Egwald Economics - Cost Functions: Diewert (Generalized Leontief) Cost Function

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Cost Functions

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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	-0.148053	0.001	-183.569
cKK	-0.18189	0	-424.452
cMM	-0.17997	0	-380.444
cLK	0.644377	0.001	555.523
cLM	0.346619	0.001	322.232
cKM	0.51906	0.001	465.723

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	-0.170946	0.002	-71.975
dLK	0.648804	0.001	509.458
dLM	0.558155	0.002	251.513

R2 = 0.9999

R2b = 0.9999

# obs = 31

QR Least Squares
Parameter Estimates
Parameter	Coefficient	std error	t-ratio
cKK	-0.007417	0.001	-7.365
dKL	0.65269	0.002	348.454
dKM	0.503302	0.002	294.154

R2 = 1

R2b = 1

# obs = 31

QR Least Squares
Parameter Estimates
Parameter	Coefficient	std error	t-ratio
cMM	-0.288209	0.003	-94.036
dML	0.550156	0.003	167.531
dMK	0.507928	0.002	278.609

R2 = 0.9999

R2b = 0.9999

# obs = 31

The three estimated factor demand functions are:

∂C/∂wL = L(q; wL, wK, wM) = q^^(1/1) * [-0.1709 + 0.6488 * (wK / wL)^^(1/2) + 0.5582 * (wM / wL)^^(1/2)]

∂C/∂wK = K(q; wL, wK, wM) = q^^(1/1) * [-0.0074 + 0.6527 * (wL / wK)^^(1/2) + 0.5033 * (wM / wK)^^(1/2)]

∂C/∂wM = M(q; wL, wK, wM) = q^^(1/1) * [-0.2882 + 0.5502 * (wL / wM)^^(1/2) + 0.5079 * (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:

ε_L,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),

ε_L,wK = ∂ln(L(q;wL,wK,wM))/∂ln(wK) = .5 * (q^^1/nu) / L(q;wL,wK,wM)) * dLK * (wK/wL)^^1/2,

ε_L,wM = ∂ln(L(q;wL,wK,wM))/∂ln(wM) = .5 * (q^^1/nu) / L(q;wL,wK,wM)) * dLM * (wM/wL)^^1/2,

ε_L,q = ∂ln(L(q;wL,wK,wM))/∂ln(q) = 1/nu, etc.

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

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

-0.5695	0.3594	0.2101	1
0.2944	-0.5046	0.2102	1
0.282	0.3548	-0.6368	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) * [-0.17095 * wL + -0.00742 * wK + -0.28821 * wM + 1.30149 * (wL*wK)^^(1/2) + 1.10831 * (wL*wM)^^(1/2) + 1.01123 * (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	-0.168229	0.002182	-77.087183
dLK	0.64952	0.001229	528.399974
dLM	0.554138	0.001656	334.655722
cKK	-0.007208	0.001119	-6.442353
dKL	0.64952	0.001229	528.399974
dKM	0.50645	0.001078	469.6359
cMM	-0.290371	0.00191	-152.045411
dML	0.554138	0.001656	334.655722
dMK	0.50645	0.001078	469.6359

R2 = 1

R2b = 1

# 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) * [-0.1682 + 0.6495 * (wK / wL)^^(1/2) + 0.5541 * (wM / wL)^^(1/2)]

∂C/∂wK = K(q; wL, wK, wM) = q^^(1/1) * [-0.0072 + 0.6495 * (wL / wK)^^(1/2) + 0.5065 * (wM / wK)^^(1/2)]

∂C/∂wM = M(q; wL, wK, wM) = q^^(1/1) * [-0.2904 + 0.5541 * (wL / wM)^^(1/2) + 0.5065 * (wK / wM)^^(1/2)]

The estimated factor demand elasticities are obtained by:

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

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

-0.5684	0.3598	0.2086	1
0.293	-0.5044	0.2115	1
0.284	0.3538	-0.6378	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) * [-0.16823 * wL + -0.00721 * wK + -0.29037 * wM + 1.29904 * (wL*wK)^^(1/2) + 1.10828 * (wL*wM)^^(1/2) + 1.0129 * (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 = ∇²_wwc(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 = ∇²_wwc =

-0.09987	0.03404	0.04275
0.03404	-0.03156	0.02867
0.04275	0.02867	-0.112

The principal minors of H are H1 = -0.09987, H2 = 0.001993, 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 = -0.1491, e2 = -0.0943, 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:

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

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

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

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

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

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

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:

