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	2.0E-6	0	0.004
lnL	0.350314	0	2688.462
lnK	0.400064	0	2825.727
lnM	0.249571	0	2062.168
lnLlnL	-0.020192	0	-885.213
lnKlnK	-0.021117	0	-947.043
lnMlnM	-0.01719	0	-758.781
lnLlnK	0.02407	0	661.054
lnLlnM	0.016274	0	448.189
lnKlnM	0.018169	0	473.163

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	15.64	17.22	11.46	8.76	8.94	9.02
2	16	16	14.4	18.03	15.38	9.8	8.6	6.48
3	17	17	15.7	20.59	14.11	8.74	7.26	7.08
4	18	18	16.75	19.58	17.43	8.92	8.48	6.08
5	19	19	18.16	22.51	15.56	9.08	8.06	7.78
6	20	20	20.93	22.73	15.43	7.08	7.34	7.24
7	21	21	23.03	23.7	15.39	6.5	7.18	7.46
8	22	22	24.25	23.33	17.59	8.06	9.64	8.4
9	23	23	22.67	24	21.94	9	9.62	6.68
10	24	24	26.05	26.24	18.69	8.26	9.36	8.72
11	25	25	21.83	36.61	17.02	9.84	6.12	9.42
12	26	26	32.03	27.38	18.19	6.46	8.88	8.98
13	27	27	29.88	28.38	21.76	7.56	9.18	7.84
14	28	28	34.34	27.99	21.31	6.66	9.68	8.34
15	29	29	30.38	30.35	25.32	8.34	9.54	7.38
16	30	30	33.05	33.86	21.85	8.12	9.02	9.44
17	31	31	33.68	35.71	22.31	6.88	7.34	7.98
18	32	32	37.83	33.43	23.87	7.02	9.28	8.62
19	33	33	36.71	34.1	27.11	7.88	9.82	8.04
20	34	34	38.91	36.96	24.92	7.84	9.52	9.46
21	35	35	34.48	39.11	30.04	8.12	8	6.82
22	36	36	31.12	43.83	32.5	9.4	7.18	6.38
23	37	37	37.3	39.11	33.52	9.16	9.9	7.42
24	38	38	37.3	48.31	27.06	8.66	7.3	9.02
25	39	39	39.98	47.37	28.02	8.46	7.92	9.18
26	40	40	40.55	48.46	29.27	7.92	7.34	8.3
27	15	15	17.79	12.48	16	7.74	13.42	6.26
28	16	16	20.04	13.21	16.06	7.94	14.82	7.36
29	17	17	22.4	13	18.09	6.36	13.78	5.84
30	18	18	20.4	14.68	21.14	7.68	12.92	5.26
31	19	19	24.06	13.99	22.81	6.76	14.62	5.14
32	20	20	27.99	15.41	19.41	5.68	13.1	6.24
33	21	21	29.16	15.6	21.9	5.5	13.12	5.5
34	22	22	24.85	17.23	27.84	7.84	13.78	4.9
35	23	23	33.97	15.32	26.68	5.1	14.88	4.84
36	24	24	32.09	18.33	25.13	5.46	12.06	5.2
37	25	25	32.89	19.84	25.07	7.24	15	7.12
38	26	26	40.69	20.36	21.29	5.2	13.42	7.96
39	27	27	36.93	22.04	24.55	6.34	13.3	7.32
40	28	28	30.83	21.4	38.39	7.54	13.24	4.16
41	29	29	36.67	25.05	26.66	6.64	11.88	6.9
42	30	30	36.34	25.85	29.34	8.38	14.3	7.7
43	31	31	37.51	25.68	32.4	7.26	12.96	6.16
44	32	32	34.45	27.34	37.37	8.32	12.48	5.4
45	33	33	50.02	26.82	26.51	5.16	12.28	7.78
46	34	34	44.3	28.18	32.15	6.72	13.08	7
47	35	35	47.5	30.09	29.57	6.22	12.16	7.76
48	36	36	52.73	29.75	29.39	5.56	12.46	7.9
