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	-0.000108	0	-0.27
lnL	0.350065	0	2692.77
lnK	0.400019	0	2861.838
lnM	0.249996	0	2180.404
lnLlnL	-0.020147	0	-884.766
lnKlnK	-0.021097	0	-920.865
lnMlnM	-0.017209	0	-724.585
lnLlnK	0.024091	0	668.444
lnLlnM	0.016233	0	430.664
lnKlnM	0.018114	0	523.871

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	13.05	19.15	12.5	9.26	6.74	6.96
2	16	16	18.65	18.72	10.29	6.82	7.76	9.8
3	17	17	18.61	18.5	13.19	6.22	7.16	6.66
4	18	18	17.85	24.09	11.75	8.44	6.78	9.86
5	19	19	21.07	22.97	12.41	6.22	6.42	8.28
6	20	20	19.77	26.14	13.56	8.34	6.86	9.28
7	21	21	24.39	22.41	15.53	6.32	7.98	7.68
8	22	22	26.78	23.52	15.28	6.28	8.36	8.68
9	23	23	24.53	24.58	18.98	7	7.98	6.76
10	24	24	22.79	26.81	21.68	9	8.5	6.82
11	25	25	26.19	29.58	18.14	6.38	6.32	7.02
12	26	26	27.47	27.24	22.41	7.14	8.24	6.48
13	27	27	28.93	28.41	22.68	6.56	7.66	6.24
14	28	28	31.28	27.64	24.55	6.4	8.46	6.08
15	29	29	26.66	40.68	19.58	9.64	6.7	9.9
16	30	30	29.31	34.04	25.45	9.58	9.18	8.08
17	31	31	30.15	33.7	28.25	8.56	8.58	6.6
18	32	32	29.07	37.6	28.45	9.12	7.7	6.68
19	33	33	28.34	44.42	25.91	9.8	6.6	7.78
20	34	34	34.84	39.17	26.42	9.04	9	8.94
21	35	35	35.65	40.96	26.79	7.56	7.34	7.56
22	36	36	44.42	37.71	25.33	6.74	9.34	9.32
23	37	37	35.35	38.46	37.1	9.16	9.48	6.18
24	38	38	44.68	40.48	27.72	6.42	8.24	8.04
25	39	39	44.66	46.56	24.91	6.58	7.16	9.34
26	40	40	48.4	40.35	30.54	6.4	9.06	7.86
27	15	15	17.7	12.1	16.95	6.68	11.94	5.02
28	16	16	20.52	14.48	13.41	6.64	11.44	7.82
29	17	17	19.65	15.48	16.19	7.56	11.44	6.78
30	18	18	24.68	14.37	16.91	6.8	14.68	7.58
31	19	19	21.83	17.64	17.68	7.52	11.04	6.88
32	20	20	24.47	14.62	25.49	6.14	12.86	4.18
33	21	21	28.08	15.61	23.02	5.34	12.18	4.82
34	22	22	27.06	15.77	28.85	6.34	13.68	4.2
35	23	23	30.63	17.64	24.02	6.5	14.22	6.18
36	24	24	25.94	18.87	32.16	8.94	14.86	4.96
37	25	25	33.27	22.07	20.76	6.22	11.52	7.74
38	26	26	34.34	20.55	26.12	6.82	14.26	6.72
39	27	27	29.89	22.8	30.91	8.68	13.64	5.96
40	28	28	37.9	22.26	26.95	6.86	14.66	7.32
41	29	28.99	29.31	23.7	40.07	8.94	13.12	4.42
42	30	30	36.53	23.51	34.14	7.24	13.9	5.6
43	31	31	36.18	25.31	34.89	7.94	13.82	5.92
44	32	32	46.31	23.92	31.26	5.06	12.58	5.74
45	33	33	43.94	24.48	36.49	6.48	14.74	5.76
46	34	34	39.01	30.33	33.82	7.6	11.68	6.42
47	35	35	44.63	28.1	35.89	6.48	12.76	5.98