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

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 —						∇²_wwc(w)
obs #	q	wL	wK	wM	L	K	M	s_LK	s_LM	s_KM	cost	est cost	est L	est K	est M	sL	sK	sM	u_LK	u_LM	u_KL	u_KM	u_ML	u_MK	e₁	e₂	e₃
1	18	5.28	13.22	4.4	24.59	12.5	21.52	0.9	0.72	0.87	389.84	389.86	24.58	12.52	21.5	0.333	0.424	0.243	0.89	0.76	0.89	0.87	0.76	0.87	-0.216	-0.125	0
2	19	7.52	12.58	5.64	21.87	15.85	21.02	0.89	0.86	0.81	482.42	482.43	21.88	15.85	21.01	0.341	0.413	0.246	0.88	0.85	0.88	0.83	0.85	0.83	-0.152	-0.093	-0
3	20	6.94	11.76	4.78	22.73	16.3	23.43	0.91	0.83	0.79	461.45	461.46	22.74	16.29	23.43	0.342	0.415	0.243	0.9	0.83	0.9	0.82	0.83	0.82	-0.181	-0.101	-0
4	21	8.78	12.66	6.56	22.88	18.86	22.17	0.88	0.92	0.8	585.14	585.15	22.9	18.86	22.14	0.344	0.408	0.248	0.88	0.88	0.88	0.82	0.88	0.82	-0.127	-0.083	-0
5	22	6.22	11	5.5	26.76	18.47	22.34	0.86	0.88	0.86	492.43	492.43	26.77	18.47	22.33	0.338	0.412	0.249	0.86	0.86	0.86	0.86	0.86	0.86	-0.163	-0.106	-0
6	23	5.46	13.02	4.46	30.74	16.31	27.34	0.9	0.73	0.86	502.14	502.16	30.72	16.33	27.33	0.334	0.423	0.243	0.89	0.77	0.89	0.86	0.77	0.86	-0.21	-0.121	-0
7	24	7.94	14.72	6.82	29.51	19.55	25.25	0.87	0.85	0.86	694.22	694.22	29.51	19.55	25.24	0.338	0.415	0.248	0.87	0.84	0.87	0.85	0.84	0.85	-0.131	-0.083	-0
8	25	7.1	13.24	5.06	29.66	19.54	29.62	0.91	0.8	0.81	619.23	619.23	29.66	19.54	29.63	0.34	0.418	0.242	0.89	0.81	0.89	0.83	0.81	0.83	-0.174	-0.097	-0
9	26	7.8	12.9	6.06	30.02	21.97	28.03	0.88	0.87	0.82	687.46	687.47	30.04	21.97	28.01	0.341	0.412	0.247	0.88	0.86	0.88	0.83	0.86	0.83	-0.142	-0.089	0
10	27	6	11.72	6.22	35.21	22.33	25.6	0.83	0.87	0.91	632.12	632.13	35.2	22.31	25.62	0.334	0.414	0.252	0.84	0.86	0.84	0.89	0.86	0.89	-0.159	-0.098	0
11	28	7.8	13.02	7.22	33.73	24.43	27.03	0.84	0.91	0.87	776.34	776.33	33.71	24.43	27.04	0.339	0.41	0.251	0.85	0.88	0.85	0.86	0.88	0.86	-0.125	-0.084	-0
12	29	8.26	11.92	5.98	31.38	25.88	31.25	0.89	0.91	0.79	754.52	754.55	31.42	25.87	31.2	0.344	0.409	0.247	0.88	0.88	0.88	0.81	0.88	0.81	-0.14	-0.088	-0
13	30	7	13	6	36.89	24.4	31.62	0.87	0.85	0.86	765.2	765.2	36.9	24.4	31.61	0.338	0.415	0.248	0.87	0.84	0.87	0.85	0.84	0.85	-0.149	-0.094	-0
14	31	7.46	11.92	6.98	36.88	27.72	29.25	0.83	0.93	0.86	809.68	809.68	36.85	27.72	29.27	0.34	0.408	0.252	0.85	0.89	0.85	0.86	0.89	0.86	-0.129	-0.089	-0
15	32	8.9	13.02	7.5	36.03	29.24	31.4	0.85	0.94	0.83	936.94	936.92	36.03	29.25	31.38	0.342	0.407	0.251	0.86	0.9	0.86	0.84	0.9	0.84	-0.113	-0.08	0
16	33	5.62	12.86	5.14	44.35	24.49	36.01	0.87	0.78	0.89	749.25	749.25	44.36	24.5	35.97	0.333	0.42	0.247	0.87	0.8	0.87	0.87	0.8	0.87	-0.186	-0.112	0
