Egwald Economics - Cost Functions: Generalized CES-Diewert Cost Function

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Cost Functions

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Elmer G. Wiens

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

This web page replicates the procedures used to estimate the parameters of a Diewert cost function to approximate a CES cost function. To determine the Diewert cost function's efficacy in estimating varying elasticities of substitution among pairs of inputs, on this web page we approximate a Generalized CES cost function with the Diewert cost function.

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

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

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

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

where the returns to scale function is:

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

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

and the unit cost function is:

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

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

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

À1: Estimate the parameters the cost function by way of its factor share functions — requires data relating output quantities to (cost minimizing) factor inputs, and input prices,

À2: Estimate the parameters of the cost function directly — requires data relating output quantities to total (minimum) cost and input prices.

We shall use the three factor Generalized CES production function to generate the required cost data, yielding a data set relating output, q, and factor prices, wL, wK, and wM, [randomized about the base prices (wL*, wK*, wM*)] to the Generalized CES cost minimizing factor inputs, L, K, and M.

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

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

The three factor Generalized CES production function is:

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

where L = labour, K = capital, M = materials and supplies, and q = product.

The parameter nu permits one to adjust the returns to scale, while the parameter rho is the geometric mean of rhoL, rhoK, and rhoM:

rho = (rhoL * rhoK * rhoM)^^1/3.

If rho = rhoL = rhoK = rhoM, we get the standard CES production function. If also, rho = 0, ie sigma = 1/(1+rho) = 1, we get the Cobb-Douglas production function.

Generalized CES Elasticity of Scale of Production:

ε_LKM = ε(L,K,M) = (nu / rho) * (alpha * rhoL * L^^-rhoL + beta * rhoK * K^^-rhoK + gamma * rhoM * M^^-rhoM) / (alpha * L^^-rhoL + beta * K^^-rhoK + gamma * M^^-rhoM).

See the Generalized CES production function.

À1: Estimate the factor demand equations separately.

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

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

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

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

∂C/∂wK = K(q; wL, wK, wM) = q^^(1/nu1) * [cKK + dKL * (wL / wK)^^(1/2) + dKM * (wM / wK)^^(1/2)]

∂C/∂wM = M(q; wL, wK, wM) = q^^(1/nu1) * [cMM + dML * (wL / wM)^^(1/2) + dMK * (wK / wM)^^(1/2)]

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

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

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

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

I. Stage 1. Generate cost data with the Generalized CES production function, yielding a data set relating output, q, and factor prices, wL, wK, and wM, [randomized about the base prices (wL*, wK*, wM*)] to the Generalized CES cost minimizing factor inputs, L, K, and M. The base factor prices and the randomizing distribution are specified in the form below.

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

II. The estimated coefficients of the Diewert cost function will vary with the parameters nu, rho, rhoL, rhoK, rhoM, alpha, beta and gamma of the Generalized CES production function.

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

The restrictions ensure that the least-cost problems can be solved to obtain the underlying Generalized CES cost function, using the parameters as specified.
Intermediate (and other) values of the parameters also work.

Restrictions:
.8 < nu < 1.1;
-.6 < rhoL = rho < .6;
-.6 < rho < -.2 → rhoK = .95 * rho, rhoM = rho/.95;
-.2 <= rho < -.1 → rhoK = .9 * rho, rhoM = rho / .9;
-.1 <= rho < 0 → rhoK = .87 * rho, rhoM = rho / .87;
0 <= rho < .1 → rhoK = .7 * rho, rhoM = rho / .7;
.1 <= rho < .2 → rhoK = .67 * rho, rhoM = rho / .67;
.2 <= rho < .6 → rhoK = .6 * rho, rhoM = rho / .6;
rho = 0 → nu = alpha + beta + gamma (Cobb-Douglas)
4 <= wL* <= 11, 7<= wK* <= 16, 4 <= wM* <= 10

Generalized CES Production Function Parameters

nu:

rho:

Base Factor Prices

wL*

wK*

wM*

Distribution to Randomize Factor Prices

Use [-2, 2] Uniform distribution
Use .25 * Normal (μ = 0, σ² = 1)

The Generalized CES production function as specified:

q = 1 * [0.35 * L^^{- 0.17647} + 0.4 * K^^{- 0.11823} + 0.25 *M^^{- 0.26339}]^^(-1/0.17647) = f(L,K,M).

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

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

III. For these coefficients of the CES Generalized production function, I generated a sequence (displayed in the "Generalized CES-Diewert Cost Function" table) of factor prices, outputs, and the corresponding cost minimizing inputs. Then I used these data to estimate the coefficients of each factor share equation separately:

QR Least Squares
Parameter Estimates
Parameter	Coefficient	std error	t-ratio
cLL	-0.113323	0.004	-27.235
dLK	0.522028	0.003	153.076
dLM	0.527648	0.004	145.577

R2 = 0.9997

R2b = 0.9997

# obs = 31

QR Least Squares
Parameter Estimates
Parameter	Coefficient	std error	t-ratio
cKK	-0.061377	0.023	-2.68
dKL	0.547743	0.028	19.654
dKM	0.383565	0.025	15.619

R2 = 0.9676

R2b = 0.9652

# obs = 31

QR Least Squares
Parameter Estimates
Parameter	Coefficient	std error	t-ratio
cMM	-0.153376	0.037	-4.202
dML	0.493736	0.043	11.506
dMK	0.416022	0.034	12.176

R2 = 0.9729

R2b = 0.971

# obs = 31

The three estimated factor demand functions are:

∂C/∂wL = L(q; wL, wK, wM) = q^^(1/0.92) * [-0.1133 + 0.522 * (wK / wL)^^(1/2) + 0.5276 * (wM / wL)^^(1/2)]

∂C/∂wK = K(q; wL, wK, wM) = q^^(1/0.92) * [-0.0614 + 0.5477 * (wL / wK)^^(1/2) + 0.3836 * (wM / wK)^^(1/2)]

∂C/∂wM = M(q; wL, wK, wM) = q^^(1/0.92) * [-0.1534 + 0.4937 * (wL / wM)^^(1/2) + 0.416 * (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/nu1) / 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/nu1) / L(q;wL,wK,wM)) * dLK * (wK/wL)^^1/2,

ε_L,wM = ∂ln(L(q;wL,wK,wM))/∂ln(wM) = .5 * (q^^1/nu1) / L(q;wL,wK,wM)) * dLM * (wM/wL)^^1/2,

ε_L,q = ∂ln(L(q;wL,wK,wM))/∂ln(q) = 1/nu1, 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

-0.5521	0.3274	0.2248	1.087
0.3343	-0.5511	0.2167	1.087
0.2687	0.3086	-0.5773	1.087

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

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

= q^^(1/0.92) * [-0.11332 * wL + -0.06138 * wK + -0.15338 * wM + 1.06977 * (wL*wK)^^(1/2) + 1.02138 * (wL*wM)^^(1/2) + 0.79959 * (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	-0.11364	0.025944	-4.380145
dLK	0.531878	0.018781	28.319699
dLM	0.513554	0.017188	29.878588
cKK	-0.062328	0.018703	-3.332602
dKL	0.531878	0.018781	28.319699
dKM	0.401743	0.01597	25.156822
cMM	-0.15343	0.02161	-7.100028
dML	0.513554	0.017188	29.878588
dMK	0.401743	0.01597	25.156822

R2 = 0.9968

R2b = 0.9965

# obs = 93

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

The three estimated, restricted factor demand functions are:

∂C/∂wL = L(q; wL, wK, wM) = q^^(1/0.92) * [-0.1136 + 0.5319 * (wK / wL)^^(1/2) + 0.5136 * (wM / wL)^^(1/2)]

∂C/∂wK = K(q; wL, wK, wM) = q^^(1/0.92) * [-0.0623 + 0.5319 * (wL / wK)^^(1/2) + 0.4017 * (wM / wK)^^(1/2)]

∂C/∂wM = M(q; wL, wK, wM) = q^^(1/0.92) * [-0.1534 + 0.5136 * (wL / wM)^^(1/2) + 0.4017 * (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

-0.5523	0.3335	0.2188	1.087
0.3248	-0.5519	0.2271	1.087
0.2794	0.2979	-0.5773	1.087

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

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

= q^^(1/0.92) * [-0.11364 * wL + -0.06233 * wK + -0.15343 * wM + 1.06376 * (wL*wK)^^(1/2) + 1.02711 * (wL*wM)^^(1/2) + 0.80349 * (wK*wM)^^(1/2)] (***)

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

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

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

2 * c(7, 13, 6) = 42.75 = 42.75 = c(14, 26,12).