48	36	36	43.22	30.18	37.28	7.74	13.5	6.58
49	37	37	52.59	24.09	47.1	5.14	14.72	4.18
50	38	38	51.33	29.51	38.15	6.26	13.72	6.34
51	39	39	50.48	32.43	36.99	6.04	11.62	6.22
52	40	40	45.12	33.36	45.55	8.72	14.22	6.16
53	15	15	18.8	22.17	6.43	5.46	5.14	13.78
54	16	16	21.83	18.29	8.77	6.38	8.98	13.32
55	17	17	25.72	21.1	7.38	4.7	6.78	14.58
56	18	18	18.11	29.45	8.8	7.78	5.02	13
57	19	19	22.18	23.5	11.27	7.4	7.9	11.72
58	20	20	21.74	28.19	10.82	7.22	6.08	11.72
59	21	21	23.89	27.5	11.87	7.66	7.42	12.46
60	22	22	31.4	23.65	12.46	5.58	8.9	11.82
61	23	23	26.33	30.67	12.59	6.64	6.34	11.3
62	24	24	27.09	32.88	12.89	6.66	6.06	11.4
63	25	25	33.67	31.93	11.94	5.36	6.52	12.96
64	26	26	36.12	32.26	12.46	5.5	7.18	13.74
65	27	27	35.44	32.65	14.32	5.42	6.82	11.24
66	28	28	33.38	37.83	14.33	7.26	7.16	14.02
67	29	29	37.91	34.36	15.9	6.78	8.7	13.46
68	30	30	34.06	40.87	16.11	7.74	7.14	13.34
69	31	31	39	43.57	14.12	6.1	6.12	14.4
70	32	32	43.81	41.33	14.77	5.72	7	14.68
71	33	33	35.74	51.08	15.84	7.3	5.48	13.58
72	34	34	42.25	44.18	17.45	6.4	6.94	12.94
73	35	34.99	56.24	34.75	19.33	4.44	8.94	11.14
74	36	36	50.2	39.88	20.09	5.3	7.94	11.12
75	37	37	47	45.62	19.87	6.54	7.74	12.86
76	38	38	64.44	42.01	16.99	4.2	7.94	14.4
77	39	39	55.49	47.84	18.46	5.6	7.62	14.6
78	40	40	43.11	58.18	20.97	7.52	6.04	12.54
79	15	15	9.89	26.71	11.49	14.02	4.98	8.4
80	16	16	10.72	30.01	11.17	12.46	4.24	8.48
81	17	17	9.62	31.94	15.15	14.72	4.1	6.16
82	18	18	12.99	23.56	18.9	11.88	6.74	5.46
83	19	19	13.62	24.81	20.25	11.16	6.3	5
84	20	20	14.6	25.93	20.96	12.72	7.4	5.94
85	21	21	14.9	32.77	17.42	13.8	6.24	8.2
86	22	22	14.61	32.6	21.71	11.34	5.04	5.08
87	23	23	15.95	33.92	21.42	12.2	5.74	6.16
88	24	24	14.93	39.49	22.39	14.34	5.22	6.36
89	25	25	17.6	41.09	19.49	13.76	5.8	8.72
90	26	26	20.66	35.84	21.98	11.14	6.66	7.4
91	27	27	18.38	45.74	21.18	13.76	5.38	8.32
92	28	28	20.32	42.46	23.44	13.12	6.3	7.92
93	29	29	18.52	42.77	30.57	14.94	6.38	5.92
94	30	30	20.88	46.24	26.07	13.38	6	7.36
95	31	31	24.58	41.27	27.68	12.14	7.54	7.54
96	32	32	21.71	52.65	26.27	13	5.24	7.42
97	33	33	24.92	46.65	28.91	13.14	7.18	7.88
98	34	34	24.92	51.9	27.8	13.44	6.48	8.44
99	35	35	22.71	65.28	25.75	14.18	4.68	8.74
100	36	36	22.59	67.76	27.45	13.06	4.1	7.42
101	37	37	29.87	52.28	29.5	12	7.1	8.7
102	38	38	24.85	68.42	29.02	14.52	5.04	8.64
103	39	39	21.83	71.88	36.32	14.58	4.1	5.72
