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	2.7E-5	0	0.081
lnL	0.350199	0	3175.946
lnK	0.399943	0	3245.624
lnM	0.249855	0	2681.155
lnLlnL	-0.020157	0	-973.895
lnKlnK	-0.021084	0	-988.17
lnMlnM	-0.017248	0	-831.103
lnLlnK	0.024029	0	756.593
lnLlnM	0.01628	0	519.502
lnKlnM	0.018178	0	561.523

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	16.18	19.12	9.41	7.22	6.78	9.76
2	16	16	18.92	19.5	9.53	6.04	6.66	9.66
3	17	17	15.44	22.07	13.04	9.94	7.46	8.66
4	18	18	19.36	21.97	12.07	6.6	6.5	8.22
5	19	19	16.6	23.22	16.8	9.12	7.02	6.42
6	20	20	20.78	20.19	18.68	8	9.46	6.48
7	21	21	28.38	20.91	14.22	6.06	9.92	9.76
8	22	22	25.52	22.82	17	6.46	8.42	7.44
9	23	23	21.49	24.66	22.65	9.38	9.12	6.3
10	24	24	24.48	27.98	18.43	7.76	7.58	7.74
11	25	25	24.63	28.68	20.61	9.42	9	8.3
12	26	26	29.92	24.84	23.06	6.68	9.5	6.48
13	27	27	24.38	38.52	18.22	9.92	6.62	9.98
14	28	28	32.15	30.72	20.16	7.1	8.56	8.78
15	29	29	31.01	31.15	23.67	6.44	7.32	6.32
16	30	30	34.58	31.92	22.5	7.58	9.52	8.98
17	31	31	33.09	35.14	23.38	8	8.52	8.6
18	32	32	34.67	40.31	20.3	7.46	7.14	10
19	33	33	35.56	34.73	27.49	7.88	9.26	7.62
20	34	34	36.04	43.22	21.91	6.5	6	8.34
21	35	35	44.15	36.05	24.56	6.08	8.82	8.66
22	36	36	31.57	50.66	25.77	9.92	6.5	9
23	37	37	34.67	44.26	30.67	7.44	6.38	6.14
24	38	38	46.71	35.19	32.45	6.02	9.6	6.6
25	39	39	40.91	41.87	32.68	6.6	7.34	6.14
26	40	40	37.02	46.84	34.84	9.44	8.18	7.24
27	15	15	19.86	11.56	15.66	6.5	14.04	6.14
28	16	16	18.53	14.09	16.05	7.14	11.26	6.04
29	17	17	18.69	13.59	21.6	8.9	14.8	5.36
30	18	18	21.38	14.17	21.03	6.04	11.2	4.4
31	19	19	19.96	17.75	19.79	8.54	11.2	6.16
32	20	20	23.48	15.27	25.04	7.52	14.26	4.98
33	21	21	23.92	17.65	23.29	8.14	13.3	6
34	22	22	24.87	20.09	21.5	8.4	12.34	7.12
35	23	23	32.9	18.8	19.71	5.96	13.16	7.78
36	24	24	27.9	22.42	21.77	8.32	12.3	7.96
37	25	25	27.29	19.57	33.26	7.74	13.08	4.38
38	26	26	33.24	23.55	21.85	6.8	11.66	7.96
39	27	27	29.18	25	27.43	8.8	12.06	6.76
40	28	28	37.7	21.89	27.91	6.88	14.9	7
41	29	29	32.81	25.39	30.31	8.8	13.6	6.9
42	30	30	42.56	20.56	34.94	5.04	13.56	4.54
43	31	31	39.99	25.51	30.05	5.98	11.6	5.98
44	32	32	33.49	28.45	36.36	8.42	11.66	5.46
45	33	33	45.97	24.66	33.93	5.72	13.6	5.84
46	34	34	43.27	29.31	31.1	6.34	11.46	6.68
47	35	35	38.67	31.45	36.23	8.22	11.98	6.34
48	36	36	44.78	32.38	31.7	6.62	11.08	7.1
