Egwald Economics: Microeconomics

Production Functions

by

Elmer G. Wiens

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Cobb-Douglas | CES | Translog | Diewert | Translog vs Diewert | Diewert vs Translog | Estimate Translog | Estimate Diewert | References and Links

With the Cobb-Douglas and CES production functions, I obtained an explicit cost function, total cost as a function of q, wL, wK, and wM, by minimizing the cost of producing a given level of output. Because the Translog production function is much more general (it has a flexible functional form permitting the partial elasticities of substitution between inputs to vary), I will use numerical analysis to obtain the cost functions associated with a given Translog production function. With the Translog production function, the elasticity of scale can vary with output and factor proportions, permitting its long run average cost curve to take the traditional U-shape.

C. Translog (Transcendental Logarithmic) Production Function

The three factor Translog production function is:

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).

where L = labour, K = capital, M = materials and supplies, and q = product.

I. Constant returns to scale:

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

II. To get estimates for the parameters of the Translog production function, I generated a set of 182 observations (output, factor inputs, factor prices) from a constant returns to scale CES production function, with the

elasticity of scale = 1.0, and

elasticity of substitution = .85.

III. Estimating the translog production function using multiple regression yielded the following coefficient estimates:

            lnA = 0 aL = 0.349891 aK = 0.399994 aM = 0.250116
            bLL = -0.019665 bKK = -0.021336 bMM = -0.016437
            bLK = 0.024565 bLM = 0.014766 bKM = 0.018108

R² = 1.0       all |t-values| >> 2

Note that this procedure can be used for any set of estimated coefficients, provided we can use the resulting production function in solving, numerically, the least-cost problem below.

IV. Least-cost combination of inputs

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 - exp(f(L,K,M))]

1. G_L = wL - µ * f_L * exp(f(L,K,M)) = 0
   
2. G_K = wK - µ * f_K * exp(f(L,K,M)) = 0
   
3. G_M = wM - µ * f_M * exp(f(L,K,M)) = 0
   
4. G_µ = q - * exp(f(L,K,M)) = 0
   

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

J =	GLL	GLK	GLM	GLµ
	GKL	GKK	GKM	GKµ
	GML	GMK	GMM	GMµ
	GµL	GµK	GµM	Gµµ

V. Suppose the firm buys its inputs at the prices:

    wL = 7   wK = 13   wM = 6

Solving the least-cost problem yields, noting that µ = marginal cost:

The Allen partial elasticities of substitution, s_LK, s_LM, and s_KM,are all approximately equal to .85 as expected.

VI. Short Run: Capital Fixed.

If we set capital at the least cost level for q = 30, then K = 25.614838677156

Using the same method as for the long run cost curves, we get:

So, now again 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.

Here I have 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

VII. Elasticity of substitution = 1

elasticity of scale = 1.0, and

elasticity of substitution = 1.0

The CES production function collapses into the Cobb-Douglas production function.

            lnA = 0 aL = 0.35 aK = 0.45 aM = 0.25
            bLL = 0 bKK = 0 bMM = 0
            bLK = 0 bLM = 0 bKM = 0

R² = 1.0

Translog Long Run Cost Data Constant Returns to Scale Elasticity of Substitution = 1.0
q	est q	L	K	M	total cost	ave. cost	marg. cost	sLK	sLM	sKM
20	20	21.16	14.67	17.71	445.17	22.26	21.21	1	1	1
22	22	23.17	16.07	19.4	487.45	22.16	21.11	1	1	1
24	24	25.16	17.47	21.09	529.74	22.07	21.02	1	1	1
26	26	27.15	18.85	22.76	571.68	21.99	20.94	1	1	1
28	28	29.14	20.23	24.42	613.47	21.91	20.86	1	1	1
30	29.99	31.11	21.59	26.07	654.88	21.83	20.81	1	1	1
32	31.99	33.08	22.97	27.71	696.4	21.76	20.75	1	1	1
34	34	35.04	24.35	29.36	738.02	21.71	20.68	1	1	1
36	35.99	37.01	25.7	30.98	779.09	21.64	20.64	1	1	1
38	38	38.94	27.09	32.62	820.48	21.59	20.58	1	1	1
40	40	40.89	28.46	34.23	861.57	21.54	20.53	1	1	1

