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	-0.000408	0	-1.154
lnL	0.35017	0	2818.272
lnK	0.400294	0	3116.149
lnM	0.249769	0	2371.935
lnLlnL	-0.020171	0	-851.029
lnKlnK	-0.021092	0	-921.034
lnMlnM	-0.017146	0	-839.397
lnLlnK	0.024066	0	650.512
lnLlnM	0.016272	0	476.256
lnKlnM	0.018038	0	572.377

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	15.4	17.55	11.37	6.92	6.78	7.06
2	16	16	15.11	20.22	12.1	9.94	8.06	9.22
3	17	17	19.28	17.12	14.16	6.76	8.88	6.94
4	18	18	18.25	21.19	13.74	7.66	7.34	7.64
5	19	19	24.81	19.9	12.45	6.06	8.98	9.74
6	20	20	18.04	23.83	17.58	9.64	7.94	7.1
7	21	21	19.7	28.44	14.5	9	6.68	9.22
8	22	22	24.66	21.79	19.1	7.28	9.62	7.02
9	23	23	23.79	23.47	21.26	8.32	9.66	6.78
10	24	24	21.99	32.5	17.11	9.92	7.16	9.52
11	25	25	29.26	25.91	19.13	7.54	9.94	8.88
12	26	26	28.46	30.95	17.66	7.84	8.12	9.82
13	27	27	29.2	31.28	19.38	7.76	8.18	8.98
14	28	28	34.55	28.58	20.48	6.34	9.06	8.38
15	29	29	27.9	38.76	19.74	9.02	7	9.68
16	30	30	30.62	38.22	20.23	7.86	6.92	9.14
17	31	31	33.96	34.77	22.96	8.8	9.78	9.96
18	32	32	32.13	34.73	27.99	9.38	9.78	7.88
19	33	33	41.69	34.25	22.85	6.72	9.68	9.74
20	34	34	31.63	42.33	26.83	9.32	7.56	8.08
21	35	35	35.08	43.1	25.39	7.18	6.44	7.5
22	36	36	39.41	41.21	25.89	7.84	8.5	9.18
23	37	37	34.73	42.03	33.1	9.42	8.6	7.12
24	38	38	44.24	48.32	21.75	6	6.18	9.88
25	39	39	42.81	41.64	31.02	8.3	9.8	8.66
26	40	40	44.76	49.2	25.21	6.2	6.34	8.7
27	15	15	17.49	13.4	14.56	8.62	13.48	7.64
28	16	16	21.48	12.38	16.3	5.26	11.5	5.2
29	17	17	18.74	13.3	22.32	8.7	14.88	5.06
30	18	18	22.48	12.6	24.1	6.32	14.28	4.16
31	19	19	24.44	15.55	18.65	6.56	12.76	6.44
32	20	20	24.97	17.27	18.72	7.44	13.12	7.46
33	21	21	22.68	20.04	20.35	8.5	11.24	6.9
34	22	22	24.67	20.5	21.03	8.84	12.56	7.62
35	23	23	23.15	19.27	30.62	8.56	12.14	4.4
36	24	24	28.52	20.8	23.85	8.26	13.68	7.28
37	25	25	25.81	23.32	26.75	8.76	11.28	6
38	26	26	30.95	21.37	28.16	6.94	12.26	5.54
39	27	27	35.09	19.82	31.42	5.14	11.5	4.18
40	28	28	32.59	24.16	28.85	8.54	13.88	7.04
41	29	29	32.11	21.73	40.83	7.54	13.64	4.06
42	30	30	41.8	22.34	31.02	5.56	13.28	5.64
43	31	31	32.07	25.82	40.03	8.54	12.6	4.7
44	32	32	35.29	28.69	33.33	8.9	12.98	6.8
45	33	33	37.67	31.06	30.32	8.2	11.76	7.56
46	34	34	38.86	29.11	36.38	7.1	11.4	5.48
47	35	35	42.56	26.13	43.38	6.3	12.78	4.4
48	36	36	39.34	32.82	36.94	8.06	11.4	6.2
