Egwald Economics: Microeconomics

Duality and the CES Production / Cost Functions

by

Elmer G. Wiens

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Duality: Production / Cost Functions:   Cobb-Douglas Duality | CES Duality | Theory of Duality | Translog Duality - CES | Translog Duality - Generalized CES | References and Links

P. Duality and the CES (Constant Elasticity of Substitution) Production / Cost Functions.

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. The theory of duality links the production function models to the cost function models by way of a minimization or maximization framework. The cost function is derived from the production function by choosing the combination of factor quantities that minimize the cost of producing levels of output at given factor prices. Conversely, the production function is derived from the cost function by calculating the maximum level of output that can be obtained from specified combinations of inputs.

I. CES (Constant Elasticity of Substitution) Production Function.

A production function is a technological relationship between the levels of output that can be produced using specific combinations of factor inputs.

The three factor CES production function is:

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 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 non-negative real numbers.

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

II. CES (Constant Elasticity of Substitution) Cost Function.

A cost function is a relationship between the total cost of producing levels of output, q, at specific values of factor input prices, wL, wK, and wM, using the cost minimizing combinations of factor inputs.

The three factor CES cost function is:

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

where the returns to scale function is:

h(q) = (q/A)^1/nu

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

and the unit cost function is:

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

III. CES Production Function Partial Derivatives.

 f(L,K,M)^(1 + rho/nu) = A^(1 + rho/nu) * [alpha * (L^-rho) + beta * (K^-rho) + gamma *(M^-rho)]^((-nu/rho)*(1 + rho/nu)), and fL(L,K,M) = nu * alpha * L ^-(1 + rho) * A * [alpha * (L^-rho) + beta * (K^-rho) + gamma *(M^-rho)]^((-nu/rho)*(1 + rho/nu)),   → fL(L,K,M) = nu * alpha * A^(-rho/nu) * L ^-(1 + rho) * f(L,K,M)^(1 + rho/nu); so similarly fK(L,K,M) = nu * beta * A^(-rho/nu) * K ^-(1 + rho) * f(L,K,M)^(1 + rho/nu); fM(L,K,M) = nu * gamma * A^(-rho/nu) * M ^-(1 + rho) * f(L,K,M)^(1 + rho/nu).

IV. CES Cost Function Partial Derivatives.

 ∂C(q;wL,wK,wM) / ∂wL = h(q) * alpha^(1/(1+rho)) * (rho/(1+rho)) * wL^(-1/(1+rho)) * ((1+rho)/rho) * [alpha^(1/(1+rho)) * wL^(rho/(1+rho)) + beta^(1/(1+rho)) * wK^(rho/(1+rho)) + gamma^(1/(1+rho)) * wM^(rho/(1+rho))]^(1/rho),   → ∂C(q;wL,wK,wM) / ∂wL = h(q) * alpha^(1/(1+rho)) * wL^(-1/(1+rho)) * c(wL, wK, wM)^(1/(1+rho)); so similarly ∂C(q;wL,wK,wM) / ∂wK = h(q) * beta^(1/(1+rho)) * wK^(-1/(1+rho)) * c(wL, wK, wM)^(1/(1+rho)); ∂C(q;wL,wK,wM) / ∂wM = h(q) * gamma^(1/(1+rho)) * wM^(-1/(1+rho)) * c(wL, wK, wM)^(1/(1+rho)).

V. Duality: Production to Cost Function.

Find the values of L, K, M, and µ that minimize the Lagrangian:

G(q;L,K,M,µ) = wL * L + wK * K + wM* M + µ * [q - f(L,K,M)]

 GL = wL - µ * fL = 0 GK = wK - µ * fK = 0 GM = wM - µ * fM = 0 Gµ = q - f(L,K,M) = 0

From equations a., b., and c. we get:

 wL / wK = fL / fK = (alpha/beta) * (K/L)^(1+rho) --> K = L * (wL*beta/(wK*alpha))^(1/(1+rho)) wL / wM = fL / fM = (alpha/gamma) * (M/L)^(1+rho) --> M = L * (wL*gamma/(wM*alpha))^(1/(1+rho)) wK / wM = fK / fM = (beta/gamma) * (M/K)^(1+rho)

Substituting equations e. and f. into the CES production function and solving for L yields;

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

Finally, substituting e., f. and h. into the cost function:

C(q;wL,wK,wM) = wL * L + wk * K + wM * M

yields the cost function, as a function of output, depending on the input prices and the parameters of the CES production function.

