b = [bL, bK, bM]T, and (-B), i.e. parameters bL, bK, bM, bLL, bKL, and bKK.

If c*(w) represents the unit cost function that we want to approximate at the set of data points {q, wL, wK, wM, c*(w)}, our objective is to use a nonlinear regression algorithm to obtain values for these six free parameters that minimize the distance between ç(w) and c*(w) at the set of data points.

Of course, if the symmetric matrix D as estimated is already a negative semidefinite matrix, the reparameterization of the unit cost function, c(w), and the reestimation of ç(w) with a nonlinear regression algorithm are unnecessary.

Equating the parameters of the normalized quadratic cost function as estimated, c(w), and its reparameterization, ç(w), we determine:

bL = cL, bK = cK, bM = cM, and

 bLL^2 = -dLL, bLL*bKL = - dKL, and bKL^2 + bKK^2 = -dKK. bLL = (-dLL)^1/2, bKL = - dKL / bLL, and bKK = (-dKK - bKL^2)^1/2.

If restricted to using real numbers, we can always transform the estimated b and -B parameters into c and D parameters, and the estimated c and D parameters into b and -B parameters if the matrix D is negative semidefinite. If the matrix D is not negative semidefinite, the transformation from the estimated c and D parameters into b and -B parameters will require the use of complex numbers.

Transforming the estimated c and D parameters into b and B parameters, we obtain:

bL = 1.233527, bK = 0.80536, bM = 1.067696, and

 bLL = (-dLL)^1/2 = 1.6571923 + 0*i, bKL = - dKL / bLL = -0.5573329 + 0*i, and bKK = (-dKK - bKL^2)^1/2 = (0.5296842 + 0*i)^1/2 = 0.7277941 + 0*i.

À2:   Estimate the factor demand equations using nonlinear least squares.

XIII. Nonlinear Estimation of the Normalized Quadratic cost function.

The reparameterized unit normalized quadratic cost function::

ç(w) = bT * w + (1/2) * wT * (-B) * w / (1T * w),

is a nonlinear function in its parameters, b = [bL, bK, bM]T, and (-B), i.e. in its free parameters bL, bK, bM, bLL, bKL, and bKK.

We will approximate the CES unit cost function, c*(w), at the set of data points {q, wL, wK, wM, c*(w)}, with the normalized quadratic cost function, ç(w), by using a nonlinear regression algorithm to obtain values for the b,and (-B) parameters that minimize the distance between ç(w) and c*(w) at the set of data points. Having obtained estimates of the six free parameters, bL, bK, bM, bLL, bKL, and bKK, we will transform them into estimates of the c, and D parameters using c = b, and D = (-B).

Shephard's lemma provides that the gradient vector of the unit cost function is the vector of unit input demand functions:

ç(w) = [l(w), k(w), m(w)]T =b + (-B) * w / ( 1T * w) - (1/2) * wT * (-B) * w * 1 / (1T * w)2.

The reparameterized factor demands as functions of q, vL, vK, and vM are:

 L(q; vL, vK, vM) / q^(1/nu1) = l(vL, vK, vM) = bL + (-bLL^2 * vL - bLL*bKL * vK - (bLL*(-bLL*vL*-bKL*vK*)/vM*) * vM - (1/2) * vT * (-B) * v K(q; vL, vK, vM) / q^(1/nu1) = k(vL, vK, vM) = bK + (-bLL*bKL * vL + (-bKL^2-bKK^2) * vK           + ((-bKL*(-bLL*vL*-bKL*vK*)/vM*)+bKK^2*vK*/vM*)* vM - (1/2) * vT * (-B) * v M(q; vL, vK, vM) / q^(1/nu1) = m(vL, vK, vM) = bM + ((-bLL*(-bLL*vL*-bKL*vK*)/vM*) * vL       + ((-bKL*(-bLL*vL*-bKL*vK*)/vM*)+(bKK^2*vK*)/vM*) * vK           + (-(-bLL*vL*-bKL*vK*)^2/vM*^2-bKK^2*vK*^2/vM*^2) * vM) - (1/2) * vT * (-B) * v

where the (constant) reference vector, v*, of base prices is:

v* = w* / (wL* + wK* + wM*) = [vL*, vK*, vM*]T = [0.2692, 0.5, 0.2308]T = [7, 13, 6]T / 26.

XIV. Newton's Nonlinear Least Squares Method.

Newton's nonlinear least squares method is a generalization of Newton's method for finding a root of the equation, g(x) = 0, where g is a function of the 1-dimensional real variable x. Newton's method calculates a sequence of approximations {xn: n = 0, 1, 2, ...} to g(xn) ~= 0, beginning with a starting value, x0. At each iteration, the new approximation, xn+1, is calculated according to the formula:

xn+1 = xn - g(xn) / g'(xn).

where g'(x) is the derivative of the function g evaluated at the value of the variable x.

Newton's method generalizes for a system of n equations:

G(x) = [g1(x), g2(x), ... , gn(x)]T = 0

where G, and thus g1, g2, ... , gn are functions of the n-dimension real variable (vector) x = [x1, x2, ..., xn]T. Newton's method calculates a sequence of vector approximations {xn: n = 0, 1, 2, ...} to G(xn) ~= 0, beginning with a vector of starting values, x0. At each iteration, the new vector, xn+1, is calculated according to the formula:

xn+1 = xn - J(xn)^-1 * G(xn)

where J(x) is the nxn Jacobian matrix of G(x) defined by:

J(x) = [∂gi / ∂xj],   i, j = 1, ..., n.

At each iteration, Newton's algorithm solves the system of linear equations for the vector y:

J(xn) * y = - G(xn),

and computes the new vector, xn+1, by:

xn+1 = xn + y.

The algorithm stops when the norm of y is less than some tolerance value, TOL, i.e, ||y|| < TOL.

We will use Newton's algorithm to obtain the "least squares fit" between the gradient of the CES unit cost function, c*(w), and the gradient of the normalized quadratic unit cost function, ç(w) at the set of data points {q, wL, wK, wM, c*(w)}.

