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	-3.5E-5	0	-0.09
lnL	0.350382	0	2776.499
lnK	0.400078	0	3039.639
lnM	0.249544	0	2233.015
lnLlnL	-0.02022	0	-873.375
lnKlnK	-0.0211	0	-873.636
lnMlnM	-0.017123	0	-815.195
lnLlnK	0.02411	0	645.673
lnLlnM	0.016257	0	509.513
lnKlnM	0.018078	0	547.78

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	17.19	15.11	12.32	6.98	9.28	7.38
2	16	16	17.6	16.19	13.78	7.66	9.66	7.3
3	17	17	17.02	22.13	11.4	7.22	6.06	8.26
4	18	18	18.97	19.9	14.34	7.96	8.6	7.9
5	19	19	17.14	23.94	15.36	8.66	6.68	7.04
6	20	20	19.48	25.78	14.1	8.42	6.92	8.8
7	21	21	24.98	23.59	13.97	6.74	8.24	9.54
8	22	22	25.76	24.61	15.01	7.18	8.66	9.68
9	23	23	26.52	28.52	13.8	6.1	6.4	9.4
10	24	24	24.41	25.81	20.91	7.96	8.52	6.82
11	25	25	23.79	30	20.2	9.98	8.68	8.64
12	26	26	25.1	29.39	22.54	8.66	8.22	7.02
13	27	27	25.02	37.6	18.22	9.1	6.44	9.44
14	28	28	28.98	38.1	16.91	7.34	6.08	9.88
15	29	29	32.57	33.68	19.76	7.48	8.22	9.62
16	30	30	31.95	36.23	20.68	8.34	8.22	9.94
17	31	31	36.07	36.75	19.61	6.78	7.58	9.92
18	32	32	36.09	33.51	25.29	7.28	9.08	7.9
19	33	33	30.17	39.96	27.79	9.94	8.16	7.82
20	34	34	44.06	34.67	23.41	6.56	9.94	9.86
21	35	35	31.56	39.49	33.47	9.66	8.48	6.44
22	36	36	37.27	42.79	26.36	8.42	8.18	9.04
23	37	37	35.35	44.84	29.29	8.82	7.62	7.86
24	38	38	34.58	52.39	26.65	9.5	6.66	9.22
25	39	39	37.85	42.4	35.65	7.96	7.96	6.1
26	40	40	46.07	42.08	30.55	7.08	9	8.2
27	15	15	19.3	12.25	14.77	7.26	14.16	7.1
28	16	16	17.72	14.04	17.16	8.2	12.32	6.08
29	17	17	20.41	12.61	21.7	6.2	12.48	4.12
30	18	18	24.13	13.42	19.55	6.12	13.94	5.6
31	19	19	21.21	15.26	23.41	7.2	12.12	4.58
32	20	20	24.49	16.9	19.9	6.8	12.02	6.2
33	21	21	27.03	17.55	19.91	5.98	11.36	6.12
34	22	22	23.82	20.62	21.86	8.3	11.24	6.56
35	23	23	27.38	18.93	24.84	7.54	13.3	6.04
36	24	24	26.55	19.27	29.98	8.98	14.96	5.56
37	25	25	31.94	20.71	24.28	7.42	14.12	7.32
38	26	26	27.9	20.06	36.4	8.46	14.26	4.42
39	27	27	35.05	21.5	27.41	5.66	11.5	5.4
40	28	28	31.74	22.28	34.29	6.96	12.06	4.54
41	29	29	37.05	24.62	27.05	6.46	11.94	6.68
42	30	30	40.97	22.89	30.58	5.16	11.7	5.2
43	31	31	33.11	25.03	40.34	7.1	11.28	4.02
44	32	32	34.72	25.8	40.79	8.8	14.26	5.2
45	33	33	41.1	25.69	36.81	7.5	14.9	6.1
46	34	34	38.12	33.44	29.85	8.4	11.2	8
47	35	35	46.52	27.62	34.95	6.9	14.56	6.9
48	36	36	45.04	24.94	48.97	6.18	14.16	4
49	37	37	46.02	31.07	36.43	6.36	11.54	5.98
50	38	38	61.82	27.86	33.01	5.14	15	7.68
51	39	39	52.23	30.07	40.07	5.74	12.56	5.6
52	40	40	42.72	34.66	46.13	7.54	11.02	4.92
53	15	15	25.46	16.57	6.7	4.34	8.22	14.9
54	16	16	22.01	18.15	8.77	5.76	8.26	12.14
55	17	17	24.37	19.55	8.71	5.78	8.56	13.84
56	18	18	27.35	19.41	9.44	4.78	8.18	11.94
57	19	19	21.14	24.86	11.06	7.2	6.8	11.02
58	20	20	25.65	26.74	9.52	6.34	6.9	14.54
59	21	21	24.58	29.64	10.37	7.22	6.62	14.24
60	22	22	34.88	23.43	11.17	4.9	8.94	13.36
61	23	23	39.8	25.55	9.99	4.02	7.74	14.6
62	24	24	30.99	29.68	12.59	5.82	7	12
63	25	25	32.8	30.78	12.95	6.82	8.4	14.54
64	26	26	35.84	35.11	11.23	4.56	5.34	12.76
65	27	27.01	36.4	39.08	10.93	5.08	5.34	14.94
66	28	28	37.2	31.74	16.03	5.72	7.88	11
67	29	29	34.6	36.78	16.17	7.86	8.36	13.74
68	30	30	39.15	38.72	14.66	6.34	7.34	14.38
69	31	31	40.8	40.14	14.93	6.42	7.48	14.96
70	32	32	36.99	41.13	18.21	6.86	6.92	11.28
71	33	33	47.33	38.66	16.48	5.6	8.12	13.84
72	34	34	42.72	43.63	17.51	6.1	6.8	12.44
73	35	35	43.63	45.06	18.12	6.78	7.46	13.62
74	36	36	49.7	48.25	15.67	5.02	5.94	13.94
75	37	37	43.28	49.63	19.57	7.34	7.14	13.34
76	38	38	50.19	51.39	17.21	4.84	5.38	12.18
77	39	39	49.57	51.09	19.27	6.22	6.86	13.5
78	40	40	57.39	49.21	18.68	5.36	7.34	14.34
79	15	15	10.4	20.09	16.17	12	6.32	5.1
80	16	16	10.72	28.43	12.02	11.08	4.02	6.92
81	17	17	12.25	23.82	16.15	14.7	7.68	7.58
82	18	18	13.29	24.27	17.47	13.86	7.8	7.18
83	19	19	14.93	24.15	18.43	12.02	7.8	6.7
84	20	20	11.93	32.52	20.26	13.84	4.86	5.3
85	21	21	13.99	33.6	18.43	14.72	6	7.6
86	22	22	14.18	38.5	17.84	14.96	5.28	8.16
87	23	23	14.46	35.37	23.33	12.78	5.1	5.2
88	24	24	18.28	28.46	27.15	11.64	7.9	5.22
89	25	25	18.26	39.98	19.22	11.66	5.3	7.84
90	26	26	18.75	42.04	20.05	12.76	5.64	8.42
91	27	27	18.5	41.35	24.23	13.12	5.82	6.82
92	28	28	18.89	37.76	31.1	13.44	6.8	5.34
93	29	29	21.18	44.14	23.93	12.12	5.84	7.5
94	30	30	20.99	51.84	22.02	12.12	4.78	8.18
95	31	31	20.75	49.89	26.79	14.64	5.96	7.74
96	32	32	23.05	38.73	38.13	12.86	7.98	5.08
97	33	33	20.38	56.36	29.51	11.64	4.02	5.38
98	34	34	26.14	43.87	33.26	11.26	7	6.06
99	35	35	24.93	54.24	29.17	11.92	5.46	7.08
100	36	36	23.89	56.22	32.91	14.08	5.88	6.9
101	37	37	26.76	52.01	34.8	14	7.32	7.34
102	38	38	27.48	64.92	27.04	11.98	4.98	8.72
103	39	39	26.05	63.83	33	11.5	4.58	6.22