17	34	6.8	12.42	5.2	40.59	27.24	38.29	0.89	0.83	0.83	813.45	813.45	40.6	27.24	38.28	0.339	0.416	0.245	0.88	0.83	0.88	0.84	0.83	0.84	-0.168	-0.1	-0
18	35	7	13.26	5.22	42.15	27.39	40.54	0.9	0.81	0.82	869.84	869.85	42.15	27.39	40.55	0.339	0.417	0.243	0.89	0.82	0.89	0.83	0.82	0.83	-0.169	-0.098	-0
19	36	7.66	13.84	7.78	45.51	30.82	33.61	0.82	0.9	0.89	1036.57	1036.58	45.48	30.81	33.66	0.336	0.411	0.253	0.84	0.87	0.84	0.88	0.87	0.88	-0.124	-0.08	-0
20	37	7.84	11.24	7.22	42.27	34.8	33.98	0.83	0.97	0.85	967.98	967.96	42.23	34.82	34	0.342	0.404	0.254	0.84	0.92	0.84	0.85	0.92	0.85	-0.121	-0.089	0
21	38	6.3	12.1	6.76	49.65	31.94	34.98	0.82	0.89	0.91	935.74	935.76	49.63	31.92	35.04	0.334	0.413	0.253	0.83	0.87	0.83	0.89	0.87	0.89	-0.15	-0.092	-0
22	39	6.52	12.5	5.96	49.17	31.65	39.89	0.86	0.85	0.87	953.99	953.99	49.18	31.65	39.88	0.336	0.415	0.249	0.86	0.84	0.86	0.86	0.84	0.86	-0.155	-0.098	0
23	40	6.08	11.9	4.94	49.59	31.33	44.43	0.89	0.81	0.85	893.86	893.86	49.6	31.33	44.42	0.337	0.417	0.245	0.88	0.82	0.88	0.85	0.82	0.85	-0.181	-0.11	-0
24	41	5.74	12.46	4.9	53.33	30.79	45.82	0.88	0.78	0.87	914.28	914.29	53.33	30.8	45.8	0.335	0.42	0.245	0.88	0.8	0.88	0.86	0.8	0.86	-0.189	-0.113	0
25	42	6.72	11.66	6.04	50.94	35.72	41.9	0.85	0.89	0.86	1011.85	1011.85	50.93	35.72	41.91	0.338	0.412	0.25	0.86	0.87	0.86	0.86	0.87	0.86	-0.148	-0.098	0
26	43	8.26	11.38	7.74	48.69	41.41	38.5	0.82	0.99	0.85	1171.49	1171.46	48.61	41.44	38.54	0.343	0.403	0.255	0.84	0.93	0.84	0.85	0.93	0.85	-0.113	-0.085	-0
27	44	5.88	11.8	4.2	53.72	33.15	53.38	0.92	0.77	0.82	931.27	931.26	53.69	33.15	53.42	0.339	0.42	0.241	0.9	0.8	0.9	0.83	0.8	0.83	-0.213	-0.116	-0
28	45	6.54	13.9	5.38	57.66	33.89	51.08	0.89	0.78	0.86	1123.02	1123.03	57.66	33.9	51.06	0.336	0.42	0.245	0.88	0.8	0.88	0.85	0.8	0.85	-0.17	-0.101	-0
29	46	5.02	12.6	5.38	65.9	33.76	46.98	0.84	0.78	0.93	1008.96	1008.92	65.99	33.75	46.92	0.328	0.421	0.25	0.85	0.8	0.85	0.9	0.8	0.9	-0.199	-0.112	-0
30	47	7.68	12.78	4.36	51.08	37.31	61.47	0.96	0.79	0.74	1137.14	1137.1	51.1	37.23	61.67	0.345	0.418	0.236	0.92	0.81	0.92	0.79	0.81	0.79	-0.205	-0.092	0
31	48	6.6	14.26	5.66	62.38	36.17	53.4	0.88	0.79	0.87	1229.72	1229.73	62.38	36.18	53.37	0.335	0.42	0.246	0.88	0.8	0.88	0.86	0.8	0.86	-0.163	-0.098	0
AVE:	33	6.93	12.6	5.84	40.51	26.94	35.2	0.87	0.85	0.85	817.34	817.34	40.51	26.94	35.2	0.338	0.415	0.247	0.87	0.84	0.87	0.85	0.84	0.85	-0.16	-0.098	-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	-0.166386	0.003	-56.431
cKK	-0.007193	0.001	-7.791
cMM	-0.285936	0.002	-117.287
2*dLK	1.300233	0.003	449.616
2*dLM	1.102446	0.004	277.888
2*dKM	1.011372	0.003	400.715

