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	0.000668	0	1.811
lnL	0.350059	0	2881.3
lnK	0.399963	0	3109.784
lnM	0.249533	0	2318.608
lnLlnL	-0.020205	0	-865.998
lnKlnK	-0.021148	0	-954.993
lnMlnM	-0.017192	0	-887.744
lnLlnK	0.024156	0	647.13
lnLlnM	0.016284	0	471.267
lnKlnM	0.018177	0	562.364

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	16.05	14.91	13.8	7.78	9.7	6.64
2	16	16	16.01	21.1	10.54	7.48	6.18	8.74
3	17	17	15.81	19.85	14.77	8.92	7.8	6.9
4	18	18	17.8	24.48	11.53	8.4	6.6	10
5	19	19	19.81	20	16.55	8.34	9.42	7.36
6	20	20	19.2	25.95	14.24	8.06	6.46	8.18
7	21	21	18.8	27.3	16.38	8.36	6.16	7.02
8	22	22	22.63	26.95	15.54	7.2	6.7	8
9	23	23	23.24	25.82	18.94	8.1	8.18	7.36
10	24	24	23.78	29.92	17.36	7.66	6.68	7.92
11	25	25	24.98	29.35	19.54	7.36	6.96	7.02
12	26	26	27.71	29.85	19.28	8.9	9.32	9.74
13	27	27	27.11	33.37	19.43	7.42	6.64	7.84
14	28	28	30.84	31.42	20.57	6.26	7	7.2
15	29	29	35.2	28.11	23.44	6.42	9.56	7.4
16	30	30	33.2	38.34	18.16	6.28	6.06	9.12
17	31	31	33.02	33.33	25.4	8.76	9.9	8.52
18	32	32	30.92	34.92	29.25	8.16	8.08	6.22
19	33	33	38.82	37.91	21.54	6.16	7.24	8.8
20	34	34	38.27	32.96	30.37	7.34	10	6.88
21	35	35	32.32	42.37	29.08	9.94	8.26	8.04
22	36	36	37.01	37.85	32.01	7.6	8.46	6.44
23	37	37	36.61	49.74	24.07	7.88	6.28	9.22
24	38	38	41.32	44.51	26.66	8.06	8.44	9.64
25	39	39	35.31	50.42	30.2	9.18	6.9	7.88
26	40	40	50.3	42.49	26.91	6.6	9.2	9.84
27	15	15	17.65	12.47	16.2	8.48	14.58	6.7
28	16	16	19.17	12.44	18.88	6.82	12.96	4.96
29	17	17	20.4	14.71	16.72	7.16	12.02	6.46
30	18	18	20.94	13.9	22.4	6.06	11.22	4
31	19	19	21.62	15.54	22.08	6.66	11.22	4.64
32	20	20	28.6	13.18	24.71	5.02	14.28	4.26
33	21	21	26.07	16.81	22.45	6.6	12.64	5.62
34	22	22	27.73	18.44	21.33	6.5	12	6.32
35	23	23	23.98	21.17	24.81	8.92	11.8	6.12
36	24	24	31.3	17.99	26.81	6.44	14.12	5.52
37	25	25	28.78	21.02	27.3	7	11.58	5.32
38	26	26	40.37	18.62	24.92	5	14.2	6.3
39	27	27	33.18	21.4	29.75	6.28	12.02	5.1
40	28	28	30.92	24.77	29.76	8.86	13.14	6.62
41	29	29	36.69	21.26	35.11	6.62	14.38	4.98
42	30	30	43.04	23.65	27.15	6.48	14.98	7.96
43	31	31	34.28	27.31	33.11	8.36	12.48	6.22
44	32	32	44.94	24.37	31.52	5.46	12.82	5.92
45	33	33	44.44	27.1	30.32	5.84	11.94	6.54
46	34	34	50.5	23.35	37.09	5.22	14.78	5.36
47	35	35	39.15	26.65	47.19	7.08	12.72	4.06
48	36	36	39.59	27.8	48.54	8.22	14.24	4.62
