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).

Elasticity of Scale of Production:

        Long Run: Capital Variable:

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).

        Short Run: Captial Fixed: K = K:

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

See the Mathematical Notes below.

Curvature:

The Generalized CES production function, f(L,K,M), is concave to the origin of the 3-dimensional (L,K, M) space if its Hessian is negative (semi)definite. Define the matrices h₁, h₂, and h₃:

The Hessian of f(L,K,M) is negative definite if the determinants |h₁|, |h₂|, and |h₃| alternate in sign, starting with negative. If one or more the determinants have a zero value, then the Hessian of f(L,K,M) is negative semidefinite, and f(L, K, M) is quasi-concave to the origin of the 3-dimensional space of (L,K,M).

II. The least-cost combination of inputs:

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 }

Equivalently: Find the values of L, K, M, and µ that minimize the Lagrangian when the factor prices are wL, wK, and wM:

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

or equivalently, that maximize the Lagrangian G(q;L,K,M,µ) = - G(q;L,K,M,µ).

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

First Order Conditions:

To solve equations 0. to 3. numerically, for given q, wL, wK, and wM, I used the first order conditions 0. to 3. and the associated Jacobian.

Jacobian of Second Order Conditions (Bordered Hessian with F(L,K,M) = f(L,K,M)):

The Lagrangian G(q;L,K,M,µ) obtains a maximum at q, L, K, and M if the the bordered, principal minor determinant |H₂| is positive, and the bordered determinant |H₃| is negative. (See the table below, and the Mathematical Notes.)

III. Long Run: Capital Variable.

Suppose the firm buys its inputs at the prices: wL = 7   wK = 13   wM = 6 . Solve the least-cost problem (noting that µ = marginal cost) to obtain the data as displayed in the following table. The terms s_LK, s_LM, and s_KM are the Allen partial elasticities of substitution.

The Lagrangian method for obtaining the least-cost combination of inputs, yields values for the solution variables µ, L, K, and M for specified values of q, wL, wK, and wM. That is, the solution variables are functions:

µ = µ(q; wL, wK, wM), L = L(q; wL, wK, wM), K = K(q; wL, wK, wM), M = M(q; wL, wK, wM).

The Lagrangian method also produces the Jacobian matrix (bordered Hessian of G = -G) of the four functions, G_µ, G_L, G_K, G_M, with respect to the choice variables, μ, L, K, and M:

The Jacobian matrix of the four solution functions, Φ = {µ, L, K, M}, with respect to the variables, q, wL, wK, and wM equals the negative of the matrix inverse of J₃, yielding the comparative statics of the solution functions.

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

The Uzawa partial elasticities of substitution at these same values of q, wL, wK, and wM:

The factor demand elasticities at these same values of q, wL, wK, and wM:

where, for example, ε_L,wK = ∂ln(L(q; wL, wK, wM)) / ∂ln(wK) = (∂ln(L(q; wL, wK, wM)) / ∂wK) * (∂wK / ∂ln(wK)) = (1 / L(q; wL, wK, wM)) * (∂L(q; wL, wK, wM) / ∂wK) * wK = L_wK * wK / L

The cost function C(q;wL,wK,wM) is a concave in factor prices if the Hessian matrix ∇²_wwC(q;wL,wK,wM) of second order partial derivatives with respect to factor prices is negative semidefinite. The matrix ∇²_wwC(q;wL,wK,wM) is negative semidefinite if its eigenvalues are nonpositive.

The eigenvalues of ∇²_wwC  are e1 = -5.3039, e2 = -3.0594, and e3 = -0.

IV. Short Run: Capital Fixed: K = K = 24.14.

Let the firm buy its inputs at the same prices: wL = 7   wK = 13   wM = 6 . Set the level of capital at the least cost level for q = 30.   Solve the least-cost problem holding capital fixed to obtain the data in the following table. (See the discussion of the short-run elasticity of substitution, s_LM, on the translog production function page.)

The Lagrangian G(q;L,K,M,µ) obtains a maximum at q, L, K, and M if the the bordered determinant |H₂| is positive. The Generalized CES production function, f(L, K, M), is concave to the origin of the 2-dimensional (L, M) space if |h₁| < 0, and |h₂| > 0. (See the table below, and the Mathematical Notes.)

The Lagrangian method for obtaining the least-cost combination of inputs, yields values for the solution variables µ, L, and M for specified values of q, wL, wK, and wM. That is, the solution variables are functions:

µ = µ(q; wL, wK, wM), L = L(q; wL, wK, wM), M = M(q; wL, wK, wM).

The Lagrangian method also produces the Jacobian matrix (bordered Hessian of G = -G) of the three functions, G_µ, G_L, G_M, with respect to the choice variables, μ, L, and M:

The Jacobian matrix of the three solution functions, Φ = {µ, L, M}, with respect to the variables, q, wL, wK, and wM equals the negative of the matrix inverse of J₂, yielding the comparative statics of the solution functions.

For example, at q = 30,   K = K = 24.14,   wL = 7,   wK = 13,  and   wM = 6 :

The factor demand elasticities at q = 30, wL = 7,   wK = 13, and   wM = 6 :

where, for example, ε_L,wM = ∂ln(L(q; wL, wK, wM)) / ∂ln(wM) = (∂ln(L(q; wL, wK, wM)) / ∂wM) * (∂wM / ∂ln(wM)) = (1 / L(q; wL, wK, wM)) * (∂L(q; wL, wK, wM) / ∂wM) * wM = L_wM * wM / L

The short run cost function C(q;wL,wK,wM) is a concave in factor prices if the Hessian matrix ∇²_wwC(q;wL,wK,wM) of second order partial derivatives with respect to factor prices is negative semidefinite. The matrix ∇²_wwC(q;wL,wK,wM) is negative semidefinite if its eigenvalues are nonpositive.

