Egwald Economics - Production Functions: Estimate the Translog Production Function

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Egwald Economics: Microeconomics

Production Functions

by

Elmer G. Wiens

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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	1.1E-5	0	0.037
lnL	0.349936	0	3279.373
lnK	0.400261	0	3723.355
lnM	0.249763	0	2529.512
lnLlnL	-0.020132	0	-1061.922
lnKlnK	-0.021084	0	-1164.384
lnMlnM	-0.017108	0	-837.98
lnLlnK	0.0241	0	787.44
lnLlnM	0.016233	0	584.732
lnKlnM	0.018	0	570.043

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	13.04	19.58	12.11	9.88	7	7.7
2	16	16	15.12	20.38	11.95	7.98	6.42	7.52
3	17	17	19.1	20.73	10.79	6.28	6.52	8.78
4	18	18	18.05	20.22	14.96	8.58	8.58	7.64
5	19	19	21.85	18.81	15.97	6.06	8.26	6.26
6	20	20	19.13	22.2	18.06	9.42	9.04	7.2
7	21	21	21.93	25.04	15.13	7.96	7.78	8.8
8	22	22	23.45	23.74	17.91	8.22	9.26	8.06
9	23	23	22.62	29.27	16.31	7.3	6.16	7.66
10	24	24	28.01	24.02	19.44	6.12	8.38	6.72
11	25	25	27.79	30.53	16.05	6.84	7	9.32
12	26	26	30.65	26.43	20.3	6.38	8.68	7.4
13	27	27	31.01	33.21	16.46	6.26	6.6	9.42
14	28	28	35.46	28.67	19.72	6.5	9.54	9.26
15	29	29	28.25	34.41	23.08	8.1	7.34	7.34
16	30	30	30.94	39.63	18.97	7.4	6.32	9.4
17	31	31	35.29	32.65	23.99	6.9	8.64	7.76
18	32	32	30.77	32.93	32.3	9.04	9.54	6.1
19	33	33	29.32	41.7	27.13	8	6.04	6.26
20	34	34	36.77	39.61	24.2	8.08	8.46	9.44
21	35	35	41.45	35.71	27.01	7.02	9.56	8.3
22	36	36	40.51	38.72	27.41	7.02	8.46	7.94
23	37	37	38.12	45.67	25.8	8.42	7.78	9.52
24	38	38	40.25	39.69	32.78	8.4	9.76	7.64
25	39	39	37.43	53.47	25.69	8.2	6.16	9.12
26	40	40	39.97	51.55	27.29	7.72	6.54	8.64
27	15	15	19.44	12.73	13.73	6.86	12.9	7.38
28	16	16	21.69	11.29	18.82	6.04	14.88	5.1
29	17	17	19.1	15.71	16.44	8.12	11.68	6.92
30	18	18	23.86	12.96	21.1	6.18	14.48	5.1
31	19	19	22.32	16.82	18.52	7.98	12.72	7.1
32	20	20	25.61	15.79	20.99	7.4	14.94	6.68
33	21	21	23.5	16.24	27.53	7.22	12.74	4.28
34	22	22	26.89	17.06	25.35	6.14	11.98	4.7
35	23	23	33.32	17.31	22.18	5.88	14.52	6.78
36	24	24	30.51	20.07	23.11	6.2	11.6	6.14
37	25	25	26.99	19.74	33.31	8.5	14.04	4.74
38	26	26	33.24	21.32	25.65	7.1	13.68	6.88
39	27	27	32.2	23.67	26.2	8.48	13.92	7.72
40	28	28	31.14	26.11	27.06	7.88	11.08	6.64
41	29	29	43.03	23.97	23.28	5.18	11.78	7.62
42	30	30	40.69	25.29	26.18	5.5	11	6.6
43	31	31	37.26	23.23	38.8	7.46	14.86	5.08