49	37	36.99	38.03	30.18	50.01	7.96	11.94	4.12
50	38	38	46.81	31.9	37.91	7.1	12.74	6.5
51	39	39	50.03	31.49	39.29	6.68	13.16	6.34
52	40	40	54.14	29.93	42.64	6.3	14.46	5.96
53	15	15	23.85	17.45	6.72	4.52	7.46	14.32
54	16	16	23.16	21.41	6.62	4.74	5.94	14.78
55	17	17	18.93	23.98	8.92	7.56	6.54	13.08
56	18	18	24.43	21.04	9.62	6.46	8.8	13.82
57	19	19	22.58	27.05	9.1	6.06	5.6	12.6
58	20	20	28.88	23.26	10.01	5.32	7.84	13.22
59	21	21	28.07	23.28	12.32	5.94	8.46	11.18
60	22	22	26.02	26.35	13.43	7.14	8.04	11.1
61	23	23	26.81	27.48	14.35	7.8	8.66	11.62
62	24	24	30.99	35.76	9.84	5.24	5.06	14.44
63	25	25	36.17	32.75	10.64	4.36	5.6	13.14
64	26	26	36.76	32.45	12.11	4.2	5.56	11.08
65	27	27	32.23	37.34	13.37	7.22	6.94	14.52
66	28	28	36.09	31.22	17.02	6.42	8.7	11.1
67	29	29	44.33	31.54	14.91	5.16	8.8	13.28
68	30	30	31.23	49.65	13.86	7.7	5.1	14.3
69	31	31	32.34	51.36	14.26	7.78	5.16	14.56
70	32	32	37.43	45.57	15.62	7.08	6.42	14.14
71	33	33	41.97	44.55	15.65	5.52	5.88	12.58
72	34	34	40.6	44.86	17.93	7.4	7.52	13.82
73	35	35	41.78	50.71	16.3	5.78	5.26	12.5
74	36	36	53.69	43.44	16.51	5.02	7.36	14.36
75	37	37	53.05	42.6	18.93	4.84	7.16	11.62
76	38	38	42.19	54.66	19.49	6.48	5.46	11.48
77	39	39	41.38	53.76	22.41	7.88	6.62	11.58
78	40	40	62.49	43.34	20.16	5.12	9	13.84
79	15	15	9.56	27.93	11.35	12.48	4.04	7.28
80	16	16	10.24	22.82	17.68	13.52	6.02	5.08
81	17	17	12.44	24.61	15.05	12.54	6.42	7.16
82	18	18	12.19	27.55	16.45	14.3	6.26	7.18
83	19	19	14.69	27.82	15.29	11.8	6.36	8.04
84	20	20	13.19	28.8	20.8	12.1	5.52	5.06
85	21	21	13.94	32.14	19.75	14.68	6.28	6.96
86	22	22	15.45	33.41	19.27	14.96	6.9	8.24
87	23	23	17.32	31.48	21.21	13.08	7.4	7.36
88	24	24	17.48	34.27	21.81	14.64	7.58	8.06
89	25	25	19.12	35.67	21.21	14.14	7.76	8.94
90	26	26	18.58	43	19.68	12.26	5.22	8.18
91	27	27	19.38	37.57	26.07	13.46	7.06	6.78
92	28	28	19.13	40.32	27.63	11.78	5.6	5.46
93	29	29	20	37.17	33.72	13.46	7.42	5.2
94	30	30	22.14	42	27.56	11.34	6.1	6.26
95	31	31	21.12	54.78	22.91	13.8	5.14	8.96
96	32	32	22.88	46.61	29.03	13.34	6.6	7.2
97	33	33	19.51	56.81	31.16	14.52	4.72	5.98
98	34	34	22.84	56.1	28.21	13.5	5.36	7.52
99	35	35	26.06	47.76	32.97	12.78	7.16	6.92
100	36	36	28.01	50.92	30.19	11.56	6.54	7.56
101	37	37	31.7	48.32	30.46	11.38	7.92	8.52
102	38	38	27.25	59.74	30.73	14.32	6.5	8.88
103	39	39	24.79	68.38	32.2	14.78	5.12	7.76
