
Egwald Economics: Microeconomics
The Duality of Production and Cost Functions
by
Elmer G. Wiens
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Duality: Production / Cost Functions:
CobbDouglas Duality
 CES Duality
 Theory of Duality
 Translog Duality  CES
 Translog Duality  Generalized CES
 References and Links
Cost Functions:
CobbDouglas Cost
 Normalized Quadratic Cost
 Translog Cost
 Diewert Cost
 Generalized CESTranslog Cost
 Generalized CESDiewert Cost
 References and Links
Production Functions:
CobbDouglas
 CES
 Generalized CES
 Translog
 Diewert
 Translog vs Diewert
 Diewert vs Translog
 Estimate Translog
 Estimate Diewert
 References and Links
Q. Duality of Production and Cost Functions Using the Implicit Function Theorem.
I. Profit (Wealth) Maximizing Firm.
Production and cost functions (and profit functions) can be used to model how a profit (wealth) maximizing firm hires or purchases inputs (factors), such as labour, capital (structures and machinery), and materials and supplies, and combines these inputs through its production process to produce the products (outputs) that the firm sells (supplies) to its customers.
II. The Production Function.
The production function describes the maximum output that can be produced from given quantities of factor inputs with the firm's existing technological expertise. Let the variables q, L, K, and M represent the quantity of output, and the input quantities of labour, capital, and materials and supplies, respectively.
Mathematically, the production function, f, relates output, q, to inputs, L, K, and M, written as:
q = f(L, K, M)
with the function f having certain desirable properties.
III. Example: The CES production function
q = A * [alpha * (L^^{rho}) + beta * (K^^{rho}) + gamma *(M^^{rho})]^^{(nu/rho)} = f(L, K, M).
The coefficients of the production function, A, alpha, beta, gamma, nu, and rho are positive, real numbers. The production function's inputs, L, K, and M, are nonnegative real numbers.
IV. The Total Cost of Production.
Let the variables wL, wK, and wM represent the unit prices of the factors L, K, and M, respectively. For any given combination of factor inputs, L, K, and M, the total cost of using these inputs is:
TC = wL * L + wK * K + wM * M
i.e. the sum of the quantities of factor inputs weighted by their respective factor prices.
V. The Cost Function.
The cost function describes the total cost of producing any given output quantity, using the cost minimizing quantity of inputs.
Mathematically, the cost function, C, relates the total cost, TC, to output, q, and factor prices wL, wK, and wM, if the cost minimizing combination of factor inputs is used, written as:
TC = C(q; wL, wK, wM)
with the function C having certain desirable properties.
VI. Example: The CES cost function:
C(q;wL,wK,wM) = h(q) * c(wL,wK,wM) = (q/A)^^{1/nu} * [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)}
The cost function's factor prices, wL, wK, and wM, are positive real numbers.
VII. The LeastCost Combination of Inputs: Production Function to Cost Function.
The entrepreneur, management, and employees of the profit maximizing firm choose the factor proportions and quantities, and output levels given the prices of factor inputs, and products. For any specified combination of positive factor prices, wL, wK, and wM, what combination of factor inputs, L, K, and M, will minimize the cost of producing any given level of positive output, q?
C^{*}(q; wL, wK, wM) = min_{L,K,M}{ wL * L + wK * K + wM * M : q  f(L,K,M) = 0, q > 0, wL > 0; wK > 0, and wM > 0 }
VIII. Constrained Optimization (Minimum): The Method of Lagrange.
Define the Langrangian function, G, of the leastcost problem of (VII):
G(q; wL, wK, wM, L, K, M, μ) = wL * L + wK * K + wM * M + μ * (q  f(L,K,M))
where the new variable, μ, is called the Lagrange multiplier.
a. First Order Necessary Conditions:
0. G_{µ}(q; wL, wK, wM, L, K, M, μ) = q  f(L, K, M) = 0
1. G_{L}(q; wL, wK, wM, L, K, M, μ) = wL  µ * f_{L}(L, K, M) = 0
2. G_{K}(q; wL, wK, wM, L, K, M, μ) = wK  µ * f_{K}(L, K, M) = 0
3. G_{M}(q; wL, wK, wM, L, K, M, μ) = wM  µ * f_{M}(L, K, M) = 0