49	37	37	38.88	33.81	39.96	8.3	11.18	5.74
50	38	38	43.03	34.99	36.54	7.74	11.28	6.7
51	39	38.99	39.51	32.9	50.79	8.24	11.68	4.38
52	40	40	50.82	30.71	44.46	6.22	12.86	5.2
53	15	15	17.64	21.07	7.41	6.06	5.62	12.02
54	16	16	17.38	23.53	8.18	8	6.4	13.86
55	17	17	18.66	24.74	8.7	7.12	5.84	12.48
56	18	18	24.9	20.64	9.66	5.74	8.18	12.5
57	19	19	29.54	21.13	9.29	5.04	8.54	14.04
58	20	20	28.57	23.4	10.05	5.26	7.6	12.84
59	21	21	29.72	26.86	9.45	5.16	6.64	14.18
60	22	22	23.67	34.32	10.54	7.26	5.36	13.44
61	23	23	28.2	31.35	11.25	6.82	6.88	14.36
62	24	24	27.26	29.69	14.72	7.74	8	11.42
63	25	25	34.55	34.43	10.49	5.14	5.9	14.92
64	26	26	38.87	30.12	12.57	4.42	6.82	11.92
65	27	27	34.31	39.44	11.56	5.34	5.18	13.72
66	28	28	36.92	37.67	12.79	5.84	6.52	14.52
67	29	29	38.39	32.71	16.79	6.32	8.72	11.94
68	30	30	35.41	35.31	18.82	7.76	8.9	11.66
69	31	31	47.92	36.85	13.93	4.6	7.16	14.06
70	32	32	41.15	38.4	17.58	6.92	8.58	13.44
71	33	33	49.09	34.52	18.51	4.96	8.58	11.16
72	34	33.99	56.22	35.22	17.11	4.2	8.32	12.16
73	35	35	40.4	47.5	18.51	6.16	5.82	11.02
74	36	36	52.13	44.95	16.3	5.1	6.94	14.3
75	37	37	49	47.16	18.07	6.34	7.58	14.64
76	38	38	58.36	43.49	18.11	4	6.46	11.32
77	39	39	42.92	59.89	18.55	7.2	5.56	13.8
78	40	40	44.55	64.73	17.53	6.9	5.08	14.76
79	15	15	9.4	24.27	14.18	14.9	5.58	6.56
80	16	16	9.75	26.79	15.01	13.44	4.68	5.78
81	17	17	12.96	23.9	14.81	11.04	6.14	6.74
82	18	18	13.28	24.06	17.74	11.34	6.44	5.76
83	19	19	12.49	29.27	17.97	12.2	5.12	5.68
84	20	20	13.17	31.36	18.38	13.84	5.7	6.68
85	21	21	15.01	32.6	17.38	14.08	6.46	8.46
86	22	22	15.64	36.63	16.62	11.34	4.76	7.54
87	23	23	14.39	37.39	21.67	14.64	5.44	6.46
88	24	24	17.44	31.43	24.93	13.68	7.82	6.42
89	25	25	16.87	39.87	21.64	13.96	5.8	7.44
90	26	26	18.31	36.8	25.16	14.12	7.1	6.94
91	27	27	18.27	40.6	25.34	11.64	5.2	5.66
92	28	28	16.8	44.24	29.3	14.76	5.4	5.48
93	29	29	22.68	40.12	24.93	12.8	7.48	8.18
94	30	30	20.37	51.06	23.47	12.9	5	7.8
95	31	31	22.57	45.17	27.37	11.56	5.84	6.58
96	32	32	22.96	56.53	21.99	11.92	4.72	8.96
97	33	33	20.99	45.52	38.76	14.92	6.86	5.18
98	34	34	20.73	62.34	28.04	13.1	4.1	6.56
99	35	35	24.45	47.43	36.61	14	7.34	6.22
100	36	35.99	21.72	72.89	26.34	14.62	4.02	8.32
101	37	37	24.14	62.06	31.3	11.48	4.32	6.04
102	38	38	28.03	57.72	31.01	11.26	5.5	7.14
103	39	39	28.59	53.28	37.53	12.96	7.12	6.72
