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	-7.0E-6	0	-0.016
lnL	0.350402	0	2536.828
lnK	0.400142	0	2808.414
lnM	0.249431	0	2089.233
lnLlnL	-0.020146	0	-812.607
lnKlnK	-0.021075	0	-843.831
lnMlnM	-0.017127	0	-755.526
lnLlnK	0.023983	0	661.909
lnLlnM	0.016234	0	474.374
lnKlnM	0.018139	0	496.292

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	15.91	16.01	12.5	7.74	8.78	7.34
2	16	16	15.9	17.1	14.53	9.5	9.96	7.54
3	17	17	17.86	20.88	11.64	7.7	7.32	9.1
4	18	18	16.87	24.11	12.65	8.66	6.5	8.68
5	19	19	19.08	23.45	13.71	8.14	7.3	8.58
6	20	20	19.7	23.89	15.53	9.9	9.02	9.36
7	21	21	23.38	23.81	14.99	7.6	8.5	9.16
8	22	22	24.44	24.19	16.48	8.54	9.88	9.7
9	23	23	23.74	26.59	17.61	9.6	9.6	9.74
10	24	24	23.62	28.05	19.27	6.92	6.46	6.28
11	25	25	26.79	24.99	22.74	7.32	9.08	6.34
12	26	26	23.1	29.91	24.64	9.7	8.18	6.42
13	27	27	29.85	33.36	17.17	6.98	7	9.56
14	28	28	31.88	31.11	20.01	8.06	9.48	9.96
15	29	29	31.44	32.55	21.74	7.82	8.58	8.62
16	30	30	37.25	31.14	21.22	6.96	9.82	9.64
17	31	31	30.98	39.06	21.83	8.46	7.36	9.12
18	32	32	31.69	36.1	26.88	7.06	6.92	6.12
19	33	33	31.12	44.28	22.93	9.38	7.08	9.6
20	34	34	39.59	34.49	27.06	7.32	9.84	8.18
21	35	35	39.8	35.8	28.37	7.54	9.76	8.02
22	36	36	33.76	43.18	29.7	9.56	8.18	7.94
23	37	37	34.94	45.56	29.07	7.94	6.64	7.04
24	38	38	39.82	43.66	28.78	8.58	8.8	8.98
25	39	39	40.74	46.72	27.89	6.58	6.4	7.34
26	40	40	44.89	38.57	36.17	7.32	10	6.74
27	15	15	17.17	11.85	18.34	6.96	12.3	4.6
28	16	16	17.9	14.15	16.7	8.64	13.02	6.7
29	17	17	25.61	12.45	16.32	5.04	13.46	6.12
30	18	18	20.31	15.18	20.09	7.38	11.88	5.34
31	19	19	21.47	14.58	24.93	7.68	13.84	4.6
32	20	20	25.92	15.53	21.23	5.4	11.28	4.88
33	21	21	28.4	15.56	22.79	6.18	14.34	5.72
34	22	22	22.5	19.36	26.28	8.74	11.92	5.2
35	23	23	35.34	17.69	19.85	5.24	13.52	7.38
36	24	24	33.02	19.15	22.48	6.5	14.1	7.3
37	25	25	26.16	21.25	30.66	8.54	12.46	5.06
38	26	26	31.85	22.94	24.11	6.72	11.3	6.66
39	27	26.99	27.09	21.92	38.15	8.84	12.96	4.22
40	28	28	36.42	21.95	29.1	6.3	13.06	5.86
41	29	29	41.66	23.89	24.4	5.64	12.4	7.56
42	30	30	32.37	26.99	32.04	8.52	12.06	6.16
43	31	31	38.35	25.52	31.76	6.84	12.62	6.1
44	32	32	39.32	26.75	32.25	7.12	12.8	6.42
45	33	33	38.27	29.3	32.58	7.3	11.42	6.3
46	34	34	35.66	28.37	42.89	8.56	12.8	4.92
47	35	35	39.12	29.94	38.68	7.46	11.68	5.4
48	36	36	38.67	30.64	42.43	7.84	11.78	5.02
