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. For these coefficients of the CES production function, I generated a sequence of factor prices, outputs, and the corresponding cost minimizing inputs. Then I estimated the Translog production function using multiple regression yielding 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 = 0.039331 =~ 0.039331 = bLK + bLM
-2*bKK = 0.042673 =~ 0.042673 = bLK + bKLM
-2*bMM = 0.032875 =~ 0.032875 = bLM + bKM

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).        

III. For these coefficients of the Translog production function, I generated a new sequence (displayed in the "Translog Cost Function" table) of factor prices, outputs, and the corresponding cost minimizing inputs.

Then I used these data to estimate the coefficients of each factor share equation separately:

The three estimated factor share functions are:

As estimated, generally dLK != dKL, dLM != dML, and dKM != dMK: Young's Theorem doesn't hold without constraints across equations.

Note: C(q;wL,wK,wM) linear homogeneous in factor prices implies:

IV. The Restricted Factor Share Equations.

Having obtained the unrestricted estimates of the coefficients of the factor share functions, we compute the restricted least squares estimates with dLK = dKL, dLM = dML, and dKM = dMK.

The three estimated, restricted factor share functions are:

Imposing the dLK = dKL, dLM = dML, and dKM = dM restrictions may distort five necessary C(q;wL,wK,wM) linear homogeneous in factor prices restraints:

Let us impose these additional five restraints and re-estimate the Restricted Factor Share Equations:

The three re-estimated, restricted factor share functions are:

V.  To obtain estimates of the remaining three parameters, c, cq, and dqq, write:

ln(C(q;wL,wK,wM)) - {cL * ln(wL) + cK * ln(wK) + cM * log(wM) + .5 * [dLL * ln(wL)^² + dKK * ln(wK)^² + dMM * ln(wM)^²]
+ .5 * [(dLK + dKL) * ln(wL)*ln(wK) + (dLM + dML) * ln(wL)*ln(wM) + (dKM + dMK) * ln(wK)*log(wM)]
+ dLq * ln(wL)*ln(q) + dKq * ln(wK)*ln(q) + dMq * ln(wM)*ln(q)}
= R(q;wL,wK,wM) = c + cq * ln(q) + .5 * dqq * ln(q)^²

Estimate the linear equation:

R(q;wL,wK,wM) = c + cq * ln(q) + .5 * dqq * ln(q)^²

to obtain c, cq, and dqq.

VI. The Translog cost function as estimated is:

ln(C(q;wL,wK,wM)) = 1.083165 + 1 * ln(q) + 0.348451 * ln(wL) + 0.389979 * ln(wK) + 0.26157 * log(wM)
+ .5 * [-0 * ln(q)^² + 0.03344 * ln(wL)^² + 0.036293 * ln(wK)^² + 0.027928 * ln(wM)^²]
+ .5 * [-0.041805 * ln(wL)*ln(wK) + -0.025075 * ln(wL)*ln(wM) + -0.030782 * ln(wK)*log(wM)]
+ 6.0E-6 * ln(wL)*ln(q) + -7.0E-6 * ln(wK)*ln(q) + 2.0E-6 * ln(wM)*ln(q)           (***)

VII. Check for linear homogeneity:

C(q;wL,wK,wM) linear homogeneous in factor prices requires for any t > 0, that C(q;wL,wK,wM) obey:

C(q;t*wL,t*wK,t*wM) = t * C(q;wL,wK,wM)

      For example, with wL = 7, wK = 13, wM = 6, and t = 2:

2 * C(25; 7, 13, 6) = 1275.29 =? 1275.29 = C(25; 14, 26,12).

2 * C(30; 7, 13, 6) = 1530.35 =? 1530.35 = C(30; 14, 26,12).

VIII. Check for homotheticity:

Can we write C(q;wL,wK,wM) as the product of the unit cost function C(1;wL,wK,wM) and a suitable increasing function h(q) with h(0) = 0, and h(1) = 1:

C(q;wL,wK,wM) =? h(q) * C(1;wL,wK,wM)

To do so it is necessary that:

1   =?   cq   =   1,     0   =?   dqq  =   -0,

0   =?   dLq  =   6.0E-6,     0   =?   dKq  =   -7.0E-6,     0   =?   dMq  =   2.0E-6.

      For example, with wL = 7, wK = 13, wM = 6, and q = 30, try h(q) = q^1/nu with nu = 1:

C(30; 7, 13, 6) = 765.17 =? 765.19 = 30 * 25.51 = 30^^1/1 * C(1; 7, 13, 6).

Try it with sigma = 1!

