Egwald Economics - Production Functions: Estimate the Translog Production Function

Egwald Web Services
Domain Names
Web Site Design

Egwald Economics: Microeconomics

Production Functions

by

Elmer G. Wiens

Egwald's popular web pages are provided without cost to users.
Follow Elmer Wiens on Twitter:

G. Generate CES Data and Estimate a Translog Production Function

1. The three factor CES production function is:

q = A * [alpha * (L^^-rho) + beta * (K^^-rho) + gamma *(M^^-rho)]^^(-nu/rho) = f(L,K,M).

where L = labour, K = capital, M = materials and supplies, and q = product. The parameter nu is a measure of the returns to scale, while the parameter rho yields the elasticity of substitution sigma = 1/(1 + rho).

2. 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.

3. The coefficients of the translog production function vary with sigma and nu.

The program will generate a set of 208 observations, and use ordinary least squares to estimate the Translog coefficients.

The observations are the CES function's cost-minimizing combinations of L, K, and M at the factor prices, wL, wK, wM for values of q from 15 to 40.

The factor prices are distributed about the base factor prices by adding a random number distributed uniformly in the domain [-2, 2]. The program generates the CES cost-minimizing values of L, K, and M for each combination of base prices for each value of q in the [15, 45] domain.

Base Factor Prices
wL	wK	wM
8	8	8
7	13	6
6	7	13
13	6	7
9	15	11
11	9	15
15	11	9

12	12	12

Set the parameters below to re-run with your own CES parameters.

Restrictions: .5 < nu < 2; .2 < sigma < 5; .1 < alpha, beta, gamma < .9
sigma = 1 → nu = alpha + beta + gamma (Cobb-Douglas)
sigma < 1 → inputs complements; sigma > 1 → inputs substitutes

Coefficient Estimates
Variable	Coefficient	std error	t-ratio
lnA	0.000142	0	0.318
lnL	0.350018	0	2506.826
lnK	0.399991	0	2536.872
lnM	0.249897	0	1971.83
lnLlnL	-0.020127	0	-898.016
lnKlnK	-0.021128	0	-787.749
lnMlnM	-0.017189	0	-719.923
lnLlnK	0.024115	0	633.654
lnLlnM	0.01618	0	432.971
lnKlnM	0.018163	0	458.59

R2 = 1

R2b = 1

# obs = 208

The table below displays the CES function's cost-minimizing combinations of L, K, and M at the factor prices, wL, wK, wM for values of q from 15 to 40.
The column est q = f(L, K, M) (using the translog function as estimated), where L, K, and M are the CES function's cost-minimizing combinations.

