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	-0.000488	0	-1.401
lnL	0.350228	0	3127.382
lnK	0.400268	0	3190.205
lnM	0.249803	0	2406.505
lnLlnL	-0.020158	0	-1023.44
lnKlnK	-0.021128	0	-930.247
lnMlnM	-0.01721	0	-812.746
lnLlnK	0.024042	0	798.685
lnLlnM	0.016256	0	530.571
lnKlnM	0.018151	0	533.787

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	14.59	15.73	14.46	9.12	9.54	6.58
2	16	16	15.36	16.89	15.56	9.1	9.3	6.4
3	17	17	15.91	20.28	14.18	9.66	8.3	7.9
4	18	18	17.16	22.66	13.51	9.32	7.68	8.82
5	19	19	16.52	25.64	14.61	9.16	6.24	7.56
6	20	20	19.97	23.83	15.3	9.78	9.08	9.56
7	21	21	24.6	24.53	13.45	6.26	7.18	9.1
8	22	22	24.18	23.67	17.27	7.22	8.46	7.66
9	23	23	27.79	26.37	14.55	6.5	7.9	9.94
10	24	24	23.18	32.58	15.9	8.46	6.48	9.42
11	25	25	26.18	26.23	21.75	8.24	9.4	7.32
12	26	26	21.81	35.04	21.04	9.26	6.06	6.9
13	27	27	25.08	31.89	23.09	7.8	6.72	6.14
14	28	28	29.14	33.47	20.19	9	8.74	9.9
15	29	29	29.57	30.46	26.12	8.88	9.8	7.34
16	30	30	31.76	35.65	21.34	8.14	8.12	9.28
17	31	31	32.66	33.4	25.68	7.26	8.08	6.88
18	32	32	36.81	37.87	20.58	6.26	6.92	8.86
19	33	33	37.16	36.89	23.7	7.72	8.9	9.36
20	34	34	34.18	42.96	23.67	7.58	6.62	8.34
21	35	35	36.7	39.66	27.05	7.88	8.22	8.06
22	36	36	34.66	41.64	30.26	9.64	8.88	8.08
23	37	37	36.46	45.72	27.32	6.92	6.06	6.94
24	38	38	34.65	48.53	29.68	7.96	6.12	6.82
25	39	39	44.74	39.11	32.22	7.44	9.96	7.82
26	40	40	37.55	44.62	36.79	9.54	8.9	6.98
27	15	15	18.26	13.1	14.25	6.8	11.48	6.5
28	16	16	18.8	12.7	18.72	6.94	12.58	4.98
29	17	17	23.2	13.2	16.84	5.86	13	6.1
30	18	18	25.44	13.76	17.47	5.74	13.52	6.38
31	19	19	23.06	15.78	19.7	6.54	11.68	5.62
32	20	20	23.41	15.4	24.79	6.98	13.06	4.66
33	21	21	24.07	16.43	26.09	6.62	11.86	4.3
34	22	22	29.16	19.79	17.83	6.2	11.18	7.9
35	23	23	27.97	18.77	24.46	6.6	12.06	5.52
36	24	24	29.77	17.17	31.18	6.86	14.98	4.64
37	25	25	34.46	21.17	21.21	5.98	12.12	7.56
38	26	26	31.4	21.22	27.92	7.56	13.7	6.2
39	27	27	31.9	23.15	27.54	7.02	11.7	5.96
40	28	28	31.1	23.83	31.48	8.52	13.32	6
41	29	29	32.58	24.27	33.01	8.22	13.28	5.78
42	30	30	32.57	24.8	36.62	7.2	11.34	4.48
43	31	31	38.28	23.24	37.34	6.5	13.36	4.78
44	32	32	38.24	29	29.36	7.14	11.3	6.96
45	33	33	39.07	30.35	29.94	7.88	12.12	7.7
46	34	34	41.08	29.59	32.82	7.46	12.54	6.94
47	35	35	39.68	33.95	30.93	8.04	11.04	7.7
48	36	36	38.87	31.46	40.3	7.86	11.52	5.38
