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	-0.000127	0	-0.346
lnL	0.350029	0	2849.172
lnK	0.400026	0	3108.313
lnM	0.25003	0	2350.189
lnLlnL	-0.020181	0	-909.809
lnKlnK	-0.021039	0	-957.648
lnMlnM	-0.017197	0	-793.172
lnLlnK	0.02408	0	812.919
lnLlnM	0.016327	0	440.673
lnKlnM	0.017996	0	532.801

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	14.88	16.63	12.92	7.66	7.68	6.46
2	16	16	14.99	21.47	11.22	9.64	7.22	9.68
3	17	17	19.28	19.53	11.62	7.48	8.42	9.7
4	18	18	15	23.37	15.54	8.84	6	6.06
5	19	19	21.31	18.75	16.58	6.94	9.22	6.66
6	20	20	19.81	21.81	17.69	8.6	8.78	7.02
7	21	21	22.87	21.12	18.51	6.9	8.66	6.32
8	22	22	20.84	29.11	15.5	8.32	6.42	8.42
9	23	23	21.55	27.82	18.76	7.54	6.38	6.34
10	24	24	24.62	30.38	16.23	7.8	6.96	9.1
11	25	25	24.87	30.73	18.37	8.98	8	9.16
12	26	26	30.54	28.6	18.11	6.98	8.62	9.22
13	27	27	31.61	26.48	22.48	6.28	8.84	6.7
14	28	28	31.84	34.34	17.37	6.12	6.4	8.92
15	29	29	26.67	34.29	25.11	9.08	7.72	6.96
16	30	30	36.71	34.15	18.88	6.3	7.84	9.84
17	31	31	31.66	35.55	24.37	7.08	7.06	6.88
18	32	32	26.77	41.61	27.41	9.76	6.64	6.78
19	33	33	39.43	38.96	20.3	6.04	7	9.42
20	34	34	36.11	36.62	27.88	8.2	9.22	7.94
21	35	35	30.21	42.84	31.42	9.5	7.2	6.48
22	36	36	36.54	41	28.83	9.28	9.26	8.76
23	37	37	29.82	48.31	33.21	9.66	6.26	6.08
24	38	38	34.6	49.17	29.17	8.52	6.44	7.44
25	39	39	36.8	40.87	39.29	9.86	9.96	6.52
26	40	40	37.92	41.51	40.65	9.42	9.68	6.2
27	15	15	18.05	11.64	17.63	7.76	14.86	5.7
28	16	16	22.23	12.43	15.47	5.96	13.5	6.52
29	17	17	22.72	12.84	18.14	5.82	13.02	5.42
30	18	18	18.35	14.7	24.58	7.94	11.78	4.02
31	19	19	21.87	15.15	22.7	7.52	13.24	5.14
32	20	20	23.23	18.97	17.73	7.76	11.26	7.62
33	21	21	22.67	17.06	26.62	7.88	12.58	4.66
34	22	22	23.52	17.55	29.2	7.44	12	4.12
35	23	23	35.29	16.75	21.77	5.36	14.72	6.76
36	24	24	29.74	17.82	29.25	6.18	12.9	4.5
37	25	25	28.96	19.2	31.57	7.22	13.38	4.66
38	26	26	33.96	19.57	28.78	5.6	12.24	4.86
39	27	27	28.66	24.21	29.66	8.28	11.54	5.68
40	28	28	42.52	21.23	25.12	5.14	13.3	6.82
41	29	29	34.94	23.28	32.13	8.12	14.96	6.4
42	30	30	35.36	23.13	36.73	7.32	13.78	5
43	31	31	35.04	24.01	39.98	6.9	12.3	4.22
44	32	32	36.49	27.39	34.36	8.74	14	6.7
45	33	33	37.4	30.71	31.17	8.52	12.28	7.54
46	34	34	39.41	27.37	39.57	7.26	12.74	5.16
47	35	35	46.95	26.54	36.91	5.76	12.88	5.46
48	36	36	39.19	28.28	47.81	7.82	13.12	4.42
