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	0.000144	0	0.4
lnL	0.350121	0	3019.434
lnK	0.399875	0	3053.788
lnM	0.249928	0	2462.512
lnLlnL	-0.020207	0	-943.515
lnKlnK	-0.021105	0	-922.85
lnMlnM	-0.017187	0	-868.856
lnLlnK	0.024172	0	682.104
lnLlnM	0.01625	0	489.048
lnKlnM	0.018087	0	570.591

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	15.19	16.91	12.23	7.68	7.74	7.08
2	16	16	15	17.27	15.52	9.36	9.06	6.42
3	17	17	18.7	19.29	12.3	8.04	8.86	9.4
4	18	18	15.89	22.93	14.75	9.54	7.08	7.44
5	19	19	20.22	21.72	14.21	8.3	8.72	8.98
6	20	20	22.54	20.68	16.14	7.84	9.92	8.3
7	21	21	21.57	24.69	15.79	6.4	6.24	6.6
8	22	22	21.62	23.95	19.71	7.66	7.76	6.1
9	23	23	26.2	24.9	17.06	7.98	9.68	9.44
10	24	24	24.29	25	22.13	8.38	9.26	6.68
11	25	25	25.28	32.64	16.48	7.28	6.16	8.6
12	26	26	24.05	33.1	20	9.58	7.52	8.5
13	27	27	23.84	32.04	24.62	9.54	7.7	6.56
14	28	28	33.82	28.98	20.61	7.02	9.62	8.98
15	29	29	29.02	30.6	26.62	8.4	9.02	6.64
16	30	30	33.38	31.48	24.07	6.06	7.42	6.36
17	31	31	37.25	31.37	23.76	6.22	8.7	7.54
18	32	32	36.4	36.6	21.93	6.34	7.2	8.22
19	33	33	41.78	33.71	23.34	6.66	9.8	9.44
20	34	34	42.48	34.93	24.24	6.02	8.66	8.32
21	35	35	33.43	41.5	28.65	8.08	7.16	6.92
22	36	36	42.97	40.54	23.74	6.34	7.76	9.1
23	37	37	47.14	39.64	24.18	6.28	8.8	9.84
24	38	38	33.57	49.65	29.97	9.68	6.98	7.9
25	39	39	44.95	36.64	35.46	6.44	9.36	6.08
26	40	40	39.23	45.86	33.22	8.98	8.54	7.8
27	15	15	22.08	11.06	14.68	5.42	13.98	6.26
28	16	16	22.67	12.98	14.05	5.52	12.16	6.92
29	17	17	21.62	11.99	21.9	6.34	14.5	4.46
30	18	18	19.36	14.53	23.18	8.58	13.74	4.96
31	19	19	25.15	15.41	18.23	6.52	13.26	6.8
32	20	20	24.34	18.34	17.57	7.1	11.32	7.44
33	21	21	32.07	15.76	19.01	5.1	13.44	6.74
34	22	22	25.51	19.12	22.51	8.8	14.12	7.28
35	23	23	26.78	20.69	22.12	8.54	13.22	7.64
36	24	24	28.5	18.5	29.1	7.66	14.56	5.34
37	25	25	26.93	19.52	34.07	8.64	14.42	4.68
38	26	26	29.75	19.37	35.28	7.46	14.12	4.36
39	27	27	27.52	21.87	37.41	8.92	13.36	4.44
40	28	28	34.59	22.3	30.39	6.9	13.22	5.74
41	29	29	40.1	21.75	29.93	5.18	12.16	5.22
42	30	30	36.76	25.45	29.63	6.28	11.06	5.78
43	31	31	38.96	22.97	37.19	5.62	11.96	4.24
44	32	32	39.63	25.43	34.74	6.74	12.98	5.62
45	33	33	41.79	24.48	39.07	6.26	13.42	4.84
46	34	34	49.72	25.61	32.36	5.98	14.92	7.08
47	35	35	47.91	27.64	33.58	6.12	13.36	6.64
48	36	36	42.46	30.3	37.94	8.12	13.8	6.62