49	37	37	40.49	29.8	46.69	7.98	13.08	4.82
50	38	38	41.25	37.74	34.31	8.7	11.04	7.72
51	39	39	45.1	34.05	39.74	7.72	12.28	6.4
52	40	40	44.22	37.74	38.23	8.54	11.76	7.24
53	15	15	18.69	20.01	7.41	6.52	6.88	13.84
54	16	16	20.72	19.38	8.6	6.6	8.16	13.26
55	17	17	24.45	20.92	7.91	4.88	6.7	13.14
56	18	18	24.48	21.39	9.38	5.94	7.96	13.12
57	19	19	23.78	23.71	10.22	6.4	7.34	12.34
58	20	20	24.97	23.68	11.61	6.56	7.98	11.54
59	21	21	30.47	25.56	9.82	5.52	7.76	14.94
60	22	22	26.06	31.03	10.71	6.92	6.44	14.06
61	23	23	26.64	30.18	12.69	7.72	7.62	13.2
62	24	24	29.95	29.48	13.24	7.18	8.36	13.4
63	25	25	36.25	31.34	11.25	4.72	6.4	13.36
64	26	26	27.81	35.46	15.01	7.92	6.8	11.68
65	27	27	37.57	32.43	13.49	5.8	7.88	13.82
66	28	28	28.59	41.87	15.13	7.84	5.72	11.84
67	29	29	33.72	34.25	18.45	7.88	8.84	11.44
68	30	29.99	52.99	31.84	13.56	4.18	8.7	14.84
69	31	31	37.22	47.02	13.51	6.22	5.4	14.64
70	32	32.01	40.2	48.23	13.34	5.68	5.24	14.86
71	33	33	40.67	43.68	16.68	6.28	6.6	12.8
72	34	34	38.5	43.5	19.97	7.88	7.8	12.18
73	35	35	44.8	43.35	18.51	6.92	8.22	13.98
74	36	36	43.23	51.89	16.76	6.4	5.9	13.94
75	37	37	39.83	52.94	19.86	7.68	6.28	12.44
76	38	38	45.34	54.68	17.86	6	5.5	12.82
77	39	39	49.4	47.82	21.18	6.72	7.98	13
78	40	40	41.02	66.6	18.68	7.8	5.04	14.06
79	15	15	10.21	26.42	11.15	12.64	4.72	8.14
80	16	16	11.43	21.69	16.18	11.42	6.14	5.42
81	17	17	12.76	24.69	14.43	13.66	7.18	8.44
82	18	18	13.05	27.88	14.64	11.16	5.22	6.96
83	19	19	13.37	26.28	19.07	13.1	6.76	6.16
84	20	20	13.37	27.88	21.42	12.38	5.96	5.08
85	21	21	12.82	38.08	17.56	14.24	4.52	7.02
86	22	22	16.41	27.38	23.79	11.44	7.16	5.28
87	23	23	13.22	43.1	20.09	14.34	4.08	6.26
88	24	24	17.09	33.14	23.71	13.04	6.84	6.34
89	25	25	16.29	45.69	18.79	12.42	4.22	7.5
90	26	26	19.02	32.06	29.38	12.94	8	5.54
91	27	27	20.93	37.43	23.45	12.9	7.44	8.06
92	28	28	20.24	39.74	25.98	11.38	5.88	6.06
93	29	29	20.81	43.41	25.15	13.72	6.6	7.84
94	30	30	23.62	39.84	27.15	12.14	7.5	7.36
95	31	31	23.55	41.22	29.49	12.04	7.12	6.6
96	32	32	23.71	39.23	35.79	12.18	7.7	5.36
97	33	33	22.35	44.15	36.92	14.66	7.52	5.8
98	34	34	22.43	63.14	24.57	12.06	4.08	7.74
99	35	35	25.96	48.75	32.15	11.2	6.1	6.22
100	36	36	25.09	57.08	29.96	13.76	5.98	7.98
101	37	37	27.41	51	34.58	14.02	7.72	7.62
102	38	38	24.13	60.21	36.33	12.42	4.84	5.48
103	39	39	29.01	56.56	33.62	13.82	7.2	8.3