b. Solution Functions:
We want to solve, simultaneously, these four equations for the variables L, K, M, and µ as continuously differentiable functions of the variables q, wL, wK, and wM, and the parameters of the production function.
L = L(q; wL, wK, wM)
K = K(q; wL, wK, wM)
M = M(q; wL, wK, wM)
µ = µ(q; wL, wK, wM)
Suppose that the point Z = (q, wL, wK, wM, L, K, M, µ) satisfies equations 0. to 3., and that each of the functions G_{µ}, G_{L}, G_{K}, and G_{M} has continuous partial derivatives with respect to each of the variables q, wL, wK, wM, L, K, M, and μ at the point Z. Also, suppose the determinant of the Jacobian matrix, defined below, when evaluated at the point Z is not equal to zero.
According to the Implicit Function Theorem, functions L, K, M, and µ exist that express the variables L, K, M, and µ as continuously differentiable functions of the variables q, wL, wK, and wM.
Moreover:
L = L(q; wL, wK, wM)
K = K(q; wL, wK, wM)
M = M(q; wL, wK, wM)
µ = µ(q; wL, wK, wM)
and:
0'. G_{µ}(q; wL, wK, wM, L, K, M, μ) = q  f(L(q; wL, wK, wM), K(q; wL, wK, wM), M(q; wL, wK, wM)) = 0
1'. G_{L}(q; wL, wK, wM, L, K, M, μ) = wL  µ(q; wL, wK, wM) * f_{L}(L(q; wL, wK, wM), K(q; wL, wK, wM), M(q; wL, wK, wM)) = 0
2'. G_{K}(q; wL, wK, wM, L, K, M, μ) = wK  µ(q; wL, wK, wM) * f_{K}(L(q; wL, wK, wM), K(q; wL, wK, wM), M(q; wL, wK, wM)) = 0
3'. G_{M}(q; wL, wK, wM, L, K, M, μ) = wM  µ(q; wL, wK, wM) * f_{M}(L(q; wL, wK, wM), K(q; wL, wK, wM), M(q; wL, wK, wM)) = 0

for points (q; wL, wK, wM) in a neighborhood of the point (q; wL, wK, wM). That is, the points (q, wL, wK, wM, L(q; wL, wK, wM), K(q; wL, wK, wM), M(q; wL, wK, wM), µ(q; wL, wK, wM)) also satisfy the first order conditions.
c. Jacobian Matrices:
The Jacobian matrix (bordered Hessian of G) of the four functions, G_{µ}, G_{L}, G_{K}, G_{M}, with respect to the choice variables, μ, L, K, and M:
J_{3} =

G_{µµ}  G_{µL}  G_{µK}  G_{µM} 
G_{Lµ}  G_{LL}  G_{LK}  G_{LM} 
G_{Kµ}  G_{KL}  G_{KK}  G_{KM} 
G_{Mµ}  G_{ML}  G_{MK}  G_{MM} 
 = 
0  f_{L}  f_{K}  f_{M} 
f_{L}  µ * f_{LL}  µ * f_{LK}  µ * f_{LM} 
f_{K}  µ * f_{KL}  µ * f_{KK}  µ * f_{KM} 
f_{M}  µ * f_{ML}  µ * f_{MK}  µ * f_{MM} 

=
J_{µ, L, K, M}

The bordered principal minor of the bordered Hessian of the Langrangian function, G:
J_{2} =

0  f_{L}  f_{K} 
f_{L}  µ * f_{LL}  µ * f_{LK} 
f_{K}  µ * f_{KL}  µ * f_{KK} 

d. Second Order Necessary Conditions:
The second order necessary conditions require that the Jacobian matrix (bordered Hessian of G) be positive definite at Z = (q, wL, wK, wM, L, K, M, µ). The Jacobian matrix, J_{3}, is a positive definite matrix if the determinants of J_{2} and J_{3} are both negative.
e. Sufficient Conditions:
If the second order necessary conditions are satisfied, then the first order necessary conditions are sufficient for a minimum at Z. Therefore:
C^{*}(q; wL, wK, wM) = wL * L(q; wL, wK, wM) + wK * K(q; wL, wK, wM) + wM * M(q; wL, wK, wM)
Moreover, since the sixteen functions that comprise the components of J_{3} are continuous, the Jacobian matrix, J_{3}, is a positive definite matrix at points (q; wL, wK, wM) in a neighborhood of the point (q; wL, wK, wM). That is, the points (q, wL, wK, wM, L(q; wL, wK, wM), K(q; wL, wK, wM), M(q; wL, wK, wM), µ(q; wL, wK, wM)) also satisfy the second order conditions.
Consequently, for points (q; wL, wK, wM) in a neighborhood of the point (q; wL, wK, wM):
C^{*}(q; wL, wK, wM) = wL * L(q; wL, wK, wM) + wK * K(q; wL, wK, wM) + wM * M(q; wL, wK, wM) = G(q; wL, wK, wM, L(q; wL, wK, wM), K(q; wL, wK, wM), M(q; wL, wK, wM), μ(q; wL, wK, wM))
IX. Marginal Cost Function.
MC^{*}(q; wL, wK, wM) = ∂C^{*}(q; wL, wK, wM)/∂q = ∂G(q; wL, wK, wM, L, K, M, μ)/∂q = μ(q; wL, wK, wM)
Example: CES Marginal Cost
MC(q; wL, wK, wM) = ∂C(q; wL, wK, wM)/∂q = h'(q) * c(wL,wK,wM) = (1/nu) * (1/A) * (q/A) ^^{(1  nu)/nu} * c(wL,wK,wM)