Let {[li, ki, mi]T = c*(wi)} be the CES factor demands at the data set {qi, wi = [wLi, wKi, wMi]T}.

Objective: Choose parameters bL, bK bM, bLL, bKL, bKK that minimize:

S(bL, bK, bM, bLL, bKL, bKK) = i [ (li - l(wi))^2 + (ki - k(wi))^2 + (mi - m(wi))^2]

over the data set {qi, wi}.

The first order conditions are the system of six equations:

[∂S/∂bL, ∂S/∂bK, ∂S/∂bM, ∂S/∂bLL, ∂S/∂bKL, ∂S/∂bKK]T = 0.

For example:

∂S/∂bL = 2 * i [ (li - l(wi)) * ∂l(wi)/∂bL + (ki - k(wi)) * ∂k(wi)/∂bL + (mi - m(wi)) * ∂m(wi)/∂bL] = 0.

The Jacobian of the first order conditions, also the Hessian matrix of the function S(bL, bK, bM, bLL, bKL, bKK), is the 6 by 6 matrix of second order partial derivatives:

J(bL, bK, bM, bLL, bKL, bKK) = [ ∂2S / ∂b_∂b_ ]

for all b_ ∈ {bL, bK, bM, bLL, bKL, bKK}.

(In practice, only the first order partial derivatives of l(wi), k(wi), and m(wi) of the Jacobian are used.)

Using Newton's nonlinear regression algorithm as discussed, we get the following estimates:

Nonlinear Least Squares: Newton's Method
Parameter Estimates
Iter #bLbKbMbLLbKLbKK
0 1111-11
11.2327820.8052371.0687561.835327-0.0577731.355983
21.2327820.8052371.0687561.645233-0.49260.963368
31.2327820.8052371.0687561.634251-0.546130.757861
41.2327820.8052371.0687561.634214-0.5465020.727839
51.2327820.8052371.0687561.634214-0.5465020.72722
61.2327820.8052371.0687561.634214-0.5465020.727219

Next, we transform these estimates of the six free parameters, bL, bK, bM, bLL, bKL, and bKK into estimates of the c, and D parameters using c = b, and D = (-B). By construction, the estimated matrix D is negative semidefinite.

XV. As estimated using nonlinear least squares, the parameters of the Normalized Quadratic cost function are:

c = [cL, cK, cM]T = [1.232782, 0.805237, 1.068756], and

 D = dLL dLK dLM dKL dKK dKM dML dMK dMM
=
 -2.67065 0.893101 1.18071 0.893101 -0.827512 0.750992 1.18071 0.750992 -3.00465

The eigenvalues of the matrix D are:

e1 = -4.03, e2 = -2.473, e3 = -0.

The matrix D is negative semidefinite, since its three eigenvalues are nonpositive.

The Normalized Quadratic cost function is:

C(q;vL,vK,vM) / q^(1/1) = c(vL,vK,vM) = 1.232782 * vL + 0.805237 * vK + 1.068756 * vM + (1/2) *
 [-2.670654 * (vL*vL) + 0.893101 * (vL*vK) + 1.180712 * (vL*vM) + 0.893101 * (vK*vL) + -0.827512 * (vK*vK) + 0.750992 * (vK*vM) + 1.180712 * (vM*vL) + 0.750992 * (vM*vK) + -3.004647 * (vM*vM)]

The factor demand functions are:

L(q;vL,vK,vM) / q^(1/1) = l(vL,vK,vM) = 1.232782 + -2.670654 * vL + 0.893101 * vK + 1.180712 * vM - (1/2) *
 [-2.670654 * (vL*vL) + 0.893101 * (vL*vK) + 1.180712 * (vL*vM) + 0.893101 * (vK*vL) + -0.827512 * (vK*vK) + 0.750992 * (vK*vM) + 1.180712 * (vM*vL) + 0.750992 * (vM*vK) + -3.004647 * (vM*vM)]

K(q;vL,vK,vM) / q^(1/1) = k(vL,vK,vM) = 0.805237 + 0.893101 * vL + -0.827512 * vK + 0.750992 * vM - (1/2) *
 [-2.670654 * (vL*vL) + 0.893101 * (vL*vK) + 1.180712 * (vL*vM) + 0.893101 * (vK*vL) + -0.827512 * (vK*vK) + 0.750992 * (vK*vM) + 1.180712 * (vM*vL) + 0.750992 * (vM*vK) + -3.004647 * (vM*vM)]

M(q;vL,vK,vM) / q^(1/1) = m(vL,vK,vM) = 1.068756 + 1.180712 * vL + 0.750992 * vK + -3.004647 * vM - (1/2) *
 [-2.670654 * (vL*vL) + 0.893101 * (vL*vK) + 1.180712 * (vL*vM) + 0.893101 * (vK*vL) + -0.827512 * (vK*vK) + 0.750992 * (vK*vM) + 1.180712 * (vM*vL) + 0.750992 * (vM*vK) + -3.004647 * (vM*vM)]

XVI. Table of Results: check that the costs, factor inputs, and Uzawa elasticities from the estimated Normalized Quadratic cost function at given levels of output agree with those obtained from the CES cost function.

