The CES production function as specified:

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

II. Estimating the Diewert production function using multiple regression yielded the following coefficient estimates:

nu = 1
aLL = -0.148053, aKK = -0.18189, aMM = -0.17997
bLK = 0.644377, bLM = 0.346619 bKM = 0.51906

aLL + aKK + aMM + bLK + bLM + bKM = 1

The estimated Diewert production function:

q^^1/1 = -0.148053 * L + -0.18189 * K + -0.17997 * M + 0.644377 * L^^1/2 * K^^1/2 + 0.346619 * L^^1/2 * M^^1/2 + 0.51906 * K^^1/2 * M^^1/2     =     f(L,K,M)

Elasticity of Scale of Production:

        Long Run: Capital Variable:

ε_LKM = ε(L,K,M) = nu * [f_L(L,K,M) * L + f_K(L,K,M) * K + f_M(L,K,M) * M] / f(L, K, M),

              = nu,   since f(L, K ,M) is linear homogeneous (see the Mathematical Notes).

        Short Run: Captial Fixed: K = K:

ε_LKM = ε(L,K,M) = nu * [ f_L(L,K,M) * L + f_M(L,K,M) * M] / f(L, K, M).

Curvature:

The Diewert production function, q = F(L,K,M) = f(L,K,M)^^nu, 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).

The curvature of the estimated Diewert production function depends on the elasticity of substitution, sigma, of the CES production function.

sigma < 1 & nu = 1   →   F(L,K,M) is quasi-concave to the origin of the 3-dimensional space of (L,K,M),

sigma = 1   & nu = alpha + beta + gamma < 1 →   F(L,K,M) is concave to the origin of the 3-dimensional space of (L,K,M),
       sigma = 1   & nu = alpha + beta + gamma = 1 →   F(L,K,M) is quasi-concave to the origin of the 3-dimensional space of (L,K,M),
     sigma = 1   & nu = alpha + beta + gamma > 1 →   F(L,K,M) is NOT concave to the origin of the 3-dimensional space of (L,K,M),

sigma > 1 & nu = 1   →   F(L,K,M) is quasi-concave to the origin of the 3-dimensional space of (L,K,M),

Check the displayed curvature conditions for other values of sigma, nu, alpha, beta, and gamma.

III. 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)^^nu],

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)^^nu].

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)^^nu):

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.)

IV. Long Run: Capital Variable.

Suppose the firm buys its inputs at the prices: wL = 7   wK = 13   wM = 6 . Solving the least-cost problem, (and noting that µ = marginal cost,) we can compare the CES cost data with the estimated Diewert cost 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 = -4.4976, e2 = -2.8335, and e3 = 0.

V. Short Run: Capital Fixed: K = K = 24.4.

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. .   Solving the least-cost problem holding capital fixed, we can compare the short-run CES cost data with the estimated Diewert short-run cost data as displayed 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 Diewert 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)

We get a U-shaped, short run average cost curve, with capital fixed. The short run average cost curve is (approx.) tangent to the long run average cost curve, at q = 30. The elasticity of scale, ε_LKM, is constant along the short run average cost curve, and equals the long run elasticity of scale, ε_LKM. The short run elasticity of scale with capital fixed at K = 24.4 is a decreasing function along the short run average cost curve, since sigma is less than 1.

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 = 24.4,   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 = -4.4964, and e2 = -0 .

Graph of Average Cost and Marginal Cost Diewert Production / Cost Functions - Capital Fixed
Returns to Scale = 1, Elasticity of Substitution = 0.85
wL = 7, wK= 13, wM = 6.
	Average cost function   Marginal cost function   L.R. Average cost function

VI. Long Run: Capital Variable.

Let us increase the price of M, materials and supplies. Now the firm buys its inputs at the prices: wL = 7   wK = 13   wM = 7 . Solving the least-cost problem, (and noting that µ = marginal cost,) we can compare the CES cost data with the estimated Diewert cost data as displayed in the following table. The terms s_LK, s_LM, and s_KM are the Allen partial elasticities of substitution.

VII. Short Run: Capital Fixed: K = K = 25.24.

Let the firm buy its inputs at the same prices: wL = 7   wK = 13   wM = 7 . Set the level of capital at the least cost level for q = 30: then .   Solving the least-cost problem holding capital fixed, we can compare the short-run CES cost data with the estimated Diewert short-run cost data as displayed in the following table. (See the discussion of the short-run elasticity of substitution, s_LM, on the translog production function page.) The Diewert production function, F(L, K, M), is concave to the origin of the 2-dimensional (L, M) space if |h₁| < 0, and |h₂| > 0.