104	40	40	29.3	56.99	36.18	11.52	6.02	6.42
105	15	15	17.65	14.18	13.13	10.16	15.02	10.28
106	16	16	19.53	14.13	14.88	9.66	16.16	9.5
107	17	17	26.32	14.47	12.36	7.16	16.54	12.44
108	18	18	25.56	15.42	14.43	7.3	15.12	10.22
109	19	19	23.98	18.6	14.39	8.72	13.44	11.36
110	20	20	27.74	18.08	15.19	8.86	16.76	12.86
111	21	21	24.76	19.74	18.51	10.38	15.48	10.44
112	22	22	30.46	20.32	16.2	7.78	14.32	11.68
113	23	23	33.6	20.88	16.26	7.32	14.64	12.28
114	24	24	32.79	21.26	19.18	7.24	13.78	9.72
115	25	25	28.24	24.42	21.96	10.16	13.78	9.76
116	26	26	33.32	22.77	22.98	9.26	16.56	10.24
117	27	27	33.69	26.88	20.22	9.86	14.7	12.84
118	28	28	40.58	25.95	19.37	7.26	14.04	12.38
119	29	29	34.53	28.72	23.26	10.2	14.48	11.6
120	30	30	37.38	28.35	24.39	9.86	15.6	11.64
121	31	31	37.81	29.57	25.53	10.46	15.96	11.86
122	32	32	42.12	30.74	23.64	8.6	14.24	12.12
123	33	33	49.99	28.83	23.68	7.5	16.38	12.9
124	34	34	46.41	30.25	27.01	8.96	16.94	12.1
125	35	35	50.23	33.03	23.85	7.32	13.7	12.56
126	36	36	45.69	33.88	28.76	8.48	13.78	10.44
127	37	37	46.67	35.29	29.18	8.54	13.56	10.6
128	38	38	55.71	33.03	28.62	7.34	15.52	11.48
129	39	39	45.86	36.02	35.47	10.68	16.22	10.32
130	40	40	46.26	39.81	33.07	11	15	11.66
131	15	15	17.4	19.8	8.19	9.72	9.54	16.84
132	16	16	15.99	23.82	8.93	11.52	8.24	16.34
133	17	17	19.37	21.85	9.84	9.68	9.6	15.34
134	18	18	18.84	24.23	10.88	10.4	8.84	14.18
135	19	19	18.65	27.94	11.03	12.36	8.78	16.38
136	20	20	20.88	29.97	10.48	9.8	7.32	15.74
137	21	21	24.17	27.03	11.98	10.18	10.2	16.6
138	22	22	21.23	34.03	12.2	10.82	7.1	14.84
139	23	23	22.19	33.4	13.89	12.8	9.04	15.86
140	24	24	23.58	33.41	15.01	10.86	8.24	13.2
141	25	25	28.78	33.03	13.76	9.24	8.98	15.72
142	26	26	28.82	32.55	16.17	10.16	10.06	14.32
143	27	27	29.44	36.48	15.42	10.18	9.04	15.56
144	28	28	27.39	43.09	15.36	11.3	7.58	15.94
145	29	29	29.11	40.62	17.49	10.98	8.48	14.28
146	30	30	32.61	37.82	18.94	10.02	9.62	13.56
147	31	31	32.75	41.41	18.71	11.14	9.66	15.38
148	32	32	33.78	43.86	18.66	11.42	9.6	16.4
149	33	33	37.37	39.11	21.6	10.08	10.92	13.72
150	34	34	35.19	50.28	18.33	9.72	7.3	14.96
151	35	35	34.76	55.01	18.32	10.66	7.1	16.18
152	36	36	40.99	44.02	22.38	9.84	10.34	14.32
153	37	37	36.14	56.48	20.56	10.86	7.34	15.06
154	38	38	36.82	56.88	21.9	12.2	8.36	16.06
155	39	39	45.97	46.42	24.12	9.7	10.96	14.8
156	40	40	46.23	50.07	23.61	9.92	10.32	15.62