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

∂²C/∂q∂wL = h'(q) * ∂c/∂wL = ∂²C/∂wL∂q,

∂²C/∂q∂wK = h'(q) * ∂c/∂wK = ∂²C/∂wK∂q,

∂²C/∂q∂wM = h'(q) * ∂c/∂wM = ∂²C/∂wM∂q,

∂²c/∂wL∂wK = .25 * (dLK+dKL)/(wL*wK)^^(1/2) = ∂²c/∂wK∂wL,

∂²c/∂wL∂wM = .25 * (dLM+dML)/(wL*wM)^^(1/2) = ∂²c/∂wM∂wL,

∂²c/∂wK∂wM = .25 * (dKM+dMK)/(wL*wK)^^(1/2) = ∂²c/∂wM∂wK, and

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

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

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

3. The cost function C(q;wL,wK,wM) is a concave in factor prices if the Hessian matrix H = ∇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 =

-0.08573	0.02788	0.03962
0.02788	-0.02551	0.02274
0.03962	0.02274	-0.0955

The principal minors of H are H1 = -0.085735, H2 = 0.00141, 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 = -0.1306, e2 = -0.0762, 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) * M(q;wL,wK,wM)),

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

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 and input amounts for a given level of output agree with the Generalized CES production function's minimized cost and inputs. Also, compare the Allen partial elasticities of substitution, s_LK, s_LM, and s_MK, with the Uzawa partial elasticities of substitution, u_LK, u_LM, and u_KM.

Generalized CES-Diewert Cost Function À1: Estimate the Translog cost function using restricted factor shares Parameters: rho = 0.17647, rhoL = 0.17647, rhoK = 0.11823, rhoM = 0.26339
					nu = 1 — Generalized CES Cost Data —								nu1 = 0.92 — Diewert Cost Data —				Factor Shares			Uzawa Elasticities						∇²_wwc(w)
obs #	q	wL	wK	wM	L	K	M	s_LK	s_LM	s_KM	ε_LKM	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	5.66	12.08	4.74	26.18	12.41	25.31	0.9	0.79	0.83	0.924	418	418.26	26.23	12.81	24.28	0.355	0.37	0.275	0.93	0.75	0.93	0.83	0.75	0.83	-0.169	-0.094	0
2	19	5.78	12.1	7.66	30.71	14.93	20.43	0.9	0.79	0.83	0.932	514.65	512.47	30.61	15.34	19.58	0.345	0.362	0.293	0.85	0.81	0.85	0.87	0.81	0.87	-0.141	-0.067	0
3	20	5.34	11.72	5.34	30.83	14.37	25.62	0.9	0.79	0.83	0.925	469.89	469.72	30.83	14.74	24.79	0.35	0.368	0.282	0.9	0.77	0.9	0.85	0.77	0.85	-0.163	-0.09	-0
4	21	6.98	13.48	5.74	29.83	15.57	29.01	0.9	0.79	0.83	0.924	584.58	584.88	29.86	15.94	28.15	0.356	0.367	0.276	0.92	0.77	0.92	0.81	0.77	0.81	-0.136	-0.077	0
5	22	7.52	14.56	4.4	29.14	15.15	37.15	0.89	0.79	0.83	0.917	603.24	605.42	29.34	15.57	35.94	0.364	0.374	0.261	0.97	0.74	0.97	0.78	0.74	0.78	-0.18	-0.076	-0
6	23	8.44	11.54	7.04	29.59	21.02	28.47	0.9	0.79	0.83	0.929	692.78	691.76	29.52	21.34	27.89	0.36	0.356	0.284	0.89	0.85	0.89	0.78	0.85	0.78	-0.104	-0.07	0
7	24	5.26	11.42	7.72	40.94	19.56	24.63	0.9	0.79	0.83	0.93	628.85	627.73	40.89	19.9	24.02	0.343	0.362	0.295	0.84	0.82	0.84	0.89	0.82	0.89	-0.156	-0.067	-0
8	25	8.46	14.48	7.04	34.76	20.37	33.13	0.9	0.79	0.83	0.923	822.25	822.53	34.75	20.65	32.6	0.357	0.364	0.279	0.91	0.8	0.91	0.8	0.8	0.8	-0.109	-0.065	-0
9	26	8.46	11.88	7.46	34.5	24.12	31.43	0.9	0.79	0.83	0.927	812.92	812.55	34.48	24.33	31.08	0.359	0.356	0.285	0.89	0.85	0.89	0.79	0.85	0.79	-0.101	-0.068	0
10	27	5.22	12.16	6.24	45.33	20.44	31.85	0.9	0.79	0.83	0.923	683.93	684.25	45.3	20.64	31.54	0.346	0.367	0.288	0.88	0.77	0.88	0.87	0.77	0.87	-0.161	-0.08	-0
11	28	5.06	12.62	5.44	47.07	20.01	35.87	0.89	0.79	0.83	0.92	685.85	686.51	47.09	20.14	35.68	0.347	0.37	0.283	0.9	0.75	0.9	0.87	0.75	0.87	-0.172	-0.089	-0
12	29	6.52	12.44	4.74	41.07	22.03	43.07	0.89	0.79	0.83	0.917	745.96	746.86	41.16	22.18	42.74	0.359	0.369	0.271	0.94	0.76	0.94	0.8	0.76	0.8	-0.162	-0.085	-0
13	30	7	13	6	43.79	24.14	40.13	0.89	0.79	0.83	0.92	861.17	861.89	43.82	24.23	40.03	0.356	0.365	0.279	0.91	0.79	0.91	0.82	0.79	0.82	-0.131	-0.076	0
14	31	6.74	11.22	7.72	46.77	28.53	33.92	0.9	0.79	0.83	0.926	897.26	897.92	46.89	28.55	33.88	0.352	0.357	0.291	0.86	0.84	0.86	0.84	0.84	0.84	-0.118	-0.069	-0
15	32	7.42	12.32	4.14	41.3	25.09	53.37	0.89	0.79	0.83	0.913	836.45	837.21	41.32	25.23	53.08	0.366	0.371	0.262	0.97	0.76	0.97	0.76	0.76	0.76	-0.187	-0.078	-0
16	33	5.4	13.48	7.38	59.45	25.56	36.87	0.9	0.79	0.83	0.921	937.74	938.69	59.35	25.56	37.07	0.341	0.367	0.291	0.86	0.78	0.86	0.89	0.78	0.89	-0.156	-0.068	-0
17	34	7.32	14.38	7.92	53.83	28.53	40.45	0.89	0.79	0.83	0.921	1124.63	1125.72	53.87	28.43	40.73	0.35	0.363	0.287	0.88	0.8	0.88	0.85	0.8	0.85	-0.113	-0.063	-0
18	35	8.68	14.92	7.56	50.68	30.17	45.39	0.89	0.79	0.83	0.919	1233.13	1234.1	50.68	30.01	45.83	0.356	0.363	0.281	0.9	0.8	0.9	0.81	0.8	0.81	-0.103	-0.062	0
19	36	7.06	13.86	5.42	53.12	28.15	52.41	0.89	0.79	0.83	0.914	1049.19	1049.25	53.17	27.95	52.86	0.358	0.369	0.273	0.93	0.76	0.93	0.81	0.76	0.81	-0.143	-0.078	0
20	37	7.12	12.86	6.72	55.69	31.87	46.51	0.89	0.79	0.83	0.919	1118.98	1119.9	55.72	31.6	47.15	0.354	0.363	0.283	0.9	0.8	0.9	0.82	0.8	0.82	-0.121	-0.072	-0
21	38	5.38	15	4.32	64.04	25.04	60.2	0.89	0.79	0.83	0.908	980.26	978.45	64.37	24.6	60.91	0.354	0.377	0.269	0.95	0.69	0.95	0.85	0.69	0.85	-0.194	-0.102	-0
22	39	7.7	14.26	4.18	52.9	29.53	68.69	0.89	0.79	0.83	0.908	1115.47	1114.03	53.02	29.28	68.95	0.366	0.375	0.259	0.98	0.74	0.98	0.77	0.74	0.77	-0.191	-0.074	-0
23	40	7.16	11.76	4.84	54.68	34.04	59.55	0.89	0.79	0.83	0.914	1079.98	1079.41	54.59	33.65	60.5	0.362	0.367	0.271	0.94	0.79	0.94	0.78	0.79	0.78	-0.155	-0.08	0
24	41	5.28	14.26	6.46	75.4	30.59	50.22	0.89	0.79	0.83	0.915	1158.8	1158.64	75.23	30.11	51.4	0.343	0.371	0.287	0.89	0.75	0.89	0.89	0.75	0.89	-0.163	-0.076	-0
25	42	6.14	12.36	4.44	62.61	32.72	64.05	0.89	0.79	0.83	0.911	1073.27	1071.49	62.65	32.17	65.15	0.359	0.371	0.27	0.94	0.75	0.94	0.81	0.75	0.81	-0.175	-0.09	-0
26	43	8.1	12.86	6.5	60.69	39.16	57.3	0.89	0.79	0.83	0.916	1367.59	1367.84	60.63	38.49	58.74	0.359	0.362	0.279	0.91	0.81	0.91	0.79	0.81	0.79	-0.115	-0.07	-0
27	44	7.18	14.1	7.5	70.71	38.02	53.61	0.89	0.79	0.83	0.917	1445.84	1446.54	70.72	37.31	55.02	0.351	0.364	0.285	0.89	0.8	0.89	0.84	0.8	0.84	-0.117	-0.066	-0
28	45	6.48	11.58	5.16	66.22	38.61	62.52	0.89	0.79	0.83	0.913	1198.84	1197.69	66.14	37.83	64.16	0.358	0.366	0.276	0.92	0.79	0.92	0.8	0.79	0.8	-0.148	-0.085	0
29	46	8.42	11.24	5.54	59.15	44.5	65.46	0.89	0.79	0.83	0.915	1360.96	1360.23	58.87	43.64	67.5	0.364	0.361	0.275	0.93	0.83	0.93	0.75	0.83	0.75	-0.13	-0.073	-0
30	47	5.28	13.46	6.56	86.15	37.06	56.17	0.89	0.79	0.83	0.914	1322.14	1321.49	85.92	36.21	57.99	0.343	0.369	0.288	0.88	0.76	0.88	0.88	0.76	0.88	-0.161	-0.075	-0
31	48	8.32	13.52	6.72	69.06	43.93	64.29	0.89	0.79	0.83	0.914	1600.62	1599.71	68.95	42.89	66.39	0.359	0.362	0.279	0.91	0.81	0.91	0.8	0.81	0.8	-0.112	-0.067	-0
AVE:	33	6.8	12.93	6.09	49.87	26.82	44.7	0.89	0.79	0.83	0.919	949.2	949.13	49.87	26.82	44.7	0.355	0.366	0.279	0.91	0.78	0.91	0.82	0.78	0.82	-0.145	-0.076	-0