IX. 1. The matrix ∇²C of second order partial derivatives of the cost function C(q;wL,wK,wM) is symmetric. To show this, write:

ln(C(q;wL,wK,wM) = C(q;wL,wK,wM), so C(q;wL,wK,wM) = exp(C(q;wL,wK,wM)).

        Since the matrix ∇²C(q;wL,wK,wM) is symmetric, the matrix ∇²C(q;wL,wK,wM) is also symmetric.

  2. The cost function C(q;wL,wK,wM) is a concave in factor prices if its Hessian matrix ∇²_wwC of second order partial derivatives with respect to factor prices is negative semidefinite.   Following the procedure suggested by Diewert and Wales (1987), write sL = sL(q;wL,wK,wM), sK = sK(q;wL,wK,wM), and sM = sM(q;wL,wK,wM), and the matrices:

Assuming C(q;wL,wK,wM) > 0, then ∇²_wwC is negative semidefinite if and only if the matrix:

H = W * ∇²_wwC * W   =   D   -   S   +   SS

is negative semidefinite. See the Mathematical Notes.

  The values of the second order partial derivatives depend on the amount of output, and on factor prices. As an example, consider the case where q = 30, wL = 7, wK = 13, and wM = 6:

The principal minors of H are H1 = -0.190142, H2 = 0.025075, and H3 = -0.

If these principal minors alternate in sign, starting with negative, with H3 <= 0, the matrix H is negative (semi)definite,
and the cost function C(q;wL,wK,wM) is a concave function in factor prices at q = 30, wL = 7, wK = 13, and wM = 6.

The eigenvalues of H are e1 = -0.3198, e2 = -0.2352, and e3 = -0.
H3 = e1 * e2 * e3 = -0.

X. The three estimated factor demand functions are obtained by:

    The estimated factor demand elasticities are obtained by:

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

XI. Uzawa Partial Elasticities of Substitution:

    where the partial derivatives of the factor demand functions are:

With the dLK = dKL, dLM = dML, and dKM = dMK restrictions in the factor share functions, we get (approximate?) equality of the cross partial elasticities of substitution:

u_LK = u_KL,    u_LM = u_ML,  and    u_KM = u_MK.

as seen in the following table.

XII. Table of Results: check that the estimated Translog cost function for a given level of output agrees with the minimized cost using the Translog production function.

À2:   Estimating the Translog cost function directly.

XIII. If we have a data set relating the input (factor) prices, wL, wK, and wM, to the total (minimum) cost of producing output for varying levels output, but we do not have data on the required levels of inputs, we can estimate the Translog cost function directly. We will use the same sequence (displayed in the "Translog Cost Function" table) of factor prices, outputs, and total (minimum) cost as used in À1. The estimated coefficients of the cost function will vary with the parameters sigma, nu, alpha, beta and gamma of the CES production function used to generate these data with the Translog production function. We impose 9 restrictions on the estimates of the parameters as specified, including the 4 necessary homothetic conditions.

XIV. The Translog cost function estimated directly is:

ln(C(q;wL,wK,wM)) = 1.083166 + 1 * ln(q) + 0.348482 * ln(wL) + 0.389946 * ln(wK) + 0.261572 * log(wM)
+ .5 * [0 * ln(q)^² + 0.033442 * ln(wL)^² + 0.036308 * ln(wK)^² + 0.027921 * ln(wM)^²]
+ .5 * [-0.041829 * ln(wL)*ln(wK) + -0.025055 * ln(wL)*ln(wM) + -0.030787 * ln(wK)*log(wM)]
+ 0 * ln(wL)*ln(q) + -0 * ln(wK)*ln(q) + -0 * ln(wM)*ln(q)           (***)

XV. Its three derived factor share functions are:

XVI. Notes:
      1. As derived, dLK = dKL, dLM = dML, and dKM = dMK: Young's Theorem holds by construction.

      2. C(q;wL,wK,wM) linear homogeneous in factor prices implies:

      3. The matrix ∇²C of second order partial derivatives of the cost function C(q;wL,wK,wM) is symmetric.

     4. The cost function C(q;wL,wK,wM) is a concave in factor prices if its Hessian matrix ∇²_wwC of second order partial derivatives with respect to factor prices is negative semidefinite.   Following the procedure suggested by Diewert and Wales (1987), write sL = sL(q;wL,wK,wM), sK = sK(q;wL,wK,wM), and sM = sM(q;wL,wK,wM), and the matrices:

Assuming C(q;wL,wK,wM) > 0, then ∇²_wwC is negative semidefinite if and only if the matrix:

H = W * ∇²_wwC * W   =   D   -   S   +   SS

is negative semidefinite. See the Mathematical Notes.