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) * [-0.16639 * wL + -0.00719 * wK + -0.28594 * wM + 1.30023 * (wL*wK)^^(1/2) + 1.10245 * (wL*wM)^^(1/2) + 1.01137 * (wK*wM)^^(1/2)] (***)

XII. Its three derived factor demand functions are:

∂C/∂wL = L(q; wL, wK, wM) = q^^(1/1) * [-0.1664 + 0.6501 * (wK / wL)^^(1/2) + 0.5512 * (wM / wL)^^(1/2)]

∂C/∂wK = K(q; wL, wK, wM) = q^^(1/1) * [-0.0072 + 0.6501 * (wL / wK)^^(1/2) + 0.5057 * (wM / wK)^^(1/2)]

∂C/∂wM = M(q; wL, wK, wM) = q^^(1/1) * [-0.2859 + 0.5512 * (wL / wM)^^(1/2) + 0.5057 * (wK / wM)^^(1/2)]

The derived factor demand elasticities are obtained by:

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

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

-0.5676	0.3602	0.2075	1
0.2932	-0.5044	0.2112	1
0.2825	0.3532	-0.6357	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 = ∇²_wwc(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 = ∇²_wwc =

-0.09974	0.03408	0.04253
0.03408	-0.03156	0.02863
0.04253	0.02863	-0.11164

The principal minors of H are H1 = -0.099735, H2 = 0.001987, 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 = -0.1486, e2 = -0.0943, 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 —						∇²_wwc(w)
obs #	q	wL	wK	wM	L	K	M	s_LK	s_LM	s_KM	cost	est cost	est L	est K	est M	sL	sK	sM	u_LK	u_LM	u_KL	u_KM	u_ML	u_MK	e₁	e₂	e₃
1	18	5.28	13.22	4.4	24.59	12.5	21.52	0.9	0.72	0.87	389.84	389.86	24.58	12.52	21.5	0.333	0.424	0.243	0.89	0.76	0.89	0.86	0.76	0.86	-0.215	-0.125	0
2	19	7.52	12.58	5.64	21.87	15.85	21.02	0.89	0.86	0.81	482.42	482.43	21.89	15.85	21.01	0.341	0.413	0.246	0.88	0.84	0.88	0.83	0.84	0.83	-0.152	-0.093	-0
3	20	6.94	11.76	4.78	22.73	16.3	23.43	0.91	0.83	0.79	461.45	461.46	22.75	16.29	23.43	0.342	0.415	0.243	0.9	0.83	0.9	0.82	0.83	0.82	-0.18	-0.101	-0
4	21	8.78	12.66	6.56	22.88	18.86	22.17	0.88	0.92	0.8	585.14	585.15	22.91	18.86	22.14	0.344	0.408	0.248	0.88	0.88	0.88	0.82	0.88	0.82	-0.127	-0.083	-0
5	22	6.22	11	5.5	26.76	18.47	22.34	0.86	0.88	0.86	492.43	492.43	26.76	18.46	22.34	0.338	0.412	0.25	0.86	0.85	0.86	0.85	0.85	0.85	-0.162	-0.106	-0
6	23	5.46	13.02	4.46	30.74	16.31	27.34	0.9	0.73	0.86	502.14	502.15	30.72	16.32	27.32	0.334	0.423	0.243	0.89	0.77	0.89	0.86	0.77	0.86	-0.209	-0.121	-0
7	24	7.94	14.72	6.82	29.51	19.55	25.25	0.87	0.85	0.86	694.22	694.22	29.51	19.55	25.24	0.338	0.414	0.248	0.87	0.84	0.87	0.85	0.84	0.85	-0.131	-0.083	-0
8	25	7.1	13.24	5.06	29.66	19.54	29.62	0.91	0.8	0.81	619.23	619.23	29.67	19.54	29.63	0.34	0.418	0.242	0.9	0.81	0.9	0.83	0.81	0.83	-0.173	-0.097	0
9	26	7.8	12.9	6.06	30.02	21.97	28.03	0.88	0.87	0.82	687.46	687.46	30.04	21.97	28.01	0.341	0.412	0.247	0.88	0.85	0.88	0.83	0.85	0.83	-0.141	-0.089	-0
10	27	6	11.72	6.22	35.21	22.33	25.6	0.83	0.87	0.91	632.12	632.13	35.19	22.31	25.64	0.334	0.414	0.252	0.84	0.85	0.84	0.88	0.85	0.88	-0.159	-0.098	-0
11	28	7.8	13.02	7.22	33.73	24.43	27.03	0.84	0.91	0.87	776.34	776.33	33.71	24.43	27.05	0.339	0.41	0.252	0.85	0.88	0.85	0.86	0.88	0.86	-0.124	-0.084	-0
12	29	8.26	11.92	5.98	31.38	25.88	31.25	0.89	0.91	0.79	754.52	754.54	31.42	25.87	31.2	0.344	0.409	0.247	0.88	0.88	0.88	0.81	0.88	0.81	-0.139	-0.088	-0