104	40	40	33.57	52.96	33.22	11.34	7.58	8.2
105	15	15	20.84	13.36	11.64	8.62	16.62	12.22
106	16	16	18.76	15.02	14.24	10.24	15.2	10.12
107	17	17	23.62	13.77	15.34	7.82	16.86	9.28
108	18	18	26.77	15.35	13.72	7.36	16.18	11.54
109	19	19	25.4	17.35	14.88	8.64	15.46	11.58
110	20	20	25.81	18.88	15.57	8.48	14	10.98
111	21	21	27.14	20.61	15.37	8.28	13.08	11.54
112	22	22	33.82	18.99	15.8	7	15.78	12.24
113	23	23	30.98	19.39	20.24	8.18	16.22	9.64
114	24	24	28.13	22.52	21.37	9.2	13.66	9.08
115	25	25	32.82	23.27	19.45	8.2	14.04	10.84
116	26	26	31.16	24.54	22.27	10.28	15.56	10.9
117	27	27	32.42	25.5	23.05	10.16	15.4	10.84
118	28	28	32.94	27.75	22.78	10.2	14.26	11.24
119	29	29	40.52	26.4	21.58	7.02	13.28	10.52
120	30	30	39.34	26.81	24.91	8.54	15.32	10.44
121	31	31	41.01	28.97	23.74	8.08	13.9	10.98
122	32	32	42.11	28.5	26.61	9.12	16.5	11.18
123	33	33	45.35	29.43	25.9	8.46	16.08	11.68
124	34	34	39.55	30.55	32.81	10.9	16.88	9.7
125	35	35	45.97	31.55	28.61	7.34	13.06	9.16
126	36	36	42.52	33.78	31.74	10.8	16.18	10.88
127	37	37	42.96	34.95	33.03	10.42	15.18	10.14
128	38	38	48.33	38.73	26.81	8.86	13.14	12.66
129	39	39	51.38	36.08	30.49	8.82	15.28	11.64
130	40	40	45.18	39.62	34.38	10.78	14.38	10.62
131	15	15	13.95	22.02	9.36	12.06	8.06	13.78
132	16	16	16.5	20.68	10.44	10.98	9.62	13.44
133	17	17	17.41	21.81	11.3	12.3	10.78	14.6
134	18	18	17.15	28.13	10	11.46	7.32	15.44
135	19	19	19.47	25.9	11.6	12.56	10.26	16.5
136	20	20	20.52	25.67	13.25	12	10.54	14.34
137	21	21	22.69	30.47	11.01	10.1	8.16	16.9
138	22	22	22.97	30.88	12.59	11.58	9.34	16.78
139	23	23	23.37	30.66	14.63	12.26	10.18	15.2
140	24	24	28.13	30.51	13.63	9.86	10.24	16.52
141	25	25	23.63	38.07	14.53	10.98	7.16	13.9
142	26	26	27.68	32.74	16.89	11.62	10.9	14.84
143	27	27	26.12	38.74	16.52	11.52	8.28	14.1
144	28	28	30.79	36.06	16.9	10.98	10.42	15.88
145	29	29	30.9	36.77	18.63	11.04	10.28	14.3
146	30	30	29.05	45.59	16.96	12.46	8.38	16.76
147	31	31	35.4	39.72	17.98	9.88	9.86	15.66
148	32	32	36.86	38.25	20.25	9.12	9.98	13.18
149	33	33	34.44	45.6	19.3	9.42	7.74	13.3
150	34	34	35.63	51	17.71	9.74	7.3	15.84
151	35	35	38.72	44.68	21.23	11.1	10.72	16.08
152	36	36	39.51	46.31	21.82	9.26	8.78	13.3
153	37	37	38.04	47.05	24.79	12.18	10.84	14.4
154	38	38	38.75	56	20.97	11.04	8.18	16.24
155	39	39	35.53	55.36	26.2	12.74	8.64	13.02
156	40	40	40.07	57.5	23.34	11.24	8.4	15.16