49	37	37	52.86	30.93	30.61	5.58	11.98	7.58
50	38	38	43.04	36.93	33.52	8.2	11.22	7.86
51	39	39	56.86	30.85	34.37	5.08	11.92	6.56
52	40	40	55.71	28.54	44.51	5.14	12.9	4.78
53	15	15	14.74	23.13	8.16	7.88	5.3	11.28
54	16	16	18.51	23.72	7.5	7.24	6.18	14.96
55	17	17	24.56	19.73	8.53	5.18	7.66	12.84
56	18	18	21.56	24.72	8.96	6	5.84	12.04
57	19	19	23.91	25.13	9.38	5.14	5.54	11.04
58	20	20	33.54	21.2	9.6	4.06	7.96	12.64
59	21	21	24.92	26.68	11.76	7.66	8.08	13.24
60	22	22	27.27	31.2	10.08	5.66	5.52	13.04
61	23	23	29.59	29.64	11.45	6.84	7.8	14.92
62	24	24	27.56	33.9	12.11	7.66	6.86	14.4
63	25	25	28.93	32.2	14.16	7.82	7.88	12.94
64	26	26	29.16	37.03	13.35	7	6.04	12.54
65	27	27	31.11	35.64	14.87	7.66	7.46	13.04
66	28	28	34.03	36.33	14.79	7.88	8.34	15
67	29	29	34.06	38.69	15.37	7.42	7.3	13.52
68	30	30	35.03	37.84	17.3	7.3	7.62	11.96
69	31	31	42.86	40.53	13.92	5.08	6.2	13.62
70	32	32	51.74	32.55	16.95	4.16	8.2	11.04
71	33	33	46.86	45.28	13.5	4.32	5.14	13.34
72	34	34	48.26	40.78	16.64	4.68	6.52	11.7
73	35	35	38.08	51.77	17.74	6.28	5	11.02
74	36	36	39.75	54.04	17.57	6.28	5	11.72
75	37	37	51.72	45.21	17.97	6.02	8.06	14.92
76	38	38	57.05	45	17.73	4.66	7.04	13.16
77	39	39	43.95	50.04	22.95	7.42	7.28	11.38
78	40	40	49.23	52.44	20.52	6.22	6.6	12.44
79	15	15	10.62	22.51	13.22	13.76	6.5	7.6
80	16	16	10.57	21.51	18.45	14.94	7.4	5.54
81	17	17	11.4	22.3	19.9	14.56	7.56	5.4
82	18	18	12.1	31.88	13.51	11.54	4.22	7.24
83	19	19	11.35	31.09	19.01	13.98	4.88	5.44
84	20	20	12.51	31.94	19.33	14.34	5.44	6.14
85	21	21	13.58	40.38	14.9	13.12	4.16	8.4
86	22	22	13.9	41.26	16.7	13.66	4.34	7.86
87	23	23	14.57	35.69	22.74	12.9	5.14	5.46
88	24	24	17.11	32.97	23.85	12	6.34	5.8
89	25	25	17.9	41.02	19.07	12.58	5.42	8.34
90	26	26	19.39	36.86	23.09	14.68	7.88	8.54
91	27	27	17.71	35.69	32.32	15	7.52	5.28
92	28	28	16.9	53.08	22.38	15	4.46	7.7
93	29	29	16.56	54.42	25.54	14.4	4.06	6.18
94	30	30	21.97	44.04	25.98	11.66	5.88	6.84
95	31	31	20.37	52.65	25.5	14.44	5.4	7.92
96	32	32	24.17	47.89	25.79	11.7	5.98	7.74
97	33	33	22.99	52.17	27.6	14.82	6.46	8.54
98	34	34	24.62	45.24	34.67	11.1	6.2	5.3
99	35	35	23.72	45.98	40.23	14.34	7.52	5.5
100	36	36	24.47	46.36	42.51	13.62	7.34	5.08
101	37	37	26.07	50.05	38.33	12.78	6.78	5.8
102	38	38	27.16	55.52	34.33	12.54	6.18	6.8
103	39	39	24.52	58.88	40.64	13.24	5.4	5.22