CES Production Function Data
obs #	q	est q	L	K	M	wL	wK	wM
1	15	15	12.81	19.16	12.82	9.56	6.8	6.82
2	16	16	17.19	19.39	10.85	7.58	7.52	9.3
3	17	17	16.51	18.91	14.98	7.94	7.74	6.36
4	18	18	19.35	21.62	12.36	7.22	7.24	8.74
5	19	19	18.66	21.32	16.28	8.82	8.62	7.4
6	20	20	20.82	21.7	16.66	7	7.62	6.5
7	21	21	18.45	26.54	17.52	8.72	6.5	6.62
8	22	22	24.77	23.39	17.03	7.64	9.34	8.48
9	23	23	23.89	24.33	19.98	7.24	8.1	6.38
10	24	24	25.73	27.03	18.17	7.66	8.26	8.24
11	25	25	20.68	32.28	21.99	9.66	6.54	6.42
12	26	26	27.58	31.03	18.33	8.1	8.06	9.36
13	27	27	31.75	29.77	18.72	7.22	8.9	9.6
14	28	28	29.18	29.04	24.97	7.62	8.76	6.54
15	29	29	28.51	36.03	21.31	7.1	6.16	7.14
16	30	30	28.07	40.67	20.76	9.5	7.02	9.68
17	31	31	30.9	36.47	24.23	9.38	8.82	8.92
18	32	32	36.72	35.82	22.39	6.9	8.12	8.82
19	33	33	34.15	38.25	25.09	7.56	7.56	7.76
20	34	34	32.3	35.98	33.39	8.82	8.88	6.06
21	35	35	33.31	45.89	24.84	8.98	7.04	9.06
22	36	36	38.32	42.12	26.01	8.84	9.04	9.96
23	37	37	39.03	41.82	28.46	7.82	8.24	8.1
24	38	38	38.66	39.92	34.32	7.74	8.52	6.36
25	39	39	41.7	39.29	35.14	7.44	9.12	6.5
26	40	40	39.23	43	36.67	9.26	9.5	7.16
27	15	15	17.53	12.35	16.63	7.66	13.22	5.82
28	16	16	18.82	12.19	20.05	6.88	13.1	4.56
29	17	17	24.13	13.34	15.71	5.76	13.22	6.82
30	18	18	25.15	14.47	16.33	5.46	11.96	6.48
31	19	19	20.57	14.56	26.56	8.02	13.76	4.24
32	20	20	25.91	15.93	20.35	5.46	11.06	5.18
33	21	21	29.11	15.13	23.13	5	12.34	4.68
34	22	22	26.03	17.69	24.92	8.22	14.8	6.18
35	23	23	32.91	16.65	24.1	5.22	13.3	5.38
36	24	24	33.79	18.97	22.15	5.98	13.48	7.02
37	25	25	33.48	21.03	22.25	6.32	12.48	7.3
38	26	26	31.81	21.79	26.26	6.66	11.88	5.96
39	27	27	27.99	23.84	31.48	8.78	12.12	5.46
40	28	28	44.06	20.19	26.07	5.06	14.48	6.7
41	29	29	36.69	22.41	32.07	6.74	13.76	5.64
42	30	30	40.27	22.89	31.27	5.24	11.64	5.04
43	31	31	36.82	26.74	31.07	7.04	11.72	6.14
44	32	32	33.35	29.28	34.88	8.56	11.4	5.8
45	33	33	36.09	31.54	31.35	8.78	11.76	7.4
46	34	34	39.37	25.07	46.18	7.06	13.72	4.18
47	35	35	50.33	23.27	42.22	5.26	14.9	4.62
48	36	36	45.77	28.44	38.08	6.36	12.72	5.64
49	37	37	51.22	29.14	35.13	6.02	13.36	6.7
50	38	38	45.22	35.28	33.73	7.24	11.08	7.3
51	39	39	40.84	33.38	47.22	9	13.04	5.42
52	40	39.99	40.04	34.73	50.46	8.6	11.62	4.68
53	15	15	17.42	20.94	7.58	7.3	6.72	13.88
54	16	16	21.42	18.02	9.16	5.94	8.32	11.52
55	17	17	23.5	22.04	7.72	5.24	6.46	13.86
56	18	18	27.91	19.81	8.96	4.36	7.46	11.86
57	19	19	29.65	21.75	8.89	4.86	8	14.32
58	20	20	25.36	29.17	8.6	5.88	5.7	14.98
59	21	21	27.12	29.89	9.13	5.14	5.24	13.22
60	22	22	28.52	30.09	10.03	5.48	5.88	13.38
61	23	23	29.51	31.09	10.78	5.08	5.46	11.86
62	24	24	30.23	34.06	10.78	6.04	6	14.52
63	25	25	38.26	28.57	11.99	4.94	7.96	13.82
64	26	26	27.34	37.89	14.01	7.22	5.62	11.32
65	27	27	28.55	40.59	13.87	7.28	5.5	12.16
66	28	28	36.86	34.8	14.25	5.2	6.36	11.36
67	29	29	37.4	39.07	13.57	6.08	6.6	14.32
68	30	30	42.06	35.84	14.97	5.48	7.56	13.2
69	31	31	52.74	35.16	13.34	4.04	7.44	14.54
70	32	31.99	55.43	32.15	15.95	4.14	8.98	12.8
71	33	33	46.5	40.63	15.71	5.54	7.42	14.18
