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	-0.000507	0	-1.208
lnL	0.350202	0	2349.837
lnK	0.400261	0	2763.774
lnM	0.249859	0	2196.077
lnLlnL	-0.020177	0	-767.976
lnKlnK	-0.02116	0	-871.308
lnMlnM	-0.017148	0	-763.658
lnLlnK	0.02415	0	645.637
lnLlnM	0.016182	0	467.586
lnKlnM	0.018102	0	515.648

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	15.8	18.53	10.15	6.84	6.48	8.22
2	16	16	19.32	17.25	11.09	7.2	9.4	9.88
3	17	17	19.93	19.93	10.83	6.16	7.04	9.02
4	18	18	21.66	19.22	12.71	7.28	9.58	9.74
5	19	19	19.1	24.59	12.78	8.08	6.86	9.26
6	20	20	19.23	25.51	14.56	8.66	7.1	8.58
7	21	21	19.2	26.1	17	9.44	7.52	7.78
8	22	22	20.39	29.43	15.71	8.84	6.56	8.58
9	23	23	24.34	26.75	16.89	8.76	8.96	9.62
10	24	24	21.87	32.21	17.46	8.72	6.32	8.12
11	25	25	26	30.99	17.11	6.52	6.06	7.62
12	26	26	25.97	31.3	19.59	8	7.34	7.96
13	27	27	26.11	30.4	23.49	9	8.6	7.28
14	28	28	26.43	33.44	23.04	8.86	7.68	7.44
15	29	29	27.98	32	26.11	9.24	9.02	7.16
16	30	30	32.07	37.56	19.54	7.02	6.66	8.98
17	31	31	30.88	33.12	28.09	8.74	9.2	6.98
18	32	32	26.22	40.07	29.89	9.8	6.8	6
19	33	33	32.87	40.76	24.03	9.22	8.18	9.52
20	34	34	38.93	36.87	25	6.5	7.92	7.82
21	35	35	42.81	36.54	25.02	7.22	9.94	9.7
22	36	36	39.49	40.21	26.78	8.74	9.78	9.86
23	37	37	39.94	42.93	26.57	8.46	8.88	9.76
24	38	38	39.34	41.38	31.72	6.52	7.02	6
25	39	39	40.82	46.23	28.25	6.32	6.24	6.96
26	40	40	38.74	43.77	36.29	9.88	9.78	7.62
27	15	15	18.09	13.04	14.54	7.36	12.36	6.8
28	16	16	20.24	13.63	15.04	6.08	11.06	6.16
29	17	17	21.55	14.21	16.42	6.08	11.34	5.98
30	18	18	25.45	12.97	19.31	5.2	13.14	5.14
31	19	19	25.21	12.92	24.61	5.96	14.96	4.38
32	20	20	21.33	16.44	25.28	7.9	12.26	4.62
33	21	21	27.43	16.01	22.74	6.56	14.12	5.84
34	22	22	29.66	18.43	19.53	5.72	11.44	6.68
35	23	23	32.67	19.69	18.45	5.72	11.86	8
36	24	24	26.06	19.72	29.58	8.16	12.94	5.02
37	25	25	26.4	20.43	32.38	7.44	11.5	4.18
38	26	26	29.11	19.61	35.6	8.16	14.84	4.6
39	27	27	27.88	23.28	32.94	8.62	12.18	5.06
40	28	28	35.39	22.29	29.47	6.3	12.4	5.58
41	29	29	32.55	27.38	27.11	8.74	12.24	7.74
42	30	30	46.54	21.92	27.84	5.34	14.8	6.98
43	31	31	43.39	25.01	27.89	6.16	13.46	7.4
44	32	32	36	27.37	35.05	8.9	14.04	6.56
45	33	32.99	34.49	25.89	46.62	8.94	14.32	4.48
46	34	34	40.72	30.03	32.42	7.56	12.36	7.06
47	35	35	46.15	26.04	39.02	6.48	14.52	5.64
48	36	36	41.14	31.23	37.69	7.2	11.38	5.7