157	15	15	12.69	19.84	12.33	13.56	9.16	10.02
158	16	16	13.43	19.63	14.9	15.7	11.48	9.92
159	17	17	13.59	23.27	14.39	15.16	9.2	10.12
160	18	18	13.64	23.85	17.28	16.42	9.72	8.88
161	19	19	15.62	23.38	18.14	15.36	10.92	9.2
162	20	20	16.93	24.84	18.06	15.14	11.02	10.02
163	21	21	16.3	27.74	19.55	16.78	10.26	9.68
164	22	22	17.03	26.98	23.06	16.48	10.96	8.24
165	23	23	17.92	30	21.71	16.1	10.04	9.18
166	24	24	22.11	29.18	19.88	13.12	10.82	10.62
167	25	25	20.53	29.73	25.2	16.18	11.96	9.08
168	26	26	20.16	34.98	23.59	15.9	9.5	9.44
169	27	27	21.06	32.82	28.34	16.04	10.88	8.08
170	28	28	20.95	37.13	27.34	15.78	9.2	8.24
171	29	29	23.95	32.92	31.18	14.86	11.68	7.78
172	30	30	26.83	32.84	30.45	13.94	12.56	8.58
173	31	31	25.08	37.08	31.64	15.72	11.34	8.54
174	32	32	29.66	36.13	29.4	13.66	12.38	9.86
175	33	33	25.1	47.37	27.97	16.7	9.04	10.5
176	34	34	30.1	41.18	29.92	15.18	12	10.92
177	35	35	27.59	43.61	34.82	15.02	10.02	8.16
178	36	36	28.95	39.62	42.3	16.1	12.72	7.36
179	37	37	29.74	48.74	32.9	16.18	10.34	10.26
180	38	38	32.33	45.93	35.49	15.84	11.98	10.14
181	39	39	29.31	49.51	40.44	14.72	9.08	7.2
182	40	40	31.27	44.63	47.95	16.62	12.5	7.18
183	15	15	14.44	19.01	11.01	13.1	10.84	12.88
184	16	16	15.45	19.75	12.15	13.6	11.64	12.88
185	17	17	18.58	19.86	11.89	10.6	11.2	12.8
186	18	18	17.9	20.89	14.41	11.04	10.52	10.18
187	19	19	18.54	21.51	16.2	12.76	12.24	10.68
188	20	20	20.99	20.94	17.41	11.64	13.34	10.36
189	21	21	19.22	27.63	15.62	13.6	10.14	12.4
190	22	22	23.22	23.13	18.87	11.76	13.5	10.72
191	23	23	23.09	26.56	18.31	11.66	11.3	10.94
192	24	24	25.9	28.35	16.79	10.86	11.16	12.92
193	25	25	26.75	28.33	18.8	11.98	12.8	12.96
194	26	26	25.52	33	18.56	12.86	10.86	13.36
195	27	27	26.54	32.71	20.59	12.4	11.08	11.94
196	28	28	28.87	32.29	21.56	11.48	11.5	11.56
197	29	29	29.05	36.2	20.64	12.92	11.4	13.8
198	30	30	33.37	30.76	24.94	10.18	12.8	10.24
199	31	31	33.21	35.12	23.29	12.54	13.42	13.6
200	32	32	37.55	34.24	23.26	10.44	13.3	13.1
201	33	33	32.76	36.95	27.95	12.36	12.26	10.64
202	34	34	35.91	36.86	27.81	11.46	12.7	11.06
203	35	35	32.72	44.15	26.9	13.94	11.2	12.54
204	36	36	36.41	39.77	30.34	11.32	11.66	10.02
205	37	37	36.98	44.39	27.99	11.24	10.36	11.14
206	38	38	42.03	41.13	29.31	10.9	12.78	11.9
207	39	39	42.36	45.76	27.33	10.1	10.54	12.08
208	40	40	46.91	41.5	30.47	10.06	13.28	11.94