72	34	34	36.53	50.07	17.6	7.86	6.2	13.26
73	35	35	48.5	41.22	18.06	5.42	7.5	12.38
74	36	36	41.75	53.01	16.98	5.84	5.04	12.02
75	37	37	48.71	50.99	16.4	5.78	6.26	14.86
76	38	38	42.41	57.17	18.25	7.56	6.08	14.56
77	39	39	45.56	51.42	21.15	6.9	6.84	12.16
78	40	39.99	70.79	39.95	19.63	4	8.96	12.92
79	15	15	8.81	26.06	14.12	14.48	4.62	5.94
80	16	16	10.15	27.73	13.48	13.02	4.56	6.66
81	17	17	10.35	26.19	18.05	14.7	5.64	5.46
82	18	18	13.5	27.41	14.3	12.08	6	8.06
83	19	19	11.2	36.18	15.63	14.88	4.28	7.18
84	20	20	13.89	26.29	22.12	13.12	7.08	5.42
85	21	21	13.27	29.99	23.65	14.48	6.34	5.24
86	22	22	13.46	41.95	17.11	14	4.2	7.54
87	23	23	16.8	32.73	20.93	13.96	7.28	7.7
88	24	24	17.58	38.2	18.51	11.34	5.2	7.62
89	25	25	18.4	34.43	23.62	12.62	6.9	6.72
90	26	26	15.98	51.67	18.99	14.34	4.12	8.36
91	27	27	20.33	36.74	25.13	13.8	7.86	7.68
92	28	28	21.87	39.91	23.09	12.18	6.86	8.16
93	29	29	18.93	51.32	22.74	11.88	4.2	6.84
94	30	30	21.57	43.4	27.26	14.42	7.24	7.82
95	31	31	19	55.17	26.44	13.92	4.54	6.74
96	32	32	26.29	42.16	27.57	12.2	8	8.24
97	33	33	25.68	40.3	34.52	11.42	7.68	5.76
98	34	34	24.5	56.08	25.53	11.96	5.16	8.14
99	35	35	23.37	57.01	29.83	14.44	5.78	7.74
100	36	36	21.89	58.09	35.77	12.52	4.54	5.02
101	37	37	25.75	45.79	44.87	13.78	8	5.12
102	38	38	25.11	57.15	36.94	12.48	5.42	5.66
103	39	39	29.5	56.45	32.92	13.86	7.38	8.7
104	40	40	25.48	63.97	37.55	14.32	5.54	6.48
105	15	15	20.9	13.51	11.38	7.74	14.78	11.3
106	16	16	22.05	13.9	13.04	7.14	14.04	9.46
107	17	17	22.14	16.08	13.03	8.78	14.62	11.7
108	18	18	21.72	17.21	14.98	8.68	13.04	9.6
109	19	19	23.32	17.23	16.8	8.74	14.26	9.18
110	20	20	27.76	17.94	15.36	8.08	15.44	11.58
111	21	21	24.57	20.89	17.11	10.28	14.22	11.24
112	22	22	29.82	18.45	19.37	7.94	15.96	9.42
113	23	23	27.54	20.9	20.96	10.4	16.44	10.24
114	24	24	27.92	22.39	21.8	9.42	13.96	9
115	25	25	31.95	21.39	23.03	8.82	16.16	9.26
116	26	26	35.98	23.96	19.22	7.82	14.42	11.68
117	27	27	32.35	25.89	22.57	10.1	15	11.02
118	28	28	38.32	24.62	22.58	7.5	14.42	9.98
119	29	29	34.41	26.34	26.76	10.02	15.68	9.62
120	30	30	40.88	28.73	21.32	8.16	14.12	12.54
121	31	31	37.66	31.06	23.79	10.52	15.08	12.9
122	32	32	38.51	32.88	23.93	10.3	14.18	12.88
123	33	33	40.86	30.02	28.71	10.24	16.82	11.08
124	34	34	44.81	31.64	26.33	7.88	13.56	10.52
125	35	35	46.02	33.95	25.51	8.42	13.76	12.04
126	36	36	42.48	34.88	30.22	9.98	14.38	10.64
127	37	37	40.64	38.43	30.67	10.8	13.18	10.74
128	38	38	45.62	37.35	30.5	9.64	13.94	11.06
129	39	39	49.36	37.34	30.44	8.72	13.84	11
130	40	40	47.94	39.08	32.47	9.88	14.36	11.16
131	15	15	16.07	18.06	10.31	10.98	10.94	13.22
132	16	16	16.62	22.43	9.23	10.68	8.58	15.24
133	17	17	17.79	23.58	9.86	9.68	7.94	13.84
134	18	18	18.93	26.41	9.61	9.32	7.2	14.78
135	19	19	21.7	24.19	11.12	10.54	10.6	16.54
136	20	20	18.33	29.92	12.4	12.3	7.9	13.92
137	21	21	23.14	29.08	11.45	9.8	8.56	16.02
138	22	22	22.39	30.05	13.5	11.06	8.94	14.32
139	23	23	21.42	36.87	12.72	12.3	7.42	16.22
140	24	24	24.38	33.59	14.28	11.1	8.7	14.88
141	25	25	28.87	32.62	13.95	9.82	9.72	16.5