49	37	37	46.37	31.25	35.72	6.94	12.62	6.74
50	38	38	53.98	31.27	32.47	6.16	13.38	8
51	39	39	50.18	31.71	38.69	6.68	13.1	6.48
52	40	40	43.33	34.43	45.71	8.44	12.64	5.66
53	15	15	19.44	20.62	6.8	6	6.4	14.76
54	16	16	18.97	20.94	8.6	6.9	7.02	12.5
55	17	17	20.02	21.33	9.77	7.94	8.42	13.18
56	18	18	25.14	20.5	9.64	5.34	7.76	11.78
57	19	19	24.04	26.32	8.76	5.14	5.28	12.04
58	20	20	24.41	26.01	10.47	5.94	6.3	11.48
59	21	21	26.7	27.46	10.4	6.6	7.3	14.3
60	22	22	31.23	29.47	9.29	4.74	5.8	14.1
61	23	23	28.44	31.32	11.15	5.92	6.04	12.72
62	24	24	36.02	26.19	12.54	4.8	7.98	11.86
63	25	25	30.85	34.65	11.87	6.48	6.46	14.24
64	26	26	30.38	39.32	11.74	6.14	5.18	13.42
65	27	27	30.89	33.44	16.39	7.8	8.12	11.74
66	28	28	37.18	35.69	13.64	6.22	7.46	14.46
67	29	29	33.98	39.15	15.17	7.36	7.12	13.58
68	30	30	45.32	33.56	15.04	4.94	8.04	12.92
69	31	31	46.96	34.94	15.33	4.92	7.96	13.12
70	32	32	40.86	38.95	17.38	6.7	8.1	13.08
71	33	33	38.63	40.91	19.44	7.34	7.84	11.76
72	34	34	46.97	42.47	16.25	5.3	6.82	13.2
73	35	35	44.88	41.35	19.72	7.12	8.96	13.38
74	36	36	50.07	46.69	16.23	5.4	6.7	14.52
75	37	37	52.8	44.31	18.03	4.5	6.32	11.38
76	38	38	42.73	48.65	22.5	7.36	7.22	11.18
77	39	39	57.37	44.06	19.88	4.58	7.14	11.38
78	40	40	46.79	52.54	21.76	7.02	7	12.34
79	15	15	10.77	19.18	16.46	13.7	7.94	5.94
80	16	16	9.92	28.94	13.11	14.92	4.84	7.68
81	17	17	11.9	25.13	15.58	14.34	6.8	7.46
82	18	18	13.66	25.33	15.75	12.08	6.68	7.3
83	19	19	12.2	35.11	14.38	13.66	4.5	8.04
84	20	20	13.11	31.08	18.76	14.64	6.06	6.86
85	21	21	13.04	34.56	19.64	14.32	5.2	6.32
86	22	22	16.28	29.6	21.36	11.6	6.56	6.02
87	23	23	13.45	45.3	18.28	14.82	4.06	7.38
88	24	24	14.34	40.82	22.68	13.06	4.36	5.44
89	25	25	16.3	45.97	18.62	14.34	4.84	8.76
90	26	26	17.54	46.8	19	11.94	4.3	7.76
91	27	27	17.47	49.12	20.63	14.64	4.96	8.6
92	28	28	16.95	56.01	20.71	14.92	4.18	8.42
93	29	29	19.14	43.05	28.8	14.4	6.34	6.36
94	30	30	17.74	50.3	29.43	13.72	4.6	5.4
95	31	31	24.27	44.79	24.99	12.2	6.78	8.42
96	32	32	23.24	42.44	32.65	13.58	7.64	6.5
97	33	33	23.06	57.19	24.17	12.28	4.82	8.3
98	34	34	24.11	46.59	34.2	11.66	6.14	5.52
99	35	35	22.94	50.75	36.39	12.96	5.82	5.38
100	36	36	26.93	51.25	31.64	13.88	7.44	8.2
101	37	37	28.29	48.99	35.09	12.12	7.26	6.72
102	38	38	28.64	51.97	35.04	11.64	6.6	6.56
103	39	39	24.09	71.47	31.57	14.28	4.54	7.42
