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	-0.000191	0	-0.512
lnL	0.350246	0	2865.106
lnK	0.400079	0	3176.733
lnM	0.249777	0	2288.4
lnLlnL	-0.020151	0	-875.522
lnKlnK	-0.021116	0	-999.304
lnMlnM	-0.017213	0	-801.486
lnLlnK	0.024046	0	682.9
lnLlnM	0.016232	0	475.331
lnKlnM	0.018187	0	503.376

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	17.76	15.66	11.18	7	9.28	8.62
2	16	16	16.59	17.62	13.1	8.1	8.62	7.64
3	17	17	15.6	22.11	12.82	8.76	6.64	7.88
4	18	18	16.01	23.49	14.09	9.56	6.96	7.94
5	19	19	22.4	22.99	11.48	6.08	6.74	9.54
6	20	20	18.6	26.74	14.23	8.66	6.46	8.48
7	21	21	22.38	21.32	18.78	8.02	9.7	7.04
8	22	22	19.01	30.09	16.74	9.16	6.1	7.6
9	23	23	25.16	24.26	18.72	7.16	8.54	7.24
10	24	24	22.52	30.22	18.41	9.3	7.52	8.42
11	25	25	24.46	26.86	23	9.22	9.44	7.08
12	26	26	22.88	35.98	18.96	9.18	6.16	8.18
13	27	27	28.41	31.04	20.32	9.4	9.68	9.96
14	28	28	29.13	35.2	18.77	7.28	6.66	8.72
15	29	29	26.83	30.75	29.48	9.82	9.56	6.28
16	30	30	31.01	31.93	25.98	8.46	9.34	7.44
17	31	31	31.66	38.68	21.53	8.04	7.26	9.04
18	32	32	33.55	35.75	25.25	9.24	9.8	9.22
19	33	33	36.22	35.19	26.3	6.36	7.52	6.62
20	34	34	32.71	41.23	26.64	8.18	7.12	7.44
21	35	35	37.33	38.08	28.1	8.6	9.6	8.58
22	36	36	38.05	43.82	24.78	6.88	6.66	8.14
23	37	37	39.14	41.5	28.7	6.9	7.36	7.1
24	38	38	39.82	42.84	29.61	6.56	6.88	6.64
25	39	39	40.26	41.36	34.03	8.96	9.92	7.8
26	40	40	38.99	42.37	37.84	8.3	8.6	6.14
27	15	15	19.47	10.5	19.01	5.88	13.9	4.32
28	16	16	19.03	13.81	16.01	8.82	14.7	7.72
29	17	17	23.54	11.95	19.57	5.7	14.46	5.06
30	18	18	27.21	12.87	17.91	5.36	14.78	6.26
31	19	19	28.08	14.23	18.01	5.38	13.68	6.48
32	20	20	25.43	15.86	21.03	7.12	14.18	6.36
33	21	21	24.49	16.34	25.68	7.46	13.72	5.04
34	22	22	22.24	18.77	28.19	7.88	11	4.26
35	23	23	28.06	18.06	26.02	6.2	11.9	4.84
36	24	24	29.89	21.02	22.02	6.88	11.9	7.04
37	25	25	31.32	19.42	27.77	7.12	14.28	5.86
38	26	26	32.27	21.25	26.85	6.54	12.22	5.8
39	27	27	33.42	21.34	29.6	5.68	11	4.68
40	28	28	34.61	24.67	25.7	7.24	12.32	7.34
41	29	29	32.96	26.58	27.95	7.84	11.54	6.8
42	30	30	39.06	24.94	28.24	6.08	11.78	6.36
43	31	31	35.06	24.27	39.18	6.54	11.52	4.1
44	32	32	36.24	25.52	39.1	7.96	13.74	5.2
45	33	33	43.32	28.42	29.04	6	11.26	6.86
46	34	34	42.19	27.08	36.66	7.5	14.44	6.32
47	35	35	44.41	26.52	39.85	6.26	13.12	5.08
48	36	36	39.92	29.11	44.27	8.16	13.52	5.16