13	30	7	13	6	36.89	24.4	31.62	0.87	0.85	0.86	765.2	765.19	36.9	24.4	31.61	0.338	0.415	0.248	0.87	0.84	0.87	0.85	0.84	0.85	-0.149	-0.094	0
14	31	7.46	11.92	6.98	36.88	27.72	29.25	0.83	0.93	0.86	809.68	809.68	36.85	27.72	29.29	0.339	0.408	0.252	0.85	0.89	0.85	0.86	0.89	0.86	-0.128	-0.089	0
15	32	8.9	13.02	7.5	36.03	29.24	31.4	0.85	0.94	0.83	936.94	936.92	36.03	29.25	31.39	0.342	0.406	0.251	0.86	0.89	0.86	0.84	0.89	0.84	-0.113	-0.08	0
16	33	5.62	12.86	5.14	44.35	24.49	36.01	0.87	0.78	0.89	749.25	749.24	44.36	24.5	35.98	0.333	0.42	0.247	0.87	0.79	0.87	0.87	0.79	0.87	-0.186	-0.112	0
17	34	6.8	12.42	5.2	40.59	27.24	38.29	0.89	0.83	0.83	813.45	813.45	40.6	27.24	38.28	0.339	0.416	0.245	0.88	0.82	0.88	0.83	0.82	0.83	-0.168	-0.1	0
18	35	7	13.26	5.22	42.15	27.39	40.54	0.9	0.81	0.82	869.84	869.84	42.15	27.39	40.54	0.339	0.417	0.243	0.89	0.81	0.89	0.83	0.81	0.83	-0.168	-0.098	-0
19	36	7.66	13.84	7.78	45.51	30.82	33.61	0.82	0.9	0.89	1036.57	1036.59	45.47	30.8	33.68	0.336	0.411	0.253	0.84	0.87	0.84	0.88	0.87	0.88	-0.124	-0.079	-0
20	37	7.84	11.24	7.22	42.27	34.8	33.98	0.83	0.97	0.85	967.98	967.97	42.22	34.82	34.02	0.342	0.404	0.254	0.84	0.91	0.84	0.85	0.91	0.85	-0.12	-0.089	0
21	38	6.3	12.1	6.76	49.65	31.94	34.98	0.82	0.89	0.91	935.74	935.77	49.61	31.92	35.06	0.334	0.413	0.253	0.84	0.86	0.84	0.89	0.86	0.89	-0.15	-0.092	-0
22	39	6.52	12.5	5.96	49.17	31.65	39.89	0.86	0.85	0.87	953.99	953.98	49.17	31.65	39.89	0.336	0.415	0.249	0.86	0.84	0.86	0.86	0.84	0.86	-0.154	-0.098	-0
23	40	6.08	11.9	4.94	49.59	31.33	44.43	0.89	0.81	0.85	893.86	893.85	49.6	31.33	44.42	0.337	0.417	0.245	0.88	0.82	0.88	0.85	0.82	0.85	-0.18	-0.11	-0
24	41	5.74	12.46	4.9	53.33	30.79	45.82	0.88	0.78	0.87	914.28	914.28	53.33	30.8	45.8	0.335	0.42	0.245	0.88	0.8	0.88	0.86	0.8	0.86	-0.188	-0.113	0
25	42	6.72	11.66	6.04	50.94	35.72	41.9	0.85	0.89	0.86	1011.85	1011.84	50.93	35.71	41.92	0.338	0.412	0.25	0.86	0.86	0.86	0.86	0.86	0.86	-0.148	-0.098	0
26	43	8.26	11.38	7.74	48.69	41.41	38.5	0.82	0.99	0.85	1171.49	1171.47	48.6	41.44	38.56	0.343	0.403	0.255	0.84	0.93	0.84	0.85	0.93	0.85	-0.113	-0.085	-0
27	44	5.88	11.8	4.2	53.72	33.15	53.38	0.92	0.77	0.82	931.27	931.26	53.7	33.15	53.41	0.339	0.42	0.241	0.9	0.79	0.9	0.83	0.79	0.83	-0.212	-0.117	-0
28	45	6.54	13.9	5.38	57.66	33.89	51.08	0.89	0.78	0.86	1123.02	1123.02	57.66	33.9	51.06	0.336	0.42	0.245	0.88	0.8	0.88	0.85	0.8	0.85	-0.169	-0.101	-0
29	46	5.02	12.6	5.38	65.9	33.76	46.98	0.84	0.78	0.93	1008.96	1008.91	65.97	33.75	46.94	0.328	0.421	0.25	0.85	0.79	0.85	0.9	0.79	0.9	-0.198	-0.112	-0
30	47	7.68	12.78	4.36	51.08	37.31	61.47	0.96	0.79	0.74	1137.14	1137.12	51.12	37.23	61.64	0.345	0.418	0.236	0.92	0.81	0.92	0.79	0.81	0.79	-0.204	-0.092	0
31	48	6.6	14.26	5.66	62.38	36.17	53.4	0.88	0.79	0.87	1229.72	1229.72	62.38	36.18	53.37	0.335	0.42	0.246	0.88	0.8	0.88	0.86	0.8	0.86	-0.163	-0.098	-0
AVE:	33	6.93	12.6	5.84	40.51	26.94	35.21	0.87	0.85	0.85	817.34	817.34	40.51	26.94	35.21	0.338	0.414	0.247	0.87	0.84	0.87	0.85	0.84	0.85	-0.16	-0.098	-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	-0.146532	0.001	-224.239
cKK	-0.180599	0.001	-248.457
cMM	-0.17977	0.001	-294.259
cLK	0.641629	0.001	501.104
cLM	0.345748	0.001	401.233
cKM	0.519716	0.001	427.518