49	37	37	44.39	32.06	36.32	6.92	11.6	6.26
50	38	38	40.21	35.55	39.1	8.48	11.2	6.26
51	39	39	41.47	34.64	43.41	7.86	11.1	5.32
52	40	40	53.35	28.42	47.57	5.48	13.14	4.48
53	15	15	21.23	16.92	8.02	5.32	7.94	11.94
54	16	16	28.36	16.75	7.34	4.2	8.92	14.7
55	17	17	27.79	18.54	8.07	4.6	8.46	14.08
56	18	18	27.98	20.33	8.61	5	8.32	14.28
57	19	19	22.08	23.48	11.35	7.6	8.08	11.88
58	20	20	32.24	23.98	8.47	4.04	6.54	13.9
59	21	21	24	31.05	10.03	6.94	5.86	13.84
60	22	22	25.81	31.79	10.49	6.06	5.42	12.48
61	23	23	31.56	27.91	11.52	5.86	7.74	13.7
62	24	24	30.87	28.43	13.43	6.78	8.54	12.9
63	25	25	28.77	36.25	12.15	6.04	5.26	11.9
64	26	26	37.98	27.87	14.39	5.46	8.98	12.22
65	27	27	42.24	32.84	11.61	4.06	6.24	13.26
66	28	28	42.13	33.54	12.83	5.18	7.74	14.98
67	29	29	31.02	46.56	13.48	7.28	5.16	13.86
68	30	30	37.33	40.06	14.82	5.4	5.68	11.44
69	31	31	39.63	43.95	13.7	5.08	5.14	12.66
70	32	32	37.15	40.85	18.28	7.22	7.38	11.88
71	33	33.01	46.87	46.39	13.08	4.46	5.16	14.3
72	34	34	38.88	51.64	15.67	6.94	5.68	14.44
73	35	35	41.95	45.94	18.5	5.9	6.06	11.04
74	36	36	50.57	44.89	16.91	5.78	7.6	14.98
75	37	36.99	66.25	39.42	16.39	4	8.42	14.78
76	38	38	47.27	54.14	17.23	6.2	6.04	14.52
77	39	39	54.78	50.07	17.62	4.44	5.64	12.04
78	40	40	58.2	53.21	16.58	4	5.08	12.52
79	15	15	12.18	21.39	11.7	11.64	6.86	8.72
80	16	16	11.87	22.11	14.86	12.84	7.06	7.04
81	17	17	11.05	27.73	15.05	12.6	4.88	6.26
82	18	18	12.78	29.06	14.22	13.52	5.88	8.52
83	19	19	12.25	31.49	16.63	13.76	5.18	6.86
84	20	20	12.32	39	14.93	14.46	4.26	8.24
85	21	21	14.06	33.92	18.05	12.78	5.18	6.8
86	22	22	14.55	38.88	16.95	13.9	5	8.3
87	23	23	15.77	36.87	19.31	14.5	6.1	8.16
88	24	24	16.22	42.2	18.07	11.86	4.4	7.46
89	25	25	17.06	40.95	20.48	13.68	5.58	7.88
90	26	26	19.51	42.28	18.82	12	5.52	8.94
91	27	27	19.78	41.03	22.23	14.2	6.88	8.84
92	28	28	20.94	37.35	27.12	12	6.94	6.32
93	29	29	19.34	47.17	24.83	14.54	5.82	7.74
94	30	30	19.91	52.4	23.39	11.2	4.1	6.62
95	31	31	22.21	48.28	25.45	12.52	5.74	7.62
96	32	32	20.42	51.54	29.65	14.98	5.76	6.9
97	33	33	20.34	63.5	25.07	14.36	4.3	8.02
98	34	34	27.93	45.64	28.51	11.82	7.58	8.24
99	35	35	24.24	53.36	31.1	14.3	6.46	7.62
100	36	36	25.53	61.48	26.33	11.12	4.52	7.66
101	37	37	24.48	52.6	39.08	14.42	6.7	5.94
102	38	38	28.19	52.52	35.3	13.28	7.3	7.28
103	39	39	25.65	70.04	29.64	11.58	4.06	6.98