To see this write:

R = [alpha^(1/(1+rho)) * wL^(rho/(1+rho)) + beta^(1/(1+rho)) * wK^(rho/(1+rho)) + gamma^(1/(1+rho)) * wM^(rho/(1+rho))].

Then:

 i.   L = (q/A)^1/nu * (alpha / wL)^(1/(1+rho)) * R^(1/rho), j.   K = (q/A)^1/nu * (beta / wK)^(1/(1+rho)) * R^(1/rho), k.   M = (q/A)^1/nu * (gamma / wM)^(1/(1+rho)) * R^(1/rho).

Therefore:

C(q;wL,wK,wM) = wL * (q/A)^1/nu * (alpha / wL)^(1/(1+rho)) * R^(1/rho) + wK * q/A)^1/nu * (beta / wK)^(1/(1+rho)) * R^(1/rho) + wM * (q/A)^1/nu * (gamma / wM)^(1/(1+rho)) * R^(1/rho),  →

Since:

1 - 1 / (1 + rho) = (1 + rho - 1) / (1 + rho) = rho / (1 + rho);   1 / rho + 1 = (1 + rho) / rho;

C(q;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))] * R^(1/rho),  →

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

VI. CES Factor Demand Functions (Cost Minimizing Factor Inputs).

 l.   ∂C(q;wL,wK,wM) / ∂wL   =   L(q;wL,wK,wM)   =   h(q) * l(wL,wK,wM)   = h(q) * [(alpha / wL) * c(wL,wK,wM)]^(1/(1+rho)) m.   ∂C(q;wL,wK,wM) / ∂wK   =   K(q;wL,wK,wM)   =   h(q) * m(wL,wK,wM)   = h(q) * [(beta / wK) * c(wL,wK,wM)]^(1/(1+rho)) n.   ∂C(q;wL,wK,wM) / ∂wM   =   M(q;wL,wK,wM)   =   h(q) * m(wL,wK,wM)   = h(q) * [(gamma / wM) * c(wL,wK,wM)]^(1/(1+rho))

Note: equations i. = l.,   j. = m.,   and k. = n.

VII. Duality: Cost to Production Function.

 l.   →   o.   wL = alpha * [h(q)/L(q;wL,wK,wM)]^(1+rho) * c(wL,wK,wM) m.   →   p.   wK = beta * [h(q)/K(q;wL,wK,wM)]^(1+rho) * c(wL,wK,wM) n.   →   q.   wM = gamma * [h(q)/M(q;wL,wK,wM)]^(1+rho) * c(wL,wK,wM)

Substituting equations o., p., and q. into the CES unit cost function yields:

c(wL,wK,wM)   =   { alpha^(1/(1+rho)) * [alpha * [h(q)/L(q;wL,wK,wM)]^(1+rho) * c(wL,wK,wM)]^(rho/(1+rho)) + beta^(1/(1+rho)) * [beta * [h(q)/K(q;wL,wK,wM)]^(1+rho) * c(wL,wK,wM)]^(rho/(1+rho))

+ gamma^(1/(1+rho)) * [gamma * [h(q)/M(q;wL,wK,wM)]^(1+rho) * c(wL,wK,wM)]^(rho/(1+rho))^ } ^((1+rho)/rho)

After factoring [h(q)^(1+rho) * c(wL,wK,wM)]^(rho/(1+rho)):

c(wL,wK,wM)   =   { [h(q)^(1+rho) * c(wL,wK,wM)]^(rho/(1+rho)) * [ alpha * L^(-rho) + beta * K^(-rho) + gamma * M^(-rho)] }^((1+rho)/rho),

c(wL,wK,wM)   =   h(q)^(1+rho) * c(wL,wK,wM) * [ alpha * L^(-rho) + beta * K^(-rho) + gamma * M^(-rho)] ^((1+rho)/rho).

Cancelling c(wL,wK,wM) and solving for h(q):

h(q)   =   [ alpha * L^(-rho) + beta * K^(-rho) + gamma * M^(-rho) ] ^(-1/rho).

Using the definition of h(q):

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

VIII. Properties of the CES Production Function, f(L,K,M).

a. If all factor inputs are zero, the level of product output is zero.

f(0, 0, 0) = 0.

b. Any increase in factor inputs will NOT decrease the level of product output.