104	40	40	31.53	51.38	38	11.72	7.54	6.72
105	15	15	18.77	13.31	13.39	8.88	15.2	9.44
106	16	16	20.69	14.16	13.74	8.94	15.96	10.34
107	17	17	22.05	15.19	14.32	9.32	16.52	11.06
108	18	18	25.53	15.56	14.25	7.88	16.12	11.18
109	19	19	25.97	16.83	15.18	7.68	14.62	10.32
110	20	20	22.84	18.63	18.66	10.84	15.74	9.82
111	21	21	26.6	20.34	16.08	8.36	13.1	10.8
112	22	22	28.85	19.27	18.85	9.2	16.9	10.84
113	23	23	26.24	20.92	22.33	10.7	15.96	9.24
114	24	24	29.64	23.68	18.46	8.92	13.28	11.12
115	25	25	36.42	22.88	17.53	7.12	14.06	12.02
116	26	26	37.83	24.22	17.77	7.02	13.56	12.2
117	27	27	36.53	25.89	19.32	8.38	14.36	12.66
118	28	28	35.37	27.03	21.64	9.36	14.68	11.92
119	29	29	33.94	28.67	23.85	9.54	13.3	10.32
120	30	30	36.66	27.78	25.82	10.42	16.5	11.24
121	31	31	43.32	26.6	25.3	7.78	15.78	10.46
122	32	32	40.66	31.42	23.92	9.42	14.58	12.56
123	33	33	43.75	28.2	29.02	7.86	15.06	9.1
124	34	34	42.1	31.17	29.22	9.24	15.04	10.14
125	35	35	42	33.7	29.02	9.48	14.04	10.46
126	36	36	47.39	34.31	26.92	9.24	15.44	12.84
127	37	37	51.53	31.5	30.72	7.16	14.6	9.4
128	38	38	48.37	32.75	34.75	8.54	15.44	9
129	39	39	47.15	36.76	33.11	10.16	15.56	11
130	40	40	55.26	37.5	28.87	8.44	15.22	12.94
131	15	15	15.58	21.31	8.5	9.64	7.62	14.04
132	16	16	17.44	20.98	9.53	11.18	10.28	16.26
133	17	17	15.08	26.01	10.67	12.56	7.56	13.48
134	18	18	19.81	22.31	11.46	10.04	9.98	13.66
135	19	19	23.01	24.26	10.3	9.16	9.84	16.84
136	20	20	22.34	25.67	11.9	10.78	10.46	16.16
137	21	21	20.99	26.61	14.67	12.12	10.48	13.2
138	22	22	22.47	27.73	15.06	11.68	10.42	13.36
139	23	23	20.67	37.98	12.8	12.96	7.24	16.26
140	24	24	25.01	30.43	15.87	12.1	10.98	14.76
141	25	25	23.27	39.44	14.13	11.46	7.04	14.72
142	26	26	27.42	34.44	15.91	9.98	8.72	13.52
143	27	27	27.2	41.34	14.38	10.74	7.5	16.24
144	28	28	28.95	36.71	17.83	12.22	10.56	15.44
145	29	29	27.08	43.52	17.47	12.66	8.28	15.14
146	30	30	35.16	39.75	16.09	9.18	9.08	16.44
147	31	31	31.6	44.83	17.56	10.8	8.18	15.4
148	32	32	34.75	46.46	16.65	9.58	7.78	16.26
149	33	33	39.8	42.82	17.59	9.04	9.48	16.88
150	34	34	35.81	44.02	21.52	10.06	9.02	13.08
151	35	35	39.8	42.16	22.27	9.7	10.36	13.72
152	36	36	43.05	42.39	22.21	9.28	10.8	14.44
153	37	37	33.55	59.77	20.97	12.98	7.52	16.12
154	38	38	36.27	58	21.74	12.1	7.96	15.78
155	39	39	43.08	51.31	22.73	9.74	9.06	14.76
156	40	40	42.56	54	23.58	9.1	7.86	13.02
