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	-0.000719	0	-1.587
lnL	0.34996	0	2387.678
lnK	0.400412	0	2582.287
lnM	0.250081	0	2005.66
lnLlnL	-0.020145	0	-844.774
lnKlnK	-0.021157	0	-782.596
lnMlnM	-0.01712	0	-727.541
lnLlnK	0.024176	0	607.654
lnLlnM	0.016163	0	413.539
lnKlnM	0.018013	0	450.213

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	14.74	18.53	11.12	7.88	6.88	7.84
2	16	16	16.99	20.02	10.52	7.22	6.8	9.06
3	17	17	20.22	18.67	11.68	7.06	8.86	9.62
4	18	18	17.35	20.18	15.84	9.66	9.24	7.68
5	19	19	17.64	24.04	14.68	9.92	7.88	8.8
6	20	20	20.44	20.74	18.32	8.76	9.84	7.12
7	21	21	22.23	22.47	17.48	7.64	8.62	7.24
8	22	22	25.14	23.98	16.09	7.4	8.94	8.94
9	23	23	23.14	23.12	22.62	8.34	9.54	6.12
10	24	24	22.01	30.78	18.48	8.78	6.76	7.7
11	25	25	28.43	24.36	21.85	6.8	9.32	6.62
12	26	26	31.86	27.93	17.78	6.16	8.22	8.74
13	27	27	31.97	30.14	18.22	6.4	7.84	8.86
14	28	28	28.46	31.01	23.35	7.3	7.54	6.58
15	29	29	24.26	37.94	24.62	9.06	6.12	6.36
16	30	30	31.34	33.81	23.49	7.1	7.42	7.12
17	31	31	29.17	35.19	27.66	8.84	8.1	6.72
18	32	32	37.02	32.36	25.81	7.2	9.64	7.86
19	33	33	35.15	36.64	25.74	7.94	8.64	8.18
20	34	34	36.62	40.04	23.96	7.6	7.82	8.94
21	35	35	40.99	35.81	27.29	7.06	9.46	8.14
22	36	36	37.87	40.96	27.53	8.12	8.46	8.44
23	37	37	36.32	42.24	30.88	9.74	9.32	8.42
24	38	38	42.94	37.46	32.88	6.26	8.4	6.12
25	39	39	38.51	43.21	33.8	9.32	9.3	7.76
26	40	40	40.14	48.96	29.23	9.64	8.72	10
27	15	15	21.22	10.71	16.32	5.78	14.76	5.62
28	16	16	19.27	12.48	18.63	6.7	12.76	4.98
29	17	17	16.36	15.04	21.99	8.8	11.1	4.44
30	18	18	23.52	13.96	18.92	6.46	13.64	5.96
31	19	19	22.98	15.4	20.61	7.76	14.2	6.3
32	20	20	22.01	16.63	23.71	8.04	12.78	5.26
33	21	21	20.94	17.06	29.89	8.98	13.06	4.22
34	22	22	24.15	18.03	26.81	7.98	12.86	5.04
35	23	23	30.92	19.32	20.41	6.2	12.32	7.22
36	24	24	31.48	18.26	25.95	5.42	11.76	4.86
37	25	25	27.38	21.17	28.92	8.3	12.84	5.56
38	26	26	34.65	22.37	22.46	5.92	11.32	7.04
39	27	27	26.57	23.09	35.84	8.96	12.08	4.5
40	28	28	41.11	20.2	28.51	5.28	13.92	5.8
41	29	29	34.47	23.83	31.47	6.44	11.36	5.12
42	30	30	42.16	22.81	29.61	5.08	11.96	5.5
43	31	31	37.64	26.98	29.71	7.44	12.58	7.02
44	32	32	34.18	26.7	39.32	7.46	11.4	4.52
45	33	33	52.18	25.47	27.32	5.14	13.66	7.86
46	34	34	45.7	29.19	29.15	6.32	12.24	7.66
47	35	35	41.6	31.91	32.04	7.66	11.96	7.44