104	40	40	26.22	68.71	32.49	12.5	4.6	6.94
105	15	15	17.97	13.59	13.72	10.62	16.86	10.42
106	16	16	18.81	15.05	14.13	11	16.34	11
107	17	17	22.64	15.76	13.07	7.94	13.9	10.82
108	18	18	23.97	16.44	14.16	8.82	15.7	11.7
109	19	19	23.25	17.63	16.28	8.78	13.9	9.54
110	20	20	23.24	18.65	18.2	10	14.8	9.52
111	21	21	30.96	17.24	17.18	7.34	16.7	10.48
112	22	22	32.61	19.75	15.55	7.16	14.76	12.22
113	23	23	33.75	20.17	17.06	7.48	15.66	11.92
114	24	24	29.06	22.57	20.41	10.4	16	11.26
115	25	25	33.53	21.18	21.96	8	15.7	9.4
116	26	26	30.8	24.69	22.41	9.32	13.82	9.68
117	27	27	32.91	26.01	21.92	10.26	15.46	11.82
118	28	28	34.47	25.47	24.54	9.82	16.02	10.46
119	29	29	41.08	23.87	24.86	7.8	16.88	10.06
120	30	30	41.61	25.83	24.59	7.18	14.38	9.52
121	31	31	36.91	31.84	23.52	9.72	13.22	11.8
122	32	32	35.74	31.48	28.22	9.84	13.06	9.28
123	33	33	51.73	27.97	23.79	7.14	16.82	12.72
124	34	34	43.16	32.06	27.07	10.24	16.6	12.66
125	35	35	50.41	29.37	28.49	7.34	15.84	10.26
126	36	36	45.41	33.08	30.09	8.16	13.54	9.46
127	37	37	48.37	32.92	31.07	8.3	14.92	9.98
128	38	38	50.76	31.64	34.51	8.5	16.94	9.56
129	39	39	50.31	36.09	31.31	9.2	15.54	11.48
130	40	40	57.92	36.04	28.94	7.18	14.34	11.6
131	15	15	15.94	20.18	8.9	9.68	8.38	13.72
132	16	16	16.94	21.56	9.51	9.9	8.52	13.94
133	17	17	16.27	24.31	10.59	12.74	9.08	15.08
134	18	18	21.04	23.36	9.97	9.1	9.2	15.66
135	19	19	19.16	28.88	10.2	10.92	7.7	16.38
136	20	20	23.08	26.05	11.19	9	8.92	15.06
137	21	21	20.56	30.44	12.48	12.88	9.28	16.56
138	22	22	22.13	32.08	12.53	10.78	7.96	15.04
139	23	23	21.83	33.53	14.11	12.38	8.54	14.78
140	24	24	22.04	38.31	13.59	12.88	7.68	16.24
141	25	25	26.49	36.11	13.51	9.98	7.92	15.74
142	26	26	24.74	38.63	15.49	11.56	7.82	14.32
143	27	27	29.02	33.64	17.58	10.68	10.26	13.76
144	28	28	27.93	42.5	15.27	10.58	7.38	15.38
145	29	29	35.03	34.08	17.71	9.2	10.86	14.66
146	30	30	31.56	43.19	16.43	10.3	8.14	15.86
147	31	31	31.18	45.51	17.5	9.72	7.12	13.7
148	32	32	32.9	42.88	19.91	12.76	10.68	16.46
149	33	33	37.71	41.51	19.62	9.74	9.94	15
150	34	34	38.73	43.23	19.99	9.58	9.62	14.9
151	35	35	38.18	48.6	19.23	9.46	8.14	15.14
152	36	36	38.24	50.12	20.38	10.42	8.66	15.6
153	37	37	43.97	46.96	20.7	9.72	10.28	16.84
154	38	38	37.15	51.4	24.9	12.64	9.86	14.46
155	39	39	42.8	47.04	25.91	10.5	10.74	13.54
156	40	40	39.48	60.68	22.12	11.2	7.72	15.82