157	15	15	12.61	17.3	15.32	16.4	12.92	9.32
158	16	16	12.27	21.01	15.29	16.9	10.26	9.32
159	17	17	13.82	22.68	14.61	16.08	10.26	10.76
160	18	18	15.48	20.64	17.97	15.42	12.56	9.24
161	19	19	14.98	25.2	17.19	14.58	9.04	8.86
162	20	20	15.24	25.84	19.79	14.66	9	7.7
163	21	21	19.1	25.84	17.38	13.28	10.64	10.6
164	22	22	18.82	24.6	23	13.72	11.44	7.74
165	23	23	19.23	25.79	24.75	15.52	12.56	8.24
166	24	24	17.52	30.05	26.57	16.64	10.08	7.28
167	25	25	22.01	29.84	22.67	14.32	11.44	9.88
168	26	26	21.63	31.79	24.63	16.54	12.02	10.14
169	27	27	24.84	31.88	23.41	14.34	12.22	10.98
170	28	28	23.09	31.46	30.66	16.14	12.82	8.26
171	29	29	23.64	34.53	29.49	16.78	12.28	9.24
172	30	30	25.68	37.46	26.44	15.88	11.64	10.96
173	31	31	28.34	36.59	27.13	14.42	12.2	10.84
174	32	32	27.74	38.54	29.25	13.32	10.34	8.94
175	33	33	27.28	36.15	37.48	14.28	11.72	7.02
176	34	34	28.36	39.02	35.41	15.74	12.36	8.66
177	35	35	29.12	39.43	37.66	15.5	12.4	8.18
178	36	36	27.85	42.33	40.33	16.84	11.76	7.78
179	37	37	32.71	44.79	32.66	13.5	10.66	9.66
180	38	38	31.43	46.52	36.23	14.1	10.16	8.52
181	39	39	31.21	43.04	45.97	16.12	12.62	7.3
182	40	40	36	44.04	39.85	14.42	13	9.14
183	15	15	16.28	17.59	10.54	10.14	10.58	12.08
184	16	16	16.74	18.26	12.26	11.2	11.56	11.54
185	17	17	17.26	20.47	12.53	11.14	10.42	11.6
186	18	18	17.37	22.44	13.5	13.72	11.6	13.18
187	19	19	18.41	23.39	14.43	12.64	10.9	12.02
188	20	20	20.99	22.75	15.35	11	11.44	11.36
189	21	21	22.89	23.99	15.23	10.82	11.7	12.48
190	22	22	23.11	23.98	17.98	11.42	12.5	10.96
191	23	23	22.92	26.25	18.82	13.28	12.94	11.96
192	24	24	26.15	26.42	18.41	11.32	12.78	12.22
193	25	25	23.01	31.45	19.71	13.98	11.06	11.98
194	26	26	24.58	31.12	21.28	13.54	11.72	11.46
195	27	27	27.55	28.82	23.7	12.22	13.24	10.42
196	28	28	26.17	35.58	21.29	12.96	10.32	11.8
197	29	29	28.08	34.22	23.47	12.9	11.68	11.38
198	30	30	28.6	36.39	23.8	12.36	10.64	10.96
199	31	31	30.05	38.33	23.37	13.12	11.26	12.6
200	32	32	33.52	40.26	21.24	11.18	10.3	13.66
201	33	33	31.56	43.92	22.77	13.22	10.24	13.86
202	34	34	36.97	38.39	25.18	11.8	12.9	13.24
203	35	35	36.44	39.67	27.29	10.58	10.94	10.62
204	36	36	38.7	43.7	24.34	10.66	10.56	13.14
205	37	37	36.25	42.36	30.83	12.82	12.2	11.08
206	38	38	43.98	41.18	27.58	11.16	13.78	13.8
207	39	39	37.22	48.35	29.91	13.12	11.02	12.12
208	40	40	42.21	50.34	26.3	10.96	10.18	13.66