49	37	37	46.98	29.41	38.8	5.68	11.26	5.08
50	38	38	42.53	33.8	39.29	8.24	12.34	6.46
51	39	39	48.05	30.19	44.59	6.18	12.2	4.82
52	40	40	51.86	32.78	38.77	7.06	13.84	7.1
53	15	15	24.1	18.03	6.36	4.24	6.82	14.52
54	16	16	20.94	22.45	6.97	5.66	5.96	14.74
55	17	17	23.01	19.16	9.59	5.74	8.14	11.48
56	18	18	21.96	25.18	8.56	5.94	5.78	12.86
57	19	19	26.63	23.84	8.87	5.5	7.16	14.32
58	20	20	27.24	23.57	10.52	6.3	8.54	13.78
59	21	21	24.26	27.05	11.92	7.38	7.42	12.16
60	22	22	37.48	21.47	11.65	4.08	8.98	11.52
61	23	23	30.22	29.98	11	4.82	5.56	11.3
62	24	24	25.13	37.61	11.81	7.96	5.66	13.82
63	25	25	34.77	29.32	12.92	5.9	8.24	13.5
64	26	26	35.02	33.68	12.19	5.8	6.94	14.34
65	27	27	32.4	33.56	15.36	7.04	7.72	12.1
66	28	28	33.51	37.18	14.6	7.26	7.34	13.78
67	29	29	34.32	34.71	17.71	7.66	8.64	11.92
68	30	30	35.92	40.02	15.54	6.32	6.36	12.1
69	31	31	41.53	43.1	13.32	5.32	5.82	14.48
70	32	32	39.33	41.77	16.54	6.5	6.92	12.86
71	33	33	37.28	40.53	20.59	7.84	8.12	11.26
72	34	34	45.57	38.51	19.28	6.26	8.72	12.3
73	35	35	42.51	47.27	17.52	6.82	6.88	13.82
74	36	36	45.97	42.94	20.12	6.8	8.42	12.84
75	37	37	39.27	53.58	19.88	7.82	6.2	12.44
76	38	38	45.29	52.65	18.81	6.58	6.3	13.22
77	39	39	48.95	47.22	21.8	7.08	8.44	13.1
78	40	40	63.08	43.97	19.55	4.2	7.34	11.9
79	15	15	9.93	25.69	12.07	11.94	4.46	6.78
80	16	16	9.38	25.18	17.45	14.76	5.28	5.08
81	17	17	11.18	29.53	13.54	12.84	4.68	7.32
82	18	18	12.57	29.2	14.46	14.76	6.26	8.94
83	19	19	15.39	25.83	15.96	12.36	7.68	8.46
84	20	20	12.73	37.81	14.85	14.86	4.72	8.86
85	21	21	13.36	34.61	18.92	14	5.22	6.64
86	22	22	14.71	33.03	21.07	12.74	5.62	5.96
87	23	23	15.46	29.43	27.89	14.74	7.9	5.26
88	24	24	17.25	38.43	18.85	11	4.9	7.08
89	25	25	15.8	46.15	19.38	12.6	4.08	7.08
90	26	26	17.7	40.98	22.58	13.72	5.84	7.36
91	27	27	18.05	34.74	32.75	14.9	7.88	5.28
92	28	28	21.5	40.27	23.35	12.56	6.86	8.14
93	29	29	19.05	43.36	28.7	14.88	6.46	6.56
94	30	30	21.97	37.1	33.69	12.32	7.6	5.32
95	31	31	21.89	48.11	26.1	14.98	6.78	8.7
96	32	32	21.85	52.38	26.22	14.78	6.04	8.52
97	33	33	20.13	56.78	29.73	12.8	4.32	5.78
98	34	34	25.24	41.94	37.56	12.02	7.56	5.38
99	35	35	21.94	65.42	27	14.62	4.62	8.18
100	36	36	25.1	53.04	33.33	13.76	6.52	7.04
101	37	37	24.86	51.27	39.68	14.56	7.1	6
102	38	38	26.59	56.42	34.6	13.36	6.3	7
103	39	39	28.51	51.5	39.67	12.14	6.92	5.88