44	32	32	34.73	25.14	42.67	7.6	12.7	4.26
45	33	33	38.89	26.68	37.25	7.64	13.6	5.74
46	34	34	35.56	30.67	37.75	8.44	11.48	5.62
47	35	35	41.48	29.81	35.94	7.9	13.32	6.68
48	36	36	43.21	30.36	36.92	6.54	11.32	5.62
49	37	37	42.63	33.59	35.55	7.96	12.04	7.04
50	38	38	53.95	29.01	36.76	6.24	14.8	7
51	39	39	44.53	35.12	38.46	8.06	12.18	6.84
52	40	40	51.18	33.84	37.47	7.12	13.24	7.34
53	15	15	17.64	19.38	8.29	6.9	7.06	11.98
54	16	16	19.29	20.52	8.67	7.32	7.78	13.4
55	17	17	20.06	22.73	8.94	7.42	7.32	13.72
56	18	18	25.72	21.89	8.58	4.4	6.08	11.44
57	19	19	20.81	26.46	10.35	7.5	6.46	12.18
58	20	20	34.02	21.1	9.5	4.04	8.1	12.94
59	21	21	29.59	25	10.47	5.74	8	13.92
60	22	22	29.13	29.4	10.09	4.44	5.02	11.04
61	23	23	26.75	30.39	12.5	7.4	7.28	12.94
62	24	24	38.07	27.85	10.82	4.36	7.2	13.68
63	25	25	30.9	31.05	13.74	6	6.82	11.12
64	26	26	31.01	32.12	15.04	7.66	8.4	12.82
65	27	27	27.19	40.84	14.61	7.54	5.34	11.18
66	28	28	32.17	33.88	17.47	7.7	8.28	11.28
67	29	29	37.78	33.68	16.43	6.36	8.32	12.1
68	30	30	42.22	37.65	13.94	5.2	6.8	13.68
69	31	31	48.73	35.04	14.64	4.38	7.38	12.88
70	32	32	44.03	37.75	16.61	6.44	8.82	14.48
71	33	33	53.12	36.13	15.85	4.46	8.02	13.22
72	34	34	54.77	37.76	15.99	4	7.08	12.16
73	35	35	37.4	51.49	18.26	7.24	5.68	12.02
74	36	36	38.17	50.7	20.11	7.38	6.04	11.2
75	37	37	47.72	42.72	21.39	6.1	7.94	11.2
76	38	38	48.23	43.57	22.62	6.94	8.94	12.08
77	39	39	41.75	60.72	18.84	7.94	5.84	14.46
78	40	40	53.61	45.98	22.22	5.96	8.16	12
79	15	15	10.8	19.52	15.96	11.66	6.64	5.26
80	16	16	11.3	28.96	10.88	11.7	4.42	8.74
81	17	17	11.42	26.66	15.16	11.76	4.96	6.02
82	18	18	14.25	26.54	13.88	11.02	6.06	8.12
83	19	19	13.2	27.77	17.88	13.68	6.52	6.84
84	20	20	14.34	28.01	19.16	11.66	6.06	5.92
85	21	21	13.96	40.16	14.44	12.56	4.14	8.62
86	22	22	14.99	35.9	18.16	13.3	5.44	7.58
87	23	23	17.32	34.39	18.64	12.94	6.6	8.48
88	24	24	14.79	40.53	21.88	14.78	5.16	6.66
89	25	25	17.21	35.46	24.94	12.86	6.28	5.94
90	26	26	18.3	40.26	22.07	14.56	6.58	8.34
91	27	27	17.22	42.59	25.8	11.98	4.72	5.32
92	28	28	18.77	39.71	29.08	14.62	6.92	6.24
93	29	29	18.19	42.48	31.76	14.24	6	5.28
94	30	30	24.38	40.64	25.2	12.58	7.88	8.64
95	31	31	20.51	40.32	37.57	14.5	7.48	5.08
96	32	32	18.91	53.83	31.27	13.66	4.56	5.4
97	33	33	24.32	52.83	25.03	12.86	5.9	8.88
98	34	34	22.1	58.4	27.97	14	5.1	7.58
99	35	35	24.44	48.47	35.44	12.14	6.2	5.6
100	36	36	24.77	58.76	29.3	12.86	5.32	7.54