We get a U-shaped, short run average cost curve, with capital fixed. The short run average cost curve is (approx.) tangent to the long run average cost curve, at q = 30. The elasticity of scale, ε_LKM, is constant along the short run average cost curve, and equals the long run elasticity of scale, ε_LKM. The short run elasticity of scale with capital fixed at K = 25.24 is a decreasing function along the short run average cost curve, since sigma is less than 1.

Graph of Average Cost and Marginal Cost Diewert Production / Cost Functions - Capital Fixed
Returns to Scale = 1, Elasticity of Substitution = 0.85
wL = 7, wK= 13, wM = 7.
	Average cost function   Marginal cost function   L.R. Average cost function

VIII. 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 Diewert production function:

q^^1/nu = aLL * L + aKK * K + aMM * M + bLK * L^^1/2 * K^^1/2 + bLM * L^^1/2 * M^^1/2 + bKM * K^^1/2 * M^^1/2   =   f(L,K,M)

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 Diewert production function as estimated above:

q^^1/1 = -0.148053 * L + -0.18189 * K + -0.17997 * M + 0.644377 * L^^1/2 * K^^1/2 + 0.346619 * L^^1/2 * M^^1/2 + 0.51906 * K^^1/2 * M^^1/2

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

(L, K, M) = (36.89, 24.4, 31.62), and

C(30) = 7 * 36.89 + 13 * 24.4 + 6 * 31.62 = 765.2.

Solving the Diewert 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 in these equations, we obtain three 2-dimensional isoquant curves.

Firstly, consider the L-K Isoquant. Set M = M, and let:

a = aKK, b = bLK * L^^(1/2) + bKM * M^^(1/2),
c = q^^(1/nu) - aLL * L - aMM * M - bLM * L^^(1/2) * M^^(1/2),

The L-K isoquant expressed as K as a function of L:

K = -(1/2/a*(-b + sqrt(b * b + 4*a*c))*b-c)/a.

Secondly, consider the L-M Isoquant. Set K = K, and let:

a = aMM, b = bLM * L^^(1/2) + bKM * K^^(1/2),
c = q^^(1/nu) - aLL * L - aKK * K - bLK * L^^(1/2) * K^^(1/2),

The L-M isoquant expressed as M as a function of L:

M = -(1/2/a*(-b + sqrt(b*b + 4*a*c))*b-c)/a.

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 = (765.2 - 6 * 31.62)/13 = 44.27,
while the L-M isocost line has a M-intercept at (C(30) - wK * K) / wM = (765.2 - 13 * 24.4)/6 = 74.66.

For q = 30, the L-K isoquant is tangent to the L-K isocost line at (L, K) = (36.89, 24.4), while the L-M isoquant is tangent to the L-M isocost line at (L, M) = (36.89, 31.62).

Diewert Production Function Isoquants
L-K Isoquants, M = M	L-M Isoquants, K = K

The red rays emanating from the origin in the diagrams intersect each isoquant at the same angle. Consequently, any isoquant is a radial projection of each other isoquant. In particular, any isoquant is a radial projection of the unit isoquant, i.e. the isoquant for q = 1. Production functions with this property are called homothetic production functions.

It can be profitable to compare the Diewert isoquants with the isoquants of the CES production function, from which the Diewert's isoquants were derived.

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

(L, K, M) = (36.89, 24.42, 31.59), and

C(30) = 7 * 36.89 + 13 * 24.42 + 6 * 31.59 = 765.17.

CES Production Function Isoquants
L-K Isoquants, M = M	L-M Isoquants, K = K

Mathematical Notes

I. The Diewert (Generalized Leontief) Production Function

Elasticity of Scale of Production:

        Long Run: Capital Variable:

ε_LKM = ε(L,K,M) = nu * [f_L(L,K,M) * L + f_K(L,K,M) * K + f_M(L,K,M) * M] / f(L, K, M),

              = nu,   since f(L, K ,M) is linear homogeneous.

        Short Run: Captial Fixed: K = K:

ε_LKM = ε(L,K,M) = nu * [ f_L(L,K,M) * L + f_M(L,K,M) * M] / f(L, K, M).

Curvature:

The Diewert production function, q = F(L,K,M) = f(L,K,M)^^nu, 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)^^nu],

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)^^nu].

First Order Conditions:

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

 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)^^nu],

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)^^nu].

First Order Conditions: K = K:

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

 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|