104	40	40	26.4	74.74	28.66	12.74	4.28	8.26
105	15	15	19.87	14.02	11.46	7.72	13.3	10.54
106	16	16	21.83	14.57	12.26	7.8	14.34	10.98
107	17	17	19.53	15.22	16.77	10.6	16.24	9.06
108	18	18	22.23	18.01	13.56	9.78	14.32	12.5
109	19	19	22.52	17.13	17.77	9.68	15.26	9.14
110	20	20	23.45	19.56	16.69	10.36	14.66	11.04
111	21	21	31.11	19.57	14.03	7.08	13.96	12.9
112	22	22	33.76	19.56	15.13	7.08	15.38	13
113	23	23	29.87	21.08	18.59	9.84	16.94	12.28
114	24	24	34	22.1	17.25	7.94	15.06	12.6
115	25	25	35.58	21.92	19.28	7.76	15.68	11.4
116	26	26	35.11	23.35	20.64	7.3	13.48	9.74
117	27	27	33.74	25.44	21.96	8.16	13	9.66
118	28	28	33.08	24.63	27.38	10.24	16.56	9.14
119	29	29	35.06	27.91	23.83	9.1	13.6	10.24
120	30	30	33.31	28.88	27.6	10.12	13.68	9.02
121	31	31	33.37	30.92	28.12	10.62	13.28	9.28
122	32	32	41.79	27.63	28.23	9.14	17	10.36
123	33	33	40.74	31.89	26.22	9.72	14.82	11.66
124	34	34	44.57	29.86	29.04	7.8	14.28	9.22
125	35	35	44.01	32.17	29.36	7.88	13.02	9.06
126	36	36	49.65	33.85	25.91	7.66	13.74	11.76
127	37	37	43.51	32.81	35.94	10.22	16.28	9.14
128	38	38	47.78	34.79	32.09	9.94	16.5	11.34
129	39	39	52.75	36.76	28.66	8.32	14.54	12.18
130	40	40	51.97	36.3	32.81	9.34	16.28	11.46
131	15	15	14.68	20.84	9.45	12.84	9.72	15.4
132	16	16	17.23	20.96	9.68	10.84	9.84	15.26
133	17	17	16.88	24.45	10.02	11.58	8.56	15.28
134	18	18	20.98	21.47	11.26	9.82	10.92	14.58
135	19	19	18.85	25.08	12.65	11.56	9.44	13.2
136	20	20	20.34	27.93	11.91	10.7	8.42	14.34
137	21	21	22.39	29.99	11.43	9.5	7.7	14.96
138	22	22	20.17	31.46	14.51	12.64	8.56	13.3
139	23	23	23.69	33.56	12.7	11.1	8.42	16.5
140	24	24	22.91	37.1	13.5	12.74	8.26	16.96
141	25	25	24.64	35.54	15.12	11.98	8.9	15.2
142	26	26	27.92	32.01	17.25	10.38	10.1	13.06
143	27	27	28.49	34.77	17.18	10.22	9.24	13.24
144	28	28	28.82	39.61	16.14	9.64	7.58	13.62
145	29	29	29.37	40.37	17.44	11.78	9.26	15.54
146	30	30	32.53	41.43	16.74	9.7	8.34	15.14
147	31	31	34.23	39.39	18.97	10.28	9.96	14.7
148	32	32	31.04	46.7	19.06	10.3	7.28	13.06
149	33	33	30.12	47.03	21.99	12.98	8.78	13.42
150	34	34	33.58	45.68	22.2	12.54	9.98	14.58
151	35	35	37	48.95	19.82	9.9	8.14	14.74
152	36	36	36.89	47.88	22.72	12.08	10.16	15.26
153	37	37	38.58	51.06	21.7	10.9	8.96	15.32
154	38	38	36.37	56.3	22.56	10.68	7.3	13.38
155	39	39	41.78	50.97	23.84	11.3	10.22	15.62
156	40	40	36.89	63.67	22.62	12.44	7.48	15.8