49	37	37	48.78	26	45.58	5.3	12.7	4.1
50	38	38	49.44	28.33	43	6.7	14.74	5.64
51	39	39	43.35	30.08	51.87	7.4	13	4.28
52	40	40	53.32	31.87	39.15	6.6	13.82	6.78
53	15	15	17.65	23.35	6.48	6.3	5.18	14.64
54	16	16	20.25	21.13	7.86	5.52	6	12
55	17	17	25.78	18.95	8.53	5.02	8.24	13.18
56	18	18	26.34	21.06	8.8	5.42	8.06	14.06
57	19	19	21.57	27.96	9.2	6.58	5.54	12.8
58	20	20	28.84	23.07	10.14	5.68	8.44	13.88
59	21	21	26.06	26.32	11.35	7.4	8.36	14.06
60	22	22	26.44	26.21	13.26	7.36	8.5	11.84
61	23	23	26.72	31.71	11.82	6.4	5.98	11.94
62	24	24	28.41	32.6	12.3	7.84	7.62	15
63	25	25	31.59	28.25	15.28	6.72	8.76	11.28
64	26	26	31.29	36.83	12.35	5.64	5.32	12.02
65	27	27	32.42	36.82	13.53	6.18	6.08	12.34
66	28	28	34.32	40.39	12.71	5.68	5.36	13.06
67	29	29	34.05	37.36	16.13	7.22	7.4	12.42
68	30	30	38.82	42.02	13.28	5.86	6.1	14.78
69	31	31	46.77	35.99	14.78	4.86	7.56	13.46
70	32	32	47.92	41.32	13.35	4.22	5.74	13.56
71	33	33	42.23	42.74	16.43	5.2	5.86	11.28
72	34	34	53.08	42.78	14.01	4.06	5.98	13.9
73	35	35	53.05	42	15.88	4.68	7.04	13.82
74	36	36	52.15	41.71	18.04	5.92	8.8	14.74
75	37	37	58.58	39.95	18.46	4.06	7.28	11.28
76	38	38	49.76	47.17	19.5	6.54	7.96	14.06
77	39	39	46.09	51.3	20.92	7.86	7.92	14.22
78	40	40	58.39	45.48	20.39	5.36	8.22	13.2
79	15	15	11.32	22.13	12.35	13.28	6.9	8.56
80	16	16	9.93	28.69	13.27	14.7	4.82	7.46
81	17	17	11.78	22.43	18.75	14.7	7.88	6.08
82	18	18	13.56	23.56	17.74	12.4	7.4	6.46
83	19	19	12.87	26.81	19.58	13.9	6.7	6.06
84	20	20	13.82	34.12	15.21	12.32	4.86	7.86
85	21	21	14.15	27.72	24.16	13.24	6.86	5.04
86	22	22	14.83	32.63	21.19	13.72	6.2	6.44
87	23	23	15.82	32.71	22.89	11.76	5.72	5.44
88	24	24	16.74	36.92	20.84	14.02	6.32	7.74
89	25	25	17.9	33.65	25.47	14.16	7.7	6.68
90	26	26	19.8	37.92	21.5	12.22	6.5	7.92
91	27	27	16.33	46.26	24.8	14.42	4.84	6.3
92	28	28	18.15	43.1	27.07	13.26	5.48	5.92
93	29	29	19.2	45.25	26.62	13.24	5.52	6.44
94	30	30	19.25	52.84	24.28	12	4.18	6.52
95	31	31	20.88	55.78	22.72	12.4	4.46	8.02
96	32	32	23.09	48.92	26.7	13.12	6.2	7.9
97	33	33	23.46	49.25	29.14	14.24	6.8	7.88
98	34	34	23.35	50.69	31.62	13.72	6.3	6.86
99	35	35	26.67	52.25	27.97	12.2	6.32	8.24
100	36	36	25.22	60.29	27.54	13.76	5.64	8.86
101	37	37	25.73	56.55	32.58	12.64	5.72	6.84
102	38	38	24.45	60.44	35.42	14.16	5.58	6.54
103	39	39	30.69	52.94	34.19	12.4	7.46	7.8