À2: Estimate the Diewert cost function directly.

XI. If we have a data set relating the input (factor) prices, wL, wK, and wM, to the total (minimum) cost of producing output for varying levels output, but do not have data on the required levels of inputs, we can estimate the Diewert cost function directly. We will use the same sequence (displayed in the "Diewert Cost Function" table) of factor prices, outputs, and total (minimum) cost as used in À1. The estimated coefficients of the cost function will vary with the parameters nu, rho, rhoL, rhoK, rhoM, alpha, beta and gamma of the Generalized CES production function used to generate these data.

QR Least Squares
Parameter Estimates
Parameter	Coefficient	std error	t-ratio
cLL	-0.013668	0.09	-0.151
cKK	0.009586	0.044	0.22
cMM	-0.074132	0.059	-1.253
2*dLK	0.935316	0.102	9.2
2*dLM	1.00084	0.103	9.688
2*dKM	0.721246	0.087	8.291

R2 = 0.9999

R2b = 0.9998

# obs = 31

The Diewert cost function as obtained by direct estimation is:

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

= q^^(1/0.92) * [-0.01367 * wL + 0.00959 * wK + -0.07413 * wM + 0.93532 * (wL*wK)^^(1/2) + 1.00084 * (wL*wM)^^(1/2) + 0.72125 * (wK*wM)^^(1/2)] (***)

XII. Its three derived factor demand functions are:

∂C/∂wL = L(q; wL, wK, wM) = q^^(1/0.92) * [-0.0137 + 0.4677 * (wK / wL)^^(1/2) + 0.5004 * (wM / wL)^^(1/2)]

∂C/∂wK = K(q; wL, wK, wM) = q^^(1/0.92) * [0.0096 + 0.4677 * (wL / wK)^^(1/2) + 0.3606 * (wM / wK)^^(1/2)]

∂C/∂wM = M(q; wL, wK, wM) = q^^(1/0.92) * [-0.0741 + 0.5004 * (wL / wM)^^(1/2) + 0.3606 * (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

-0.5063	0.2932	0.2131	1.087
0.287	-0.492	0.2049	1.087
0.271	0.2662	-0.5372	1.087

XIII. Notes:
1. As derived, dLK = dKL, dLM = dML, and dKM = dMK: Young's Theorem holds by construction.

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

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

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

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

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

4. The cost function C(q;wL,wK,wM) is a concave in factor prices if the Hessian matrix H = ∇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 =

-0.07861	0.02451	0.03861
0.02451	-0.02262	0.02042
0.03861	0.02042	-0.08928

The principal minors of H are H1 = -0.078615, H2 = 0.001178, 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 = -0.1229, e2 = -0.0676, and e3 = -0.
H3 = e1 * e2 * e3 = -0.

XIV. Table of Results: check that the estimated Diewert cost function and input amounts for a given level of output agree with the Generalized CES production function's minimized cost and inputs. Also, compare the Allen partial elasticities of substitution, s_LK, s_LM, and s_MK, with the Uzawa partial elasticities of substitution, u_LK, u_LM, and u_KM.

