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	0.000502	0	1.432
lnL	0.350103	0	3099.198
lnK	0.400102	0	3244.783
lnM	0.249458	0	2430.154
lnLlnL	-0.0202	0	-964.403
lnKlnK	-0.02116	0	-868.798
lnMlnM	-0.017265	0	-889.969
lnLlnK	0.02407	0	738.995
lnLlnM	0.016362	0	496.142
lnKlnM	0.018249	0	547.033

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	14.49	17.52	12.37	9.04	8.26	7.78
2	16	16	17.83	15.62	14.32	6.92	9.24	6.4
3	17	17	20.02	19.81	10.85	6.76	7.82	9.92
4	18	18	16.73	20.19	16.65	9.8	8.98	7.04
5	19	19	18.94	24.47	13.01	7.76	6.56	8.62
6	20	20	19.41	23.33	16.42	9.8	9.02	8.52
7	21	21	23.68	24.05	14.53	7.52	8.44	9.54
8	22	22	21.96	27.42	15.77	8.84	7.78	9.32
9	23	23	19.87	27.19	21.74	9.74	7.7	6.26
10	24	24	24.01	29.49	17.51	8.56	7.68	8.86
11	25	25	27.18	29.47	17.38	8.16	8.48	9.86
12	26	26	25.3	30.84	20.74	7.58	6.86	6.84
13	27	27	30.28	28.78	20.93	7.78	9.44	8.58
14	28	28	32.07	29.72	21.25	7.04	8.8	8.16
15	29	29	31.79	30.24	23.95	6.22	7.54	6.2
16	30	30	32.07	38.39	18.94	6.66	6.16	8.84
17	31	31	39.08	30.44	23.4	6.34	9.72	8.28
18	32	32	33.91	34.2	26.63	7.46	8.44	7.08
19	33	33	33.86	40.06	23.7	9	8.44	9.78
20	34	34	31.5	44.79	24.84	9.72	7.34	9.18
21	35	35	41.81	38.24	24.11	6.9	8.76	9.42
22	36	36	32.64	43.32	30.96	9.84	8.06	7.48
23	37	37	37.53	42.54	29.23	7.26	7.16	6.96
24	38	38	46.14	37.03	30.46	6.7	9.92	7.8
25	39	39	44.91	40.72	30.13	7.72	9.9	8.82
26	40	40	37.31	45.88	35.57	9.24	8.28	6.98
27	15	15	20.45	12.45	13.34	6.4	13.12	7.56
28	16	16	17.41	13.62	18.5	8	12.2	5.32
29	17	17	24.71	13.46	15.02	6.22	14.52	7.98
30	18	18	18.75	16.9	18.82	8.52	11	6.06
31	19	19	25.58	15.43	17.78	6.48	13.42	7.1
32	20	20	29.84	16.12	16.6	5.52	13.02	7.86
33	21	21	24.78	19.37	19.06	7.32	11.18	7.12
34	22	22	23.68	18.27	26.95	7.3	11.32	4.48
35	23	23	27.71	17.89	26.91	6.68	12.78	4.94
36	24	24	30.3	19.05	25.44	7.34	14.48	6.44
37	25	25	32.01	19	27.97	6.64	14.02	5.56
38	26	26	33.58	19.05	30.63	5.98	13.32	4.76
39	27	27	37.26	20.74	26.84	5.92	13.48	6.22
40	28	28	35.18	24.28	25.8	6.98	12.34	7.18
41	29	29	39.14	23.21	27.72	5.3	11.2	5.68
42	30	30	37.41	23.54	32.95	7.5	14.78	6.22
43	31	31	43.25	23.67	30.67	5.4	12.54	5.78
44	32	32	41.46	27.86	28.14	6.14	11.2	6.92
45	33	33	37.73	29.41	33.03	8.5	13.02	7.1
46	34	34	47.55	27.01	31.4	5.74	12.76	6.68
47	35	35	43.81	25.59	43.21	6.86	14.76	4.98
48	36	36	51.66	30.07	29.65	5.26	11.36	7.22