X. Factor Demand Functions.
If the cost function C^{*}(q; wL, wK, wM) satisfies certain properties (HotellingShephard's lemma), properties derived from the properties of the production function f(L, K, M):
L(q; wL, wK, wM) = ∂C^{*}(q; wL, wK, wM)/∂wL = L
K(q; wL, wK, wM) = ∂C^{*}(q; wL, wK, wM)/∂wK = K
M(q; wL, wK, wM) = ∂C^{*}(q; wL, wK, wM)/∂wM = M
where (L, K, M) are the factor proportions that minimize the cost of producing q units of output at specific factor prices (wL, wK, wM) for a given output quantity q.
Example: CES Factor Demand Functions
L(q;wL,wK,wM) = h(q) * l(wL,wK,wM) = h(q) * [(alpha / wL) * c(wL,wK,wM)]^^{(1/(1+rho))}
K(q;wL,wK,wM) = h(q) * m(wL,wK,wM) = h(q) * [(beta / wK) * c(wL,wK,wM)]^^{(1/(1+rho))}
M(q;wL,wK,wM) = h(q) * m(wL,wK,wM) = h(q) * [(gamma / wM) * c(wL,wK,wM)]^^{(1/(1+rho))}

If the cost function C^{*}(q; wL, wK, wM) satisfies the properties required by the HotellingShephard's lemma, then the factor demand functions satisfy:
wL * ∂ L(q;wL,wK,wM) / ∂wL + wK * ∂ K(q;wL,wK,wM) / ∂wL + wM * ∂ M(q;wL,wK,wM) / ∂wL = 0
wL * ∂ L(q;wL,wK,wM) / ∂wK + wK * ∂ K(q;wL,wK,wM) / ∂wK + wM * ∂ M(q;wL,wK,wM) / ∂wK = 0
wL * ∂ L(q;wL,wK,wM) / ∂wM + wK * ∂ K(q;wL,wK,wM) / ∂wM + wM * ∂ M(q;wL,wK,wM) / ∂wM = 0
If the firstorder conditions are sufficient for minimum, then;
q ≡ f(L(q; wL, wK, wM), K(q; wL, wK, wM), M(q; wL, wK, wM)) →
0 ≡ f_{L} * ∂L(q; wL, wK, wM)/∂wL + f_{K} * ∂K(q; wL, wK, wM)/∂wL + f_{M} * ∂M(q; wL, wK, wM)/∂wL, and
wL = μ(q; wL, wK, wM) * f_{L}(L(q; wL, wK, wM), K(q; wL, wK, wM), M(q; wL, wK, wM))
wK = μ(q; wL, wK, wM) * f_{K}(L(q; wL, wK, wM), K(q; wL, wK, wM), M(q; wL, wK, wM))
wM = μ(q; wL, wK, wM) * f_{M}(L(q; wL, wK, wM), K(q; wL, wK, wM), M(q; wL, wK, wM))
so:

∂C^{*}(q; wL, wK, wM)/∂wL

= ∂ { wL * L(q; wL, wK, wM) + wK * K(q; wL, wK, wM) + wM * M(q; wL, wK, wM) } / ∂wL


= L(q; wL, wK, wM) + wL * ∂L(q; wL, wK, wM)/∂wL + wK * ∂K(q; wL, wK, wM)/∂wL + wM * ∂M(q; wL, wK, wM)/∂wL


= L(q; wL, wK, wM) + μ(q; wL, wK, wM) * (f_{L} * ∂L(q; wL, wK, wM)/∂wL + f_{K} * ∂K(q; wL, wK, wM)/∂wL + f_{M} * ∂M(q; wL, wK, wM)/∂wL)


= L(q; wL, wK, wM)

XI. Example. Duality: CES production function to CES cost function.
XII. The Solution Functions' Comparative Statics.
The Jacobian matrix of the four functions, G_{µ}, G_{L}, G_{K}, G_{M}, with respect to the variables, q, wL, wK, and wM:
J_{q, wL, wK, wM;} =

G_{µq}  G_{µwL}  G_{µwK}  G_{µwM} 
G_{Lq}  G_{LwL}  G_{LwK}  G_{LwM} 
G_{Kq}  G_{KwL}  G_{KwK}  G_{KwM} 
G_{Mq}  G_{MwL}  G_{MwK}  G_{MwM} 
 = 

The Jacobian matrix of the four solution functions, Φ = {µ, L, K, M}, with respect to the variables, q, wL, wK, and wM:
J_{Φ} =

µ_{q}  µ_{wL}  µ_{wK}  µ_{wM} 
L_{q}  L_{wL}  L_{wK}  L_{wM} 
K_{q}  K_{wL}  K_{wK}  K_{wM} 
M_{q}  M_{wL}  M_{wK}  M_{wM} 