  The values of the second order partial derivatives depend on the amount of output, and on factor prices. As an example, consider the case where q = 30, wL = 7, wK = 13, and wM = 6:

The principal minors of H are H1 = -0.190141, H2 = 0.025075, and H3 = -0.

If these principal minors alternate in sign, starting with negative, with H3 <= 0, the matrix H is negative (semi)definite,
and the cost function C(q;wL,wK,wM) is a concave function in factor prices at q = 30, wL = 7, wK = 13, and wM = 6.

The eigenvalues of H are e1 = -0.3198, e2 = -0.2352, and e3 = 0.
H3 = e1 * e2 * e3 = 0.

XVII. Check for linear homogeneity:

C(q;wL,wK,wM) linear homogeneous in factor prices requires for any t > 0, that C(q;wL,wK,wM) obey:

C(q;t*wL,t*wK,t*wM) = t * C(q;wL,wK,wM)

      For example, with wL = 7, wK = 13, wM = 6, and t = 2:

2 * C(25; 7, 13, 6) = 1275.29 =? 1275.29 = C(25; 14, 26,12).

2 * C(30; 7, 13, 6) = 1530.34 =? 1530.34 = C(30; 14, 26,12).

XVIII. Check for homotheticity:

Can we write C(q;wL,wK,wM) as the product of the unit cost function C(1;wL,wK,wM) and a suitable increasing function h(q) with h(0) = 0, and h(1) = 1:

C(q;wL,wK,wM) =? h(q) * C(1;wL,wK,wM)

To do so it is necessary that:

1   =?   cq   =   1,     0   =?   dqq  =   0,

0   =?   dLq  =   0,     0   =?   dKq  =   -0,     0   =?   dMq  =   -0.

      As examples:

          a. with wL = 7, wK = 13, wM = 6, and q = 25, try h(q) = q^1/nu with nu = 1:

C(25; 7, 13, 6) = 637.64 =? 637.64 = 25 * 25.51 = 25^^1/1 * C(1; 7, 13, 6).

          b. with wL = 7, wK = 13, wM = 6, and q = 30, try h(q) = q^1/nu with nu = 1:

C(30; 7, 13, 6) = 765.17 =? 765.17 = 30 * 25.51 = 30^^1/1 * C(1; 7, 13, 6).

Try it with sigma = 1!

XIX. The three derived factor demand functions are obtained by:

    The factor demand elasticities are obtained by:

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

XX. Table of Results: check that the estimated Translog cost function for a given level of output agrees with the minimized cost using the Translog production function.

À3:   Re-estimate the Translog Production Function.

XXI. Assuming we had a data set relating the input (factor) prices, wL, wK, and wM, to the total (minimum) cost of producing output for varying levels output, but that we did not have data on the cost minimizing levels of inputs, we estimated the Translog cost function directly. With the derived factor demand functions to determine cost minimizing inputs, we can estimate a Translog production function using the data in the Translog Cost Function table.

The estimated Translog production function using the cost data is:

ln(q) = 2.1E-5 + 0.349871 * ln(L) + 0.400024 * ln(K) + 0.2501 * ln(M) + -0.019663 * ln(L)*ln(L) + -0.021333 * ln(K)*ln(K) + -0.016435 * ln(M)*ln(M)
+ 0.024557 * ln(L)*ln(K) + 0.014773 * ln(L)*ln(M) + 0.018102 * ln(K)*ln(M)   =   f(L,K,M).        

The coefficients of this production function are approximately? equal to the coefficients of the original Translog production function.

XXII. Knowing the Translog production function permits us to investigate further properties of the production / cost technology. For example, we can determine the short-run average and marginal cost functions in relation to the long-run average and marginal cost functions obtained from the Translog cost function, and the short-run Allen elasticities of substitution.

For example, if we set capital at the least cost level for q = 30, then K = 24.41, and we can solve for the least cost inputs of labour, L, and materials and supplies, M, for various levels of output, q.   (See the Translog Production Function web page.)

We get a U-shaped, short run average cost curve, with capital fixed.

As seen in the diagram below, the short run average cost curve is (approximately) tangent to the long run average cost curve, at q = 30.

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

Mathematical Notes

1. The Translog Cost Function:

2. The Factor Share Functions:

3. The Factor Demand Functions:

4. The Factor Demand Elasticities:

5. Cost Function C(q;wL,wK,wM) Concave in Factor Prices: