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	9.3E-5	0	0.254
lnL	0.350311	0	2855.307
lnK	0.400048	0	3105.855
lnM	0.249593	0	2480.376
lnLlnL	-0.020223	0	-854.879
lnKlnK	-0.021084	0	-976.302
lnMlnM	-0.017157	0	-781.144
lnLlnK	0.024092	0	757.651
lnLlnM	0.016302	0	521.513
lnKlnM	0.018074	0	558.558

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	16.84	16.86	10.73	6.62	7.56	8.04
2	16	16	16.88	19.88	10.72	8.04	7.58	9.8
3	17	17	19.42	18.2	12.78	7.98	9.84	9.32
4	18	18	15.42	24.19	14.21	9.72	6.54	7.64
5	19	19	23.73	20.1	12.98	6.58	9.14	9.56
6	20	20	19.31	22.56	17.39	9.46	9	7.64
7	21	21	19.26	23.06	20.46	9.56	8.84	6.36
8	22	22	19.97	27.62	17.72	9.1	7.1	7.48
9	23	23	23.48	29.62	15.27	7.66	6.66	9.08
10	24	24	27.68	24.02	19.73	6.28	8.48	6.68
11	25	25	21.75	29.52	23.43	9.2	7.34	6.02
12	26	26	27.02	26.63	23.72	8.24	9.58	6.86
13	27	27	26.74	30.92	22.17	7.14	6.88	6.36
14	28	28	36.03	28.09	19.93	6.32	9.68	9.06
15	29	29	26.5	33.25	26.54	8.64	7.56	6.16
16	30	30	28.34	33.31	27.55	9.48	8.96	7
17	31	31	34.12	33.71	23.9	6.54	7.58	7.1
18	32	32	33.98	41.32	20.12	6.94	6.3	9.18
19	33	33	29.8	39.61	28.64	8.66	7.08	6.48
20	34	34	39.26	37.54	24.08	6.16	7.42	7.82
21	35	35	33.17	42.65	27.8	8.28	7.04	7.28
22	36	36	30.97	48.47	28.19	9.1	6.14	7.26
23	37	37	34.06	44.51	31.17	8.68	7.24	6.88
24	38	38	35.99	42.89	33.9	7.96	7.4	6.1
25	39	39	35.92	49.29	30.5	9.98	7.86	8.64
26	40	40	48.22	37.54	34.32	6.4	9.82	6.82
27	15	15	20.53	12.31	13.51	6.06	12.64	7.08
28	16	16	19.33	13.27	16.72	6.8	12.1	5.76
29	17	17	25.66	12.71	15.73	5.48	14.32	6.96
30	18	18	19.89	14.23	23.14	7.06	11.96	4.22
31	19	19	24.15	13.57	23.94	5.96	13.42	4.3
32	20	20	20.55	16.06	27.82	8.76	13.38	4.38
33	21	21	25.93	17.04	22.1	7.24	13.56	6.24
34	22	22	30.63	14.88	26.9	5.36	14.32	4.46
35	23	23	25.21	18.01	30.42	8.14	13.82	4.66
36	24	24	28.96	21.15	22.74	7.46	12.34	7.08
37	25	25	39.08	18.56	22.38	5.42	14.88	7.46
38	26	26	28.95	21.82	29.82	7.48	11.92	5.16
39	27	27	35.24	23.23	23.96	6.58	12.28	7.4
40	28	28	31.41	25.44	27.87	7.66	11.22	6.3
41	29	29	33.24	25.03	30.49	8.7	13.88	6.88
42	30	30	34.48	23.35	37.46	7.5	13.56	4.86
43	31	31	37.74	26.37	30.75	8.56	14.92	7.78
44	32	32	48.45	23.02	31.45	5.22	14.32	6.2
45	33	33	50.21	24.65	30.26	5.54	14.62	7.18
46	34	34	37.56	27.57	41.82	8.26	13.58	5.2
47	35	35	46.16	27.87	34.78	5.82	12.04	5.8
48	36	36	52.03	28.13	32.73	5.34	12.58	6.58
49	37	37	45.49	31.11	36.92	7.14	12.76	6.52
50	38	38	51.7	31.91	33.23	6.26	12.62	7.52
51	39	39	52.45	29.3	41.64	6.06	13.74	5.68
52	40	40	55.1	30.35	40.66	5.64	13	5.76
53	15	15	24.7	15.76	7.43	4.56	8.84	13.38
54	16	16	17.19	22.23	8.95	7.98	6.74	12.28
55	17	17	24.18	19.68	8.71	5.08	7.4	12.06
56	18	18	22.29	21.71	10.28	6.48	7.64	11.5
57	19	19	27.03	25.17	8.12	4.44	5.52	13.06
58	20	20	29.32	23.57	9.66	5.24	7.74	13.82
59	21	21	28.02	24.48	11.49	6.24	8.36	12.72
60	22	22	28.97	26.1	11.94	6.1	7.88	12.36
61	23	23	31.49	26.3	12.54	6.2	8.76	13.08
62	24	24	38.55	26.88	11.19	4.34	7.58	13.28
63	25	25	31.05	31.75	13.25	7.4	8.24	14.4
64	26	26	32.43	33.21	13.6	7.2	8	14.3
65	27	27	30.29	34.53	16.05	7.88	7.72	11.88
66	28	28	40.21	32.23	14.3	5.76	8.54	13.88
67	29	29	38.94	35.87	14.52	5.72	7.2	13.04
68	30	30	37.49	36.66	16.64	6	7.04	11.14
69	31	31	44.02	42.71	12.61	4.78	5.66	14.86
70	32	32	34.81	49.47	15.31	6.88	5.2	12.92
71	33	33	39.16	46.96	15.85	6.76	6.24	14
72	34	34	46.54	44.14	15.6	5.64	6.86	14.58
73	35	35	50.54	44.79	15.39	5.04	6.64	14.58
74	36	36	44.36	47.55	18.26	6	6.32	12.18
75	37	37	43.32	51.5	18.59	7.1	6.62	13.72
76	38	38	41.28	53.78	20.46	6.88	5.76	11.22
77	39	39	41.25	54.07	22.32	7.94	6.6	11.68
78	40	40	48.93	55.24	19.27	6.54	6.48	13.98
79	15	15	10.57	26.05	10.82	12.08	4.78	8.4
80	16	16	12.15	19.93	16.83	12.12	7.74	5.9
81	17	17	10.7	29.54	14.42	13	4.5	6.54
82	18	18	12.79	29.89	13.66	13.5	5.68	8.92
83	19	19	12.47	35.06	13.96	12.76	4.32	7.98
84	20	20	13.71	27.25	21.35	14.1	7.18	5.98
85	21	21	16.06	29.4	18.32	12.58	7.06	7.7
86	22	22	18.14	28.12	19.73	11.28	7.7	7.3
87	23	23	17.05	30.47	22.8	11.78	6.8	5.98
88	24	24	14.56	43.82	20.07	13.44	4.2	6.58
89	25	25	18.57	37.14	20.86	12.62	6.38	7.86
90	26	26	17.06	43.64	21.79	11.2	4.24	6
91	27	27	16.97	48.54	21.89	12.96	4.3	6.86
92	28	28	18.9	37.59	31.31	14.5	7.38	5.72
93	29	29	21.92	35.32	31.8	11.32	7.38	5.22
94	30	30	21.1	42.75	28.81	14.18	7.06	7.02
95	31	31	21.96	42.17	31.59	14.52	7.7	6.76
96	32	32	19.97	52.01	30.26	14.52	5.38	6.36
97	33	33	21.77	61.72	23.64	13.42	4.5	8.7
98	34	34	20.95	53.53	34.33	12.66	4.8	5.06
99	35	35	22.83	61.18	27.95	14.96	5.36	8.42
100	36	36	24.09	54.94	33.62	14.22	6.16	6.86
101	37	37	28.09	50.07	34.31	12.54	7.26	7.08
102	38	38	27.06	52.05	37.99	11.56	6.12	5.54
103	39	39	28.41	48.72	43.45	12.74	7.72	5.52