49	37	37	42.87	32.25	37.72	8.84	14.12	7.34
50	38	38	44.99	35.84	33.13	7.62	11.38	7.8
51	39	39	45.45	34	39.43	8.22	13.22	6.94
52	40	40	59.34	29.44	38.85	5.7	14.86	6.7
53	15	15	17.66	22.02	6.98	6.94	6.12	14.78
54	16	16	22.3	20.82	7.16	5.28	6.54	14.34
55	17	17	19.89	23.96	8.41	6.12	5.62	12.04
56	18	18	18.43	27.68	9.34	7.4	5.24	11.76
57	19	19	23.75	24.31	9.89	6.76	7.52	13.54
58	20	20	23.23	27.31	10.41	7.62	7.2	14
59	21	21	33.96	22.01	10.66	4.12	7.84	11.5
60	22	22	29.55	28.21	10.48	6.08	7.34	14.7
61	23	23	27.67	29.19	12.68	6.82	7.32	12.2
62	24	24	26.18	33.86	12.91	7.06	5.96	11.58
63	25	25	27.87	37.15	12.22	6.16	5.02	11.6
64	26	26	37.82	34.82	10.68	4.08	5.14	12.9
65	27	27	28.22	38.17	15.28	7.74	6.2	11.38
66	28	28	31.89	36.76	15.75	7.24	7	11.86
67	29	29	34.99	40.59	13.96	5.46	5.24	11.5
68	30	30	45.44	32.6	15.61	4.5	7.6	11.3
69	31	31	44	34.82	16.61	5.9	8.88	13.26
70	32	32	43.59	42.97	14.11	4.56	5.3	12.28
71	33	33	39.13	40.3	19.53	7.32	8.08	11.84
72	34	34	52.84	41.23	14.79	4.3	6.58	13.74
73	35	35	42.2	53.51	15.02	6.04	5.22	14.54
74	36	36	46.09	45.54	18.49	5.9	6.84	12.34
75	37	37	56.82	42.21	17.71	4.86	7.88	13.68
76	38	38	59.94	42.87	17.92	4.2	7.12	12.42
77	39	39	49.52	50.68	19.5	6.96	7.74	14.88
78	40	40	49.15	52.36	20.6	6.24	6.62	12.4
79	15	15	11.37	22.25	12.19	12.34	6.4	8.12
80	16	16	10.4	25.66	14.52	13.98	5.52	6.74
81	17	17	10.42	32.52	13.12	13.62	4.08	7.42
82	18	18	14.23	23.73	16.37	11.66	7.3	7.06
83	19	19	14.48	23.34	20.31	11.38	7.42	5.46
84	20	20	14.04	26.83	21.1	13.84	7.38	6.12
85	21	21	16.11	30	17.71	13.24	7.28	8.46
86	22	22	14.04	36.67	19.4	11.1	4.1	5.42
87	23	23	18.14	30.48	20.83	11.76	7.3	7.14
88	24	24	15.81	39.89	20.27	13.88	5.34	7.4
89	25	25	18	43.01	17.74	11.56	4.74	8.4
90	26	26	20.53	38.86	19.73	11.38	6.14	8.52
91	27	27	20.95	33.29	27.96	11.68	7.74	5.94
92	28	28	22.4	37.75	24.22	11.48	7.1	7.48
93	29	29	18.13	54.95	22.04	13.74	4.26	7.8
94	30	30	21.45	43.75	27.16	11.94	5.9	6.46
95	31	31	21.63	47.35	27.2	11.44	5.2	6.24
96	32	32	23.74	46.28	27.81	14.2	7.4	8.42
97	33	33	22.97	47.4	31.79	11	5.36	5.36
98	34	34	26.93	49.22	26.88	12.24	6.88	8.76
99	35	35	23.1	62.97	26.4	11.5	4.04	7.02
100	36	36	21.76	64.99	30.73	14.96	4.72	7.12
101	37	37	24.73	60	31.69	12.86	5.18	6.86
102	38	38	26.1	49.83	42.92	12.92	6.9	5.14
103	39	39	29.62	59.57	30.28	12.5	6.28	8.7