CES Production Function:  Returns to Scale = 1, Elasticity of Substitution = 0.85
Normalized Quadratic:  Returns to Scale = 1;   Base Prices: wL* = 7, wK* = 13, wM* = 6
—   —   CES Cost Data   —   —     —   Normalized Quadratic Cost Data   —
Factor PricesFactor InputsFactor SharesUzawa Elasticities  Factor InputsFactor SharesUzawa Elasticities
obs #qwLwKwM LKMsLsKsM uLKuLMuKLuKMuMLuMKcostest costest Lest Kest MsLsKsMuLKuLMuKLuKMuMLuMK
1176.54 13.865.64 21.9312.98 18.690.3350.420.2460.850.850.850.850.850.85428.69428.522.0112.8118.970.3360.4140.250.880.720.880.870.720.87
2185 14.86.8 28.8412.84 16.680.3220.4250.2530.850.850.850.850.850.85447.7448.3927.8312.9817.220.310.4280.2610.560.620.561.280.621.28
3197.2 14.646.04 24.0214.72 20.950.3360.4180.2460.850.850.850.850.850.85514.89514.7224.1114.5421.240.3370.4130.2490.90.750.90.840.750.84
4205.66 11.26.12 26.4216.57 18.570.3330.4140.2530.850.850.850.850.850.85448.82449.0426.5716.5118.580.3350.4120.2530.690.970.691.110.971.11
5217.42 13.244.22 23.416.02 28.40.3430.420.2370.850.850.850.850.850.85505.53505.5223.5616.1427.740.3460.4230.2321.170.681.170.510.680.51
6227.22 13.345.3 26.1417.38 25.540.340.4170.2430.850.850.850.850.850.85555.96555.7826.2117.2325.780.3410.4140.24610.7810.710.780.71
7235.26 12.466.14 32.9217.72 21.680.3290.4190.2530.850.850.850.850.850.85527.06527.2732.6817.721.950.3260.4180.2560.650.780.651.180.781.18
8246.54 13.287.52 32.4219.89 21.630.3320.4130.2550.850.850.850.850.850.85638.9639.1632.619.9121.490.3340.4140.2530.640.980.641.180.981.18
9256.76 12.964.34 29.2818.87 32.060.340.420.2390.850.850.850.850.850.85581.61581.3629.5718.8131.740.3440.4190.2371.080.661.080.610.660.61
10267.76 11.025.6 28.0723.34 27.830.3450.4080.2470.850.850.850.850.850.85630.91631.0627.6623.2528.60.340.4060.2541.021.121.020.641.120.64
11277.52 13.184.2 29.7820.7 36.710.3440.4190.2370.850.850.850.850.850.85650.9965129.9320.935.820.3460.4230.2311.180.691.180.490.690.49
12285.48 12.646.38 39.6821.84 26.190.3290.4180.2530.850.850.850.850.850.85660.63660.939.4721.8326.430.3270.4180.2550.650.820.651.180.821.18
13296.82 14.547.98 40.0123.55 26.30.3310.4150.2540.850.850.850.850.850.85825.12825.4640.0723.5826.230.3310.4150.2540.640.930.641.20.931.2
14307 136 36.8924.42 31.590.3370.4150.2480.850.850.850.850.850.85765.17765.336.9824.1632.060.3380.410.2510.880.880.880.860.880.86
15317.18 11.027.78 37.5129.19 26.320.3380.4040.2570.850.850.850.850.850.85795.84796.4938.4529.1225.640.3470.4030.250.641.360.641.091.361.09
16325.16 13.327.8 49.9524.99 26.410.3240.4180.2590.850.850.850.850.850.85796.63797.1848.9925.7125.880.3170.430.2530.460.870.461.480.871.48
17338.78 13.026.94 36.8729.55 33.830.3430.4080.2490.850.850.850.850.850.85943.21943.8736.6629.3334.60.3410.4050.2540.941.130.940.731.130.73
18345.1 12.567.64 52.0927.12 27.760.3250.4160.2590.850.850.850.850.850.85818.41818.5751.4727.926.920.3210.4280.2510.460.930.461.480.931.48
19357.34 11.387.6 42.0832.47 30.690.3390.4050.2560.850.850.850.850.850.85911.63912.5642.8832.3230.280.3450.4030.2520.691.30.691.041.31.04
20368.1 11.124.3 36.0530.85 46.40.350.4110.2390.850.850.850.850.850.85834.64833.3934.8331.5146.710.3390.420.2411.260.951.260.390.950.39
21375.8 14.344.86 50.1526.03 43.790.3320.4260.2430.850.850.850.850.850.85876.95875.7150.4725.6344.340.3340.420.2460.880.530.880.840.530.84
22388.6 13.065.22 40.5231.82 46.530.3460.4130.2410.850.850.850.850.850.851006.971006.3639.8832.0246.960.3410.4160.2441.160.911.160.520.910.52
23396.82 12.925.12 46.9730.57 45.030.3390.4180.2440.850.850.850.850.850.85945.88945.5547.1530.2845.460.340.4140.2460.980.770.980.740.770.74
24407.02 13.824.78 47.8730.15 49.860.3390.420.240.850.850.850.850.850.85991.09990.5348.3229.9549.670.3420.4180.241.040.671.040.660.670.66
25416.46 11.945.76 50.7533.73 42.030.3370.4140.2490.850.850.850.850.850.85972.64972.9650.9133.3942.620.3380.410.2520.850.910.850.90.910.9
26426.62 13.44.16 49.9530.72 55.690.3390.4230.2380.850.850.850.850.850.85974.04973.6450.7630.6354.610.3450.4220.2331.080.591.080.60.590.6
27438.9 14.685.16 46.7734.24 55.840.3450.4160.2390.850.850.850.850.850.851206.961206.5346.5534.5155.350.3430.420.2371.180.781.180.510.780.51
28447.74 14.47.34 55.3136.55 43.470.3360.4130.2510.850.850.850.850.850.851273.481274.155.5636.2343.930.3380.4090.2530.80.950.80.960.950.96
AVE:30.56.85 13.045.96 37.9324.6 33.10.3370.4160.2470.850.850.850.850.850.85768.94768.9637.9324.633.10.3360.4160.2480.870.860.870.880.860.88

XVII. Table of Results: check that second order partial derivatives of the estimated Normalized Quadratic cost function for a given level of output agree with the second order partial derivatives of the CES cost function. If all of the eigenvalues of 2c*(w) and 2c(w) are nonpositive, the cost functions c*(w*) and c(w*) are concave in factor prices.