From the Implicit Function Theorem:
J_{q, wL, wK, wM;} + J_{µ, L, K, M} * J_{Φ} = 0 (zero matrix) →
J_{Φ} =  (J_{µ, L, K, M})^{1}
for points (q; wL, wK, wM) in a neighborhood of the point (q; wL, wK, wM), and L = L(q; wL, wK, wM), K = K(q; wL, wK, wM),
M = M(q; wL, wK, wM), and µ = µ(q; wL, wK, wM).
XIII.Numerical Example: Production Function to Cost Function.
The CES production function as specified:
f(L, K, M) = 1 * [0.35 * (L^^{ 0.17647}) + 0.4 * (K^^{ 0.17647}) + 0.25 *(M^^{ 0.17647})]^^{(1/0.17647)}
with A = 1, alpha = 0.35, beta = 0.4, gamma = 0.25, rho = 0.17647 (sigma = 0.85), and nu = 1. The CES production function has continuous first and second order partial derivatives with respect to its arguments.
f_{L}(L,K,M) = 1 * 0.35 * 1^^{(0.17647/1)} * L ^^{(1 + 0.17647)} * f(L,K,M)^^{(1 + 0.17647/1)},
f_{K}(L,K,M) = 1 * 0.4 * 1^^{(0.17647/1)} * K ^^{(1 + 0.17647)} * f(L,K,M)^^{(1 + 0.17647/1)},
f_{M}(L,K,M) = 1 * 0.25 * 1^^{(0.17647/1)} * M ^^{(1 + 0.17647)} * f(L,K,M)^^{(1 + 0.17647/1)}.
The factor prices are set: wL = 7, wK = 13, wM = 6. Output is set: q = 30.

The LeastCost Combination of Inputs: Production Function to Cost Function.
The dual cost function, C^{*}, is obtained from the production function, f, by:
C^{*}(q; wL, wK, wM) = min_{L,K,M}{ wL * L + wK * K + wM * M : q  f(L,K,M) = 0, q > 0, wL > 0; wK > 0, and wM > 0 }
C^{*}(30; 7, 13, 6) = min_{L,K,M}{ 7 * L + 13 * K + 6 * M : 30  f(L,K,M) = 0 }
Constrained Optimization (Minimum): The Method of Lagrange.
G(q; wL, wK, wM, L, K, M, μ) = wL * L + wK * K + wM * M + μ * (q  f(L,K,M))
G(30; 7, 13, 6, L, K, M, μ) = 7 * L + 13 * K + 6 * M + μ * (30  f(L,K,M))
First Order Necessary Conditions:
0. G_{µ}(30; 7, 13, 6, L, K, M, μ) = 30  f(L, K, M) = 0
1. G_{L}(30; 7, 13, 6, L, K, M, μ) = 7  µ * f_{L}(L, K, M) = 0
2. G_{K}(30; 7, 13, 6, L, K, M, μ) = 13  µ * f_{K}(L, K, M) = 0
3. G_{M}(30; 7, 13, 6, L, K, M, μ) = 6  µ * f_{M}(L, K, M) = 0

Solve these four equations simultaneously (say, using Newton's Method) for L, K, M, and µ. With L = 36.89, K = 24.42, M = 31.59, and µ = 25.506,
f(L,K,M) = 30, f_{L}(L,K,M) = 0.2744, f_{K}(L,K,M) = 0.5097, f_{M}(L,K,M) = 0.2352,
0. G_{µ}(30; 7, 13, 6, L, K, M, μ) = 30  30 = 0
1. G_{L}(30; 7, 13, 6, L, K, M, μ) = 7  25.506 * 0.2744 ≈ 0
2. G_{K}(30; 7, 13, 6, L, K, M, μ) = 13  25.506 * 0.5097 ≈ 0
3. G_{M}(30; 7, 13, 6, L, K, M, μ) = 6  25.506 * 0.2352 ≈ 0

Second Order Necessary Conditions:
The second order necessary conditions require that the Jacobian matrix (bordered Hessian of G) be positive definite at Z = (q, wL, wK, wM, L, K, M, µ) = (30, 7, 13, 6, 36.89, 24.42, 31.59, 25.506). The Jacobian matrix, J_{3}, is a positive definite matrix if the determinants of J_{2} and J_{3} are both negative.
J_{µ, L, K, M} =

J_{3} =

0  f_{L}  f_{K}  f_{M} 
f_{L}  µ * f_{LL}  µ * f_{LK}  µ * f_{LM} 
f_{K}  µ * f_{KL}  µ * f_{KK}  µ * f_{KM} 
f_{M}  µ * f_{ML}  µ * f_{MK}  µ * f_{MM} 

= 
0  0.274  0.51  0.235 
0.274  0.148  0.14  0.065 
0.51  0.14  0.367  0.12 
0.235  0.065  0.12  0.168 