104	40	40	27.82	53.44	43.06	13.2	7	5.64
105	15	15	21.23	14.08	10.48	7.74	14.34	12.68
106	16	16	19.83	16.05	11.95	9.8	14.36	12.7
107	17	17	19.74	16.9	14.01	10.1	13.86	10.8
108	18	18	22.41	17.62	13.88	10.02	15.2	12.58
109	19	19	24.86	17.48	15.12	8.58	14.84	11
110	20	20	30.35	16.77	15.27	7.4	17	11.86
111	21	21	24.43	20.91	17.21	10.48	14.38	11.3
112	22	22	26.76	19.59	20.28	9.7	16	9.6
113	23	23	31.19	20.58	18.26	7.22	13.46	9.68
114	24	24	34.35	19.47	20.8	7.34	16.36	9.46
115	25	25	35.37	23.03	17.98	7.1	13.44	11.24
116	26	26	36.37	23.82	19.14	7.34	13.8	11.16
117	27	27	36.12	24.21	21.74	8.46	15.48	10.98
118	28	28	31.23	29.74	21.98	10.74	13	11.6
119	29	29	37.86	27.44	22.16	7.86	13.12	10.54
120	30	30	42.29	27.51	21.83	7.64	14.48	11.88
121	31	31	37.83	30.3	24.57	10.38	15.4	12.32
122	32	32	36.71	29.59	30.03	10.88	16.02	9.84
123	33	33	36.63	33.69	27.7	10.5	13.24	10.42
124	34	34	45.43	32.4	24.95	8.3	14.12	12
125	35	35	44.76	31.85	29.18	8.82	15.04	10.42
126	36	36	48.62	33.86	26.59	8.3	14.52	12.06
127	37	37	47.64	35.78	27.82	8.76	14.02	11.78
128	38	38	53.52	36.34	25.97	7.38	13.3	12.34
129	39	39	54.47	34.24	30.73	7.58	14.96	10.62
130	40	40	50.27	38.51	31.24	8.48	13.26	10.6
131	15	15	13.34	24.01	8.82	12.96	7.42	15.06
132	16	16	14.71	24.94	9.34	12.54	7.7	15.28
133	17	17	18.62	23.17	9.54	9.42	8.32	14.78
134	18	18	19.77	24.16	10.28	10.66	9.62	16.44
135	19	19	20.85	27.88	9.63	9.14	7.42	16.2
136	20	20	23.8	24.35	11.84	9.74	10.84	15.82
137	21	21	19.61	33.62	11.6	12.54	7.6	16.62
138	22	22	23.65	29.64	12.83	10.34	9.06	15.16
139	23	23	22.29	32.89	14.1	11.7	8.46	14.32
140	24	24	22.24	33.21	16.34	12.82	9.14	13.16
141	25	25	25.95	36	13.92	9.9	7.7	14.72
142	26	26	25.52	38.35	15.04	11.56	8.18	15.38
143	27	27	29.48	37.2	14.98	9.32	8.1	14.76
144	28	28	28.62	38.1	17.18	11.86	9.68	15.44
145	29	29	32.31	36.07	18.09	10.12	10.16	14.3
146	30	30	28.37	41.53	19.87	12.74	9.3	13.84
147	31	31	32.46	41.07	19.14	12.14	10.52	16.14
148	32	32	32.05	48.93	17.19	10.94	7.6	16.26
149	33	33	31.51	47.35	20.54	11.64	8.24	13.76
150	34	34	33.85	49.32	19.75	10.52	7.72	14.16
151	35	35	33.48	54.75	19.33	11.58	7.42	15.78
152	36	36	37.7	46.99	22.69	11.68	10.3	15.16
153	37	37	41.93	48.39	21.06	9.28	8.96	14.9
154	38	38	39.21	48.97	24.85	10.66	9.38	13.02
155	39	39	38.94	59.42	21.13	11.02	7.66	16.16
156	40	40	38.31	63.48	21.64	11.54	7.28	16.14