CES Production Function:  Returns to Scale = 1, Elasticity of Substitution = 0.85
Normalized Quadratic:  Returns to Scale = 1;   Base Prices: wL* = 7, wK* = 13, wM* = 6
—     —   CES Cost Data   —     —     —     —   Normalized Quadratic Cost Data   —     —
Factor Prices2c*(w)Eigenvalues 2c(w)Eigenvalues
obs #qwLwKwM cLLcLKcLMcKKcKMcMMe1e2e3 cLLcLKcLMcKKcKMcMMe1e2e3
1176.5413.865.64 -0.1120.0330.048-0.0270.028-0.125-0.167-0.097-0 -0.1070.0340.041-0.0280.029-0.119-0.155-0.0990
218514.86.8 -0.1850.0390.051-0.0240.023-0.086-0.21-0.0840 -0.1240.0250.037-0.0250.036-0.105-0.152-0.1010
3197.214.646.04 -0.0990.0310.044-0.0260.027-0.117-0.153-0.09-0 -0.0980.0320.039-0.0270.027-0.111-0.145-0.0920
4205.6611.26.12 -0.1320.0410.046-0.0370.029-0.096-0.167-0.098-0 -0.1250.0340.053-0.0380.038-0.119-0.175-0.106-0
5217.4213.244.22 -0.0840.030.053-0.0280.036-0.208-0.23-0.09-0 -0.0980.0420.042-0.030.022-0.141-0.167-0.1030
6227.2213.345.3 -0.0920.0320.046-0.0290.031-0.141-0.17-0.0930 -0.10.0370.043-0.030.026-0.124-0.157-0.0970
7235.2612.466.14 -0.1550.0410.05-0.0310.027-0.098-0.188-0.096-0 -0.1280.0310.046-0.0320.038-0.116-0.169-0.1070
8246.5413.287.52 -0.1170.0360.039-0.0310.024-0.076-0.144-0.08-0 -0.1070.0270.045-0.0320.033-0.097-0.147-0.089-0
9256.7612.964.34 -0.0970.0320.055-0.0290.035-0.191-0.218-0.0990 -0.1070.0410.043-0.030.025-0.142-0.17-0.108-0
10267.7611.025.6 -0.0770.0340.04-0.0410.034-0.122-0.148-0.093-0 -0.0960.040.054-0.0410.026-0.126-0.167-0.096-0
11277.5213.184.2 -0.0820.030.053-0.0290.037-0.21-0.232-0.088-0 -0.0970.0420.042-0.0310.021-0.141-0.167-0.102-0
12285.4812.646.38 -0.1470.040.048-0.0310.026-0.093-0.179-0.092-0 -0.1230.030.046-0.0320.037-0.112-0.164-0.103-0
13296.8214.547.98 -0.1150.0330.037-0.0280.022-0.072-0.14-0.0750 -0.1010.0250.041-0.0290.031-0.091-0.137-0.0840
14307136 -0.0990.0330.043-0.0310.029-0.112-0.149-0.093-0 -0.1030.0340.045-0.0320.029-0.116-0.155-0.095-0
15317.1811.027.78 -0.0950.0380.034-0.0430.026-0.069-0.123-0.0840 -0.1030.0290.054-0.0420.033-0.097-0.155-0.0880
16325.1613.327.8 -0.1740.0420.044-0.0290.022-0.067-0.196-0.0740 -0.1240.0230.043-0.0310.039-0.095-0.156-0.0950
17338.7813.026.94 -0.0710.030.034-0.0350.027-0.094-0.119-0.0810 -0.0840.0320.046-0.0350.024-0.103-0.141-0.081-0
18345.112.567.64 -0.1720.0430.044-0.0320.023-0.067-0.195-0.0760 -0.1280.0240.047-0.0340.04-0.097-0.162-0.097-0
19357.3411.387.6 -0.0920.0360.034-0.0410.027-0.073-0.121-0.085-0 -0.1010.030.053-0.040.032-0.099-0.153-0.087-0
20368.111.124.3 -0.0680.0310.047-0.0390.04-0.194-0.215-0.0860 -0.0910.0460.052-0.0410.019-0.147-0.178-0.1010
21375.814.344.86 -0.1330.0340.058-0.0240.03-0.157-0.204-0.11-0 -0.1170.0350.037-0.0240.029-0.131-0.161-0.1110
22388.613.065.22 -0.0690.0290.042-0.0320.033-0.151-0.172-0.08-0 -0.0860.0390.045-0.0340.02-0.124-0.154-0.09-0
23396.8212.925.12 -0.0990.0330.049-0.030.032-0.145-0.177-0.098-0 -0.1060.0380.045-0.0310.027-0.129-0.163-0.1020
24407.0213.824.78 -0.0960.0310.051-0.0270.032-0.168-0.196-0.095-0 -0.1020.0380.04-0.0280.025-0.131-0.159-0.1020
25416.4611.945.76 -0.1080.0360.045-0.0340.03-0.114-0.156-0.0990 -0.1110.0360.05-0.0350.032-0.122-0.167-0.1020
26426.6213.44.16 -0.1010.0320.058-0.0270.036-0.206-0.234-0.10 -0.1080.0410.04-0.0280.025-0.143-0.17-0.110
27438.914.685.16 -0.0680.0260.043-0.0270.031-0.163-0.182-0.0760 -0.0820.0360.039-0.0290.019-0.12-0.144-0.087-0
28447.7414.47.34 -0.0920.0310.036-0.0290.024-0.086-0.126-0.08-0 -0.0930.0290.041-0.0290.027-0.097-0.136-0.0830
AVE:30.56.85 13.045.96 -0.1080.0340.045-0.0310.029-0.125-0.175-0.089-0 -0.1050.0340.045-0.0320.029-0.118-0.158-0.0970

XVIII. Table of Results: check that factor demand price elasticities of the estimated Normalized Quadratic cost function for a given level of output and factor prices agree with the factor demand price elasticities of the CES cost function.