The eigenvalues of ∇²_wwC  are e1 = -5.3017, and e2 = -0 .

Graph of Average Cost and Marginal Cost Generalized CES Production / Cost Functions - Capital Fixed
wL = 7, wK= 13, wM = 6.
	Average cost function   Marginal cost function   L.R. Average cost function

V. Isoquants

A given level of output, q = q, can be produced by different combinations of factor inputs, L, K, and M. Fixing the level of product output at q = q, we obtain an equation from the CES production function:

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

for the 3-dimensional isoquant surface, when q = q.

The isoquant surface is tangent to the isocost plane:

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

at the cost minimizing combination of factor inputs, (L, K, M ) = (L, K, M).

Consider the Generalized CES production function as specified above:

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

When q = 30 and (wL, wK, wM) = (7, 13, 6), the cost minimizing inputs are:

(L, K, M) = (43.79, 24.14, 40.13), and

C(30) = 7 * 43.79 + 13 * 24.14 + 6 * 40.13 = 861.17.

Solving the CES equation for L, K, and M in turn, we get:

three equations for the 3-dimensional isoquant surface.

By fixing the amount of input for one factor, we obtain a 2-dimensional isoquant curve. As examples, fixing M = M in equation 2, and K = K in equation 3, we get:

with K and M as functions of one variable, L. The following diagrams graph, in blue, the L-K and L-M isoquants for q = 24, 30, and 36.

The yellow lines represent the isocost lines, combinations of L, K, and M that can be purchased at a constant total cost at the prices wL = 7, wK = 13, and wM = 6.

The slope of an L-K isocost line is m_K = -wL / wK = -7 / 13; the slope of an L-M isocost line is m_M = -wL / wM = -7 / 6.

For q = 30, the L-K isocost line has a K-intercept at (C(30) - wM * M) / wK = (861.17 - 6 * 40.13)/13 = 47.72,
while the L-M isocost line has a M-intercept at (C(30) - wK * K) / wM = (861.17 - 13 * 24.14)/6 = 91.22.

For q = 30, the L-K isoquant is tangent to the L-K isocost line at (L, K) = (43.79, 24.14), while the L-M isoquant is tangent to the L-M isocost line at (L, M) = (43.79, 40.13).

L-K Isoquants, M = M	L-M Isoquants, K = K

Mathematical Notes

I. The Generalized CES Production Function

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.

Elasticity of Scale of Production:

        Long Run: Capital Variable:

Using the partial derivatives of f(L,K,M) below we get:
dq/q = (nu / rho) * (alpha * rhoL * L^^-rhoL * dL / L + beta * rhoK * K^^-rhoK * dK / K + gamma * rhoM * M^^-rhoM * dM / M) / (alpha * L^^-rhoL + beta * K^^-rhoK + gamma * M^^-rhoM)

Substitute du/u = dL/L = dK/K = dM/M in the equation above, and divide by du/u.

Then,   (dq/q)/(du/u) = ε(L,K,M),   and

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).

If rho = 0, ε_LKM = ε(L,K,M) = alpha + beta + gamma.

        Short Run: Captial Fixed: K = K:

Set dK = 0 in the equation for dq/q with du/u = dL/L = dM/M, and divide by du/u.

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

If rho = 0, ε_LKM = ε(L,K,M) = alpha + gamma.

Curvature:

The Generalized CES production function, f(L,K,M), is concave to the origin of the 3-dimensional (L,K, M) space if its Hessian is negative (semi)definite. Define the matrices h₁, h₂, and h₃:

The Hessian of f(L,K,M) is negative definite if the determinants |h₁|, |h₂|, and |h₃| alternate in sign, starting with negative. If one or more the determinants have a zero value, then the Hessian of f(L,K,M) is negative semidefinite, and f(L, K, M) is quasi-concave to the origin of the 3-dimensional space of (L,K,M).

II. The partial derivatives of f(L,K,M)

III. The least-cost combination of inputs:

Find the values of L, K, M, and µ that minimize the Lagrangian when the factor prices are wL, wK, and wM:

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

or equivalently, that maximize the Lagrangian G(q;L,K,M,µ) = - G(q;L,K,M,µ).

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

First Order Conditions:

Jacobian of Second Order Conditions (Bordered Hessian with F(L,K,M) = f(L,K,M)):

 where:

IV. The least-cost combination of inputs: Capital fixed:

With the value of the capital value fixed at K = K, find the values of L, M, and µ that minimize the Lagrangian when the factor prices are wL, wK, and wM:

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

or equivalently, that maximize the Lagrangian G(q;L,K,M,µ) = - G(q;L,K,M,µ).

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

First Order Conditions: K = K:

Jacobian of Second Order Conditions: K = K: (Bordered Hessian with F(L,K,M) = f(L,K,M)):

 where:

V. Allen partial elasticity of substitution

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

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

s_LK = ((F_L * L + F_K * K + F_M *M) / (L * K)) * (|F_LK|/|F|)

VI. 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:

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

s_LM = - (F_L * L + F_M * M) / (L * M ) * (F_L * F_M) / |F|