157	15	15	11.38	19.82	14.42	15.46	9.2	8.36
158	16	16	13.18	19.21	15.82	16.28	11.94	9.38
159	17	17	15.11	19.24	16.5	13.88	11.94	8.94
160	18	18	16.87	20.59	15.97	13.52	12.22	10.3
161	19	19	15.48	25.03	16.57	15.74	10.22	10.38
162	20	20	17.99	21.42	20.84	13.52	12.58	8.12
163	21	21	18.93	25.7	17.75	13.14	10.48	10.12
164	22	22	17.11	27.61	22.07	15.3	9.96	8.1
165	23	23	20.28	28.79	19.37	13.66	10.34	10.3
166	24	24	19.94	32.68	19.4	14.58	9.32	10.76
167	25	25	21.21	28.81	25.23	13.6	10.84	7.92
168	26	26	19.9	31.11	28.78	16.64	11.24	7.7
169	27	27	21.23	35.41	24.95	16.36	10.24	9.66
170	28	28	24.54	34.02	24.88	14.88	11.58	10.46
171	29	29	23.56	35.82	27.99	16.36	11.42	9.54
172	30	30	24.5	41.03	24.68	14.8	9.22	10.48
173	31	31	26.02	35.11	32.66	14.56	11.7	7.96
174	32	32	28.33	39.26	27.61	14.18	11.04	10.44
175	33	33	26.46	43.79	29.12	15.54	9.82	9.92
176	34	34	28.81	40.22	33.02	13.94	10.76	8.48
177	35	35	28.07	48.73	28.77	15.54	9.28	10.78
178	36	36	30.06	43.45	34.62	14.58	10.8	8.82
179	37	37	29.64	44.04	38.6	16.2	11.62	8.48
180	38	38	34.99	40.24	38.97	13.16	12.76	8.28
181	39	39	28.02	52.73	39.26	16.86	9.16	8.1
182	40	40	33.27	45.28	42.74	15.64	12.44	8.32
183	15	15	13.89	17.67	12.94	13.22	11.38	10.26
184	16	16	16.98	18.62	11.69	11.8	12.1	13.08
185	17	17	16.66	20.79	12.84	11.54	10.16	11.2
186	18	18	17.4	23.9	12.29	12.96	10.2	13.94
187	19	19	22.02	20.4	13.96	10.96	13.7	13.38
188	20	20	22.52	21.57	15.16	10.86	13.06	12.36
189	21	21	18.91	25.3	18.2	13.68	11.1	10.22
190	22	22	19.88	26.55	18.93	13.38	10.88	10.12
191	23	23	21.39	28.16	18.61	12.58	10.4	10.58
192	24	24	28.09	24.32	18.99	10	13.54	11.32
193	25	25	25.15	27.54	21.32	12.68	13.02	11
194	26	26	24.24	32.31	20.51	13.5	11	11.74
195	27	27	29.63	32.01	18.39	11.02	11.5	13.8
196	28	28	26.53	30.89	25.86	13.86	13.24	10.2
197	29	29	30.91	31.69	23.15	10.76	11.94	10.8
198	30	30	31.78	35.26	21.66	10.54	10.66	11.82
199	31	31	30.48	37.65	23.55	11.8	10.52	11.42
200	32	32	34.39	35.6	24.6	12.52	13.74	13.26
201	33	33	35.69	37.15	24.72	12.2	13.3	13.42
202	34	34	36.21	34.63	30.28	11.46	13.8	10.1
203	35	35	41.18	38.03	24.77	10.52	13.2	13.66
204	36	36	38.02	44.3	24.41	11.52	11	13.86
205	37	37	38.73	40.51	30.17	12.4	13.44	11.88
206	38	38	38.99	47.21	26.38	12.36	11.28	13.98
207	39	39	39.33	45.08	30.82	11.98	11.66	11.4
208	40	40	39.73	45.71	32.82	12.28	11.9	10.98