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 Translog production function using multiple regression yielded the following coefficient estimates:

lnA = 6.0E-6 aL = 0.349891 aK = 0.399994 aM = 0.250116
bLL = -0.019666 bKK = -0.021337 bMM = -0.016437
bLK = 0.024565 bLM = 0.014766 bKM = 0.018108

aL + aK + aM = 1
2 * bLL + bLK + bLM = 0
2 * bKK + bLK + bKM = 0
2 * bMM + bLM + bKM = 0

The estimated Translog production function:

ln(q) = 6.0E-6 + 0.349891 * ln(L) + 0.399994 * ln(K) + 0.250116 * ln(M) + -0.019666 * ln(L)*ln(L) + -0.021337 * ln(K)*ln(K) + -0.016437 * ln(M)*ln(M)
+ 0.024565 * ln(L)*ln(K) + 0.014766 * ln(L)*ln(M) + 0.018108 * ln(K)*ln(M)   =   f(L,K,M).        

Elasticity of Scale of Production:

        Long Run: Capital Variable:

ε_LKM = ε(L,K,M) = aL + aK + aM + (2*bLL + bLK + bLM) * ln(L) + (2*bKK + bLK + bKM) * ln(K) + (2*bMM + bLM + bKM) * ln(M).

        Short Run: Captial Fixed: K = K:

ε_LKM = ε(L,K,M) = aL + aM + (2*bLL + bLM) * ln(L) + (bLK + bKM) * ln(K) + (2*bMM + bLM) * ln(M).

See the Mathematical Notes below.

Curvature:

The Translog production function, q = F(L,K,M) = exp(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).

III. 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 - exp(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 - exp(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) = exp(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.)

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 Translog cost data as displayed in the following table. The terms s_LK, s_LM, and s_KM are the Allen partial elasticities of substitution.

V. Short Run: Capital Fixed: K = K = 25.6.

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 Translog short-run cost data as displayed in the following table. (See the discussion of the short-run elasticity of substitution, s_LM, below.)

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 Translog 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 = 25.6 is a decreasing function along the short run average cost curve, since sigma is less than 1.

Graph of Average Cost and Marginal Cost Translog Cost / Production Functions - Capital Fixed
	Average cost function   Marginal cost function   L.R. Average cost function

Here I listed two measures of the short-run elasticity of substitution between L and M. The 3-factor measure of s_LM uses the Allen partial elasticity of substitution formula. The 2-factor measure of s_LM uses the standard short-run formula, which assumes that only L and M can vary with output, with a fixed amount of capital present. Another option would be to estimate a two factor production function q = F(L,M), and then to compute s_LM. But then capital, K, is a missing variable from the estimation, skewing the estimates of the coefficients of the production function F

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 Translog 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 = 26.41.

Let the firm buy its inputs at the same (wM increased) 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 Translog short-run cost data as displayed in the following table. (See the discussion of the short-run elasticity of substitution, s_LM, above.)

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 = 26.41 is a decreasing function along the short run average cost curve, since sigma is less than 1.

Graph of Average Cost and Marginal Cost Translog Cost / Production Functions - Capital Fixed
	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 Translog production function:

ln(q) = ln(A) + aL*ln(L) + aK*ln(K) + aM*ln(M) + bLL*ln(L)*ln(L) + bKK*ln(K)*ln(K) + bMM*ln(M)*ln(M)
+ bLK*ln(L)*ln(K) + bLM*ln(L)*ln(M) + bKM*ln(K)*ln(M)   =   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 Translog production function as estimated above:

ln(q) = 6.0E-6 + 0.349891 * ln(L) + 0.399994 * ln(K) + 0.250116 * ln(M) + -0.019666 * ln(L)*ln(L) + -0.021337 * ln(K)*ln(K) + -0.016437 * ln(M)*ln(M)
+ 0.024565 * ln(L)*ln(K) + 0.014766 * ln(L)*ln(M) + 0.018108 * ln(K)*ln(M)

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

(L, K, M) = (33.66, 25.6, 33.12), and

C(30) = 7 * 33.66 + 13 * 25.6 + 6 * 33.12 = 767.09.

Solving the Translog 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 = bKK, b = aK + bLK*log(L) + bKM * log(M),
c = log(q) - log(A) - aL * log(L) - aM * log(M) - bLL * log(L) * log(L) - bMM * log(M) * log(M) - bLM * log(L) * log(M).

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

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

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

a = bMM, b = aM + bLM*log(L) + bKM * log(K),
c = log(q) - log(A) - aL * log(L) - aK * log(K) - bLL * log(L) * log(L) - bKK * log(K) * log(K) - bLK * log(L) * log(K).

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

M = exp( (-b + sqrt(b*b+4*a*c)) / (2*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 = (767.09 - 6 * 33.12)/13 = 43.72,
while the L-M isocost line has a M-intercept at (C(30) - wK * K) / wM = (767.09 - 13 * 25.6)/6 = 72.39.

For q = 30, the L-K isoquant is tangent to the L-K isocost line at (L, K) = (33.66, 25.6), while the L-M isoquant is tangent to the L-M isocost line at (L, M) = (33.66, 33.12).
(Note: for values of sigma ~= .5, the L-K isocost line intersects the L-K isoquant at (L, K),
while the L-M isocost line intersects the L-M isoquant at (L, M). The curvature of the Translog function, as estimated, does not provide tangency.)

Translog Production Function isoquants
L-K Isoquants, M = M	L-M Isoquants, K = K

It can be profitable to compare the Translog isoquants with the isoquants of the CES production function, from which the Translog'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 Translog (Transcendental Logarithmic) Production Function

Elasticity of Scale of Production:

        Long Run: Capital Variable:

ε_LKM = ε(L,K,M) = aL + aK + aM + (2*bLL + bLK + bLM) * ln(L) + (2*bKK + bLK + bKM) * ln(K) + (2*bMM + bLM + bKM) * ln(M).

        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) = aL + aM + (2*bLL + bLM) * ln(L) + (bLK + bKM) * ln(K) + (2*bMM + bLM) * ln(M).

Curvature:

The Translog production function, q = F(L,K,M) = exp(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 - exp(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 - exp(f(L,K,M))].

First Order Conditions:

Jacobian of Second Order Conditions (Bordered Hessian with F(L,K,M) = exp(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 - exp(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 - exp(f(L,K,M))].

First Order Conditions: K = K

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