R2 = 1

R2b = 1

# obs = 31

The estimated Diewert production function using the cost data is:

q^^1/1 = -0.146532 * L + -0.180599 * K + -0.17977 * M + 0.641629 * L^^1/2 * K^^1/2 + 0.345748 * L^^1/2 * M^^1/2 + 0.519716 * 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
q	est q	L	K	M	total cost	ave. cost	marg. cost	3 factor s_LM	2 factor s_LM
14	14	10.72	24.4	9.29	447.98	32	14.04	1.39	1.15
16	16	13.15	24.4	11.37	477.49	29.84	15.51	1.31	1.1
18	18	15.82	24.4	13.67	509.97	28.33	16.97	1.23	1.06
20	20	18.72	24.4	16.17	545.32	27.27	18.42	1.16	1.02
22	22	21.88	24.4	18.87	583.59	26.53	19.86	1.09	0.98
24	24	25.27	24.4	21.77	624.74	26.03	21.29	1.03	0.94
26	26	28.91	24.4	24.87	668.82	25.72	22.7	0.96	0.91
28	28	32.76	24.4	28.16	715.51	25.55	24.12	0.91	0.88
30	30	36.87	24.4	31.66	765.27	25.51	25.5	0.85	0.85
32	32	41.2	24.4	35.34	817.69	25.55	26.89	0.8	0.83
34	34	45.76	24.4	39.21	872.86	25.67	28.27	0.75	0.8
36	36	50.56	24.4	43.27	930.78	25.85	29.63	0.7	0.78
38	38	55.57	24.4	47.5	991.21	26.08	31.01	0.66	0.76
40	40	60.82	24.4	51.95	1054.67	26.37	32.36	0.62	0.74
42	42	66.28	24.4	56.59	1120.71	26.68	33.7	0.58	0.72
44	44	71.96	24.4	61.41	1189.43	27.03	35.03	0.54	0.7
46	46	77.87	24.4	66.42	1260.81	27.41	36.36	0.51	0.69
48	48	83.99	24.4	71.61	1334.82	27.81	37.68	0.47	0.67
50	50	90.33	24.4	76.98	1411.4	28.23	38.98	0.44	0.66
52	52	96.89	24.4	82.52	1490.52	28.66	40.28	0.41	0.64

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:

3. The Factor Share Functions:

4. The Factor Demand Elasticities:

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).