Determinant(J_{3}) = 0.0312

J_{2} =

0  f_{L}  f_{K} 
f_{L}  µ * f_{LL}  µ * f_{LK} 
f_{K}  µ * f_{KL}  µ * f_{KK} 

= 
0  0.274  0.51 
0.274  0.148  0.14 
0.51  0.14  0.367 


Determinant(J_{2}) = 0.1052

The LeastCost Combination of Inputs.
C^{*}(30; 7, 13, 6) =
7 * L + 13 * K + 6 * M =
7 * 36.89 + 13 * 24.42 + 6 * 31.59 = 765.17
The Solution Functions' Comparative Statics.
From the Implicit Function Theorem:
J_{Φ} =  (J_{µ, L, K, M})^{1}
J_{Φ} =

µ_{q}  µ_{wL}  µ_{wK}  µ_{wM} 
L_{q}  L_{wL}  L_{wK}  L_{wM} 
K_{q}  K_{wL}  K_{wK}  K_{wM} 
M_{q}  M_{wL}  M_{wK}  M_{wM} 

= 
0  1.23  0.814  1.053 
1.23  2.968  1  1.295 
0.814  1  0.934  0.857 
1.053  1.295  0.857  3.367 

XIV. Maximum Output: Cost Function to Production Function.
The entrepreneur, management, and employees of the profit maximizing firm can investigate the technology (production function) available in the firm's cost function, C(q; wL, wK, wM), by determining the factor prices, wL, wK, and wM, consistent with the maximum level of output, q, for a given combination of factor inputs, L, K, and M.
f^{*}(L, K, M) = max_{q} {q : C(q; wL, wK, wM) <= wL * L + wK * K + wM * M, L > 0, K > 0, M > 0, for all wL >= 0 , wK >= 0, wM >= 0}
Question: f^{*} == f (original production function)?
Consider the case where the cost function, C(q; wL, wK, wM), factors:
C(q; wL, wK, wM) = q^^{1/nu} * c(wL, wK, wM)
Setting:
f^{*}(L, K, M) = max_{q} {q : q^^{1/nu} * c(wL, wK, wM) <= wL * L + wK * K + wM * M, L > 0, K > 0, M > 0, for all wL >= 0 , wK >= 0 , wM >= 0}
With c(wL, wK, wM) and wL * L + wK * K + wM * M linear homogeneous:
f^{*}(L, K, M) = max_{q} {q : q^^{1/nu} * c(wL, wK, wM) <= 1, L > 0, K > 0, M > 0, wL * L + wK * K + wM * M = 1}
f^{*}(L, K, M) = max_{q} {q : q^^{1/nu} <= 1 / c(wL, wK, wM), L > 0, K > 0, M > 0, wL * L + wK * K + wM * M = 1}
Rewrite this as (Diewert, 1974, 157):
f^{*}(L, K, M)^^{1/nu} = min_{wL,wK,wM} {1 / c(wL, wK, wM) : wL * L + wK * K + wM * M = 1, wL >= 0, wK >= 0, wM >= 0}
f^{*}(L, K, M)^^{1/nu} = 1 / max_{wL,wK,wM} {c(wL, wK, wM) : wL * L + wK * K + wM * M = 1, wL >= 0, wK >= 0, wM >= 0}, since c(wL, wK, wM) >= 0
XV. Constrained Optimization (Maximum): The Method of Lagrange.
Define the Langrangian function, H, of the output maximization problem of (XIV):
H(L, K, M, wL, wK, wM, λ) = c(wL, wK, wM) + λ * (1  (wL * L + wK * K + wM * M))
where the new variable, λ, is called the Lagrange multiplier.
a. First Order Necessary Conditions:
0. H_{λ}(L, K, M, wL, wK, wM, λ) = 1  (wL * L + wK * K + wM * M) = 0
1. H_{wL}(L, K, M, wL, wK, wM, λ) = ∂c(wL, wK, wM)/∂wL  λ * L = 0
2. H_{wK}(L, K, M, wL, wK, wM, λ) = ∂c(wL, wK, wM)/∂wK)  λ * K = 0
3. H_{wM}(L, K, M, wL, wK, wM, λ) = ∂c(wL, wK, wM)/∂wM  λ * M = 0

b. Solution Functions:
We want to solve, simultaneously, these four equations for the variables wL, wK, and wM, and λ as continuously differentiable functions of the variables L, K, and M, and the parameters of the cost function.
λ = λ(L, K, M)
wL = wL(L, K, M)
wK = wK(L, K, M)
wM = wM(L, K, M)
Suppose that the point W = (L, K, M, wL, wK, wM, λ) satisfies equations 0. to 3., and that each of the functions H_{λ}, H_{wL}, H_{wK}, and H_{wM} has continuous partial derivatives with respect to each of the variables L, K, M, wL, wK, wM, and λ at the point W. Also, suppose the determinant of the Jacobian matrix, defined below, when evaluated at the point W is not equal to zero.
According to the Implicit Function Theorem, functions wL, wK, wM, and λ exist that express the variables wL, wK, wM, and λ as continuously differentiable functions of the variables L, K, and M.
Moreover:
λ = λ(L, K, M)
wL = wL(L, K, M)
wK = wK(L, K, M)
wM = wM(L, K, M)
and:
0'. H_{λ}(L, K, M, wL, wK, wM, λ) = 1  ( wL(L, K, M) * L + wK(L, K, M) * K + wM(L, K, M) * M ) = 0
1'. H_{wL}((L, K, M, wL, wK, wM, λ) = ∂c(wL(L, K, M), wK(L, K, M), wM(L, K, M))/∂wL  λ(L, K, M) * L = 0
2'. H_{wK}(L, K, M, wL, wK, wM, λ) = ∂c(wL(L, K, M), wK(L, K, M), wM(L, K, M))/∂wK  λ(L, K, M) * K = 0
3'. H_{wM}((L, K, M, wL, wK, wM, λ) = ∂c(wL(L, K, M), wK(L, K, M), wM(L, K, M))/∂wM  λ(L, K, M) * M = 0