49	37	37	49.86	26.55	42.64	5.52	13.24	4.74
50	38	38	40.76	36.75	36.38	8.94	11.54	7.3
51	39	39	50.33	30.22	41.77	5.8	12.08	5.16
52	40	40	54.41	32.97	36.05	6.28	12.94	7.28
53	15	15	19.45	19.57	7.28	5.18	5.88	11.76
54	16	16	25	19.11	7.06	4.58	7.18	14.48
55	17	17	27.64	18.58	8.09	4.66	8.5	14.12
56	18	18	23.71	24.09	8.29	4.6	5.16	11.32
57	19	19	31.04	21.1	8.8	4.5	8.1	14.16
58	20	20	26.74	25.42	9.7	6.2	7.52	14.6
59	21	21	25.32	27.77	10.91	6.4	6.56	12.3
60	22	22	25.93	27.87	12.47	6.84	7.18	11.56
61	23	23	27.19	33.19	10.89	6.66	6.02	13.96
62	24	24	33.5	30.87	10.9	4.88	6.14	13.06
63	25	25	37.14	28.15	12.66	4.56	7.22	11.56
64	26	26	30.38	35.27	13.55	7.28	6.98	13.44
65	27	27	42.89	31.23	12.21	4.06	6.74	12.72
66	28	28	38.4	31.64	15.5	6.02	8.64	12.5
67	29	29	33.4	37.97	16.15	7.04	6.92	11.82
68	30	30	33.28	43.64	15.18	6.86	5.7	12.34
69	31	31	42.58	38.23	15.17	5.92	7.68	14.24
70	32	32	49.09	32.75	17.86	4.74	8.72	11.12
71	33	33	40.62	50.03	13.98	5.86	5.24	14.68
72	34	33.99	56.84	35.93	16.43	4	7.84	12.3
73	35	35	42.01	48.45	17.2	7.12	6.88	14.54
74	36	36	35.87	56.6	18.76	7.9	5.28	12.1
75	37	37	41.5	53.42	18.65	6.5	5.52	11.9
76	38	38	47.87	55.3	16.51	5.62	5.42	14.04
77	39	39	47.52	56.52	17.72	5.58	5.2	12.72
78	40	40	43.2	55.61	22.23	7.42	6.3	11.58
79	15	15	10.91	20.82	14.25	12.54	6.7	6.54
80	16	16	10.96	26.94	12.54	13.16	5.22	8.02
81	17	17	12.6	29.65	11.37	11.02	4.6	8.88
82	18	18	14.15	24.99	15.29	12.3	7.2	8.02
83	19	19	14.72	24.91	17.94	12.58	7.74	7.12
84	20	20	13.53	26.91	22.2	13.28	6.76	5.3
85	21	21	14.48	28.86	21.93	11.54	5.86	5.06
86	22	22	13.11	38.32	20.18	12.98	4.2	5.58
87	23	23	13.57	43.36	19.17	14.76	4.3	7.02
88	24	24	18.5	30.02	24.53	12.06	7.8	6.18
89	25	25	18.18	34.09	24.41	11.88	6.48	6
90	26	26	17.24	35.51	29.09	14.04	6.86	5.42
91	27	27	17.31	38.97	29.22	13.32	5.86	5.14
92	28	28	19.94	36.24	30.54	12.76	7.22	5.52
93	29	29	17.37	49.01	27.56	14.94	5.04	6.2
94	30	30	19.24	51.08	25.48	14.96	5.42	7.68
95	31	31	21.28	47.67	27.58	12.16	5.38	6.4
96	32	32	24.64	48.85	24.39	11.04	5.64	7.98
97	33	33	23.21	44.49	34.43	14.56	7.74	6.54
98	34	34	22.39	61.65	25.45	14	4.86	8.6
99	35	35	26.48	47.49	32.48	13.6	7.82	7.64
100	36	36	22.65	62.35	30.7	13.54	4.7	6.76
101	37	37	27.62	53.13	32.19	13.98	7.4	8.34
102	38	38	22.81	63.72	36.42	13.6	4.64	5.6
103	39	39	25.06	62.97	35.62	13.76	5.32	6.5