104	40	40	25.06	57.7	44.92	14.52	6.22	5.22
105	15	15	20.69	13.17	12.01	8.18	15.9	11.08
106	16	16	22.52	14.08	12.43	8.28	16.44	11.9
107	17	17	21.32	15.28	14.84	9.98	16.88	10.92
108	18	18	21.14	18.29	14.13	10.7	14.5	12.28
109	19	19	21.2	19.63	15.55	10.42	13.04	10.72
110	20	20	24.23	17.57	18.94	9.98	16.64	9.52
111	21	21	28.8	18.77	16.47	8.08	15.28	11.14
112	22	22	28.06	20.77	17.38	10.12	16.48	12.7
113	23	23	26.43	21.91	20.54	10.68	15.22	10.26
114	24	24	30.94	20.78	21.42	9.14	16.68	10.06
115	25	25	32.43	22.64	20.62	8.84	15.42	10.76
116	26	26	34.77	23.92	20.12	7.9	14.02	10.74
117	27	27	35.91	23.86	22.41	8.2	15.16	10.2
118	28	28	30.15	27.77	25.62	10.96	13.8	9.48
119	29	29	40.78	24.93	23.42	7.64	15.58	10.48
120	30	30	41.99	26.1	23.91	7.06	14.12	9.78
121	31	31	36.97	29.34	26.62	9	13.5	9.46
122	32	32	40.34	28.97	27.4	9.2	15.52	10.36
123	33	33	39.12	30.53	29.62	10.76	16.46	10.66
124	34	34	48.19	31.58	24.11	7.3	13.72	11.78
125	35	35	44.31	34.09	26.6	9	14	11.72
126	36	36	48.64	31.85	29.23	8.72	16.4	11.34
127	37	37	50.01	33.89	28.45	7.44	13.44	10.32
128	38	38	46.95	38.24	28.35	9.76	14.2	12.62
129	39	39	48.53	36.8	31.83	10.06	15.92	11.8
130	40	40	52.38	35.73	33.3	8.76	15.7	10.66
131	15	15	15.26	21.16	8.81	9.72	7.56	13.24
132	16	16	16.85	21.75	9.46	10.94	9.26	15.42
133	17	17	17.12	22.15	11.3	12.82	10.82	14.92
134	18	18	18.73	24.68	10.69	12.3	10.16	17
135	19	19	19.33	26.05	11.61	12.28	9.88	15.98
136	20	20	19.8	28.04	12.26	12.2	9.26	15.32
137	21	21	21.79	29.95	11.85	9.72	7.64	14.22
138	22	22	21.08	29.59	14.92	12.94	9.92	13.88
139	23	23	24.23	32.67	12.8	9.4	7.56	14.22
140	24	24	23.18	34.49	14.68	12.68	9.08	15.5
141	25	25	26.87	33.92	14.45	10.66	9.26	15.8
142	26	26	27.52	35.02	15.47	11.62	10	16.34
143	27	27	28.51	37.02	15.73	10.54	8.86	15.16
144	28	28	27.18	41.87	16.12	10.24	7.04	13.52
145	29	29	27.87	45.03	16.08	12.4	8.06	16.92
146	30	30	28.79	49.18	15.51	11.5	7	17
147	31	31	32.97	40.47	19.16	11.56	10.38	15.64
148	32	32	29.96	49.11	18.64	12.24	7.82	15.28
149	33	33	34.19	47.32	18.52	10.44	8.14	15.34
150	34	34	34.47	48.29	19.87	11.76	9.04	16.06
151	35	35	37.7	50.46	18.58	10.16	8.24	16.68
152	36	36	36.28	53.76	19.81	10.62	7.64	15.46
153	37	37	40.64	48.92	21.57	9.62	8.84	14.48
154	38	38	37.2	56.47	21.83	12.04	8.42	16.1
155	39	39	35.95	64.08	21.16	12.16	7.04	16.2
156	40	40	40.02	59.38	22.38	11.08	7.96	15.68
