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	-0.000292	0	-0.641
lnL	0.35005	0	2283.005
lnK	0.400287	0	2571.589
lnM	0.249813	0	1783.123
lnLlnL	-0.020194	0	-788.293
lnKlnK	-0.021163	0	-795.367
lnMlnM	-0.017222	0	-686.902
lnLlnK	0.024134	0	564.864
lnLlnM	0.016298	0	389.711
lnKlnM	0.018128	0	427.076

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	14.4	18.2	11.78	9.06	7.86	8.2
2	16	16	16.04	17.49	13.86	7.38	7.62	6.26
3	17	17	18.4	19.6	12.29	7.54	8	8.66
4	18	18	21.28	20.4	11.9	6.18	7.42	8.74
5	19	19	23.68	18.45	14.8	6.04	9.26	7.5
6	20	20	19.88	23.11	16.11	8.42	8.06	7.7
7	21	21	20.53	24.77	16.79	8.84	8.1	8
8	22	22	25.88	26.84	13.18	6.1	6.68	9.64
9	23	23	29.43	23.64	15.91	6.68	9.88	9.84
10	24	24	30.3	25.71	15.87	6.36	8.82	9.72
11	25	25	25.45	26.88	21.76	7.8	8.36	6.7
12	26	26	30.49	24.26	23.35	6.26	9.36	6.12
13	27	27	32	29.27	19	6.84	8.68	9.02
14	28	28	29.12	29.08	24.99	7.6	8.7	6.5
15	29	29	25.88	38.61	21.95	9.78	6.98	8.48
16	30	30	25.4	37.13	27.21	9.38	6.86	6.18
17	31	31	35.54	32.28	24.19	7.64	9.78	8.58
18	32	32	32.27	36.3	26.01	8.58	8.54	7.9
19	33	33	40.79	34.33	23.4	6.7	9.38	9.2
20	34	34	32.69	41.57	26.35	9.82	8.46	9.04
21	35	35	32.28	41.78	29.75	9.74	8.22	7.66
22	36	36	36.16	45.53	25.06	8.24	7.18	9.06
23	37	37	37.3	37.63	35.62	8.7	9.84	6.56
24	38	38	34.93	41.91	36.63	9.8	9.04	6.62
25	39	39	46.42	39.8	29.89	6.44	8.82	7.72
26	40	40	35.8	52.7	30.62	9.9	7.18	8.5
27	15	15	21.02	10.41	17.36	5.5	14.36	4.92
28	16	16	19.36	12.15	19.39	7.18	14.2	5.12
29	17	17	22.24	14.87	14.65	6.46	11.86	7.54
30	18	18	19.32	15.32	21.24	8.7	13.06	5.56
31	19	19	25.59	14.64	19.4	5.48	12.08	5.42
32	20	20	25.92	16.06	20.06	6.34	12.72	6.12
33	21	21	27.91	16.68	20.73	5.96	12.48	6.04
34	22	22	29.41	16.18	24.55	5.32	12.28	4.7
35	23	23	32.39	17.29	23.07	5.56	13.3	5.92
36	24	24	25.97	20.69	27.4	8.26	12.34	5.54
37	25	25	34.91	17.68	28.16	5.22	13.28	4.8
38	26	26	30.42	21.39	28.78	7.32	12.66	5.58
39	27	27	35.84	22.61	24.49	6.26	12.3	7
40	28	28	41.99	21.48	25.03	5.24	13.18	6.88
41	29	29	39.63	23.2	27.28	6.46	13.86	7.16
42	30	30	34.92	23.62	36.05	6.22	11.26	4.28
43	31	31	42.61	22.79	33.36	5.48	13.08	5.22
44	32	32	39	28.01	30.25	8.12	13.7	7.82
45	33	33	33.04	28.44	42.15	8.54	11.64	4.58
46	34	34	42.06	24.82	42.81	6.66	14.16	4.66
47	35	35	39.92	28.4	41.09	6.46	11.02	4.46
48	36	36	37.39	30.17	45.74	8.66	12.74	4.88