48	36	36	54.18	26.71	33.86	5.46	14.34	6.78
49	37	37	41.03	29.72	46.02	7.82	13.06	4.88
50	38	38	46.09	28.91	45.75	7.3	14.44	5.26
51	39	38.99	38.16	32.42	54.85	8.58	11.88	4
52	40	40	61.44	32.84	31.09	5	11.94	7.96
53	15	15	19.14	21.92	6.39	5.44	5.3	14.12
54	16	16	20.3	20.15	8.36	6.78	7.82	13.76
55	17	17	21.77	22.7	8.12	6.36	6.92	14.5
56	18	18	24.61	22.15	8.89	4.9	6.34	11.6
57	19	19	22.16	26.24	9.69	7.56	7.08	14.28
58	20	20	26.27	25.15	10.05	6.6	7.94	14.6
59	21	21	22.9	29.56	11.34	6.76	5.72	11.04
60	22	22	28.9	27.81	10.97	6.02	7.2	13.44
61	23	23	36.28	26.5	10.54	4.04	6.68	12.36
62	24	24	35.71	26.9	12.21	4.54	7.24	11.46
63	25	25	29.88	37.28	11.2	6.38	5.62	14.46
64	26	26	29.55	36.13	13.57	6.94	6.26	12.38
65	27	27	36.47	31.33	14.66	5.3	7.24	11.06
66	28	28	31.11	43.69	12.9	6.68	5.12	13.44
67	29	29	37.18	33.8	16.66	6.96	8.9	12.78
68	30	29.99	51.36	30.56	14.86	4.18	8.8	12.84
69	31	31	38.7	39.13	16.46	6.4	7.22	12.5
70	32	32	36.71	40.27	18.92	7.96	8.16	12.4
71	33	33	51.68	34.86	17.19	4.9	8.9	12.78
72	34	34	40.38	51.84	14.91	6.34	5.4	14.62
73	35	35	53.37	43.31	15.13	4.64	6.78	14.6
74	36	36	48.72	44.7	17.76	5.52	6.98	12.92
75	37	37	58.84	41.46	17.45	4.08	7.04	12.18
76	38	38	44.01	54.36	18.65	7.18	6.4	14.08
77	39	39	64.01	44.04	17.57	4.52	8.02	14.78
78	40	40	49.3	58.37	17.77	5.7	5.34	13.52
79	15	15	9.82	26.07	12.02	13.14	4.76	7.4
80	16	16	9.53	26.96	15.4	14.04	4.72	5.7
81	17	17	11.27	28.08	14.37	12.3	4.8	6.6
82	18	18	14.43	27.11	13.22	11.32	6.16	8.96
83	19	19	12.59	34.5	14.08	13.92	4.86	8.72
84	20	20	15.02	27.62	18.28	11.72	6.54	6.64
85	21	21	14.72	25.9	25.32	13.3	7.82	5.02
86	22	22	16.79	30.92	19.14	14.14	7.88	8.66
87	23	23	14.79	35.97	22	14.84	5.96	6.64
88	24	24	16.62	42.99	17.02	11.46	4.28	7.96
89	25	25	15.99	38.15	24.98	12.02	4.94	5.08
90	26	26	19.01	38.11	22.62	11.3	5.7	6.58
91	27	27	20.48	36.94	24.66	13.24	7.56	7.6
92	28	28	19.19	38.83	29.11	14.4	7.18	6.3
93	29	29	19.16	53.52	21.09	13.42	4.58	8.56
94	30	30	16.96	57.4	26.11	14.84	4.04	6.38
95	31	31	23.04	41.82	29.79	11.86	6.72	6.26
96	32	32	23.3	40.56	34.87	13	7.74	5.78
97	33	33	26.63	43.57	29.09	11.9	7.62	7.66
98	34	34	22.15	52.94	32.08	13.46	5.52	6.22
99	35	35	25.92	49.5	31.51	13.6	7.26	7.72
100	36	36	24.01	52.92	35.7	13.08	5.9	5.86
101	37	37	24.64	63.22	29.6	13.1	4.94	7.54
102	38	38	26.01	60.23	32.49	14.8	6.3	8.14
103	39	39	25.4	57.33	40.11	13.18	5.78	5.5