for points (L, K, M) in a neighborhood of the point (L, K, M). That is, the points (L, K, M, wL(L, K, M), wK(L, K, M), wM(L, K, M), λ(L, K, M)) also satisfy the first order conditions.
c. Jacobian Matrices:
The Jacobian matrix (bordered Hessian of H) of the four functions, H_{λ}, H_{wL}, H_{wK}, H_{wM}, with respect to the choice variables, λ, wL, wK, wM :
J_{3} =

H_{λλ}  H_{λwL}  H_{λwK}  H_{λwM} 
H_{wLλ}  H_{wLwL}  H_{wLwK}  H_{wLwM} 
H_{wKλ}  H_{wKwL}  H_{wKwK}  H_{wKwM} 
H_{wMλ}  H_{wMwL}  H_{wMwK}  H_{wMwM} 
 = 
0  L  K  M 
L  c_{wLwL}  c_{wLwK}  c_{wLwM} 
K  c_{wKwL}  c_{wKwK}  c_{wKwM} 
M  c_{wMwL}  c_{wMwK}  c_{wMwM} 

= J_{λ, wL, wK, wM}

The bordered principal minor of the bordered Hessian of the Langrangian function, H:
J_{2} =

0  L  K 
L  c_{wLwL}  c_{wLwK} 
K  c_{wKwL}  c_{wKwK} 

d. Second Order Necessary Conditions:
The second order necessary conditions require that the Jacobian matrix (bordered Hessian of H) be negative definite at W = (L, K, M, wL, wK, wM, λ). The Jacobian matrix, J_{3}, is a negative definite matrix if the determinant of J_{2} is positive, and the determinant of J_{3} is negative.
e. Sufficient Conditions:
If the second order necessary conditions are satisfied, then the first order necessary conditions are sufficient for a maximum at W. Therefore:
f^{*}(L, K, M)^^{1/nu} = q(L, K, M)^^{1/nu} = 1 / c(wL(L, K, M, wK(L, K, M), wM(L, K, M))
Moreover, since the sixteen functions that comprise the components of J_{3} are continuous, the Jacobian matrix, J_{3}, is a negative definite matrix at points (L, K, M) in a neighborhood of the point (L, K, M). That is, the points (L, K, M, wL(L, K, M), wK(L, K, M), wM(L, K, M), λ(L, K, M)) also satisfy the second order conditions.
Consequently, for points (L, K, M) in a neighborhood of the point (L, K, M):
f^{*}(L, K, M)^^{1/nu} = q(L, K, M)^^{1/nu} = 1 / H(L, K, M, wL(L, K, M), wK(L, K, M), wM(L, K, M), λ(L, K, M))
XVI. The Lagrange Multiplier.
The first order necessary conditions (XV. a. 13.) imply:
λ(wL, wK, wM) = c(wL, wK, wM)/∂wL / L = c(wL, wK, wM)/∂wK / K = c(wL, wK, wM)/∂wM / M
where:
wL = wL(L, K, M)
wK = wK(L, K, M)
wM = wM(L, K, M)
for points (L, K, M) in a neighborhood of the point (L, K, M).
XVII. Factor Demand Functions.
If the specified (or derived) cost function, C(q; wL, wK, wM) = q^^{1/nu} * c(wL, wK, wM), satisfies the HotellingShephard properties, then the factor demand functions are given by:
L(q; wL, wK, wM) = ∂C(q; wL, wK, wM)/∂wL = q^^{1/nu} * ∂c(wL, wK, wM)/∂wL = q^^{1/nu} * l(wL, wK, wM)
K(q; wL, wK, wM) = ∂C(q; wL, wK, wM)/∂wK = q^^{1/nu} * ∂c(wL, wK, wM)/∂wK = q^^{1/nu} * k(wL, wK, wM)
M(q; wL, wK, wM) = ∂C(q; wL, wK, wM)/∂wM = q^^{1/nu} * ∂c(wL, wK, wM)/∂wM = q^^{1/nu} * m(wL, wK, wM)
XVIII. Example. Duality: CES cost function to CES production function.
XIX. The Solution Functions' Comparative Statics.
The Jacobian matrix of the four functions, H_{λ}, H_{wL}, H_{wK}, H_{wM}, with respect to the variables, L, K, M:
J_{L, K, M} =