49	37	37	47.69	31.13	34.62	7.3	13.78	7.6
50	38	38	43.66	30.06	46.14	7.5	13.3	5.02
51	39	39	44.61	29.32	52.08	7.66	14.34	4.56
52	40	40	44.65	33.36	46.21	8.98	14.46	6.16
53	15	15	19.75	19.3	7.28	6.22	7.3	14.36
54	16	16	23.23	18.04	8.3	5.6	8.62	13.42
55	17	17	25.18	18.49	9.07	5.44	8.94	12.92
56	18	18	21.67	27.17	7.87	6.28	5.5	14.76
57	19	19	24.81	23.95	9.58	5.54	6.6	12.12
58	20	20	24.86	26.06	10.22	7.38	7.98	15
59	21	21	23.43	29.89	10.86	7.32	6.28	12.92
60	22	22	27.5	26.89	12.19	6.12	7.18	11.38
61	23	23	27.04	31.22	11.89	6.86	6.62	12.88
62	24	24	28.2	33.3	12.06	7.64	7.18	14.82
63	25	25	36.77	27.07	13.52	4.98	8.16	11.54
64	26	26	28.09	40.39	12.45	7.86	5.86	14.62
65	27	27	40.61	35.66	10.88	4.4	5.86	14.8
66	28	28	45.4	29.99	13.75	4.34	8.08	12.64
67	29	29	42.78	37.23	12.4	4.74	6.38	14.54
68	30	30	40.13	37.9	14.66	6.2	7.58	14.48
69	31	31	45.43	42.03	12.43	4.2	5.26	13.78
70	32	32	43.82	46.49	12.67	4.84	5.16	14.88
71	33	33	41.36	45.72	15.39	6.36	6.46	14.54
72	34	34	48.79	38.17	18	5.9	9	13.62
73	35	35	42.85	40.69	21.36	6.9	8.38	11.18
74	36	36	47.56	50.34	15.6	5.22	5.58	13.84
75	37	37	45.75	46.88	19.78	6.88	7.64	13.18
76	38	38	50.56	49.98	17.7	5.18	6	12.72
77	39	39	59.14	43.52	19.53	4.6	7.54	12.1
78	40	40	48.52	56.4	18.94	5.62	5.38	12.14
79	15	15	8.91	25.01	14.74	14.74	5	5.82
80	16	16	10.93	27.59	12.18	12.84	4.94	8.08
81	17	17	11.79	23.28	17.68	13.44	6.9	5.96
82	18	18	12.71	25.79	17.04	13.36	6.64	6.76
83	19	19	15.83	24.89	16.19	11.42	7.66	7.94
84	20	20	14.97	29.08	17.02	12.54	6.56	7.7
85	21	21	12.13	40.42	17.56	14.5	4.02	6.7
86	22	22	19.07	29.39	17.24	11.12	7.64	8.94
87	23	23	14.68	38.46	20.18	12.66	4.66	6.22
88	24	24	17.18	32.44	24.31	11.5	6.22	5.46
89	25	25	15.57	43.59	21.43	13.86	4.72	6.8
90	26	26	19.24	35.98	24.2	11.88	6.5	6.48
91	27	27	19.57	38.28	24.99	13.18	6.84	7.06
92	28	28	19.02	43.12	25.24	14.26	6.22	7.3
93	29	29	23.91	37.09	25.99	11.06	7.54	7.16
94	30	30	21.19	42.68	28.69	13.48	6.76	6.74
95	31	31	21.23	54.22	23.06	12.34	4.68	8
96	32	32	21.78	44.29	33.71	12.54	6.22	5.36
97	33	33	22.68	50.74	29.29	12.86	5.7	6.8
98	34	34	25.34	49.11	29.41	12.58	6.6	7.54
99	35	35	26.42	55.59	25.95	11.84	5.64	8.64
100	36	36	23.27	52.16	38.26	14.02	6.2	5.58
101	37	37	26.49	56.87	30.98	11.18	5.2	6.64
102	38	38	29.07	55.65	31.03	11.46	6.1	7.58
103	39	39	25.36	64.33	33.95	13.5	5.16	6.84