101	37	37	27.87	48.93	35.93	11.74	6.92	6.22
102	38	38	22.39	72.67	31.09	14.54	4.16	7.06
103	39	39	23.12	73.21	32.45	14.6	4.3	7
104	40	40	27.05	66.25	32.69	12.7	5.06	7.26
105	15	15	20.34	12.61	13.15	8.2	16.44	9.78
106	16	16	20.56	15.5	12.03	9.12	14.54	12.24
107	17	17	21.73	15.81	13.71	9.88	16.42	12.14
108	18	18	22.13	16.81	15.16	9.42	14.88	10.5
109	19	19	24.13	16.87	16.63	9.74	16.96	10.78
110	20	20	22.4	20.13	16.96	10.88	14.1	10.78
111	21	21	26.14	19.83	17.11	8.52	13.48	10.02
112	22	22	28.82	20.07	17.71	8.78	15.36	11.12
113	23	23	31.22	19.81	19.37	7.36	14.36	9.22
114	24	24	31.77	21.61	19.46	9.14	16.44	11.62
115	25	25	30.25	23.92	20.72	9.04	13.62	10.08
116	26	26	34.86	22.7	21.77	8.8	16.66	10.94
117	27	27	37.72	22.66	22.85	8.04	16.74	10.36
118	28	28	33.65	28.31	21.49	10.24	14.34	12.4
119	29	29	35.49	26.02	26.19	9.44	15.54	9.64
120	30	30	38.36	28.86	22.95	9.64	15.4	12.6
121	31	31	39.84	29.17	24.36	8.32	13.72	10.6
122	32	32	40.45	28.85	27.49	9.1	15.48	10.24
123	33	33	41.91	31.02	26.38	9.68	15.76	11.92
124	34	34	48.27	28.4	28.39	7.38	15.74	9.84
125	35	35	40.87	35.74	27.47	10.58	14.16	12.06
126	36	36	43.15	34.55	30.06	10.98	16.3	12
127	37	37	42.06	35.37	33.34	10.12	14.18	9.5
128	38	38	46.79	36.91	30.06	10.38	15.68	12.48
129	39	39	59.57	33.79	28.06	7.32	16.3	12.68
130	40	40	48.43	38.12	33.32	9.72	14.72	10.78
131	15	15	16.8	19.52	8.73	10.42	9.98	16.08
132	16	16	15.79	20.42	11.25	13	10.98	13.84
133	17	17	16.46	24.96	10.07	11.54	8.08	14.7
134	18	18	18.15	27.11	9.79	10.02	7.14	14.8
135	19	19	21.14	25.04	10.95	9.74	9.12	15.08
136	20	20	18.94	28.45	12.73	12.4	8.78	14.14
137	21	21	23.63	27.07	12.31	10.8	10.52	16.62
138	22	22	22.51	30.63	13.06	10.36	8.24	14.04
139	23	23	24.13	31.36	13.61	11.98	10.06	16.78
140	24	24	25.6	30.43	15.4	11.02	10.28	14.32
141	25	25	22.39	39.79	14.68	12.22	7.1	14.34
142	26	26	25.34	38.53	15.08	10.86	7.58	14.28
143	27	27	27.36	35.74	17.41	12.8	10.68	15.56
144	28	28	29.33	40.03	15.56	9.84	7.8	14.82
145	29	29	29.76	41.83	16.34	9.56	7.32	13.82
146	30	30	29.64	47.09	15.83	10.74	7.12	16.04
147	31	31	32.43	42.76	18.12	11.7	9.66	16.58
148	32	32	33.21	40.36	21.43	10.92	9.92	13.06
149	33	33	32.6	44.62	21.34	12.48	9.86	14.68
150	34	34	39.83	41.49	20.46	10.06	10.96	15.74
151	35	35	41.21	47.04	18.31	9.1	8.9	16.88
152	36	36	36.59	48.47	22.57	12.96	10.64	16.34
153	37	37	40.26	50.1	21.12	9.62	8.5	14.68
154	38	38	39.76	48.45	24.79	11.9	10.78	14.82
