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

F. Comparing the Translog and Diewert Production Functions

To obtain the estimates of the Diewert 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 Diewert 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 Translog 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.

nu = 1
aLL = -0.4669 aKK = -0.5905 aMM = -0.4703
bLK = 1.0794 bLM = 0.5468 bKM = 0.9017

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

R² = 1.0       all |t-values| >> 2

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

Short Run: Capital Fixed.

If we set capital at the least cost level for q = 30, then K = 26.667327433218

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

Diewert 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	17.88	26.67	16.44	570.49	28.52	15.41	0.85	0.72
22	22	20.49	26.67	18.8	602.91	27.4	17.04	0.79	0.68
24	23.99	23.36	26.67	21.41	638.69	26.61	18.84	0.73	0.63
26	25.99	26.55	26.67	24.31	678.36	26.09	20.86	0.66	0.59
28	28	30.09	26.67	27.52	722.38	25.8	23.11	0.58	0.55
30	30	34.01	26.67	31.07	771.2	25.71	25.71	0.5	0.51
32	32	38.38	26.67	35.02	825.45	25.8	28.66	0.42	0.47
34	33.99	43.25	26.67	39.43	886.01	26.06	32.12	0.33	0.43
36	36	48.76	26.67	44.38	954.32	26.51	36.21	0.24	0.39
38	37.99	54.95	26.67	50.01	1031.39	27.14	41.19	0.15	0.35
40	39.98	62.07	26.67	56.4	1119.57	27.99	47.39	0.05	0.31

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.

2. Elasticity of substitution > > 1.

elasticity of scale = 1.0, and

elasticity of substitution = 2.0.

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

nu = 1
aLL = 0.1225 aKK = 0.16 aMM = 0.0625
bLK = 0.28 bLM = 0.175 bKM = 0.2

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

R² = 1.0

Diewert 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	30.9	11.7	21.46	497.21	24.86	24.86	2	2	2
22	22	33.99	12.87	23.61	546.93	24.86	24.86	2	2	2
24	24	37.08	14.04	25.75	596.65	24.86	24.86	2	2	2
26	26	40.17	15.21	27.9	646.37	24.86	24.86	2	2	2
28	28	43.26	16.38	30.04	696.1	24.86	24.86	2	2	2
30	30	46.35	17.55	32.19	745.82	24.86	24.86	2	2	2
32	32	49.44	18.72	34.34	795.54	24.86	24.86	2	2	2
34	34	52.53	19.89	36.48	845.26	24.86	24.86	2	2	2
36	36	55.62	21.06	38.63	894.98	24.86	24.86	2	2	2
38	38	58.71	22.24	40.77	944.7	24.86	24.86	2	2	2
40	40	61.8	23.41	42.92	994.42	24.86	24.86	2	2	2

Short Run: Capital Fixed.

If we set capital at the least cost level for q = 30, then K = 17.554007079253

Diewert 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	19.99	24.97	17.55	17.52	508.13	25.41	22.42	2	2
22	21.99	29.04	17.55	20.34	553.53	25.16	23.04	2	2
24	24.01	33.18	17.55	23.34	600.5	25.02	23.55	2	2
26	25.99	37.69	17.55	25.96	647.78	24.91	24.06	2	2
28	28	41.74	17.55	29.36	696.52	24.88	24.48	2	2
30	30.01	46.04	17.55	32.58	745.96	24.87	24.86	2	2
32	32	50.45	17.55	35.74	795.78	24.87	25.23	2	2
34	34	55.03	17.55	38.86	846.57	24.9	25.54	2	2
36	36.01	59.73	17.55	41.99	898.25	24.95	25.79	2	2
38	38.01	64.37	17.55	45.22	950.16	25	26.06	2	2
40	40	68.98	17.55	48.54	1002.26	25.06	26.34	2	2

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