104	40	40	22.25	70.24	40.32	14.82	4.38	5.26
105	15	15	22.01	11.85	13.14	7.14	16.9	9.36
106	16	16	19.2	16.77	11.66	10.1	13.54	12.98
107	17	17	24.85	15.12	12.39	7.62	15.62	12.34
108	18	18	23.76	16.62	14.09	8.1	14.1	10.7
109	19	19	23.96	17.21	16.26	9.76	16.46	11
110	20	20	26.02	18.24	16.25	8.16	14.16	10.14
111	21	21	25.14	20.22	17.47	9.16	13.52	10.04
112	22	22	29.31	20.84	16.35	9.02	15.4	12.8
113	23	23	28.29	21.6	19.19	10.18	15.98	11.48
114	24	24	27.03	25.44	18.66	10.9	13.38	12.04
115	25	25	31.75	22.92	20.79	9.1	15.26	10.7
116	26	26	31.27	25.34	21.1	9.66	14.14	10.96
117	27	27	33.99	23.36	24.9	9.32	16.56	9.6
118	28	28	38	26.08	20.86	8.16	14.52	11.8
119	29	29	34.67	27.11	25.31	9.8	14.96	10.14
120	30	30	40.25	27.41	23.37	8.02	14.4	10.86
121	31	31	38.36	30.56	23.82	9.32	13.92	11.66
122	32	32	42.56	31.29	22.72	8.2	13.46	12.26
123	33	33	38.28	32.67	27.4	10.6	14.6	11.22
124	34	34	50.87	28.62	26.25	7.16	16.1	11.14
125	35	35	44.98	31.65	29.27	9.78	16.9	11.58
126	36	36	42.91	35.25	29.34	9.04	13.02	10.1
127	37	37	48.65	35.52	27.4	8.28	13.7	11.62
128	38	38	48.01	37.66	28.2	9.18	13.96	12.26
129	39	39	52.99	35.97	29.46	8.42	15.18	12
130	40	40	47.08	36.07	37.76	9.72	15.2	9
131	15	15	18.29	18.59	8.41	9.44	10.58	16.82
132	16	16	16.25	22.17	9.66	11.78	9.34	15.52
133	17	17	18.28	21.24	11.02	10.46	10.02	13.56
134	18	18	17.6	26.53	10.47	12	8.46	15.78
135	19	19	18.92	24.96	12.67	12.8	10.56	14.66
136	20	20	20.17	28.25	11.85	11.68	8.98	15.6
137	21	21	20.54	29.44	13.08	10.72	8.02	13.02
138	22	22	21.86	29.47	14.31	11.74	9.44	13.8
139	23	23	21.67	32.97	14.58	12.3	8.58	14
140	24	24	24.91	31.54	15.16	12.52	10.84	16.04
141	25	25	26.02	31.94	16.37	10.58	9.5	13.04
142	26	26	28.01	35.67	14.76	11	9.46	16.7
143	27	27	24.75	38.6	17.82	12.78	8.66	13.44
144	28	28	30.91	39	15.1	9.96	8.66	16.52
145	29	29	27.9	44.36	16.38	12.14	8.04	16.22
146	30	30	31.88	41.88	16.92	11	9.12	16.56
147	31	31	34.44	40.68	18	10.26	9.64	15.72
148	32	32	30.32	48.59	18.61	12.74	8.36	16.16
149	33	33	37.08	41.54	20.02	9.94	9.94	14.66
150	34	34	33.24	52.49	18.59	11.74	7.84	16.62
151	35	35	32.1	49.59	23.36	12.58	8.62	13.06
152	36	36	35.92	51.81	21.08	10.5	7.8	14.04
153	37	37	41.14	47.57	22.08	10.88	10.48	16.16
154	38	38	40.88	51.89	21.75	10.38	8.96	15.58
155	39	39	43.06	49.32	24.03	10.04	9.78	14.24
156	40	40	45.58	53.75	21.79	9.54	8.98	16.24