104	40	40	27.15	52.16	46.4	14.94	7.92	5.68
105	15	15	17.45	15.07	12.13	10.12	13.74	11.08
106	16	16	18.28	15.4	14.16	10.3	14.4	9.94
107	17	17	21.15	15.52	14.61	9.5	15.62	10.48
108	18	18	20.5	18.78	14.12	10.78	13.66	11.94
109	19	19	23.16	18.62	15.02	10.6	15.66	12.6
110	20	20	25.6	18.42	16.35	9.8	16.5	11.86
111	21	21	25.95	21.12	15.68	8.98	13.08	11.6
112	22	22	24.97	21.88	18.67	10.52	14.04	10.58
113	23	23	30.81	18.83	21.39	8.22	16.76	9.02
114	24	24	31.49	22.29	18.76	8.6	14.76	11.3
115	25	25	32.43	25.16	17.53	8.8	13.56	12.96
116	26	26	33.66	23.98	20.89	9.28	15.8	11.62
117	27	27	30	28.02	22.06	10.98	13.6	11.26
118	28	28	36.51	23.85	25.3	8.26	15.58	9.08
119	29	29	42.7	25.75	21.02	7.82	16.2	12.86
120	30	30	35.15	27.41	27.9	9.9	15.16	9.28
121	31	31	36.62	30.69	25.13	9.24	13	10.28
122	32	32	36.41	32.63	26.04	10.4	13.52	11.02
123	33	33	41.22	32.54	25.04	8.88	13.4	11.4
124	34	34	41	31.99	29.04	10.34	15.82	11.08
125	35	35	48.16	29.65	29.74	8.18	16.54	10.3
126	36	36	46.2	33.7	28.58	10.06	16.66	12.64
127	37	37	54.42	32.27	27.6	7.76	16.4	12.32
128	38	38	54.95	32.66	29.63	8	16.86	11.82
129	39	39	52.98	33.36	33.17	7.86	15.48	9.74
130	40	40	46.64	41.55	30.64	10.54	13.8	12.34
131	15	15	15.57	21.26	8.53	10.12	8.02	14.68
132	16	16	16.76	21.58	9.63	11.92	10.12	16.34
133	17	17	17.57	21.79	11.18	11.2	9.94	13.62
134	18	18	16.12	27.5	11.18	12.14	7.4	13.34
135	19	19	22.24	24.34	10.7	10.06	10.34	17
136	20	20	19.87	27.85	12.33	12.94	9.94	16.2
137	21	21	21.24	29.81	12.32	9.96	7.64	13.5
138	22	22	20.74	33.14	13.02	11.24	7.4	13.88
139	23	23	24.77	28.61	14.98	10.12	9.76	13.06
140	24	24	25.34	31.88	14.61	12.08	10.54	16.5
141	25	25	23.89	37.49	14.64	12.4	8.34	15.76
142	26	26	29.21	35.72	13.98	9.38	8.46	15.94
143	27	27	26.66	40.74	15.03	10.2	7.08	14.3
144	28	28	27.7	40.21	16.63	12.18	8.98	15.86
145	29	29	30.1	41.14	16.48	10.92	8.64	15.84
146	30	30	28.67	46.41	16.83	12.18	7.9	16.28
147	31	31	28.86	46.31	18.86	11.66	7.64	13.74
148	32	32	35.63	44.45	17.14	9.6	8.46	16.22
149	33	33	31.9	47	20.42	11.68	8.46	14.1
150	34	34	31.75	52.59	19.66	11.6	7.32	14.56
151	35	35	31.16	54.4	21.39	12.88	7.64	14.32
152	36	36	33.59	55	21.19	12.44	7.96	15.28
153	37	37	38.88	51.05	21.5	10.56	8.76	15.14
154	38	38	40.47	50.91	22.61	10.98	9.58	15.56
155	39	39	43.83	49.88	23.14	9.76	9.58	14.78
156	40	40	47.25	53.32	21.08	9.1	9.02	16.8