Generalized CES-Diewert Cost Function À2: Estimate the Diewert cost function directly Parameters: rho = 0.17647, rhoL = 0.17647, rhoK = 0.11823, rhoM = 0.26339
					nu = 1 — Generalized CES Cost Data —								nu1 = 0.92 — Diewert Cost Data —				Factor Shares			Uzawa Elasticities						∇²_wwc(w)
obs #	q	wL	wK	wM	L	K	M	s_LK	s_LM	s_KM	ε_LKM	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	5.66	12.08	4.74	26.18	12.41	25.31	0.9	0.79	0.83	0.924	418	418.03	26.09	12.86	24.26	0.353	0.372	0.275	0.82	0.74	0.82	0.74	0.74	0.74	-0.159	-0.083	0
2	19	5.78	12.1	7.66	30.71	14.93	20.43	0.9	0.79	0.83	0.932	514.65	512.83	30.41	15.21	19.97	0.343	0.359	0.298	0.76	0.78	0.76	0.78	0.78	0.78	-0.131	-0.061	-0
3	20	5.34	11.72	5.34	30.83	14.37	25.62	0.9	0.79	0.83	0.925	469.89	469.54	30.61	14.76	24.93	0.348	0.368	0.283	0.8	0.75	0.8	0.75	0.75	0.75	-0.152	-0.08	-0
4	21	6.98	13.48	5.74	29.83	15.57	29.01	0.9	0.79	0.83	0.924	584.58	584.52	29.83	15.91	28.2	0.356	0.367	0.277	0.81	0.75	0.81	0.73	0.75	0.73	-0.128	-0.068	0
5	22	7.52	14.56	4.4	29.14	15.15	37.15	0.89	0.79	0.83	0.917	603.24	605.26	29.36	15.66	35.58	0.365	0.377	0.259	0.85	0.73	0.85	0.7	0.73	0.7	-0.169	-0.067	-0
6	23	8.44	11.54	7.04	29.59	21.02	28.47	0.9	0.79	0.83	0.929	692.78	692.38	29.91	20.88	28.26	0.365	0.348	0.287	0.79	0.8	0.79	0.71	0.8	0.71	-0.099	-0.062	0
7	24	5.26	11.42	7.72	40.94	19.56	24.63	0.9	0.79	0.83	0.93	628.85	628.49	40.55	19.73	24.6	0.339	0.358	0.302	0.75	0.78	0.75	0.79	0.78	0.79	-0.145	-0.061	0
8	25	8.46	14.48	7.04	34.76	20.37	33.13	0.9	0.79	0.83	0.923	822.25	822.22	34.88	20.46	32.8	0.359	0.36	0.281	0.81	0.77	0.81	0.72	0.77	0.72	-0.102	-0.057	-0
9	26	8.46	11.88	7.46	34.5	24.12	31.43	0.9	0.79	0.83	0.927	812.92	813.28	34.88	23.82	31.54	0.363	0.348	0.289	0.79	0.8	0.79	0.72	0.8	0.72	-0.095	-0.06	0
10	27	5.22	12.16	6.24	45.33	20.44	31.85	0.9	0.79	0.83	0.923	683.93	684.3	44.85	20.65	31.9	0.342	0.367	0.291	0.78	0.75	0.78	0.77	0.75	0.77	-0.15	-0.071	-0
11	28	5.06	12.62	5.44	47.07	20.01	35.87	0.89	0.79	0.83	0.92	685.85	686.48	46.53	20.29	35.83	0.343	0.373	0.284	0.8	0.73	0.8	0.77	0.73	0.77	-0.16	-0.08	-0
12	29	6.52	12.44	4.74	41.07	22.03	43.07	0.89	0.79	0.83	0.917	745.96	746.39	41.16	22.18	42.64	0.36	0.37	0.271	0.82	0.74	0.82	0.72	0.74	0.72	-0.153	-0.076	-0
13	30	7	13	6	43.79	24.14	40.13	0.89	0.79	0.83	0.92	861.17	861.43	43.83	24.1	40.21	0.356	0.364	0.28	0.81	0.76	0.81	0.73	0.76	0.73	-0.123	-0.068	-0
14	31	6.74	11.22	7.72	46.77	28.53	33.92	0.9	0.79	0.83	0.926	897.26	898.8	47.02	28.05	34.61	0.353	0.35	0.297	0.77	0.8	0.77	0.75	0.8	0.75	-0.11	-0.062	-0
15	32	7.42	12.32	4.14	41.3	25.09	53.37	0.89	0.79	0.83	0.913	836.45	837	41.64	25.16	52.68	0.369	0.37	0.261	0.85	0.75	0.85	0.69	0.75	0.69	-0.176	-0.069	-0
16	33	5.4	13.48	7.38	59.45	25.56	36.87	0.9	0.79	0.83	0.921	937.74	939.25	58.6	25.6	37.63	0.337	0.367	0.296	0.77	0.76	0.77	0.79	0.76	0.79	-0.145	-0.061	0
17	34	7.32	14.38	7.92	53.83	28.53	40.45	0.89	0.79	0.83	0.921	1124.63	1125.65	53.7	28.22	41.25	0.349	0.361	0.29	0.78	0.77	0.78	0.75	0.77	0.75	-0.106	-0.056	-0
18	35	8.68	14.92	7.56	50.68	30.17	45.39	0.89	0.79	0.83	0.919	1233.13	1233.73	50.85	29.7	46.19	0.358	0.359	0.283	0.8	0.77	0.8	0.73	0.77	0.73	-0.097	-0.055	-0
19	36	7.06	13.86	5.42	53.12	28.15	52.41	0.89	0.79	0.83	0.914	1049.19	1048.59	53.1	27.97	52.78	0.357	0.37	0.273	0.82	0.74	0.82	0.73	0.74	0.73	-0.135	-0.069	0
20	37	7.12	12.86	6.72	55.69	31.87	46.51	0.89	0.79	0.83	0.919	1118.98	1119.61	55.76	31.31	47.6	0.355	0.36	0.286	0.79	0.77	0.79	0.74	0.77	0.74	-0.113	-0.064	-0
21	38	5.38	15	4.32	64.04	25.04	60.2	0.89	0.79	0.83	0.908	980.26	979.32	63.38	25.19	60.29	0.348	0.386	0.266	0.83	0.69	0.83	0.75	0.69	0.75	-0.182	-0.09	-0
22	39	7.7	14.26	4.18	52.9	29.53	68.69	0.89	0.79	0.83	0.908	1115.47	1113.86	53.17	29.42	68.17	0.368	0.377	0.256	0.85	0.73	0.85	0.7	0.73	0.7	-0.179	-0.066	-0
23	40	7.16	11.76	4.84	54.68	34.04	59.55	0.89	0.79	0.83	0.914	1079.98	1078.95	54.97	33.4	60.46	0.365	0.364	0.271	0.83	0.76	0.83	0.7	0.76	0.7	-0.146	-0.071	0
24	41	5.28	14.26	6.46	75.4	30.59	50.22	0.89	0.79	0.83	0.915	1158.8	1159.11	74.09	30.4	51.76	0.338	0.374	0.288	0.79	0.73	0.79	0.78	0.73	0.78	-0.152	-0.069	0
25	42	6.14	12.36	4.44	62.61	32.72	64.05	0.89	0.79	0.83	0.911	1073.27	1070.88	62.51	32.28	64.88	0.358	0.373	0.269	0.83	0.74	0.83	0.72	0.74	0.72	-0.165	-0.08	-0
26	43	8.1	12.86	6.5	60.69	39.16	57.3	0.89	0.79	0.83	0.916	1367.59	1367.64	61.06	38	59.14	0.362	0.357	0.281	0.81	0.78	0.81	0.72	0.78	0.72	-0.109	-0.062	-0
27	44	7.18	14.1	7.5	70.71	38.02	53.61	0.89	0.79	0.83	0.917	1445.84	1446.28	70.51	37.07	55.64	0.35	0.361	0.289	0.79	0.77	0.79	0.75	0.77	0.75	-0.109	-0.059	-0
28	45	6.48	11.58	5.16	66.22	38.61	62.52	0.89	0.79	0.83	0.913	1198.84	1197.04	66.29	37.6	64.34	0.359	0.364	0.277	0.81	0.76	0.81	0.72	0.76	0.72	-0.14	-0.075	0
29	46	8.42	11.24	5.54	59.15	44.5	65.46	0.89	0.79	0.83	0.915	1360.96	1360.96	59.84	42.84	67.8	0.37	0.354	0.276	0.82	0.79	0.82	0.69	0.79	0.69	-0.123	-0.064	0
30	47	5.28	13.46	6.56	86.15	37.06	56.17	0.89	0.79	0.83	0.914	1322.14	1321.88	84.79	36.41	58.55	0.339	0.371	0.291	0.78	0.74	0.78	0.78	0.74	0.78	-0.15	-0.068	0
31	48	8.32	13.52	6.72	69.06	43.93	64.29	0.89	0.79	0.83	0.914	1600.62	1599.34	69.38	42.39	66.82	0.361	0.358	0.281	0.81	0.78	0.81	0.72	0.78	0.72	-0.106	-0.06	-0
AVE:	33	6.8	12.93	6.09	49.79	26.69	44.88	0.89	0.79	0.83	0.919	949.2	949.13	49.79	26.69	44.88	0.354	0.365	0.281	0.8	0.76	0.8	0.74	0.76	0.74	-0.136	-0.068	-0