142	26	26	27.26	34.71	15.86	12.44	10.7	16.8
143	27	27	24.21	42.82	15.91	11.98	7	14.02
144	28	28	28.37	35.3	19.36	11.86	10.48	13.28
145	29	29	29.74	38.31	18.46	11.36	9.64	14.22
146	30	30	28.96	44.55	17.56	11.5	7.92	14.8
147	31	31	32.09	43.28	18.06	11.92	9.58	16.74
148	32	32	36.78	38.06	20.45	9.42	10.34	13.42
149	33	33	34.82	42.95	20.69	11.02	9.84	14.52
150	34	34	36.74	47.68	18.66	9.68	8.14	15.34
151	35	35	42.45	43.76	19.5	9.16	10.1	16.34
152	36	36	39.92	46.11	21.67	11.3	10.9	16.56
153	37	37	37.44	52.01	21.98	10.82	8.4	14.46
154	38	38	37.39	55.21	22.36	12.54	9.06	16.4
155	39	39	36.27	54.32	26.18	12.92	9.18	13.54
156	40	40	42.16	51.59	25.44	11.96	10.78	15.48
157	15	15	11.87	18.62	14.92	14.18	9.54	7.74
158	16	16	12.45	19.7	16.53	14.78	9.84	7.56
159	17	17	12.32	21.79	18.35	16.78	9.8	7.5
160	18	18	14.55	20.36	20.09	16.08	12.38	7.86
161	19	19	16.37	20.15	21.41	14.32	12.82	7.46
162	20	20	16.9	23.69	19.45	16.74	12.86	10.14
163	21	21	16.37	27.88	19.29	15.56	9.5	9.16
164	22	22	18.3	25.14	23.16	15.66	12.32	8.48
165	23	23	20.81	26.26	21.49	14.22	12.36	9.78
166	24	24	18.57	30.57	23.72	16.54	10.52	8.86
167	25	25	21.09	31.08	22.64	16.04	11.62	10.54
168	26	26	22.56	30.18	25.14	13.58	11.02	8.54
169	27	27	23.24	32.45	25.02	15.42	11.9	10.1
170	28	28	20.69	37.63	27.29	16.48	9.32	8.5
171	29	29	23.91	38.37	24.72	14.5	9.5	9.96
172	30	30	23.53	38.69	28.53	16.4	10.44	9.34
173	31	31	27.07	36.93	28.53	15.46	12.26	10.38
174	32	32	24.97	40.62	31.41	15.88	10.24	8.66
175	33	33	24.28	40.65	37.01	16.78	10.46	7.3
176	34	34	23.96	45.33	36	16.86	9.1	7.46
177	35	35	27.91	45.75	31.84	14.68	9.38	8.98
178	36	36	26.8	44.55	39.39	15.94	10.02	7.24
179	37	37	31.81	43.5	35.53	15.02	11.88	9.42
180	38	38	34.82	46.06	31.86	13.62	11.2	10.8
181	39	39	32.94	45.23	39.23	13.72	10.8	7.98
182	40	40	32.15	48.69	40.1	16.88	11.84	9.3
183	15	15	16.26	16.37	11.74	11.68	13.24	12.24
184	16	16	16.82	17.17	13.39	12.46	13.9	11.64
185	17	17	18.7	20.81	11.02	10.1	10.18	13.44
186	18	18	18.23	21.82	13.19	12.8	11.84	13.38
187	19	19	20.12	21.43	14.59	13.18	13.98	13.74
188	20	20	20.85	22.26	16.01	12.7	13.44	12.38
189	21	21	23.95	22.51	15.79	10.46	12.86	12.2
190	22	22	22.02	27.53	15.62	12.36	10.86	13.22
191	23	23	25.84	26.35	15.97	10.78	12.04	13.56
192	24	24	25.85	24.85	20.52	11.48	13.74	10.76
193	25	25	26.41	26.27	21.45	11.62	13.36	10.6
194	26	26	28.8	29.72	18.45	11.06	12.18	13.34
195	27	27	27.78	31.72	20.28	11.6	11.34	12
196	28	28	29.5	32.69	20.56	12.34	12.5	13.48
197	29	29	35.28	30.46	20.68	10.22	13.88	13.68
198	30	30	30.25	37.32	21.28	12.66	11.3	13.68
199	31	31	29.66	38.15	23.95	12.8	10.88	11.76
200	32	32	29.3	38.46	27.19	13.18	10.94	10.28
201	33	33	30.95	41.37	25.49	12.68	10.3	11.38
202	34	34	31.4	40.85	28.56	13.5	11.32	10.78
203	35	35	36.96	39.46	27	13.02	13.78	13.46
204	36	36	38.01	41.74	26.64	11.76	12.04	12.76
205	37	37	35.14	47.29	27.32	13.18	10.62	12.66
206	38	38	40.79	42.49	29.03	12.78	13.92	13.62
207	39	39	43.72	41.97	29.82	10.46	12.54	11.72
208	40	40	37.93	46.75	33.79	12.98	11.6	10.62