104	40	40	29.72	52.38	40.24	12.92	7.58	6.46
105	15	15	21.27	12.21	13.08	7.68	16.86	9.72
106	16	16	23.04	13.64	12.71	7.12	15.08	10.24
107	17	17	21.7	14.88	15.12	8.82	15.72	9.64
108	18	18	21.49	16.27	16.6	9.32	14.78	9.02
109	19	19	22.31	17.52	17.37	9.72	14.76	9.32
110	20	20	27.79	17.27	16.28	7.78	15.56	10.42
111	21	21	30.58	16.83	18.15	7.14	16.48	9.42
112	22	22	30.48	19.5	17.25	7.68	14.84	10.72
113	23	23	30.46	21.35	17.78	7.58	13.16	10.2
114	24	24	26.88	24.84	19.49	10.88	13.64	11.34
115	25	25	31.25	23.24	20.77	10.3	16.68	11.9
116	26	26	35.56	23.86	19.64	8.02	14.66	11.52
117	27	27	36.76	25.72	19.37	8.3	14.44	12.6
118	28	28	40.11	24.69	21.22	8.06	16.3	12.18
119	29	29	37.83	26.68	23.17	7.96	13.72	10.12
120	30	30	38.95	27.05	24.89	8.58	15.06	10.38
121	31	31	45.66	26.21	24.23	7.36	16.16	11.08
122	32	32	37.69	31.75	25.96	10.04	14.04	11.12
123	33	33	43.94	32.86	22.76	8.38	13.48	12.98
124	34	34	49.09	30.18	25.26	7.78	15.76	12.14
125	35	35	48.01	34.49	23.62	7.78	13.12	12.8
126	36	36	49.85	32.79	27.08	7.92	14.82	11.6
127	37	37	44.06	33.59	34.02	10.72	16.86	10.38
128	38	38	56.56	30.95	31.11	7	16.26	10.1
129	39	39	47.61	36.9	32.5	9.22	14.22	10.32
130	40	40	47.07	37.51	35.47	10.92	16.3	10.88
131	15	15	13.37	22.77	9.45	12.96	7.92	13.92
132	16	16	15	22.83	10.28	12.48	8.7	13.9
133	17	17	16.83	25.55	9.48	10.04	7.02	14.08
134	18	18	18.58	23.48	11.57	12.42	10.78	15.5
135	19	19	18.13	29.77	10.51	11.76	7.5	15.96
136	20	20	21.22	26.52	12.13	11.58	10.18	15.96
137	21	21	23.29	26.25	13.09	11.06	10.98	15.56
138	22	22	22.33	30.68	13.17	10.68	8.4	14.2
139	23	23	24.62	29.91	14.17	11.88	10.8	16.26
140	24	24	28.63	29.41	14.03	9.9	10.96	16.36
141	25	25	26.86	30.23	17	10.82	10.76	13.24
142	26	26	22.94	40.32	16.13	12.88	7.58	13.92
143	27	27	32.7	34.2	14.8	9.3	10.08	16.88
144	28	28	26.13	39.7	18.26	12.14	8.48	13.22
145	29	29	31.43	38.3	17.22	10.4	9.42	15.08
146	30	30	30.11	46.61	15.74	10.3	7.04	15.78
147	31	31	29.31	45.89	18.72	11.42	7.7	13.82
148	32	32	35.83	38.67	20.66	9.86	10.3	13.46
149	33	33	38.75	40.95	19.33	9.02	9.66	14.6
150	34	34	42.89	41.01	18.98	9.08	10.94	16.92
151	35	35	39.14	44.46	21.09	11.04	10.86	16.32
152	36	36	33.45	56.03	20.78	12.36	7.7	15.46
153	37	37	42.15	49.28	20.41	9.34	8.88	15.66
154	38	38	43.9	48.45	21.87	10.24	10.42	16.6
155	39	39	43.22	50.75	22.98	9.28	8.78	13.94
156	40	40	45.2	53.71	22.04	9.56	8.92	15.9