The Generalized CES production function as specified:

q = 1 * [0.35 * L^^{- 0.17647} + 0.4 * K^^{- 0.11823} + 0.25 *M^^{- 0.26339}]^^(-1/0.17647) = f(L,K,M).

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

XVI. For these coefficients of the CES Generalized production function, I generated a sequence (displayed in the "Generalized CES-Diewert Cost Function" table) of factor prices, outputs, and the corresponding cost minimizing inputs. Then I used these data to estimate the coefficients of each factor share equation separately:

QR Least Squares
Parameter Estimates
Parameter	Coefficient	std error	t-ratio
cLL	-0.12489	0.079	-1.573
dLK	0.727488	0.065	11.177
dLM	0.650253	0.069	9.4

R2 = 0.9413

R2b = 0.9371

# obs = 31

QR Least Squares
Parameter Estimates
Parameter	Coefficient	std error	t-ratio
cKK	-0.077747	0.077	-1.005
dKL	0.781906	0.094	8.303
dKM	0.466778	0.083	5.625

R2 = 0.825

R2b = 0.8125

# obs = 31

QR Least Squares
Parameter Estimates
Parameter	Coefficient	std error	t-ratio
cMM	-0.242785	0.01	-24.347
dML	0.708749	0.012	60.459
dMK	0.555983	0.009	59.564

R2 = 0.9989

R2b = 0.9988

# obs = 31

The three estimated factor demand functions are:

∂C/∂wL = L(q; wL, wK, wM) = q^^(1/1) * [-0.1249 + 0.7275 * (wK / wL)^^(1/2) + 0.6503 * (wM / wL)^^(1/2)]

∂C/∂wK = K(q; wL, wK, wM) = q^^(1/1) * [-0.0777 + 0.7819 * (wL / wK)^^(1/2) + 0.4668 * (wM / wK)^^(1/2)]

∂C/∂wM = M(q; wL, wK, wM) = q^^(1/1) * [-0.2428 + 0.7087 * (wL / wM)^^(1/2) + 0.556 * (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

-0.5425	0.3375	0.205	1
0.3528	-0.5478	0.195	1
0.2854	0.3051	-0.5905	1

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

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

= q^^(1/1) * [-0.12489 * wL + -0.07775 * wK + -0.24279 * wM + 1.50939 * (wL*wK)^^(1/2) + 1.359 * (wL*wM)^^(1/2) + 1.02276 * (wK*wM)^^(1/2)] (***)

XVIII. The Restricted Factor Demand Equations.

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

QR Restricted Least Squares
Parameter Estimates
Parameter	Coefficient	std error	t-ratio
cLL	-0.14767	0.046934	-3.146336
dLK	0.720037	0.033976	21.192639
dLM	0.685125	0.031094	22.0342
cKK	-0.093216	0.033833	-2.755151
dKL	0.720037	0.033976	21.192639
dKM	0.554851	0.028889	19.206047
cMM	-0.215851	0.039093	-5.521503
dML	0.685125	0.031094	22.0342
dMK	0.554851	0.028889	19.206047

R2 = 0.9944

R2b = 0.9938

# 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) * [-0.1477 + 0.72 * (wK / wL)^^(1/2) + 0.6851 * (wM / wL)^^(1/2)]

∂C/∂wK = K(q; wL, wK, wM) = q^^(1/1) * [-0.0932 + 0.72 * (wL / wK)^^(1/2) + 0.5549 * (wM / wK)^^(1/2)]

∂C/∂wM = M(q; wL, wK, wM) = q^^(1/1) * [-0.2159 + 0.6851 * (wL / wM)^^(1/2) + 0.5549 * (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

-0.5503	0.3342	0.2161	1
0.3253	-0.5574	0.2321	1
0.2759	0.3045	-0.5805	1

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

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

= q^^(1/1) * [-0.14767 * wL + -0.09322 * wK + -0.21585 * wM + 1.44007 * (wL*wK)^^(1/2) + 1.37025 * (wL*wM)^^(1/2) + 1.1097 * (wK*wM)^^(1/2)] (***)

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

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

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

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

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

H = ∇²_wwc =

-0.1154	0.03774	0.05286
0.03774	-0.03482	0.03141
0.05286	0.03141	-0.12973

The principal minors of H are H1 = -0.115396, H2 = 0.002594, 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 = -0.1759, e2 = -0.104, and e3 = -0.
H3 = e1 * e2 * e3 = -0.

XXI. The factor share functions are:

XXII. Uzawa Partial Elasticities of Substitution:

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.

XXIII. Table of Results: check that the estimated Diewert cost function and input amounts for a given level of output agree with the Generalized CES production function's minimized cost and inputs. Also, compare the Allen partial elasticities of substitution, s_LK, s_LM, and s_MK, with the Uzawa partial elasticities of substitution, u_LK, u_LM, and u_KM.