157	15	15	14.1	17.72	12.62	13.44	11.74	10.94
158	16	16	13.15	18.64	16.63	14.88	11.28	8.06
159	17	17	15.27	21.22	14.01	13.92	10.8	11
160	18	18	15.85	23.26	14.48	13.82	10.06	10.98
161	19	19	15.12	25.99	16.21	16.38	9.9	10.78
162	20	20	16.58	25.52	17.85	15.72	10.82	10.3
163	21	21	16.72	24.62	22.64	15.4	11.16	7.7
164	22	22	18.14	28.41	19.44	15.46	10.42	10.18
165	23	23	16.53	29.77	24.73	16.64	9.52	7.4
166	24	24	18.5	29.07	25.79	15.6	10.48	7.54
167	25	25	21.95	26.16	27.99	13.08	12.16	7.02
168	26	26	21.21	32.72	24.24	16.64	11.42	10.16
169	27	27	25.54	29.33	25.61	13.26	12.88	9.44
170	28	28	22.84	32.22	29.99	16.08	12.26	8.34
171	29	29	21.51	39.91	27.12	16.48	9.1	8.96
172	30	30	23.15	39.01	28.85	15.16	9.38	8.36
173	31	31	25.98	39.63	27.17	14.64	10.18	9.92
174	32	32	26.86	34.34	36.75	15.02	12.86	7.42
175	33	33	28.02	40.68	29.99	16.5	12.16	10.88
176	34	34	29.48	37.61	35.47	13.54	11.62	7.78
177	35	35	26.8	48.18	31.25	15.7	9	9.36
178	36	36	33.11	40.34	33.83	14.24	12.9	9.92
179	37	37	33.5	42.99	33.62	14.06	11.98	10
180	38	38	31.96	45.65	36.42	13.68	10.28	8.38
181	39	39	32.83	48.75	35.14	13.68	9.82	9.02
182	40	40	28.23	53.53	42.01	16.94	9.12	7.58
183	15	15	15.3	17.06	11.96	13.7	13.78	13.08
184	16	16	16.23	18.38	12.66	13.82	13.64	13.22
185	17	17	15.71	21.04	13.65	13.84	11.22	11.66
186	18	18	18.46	21.75	13.03	11.5	10.84	12.38
187	19	19	19.67	21.06	15.43	13.26	13.98	12.6
188	20	20	19.59	21.72	18.08	13.4	13.56	10.52
189	21	21	22.19	24.99	14.94	10.14	10.08	11.54
190	22	22	25	25.76	14.59	10.3	11.36	13.86
191	23	23	22.74	26.51	18.74	13.56	12.94	12.16
192	24	24	25.83	27.47	17.65	10.46	11.12	11.7
193	25	25	26.18	27.69	20.03	10.34	11.06	10.12
194	26	26	25.93	29.12	21.86	13.88	13.84	12.12
195	27	27	26.18	33.83	19.96	12.92	10.92	12.7
196	28	28	31.07	29.8	22.06	10.48	12.58	11.2
197	29	29	32.71	32.29	20.91	11.48	13.32	13.88
198	30	30	32.33	37.1	19.68	10.76	10.46	13.78
199	31	31	34.57	33.65	23.55	11.14	13.14	12.5
200	32	32	32.56	35.82	26.22	12.22	12.48	11.26
201	33	33	34.95	35.4	27.33	11.76	13.24	11.22
202	34	34	35.2	38.27	26.99	12.76	13.22	12.46
203	35	35	37.08	39.45	26.89	10.84	11.52	11.3
204	36	36	35.26	45.6	25.84	12.1	10.22	12.46
205	37	37	43	39.57	27.25	10.92	13.76	13.34
206	38	38	42.42	39.57	30.71	10.82	13.42	11.3
207	39	39	36.96	44.91	33.71	12.94	11.76	10.3
208	40	40	41.54	47.56	29.15	12.68	12.36	13.74