104	40	40	26.35	68.97	32.08	14.22	5.24	8.06
105	15	15	17.48	14.02	13.56	9.8	14.52	9.44
106	16	16	20.03	14.86	13.29	10.04	16.3	11.62
107	17	17	19.97	14.87	16.9	10.36	16.74	9
108	18	18	20.76	17.7	15.22	9.72	13.4	10
109	19	19	22.64	17.61	16.89	9.78	15.02	9.86
110	20	20	22.93	19.75	16.93	10.94	14.9	11.16
111	21	21	27.24	18.81	17.62	7.7	13.6	9.18
112	22	22	25.52	22.73	17.11	10.08	13.2	11.52
113	23	23	33	19.94	17.87	8	16.54	11.76
114	24	24	34.83	20.16	19.31	7.82	17	11.18
115	25	25	27.74	25.12	21.52	10.78	13.84	10.38
116	26	26	39.57	22.17	19.27	7.24	16.36	12.06
117	27	27	39.19	24.38	19.4	7.48	14.94	12.22
118	28	28	38.55	23.45	24.22	8.2	16.82	10.12
119	29	29	38.53	24.26	26.32	8.44	16.62	9.44
120	30	30	36.18	28.55	25.17	10.16	15.34	11.12
121	31	31	41.1	29.28	23.29	7.82	13.32	10.9
122	32	32	42.23	31.16	23.08	8.64	14.12	12.56
123	33	33	43.81	28.86	27.91	8.6	16.06	10.44
124	34	34	47.21	29.88	26.94	7.44	14.56	10.28
125	35	35	41.78	35.1	27.44	10.52	14.76	12.32
126	36	36	42.53	35.03	29.98	9.5	13.64	10.24
127	37	37	46.73	33.36	31.81	9.76	16.58	10.96
128	38	38	57.45	32.43	28.34	7.28	16.3	11.94
129	39	39	47.87	36.48	32.84	8.34	13.12	9.28
130	40	40	48.01	38.27	33.49	9.02	13.46	9.84
131	15	15	16.6	18.86	9.29	9.44	9.28	13.34
132	16	16	16.59	23.93	8.47	10.5	7.8	16.54
133	17	17	15.85	24.84	10.64	11.58	7.8	13.22
134	18	18	18.71	23.58	11.4	10.66	9.28	13.64
135	19	19	22.58	22.74	11.54	9.28	10.52	14.6
136	20	20	19.56	30.13	11.29	10.56	7.26	14.4
137	21	21	22.33	28.57	12.26	10.92	9.34	15.8
138	22	22	25.75	27.13	13.05	9.88	10.62	15.7
139	23	23	25.85	31.69	12.32	9.12	8.2	15.58
140	24	24	25.52	36.9	11.88	9.56	7.08	16.78
141	25	25	24.47	38.05	13.91	10.44	7.1	14.5
142	26	26	27.45	39.28	13.29	9.82	7.36	16.46
143	27	27	27.49	35.14	17.72	12.78	10.94	15.3
144	28	28	28.9	35.56	18.69	11.84	10.6	14.12
145	29	29	31.24	39.5	16.63	9.48	8.22	14.22
146	30	30	29.69	42.47	18.16	11.84	8.88	15.08
147	31	31	29.27	46.51	18.42	11.92	7.9	14.68
148	32	32	31.64	46.5	18.71	11.56	8.4	15.32
149	33	33	36.85	41.79	20.01	11.14	10.98	16.32
150	34	34	35.62	44.01	21.68	11.02	9.82	14.12
151	35	35	36.02	46.13	22.25	11.12	9.5	14
152	36	36	35.23	49.28	23.16	12.7	9.78	14.86
153	37	37	39.23	49.66	22.09	11.94	10.34	16.76
154	38	38	38.09	54.74	22.09	10	7.46	13.56
155	39	39	44.88	53.17	20.58	9.42	8.82	16.84
156	40	40	41.63	52.41	25.28	11.38	9.92	14.62