Generalized CES-Diewert Cost Function À3: Estimate the Translog cost function using restricted factor shares Parameters: rho = 0.17647, rhoL = 0.17647, rhoK = 0.11823, rhoM = 0.26339
					nu = 1 — Generalized CES Cost Data —								nu1 = 1 — Diewert Cost Data —				Factor Shares			Uzawa Elasticities						∇²_wwc(w)
obs #	q	wL	wK	wM	L	K	M	s_LK	s_LM	s_KM	ε_LKM	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	5.66	12.08	4.74	26.18	12.41	25.31	0.9	0.79	0.83	0.924	418	439.51	27.56	13.45	25.53	0.355	0.37	0.275	0.93	0.74	0.93	0.84	0.74	0.84	-0.228	-0.128	-0
2	19	5.78	12.1	7.66	30.71	14.93	20.43	0.9	0.79	0.83	0.932	514.65	535.98	31.97	16.07	20.46	0.345	0.363	0.292	0.85	0.8	0.85	0.89	0.8	0.89	-0.189	-0.092	-0
3	20	5.34	11.72	5.34	30.83	14.37	25.62	0.9	0.79	0.83	0.925	469.89	489.1	32.08	15.35	25.83	0.35	0.368	0.282	0.9	0.76	0.9	0.87	0.76	0.87	-0.219	-0.123	0
4	21	6.98	13.48	5.74	29.83	15.57	29.01	0.9	0.79	0.83	0.924	584.58	606.42	30.96	16.53	29.19	0.356	0.367	0.276	0.92	0.76	0.92	0.83	0.76	0.83	-0.183	-0.105	-0
5	22	7.52	14.56	4.4	29.14	15.15	37.15	0.89	0.79	0.83	0.917	603.24	625.13	30.32	16.04	37.16	0.365	0.374	0.262	0.97	0.73	0.97	0.8	0.73	0.8	-0.243	-0.103	-0
6	23	8.44	11.54	7.04	29.59	21.02	28.47	0.9	0.79	0.83	0.929	692.78	711.5	30.36	21.99	28.63	0.36	0.357	0.283	0.89	0.84	0.89	0.8	0.84	0.8	-0.141	-0.096	-0
7	24	5.26	11.42	7.72	40.94	19.56	24.63	0.9	0.79	0.83	0.93	628.85	643.32	41.84	20.44	24.59	0.342	0.363	0.295	0.84	0.81	0.84	0.91	0.81	0.91	-0.209	-0.092	0
8	25	8.46	14.48	7.04	34.76	20.37	33.13	0.9	0.79	0.83	0.923	822.25	839.97	35.48	21.1	33.27	0.357	0.364	0.279	0.91	0.79	0.91	0.82	0.79	0.82	-0.146	-0.088	0
9	26	8.46	11.88	7.46	34.5	24.12	31.43	0.9	0.79	0.83	0.927	812.92	826.87	35.07	24.81	31.56	0.359	0.356	0.285	0.89	0.84	0.89	0.81	0.84	0.81	-0.136	-0.093	-0
10	27	5.22	12.16	6.24	45.33	20.44	31.85	0.9	0.79	0.83	0.923	683.93	694.14	45.91	20.95	32	0.345	0.367	0.288	0.88	0.77	0.88	0.89	0.77	0.89	-0.216	-0.109	-0
11	28	5.06	12.62	5.44	47.07	20.01	35.87	0.89	0.79	0.83	0.92	685.85	694.22	47.6	20.36	36.12	0.347	0.37	0.283	0.9	0.74	0.9	0.89	0.74	0.89	-0.231	-0.122	-0
12	29	6.52	12.44	4.74	41.07	22.03	43.07	0.89	0.79	0.83	0.917	745.96	752.92	41.5	22.35	43.11	0.359	0.369	0.271	0.94	0.75	0.94	0.82	0.75	0.82	-0.219	-0.116	-0
13	30	7	13	6	43.79	24.14	40.13	0.89	0.79	0.83	0.92	861.17	866.33	44.04	24.36	40.23	0.356	0.366	0.279	0.91	0.78	0.91	0.83	0.78	0.83	-0.176	-0.104	-0
14	31	6.74	11.22	7.72	46.77	28.53	33.92	0.9	0.79	0.83	0.926	897.26	899.85	46.95	28.68	33.89	0.352	0.358	0.291	0.86	0.83	0.86	0.86	0.83	0.86	-0.159	-0.095	-0
15	32	7.42	12.32	4.14	41.3	25.09	53.37	0.89	0.79	0.83	0.913	836.45	836.82	41.34	25.19	53.07	0.367	0.371	0.263	0.97	0.75	0.97	0.78	0.75	0.78	-0.254	-0.106	0
16	33	5.4	13.48	7.38	59.45	25.56	36.87	0.9	0.79	0.83	0.921	937.74	935.81	59.1	25.51	36.96	0.341	0.367	0.291	0.86	0.77	0.86	0.91	0.77	0.91	-0.209	-0.093	0
17	34	7.32	14.38	7.92	53.83	28.53	40.45	0.89	0.79	0.83	0.921	1124.63	1119.27	53.52	28.3	40.48	0.35	0.364	0.286	0.88	0.79	0.88	0.86	0.79	0.86	-0.152	-0.086	0
18	35	8.68	14.92	7.56	50.68	30.17	45.39	0.89	0.79	0.83	0.919	1233.13	1223.92	50.25	29.78	45.42	0.356	0.363	0.281	0.91	0.79	0.91	0.83	0.79	0.83	-0.138	-0.085	0
19	36	7.06	13.86	5.42	53.12	28.15	52.41	0.89	0.79	0.83	0.914	1049.19	1038.06	52.61	27.64	52.32	0.358	0.369	0.273	0.94	0.75	0.94	0.83	0.75	0.83	-0.193	-0.106	-0
20	37	7.12	12.86	6.72	55.69	31.87	46.51	0.89	0.79	0.83	0.919	1118.98	1105.31	54.97	31.21	46.51	0.354	0.363	0.283	0.9	0.79	0.9	0.84	0.79	0.84	-0.162	-0.099	0
21	38	5.38	15	4.32	64.04	25.04	60.2	0.89	0.79	0.83	0.908	980.26	963.31	63.4	24.16	60.14	0.354	0.376	0.27	0.96	0.68	0.96	0.87	0.68	0.87	-0.262	-0.139	-0
22	39	7.7	14.26	4.18	52.9	29.53	68.69	0.89	0.79	0.83	0.908	1115.47	1094.45	52.14	28.72	67.82	0.367	0.374	0.259	0.98	0.73	0.98	0.79	0.73	0.79	-0.259	-0.101	-0
23	40	7.16	11.76	4.84	54.68	34.04	59.55	0.89	0.79	0.83	0.914	1079.98	1058.18	53.54	32.98	59.29	0.362	0.367	0.271	0.94	0.78	0.94	0.8	0.78	0.8	-0.209	-0.109	0
24	41	5.28	14.26	6.46	75.4	30.59	50.22	0.89	0.79	0.83	0.915	1158.8	1133.48	73.53	29.45	50.34	0.343	0.371	0.287	0.89	0.74	0.89	0.91	0.74	0.91	-0.219	-0.105	0
25	42	6.14	12.36	4.44	62.61	32.72	64.05	0.89	0.79	0.83	0.911	1073.27	1045.93	61.17	31.37	63.65	0.359	0.371	0.27	0.95	0.74	0.95	0.82	0.74	0.82	-0.236	-0.123	-0
26	43	8.1	12.86	6.5	60.69	39.16	57.3	0.89	0.79	0.83	0.916	1367.59	1332.49	59.05	37.53	57.16	0.359	0.362	0.279	0.91	0.8	0.91	0.81	0.8	0.81	-0.155	-0.095	-0
27	44	7.18	14.1	7.5	70.71	38.02	53.61	0.89	0.79	0.83	0.917	1445.84	1406.37	68.71	36.31	53.47	0.351	0.364	0.285	0.89	0.79	0.89	0.86	0.79	0.86	-0.157	-0.09	-0
28	45	6.48	11.58	5.16	66.22	38.61	62.52	0.89	0.79	0.83	0.913	1198.84	1162.16	64.18	36.71	62.24	0.358	0.366	0.276	0.92	0.78	0.92	0.82	0.78	0.82	-0.2	-0.116	-0
29	46	8.42	11.24	5.54	59.15	44.5	65.46	0.89	0.79	0.83	0.915	1360.96	1317.35	57.04	42.3	65.28	0.365	0.361	0.275	0.93	0.82	0.93	0.77	0.82	0.77	-0.176	-0.099	0
30	47	5.28	13.46	6.56	86.15	37.06	56.17	0.89	0.79	0.83	0.914	1322.14	1277.54	82.98	35.02	56.1	0.343	0.369	0.288	0.88	0.75	0.88	0.9	0.75	0.9	-0.216	-0.103	0
31	48	8.32	13.52	6.72	69.06	43.93	64.29	0.89	0.79	0.83	0.914	1600.62	1543.54	66.52	41.41	64.01	0.359	0.363	0.279	0.91	0.8	0.91	0.81	0.8	0.81	-0.151	-0.092	-0
AVE:	33	6.8	12.93	6.09	49.54	26.65	44.38	0.89	0.79	0.83	0.919	949.2	942.56	49.54	26.65	44.38	0.355	0.366	0.279	0.91	0.77	0.91	0.84	0.77	0.84	-0.195	-0.104	-0