157	15	15	11.11	18.53	16.59	16.2	10.14	7.22
158	16	16	12.61	19.12	17	15.6	10.92	7.84
159	17	17	12.95	21.67	17.18	15.62	9.74	8
160	18	18	16.06	23.21	14.28	13.36	9.9	10.96
161	19	19	16.55	22.56	17.64	15.4	12.22	10.2
162	20	20	16.79	24.75	18.37	14.72	10.66	9.46
163	21	21	17.59	23.42	22.73	15.78	12.88	8.34
164	22	22	18.26	23.78	25.39	14.44	12.1	7
165	23	23	17.43	27.47	25.92	16.08	10.76	7.2
166	24	24	21	27.4	23.52	14.52	12.14	9.08
167	25	25	21.96	31.07	21.4	14.48	11	10.66
168	26	26	20.59	29.56	29.66	16.6	12.4	7.72
169	27	27	20.77	32.48	29.42	16.62	11.22	7.88
170	28	28	25.59	31.58	26.29	13.76	12.28	9.52
171	29	29	23.26	35.85	28.48	15.78	10.84	8.88
172	30	30	25.56	40.62	23.61	14	9.28	10.98
173	31	31	25.9	34.46	33.88	14.94	12.2	7.78
174	32	32	30.07	35.14	30.11	13.18	12.54	9.4
175	33	33	25.75	36.85	39.64	16.7	12.52	7.18
176	34	34	28.43	39.01	35.3	14.88	11.72	8.24
177	35	35	30.35	40.75	33.69	13.96	11.28	8.82
178	36	36	32.41	43.92	30.62	13.96	11.16	10.66
179	37	37	30.63	44.49	36.22	14.12	10.4	8.28
180	38	38	30.16	49.81	34.66	15.66	9.92	9.5
181	39	39	31.78	43.82	43.45	16.14	12.64	7.98
182	40	40	32.39	51.25	36.69	15.04	10.02	9.28
183	15	15	16.11	17	11.22	10.68	11.46	11.68
184	16	16	15.86	20.4	11.2	12.96	11.02	13.94
185	17	17	18.33	19.25	12.67	10.7	11.54	11.8
186	18	18	18.8	20.55	13.82	13.04	13.42	13.38
187	19	19	19.42	21.6	15.11	12.3	12.4	11.8
188	20	20	21.13	22.88	15.08	13.14	13.68	13.96
189	21	21	23.4	22.19	16.64	10.52	12.8	11.22
190	22	22	22.21	25.97	16.82	11.02	10.48	10.92
191	23	23	22.87	26.05	19.09	13.42	13.16	11.86
192	24	24	22.13	28.79	20.26	13	10.9	10.3
193	25	25	25.95	27.85	20.08	11.66	12.26	11.26
194	26	26	26.44	27.2	23.65	12.52	13.84	10.2
195	27	27	27.34	30.21	22.29	12.42	12.62	11.28
196	28	28	30.16	33.58	19.21	11.2	11.28	13.6
197	29	29	28.33	37.56	20.24	13.12	10.76	13.92
198	30	30	29.18	34.05	25.57	13.64	13	11.38
199	31	31	30.38	38.36	23.01	12.64	10.98	12.52
200	32	32	33.79	36.55	24.21	10.48	10.92	11.08
201	33	33	34.06	36.98	26.47	12.3	12.76	11.82
202	34	34	38.74	34.87	27.38	10.22	13.22	10.98
203	35	35	39.52	36.38	27.94	10.78	13.58	11.58
204	36	36	37.28	40.98	28.09	12.4	12.68	12.36
205	37	37	40.04	38.03	31.79	11.4	13.84	10.68
206	38	38	45.32	39.31	28.46	10.14	13.7	12.52
207	39	39	39.08	46.69	29.5	13.7	12.7	13.62
208	40	40	39.91	47.45	30.85	13.88	12.94	13.42