Egwald Economics: Microeconomics

Production Functions

by

Elmer G. Wiens

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

Cost Functions:   Cobb-Douglas Cost | Normalized Quadratic Cost | Translog Cost | Diewert Cost | Generalized CES-Translog Cost | Generalized CES-Diewert Cost | References and Links

Duality: Production / Cost Functions:   Cobb-Douglas Duality | CES Duality | Theory of Duality | Translog Duality - CES | Translog Duality - Generalized CES

E. Comparing the Translog and Diewert Production Functions

To obtain estimates of the Translog production function, I used the CES production function to generate a sequence of observations relating the CES least cost factor inputs to factor prices and levels of output.

Estimating the Translog production function using multiple regression yielded the specified coefficient estimates for sigma = .5 and 2.

For values of sigma significantly greater than 1, the Diewert production function provides better estimates of the Allen elasticites than the Translog production function. See the Diewert production function.

1. Elasticity of substitution < < 1

elasticity of scale = 1.0, and

elasticity of substitution = .5

and alpha = .35, beta = .4, and gamma = .25.

lnA = 0 aL = 0.348648 aK = 0.400012 aM = 0.25134
bLL = -0.105433 bKK = -0.122478 bMM = -0.091295
bLK = 0.136617 bLM = 0.07425 bKM = 0.10834

aL + aK + aM = 1
-2*bLL = 0.210866 =~ 0.210867 = bLK + bLM
-2*bKK = 0.244956 =~ 0.244957 = bLK + bKLM
-2*bMM = 0.18259 =~ 0.18259 = bLM + bKM

R² = 1.0       all |t-values| >> 2

The firm buys its inputs at the prices:     wL = 7   wK = 13   wM = 6

Translog Long Run Cost Data Constant Returns to Scale Elasticity of Substitution = .5
q	est q	L	K	M	total cost	ave. cost	marg. cost	sLK	sLM	sKM
20	20	22.68	17.79	20.71	514.19	25.71	25.71	0.5	0.5	0.5
22	22	24.94	19.56	22.78	565.61	25.71	25.71	0.5	0.5	0.5
24	24	27.21	21.34	24.85	617.03	25.71	25.71	0.5	0.5	0.5
26	26	29.48	23.12	26.92	668.45	25.71	25.71	0.5	0.5	0.5
28	28	31.75	24.9	28.99	719.86	25.71	25.71	0.5	0.5	0.5
30	30	34.01	26.68	31.06	771.28	25.71	25.71	0.5	0.5	0.5
32	32	36.28	28.46	33.13	822.7	25.71	25.71	0.5	0.5	0.5
34	34	38.55	30.24	35.2	874.12	25.71	25.71	0.5	0.5	0.5
36	36	40.82	32.01	37.27	925.54	25.71	25.71	0.5	0.5	0.5
38	38	43.09	33.79	39.34	976.96	25.71	25.71	0.5	0.5	0.5
40	40	45.35	35.57	41.41	1028.38	25.71	25.71	0.5	0.5	0.5

Short Run: Capital Fixed.

If we set capital at the least cost level for q = 30, then K = 26.678769826985

See the discussion of the short-run elasticity of substitution, s_LM, on the translog production function page.

Translog Short Run Cost Data Constant Returns to Scale Elasticity of Substitution = .5
q	est q	L	K	M	total cost	ave. cost	marg. cost	3 factor sLM	2 factor sLM
20	20	17.84	26.68	16.32	569.65	28.48	15.73	0.61	0.56
22	22	20.48	26.68	18.73	602.58	27.39	17.23	0.59	0.55
24	24	23.38	26.68	21.37	638.67	26.61	18.94	0.57	0.54
26	25.99	26.57	26.68	24.28	678.43	26.09	20.89	0.55	0.53
28	28	30.1	26.68	27.49	722.46	25.8	23.1	0.53	0.51
30	30	34.01	26.68	31.05	771.24	25.71	25.7	0.5	0.5
32	32	38.37	26.68	35.02	825.57	25.8	28.71	0.47	0.49
34	33.99	43.25	26.68	39.46	886.32	26.07	32.28	0.44	0.47
36	36	48.79	26.68	44.49	955.29	26.54	36.54	0.41	0.46
38	37.99	55.05	26.68	50.18	1033.27	27.19	41.73	0.36	0.44
40	39.99	62.26	26.68	56.73	1123.01	28.08	48.18	0.32	0.42

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

2. Elasticity of substitution > > 1

elasticity of scale = 1.0, and

elasticity of substitution = 2.0

lnA = 0.000348 aL = 0.347346 aK = 0.398656 aM = 0.253994
bLL = 0.062107 bKK = 0.055418 bMM = 0.047515
bLK = -0.070009 bLM = -0.054203 bKM = -0.040827

aL + aK + aM = 1
-2*bLL = -0.124214 =~ -0.124212 = bLK + bLM
-2*bKK = -0.110836 =~ -0.110836 = bLK + bKLM
-2*bMM = -0.09503 =~ -0.09503 = bLM + bKM

R² = 1.0

Translog Long Run Cost Data Constant Returns to Scale Elasticity of Substitution = 2.0
q	est q	L	K	M	total cost	ave. cost	marg. cost	sLK	sLM	sKM
20	20	30.91	11.7	21.47	497.29	24.86	24.86	2.11	1.92	2.06
22	22	34	12.87	23.61	547.02	24.86	24.86	2.11	1.92	2.06
24	24	37.1	14.04	25.76	596.75	24.86	24.86	2.11	1.92	2.06
26	26	40.19	15.21	27.91	646.48	24.86	24.86	2.11	1.92	2.06
28	28	43.28	16.38	30.05	696.2	24.86	24.86	2.11	1.92	2.06
30	30	46.37	17.55	32.2	745.93	24.86	24.86	2.11	1.92	2.06
32	32	49.46	18.72	34.35	795.66	24.86	24.86	2.11	1.92	2.06
34	34	52.55	19.89	36.49	845.39	24.86	24.86	2.11	1.92	2.06
36	36	55.64	21.06	38.64	895.12	24.86	24.86	2.11	1.92	2.06
38	38	58.73	22.23	40.79	944.85	24.86	24.86	2.11	1.92	2.06
40	40	61.83	23.4	42.93	994.58	24.86	24.86	2.11	1.92	2.06

Short Run: Capital Fixed.

If we set capital at the least cost level for q = 30, then K = 17.549858968771

Translog Short Run Cost Data Constant Returns to Scale Elasticity of Substitution = 2.0
q	est q	L	K	M	total cost	ave. cost	marg. cost	3 factor sLM	2 factor sLM
20	20	24.92	17.55	17.64	508.4	25.42	22.37	2.38	2.2
22	21.99	28.98	17.55	20.42	553.49	25.16	23.05	2.25	2.13
24	24.01	33.14	17.55	23.4	600.51	25.02	23.57	2.14	2.08
26	26	37.71	17.55	25.99	648.07	24.93	24.06	2.06	2.04
28	27.99	41.72	17.55	29.35	696.31	24.87	24.53	1.98	2
30	30	46.04	17.55	32.61	746.07	24.87	24.86	1.92	1.97
32	32.01	50.53	17.55	35.73	796.21	24.88	25.18	1.86	1.94
34	34	55.07	17.55	38.82	846.61	24.9	25.5	1.82	1.92
36	36	59.68	17.55	41.99	897.86	24.94	25.76	1.77	1.9
38	38	64.38	17.55	45.13	949.6	24.99	25.99	1.73	1.88
40	40.01	69.12	17.55	48.35	1002.08	25.05	26.16	1.69	1.86

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