157	15	15	12.03	19.71	13.43	15.1	9.66	9.48
158	16	16	11.99	20.58	16.33	16.22	9.82	8.06
159	17	17	13.89	20.93	16.36	13.1	9.24	7.72
160	18	18	14.45	21.81	18.2	15.9	11.2	8.66
161	19	19	16.78	21.93	18.07	15.28	12.74	10
162	20	20	16.34	23.29	20.98	16.76	12.62	8.92
163	21	21	17.85	24.68	20.51	15.5	12.1	9.4
164	22	22	18.07	27.95	20.03	16.78	11.48	10.62
165	23	23	19.03	29.5	20.43	16.44	11.22	10.8
166	24	24	19.38	31.48	21.34	15.08	9.74	9.62
167	25	25	21.77	29.7	23.2	14	11.1	9.28
168	26	26	18.47	35.16	26.6	16.94	9.08	7.88
169	27	27	20.64	36.94	24.38	16.24	9.36	9.54
170	28	28	24.04	30.15	30.94	13.34	11.68	7.08
171	29	29	26.49	36.04	23.52	13.02	10.36	10.7
172	30	30	24.4	33.55	33.77	15.96	12.54	7.78
173	31	31	24.71	36.05	33.79	16.02	11.74	7.92
174	32	32	28.94	34.46	32.78	13.84	12.88	8.54
175	33	33	28.6	36.13	35.01	14.42	12.52	8.12
176	34	34	28.53	38.87	35.31	15.18	12.06	8.44
177	35	35	27.42	45.5	32.94	16.26	10.24	9.36
178	36	36	28.45	49.93	30.39	16.52	9.74	10.92
179	37	37	31.29	46.77	32.55	15.78	11.24	10.76
180	38	38	34.22	40.75	39.43	13.5	12.56	8.16
181	39	39	34.19	46.6	35.53	13.68	10.86	9.34
182	40	40	35.54	49.05	34.38	14.7	11.5	10.92
183	15	15	16.57	15.67	12.23	11.46	13.98	11.7
184	16	16	15.87	16.95	14.78	13.16	13.92	10.22
185	17	17	19.26	19.65	11.53	10.12	11.3	13.22
186	18	18	18.89	20.27	14.03	11.58	12.18	11.74
187	19	19	23.07	20.25	13.3	10.16	13.54	13.88
188	20	20	20.53	21.25	17.54	11.86	13.02	10.2
189	21	21	23.67	24.85	13.86	10.34	11.16	13.86
190	22	22	22.5	24.75	17.77	11.16	11.4	10.52
191	23	23	24.17	27.26	16.58	12.36	12.26	13.76
192	24	24	24.95	28.76	17.26	12.6	12.18	13.88
193	25	25	26.79	26.31	20.99	11.7	13.66	11.14
194	26	26	27.65	30.15	19.05	12.54	12.94	13.88
195	27	27	28.16	29.54	22.14	10.72	11.58	10.16
196	28	28	28.79	30.29	23.83	12.04	12.96	10.74
197	29	29	28.21	32.65	25.04	12.24	11.78	10.06
198	30	30	33.45	32.25	23.14	10.98	13.1	12.1
199	31	31	30.99	37.1	23.53	11.16	10.32	11.02
200	32	32	32.98	36.19	25.38	11.48	11.76	11.16
201	33	33	32.04	37.81	27.81	12.8	12.04	10.8
202	34	34	35.86	41.12	23.69	11.02	10.72	12.82
203	35	35	37.54	38.44	27.51	11.48	12.76	11.82
204	36	36	36.02	44.34	26.17	12	10.74	12.48
205	37	37	36.75	39.81	33.29	12.94	13.46	10.38
206	38	38	37.84	44.68	29.76	13.3	12.5	12.6
207	39	39	40.64	45.38	29.21	10.88	10.92	11.46
208	40	40	42.97	41.39	34.36	11.02	13.16	10.24