H_{λ,L;}  H_{λ,K}  H_{λ,M} 
H_{wL,L}  H_{wL,K}  H_{wL,M} 
H_{wK,L}  H_{wK,K}  H_{wK,M} 
H_{wM,L}  H_{wM,K}  H_{wM,M} 
 = 

The Jacobian matrix of the four solution functions, Φ = {λ, wL, wK, wM}, with respect to the variables, L, K, and M:
J_{Φ} =

λ_{L}  λ_{K}  λ_{M} 
wL_{L}  wL_{K}  wL_{M} 
wK_{L}  wK_{K}  wK_{M} 
wM_{L}  wM_{K}  wM_{M} 

From the Implicit Function Theorem:
J_{L, K, M;} + J_{λ, wL, wK, wM, q} * J_{Φ} = 0 (zero matrix) →
J_{Φ} =  (J_{λ, wL, wK, wM, q})^{1} * J_{L, K, M}
for points (L, K, M) in a neighborhood of the point (L, K, M), and wL = wL(L, K, M), wK = wK(L, K, M), wM = wM(L, K, M), and λ = λ(L, K, M),
with:
f^{*}(L, K, M) = (1 / c(wL(L, K, M), wK(L, K, M), wM(L, K, M))^^{nu}.
XX. Numerical Example: Cost Function to Production Function.
The CES cost function as specified:
C(q;wL,wK,wM) = h(q) * c(wL,wK,wM) = (q/1)^^{1/1} * [0.35^^{(1/(1+0.17647)))} * wL^^{(0.17647)/(1+0.17647)))} + 0.4^^{(1/(1+0.17647)))} * wK^^{(0.17647)/(1+0.17647)))} + 0.25^^{(1/(1+0.17647)))} * wM^^{(0.17647)/(1+0.17647)))}]^^{((1+0.17647))/0.17647))}
with A = 1, alpha = 0.35, beta = 0.4, gamma = 0.25, rho = 0.17647 (sigma = 0.85), and nu = 1. The CES cost function has continuous first and second order partial derivatives with respect to their arguments.
∂C(q;wL,wK,wM) / ∂wL = h(q) * 0.35^^{(1/(1+0.17647))} * wL^^{(1/(1+0.17647))} * c(wL, wK, wM)^^{(1/(1+0.17647))};
∂C(q;wL,wK,wM) / ∂wK = h(q) * 0.4^^{(1/(1+0.17647))} * wK^^{(1/(1+0.17647))} * c(wL, wK, wM)^^{(1/(1+0.17647))};
∂C(q;wL,wK,wM) / ∂wM = h(q) * 0.25^^{(1/(1+0.17647))} * wM^^{(1/(1+0.17647))} * c(wL, wK, wM)^^{(1/(1+0.17647))}.
The factor inputs are set: L = 36.89, K = 24.42, M = 31.59.

Maximum Output: Cost Function to Production Function.
f^{*}(L, K, M)^^{1/nu} = 1 / max_{wL,wK,wM} {c(wL, wK, wM) : wL * L + wK * K + wM * M = 1, wL >= 0, wK >= 0, wM >= 0}
f^{*}(36.89, 24.42, 31.59)^^{1/1} = 1 / max_{wL,wK,wM} {c(wL, wK, wM) : wL * 36.89 + wK * 24.42 + wM * 31.59 = 1, wL >= 0, wK >= 0, wM >= 0}
Constrained Optimization (Maximum): The Method of Lagrange.
H(L, K, M, wL, wK, wM, λ) = c(wL, wK, wM) + λ * (1  (wL * L + wK * K + wM * M))
H(36.89, 24.42, 31.59, wL, wK, wM, λ) = c(wL, wK, wM) + λ * (1  (wL * 36.89 + wK * 24.42 + wM * 31.59))
First Order Necessary Conditions:
0. H_{λ}(36.89, 24.42, 31.59, wL, wK, wM, λ) = 1  (wL * 36.89 + wK * 24.42 + wM * 31.59) = 0
1. H_{wL}(36.89, 24.42, 31.59, wL, wK, wM, λ) = ∂c(wL, wK, wM)/∂wL  λ * 36.89 = 0
2. H_{wK}(36.89, 24.42, 31.59, wL, wK, wM, λ) = ∂c(wL, wK, wM)/∂wK)  λ * 24.42 = 0
3. H_{wM}(36.89, 24.42, 31.59, wL, wK, wM, λ) = ∂c(wL, wK, wM)/∂wM  λ * 31.59 = 0