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

XXIV. If we have a data set relating the input (factor) prices, wL, wK, and wM, to the total (minimum) cost of producing output for varying levels output, but do not have data on the required levels of inputs, we can estimate the Diewert cost function directly. We will use the same sequence (displayed in the "Diewert Cost Function" table) of factor prices, outputs, and total (minimum) cost as used in À1. The estimated coefficients of the cost function will vary with the parameters nu, rho, rhoL, rhoK, rhoM, alpha, beta and gamma of the Generalized CES production function used to generate these data.

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

SVD Least Squares
Parameter Estimates
Parameter	Coefficient	std error	t-ratio
cLL	2.712501	2.323	1.168
cKK	-0.087696	1.121	-0.078
cMM	-1.987101	1.522	-1.306
2*dLK	-1.62063	2.615	-0.62
2*dLM	-0.070591	2.657	-0.027
2*dKM	4.405904	2.238	1.969

R2 = 0.9529	R2b = 0.9435	# obs = 31
Observation Matrix Rank: 6

The Diewert cost function as obtained by direct estimation is:

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

= q^^(1/1) * [2.7125 * wL + -0.0877 * wK + -1.9871 * wM + -1.62063 * (wL*wK)^^(1/2) + -0.07059 * (wL*wM)^^(1/2) + 4.4059 * (wK*wM)^^(1/2)] (***)

XXVI. Its three derived factor demand functions are:

∂C/∂wL = L(q; wL, wK, wM) = q^^(1/1) * [2.7125 + -0.8103 * (wK / wL)^^(1/2) + -0.0353 * (wM / wL)^^(1/2)]

∂C/∂wK = K(q; wL, wK, wM) = q^^(1/1) * [-0.0877 + -0.8103 * (wL / wK)^^(1/2) + 2.203 * (wM / wK)^^(1/2)]

∂C/∂wM = M(q; wL, wK, wM) = q^^(1/1) * [-1.9871 + -0.0353 * (wL / wM)^^(1/2) + 2.203 * (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

0.3608	-0.3504	-0.0104	1
-0.3651	-0.5538	0.9189	1
-0.0157	1.3318	-1.3161	1

XXVII. Notes:
1. As derived, dLK = dKL, dLM = dML, and dKM = dMK: Young's Theorem holds by construction.

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

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

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

2 * c(7, 13, 6) = 57.84 = 57.84 = 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.

H = ∇²_wwc =

0.08121	-0.04247	-0.00272
-0.04247	-0.03469	0.12472
-0.00272	0.12472	-0.26704

The principal minors of H are H1 = 0.081211, H2 = -0.004621, 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 = -0.3218, e2 = 0.1013, and e3 = 0.
H3 = e1 * e2 * e3 = -0.

XXVIII. Table of Results: check that the estimated Diewert cost function and input amounts for a given level of output agree with the Generalized CES production function's minimized cost and inputs. Also, compare the Allen partial elasticities of substitution, s_LK, s_LM, and s_MK, with the Uzawa partial elasticities of substitution, u_LK, u_LM, and u_KM.