104	40	40	30.6	57	33.99	13.5	7.42	8.52
105	15	15	17.77	14.38	12.74	10.24	15.02	10.82
106	16	16	18.97	15.78	12.98	9.26	13.14	10.34
107	17	17	22.88	14.31	15.02	8	15.88	9.38
108	18	18	21.48	16.57	16.14	10.02	15.54	10.02
109	19	19	22.05	19.08	15.43	10.3	13.96	11.2
110	20	20	23.04	17.85	19.77	10.78	16.64	9.22
111	21	21	25.78	19.87	17.38	10.46	16.24	11.88
112	22	22	25.57	21.77	18.23	9.76	13.48	10.38
113	23	23	31.55	20.86	17.62	7.24	13.46	10.26
114	24	24	30.77	22.46	19.1	10.16	16.82	12.72
115	25	25	34.09	23.96	17.73	7.72	13.36	11.9
116	26	26	37.13	21.33	22.17	7.74	16.98	10.14
117	27	27	32.86	28.2	19.42	9.5	13	12.6
118	28	28	40.96	24.47	20.97	7.96	16.68	12.5
119	29	29	39.99	27.67	20.42	8.1	14.28	12.76
120	30	30	35.84	27.88	26.45	10.38	15.94	10.6
121	31	31	36.39	30.25	25.92	10.46	14.86	11.14
122	32	32	40.56	30.2	25.49	9.86	15.94	12.16
123	33	33	40.29	31.65	26.92	9.34	14.18	10.72
124	34	34	48.67	28.57	27.83	7.08	15.14	9.76
125	35	35	41.54	32.64	30.96	10.86	16.48	10.96
126	36	36	43.66	36.46	27.24	9.5	13.42	11.82
127	37	37	46.86	33.97	30.81	9.66	16.12	11.3
128	38	38	46.16	35.87	31.98	10.42	16.02	11.46
129	39	39	52.21	34.83	31.56	8.86	16.3	11.44
130	40	40	56.74	36.04	29.68	7.08	13.8	10.84
131	15	15	14.21	23.66	8.3	12.18	7.64	16.38
132	16	16	15.16	23.37	9.82	12.58	8.64	14.98
133	17	17	17.58	23.61	9.99	11.34	9.16	15.74
134	18	18	16.54	28.19	10.45	12.32	7.52	15.1
135	19	19	20.36	26.73	10.49	9.64	8	15.02
136	20	20	18.92	31.25	11.22	12	7.6	15.86
137	21	21	22.37	27.05	13.2	12.04	11	16
138	22	22	24.32	30.89	11.72	9.76	8.42	16.46
139	23	23	25.05	29.07	14.43	9.88	9.48	13.5
140	24	24	23.3	34.12	14.81	12.5	9.12	15.22
141	25	25	24.89	37.99	13.64	10.16	7.06	14.72
142	26	26	26.93	34.43	16.29	12.62	10.8	16.28
143	27	27	27.53	37.12	16.38	11.42	9.18	15.02
144	28	28	31.41	35.67	16.74	9.96	9.8	14.92
145	29	29	30.06	38.48	18.1	12.38	10.58	16.06
146	30	30	26.24	46.5	18.84	12.8	7.46	13.5
147	31	31	32.12	46.16	16.53	10.24	7.64	15.98
148	32	32	31.84	46.98	18.31	12.06	8.72	16.52
149	33	33	36.75	39.7	21.6	10.1	10.54	13.48
150	34	34	41.29	41.02	19.88	9.1	10.48	15.36
151	35	35	33.37	52.59	20.49	10.52	7.04	13.34
152	36	36	36.55	53.94	19.53	10.68	7.72	15.94
153	37	37	35.51	55.14	21.72	12.92	8.8	16.46
154	38	38	44.25	48.48	21.64	9.9	10.16	16.4
155	39	39	36.11	60.05	22.94	12	7.54	14.62
156	40	40	34.88	61.28	25.63	12.94	7.62	13.28