104	40	40	27.75	64.25	32.91	13.72	5.84	8.02
105	15	15	19.71	14.55	10.94	8.02	13.1	11.46
106	16	16	22.33	14.41	12.13	8.38	16.04	12.28
107	17	17	20.1	15.44	15.76	10.56	16.46	10.04
108	18	18	21.42	17.58	14.77	10.36	14.94	11.46
109	19	19	24.13	18.19	14.77	9.6	15.3	12.22
110	20	20	27.59	18.36	14.93	7.76	14.32	11.42
111	21	21	29.68	19.08	15.45	7.12	13.68	10.96
112	22	22	27.68	20.68	17.8	9.76	15.72	11.72
113	23	23	32.31	20.56	17.48	7.3	14.2	10.74
114	24	24	28.01	25.22	18.05	10.72	13.86	12.84
115	25	25	31.34	24.14	19.49	8.6	13.36	10.74
116	26	26	39.09	21.96	19.85	7.38	16.62	11.7
117	27	27	32.73	26.47	21.48	10.66	15.64	12.5
118	28	28	32.81	28.58	21.89	10.16	13.66	11.68
119	29	29	38.52	24.28	26.29	8.38	16.48	9.38
120	30	30	35.73	30.71	22.86	10.56	14.42	12.76
121	31	31	37.18	27.6	29.13	9.6	15.58	9.14
122	32	32	38.4	29.25	28.8	10.2	16.06	10.22
123	33	33	45.21	31.53	23.38	7.72	13.48	11.98
124	34	34	46.93	33.3	22.99	7.68	13.14	12.7
125	35	35	39.42	34.96	29.81	10.06	13.24	9.98
126	36	36	41.06	32.83	34.81	10.56	15.7	9.16
127	37	37	46.96	35.28	28.96	8.5	13.6	10.72
128	38	38	43.57	35.51	35.09	9.92	14.42	9.14
129	39	39	45.6	35.1	37.27	10.16	15.8	9.2
130	40	40	46.54	37.32	36.32	10.08	14.94	9.64
131	15	15	17.39	18.94	8.71	9.48	9.8	15.28
132	16	16	18.67	20.2	9.23	9.04	9.42	14.8
133	17	17	16.5	22.8	11.37	11.8	9.22	13.06
134	18	18	16.06	28.23	10.84	12.68	7.46	14.38
135	19	19	21.33	27.58	9.5	9	7.6	16.64
136	20	20	20.98	26.71	12.18	12.48	10.74	16.9
137	21	21	24.19	26.71	12.17	9.16	9.32	14.68
138	22	22	22.14	29.22	14.24	10.96	9.04	13.16
139	23	23	22.61	31.7	14.58	12.58	9.66	15.06
140	24	24	26.01	30.96	14.72	10.78	10.04	15.04
141	25	25	24.36	38.96	13.56	10.92	7.18	15.54
142	26	26	26.11	37.88	14.85	11.58	8.54	16.06
143	27	27	26.09	38.18	16.88	11.12	8.12	13.26
144	28	28	32.25	35.26	16.46	10.28	10.58	16.2
145	29	29	32.28	35.87	18.25	10.46	10.56	14.62
146	30	30	33.08	38.67	18.03	10.64	10.12	15.52
147	31	31	33.49	43.09	17.22	10.78	9.16	16.84
148	32	32	32.28	45.13	19.01	9.76	7.52	13
149	33	33	32.08	48.72	19.3	10.5	7.34	13.64
150	34	34	35.12	46.09	20.69	10.7	8.88	14.24
151	35	35	31.46	56.46	20.09	12.64	7.26	15.3
152	36	36	39.47	49.03	20.19	9.96	8.82	15.66
153	37	37	36.2	48.95	25	12.28	9.84	13.56
154	38	38	40.87	54.95	20.13	10.04	8.1	16.5
155	39	39	41.35	52.63	23.1	10.32	8.88	14.62
156	40	40	39.42	60.01	22.49	10.1	7.04	13.96