104	40	40	28.81	54.47	39.69	13.92	7.52	6.82
105	15	15	16.89	15.38	12.27	10.92	13.94	11.36
106	16	16	21.92	14.55	12.23	7.4	13.7	10.5
107	17	17	23.54	14.57	14.07	8.1	16.28	10.6
108	18	18	27	16.03	12.69	7.08	14.94	12.3
109	19	19	22.75	19.4	14.43	9.78	13.48	11.94
110	20	20	24.43	18.8	16.81	10.48	16.3	11.62
111	21	21	26.84	19.29	17.26	8.28	13.96	9.94
112	22	22	26.26	20.39	19.51	10.58	16.28	10.72
113	23	23	27.83	21.58	19.65	10.3	15.88	11.08
114	24	24	30.99	21.55	20.18	7.76	13.6	9.18
115	25	25	32.8	22.24	20.91	7.6	13.72	9.22
116	26	26	28.71	26.13	22.52	10.98	14.02	10.44
117	27	27	29	27.3	24.05	10.94	13.42	9.74
118	28	28	34.45	26.24	23.44	9.72	15.3	10.92
119	29	29	38.18	26.77	22.77	7.94	13.78	10.42
120	30	30	37.07	27.32	26.13	9.24	15.12	9.96
121	31	31	40.53	28.82	24.3	9.28	15.84	12.1
122	32	32	47.77	26.61	25.25	7.42	16.88	11.22
123	33	33	46.27	31.91	22.31	7.56	13.38	12.74
124	34	34	41.66	31.34	29.38	10.34	16.52	11.14
125	35	35	40.07	36.03	27.84	10.16	13.16	11.14
126	36	36	45.74	32.96	29.98	10	16.8	11.74
127	37	37	48.13	35.97	27.25	8.56	13.78	11.94
128	38	38	47.76	34.09	33.14	9.38	15.94	10.3
129	39	39	50.99	35.46	31.64	8.16	14.3	10.22
130	40	40	47.73	35.95	37.28	10.28	16.4	9.82
131	15	15	14.56	23.33	8.2	10.7	7.02	15.02
132	16	16	14.69	25.03	9.31	11.5	7.02	14.04
133	17	17	20.28	20.94	9.87	9.58	10.54	15.96
134	18	18	17.59	26.13	10.7	12.1	8.68	15.52
135	19	19	18.73	30.22	9.88	10.94	7.12	16.58
136	20	20	19.81	28.77	11.83	10.78	7.94	14.12
137	21	21	18.87	32.32	12.85	11.6	7.04	13.02
138	22	22	21.95	31.58	12.93	11.7	8.72	15.58
139	23	23	25.16	31.51	12.83	10.56	9.26	16.66
140	24	24	25.89	31.98	14.16	10.12	9.02	14.7
141	25	25	25.76	31.1	17.23	11.9	10.9	13.64
142	26	26	27.1	37.78	14.22	9.34	7.22	14.24
143	27	27	25.85	39.09	16.54	12.98	9.12	15.68
144	28	28	30.08	39.32	15.45	9.64	8.04	15.08
145	29	29	29.63	38.62	18.35	12.38	10.36	15.54
146	30	30	26.77	45.23	19.06	12.26	7.56	13.06
147	31	31	33.24	41.99	18.01	11.52	10	16.92
148	32	32	31.06	42.8	21.5	12.48	9.78	13.74
149	33	33	32.23	50.46	18.3	11.24	7.58	15.62
150	34	34	33.92	48.96	19.91	12.64	9.38	16.9
151	35	35	35.61	50.57	19.89	11.08	8.38	15.7
152	36	36	38.59	44.47	23.82	11	10.64	13.86
153	37	37	35.9	47.75	26.18	12.9	10.54	13.36
154	38	38	34.71	58.34	22.83	12.54	7.78	14.66
155	39	39	40.01	59.09	20.58	10.66	7.7	16.64
156	40	40	44.49	50.54	24.46	9.78	9.62	14.12