157	15	15	11.55	17.33	17.4	16.96	12.02	7.48
158	16	16	13.43	19.55	14.99	13.8	10.14	8.66
159	17	17	14.79	20.13	15.87	13.48	10.72	8.86
160	18	18	14.44	20.41	20.25	16.38	12.46	7.86
161	19	19	15.08	23.42	19.03	16.68	11.36	9.06
162	20	20	18.43	22.98	18.04	13.16	11.6	9.64
163	21	21	17.75	23.51	22.29	13.98	11.48	7.64
164	22	22	17.79	27.28	21.25	16.7	11.54	9.68
165	23	23	19.66	24.47	26.07	13.7	12.1	7.02
166	24	24	20.6	28.79	22.39	16.52	12.74	10.7
167	25	25	21.74	27.82	25.72	14.36	12.28	8.42
168	26	26	22.67	30.06	25.1	15.56	12.76	9.86
169	27	27	23.5	35.48	21.56	13.84	9.74	10.94
170	28	28	21.3	32.73	32.47	16.42	11.32	7.14
171	29	29	25.62	34.48	26.34	14.32	11.54	9.9
172	30	30	25.72	32.72	32.53	13.1	11.28	7.1
173	31	31	26.28	35.89	31.1	14.54	11.52	8.52
174	32	32	27.63	37.96	30.12	14.16	11.14	9.14
175	33	33	27.3	38.23	34.28	14.02	10.78	7.66
176	34	34	26.64	43.57	32.69	15.64	10.02	8.78
177	35	35	29.85	37.86	38.77	13.98	12.08	7.34
178	36	36	29.78	45.68	32.51	15.46	10.68	9.96
179	37	37	31.46	42.06	38.04	14.28	11.6	8.16
180	38	38	34.6	41.61	37.56	13.2	12.14	8.56
181	39	39	32.14	52.93	32.04	14.48	9.2	10.38
182	40	40	31.67	50.24	39.06	16.02	10.64	8.94
183	15	15	15.82	16.97	11.52	11.96	12.58	12.4
184	16	16	17.21	18.57	11.54	11.04	11.54	12.62
185	17	17	17.67	20.43	12.17	10.96	10.56	12.14
186	18	18	17.4	23.45	12.63	13.1	10.54	13.64
187	19	19	19.72	20.64	15.86	11.94	12.94	11.02
188	20	20	22.8	23.12	13.46	10.14	11.4	13.46
189	21	21	22.87	26.09	13.5	10.36	10.14	13.76
190	22	22	22.07	26.08	16.86	11.16	10.48	10.94
191	23	23	24.2	26.11	17.64	13.26	13.86	13.74
192	24	24	22.61	32.21	16.7	13.72	10.34	14
193	25	25	27.05	26.25	20.79	10.46	12.38	10.18
194	26	26	27.23	27.12	22.82	11.6	13.32	10.2
195	27	27	24.46	35.75	20.18	14	10.24	12.54
196	28	28	26.38	36.24	20.52	12.92	10.16	12.4
197	29	29	32.15	31.37	22.32	11.9	14	13.06
198	30	30	33.66	35.18	20.18	10.32	11.2	13.46
199	31	31	34.01	35.14	22.55	10.22	11.24	11.84
200	32	32	31.93	39.13	23.6	11.34	10.2	11.56
201	33	33	36.74	36.68	24.25	11.32	12.96	13.18
202	34	34	35.56	35.55	29.8	11.44	13.08	10.06
203	35	35	34.39	38.43	30.97	13.34	13.38	10.78
204	36	36	37.44	41.62	27.3	13.4	13.52	13.88
205	37	37	35.58	43.22	30.68	11.9	10.82	10.12
206	38	38	41.52	39.16	32.11	11.4	13.96	11.02
207	39	39	45.35	40.43	30.08	10.64	13.92	12.32
208	40	40	44.36	44.56	29.44	11.98	13.62	13.86