CES Production Function:  Returns to Scale = 1, Elasticity of Substitution = 0.85
Normalized Quadratic:  Returns to Scale = 1;   Base Prices: wL* = 7, wK* = 13, wM* = 6
—     —   CES Cost Data   —     —       —     —   Normalized Quadratic Cost Data   —     —
Factor PricesFactor Demand Price ElasticitiesFactor Demand Price Elasticities
obs #qwLwKwM εL,wLεL,wKεL,wMεK,wKεK,wMεM,wM εL,wLεL,wKεL,wMεK,wKεK,wMεM,wM
1176.5413.865.64 -0.5660.3570.209-0.4930.209-0.641 -0.5420.3630.179-0.2980.126-0.518
218514.86.8 -0.5760.3610.215-0.4890.215-0.635 -0.4010.2390.162-0.2370.157-0.46
3197.214.646.04 -0.5650.3560.209-0.4940.209-0.641 -0.5580.3710.187-0.3090.127-0.53
4205.6611.26.12 -0.5670.3510.215-0.4990.215-0.635 -0.5310.2850.246-0.3190.174-0.546
5217.4213.244.22 -0.5580.3570.201-0.4930.201-0.649 -0.6510.4940.157-0.3580.081-0.531
6227.2213.345.3 -0.5610.3540.207-0.4960.207-0.643 -0.6060.4140.192-0.3390.115-0.552
7235.2612.466.14 -0.5710.3560.215-0.4940.215-0.635 -0.4730.2720.201-0.2780.163-0.503
8246.5413.287.52 -0.5680.3510.216-0.4990.216-0.634 -0.5150.2660.249-0.3130.182-0.538
9256.7612.964.34 -0.5610.3570.203-0.4930.203-0.647 -0.6110.4540.157-0.3290.092-0.519
10267.7611.025.6 -0.5570.3470.21-0.5030.21-0.64 -0.6980.4130.285-0.4270.136-0.662
11277.5213.184.2 -0.5580.3560.201-0.4940.201-0.649 -0.660.5010.159-0.3650.08-0.536
12285.4812.646.38 -0.570.3550.215-0.4950.215-0.635 -0.480.2710.208-0.2840.166-0.509
13296.8214.547.98 -0.5690.3530.216-0.4970.216-0.634 -0.4990.2640.235-0.3020.179-0.527
14307136 -0.5630.3530.211-0.4970.211-0.639 -0.5830.3620.221-0.3360.141-0.562
15317.1811.027.78 -0.5620.3440.219-0.5060.219-0.631 -0.5980.2570.341-0.3750.207-0.609
16325.1613.327.8 -0.5750.3550.22-0.4950.22-0.63 -0.4190.1980.221-0.2730.196-0.482
17338.7813.026.94 -0.5580.3470.212-0.5030.212-0.638 -0.6670.380.288-0.4060.149-0.644
18345.112.567.64 -0.5740.3540.22-0.4960.22-0.63 -0.430.1960.235-0.2810.202-0.489
19357.3411.387.6 -0.5620.3450.217-0.5050.217-0.633 -0.6040.2760.328-0.3750.197-0.612
20368.111.124.3 -0.5530.3490.203-0.5010.203-0.647 -0.7620.5320.23-0.4730.085-0.654
21375.814.344.86 -0.5680.3620.206-0.4880.206-0.644 -0.4990.3680.131-0.2530.105-0.465
22388.613.065.22 -0.5560.3510.205-0.4990.205-0.645 -0.7020.4810.221-0.4180.101-0.618
23396.8212.925.12 -0.5620.3550.207-0.4950.207-0.643 -0.5950.4070.189-0.3310.116-0.545
24407.0213.824.78 -0.5620.3570.204-0.4930.204-0.646 -0.5950.4350.159-0.3190.098-0.517
25416.4611.945.76 -0.5630.3520.212-0.4980.212-0.638 -0.5790.3480.231-0.3370.148-0.566
26426.6213.44.16 -0.5610.3590.202-0.4910.202-0.648 -0.5940.4550.139-0.310.085-0.494
27438.914.685.16 -0.5570.3540.203-0.4960.203-0.647 -0.6780.4940.185-0.3880.089-0.571
28447.7414.47.34 -0.5640.3510.213-0.4990.213-0.637 -0.5680.3270.241-0.3340.158-0.565
AVE:30.56.85 13.045.96 -0.5640.3540.21-0.4960.21-0.64 -0.5750.3610.213-0.3350.138-0.547

Mathematical Notes

c(w) = cT * w + (1/2) * wT * D * w / (1T * w),

a linear function in its parameters, c, and D.

 c(wL,wK,wM) = cL * wL + cK * wK + cM * wM + (1/2) * [dLL * (wL*wL) + dLK*(wL*wK) + dLM * (wL*wM)                  + dKL * (wK*wL) + dKK*(wK*wK) + dKM * (wK*wM)              + dML * (wM*wL) + dMK*(wM*wK) + dMM * (wM*wM)] * (wL + wK + wM)^-1

Returns to Scale Function:

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

a continuous, increasing function of q (q >= 1), with h(0) = 0 and h(1) = 1,

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

Define the variable w as:

w = 1 / (wL + wM + wK) = 1 / (1T * w).