49	37	36.99	38.09	30.73	48.35	7.6	11.18	4.1
50	38	38	42.67	35.29	36.46	8.96	12.8	7.7
51	39	39	49.56	32.96	36.91	7.7	14.22	7.78
52	40	40	59.21	30.65	36.42	5.96	14.78	7.54
53	15	15	23.43	16.29	7.54	4.6	8.06	12.48
54	16	16	20.4	22.03	7.37	5.9	6.16	13.96
55	17	17	23.08	20.16	8.9	6.7	8.98	14.68
56	18	18	26.46	24.54	7.14	4.1	5.12	13.68
57	19	19	25.76	26.63	7.97	4.64	5.1	13.18
58	20	20	23.35	30.05	9.12	6.52	5.54	14.08
59	21	21	33.22	23.69	9.87	4.42	7.52	13.16
60	22	22	23.4	32.52	11.47	7.94	6.16	13.12
61	23	23	27.43	29.96	12.36	7.94	8.18	14.48
62	24	24	40.21	24.64	12.05	4.16	8.46	12.26
63	25	25	36.68	28.24	12.78	5.34	8.3	13.18
64	26	26	32.06	31.93	14.56	6.34	7.28	11.46
65	27	27	36.9	31.93	14.08	6.2	8.4	13.76
66	28	28	34.19	36.98	14.35	7.14	7.44	14.16
67	29	29	31.25	43.7	14.51	6.96	5.36	12.26
68	30	30	38.44	39.87	14.4	4.86	5.32	11.02
69	31	31	45.29	40.96	12.9	4.12	5.3	12.9
70	32	32	40.24	37.41	18.75	6.6	8.22	11.58
71	33	33	46.2	39.15	16.65	6.28	8.72	14.9
72	34	33.99	56.99	35.35	16.76	4.46	8.94	13.44
73	35	35	43.37	47.26	17.12	6.04	6.24	12.88
74	36	36	47.47	42.79	19.46	6.52	8.42	13.3
75	37	37	48.35	45.61	19.22	6.78	8.3	14.34
76	38	38	61.76	40.06	19.03	4.08	7.76	11.64
77	39	39	48.65	48.38	21.23	7.18	8.26	13.6
78	40	40	42.95	63.45	18.79	7.56	5.46	14.28
79	15	15	10.95	21.51	13.51	14.98	7.74	8.36
80	16	16	10.75	24.69	14.62	14.76	6.34	7.34
81	17	17	11.22	28.53	14.13	14.8	5.64	8.06
82	18	18	11.03	28.57	18.11	12.54	4.68	5
83	19	19	13.98	30.49	14.39	12.66	5.78	8.74
84	20	20	15.72	24.91	19.98	11.88	7.9	6.4
85	21	21	15.02	31.2	18.49	12.94	6.26	7.24
86	22	22	16.3	31.34	19.57	13.4	7.1	7.72
87	23	23	12.72	41.15	22.77	14.56	4.18	5.24
88	24	24	19.22	31.5	21.57	11.64	7.44	7.26
89	25	25	18.72	32.34	25.32	12.58	7.56	6.3
90	26	26	18.99	42.66	19.29	12.38	5.46	8.68
91	27	27	20.27	42.71	20.29	11.02	5.24	7.86
92	28	28	20.88	38.89	25.63	12.3	6.76	6.9
93	29	29	20.18	47.56	23.07	14.06	5.86	8.58
94	30	30	18.97	44.19	32.18	13.82	5.84	5.3
95	31	31	21.15	54.43	23.06	13.14	4.94	8.48
96	32	31.99	20.57	63.47	21.98	13.44	4.08	8.88
97	33	33	21.96	49.02	32.35	14.18	6.3	6.42
98	34	34	18.9	61.18	33.11	14.08	4.04	5.2
99	35	35	29.55	47.29	27.98	11.5	7.56	8.76
100	36	36	25.32	52.9	33.02	12.28	5.9	6.42
101	37	37	22.51	54.76	41.82	14.74	5.92	5.08
102	38	38	21.44	68.91	35.76	14.72	4.26	5.76
103	39	39	25.97	61.48	34.99	13.36	5.54	6.72