Generalized CES-Diewert Cost Function À4: Estimate the Diewert cost function directly Parameters: rho = 0.17647, rhoL = 0.17647, rhoK = 0.11823, rhoM = 0.26339
					nu = 1 — Generalized CES Cost Data —								nu1 = 1 — Diewert Cost Data —				Factor Shares			Uzawa Elasticities						∇²_wwc(w)
obs #	q	wL	wK	wM	L	K	M	s_LK	s_LM	s_KM	ε_LKM	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	5.66	12.08	4.74	26.18	12.41	25.31	0.9	0.79	0.83	0.924	418	440.06	26.94	13.28	26.84	0.346	0.364	0.289	-1.09	-0.04	-1.09	3.24	-0.04	3.24	-0.422	0.128	0
2	19	5.78	12.1	7.66	30.71	14.93	20.43	0.9	0.79	0.83	0.932	514.65	528.02	28.49	21	14.27	0.312	0.481	0.207	-0.81	-0.07	-0.81	3.83	-0.07	3.83	-0.247	0.124	0
3	20	5.34	11.72	5.34	30.83	14.37	25.62	0.9	0.79	0.83	0.925	469.89	490.07	29.53	17.05	24.82	0.322	0.408	0.27	-1	-0.04	-1	3.22	-0.04	3.22	-0.363	0.136	-0
4	21	6.98	13.48	5.74	29.83	15.57	29.01	0.9	0.79	0.83	0.924	584.58	607.61	32.64	16.1	28.35	0.375	0.357	0.268	-1.01	-0.04	-1.01	3.5	-0.04	3.5	-0.342	0.102	0
5	22	7.52	14.56	4.4	29.14	15.15	37.15	0.89	0.79	0.83	0.917	603.24	622.13	34.28	11.9	43.43	0.414	0.279	0.307	-1.3	-0.03	-1.3	3.64	-0.03	3.64	-0.491	0.096	0
6	23	8.44	11.54	7.04	29.59	21.02	28.47	0.9	0.79	0.83	0.929	692.78	714.53	39.85	21.62	18.28	0.471	0.349	0.18	-0.78	-0.05	-0.78	5.08	-0.05	5.08	-0.266	0.082	0
7	24	5.26	11.42	7.72	40.94	19.56	24.63	0.9	0.79	0.83	0.93	628.85	630.83	35.42	28.17	15.91	0.295	0.51	0.195	-0.79	-0.07	-0.79	3.96	-0.07	3.96	-0.246	0.137	0
8	25	8.46	14.48	7.04	34.76	20.37	33.13	0.9	0.79	0.83	0.923	822.25	842.27	40.5	20.72	28.34	0.407	0.356	0.237	-0.92	-0.04	-0.92	3.91	-0.04	3.91	-0.272	0.083	0
9	26	8.46	11.88	7.46	34.5	24.12	31.43	0.9	0.79	0.83	0.927	812.92	825.54	44.7	25.33	19.64	0.458	0.364	0.177	-0.77	-0.05	-0.77	5.05	-0.05	5.05	-0.251	0.082	0
10	27	5.22	12.16	6.24	45.33	20.44	31.85	0.9	0.79	0.83	0.923	683.93	695.46	38.8	25.91	28.51	0.291	0.453	0.256	-0.95	-0.05	-0.95	3.22	-0.05	3.22	-0.306	0.141	0
11	28	5.06	12.62	5.44	47.07	20.01	35.87	0.89	0.79	0.83	0.92	685.85	699.83	39.09	23.68	37.36	0.283	0.427	0.29	-1.07	-0.05	-1.07	2.95	-0.05	2.95	-0.361	0.148	0
12	29	6.52	12.44	4.74	41.07	22.03	43.07	0.89	0.79	0.83	0.917	745.96	754.59	45.33	19.88	44.67	0.392	0.328	0.281	-1.09	-0.03	-1.09	3.53	-0.03	3.53	-0.425	0.11	-0
13	30	7	13	6	43.79	24.14	40.13	0.89	0.79	0.83	0.92	861.17	867.58	47.27	24.43	36.52	0.381	0.366	0.253	-0.96	-0.04	-0.96	3.64	-0.04	3.64	-0.322	0.101	0
14	31	6.74	11.22	7.72	46.77	28.53	33.92	0.9	0.79	0.83	0.926	897.26	879.18	50.51	34.46	19.71	0.387	0.44	0.173	-0.73	-0.07	-0.73	4.75	-0.07	4.75	-0.245	0.102	0
15	32	7.42	12.32	4.14	41.3	25.09	53.37	0.89	0.79	0.83	0.913	836.45	844.78	52.54	17.94	56.51	0.462	0.262	0.277	-1.22	-0.03	-1.22	4.11	-0.03	4.11	-0.503	0.096	0
16	33	5.4	13.48	7.38	59.45	25.56	36.87	0.9	0.79	0.83	0.921	937.74	939.6	45.9	33.97	31.68	0.264	0.487	0.249	-0.94	-0.06	-0.94	3.18	-0.06	3.18	-0.257	0.139	0
17	34	7.32	14.38	7.92	53.83	28.53	40.45	0.89	0.79	0.83	0.921	1124.63	1112.18	52.36	32.95	32.21	0.345	0.426	0.229	-0.87	-0.05	-0.87	3.68	-0.05	3.68	-0.239	0.097	0
18	35	8.68	14.92	7.56	50.68	30.17	45.39	0.89	0.79	0.83	0.919	1233.13	1224.71	56.6	30.18	37.44	0.401	0.368	0.231	-0.89	-0.04	-0.89	3.93	-0.04	3.93	-0.252	0.081	0
19	36	7.06	13.86	5.42	53.12	28.15	52.41	0.89	0.79	0.83	0.914	1049.19	1039.82	55.66	25.62	53.83	0.378	0.341	0.281	-1.08	-0.04	-1.08	3.45	-0.04	3.45	-0.369	0.102	-0
20	37	7.12	12.86	6.72	55.69	31.87	46.51	0.89	0.79	0.83	0.919	1118.98	1102.38	58.8	33.37	37.89	0.38	0.389	0.231	-0.88	-0.05	-0.88	3.82	-0.05	3.82	-0.282	0.099	0
21	38	5.38	15	4.32	64.04	25.04	60.2	0.89	0.79	0.83	0.908	980.26	959.93	50.46	23.15	78.98	0.283	0.362	0.355	-1.41	-0.03	-1.41	2.73	-0.03	2.73	-0.509	0.144	0
22	39	7.7	14.26	4.18	52.9	29.53	68.69	0.89	0.79	0.83	0.908	1115.47	1090.56	61.77	19.87	79.32	0.436	0.26	0.304	-1.34	-0.03	-1.34	3.85	-0.03	3.85	-0.522	0.093	0
23	40	7.16	11.76	4.84	54.68	34.04	59.55	0.89	0.79	0.83	0.914	1079.98	1069.04	65.8	27.73	56.15	0.441	0.305	0.254	-1.03	-0.03	-1.03	4.01	-0.03	4.01	-0.408	0.099	-0
24	41	5.28	14.26	6.46	75.4	30.59	50.22	0.89	0.79	0.83	0.915	1158.8	1149.95	55.01	36.98	51.41	0.253	0.459	0.289	-1.08	-0.05	-1.08	2.85	-0.05	2.85	-0.301	0.145	0
25	42	6.14	12.36	4.44	62.61	32.72	64.05	0.89	0.79	0.83	0.911	1073.27	1045.81	64.38	27.78	69.17	0.378	0.328	0.294	-1.14	-0.03	-1.14	3.4	-0.03	3.4	-0.461	0.117	0
26	43	8.1	12.86	6.5	60.69	39.16	57.3	0.89	0.79	0.83	0.916	1367.59	1339.74	71.37	35.92	46.1	0.432	0.345	0.224	-0.89	-0.04	-0.89	4.19	-0.04	4.19	-0.293	0.086	0
27	44	7.18	14.1	7.5	70.71	38.02	53.61	0.89	0.79	0.83	0.917	1445.84	1400.07	67.8	41.39	43.95	0.348	0.417	0.235	-0.88	-0.05	-0.88	3.63	-0.05	3.63	-0.253	0.099	0
28	45	6.48	11.58	5.16	66.22	38.61	62.52	0.89	0.79	0.83	0.913	1198.84	1166.35	71.9	34.95	57.31	0.399	0.347	0.254	-0.98	-0.04	-0.98	3.73	-0.04	3.73	-0.376	0.109	0
29	46	8.42	11.24	5.54	59.15	44.5	65.46	0.89	0.79	0.83	0.915	1360.96	1350.76	80.39	34.85	50.93	0.501	0.29	0.209	-0.92	-0.04	-0.92	4.89	-0.04	4.89	-0.344	0.084	0
30	47	5.28	13.46	6.56	86.15	37.06	56.17	0.89	0.79	0.83	0.914	1322.14	1289.18	64.83	44.31	53.43	0.266	0.463	0.272	-1.01	-0.05	-1.01	3	-0.05	3	-0.293	0.143	0
31	48	8.32	13.52	6.72	69.06	43.93	64.29	0.89	0.79	0.83	0.914	1600.62	1550.83	79.1	39.83	52.72	0.424	0.347	0.228	-0.9	-0.04	-0.9	4.1	-0.04	4.1	-0.284	0.084	0
AVE:	33	6.8	12.93	6.09	51.23	27.24	41.15	0.89	0.79	0.83	0.919	949.2	944.3	51.23	27.24	41.15	0.372	0.377	0.252	-0.99	-0.04	-0.99	3.74	-0.04	3.74	-0.339	0.109	0

Mathematical Notes

1. The Diewert (Generalized Leontief) cost function

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

where the returns to scale function is: h(q) = q^^(1/nu), a continuous, increasing function of q (q >= 1), with h(0) = 0 and h(1) = 1,

and the unit cost function is: c(wL,wK,wM) = cLL * wL + cKK * wK + cMM * wM + (dLK+dKL) * (wL*wK)^^(1/2) + (dLM +dML) * (wL*wM)^^(1/2) + (dKM+dMK) * (wK*wM)^^(1/2),

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

2. The Factor Demand Functions:

∂C/∂wL = L(q; wL, wK, wM) = q^^(1/nu) * [cLL + dLK * (wK / wL)^^(1/2) + dLM * (wM / wL)^^(1/2)]

∂C/∂wK = K(q; wL, wK, wM) = q^^(1/nu) * [cKK + dKL * (wL / wK)^^(1/2) + dKM * (wM / wK)^^(1/2)]

∂C/∂wM = M(q; wL, wK, wM) = q^^(1/nu) * [cMM + dML * (wL / wM)^^(1/2) + dMK * (wK / wM)^^(1/2)]

3. The Factor Share Functions:

4. The Factor Demand Elasticities:

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.

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).