157	15	15	11.63	17.28	17.29	16.92	12.14	7.58
158	16	16	13.58	17.68	17.24	13.08	10.96	7.06
159	17	17	14.43	21.23	15.16	15.88	11.52	10.7
160	18	18	14.06	22.75	17.74	14.98	9.72	8.14
161	19	19	15.88	21.72	19.84	14.62	11.56	8.04
162	20	20	16.81	25.61	17.42	15.86	11.04	10.86
163	21	21	16.45	28.76	18.28	16.04	9.5	10.12
164	22	22	18.13	26.21	22	14.1	10.44	8.02
165	23	23	18.79	29.17	21.16	16	10.9	9.94
166	24	24	21.3	30.38	19.71	13.98	10.52	10.94
167	25	25	20.18	30.16	25.26	15.64	11.14	8.58
168	26	26	20.33	30.59	28.63	15.96	11.28	7.62
169	27	27	21.9	36.83	22.51	15.68	9.72	10.84
170	28	28	22.33	37.58	24.48	16.66	10.32	10.68
171	29	29	24.81	36.97	24.8	13.04	9.32	9.32
172	30	30	25.34	37.63	26.75	15.1	10.84	10.12
173	31	31	24.3	36.72	33.66	15.9	11.18	7.74
174	32	32	27.13	36.48	32.9	14.58	11.76	8.3
175	33	33	29.93	38.56	29.66	13.44	11.4	9.7
176	34	34	24.97	46.36	32.71	16.96	9.36	8.82
177	35	35	29.01	43.8	32.19	16.14	11.36	10.2
178	36	36	32.33	42.71	32.05	15.08	12.42	10.88
179	37	37	32.72	38.81	40.83	13.84	12.94	7.62
180	38	38	30.04	52	32.7	16.52	9.9	10.68
181	39	39	28.88	48.17	43.16	16.04	10.04	7.14
182	40	40	33.25	45.84	41.95	15.42	12.08	8.38
183	15	15	17.4	16.2	10.91	10.14	12.6	12.54
184	16	16	16.96	18.65	11.68	11.9	12.16	13.18
185	17	17	18.68	19.05	12.55	12.12	13.54	13.82
186	18	18	20.34	18.24	14.93	10.3	13.38	10.58
187	19	19	20.97	21.83	13.44	11.12	12.12	13.4
188	20	20	22.06	21.47	15.68	11.68	13.78	12.46
189	21	21	22.33	23.75	15.97	10.76	11.44	11.4
190	22	22	21.98	24.97	18.1	13.48	13.26	12.1
191	23	23	22.23	29.16	16.78	13.44	11.16	13.36
192	24	24	23.22	29.8	18.04	11.9	10.14	11.44
193	25	25	25.83	29.15	18.88	13.42	13.3	13.86
194	26	26	29.99	28.57	18.57	10.94	13.24	13.74
195	27	27	27.85	32.36	19.62	12.8	12.26	13.8
196	28	28	27.85	31.96	22.97	13.46	13.08	12.06
197	29	29	34.78	30.19	21.35	10	13.5	12.68
198	30	30	33.18	33.7	21.89	11.96	13.42	13.94
199	31	31	34.35	35.44	21.99	10.62	11.7	12.82
200	32	32	32.09	38.19	24.29	11.38	10.6	11.28
201	33	33	38.44	34.93	24.61	10.24	13.1	12.36
202	34	34	37.83	38.47	24.36	10.96	12.28	13.14
203	35	35	39.11	38.8	25.7	11.96	13.8	14
204	36	36	33.11	43.84	29.81	13.78	11.32	11.14
205	37	37	43.01	40.74	26.08	10.26	12.5	13.2
206	38	38	34.39	44.86	33.72	13.9	11.62	10.16
207	39	39	38.89	47.38	29.05	12.32	11.16	12.4
208	40	40	38.14	47.26	32.99	12.68	11.26	10.74