104	40	40	31.4	54.99	34.53	12.62	7.46	8.06
105	15	15	16.99	14.33	13.6	10.2	14.24	9.46
106	16	16	20.89	13.56	14.55	8.78	16.68	9.6
107	17	17	19.19	16.04	15.79	10.06	14.2	9.04
108	18	18	23.29	16.59	14.49	9.92	16.9	12.38
109	19	19	20.77	18.23	17.94	10.84	14.44	9.2
110	20	20	25.67	18.34	16.4	9.06	15.38	10.96
111	21	21	27.86	19.23	16.54	9.46	16.72	12.48
112	22	22	28.86	19.35	18.72	8.28	15.14	9.84
113	23	23	25.87	22.13	20.8	10.18	13.98	9.4
114	24	24	29.54	23.05	19.33	10.76	16.46	12.66
115	25	25	30.17	23.82	20.91	10.26	15.48	11.28
116	26	26	31.24	24.41	22.39	10.92	16.68	11.54
117	27	27	38.97	23.86	20.2	8	16.28	12.38
118	28	28	37.9	25.42	21.79	8.38	15.32	11.48
119	29	29	36.13	28.34	22.38	8.72	13.26	10.94
120	30	30	42.32	26.73	22.81	8.46	16.6	12.5
121	31	31	37.57	27.64	28.66	10.06	16.5	9.88
122	32	32	42.08	30.36	24.12	9.24	15.5	12.7
123	33	33	44.9	31.06	24.12	8.44	14.88	12.52
124	34	34	49.15	30.08	25.35	7.62	15.52	11.86
125	35	35	43.39	33.36	28.27	9.98	15.54	11.8
126	36	36	49.8	32.53	27.44	7.72	14.56	11.12
127	37	37	42.86	37.5	29.68	9.96	13.32	10.96
128	38	38	53.08	34.31	28.65	7.32	13.98	10.8
129	39	39	48.45	37.15	31.44	9.68	15.12	11.5
130	40	40	49.82	37.22	33.32	9.7	15.62	11.12
131	15	15	15.6	19.63	9.51	12.24	10.68	15.66
132	16	16	17.05	23.24	8.52	10	7.94	16.16
133	17	17	15.73	25.47	10.38	12.28	7.96	14.3
134	18	18	19.58	24.32	10.31	10.3	9.12	15.64
135	19	19	17.12	31.63	10.43	12.72	7.06	16.28
136	20	20	20.53	28.07	11.7	12.06	9.54	16.7
137	21	21	23.36	28.52	11.62	9.54	8.62	15.5
138	22	22	23.87	29.87	12.56	10.14	8.9	15.42
139	23	23	25.97	29.81	13.32	10.68	10.38	16.74
140	24	24	25.16	31.74	14.83	12.42	10.8	16.52
141	25	25	24.74	37.48	14	10.1	7.08	14.1
142	26	26	27.79	35.32	15.11	9.42	8.12	13.78
143	27	27	28.31	39.25	14.65	10.82	8.42	16.78
144	28	28	27.97	35.68	19.41	11.86	10.18	13.02
145	29	29	28.35	42.19	17.18	11.68	8.36	15.04
146	30	30	27.94	44.17	18.61	11.58	7.72	13.34
147	31	31	32.95	42.76	17.76	11.06	9.3	16.34
148	32	32	32.81	44.67	18.88	12.2	9.7	16.7
149	33	33	30.76	48.51	20.49	12.74	8.52	14.68
150	34	34	34.22	45.09	22.06	12.76	10.54	15.28
151	35	35	31.81	52.75	21.71	12.88	8.12	14.42
152	36	36	38.55	48.05	21.38	11.7	10.32	16.72
153	37	37	38.14	45.93	25.58	11.76	10.8	13.44
154	38	38	35.58	56.38	23.17	12.54	8.34	14.84
155	39	39	38.2	50.57	27.06	12.56	10.32	13.46
156	40	40	43.72	51.9	24.08	10.3	9.62	14.84