157	15	15	13.26	18.58	12.79	13.92	10.7	10.38
158	16	16	14.73	18.72	14.06	13.76	11.86	10.38
159	17	17	14	18.72	19.27	15.98	12.98	7.84
160	18	18	13.79	21.48	19.98	15.68	10.64	7.24
161	19	19	15.63	23.53	17.93	15.6	11.02	9.48
162	20	20	16.59	22.68	21.41	15.12	11.96	8
163	21	21	17.24	23	24.12	15.8	12.86	7.6
164	22	22	17.22	25.78	24.34	15.1	10.74	7.18
165	23	23	17.62	28.25	24.4	15.52	10.18	7.56
166	24	24	19.89	25.45	28.63	15.04	12.86	7
167	25	25	22.22	27.28	25.72	13.5	12.12	8.12
168	26	26	23.22	31.06	23.09	13.6	11.04	9.78
169	27	27	22.41	32.15	26.75	15.52	11.6	9
170	28	28	24.26	35.04	24.18	13.86	10.28	9.94
171	29	29	23.55	39.65	24.05	15.44	9.56	10.76
172	30	30	26.98	35.84	26.38	14.18	11.6	10.4
173	31	31	25.62	36.18	31.87	14.66	11.16	8.1
174	32	32	24.98	40.48	31.56	14.64	9.48	7.94
175	33	33	24.73	46.14	29.69	16.8	9.22	9.68
176	34	34	27.93	38.33	37.25	14.5	11.42	7.38
177	35	35	31.26	43.81	28.97	13.9	10.68	10.86
178	36	36	27.68	42.55	40.34	15.96	11	7.32
179	37	37	31.89	43.99	34.78	15.1	11.82	9.74
180	38	38	32.68	48.47	32.22	13.44	9.66	9.76
181	39	39	31.57	43.13	45.02	15.26	12.08	7.18
182	40	40	36.97	45.76	36.16	13.72	12.2	10.06
183	15	15	15.99	16.88	11.46	10.76	11.54	11.38
184	16	16	15.12	19.94	12.34	12.16	10.04	11.04
185	17	17	18.01	19.14	13.09	10.66	11.34	11.08
186	18	18	17.46	20	15.92	13.74	13.38	10.94
187	19	19	20.59	22.01	13.6	11.62	12.28	13.52
188	20	20	23.7	21	14.77	10.32	13.6	12.86
189	21	21	21.19	26.4	14.66	11.56	10.2	12.74
190	22	22	20.57	26.13	18.49	12.5	10.78	10.12
191	23	23	25.25	27.05	15.83	10.38	10.94	12.84
192	24	24	22.93	30.21	17.97	13.58	11.22	12.92
193	25	25	25.01	30.49	18.44	12.44	11.26	12.72
194	26	26	25.76	32.51	18.74	11.92	10.36	12.38
195	27	27	27.42	31.53	20.81	13.16	12.76	13
196	28	28	27.16	31.77	23.99	13.26	12.6	10.96
197	29	29	31.65	30.11	24.26	10.38	12.58	10.14
198	30	30	31.58	35.62	21.52	12.32	12.22	13.82
199	31	31	32.67	34.7	24.22	11.46	12.2	11.64
200	32	32	33.11	34.03	27.72	12.02	13.3	10.58
201	33	33	32.42	41.43	23.91	13.64	11.68	13.94
202	34	34	35.5	40.43	24.6	11.42	11.2	12.56
203	35	35	36.61	41.41	25.45	10.44	10.32	11.44
204	36	36	32.47	45.82	28.67	13.28	10.12	10.98
205	37	37	37.5	45.92	26.16	12.48	11.24	13.62
206	38	38	35.3	51.33	26.69	13.78	10.14	13.68
207	39	39	38.66	43.54	33.23	12.42	12.34	10.6
208	40	40	38.96	48.87	30.49	12.36	10.82	11.78