L1 <= L2, K1 <= K2, and M1 <= M2   →   f(L1, K1, M1) <= f(L2, K2, M2).

c. The CES production function's elasticity of scale is determined by the parameter nu when (alpha + beta + gamma) = 1:

nu < 1   →   decreasing returns to scale;   nu = 1   →   constant returns to scale;   nu > 1   →   increasing returns to scale.

d. The CES production function, f, is continuous and twice-continuously differentiable for L > 0, K > 0, and M > 0.

e. The CES production function, f, is is concave to the origin of the 3-dimensional (L,K, M) space for L > 0, K > 0, and M > 0.

f. The CES production function becomes the Cobb-Douglas production function, in the limit, as rho   →   0:

q = A * [alpha * L^-rho + beta * K^-rho + gamma * M^-rho]^(-nu/rho)   →
ln((q/A)^1/nu) = ln[alpha * (L^-rho) + beta * (K^-rho) + gamma *(M^-rho)] / (-rho) = n(rho) / d(rho).

Assuming (alpha + beta + gamma) = 1, since L^0 = 1, and ln(1) = 0, as rho   →   0, the RHS takes the indeterminate form of 0 / 0.

Using l'Hopital's rule, take the derivative of the denominator and numerator with respect to rho:
d'(rho) = -1; and n'(rho) = [-alpha*ln(L)*L^-rho - beta*ln(K)*K^-rho - gamma*ln(M)*M^-rho)] / [alpha*L^-rho + beta*K^-rho + gamma*M^-rho], and
n'(0) = [-alpha*ln(L) - beta*ln(K) - gamma*ln(M)] / [alpha + beta + gamma] = -[alpha*ln(L) + beta*ln(K) + gamma*ln(M)].

Therefore, as rho   →   0,
ln((q/A)^1/nu) = [alpha*ln(L) + beta*ln(K) + gamma*ln(M)], and
exp(ln((q/A)^1/nu)) = exp([alpha*ln(L) + beta*ln(K) + gamma*ln(M)]), and
(q/A)^1/nu = L^alpha * K^beta * M^gamma,   →
q = A * [L^alpha * K^beta * M^gamma]^nu = A * L^alpha*nu * K^beta*nu * M^gamma*nu

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

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

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

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

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

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

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

e. The marginal cost function can be obtained by differentiating the CES cost function with respect to output q:

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

f. The supply function, for competitive product market conditions, can be obtained by setting marginal cost to P, the price of the output:

P = MC(q;wL,wK,wM) = h'(q) * c(wL,wK,wM) = (1/nu) * (1/A) * (q/A) ^(1 - nu)/nu * c(wL,wK,wM)   →
q = S(P;wL,wK,wM) = A * [nu * A * P / c(wL,wK,wM)]^nu/(1 - nu)

X. Elasticity of substitution between inputs (sigma).

From equation e. of Part VI we get:

K/L = [(beta / alpha)* (wl / wK)]^1/(1+rho) → ln(K/L) = (1/(1+rho))*ln(beta/alpha) + (1/(1+rho))*ln(wL/wK)

sigma = d(ln(K/L))/d(ln(wL/wK)) = 1/(1+rho)

a. K and L substitutes:

-1 < rho < 0, then 1 < sigma < infinity

b. K and L complements:

0 < rho < infinity, then 0 < sigma < 1

XI. Allen partial elasticity of substitution

Writing the production function as q = F(L,K,M), let the bordered Hessian be:

 F = 0 FL FK FM FL FLL FLK FLM FK FKL FKK FKM FM FML FMK FMM

If |F| is the determinant of the bordered Hessian and |FLK| is the cofactor associated with FLK, then the Allen elasticity of substitution is defined as:

sLK = ((FL * L + FK * K + FM *M) / (L * K)) * (|FLK|/|F|)

XII. Two factor elasticity of substitution

Let the production function be q = F(L,K,M), where K is underlined to indicate it is constant in the short-run. Then the two factor (L and M) bordered Hessian is:

 F = 0 FL FM FL FLL FLM FM FML FMM

The 2-factor elasticity of substitution between L and M is:

sLM = - (FL * L + FM * M) / (L * M ) * (FL * FM) / |F|

XIII. Numerical examples:

The web page, "The Duality of Production and Cost Functions," permits one to specify the parameters of the CES (or Cobb-Douglas) production function, and to ascertain the curvature of the production function and corresponding cost function.