104	40	40	28.14	55.44	40.02	15	7.72	7.08
105	15	15	19.23	14.48	11.38	9.44	15.06	12.5
106	16	16	20.09	15.71	12.14	9.64	14.72	12.46
107	17	17	21.88	16.3	12.95	9.64	15.58	12.76
108	18	18	24.71	16.86	13.11	7.32	13.12	11.02
109	19	19	24.53	16.73	16.48	8.7	15.6	9.92
110	20	20	30.51	16.85	15.05	7.28	16.72	11.94
111	21	21	25.06	19.27	18.92	10.34	16.1	10.28
112	22	22	29.86	20.48	16.41	8.14	14.5	11.76
113	23	23	28.43	22.67	17.69	9.52	14.2	11.88
114	24	24	29.86	21.97	20.54	8.26	13.54	9.16
115	25	25	31.65	23.83	19.64	9.9	15.8	12.4
116	26	26	30.93	23.54	24.04	9.72	15.32	9.34
117	27	27	30.27	25.1	25.93	10.88	15.5	9.32
118	28	28	37.46	25.02	22.67	8.76	16.1	11.3
119	29	29	40.76	23.93	25.01	7.52	16.08	9.54
120	30	30	40.07	29.45	21.06	8.38	13.76	12.76
121	31	31	35.94	30.39	26.15	9.84	13.7	10.22
122	32	32	39.61	29.87	26.74	8.36	13.32	9.48
123	33	33	40.52	30.63	28.12	10.44	16.58	11.46
124	34	34	44.46	30.03	28.88	9.22	16.72	10.94
125	35	35	39.28	33.99	31.3	10.48	14.2	9.78
126	36	36	45.3	31.42	32.76	9.64	16.94	10.08
127	37	37	49.29	33.17	29.96	8.06	14.68	10.34
128	38	38	50.8	34.75	29.7	7.98	14.26	10.72
129	39	39	55.48	35.93	27.86	7.82	14.9	12.56
130	40	40	55.89	35.82	30.53	8.4	16.2	12.22
131	15	15	15.06	23.03	7.98	10.96	7.6	16.52
132	16	16	16.62	20.79	10.25	10.34	9.08	13.04
133	17	17	18.78	22.18	10.02	9.28	8.72	13.88
134	18	18	18.54	23.55	11.55	10.9	9.4	13.58
135	19	19	20.34	23.73	12.39	11.52	10.98	14.74
136	20	20	22.82	23.4	13.21	9.7	10.76	13.18
137	21	21	22.1	30.59	11.31	10.44	8.14	16.4
138	22	22	22.96	28.19	14.3	11.16	10.02	13.92
139	23	23	26.04	29.45	13.5	10.64	10.52	16.46
140	24	24	22.84	34.01	15.26	11.94	8.54	13.7
141	25	25	24.45	36.73	14.6	11.86	8.4	15.54
142	26	26	26.49	34.92	16.31	12.16	10.04	15.36
143	27	27	28.07	36.07	16.63	12.34	10.5	16.32
144	28	28	26.82	38.63	18.32	12.66	9.42	14.16
145	29	29	28.29	37.78	20.11	12.2	9.92	13.02
146	30	30	30.9	43.6	16.66	9.84	7.5	14.54
147	31	31	32.98	38.42	20.61	11.5	10.98	14.28
148	32	32	35.97	37.77	21.27	10.14	10.94	13.44
149	33	33	29.85	53.51	18.67	12.38	7.12	15.36
150	34	34	35.46	47.11	19.83	11.22	9.18	15.88
151	35	35	33.91	53.95	19.4	10.64	7.04	14.66
152	36	36	39.72	49.9	19.56	9.82	8.58	16.14
153	37	37	37.29	47.18	25.34	11.98	10.38	13.48
154	38	38	37.04	49.73	26.19	12.2	9.86	13.1
155	39	39	39.03	55.69	23	11.62	8.74	15.46
156	40	40	38.95	58.52	23.64	11.64	8.24	14.96