Unit Factor Demand Functions:

 l(wL, wK, wM) = cL + (dLL * wL + dLK * wK + dLM * wM) * w - (1/2) * [dLL * wL * wL + dLK * wvL * wvK + dLM * wL * wM           + dKL * wK * wL + dKK * wK * wK + dKM * wK * wM           + dML * wM * wL + dMK * wM * wK + dMM * wM * wM] * w * w k(wL, wK, wM) = cK + (dKL * wL + dKK * wK + dKM * wM) * w - (1/2) * [dLL * wL * wL + dLK * wL * wK + dLM * wL * wM           + dKL * wK * wL + dKK * wK * wK + dKM * wK * wM           + dML * wM * wL + dMK * wM * wK + dMM * wM * wM] * w * w m(wL, wK, wM) = cM + (dML * wL + dMK * wK + dMM * wM) * w - (1/2) * [dLL * wL * wL + dLK * wL * wK + dLM * wL * wM) * w           + dKL * wK * wL + dKK * wK * wK + dKM * wK * wM           + dML * wM * wL + dMK * wM * wK + dMM * wM * wM] * w * w

Unit Factor Demand Functions' Partial Derivatives:

 ∂l/∂wL = dLL * w - (dLL * wL + dLK * wK + dLM * wM + dLL * wL + dKL * wK + dML * wM) * w * w + [dLL * wL * wL + dLK * wL * wK + dLM * wL * wM   + dKL * wK * wL + dKK * wK * wK + dKM * wK * wM   + dML * wM * wL + dMK * wM * wK + dMM * wM * wM] * w * w * w ∂l/∂wK = dLK * w - (dLL * wL + dLK * wK + dLM * wM + dLK * wL + dKK * wK + dMK * wM) * w * w + [dLL * wL * wL + dLK * wL * wK + dLM * wL * wM   + dKL * wK * wL + dKK * wK * wK + dKM * wK * wM   + dML * wM * wL + dMK * wM * wK + dMM * wM * wM] * w * w * w ∂l/∂wM = dLM * w - (dLL * wL + dLK * wK + dLM * wM + dLM * wL + dKM * wK + dMM * wM) * w * w + [dLL * wL * wL + dLK * wL * wK + dLM * wL * wM   + dKL * wK * wL + dKK * wK * wK + dKM * wK * wM   + dML * wM * wL + dMK * wM * wK + dMM * wM * wM] * w * w * w ∂k/∂wL = dKL * w - (dKL * wL + dKK * wK + dKM * wM + dLL * wL + dKL * wK + dML * wM) * w * w + [dLL * wL * wL + dLK * wL * wK + dLM * wL * wM   + dKL * wK * wL + dKK * wK * wK + dKM * wK * wM   + dML * wM * wL + dMK * wM * wK + dMM * wM * wM] * w * w * w ∂k/∂wK = dKK * w - (dKL * wL + dKK * wK + dKM * wM + dLK * wL + dKK * wK + dMK * wM) * w * w + [dLL * wL * wL + dLK * wL * wK + dLM * wL * wM   + dKL * wK * wL + dKK * wK * wK + dKM * wK * wM   + dML * wM * wL + dMK * wM * wK + dMM * wM * wM] * w * w * w ∂k/∂wM = dKM * w - (dKL * wL + dKK * wK + dKM * wM + dLM * wL + dKM * wK + dMM * wM) * w * w + [dLL * wL * wL + dLK * wL * wK + dLM * wL * wM   + dKL * wK * wL + dKK * wK * wK + dKM * wK * wM   + dML * wM * wL + dMK * wM * wK + dMM * wM * wM] * w * w * w ∂m/∂wL = dML * w - (dML * wL + dMK * wK + dMM * wM + dLL * wL + dKL * wK + dML * wM) * w * w + [dLL * wL * wL + dLK * wL * wK + dLM * wL * wM   + dKL * wK * wL + dKK * wK * wK + dKM * wK * wM   + dML * wM * wL + dMK * wM * wK + dMM * wM * wM] * w * w * w ∂m/∂wK = dMK * w - (dML * wL + dMK * wK + dMM * wM + dLK * wL + dKK * wK + dMK * wM) * w * w + [dLL * wL * wL + dLK * wL * wK + dLM * wL * wM   + dKL * wK * wL + dKK * wK * wK + dKM * wK * wM   + dML * wM * wL + dMK * wM * wK + dMM * wM * wM] * w * w * w ∂m/∂wM = dMM * w - (dML * wL + dMK * wK + dMM * wM + dLM * wL + dKM * wK + dMM * wM) * w * w + [dLL * wL * wL + dLK * wL * wK + dLM * wL * wM   + dKL * wK * wL + dKK * wK * wK + dKM * wK * wM   + dML * wM * wL + dMK * wM * wK + dMM * wM * wM] * w * w * w

Normalized Quadratic Cost Function's Uzawa Elasticities:

 uLK = c(wL,wK,wM) * ∂l/∂wK / (l * k); uLM = c(wL,wK,wM) * ∂l/∂wM / (l * m); uKL = c(wL,wK,wM) * ∂k/∂wL / (k * l); uKM = c(wL,wK,wM) * ∂k/∂wM / (k * m); uML = c(wL,wK,wM) * ∂m/∂wL / (m * l); uMK = c(wL,wK,wM) * ∂m/∂wK / (m * k);

Reparameterized Normalized Quadratic Unit Cost Function:

ç(w) = bT * w + (1/2) * wT * (-B) * w / (1T * w),

is a nonlinear function in its parameters, b = [bL, bK, bM]T, and (-B), i.e. in its free parameters bL, bK, bM, bLL, bKL, and bKK.

Reparameterized Normalized Quadratic Unit Cost Function:

 ç(wL,wK,wM) = bL * wL + bK * wK + bM * wM + .5 * (-wL*bLL^2-wK*bLL*bKL-wM*bLL*(-bLL*uL-bKL*uK)/uM) * wL * w + .5 * (-wL*bLL*bKL+wK*(-bKL^2-bKK^2)+wM*(-bKL*(-bLL*uL-bKL*uK)/uM+bKK^2*uK/uM))* wK * w + .5 * (-wL*bLL*(-bLL*uL-bKL*uK)/uM+wK*(-bKL*(-bLL*uL-bKL*uK)/uM+bKK^2*uK/uM)+wM*(-(-bLL*uL-bKL*uK)^2/uM^2-bKK^2*uK^2/uM^2))* wM * w);

where u = w* / (wL* + wK* + wM*) = [uL, uK, uM]T = [vL*, vK*, vM*] (redefined), the vector of base prices.