155	39	39	36.61	58.16	23.53	12.52	8.3	15.04
156	40	40	38.48	61.52	22.44	11.46	7.54	15.44
157	15	15	12.8	17.75	14.4	16.66	12.96	10.36
158	16	16	12.8	19.82	15.72	16.94	11.58	9.5
159	17	17	13.24	20.96	17.51	15.56	10.36	8
160	18	18	14.89	22.76	16.33	13.14	9.12	8.42
161	19	19	15.85	22.26	19.17	13.52	10.36	7.72
162	20	20	15.32	26.64	18.75	16.98	10.12	9.56
163	21	21	15.55	26.14	22.97	16.8	10.42	7.58
164	22	22	19.08	29.49	17.14	13.2	9.04	10.7
165	23	23	18.55	28.63	22.17	15.16	10.4	8.78
166	24	24	21.36	27.02	23.46	14.44	12.52	9.24
167	25	25	22.45	30.13	21.74	13.8	11.16	10.24
168	26	26	22.25	30.55	25.16	16.34	12.86	10.1
169	27	27	22.07	35.45	23.56	14	9.16	9.26
170	28	28	22.86	37.4	23.86	15.4	9.86	10.46
171	29	29	22.65	40.38	24.74	16.96	9.82	10.92
172	30	30	25.16	33.76	31.98	14	11.32	7.54
173	31	31	24.23	38.82	30.99	15.3	10.04	8.18
174	32	32	26.68	38.77	30.64	16.54	12.18	10.04
175	33	33	27.3	40.43	31.42	14.5	10.44	8.78
176	34	34	27.33	41.64	33.77	15.48	10.78	8.62
177	35	35	32.25	39.21	32.82	13.92	12.64	9.74
178	36	36	32.79	43.55	30.51	13.66	11.18	10.62
179	37	37	28.34	51.01	32.94	16.14	9.24	9.66
180	38	38	30.79	42.91	42.37	15.98	12.36	7.84
181	39	39	33.01	44.77	39.74	14.8	11.82	8.5
182	40	40	33.57	48.32	38.13	15.42	11.48	9.48
183	15	15	15.75	16.36	12.25	11.4	12.46	10.94
184	16	16	16.12	18.62	12.52	13.04	12.58	12.54
185	17	17	17.55	21.13	11.71	11.08	10.18	12.74
186	18	18	18	22.46	12.85	11.58	10.2	12.3
187	19	19	17.78	21.56	17.1	13.94	12.7	10.42
188	20	20	19.72	23.46	15.93	13.78	12.84	12.66
189	21	21	20.81	24.03	17.25	13.16	12.7	11.72
190	22	22	22.18	24.49	18.4	11.82	12.02	10.52
191	23	23	21.49	28.21	18.44	13.52	11.22	11.56
192	24	24	26.15	25.23	19.74	11.34	13.52	11.28
193	25	25	26.78	28.05	19.06	12.1	13.1	12.9
194	26	26	26.99	29.43	20.39	12.48	12.88	12.4
195	27	27	23.68	33.85	22.86	13.56	10.18	10.1
196	28	28	25.7	35.41	21.98	12.86	10.08	11.04
197	29	29	27.75	35.07	22.99	13.46	11.68	12
198	30	30	31.19	32.97	24.55	12.72	13.62	12.04
199	31	31	32.46	35.67	23.45	11.54	11.8	12.08
200	32	32	31.75	36.41	26.46	11.84	11.52	10.48
201	33	33	33.11	37.09	27.39	13.82	13.82	12.34
202	34	34	38.6	35.69	26.55	10.58	13.26	11.74
203	35	35	34.58	39.75	29.19	12.8	12.42	11.16
204	36	36	35.59	43.32	27.51	13.96	12.66	13.5
205	37	37	36.83	46.48	26.31	12.52	10.88	13.28
206	38	38	38.5	47.25	26.8	12.38	11.12	13.54
207	39	39	36.63	48.74	30.2	13.54	11.06	12.14
208	40	40	45.29	41.25	32.2	10.16	12.96	10.84