157	15	15	13.33	18.35	12.94	14.76	11.58	10.92
158	16	16	13.32	20.82	13.79	13.74	9.28	9.42
159	17	17	14.24	21.35	15.32	15.5	11	10.16
160	18	18	13.77	21.93	19.39	16.46	10.88	7.86
161	19	19	13.95	23.65	21.03	16.74	10.28	7.38
162	20	20	18.46	23.76	17.11	13.8	11.72	10.78
163	21	21	18.96	24.18	19.42	14	12.02	9.72
164	22	22	20.41	26.39	18.41	13	10.98	10.48
165	23	23	19.22	31.13	18.6	14.2	9.2	10.54
166	24	24	19.77	29.05	23.42	14.7	10.68	8.6
167	25	25	22.09	29.5	22.96	15	12.2	10.24
168	26	26	20.62	29.55	29.61	15.9	11.9	7.42
169	27	27	23.36	29.61	28.64	14.48	12.52	8.14
170	28	28	21.03	38.07	26.18	16.92	9.62	9.34
171	29	29	24.06	35.41	27.64	14.34	10.4	8.7
172	30	30	28	35.3	25.64	13.86	12.06	10.98
173	31	31	22.19	40.2	33.45	16.02	9.1	7.06
174	32	32	24.61	39.41	33.62	15.56	10.22	7.7
175	33	33	27.17	40.43	31.64	14.3	10.24	8.54
176	34	34	26.79	43.59	32.4	16.88	10.88	9.64
177	35	35	28.7	43.92	32.55	15.36	10.64	9.46
178	36	36	26.72	44.74	39.3	16.4	10.22	7.44
179	37	37	28.96	49.76	33.15	16.94	10.24	10.32
180	38	38	28.38	49.79	37.89	15.46	9.12	7.86
181	39	39	31.91	41.72	46.75	15.4	12.84	7.02
182	40	40	35.31	52.54	31.33	13.38	9.58	11
183	15	15	16.56	17.97	9.99	10.48	10.88	13.56
184	16	16	16.77	17.74	12.78	10.36	11.08	10.18
185	17	17	17.11	21.22	12.02	11.7	10.38	12.66
186	18	18	17.8	21.67	13.74	12.88	11.68	12.48
187	19	19	20.39	22.27	13.54	11.88	12.24	13.74
188	20	20	21.04	22.17	15.91	10.7	11.5	10.62
189	21	21	23.37	23.7	15.1	10.5	11.8	12.54
190	22	22	22.83	24.6	17.58	11.96	12.52	11.62
191	23	23	20.81	29.62	17.93	13.26	10	11.28
192	24	24	22.26	27.76	21.23	13.82	12.18	10.44
193	25	25	24.18	27.92	22.02	13.74	13.26	10.96
194	26	26	23.81	30.82	22.55	13.16	11.1	10.02
195	27	27	23.7	33.95	22.73	14	10.48	10.5
196	28	28	29.09	32.19	21.43	11	11.16	11.26
197	29	29	32.33	32.47	21.06	11.38	12.94	13.46
198	30	30	28.86	37.26	22.69	13.78	11.66	13.06
199	31	31	30.98	37.58	23.1	11.62	10.58	11.72
200	32	32	34.31	34.95	25.37	11.66	13.04	11.88
201	33	33	36.73	36.68	24.26	11.74	13.44	13.66
202	34	34	35.88	39.96	24.68	11.54	11.62	12.8
203	35	35	34.3	39.65	29.63	13.88	13.38	11.78
204	36	36	35.07	44.3	27.14	13.96	12.12	13.48
205	37	37	34.58	47.85	27.44	13.82	10.78	12.96
206	38	38	40.67	40.94	30.83	11.48	13.02	11.36
207	39	39	43.73	44.59	27.24	11.1	12.4	13.84
208	40	40	39.11	48.77	30.42	11.98	10.56	11.5