Shephard's lemma provides that the gradient vector of the unit cost function is the vector of unit input demand functions:

ç(w) = [l(w), k(w), m(w)]T =b + (-B) * w / ( 1T * w) - (1/2) * wT * (-B) * w * 1 / (1T * w)2.

Reparameterized Normalized Quadratic Unit Factor Demand Functions:

 l(wL,wK,wM) = bL + (-bLL^2*wL - bLL*bKL*wK - bLL*(-bLL*uL-bKL*uK)/uM*wM) * w - .5 * (-wL*bLL^2-wK*bLL*bKL-wM*bLL*(-bLL*uL-bKL*uK)/uM) * wL * w * w - .5 * (-wL*bLL*bKL+wK*(-bKL^2-bKK^2)+wM*(-bKL*(-bLL*uL-bKL*uK)/uM+bKK^2*uK/uM))* wK * w * w - .5 * (-wL*bLL*(-bLL*uL-bKL*uK)/uM+wK*(-bKL*(-bLL*uL-bKL*uK)/uM+bKK^2*uK/uM)+wM*(-(-bLL*uL-bKL*uK)^2/uM^2-bKK^2*uK^2/uM^2))* wM * w * w ); k(wL,wK,wM) = bK + (-bLL*bKL*wL + (-bKL^2-bKK^2)*wK + (-bKL*(-bLL*uL-bKL*uK)/uM+bKK^2*uK/uM)* wM) * w - .5 * (-wL*bLL^2-wK*bLL*bKL-wM*bLL*(-bLL*uL-bKL*uK)/uM) * wL * w * w - .5 * (-wL*bLL*bKL+wK*(-bKL^2-bKK^2)+wM*(-bKL*(-bLL*uL-bKL*uK)/uM+bKK^2*uK/uM))* wK * w * w - .5 * (-wL*bLL*(-bLL*uL-bKL*uK)/uM+wK*(-bKL*(-bLL*uL-bKL*uK)/uM+bKK^2*uK/uM)+wM*(-(-bLL*uL-bKL*uK)^2/uM^2-bKK^2*uK^2/uM^2))* wM * w * w ); m(wL,wK,wM) = bM + (-bLL*(-bLL*uL-bKL*uK)/uM*wL + (-bKL*(-bLL*uL-bKL*uK)/uM+bKK^2*uK/uM)*wK + (-(-bLL*uL-bKL*uK)^2/uM^2-bKK^2*uK^2/uM^2)*wM) * w - .5 * (-wL*bLL^2-wK*bLL*bKL-wM*bLL*(-bLL*uL-bKL*uK)/uM) * wL * w * w - .5 * (-wL*bLL*bKL+wK*(-bKL^2-bKK^2)+wM*(-bKL*(-bLL*uL-bKL*uK)/uM+bKK^2*uK/uM))* wK * w * w - .5 * (-wL*bLL*(-bLL*uL-bKL*uK)/uM+wK*(-bKL*(-bLL*uL-bKL*uK)/uM+bKK^2*uK/uM)+wM*(-(-bLL*uL-bKL*uK)^2/uM^2-bKK^2*uK^2/uM^2))* wM * w * w );

Reparameterized Unit Labour Demand Function Partial Derivatives:

 ∂l/∂bL = 1 ∂l/∂bK = 0 ∂l/∂bM = 0
 ∂l/∂bLL = (-2*wL*bLL-wK*bKL-wM*(-bLL*uL-bKL*uK)/uM+wM*bLL*uL/uM) * w -((-wL*bLL-1/2*wK*bKL-1/2*wM*(-bLL*uL-bKL*uK)/uM+1/2*wM*bLL*uL/uM)*wL + (-1/2*wL*bKL+1/2*wM*bKL*uL/uM)*wK + (-1/2*wL*(-bLL*uL-bKL*uK)/uM+1/2*wL*bLL*uL/uM+1/2*wK*bKL*uL/uM+wM*(-bLL*uL-bKL*uK)/uM^2*uL)*wM)* w * w ∂l/∂bKL = (-wK*bLL+wM*bLL*uK/uM) * w -((-1/2*wK*bLL+1/2*wM*bLL*uK/uM)*wL + (-1/2*wL*bLL-wK*bKL+1/2*wM*(-(-bLL*uL-bKL*uK)/uM+bKL*uK/uM))*wK + (1/2*wL*bLL*uK/uM+1/2*wK*(-(-bLL*uL-bKL*uK)/uM+bKL*uK/uM)+wM*(-bLL*uL-bKL*uK)/uM^2*uK)*wM)* w * w ∂l/∂bKK = -((-wK*bKK+wM*bKK*uK/uM)*wK + (wK*bKK*uK/uM-wM*bKK*uK^2/uM^2)*wM)* w * w

Reparameterized Unit Capital Demand Function Partial Derivatives:

 ∂k/∂bL = 0 ∂k/∂bK = 1 ∂k/∂bM = 0
 ∂k/∂bLL = (-wL*bKL+wM*bKL*uL/uM) * w -((-wL*bLL-1/2*wK*bKL-1/2*wM*(-bLL*uL-bKL*uK)/uM+1/2*wM*bLL*uL/uM)*wL + (-1/2*wL*bKL+1/2*wM*bKL*uL/uM)*wK + (-1/2*wL*(-bLL*uL-bKL*uK)/uM+1/2*wL*bLL*uL/uM+1/2*wK*bKL*uL/uM+wM*(-bLL*uL-bKL*uK)/uM^2*uL)*wM)* w * w ∂k/∂bKL = (-wL*bLL-2*wK*bKL+wM*(-(-bLL*uL-bKL*uK)/uM+bKL*uK/uM)) * w -((-1/2*wK*bLL+1/2*wM*bLL*uK/uM)*wL + (-1/2*wL*bLL-wK*bKL+1/2*wM*(-(-bLL*uL-bKL*uK)/uM+bKL*uK/uM))*wK + (1/2*wL*bLL*uK/uM+1/2*wK*(-(-bLL*uL-bKL*uK)/uM+bKL*uK/uM)+wM*(-bLL*uL-bKL*uK)/uM^2*uK)*wM)* w * w ∂k/∂bKK = (-2*wK*bKK+2*wM*bKK*uK/uM)* w -((-wK*bKK+wM*bKK*uK/uM)*wK + (wK*bKK*uK/uM-wM*bKK*uK^2/uM^2)*wM)* w * w