157	15	15	13.19	16.79	15.06	13.88	11.94	8.48
158	16	16	14.21	18.04	15.64	13.88	11.98	8.86
159	17	17	12.81	21.5	17.67	15.74	9.78	7.7
160	18	18	15.17	21.34	17.57	13.18	10.08	7.92
161	19	19	14.84	23.05	19.98	14.9	10.14	7.5
162	20	20	17.49	23.83	18.37	15.64	12.42	10.54
163	21	21	17.95	24.87	20.1	13.3	10.36	8.32
164	22	22	19.84	22.99	23.75	13.42	12.9	7.76
165	23	23	18.69	29.4	21.07	16.22	10.88	10.06
166	24	24	19.71	28.75	23.91	14.02	10.28	7.98
167	25	25	21.45	31.23	21.94	14.78	10.86	10.28
168	26	26	20.72	35.41	22.28	16.44	10	10.78
169	27	27	21.98	31.98	27.72	15.34	11.28	8.34
170	28	28	25.72	32.49	24.99	14.7	12.76	10.86
171	29	29	24.65	34.41	27.91	15.52	11.98	9.58
172	30	30	23.11	36.5	32.04	16.78	11.2	8.16
173	31	31	26.39	35.69	31.19	14.18	11.36	8.32
174	32	32	24.61	38.07	35.49	15.12	10.34	7.02
175	33	33	27.35	43.54	28.04	14.94	9.88	10.36
176	34	34	30.07	41.48	29.65	14.82	11.6	10.76
177	35	35	28.82	45.18	31	16.72	11.26	10.96
178	36	36	27.96	44.97	36.47	15.58	10.18	8.14
179	37	37	28.48	48.95	34.78	16.08	9.72	9.08
180	38	38	30.2	52.07	32.39	16.14	9.72	10.62
181	39	39	35.81	44.25	36.03	13.76	12.26	9.76
182	40	40	32.48	46.91	41.87	16.8	12.46	8.9
183	15	15	16.37	18.05	10.08	10.7	10.9	13.52
184	16	16	15.12	20.55	11.81	12.98	10.34	12.4
185	17	17	16.64	19.27	14.4	12.4	11.92	10.5
186	18	18	18.36	21.37	13.46	12.56	12	12.92
187	19	19	18.38	21.57	16.32	13.78	13.04	11.32
188	20	20	19.83	23.23	16.05	13.58	12.88	12.44
189	21	21	21.18	22.21	18.99	12.66	13.68	10.28
190	22	22	24.68	24.38	16.08	11.54	13.38	13.64
191	23	23	21.74	26.59	19.83	12.64	11.4	10.06
192	24	24	26.44	27.61	17	10.42	11.32	12.52
193	25	25	23.59	28.98	21.53	13	11.66	10.34
194	26	26	25.78	30.69	20.37	12.48	11.62	11.76
195	27	27	25.76	33.14	21.02	13.86	11.78	12.58
196	28	28	31.87	30.19	20.93	11.18	13.62	13.1
197	29	29	30.22	32.2	23.29	12.88	13.66	12.5
198	30	30	32.95	31.36	24.63	10.5	12.72	10.56
199	31	31	29.28	34.35	28.56	14	13.26	10.3
200	32	32	29.9	40.44	24.55	12.86	10.3	11.58
201	33	33	33.26	39.62	24.66	13.14	12.22	13.34
202	34	34	35.87	38.36	26.24	13.2	13.94	13.62
203	35	35	41.35	37.04	25.63	10.16	13.22	12.74
204	36	36	35.32	43.07	28.03	13.48	12.2	12.64
205	37	37	35.51	42.45	31.62	13.58	12.58	11.12
206	38	38	39.13	44.21	28.92	10.14	10.04	10.34
207	39	39	39.08	43.15	33.21	12.9	13.12	11.16
208	40	40	41.78	43.86	32.65	12.34	13.32	11.78