157	15	15	13.54	17.6	13.49	14.2	11.92	10.18
158	16	16	14.22	17.83	15.93	14.82	12.98	9.26
159	17	17	13.88	22.06	15.11	15.78	10.46	10.2
160	18	18	16.39	21.27	15.82	14.48	12.18	10.78
161	19	19	14.14	25.19	18.71	15.92	9.22	8.18
162	20	20	16.52	25	18.52	15.76	11.06	9.84
163	21	21	16.9	24.9	21.89	16.62	12.04	8.76
164	22	22	17.24	26.01	23.96	15.3	10.78	7.42
165	23	23	19.59	29.3	19.81	14.44	10.28	10.18
166	24	24	19.35	31.89	20.97	16.16	10.26	10.5
167	25	25	21.31	27.27	27.34	13.66	11.68	7.28
168	26	26	22	30.89	25.13	15.26	11.7	9.32
169	27	27	23.83	33.78	22.73	14.46	10.96	10.92
170	28	28	23.82	32.02	28.5	15.32	12.36	8.86
171	29	29	27.52	31.81	26.97	13.34	12.86	9.76
172	30	30	25.47	34.98	29.71	15.94	12.54	9.5
173	31	31	25.24	42.65	25.38	15.44	9.52	10.96
174	32	32	26.44	38.21	31.76	16.92	12.54	9.74
175	33	33	25.95	39.09	35.64	15.98	11.28	7.86
176	34	34	25.82	41.52	36.87	16.22	10.6	7.62
177	35	35	29.08	42.73	33.3	16.16	11.74	9.84
178	36	36	30.3	44.86	32.59	13.94	10.04	9.14
179	37	37	29.85	48.66	32.82	14.18	9.12	9.06
180	38	38	31.02	46.09	37.47	15	10.76	8.58
181	39	39	35.19	48.68	31.96	13.2	10.3	10.56
182	40	40	31.98	51.57	37.02	14.8	9.64	8.9
183	15	15	17.15	15.65	11.72	10.7	13.62	11.96
184	16	16	17.17	18.15	11.97	11.38	12.18	12.42
185	17	17	18.12	18.45	13.71	11.94	13.36	11.84
186	18	18	16.14	23.03	14.35	13.8	10.38	11.32
187	19	19	20.7	21.04	14.45	12.22	13.7	13.32
188	20	20	19.89	21.19	18.39	13.1	13.9	10.26
189	21	21	21.02	26.63	14.63	11.86	10.26	12.98
190	22	22	23.1	26.69	15.34	10.54	10.16	12.18
191	23	23	22.88	30.04	15.48	12.08	10.02	13.66
192	24	24	23.7	27.25	20.04	13.08	12.68	11.38
193	25	25	27.55	27.78	18.63	11.7	13.24	13.24
194	26	26	25.23	33.54	18.4	13.04	10.66	13.5
195	27	27	25.84	31.97	22.08	13.94	12.4	11.98
196	28	28	27.98	33.25	21.52	13.96	13.02	13.58
197	29	29	29.74	31.14	25.05	11.88	12.86	10.38
198	30	30	34.94	34.38	19.9	10.08	11.74	13.96
199	31	31	30.61	34.72	26.43	12.18	12	10.34
200	32	32	36.11	35.76	22.93	11.42	13.2	13.92
201	33	33	38.67	35.55	23.78	10.4	13.12	13.16
202	34	34	32.35	41.46	26.84	13.24	11.3	11.78
203	35	35	32.81	44.59	26.42	13.28	10.58	12.24
204	36	36	43.13	37.21	26.82	10.12	13.76	12.64
205	37	37	36.36	44.26	28.76	11.62	10.54	10.94
206	38	38	36.75	43.63	32.09	12.7	11.86	10.64
207	39	39	41.48	42.17	31.74	10.26	11.5	10.04
208	40	40	42.29	43.73	32.25	12.56	13.8	12.34