157	15	15	11.94	20.12	13.17	15.72	9.72	10
158	16	16	14.45	18.8	14.34	14.48	12.14	10.44
159	17	17	15.14	19.36	16.32	14.96	12.8	9.78
160	18	18	14.28	22.95	17.13	14.74	9.64	8.5
161	19	19	16.39	21.95	18.66	13.7	11.1	8.4
162	20	20	17.2	25.42	17.05	14.74	10.64	10.64
163	21	21	18.93	22.57	21.68	13.24	12.3	8.06
164	22	22	17.6	28.04	20.69	16.8	11.1	9.92
165	23	23	21.28	27.66	19.24	13.58	11.4	10.92
166	24	24	20.29	27.41	24.71	15.36	12.32	8.7
167	25	25	20.91	31.61	22.35	15.14	10.64	10
168	26	26	24.23	29.52	23.51	14.24	12.9	10.54
169	27	27	23.56	32.78	24.17	14.2	11	9.84
170	28	28	22.29	33.76	28.9	14.4	10.1	7.58
171	29	29	23.18	31.84	34.51	16.32	12.84	7.3
172	30	30	26.13	36.01	27.39	16.1	12.62	10.88
173	31	31	24.9	34.61	35.65	15.16	11.76	7.1
174	32	32	26.52	36.35	34.19	14.76	11.64	7.82
175	33	33	28.8	36.11	34.69	14.64	12.82	8.4
176	34	34	29.41	38.73	33.99	14.18	11.72	8.54
177	35	35	33.29	39.32	31.26	13.6	12.78	10.46
178	36	36	27.94	43.62	38.28	16.02	10.84	7.9
179	37	37	30.21	48.72	32.19	15.78	10.28	10.46
180	38	38	30.2	52.94	31.62	16.26	9.6	11
181	39	39	30.99	52.69	33.95	16.24	9.94	10.42
182	40	40	33.57	51.39	34.74	15.22	10.54	10.44
183	15	15	15.5	17.42	11.39	12.1	12.06	12.42
184	16	16	14.22	20.73	12.67	13.66	10.02	11.18
185	17	17	17.1	19.02	14.16	11.62	11.72	10.36
186	18	18	18.89	20.9	13.39	12.12	12.3	12.98
187	19	19	19.45	22.25	14.43	12.28	11.98	12.46
188	20	20	21.51	23.56	14.12	10.46	10.74	12.26
189	21	21	24.06	23.1	15.1	10.26	12.3	12.68
190	22	22	22.88	23.62	18.65	12.66	13.94	11.5
191	23	23	23.48	27.4	17.1	10.68	10.18	11.08
192	24	24	25.3	27.52	18.08	10.28	10.64	10.9
193	25	25	28.53	26.35	19.27	10.98	13.78	12.44
194	26	26	28.02	29.51	19.32	10.38	11.16	11.48
195	27	27	25.96	33.74	20.27	13.1	11	12.52
196	28	28	25.88	34.15	22.98	12.32	10.16	10.12
197	29	29	30.3	33.57	21.81	10.88	11.02	11.44
198	30	30	31.53	31.87	25.49	12.12	13.68	11.12
199	31	31	33.06	37.72	21.1	10.4	10.18	12.6
200	32	32	31.94	37.94	24.68	12.94	12.08	12.52
201	33	33	31.34	40.32	26.03	13.44	11.42	11.94
202	34	34	38.56	36.14	26.09	11.14	13.74	12.6
203	35	35	32.61	43.46	27.67	13.2	10.76	11.44
204	36	36	41.93	38.18	26.76	10.22	13.04	12.38
205	37	37	39.05	42.97	27.33	12.42	12.68	13.5
206	38	38	45.58	40.23	27.28	10.32	13.66	13.48
207	39	39	43.41	39.34	33.22	10.24	13.14	10.02
208	40	40	40.46	46.38	31.33	11.98	11.66	11.56