157	15	15	11.52	17.75	16.82	16.48	11.32	7.54
158	16	16	13.9	19.17	14.71	13.44	10.52	8.98
159	17	17	13.29	19.74	19.13	16.34	11.72	7.6
160	18	18	15.31	20.6	18.3	14.88	12	8.62
161	19	19	16.97	21.84	17.89	14.48	12.3	9.72
162	20	20	15.9	26.33	18.09	16.92	10.68	10.38
163	21	21	16.19	27	20.57	16.26	10.18	8.76
164	22	22	19.37	25.96	20.3	15.68	12.7	10.6
165	23	23	20.64	25.36	22.95	14.18	12.72	8.94
166	24	24	20.14	29.67	22.09	14.08	10.2	9.02
167	25	25	20.51	33.98	20.62	15.24	9.62	10.82
168	26	26	22.37	29.76	25.98	13.22	10.8	7.92
169	27	27	21.17	35.09	25.39	15.6	9.84	9
170	28	28	23.03	36.4	24.58	15.02	10.02	9.94
171	29	29	23.13	36.03	28.5	16.86	11.44	9.42
172	30	30	23.74	34.73	33.3	16.76	12.24	8.04
173	31	31	24.89	39.94	28.54	15.72	10.3	9.56
174	32	32	29.04	34.83	32.08	13.44	12.4	8.54
175	33	33	28.89	38.66	31.05	15.88	12.88	10.42
176	34	34	31.22	35.81	35.31	13.14	12.78	8.12
177	35	35	30.06	40.88	33.98	15.2	12.1	9.4
178	36	36	27.72	42.48	40.37	16.6	11.48	7.62
179	37	37	30.92	40.89	40.8	14.2	11.68	7.32
180	38	38	31.26	49.21	33.55	15.7	10.52	10.32
181	39	39	33.21	43.34	41.46	14.58	12.18	8.02
182	40	40	30.51	49.24	42.56	16.44	10.7	7.94
183	15	15	15.69	17.48	11.15	12.18	12.26	13
184	16	16	16.13	18.21	12.95	12.8	12.68	11.84
185	17	17	18.88	19.96	11.56	10.44	11.18	13.28
186	18	18	19.92	19.66	13.68	10.58	12.28	11.76
187	19	19	19.68	22.1	14.35	12.94	12.9	13.4
188	20	20	21.98	23.38	13.88	10.86	11.54	13.32
189	21	21	20.94	26.7	14.65	12.6	10.82	13.7
190	22	22	22.71	26.89	15.52	10.72	10.04	11.98
191	23	23	21.71	26.09	20.46	13.06	12.02	10
192	24	24	26.44	28.59	16.15	10.38	10.82	13.24
193	25	25	26.44	27.19	20.31	12.48	13.8	12.16
194	26	26	30.28	28.01	18.89	10.3	12.9	12.82
195	27	27	24.63	34.46	21.09	13.46	10.36	11.54
196	28	28	28.11	32.48	22.14	11.14	10.74	10.54
197	29	29	28.43	37.3	20.34	12.5	10.38	13.24
198	30	30	30.72	32.73	25.34	11.52	12.22	10.32
199	31	31	34.66	33.01	24.17	10.46	12.66	11.42
200	32	32	32.43	36.05	26.11	13.06	13.18	12.04
201	33	33	37.02	37.69	23.07	10.08	11.28	12.56
202	34	34	35.9	38.73	25.83	11.46	11.98	12.06
203	35	35	34.47	40.83	28.16	13.18	12.34	11.94
204	36	36	37.25	43.61	25.65	12.6	11.96	13.96
205	37	37	38.17	44.4	26.83	11.06	10.58	11.96
206	38	38	35.12	50.43	27.56	13.58	10.14	12.9
207	39	39	38.43	50	27.3	12.62	10.58	13.48
208	40	40	44.61	42.4	31.5	10.88	13.2	11.7