VIII. Short Run: Capital Fixed.

If we set capital at the least cost level for q = 30, then K = 21.59291216252

Translog Short Run Cost Data Constant Returns to Scale Elasticity of Substitution = 1.0
q	est q	L	K	M	total cost	ave. cost	marg. cost	3 factor sLM	2 factor sLM
20	20	15.87	21.59	13.24	471.22	23.56	15.87	1	1
22	22	18.6	21.59	15.51	503.96	22.91	16.91	1	1
24	24	21.5	21.59	17.95	538.88	22.45	17.92	1	1
26	26	24.56	21.59	20.52	575.74	22.14	18.9	1	1
28	28	27.78	21.59	23.2	614.37	21.94	19.87	1	1
30	30	31.16	21.59	26.03	655.04	21.83	20.8	1	1
32	32	34.71	21.59	29.01	697.69	21.8	21.7	1	1
34	34	38.38	21.59	32.09	741.86	21.82	22.62	1	1
36	35.99	42.19	21.59	35.31	787.89	21.89	23.5	1	1
38	37.99	46.18	21.59	38.62	835.71	21.99	24.36	1	1
40	40	50.29	21.59	42.13	885.52	22.14	25.19	1	1

The short run average cost curve is (approx.) tangent to the long run average cost curve, at q = 30.

IX. Elasticity of substitution > 1

elasticity of scale = 1.0, and

elasticity of substitution = 1.15

            lnA = 0 aL = 0.349902 aK = 0.399987 aM = 0.25011
            bLL = 0.015116 bKK = 0.015516 bMM = 0.012292
            bLK = -0.01834 bLM = -0.011892 bKM = 0.012693

R² = 1.0

Translog Long Run Cost Data Constant Returns to Scale Elasticity of Substitution = 1.15
q	est q	L	K	M	total cost	ave. cost	marg. cost	sLK	sLM	sKM
20	20	20.06	12.68	19.07	419.74	20.99	17.87	1.15	1.17	0.98
22	22	21.74	13.8	20.77	456.21	20.74	17.58	1.15	1.17	0.98
24	23.99	23.42	14.88	22.44	491.98	20.5	17.33	1.15	1.17	0.98
26	26	25.07	15.98	24.08	527.66	20.29	17.08	1.15	1.17	0.98
28	28	26.71	17.04	25.7	562.69	20.1	16.86	1.15	1.17	0.98
30	29.99	28.32	18.09	27.3	597.16	19.91	16.64	1.15	1.17	0.98
32	31.99	29.9	19.13	28.88	631.35	19.73	16.45	1.15	1.17	0.98
34	33.99	31.48	20.17	30.45	665.17	19.56	16.27	1.15	1.17	0.98
36	36	33.04	21.18	32.01	698.64	19.41	16.1	1.15	1.16	0.98
38	38	34.59	22.17	33.58	731.81	19.26	15.93	1.15	1.16	0.98
40	39.99	36.12	23.15	35.13	764.54	19.11	15.78	1.15	1.16	0.98

X. Short Run: Capital Fixed.

If we set capital at the least cost level for q = 30, then K = 18.086453619194

Translog Short Run Cost Data Constant Returns to Scale Elasticity of Substitution = 1.15
q	est q	L	K	M	total cost	ave. cost	marg. cost	3 factor sLM	2 factor sLM
20	20	15.38	18.09	15.22	434.13	21.71	14.19	1.18	1.13
22	22	17.78	18.09	17.49	464.52	21.11	14.75	1.17	1.12
24	24	20.27	18.09	19.86	496.16	20.67	15.25	1.17	1.12
26	26	22.85	18.09	22.28	528.75	20.34	15.75	1.17	1.12
28	28	25.56	18.09	24.76	562.59	20.09	16.19	1.17	1.12
30	30	28.4	18.09	27.24	597.33	19.91	16.62	1.17	1.12
32	32	31.26	18.09	29.85	633.03	19.78	17.03	1.17	1.12
34	34.01	34.18	18.09	32.54	669.6	19.69	17.39	1.16	1.12
36	36	37.18	18.09	35.23	706.79	19.63	17.77	1.16	1.12
38	38.01	40.26	18.09	38.02	745.08	19.61	18.09	1.16	1.12
40	40	43.4	18.09	40.81	783.82	19.6	18.45	1.16	1.12

The short run average cost curve is (approx.) tangent to the long run average cost curve, at q = 30.

XI. Allen partial elasticity of substitution

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

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