Reparameterized Unit Materials and Supplies Demand Function Partial Derivatives:

 ∂m/∂bL = 0 ∂m/∂bK = 0 ∂k/∂bM = 1
 ∂m/∂bLL = (-wL*(-bLL*uL-bKL*uK)/uM+wL*bLL*uL/uM+wK*bKL*uL/uM+2*wM*(-bLL*uL-bKL*uK)/uM^2*uL) * w -((-wL*bLL-1/2*wK*bKL-1/2*wM*(-bLL*uL-bKL*uK)/uM+1/2*wM*bLL*uL/uM)*wL + (-1/2*wL*bKL+1/2*wM*bKL*uL/uM)*wK + (-1/2*wL*(-bLL*uL-bKL*uK)/uM+1/2*wL*bLL*uL/uM+1/2*wK*bKL*uL/uM+wM*(-bLL*uL-bKL*uK)/uM^2*uL)*wM)* w * w ∂m/∂bKL = (wL*bLL*uK/uM+wK*(-(-bLL*uL-bKL*uK)/uM+bKL*uK/uM)+2*wM*(-bLL*uL-bKL*uK)/uM^2*uK) * w -((-1/2*wK*bLL+1/2*wM*bLL*uK/uM)*wL + (-1/2*wL*bLL-wK*bKL+1/2*wM*(-(-bLL*uL-bKL*uK)/uM+bKL*uK/uM))*wK + (1/2*wL*bLL*uK/uM+1/2*wK*(-(-bLL*uL-bKL*uK)/uM+bKL*uK/uM)+wM*(-bLL*uL-bKL*uK)/uM^2*uK)*wM)* w * w ∂m/∂bKK = (2*wK*bKK*uK/uM-2*wM*bKK*uK^2/uM^2) * w -((-wK*bKK+wM*bKK*uK/uM)*wK + (wK*bKK*uK/uM-wM*bKK*uK^2/uM^2)*wM)* w * w

CES Production/Cost Functions:

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).

The efficiency parameter, A, changes output proportionally for changes in factor inputs, while the distribution parameters, alpha, beta, and gamma, determine the relative shares of the factors in the total cost of producing levels of outputs.

CES Cost Function:

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

where the returns to scale function is:

h(q) = (q/A)^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) = [alpha^(1/(1+rho)) * wL^(rho/(1+rho)) + beta^(1/(1+rho)) * wK^(rho/(1+rho)) + gamma^(1/(1+rho)) * wM^(rho/(1+rho))]^((1+rho)/rho)

CES Unit Demand Functions:

 l(wL,wK,wM) = [(alpha / wL) * c(wL,wK,wM)]^(1/(1+rho)) k(wL,wK,wM) = [(beta / wK) * c(wL,wK,wM)]^(1/(1+rho)) m(wL,wK,wM) = [(gamma / wM) * c(wL,wK,wM)]^(1/(1+rho))

Properties of the Unit CES Cost Function, c(wL,wK,wM).

a. c(wL,wK,wM) is linear homogeneous in factor prices.

c(t*wL,t*wK,t*wM) = [alpha^(1/(1+rho)) * (t*wL)^(rho/(1+rho)) + beta^(1/(1+rho)) * (t*wK)^(rho/(1+rho)) + gamma^(1/(1+rho)) * (t*wM)^(rho/(1+rho))]^((1+rho)/rho),
= [t^(rho/(1+rho)) * (alpha^(1/(1+rho)) * wL^(rho/(1+rho)) + beta^(1/(1+rho)) * wK^(rho/(1+rho)) + gamma^(1/(1+rho)) * wM^(rho/(1+rho)))]^((1+rho)/rho),
= t * [alpha^(1/(1+rho)) * wL^(rho/(1+rho)) + beta^(1/(1+rho)) * wK^(rho/(1+rho)) + gamma^(1/(1+rho)) * wM^(rho/(1+rho)))]^((1+rho)/rho),   →
c(t*wL,t*wK,t*wM) = t * c(wL,wK,wM)

b. The matrix ∇2c of second order partial derivatives of the unit cost function c(wL,wK,wM) is symmetric.

 ∂(∂c(wL,wK,wM)/∂wL)/∂wK = ∂([(alpha / wL) * c(wL,wK,wM)]^(1/(1+rho)))/∂wK = (1/(1+rho)) * (alpha/wL) * ∂c(wL,wK,wM)/∂wK * [(alpha / wL) * c(wL,wK,wM)]^(1/(1+rho)) / [(alpha / wL) * c(wL,wK,wM)] = (1/(1+rho)) * ∂c(wL,wK,wM)/∂wK * ∂c(wL,wK,wM)/∂wL / c(wL,wK,wM) = ∂(∂c(wL,wK,wM)/∂wK)/∂wL ∂(∂c(wL,wK,wM)/∂wL)/∂wM = ∂([(alpha / wL) * c(wL,wK,wM)]^(1/(1+rho)))/∂wM = (1/(1+rho)) * (alpha/wL) * ∂c(wL,wK,wM)/∂wM * [(alpha / wL) * c(wL,wK,wM)]^(1/(1+rho)) / [(alpha / wL) * c(wL,wK,wM)] = (1/(1+rho)) * ∂c(wL,wK,wM)/∂wM * ∂c(wL,wK,wM)/∂wL / c(wL,wK,wM) = ∂(∂c(wL,wK,wM)/∂wM)/∂wL etc.

c. c(wL,wK,wM) is concave in factor prices.

d. The CES production function is called homothetic, because the CES cost function can be separated (factored) into a function of output, q, times a function of input prices, wL, wK, and wM.