Solve these four equations simultaneously (say, using Newton's Method) for wL, wK, wM, and λ. With wL = 0.00915, wK = 0.01699, wM = 0.00784, and λ = 0.0333,
0. H_{λ}(36.89, 24.42, 31.59, wL, wK, wM, λ) = 1  (wL * 36.89 + wK * 24.42 + wM * 31.59) ≈ 0
1. H_{wL}(36.89, 24.42, 31.59, wL, wK, wM, λ) = 1.23  0.0333 * 36.89 ≈ 0
2. H_{wK}(36.89, 24.42, 31.59, wL, wK, wM, λ) = 0.81  0.0333 * 24.42 ≈ 0
3. H_{wM}(36.89, 24.42, 31.59, wL, wK, wM, λ) = 1.05  0.0333 * 31.59 ≈ 0

Second Order Necessary Conditions:
The second order necessary conditions require that the Jacobian matrix (bordered Hessian of H) be negative definite at W = (L, K, M, wL, wK, wM, λ) = (36.89, 24.42, 31.59, 0.00915, 0.01699, 0.00784, 0.0333). The Jacobian matrix, J_{3}, is a negative definite matrix if the determinant of J_{2} is positive, and the determinant of J_{3} is negative.
J_{λ, wL, wK, wM} =

J_{3} =

0  L  K  M 
L  c_{wLwL}  c_{wLwK}  c_{wLwM} 
K  c_{wKwL}  c_{wKwK}  c_{wKwM} 
M  c_{wMwL}  c_{wMwK}  c_{wMwM} 
 = 
0  36.888  24.415  31.592 
36.888  75.693  25.518  33.019 
24.415  25.518  23.827  21.855 
31.592  33.019  21.855  85.874 


Determinant(J_{3}) = 18741780.5534

J_{2} =

0  36.888  24.415 
36.888  75.693  25.518 
24.415  25.518  23.827 


Determinant(J_{2}) = 123508.5984

Maximum Output:
With L = 36.89, K = 24.42, M = 31.59, set wL = 0.00915, wK = 0.01699, wM = 0.00784,
f^{*}(L, K, M) = (1 / c(wL, wK, wM))^^{(nu)}, →
f^{*}(36.89, 24.42, 31.59) = (1 / c(0.00915, 0.01699, 0.00784))^^{1} = (1 / 0.0333)^^{1} = 30
Compare f^{*}(L,K,M) with f(L,K,M) at L = 36.89, K = 24.42, M = 31.59:
f^{}(36.89, 24.42, 31.59) = 30
The Solution Functions' Comparative Statics.
J_{L, K, M} =


= 
0.00915  0.01699  0.00784 
0.0333  0  0 
0  0.0333  0 
0  0  0.0333 

From the Implicit Function Theorem:
J_{Φ} =  (J_{λ, wL, wK, wM, q})^{1} * J_{L, K, M}
J_{Φ} =

λ_{L}  λ_{K}  λ_{M} 
wL_{L}  wL_{K}  wL_{M} 
wK_{L}  wK_{K}  wK_{M} 
wM_{L}  wM_{K}  wM_{M} 

= 
0.000305  0.000566  0.000261 
0.00028  3.0E5  1.0E5 
3.0E5  0.00077  2.0E5 
1.0E5  2.0E5  0.00028 

The Partial Derivates of f*(L,K,M).
Since f*(L,K,M) = (1 / c(wL,wK,wM))^^{nu} = (1 / λ)^^{nu}:
f*_{L}(L,K,M) = nu * (λ_{L} / λ^^{2}) * (1 / λ)^^{(nu1)},
f*_{K}(L,K,M) = nu * (λ_{K} / λ^^{2}) * (1 / λ)^^{(nu1)},
f*_{M}(L,K,M) = nu * (λ_{M} / λ^^{2}) * (1 / λ)^^{(nu1)}, →
f*_{L}(36.89,24.42,31.59) = 1 * (0.000305 / 0.0333^^{2}) * (1 / 0.0333)^^{(11)} = 0.274,
f*_{K}(36.89,24.42,31.59) = 1 * (0.000566 / 0.0333^^{2}) * (1 / 0.0333)^^{(11)} = 0.51,
f*_{M}(36.89,24.42,31.59) = 1 * (0.000261 / 0.0333^^{2}) * (1 / 0.0333)^^{(11)} = 0.235.
Partial Derivatives of f(L,K,M).
f_{L}(36.89,24.42,31.59) = 0.274,
f_{K}(36.89,24.42,31.59) = 0.51,
f_{M}(36.89,24.42,31.59) = 0.235.
Conclude: f* == f.
CES Production/ Cost Functions Numerical Example
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 CES Cost Function:
C(q;wL,wK,wM) = h(q) * c(wL,wK,wM) = (q/A)^^{1/nu} * [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)}
The cost function's factor prices, wL, wK, and wM, are positive real numbers.
Set the parameters below to rerun 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 (CobbDouglas)
sigma < 1 → inputs complements; sigma > 1 → inputs substitutes
15 < q < 45;
4 <= wL* <= 11, 7<= wK* <= 16, 4 <= wM* <= 10