157	15	15	12.35	19.09	13.57	13.24	9.06	8.46
158	16	16	13.15	18.68	16.57	13.12	9.92	7.14
159	17	17	15.87	17.89	17.27	13.02	12.92	8.42
160	18	18	14.52	21.99	17.85	16.28	11.42	9.12
161	19	19	16.22	21.91	18.98	14.38	11.54	8.54
162	20	20	17.34	24.46	17.86	14.66	11.18	10.12
163	21	21	17.47	26.51	18.95	14.2	9.94	9.22
164	22	22	18	28.44	19.63	16.22	10.82	10.46
165	23	23	19.75	26.21	23.22	15.38	12.6	9.08
166	24	24	20.76	28.8	22.14	16.16	12.56	10.7
167	25	25	22.1	28.28	24.49	15.14	12.94	9.58
168	26	26	21.21	36.25	20.82	14.96	9.1	10.92
169	27	27	20.66	32.66	29.4	16.2	10.8	7.64
170	28	28	23.7	33.76	26.44	14.78	11.14	9.28
171	29	29	24.39	32.45	31.06	15.82	12.92	8.5
172	30	30	23.62	40.42	26.56	16.3	9.9	10.14
173	31	31	25.8	41.03	26.06	15.52	10.28	10.96
174	32	32	26.72	41.03	28.07	15.22	10.5	10.26
175	33	33	28.06	41.98	28.54	14.14	10.06	9.9
176	34	34	24.83	45.66	33.76	16.84	9.4	8.38
177	35	35	29.91	41.45	33.51	13.28	10.34	8.3
178	36	36	27.98	42.23	40.18	16.84	11.86	7.86
179	37	37	30.83	45.92	34.19	16.92	12.1	10.7
180	38	38	33.72	42.64	37.49	13.1	11.36	8.26
181	39	39	34.87	44.66	36.89	14.54	12.42	9.72
182	40	40	35.59	44.81	39.42	14.78	12.88	9.36
183	15	15	14.07	18.08	12.28	12.64	10.76	10.6
184	16	16	14.74	19.9	12.81	13.3	10.68	11.2
185	17	17	17.54	19.52	13.16	13.82	13.92	13.84
186	18	18	19.82	19.85	13.58	11.48	13.1	12.8
187	19	19	18.66	22.39	15.12	13.6	12.54	12.44
188	20	20	21.13	21.67	16.36	11.5	12.76	11.1
189	21	21	25.23	22	15.28	10.2	13.7	13.14
190	22	22	22.34	27.9	15.03	11.68	10.28	13.3
191	23	23	21.69	29.48	17.07	13.06	10.4	12.36
192	24	24	27.17	27.15	16.82	10.16	11.62	12.76
193	25	25	26.05	29.1	18.72	13.02	13.06	13.72
194	26	26	27.04	30.47	19.31	12.06	11.98	12.8
195	27	27	25.22	33	21.77	13.54	11.28	11.5
196	28	28	29.24	30.98	22.55	11.26	12.02	10.92
197	29	29	29.47	32.09	24.22	12.36	12.78	11.12
198	30	30	29.21	32.32	27.69	13.46	13.66	10.24
199	31	31	29.58	38.98	23.28	12.42	10.26	11.76
200	32	32	35.73	36.15	22.87	10.04	11.32	12.12
201	33	33	33.14	37.22	27.2	12.76	12.72	11.5
202	34	34	30.92	40.6	29.45	13.38	11.1	10.12
203	35	35	34.19	44.85	24.8	13.32	11.06	13.88
204	36	36	34.25	43.38	28.92	13.22	11.44	11.52
205	37	37	36.94	42.42	30	13.28	12.9	12.12
206	38	38	40.16	39.98	32.51	11.42	13.12	10.46
207	39	39	42.29	41.86	31.28	12	13.88	12.22
208	40	40	44.03	43.63	30.7	11.34	13.1	12.38