157	15	15	10.76	19.42	16.19	16.76	9.56	7.4
158	16	16	12.79	21.24	14.17	14.78	9.3	9.36
159	17	17	13.14	23.78	14.62	16.28	9.26	10.26
160	18	18	14.78	22.94	16.32	17	11.58	10.8
161	19	19	14.89	22.26	20.99	16.06	11.44	7.66
162	20	20	16.59	22.71	21.36	15.8	12.48	8.38
163	21	21	17.52	27.12	18.25	15.48	10.58	10.54
164	22	22	19.01	26.34	20.39	16.08	12.52	10.58
165	23	23	17.97	31.84	19.78	16.36	9.54	10.44
166	24	24	19.65	27.73	25.41	16.4	12.5	8.66
167	25	25	19.9	32.61	22.88	14.58	9.32	8.84
168	26	26	19.25	34.34	25.93	15.9	9.2	8
169	27	27	25.86	30.69	23.47	13.64	12.74	10.92
170	28	28	23.85	32.36	27.96	15.46	12.34	9.16
171	29	29	22.31	36.66	29.25	15.48	9.86	8.04
172	30	30	21.82	38.93	31.56	15.56	9	7.2
173	31	31	26.24	39.79	26.63	13.08	9.16	9.18
174	32	32	26.19	38.22	32.18	16.98	12.44	9.52
175	33	33	30	37.76	30.51	14.68	12.8	10.28
176	34	34	26.91	40.94	35.46	14.68	10.24	7.58
177	35	35	27.66	40.14	39.5	16.14	11.9	7.58
178	36	36	26.97	43.73	40.19	15.76	10.2	7.04
179	37	37	29.53	47.59	34.45	13.9	9.06	8.28
180	38	38	29.13	46.08	41.11	16.42	10.94	7.82
181	39	39	32.02	46.31	39.41	16.48	12.2	9.22
182	40	40	35.69	46.86	36.64	14.44	11.98	10
183	15	15	15.45	19.06	10.03	11.4	10.18	13.54
184	16	16	15.73	17.45	14.3	13.62	13.78	10.88
185	17	17	18.34	19.72	12.22	10.5	11.02	12.1
186	18	18	18.73	20.59	13.85	10.74	10.98	10.94
187	19	19	18.42	22.26	15.52	13.62	12.46	11.9
188	20	20	19.6	25.78	13.99	13.14	10.88	13.96
189	21	21	19.87	25.32	16.98	11.94	10.26	10.26
190	22	22	25.55	23.47	16.28	10.96	13.84	13.3
191	23	23	23.67	27.36	16.95	10.98	10.58	11.62
192	24	24	23.87	27.89	19.16	12.84	12.22	11.88
193	25	25	24.35	27.88	21.87	13.3	12.96	10.78
194	26	26	27.28	31.82	17.92	11.7	11.16	13.7
195	27	27	28.03	30.09	21.68	12.48	13.12	12.06
196	28	28	26.47	31.95	24.63	13.58	12.44	10.56
197	29	29	30.46	32.31	22.92	10.24	10.92	10.22
198	30	30	30.46	33.74	24.48	13.62	13.8	12.58
199	31	31	31.14	38.82	21.88	12	10.58	12.98
200	32	32	30.45	40.12	24.24	13.34	11.02	12.46
201	33	33	31.49	39.37	26.78	13.7	12.04	11.84
202	34	34	33.91	41.74	24.94	11.8	10.56	12.1
203	35	35	39.83	34.61	29.87	10.34	13.94	10.36
204	36	36	39.9	42.59	24.28	10.24	10.84	13.12
205	37	37	39.76	43.71	26.02	11.58	11.84	13.62
206	38	38	36.73	41.82	34.27	13.6	13.34	10.54
207	39	39	42.52	41.25	31.75	11.5	13.62	11.58
208	40	40	40.07	44.11	34.23	12.54	12.8	10.78