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	0.000853	0	2.14
lnL	0.350185	0	2805.271
lnK	0.399709	0	2870.426
lnM	0.249573	0	2173.238
lnLlnL	-0.020147	0	-858.694
lnKlnK	-0.021058	0	-872.196
lnMlnM	-0.017203	0	-823.639
lnLlnK	0.024026	0	694.478
lnLlnM	0.016265	0	452.135
lnKlnM	0.018199	0	563.354

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	13.78	18.27	12.44	9.76	8	7.86
2	16	16	15.16	19.08	13.12	7.82	6.82	6.62
3	17	17	19.54	19.1	11.8	6.9	8.1	8.92
4	18	18	19.58	20	13.65	7.14	7.96	7.8
5	19	19	20.45	19.15	16.95	7.68	9.48	6.84
6	20	20	20.69	23.35	15.05	6.78	6.72	7.04
7	21	21	23.23	21.58	17.54	6.9	8.6	6.86
8	22	22	19.48	28.17	17.82	9.86	7.3	7.82
9	23	23	22.57	28.71	16.82	8.22	7.08	8.3
10	24	24	28.27	28.26	15.09	6.42	7.34	9.6
11	25	25	27.37	27.66	18.91	7.92	8.94	8.74
12	26	26	22.46	32.42	22.68	10	7.42	7.06
13	27	27	29.41	29.67	20.77	8.22	9.3	8.84
14	28	28	26.74	34.02	22.11	8.62	7.42	7.7
15	29	29	26.83	36.34	22.83	9.7	7.76	8.38
16	30	30	27.84	42.03	20.04	9.32	6.56	9.8
17	31	31	31.31	35.76	24.51	8.9	8.7	8.48
18	32	32	33.86	36.91	23.8	7.36	7.6	7.96
19	33	33	38.81	33.41	26	6.66	9.08	7.62
20	34	34	32.9	44.68	23.52	8.98	7.16	9.52
21	35	35	34.73	43.21	25.63	7.58	6.7	7.74
22	36	36	35.41	40.61	30.52	8.86	8.62	7.54
23	37	37	46.8	37.14	26.9	6.48	9.72	8.88
24	38	38	46.17	43.31	24.09	6.46	7.96	9.92
25	39	39	40.64	48.88	26.21	8.24	7.58	9.86
26	40	40	44.52	51.14	24.03	6.18	6	9.12
27	15	15	16.86	11.67	19.34	7.4	13.04	4.5
28	16	16	22.71	13.87	12.59	5.4	11.02	7.72
29	17	17	20.48	13.13	20.12	6.36	12.26	4.64
30	18	18	21.37	12.79	25.22	6.84	14.3	4.02
31	19	19	22.38	15.55	21.04	7.24	12.7	5.56
32	20	20	28.73	16.05	17.56	5.38	12.2	6.86
33	21	21	24.43	19.62	19.02	7.76	11.48	7.44
34	22	22	25.97	17.47	25.55	7.28	13.26	5.3
35	23	23	29.65	18.02	24.23	6.36	13.06	5.76
36	24	24	36.3	18.48	21.09	5.44	13.76	7.36
37	25	25	35.22	18.27	26.3	5.78	14.3	5.82
38	26	26	34.67	19.13	29.12	5.52	12.7	4.84
39	27	27	28.1	23.12	32.95	8.14	11.7	4.82
40	28	28	37.14	22.24	27.73	5.58	11.66	5.62
41	29	29	35.4	20.8	38.39	6.9	14.74	4.48
42	30	30	41.38	21.42	33.8	5.54	13.74	5.02
43	31	31	45.23	24.61	27.14	6.02	14.08	7.84
44	32	32	40.2	23.95	37.74	6.68	14.04	5.14
45	33	33	36.71	29.2	34.71	8.1	12.12	6.18
46	34	34	45.9	25.59	35.99	6.1	13.86	5.8
47	35	35	51.57	28.65	28.8	5.6	12.78	7.94
48	36	36	39.01	28.59	47.2	8.9	14.66	5.08
49	37	37	44.78	30.4	39.18	6.58	11.86	5.5
50	38	38	52.83	27.76	40.71	6.1	14.86	5.92
51	39	39	41.87	31.93	49.18	8.46	13.3	5
52	40	40	48.9	32.27	43.09	6.9	12.86	5.72
53	15	15	19.95	17.47	8.25	5.9	7.88	11.9
54	16	16	27.44	16.91	7.51	4.24	8.56	13.9
55	17	17	28.65	19.22	7.42	4.18	7.64	14.62
56	18	18	21.33	22.79	10.14	6.96	7.36	11.92
57	19	19	23.52	28.12	8.25	5.42	5.02	13.28
58	20	20	24.43	29.67	8.79	6.16	5.6	14.64
59	21	21	28.46	28.79	9.06	4.86	5.48	13.34
60	22	22	31.62	26.3	10.66	4.58	6.5	11.76
61	23	23	27.08	34.91	10.25	5.9	5	13.22
62	24	24	36.5	28.84	10.82	4.02	6.06	12
63	25	25	31.66	33.93	11.83	6.34	6.68	14.42
64	26	26	32.28	32.76	13.93	6.66	7.48	12.78
65	27	27	40.24	32.33	12.52	4.14	6.12	11.68
66	28	28	35.64	34.37	15.09	6.54	7.8	12.84
67	29	29	35.79	38.61	14.52	6.18	6.46	12.76
68	30	30	38.33	38.51	15.14	5.72	6.5	12.18
69	31	31	37.87	35.67	19.25	7.16	8.78	11.34
70	32	32	40.23	41.37	16.31	6.62	7.32	13.68
71	33	33	43.3	40.32	17.26	6.34	7.88	13.36
72	34	34	33.82	54	17.52	7.8	5.14	12.08
73	35	35	40.06	43.45	21.15	7.74	8.04	11.72
74	36	36.01	47.03	52.5	14.97	4.98	5	13.68
75	37	37	52.93	43.32	18.54	5.6	8.1	13.74
76	38	37.99	64.02	40.69	17.88	4.6	8.96	14.74
77	39	39	55.15	51.31	16.93	5.08	6.32	14.56
78	40	40	51.2	51.04	20.31	6.66	7.64	14.12
79	15	15	10.31	19.76	16.8	14.22	7.56	5.72
80	16	16	9.57	28.62	14.06	14.1	4.44	6.4
81	17	17	13.43	24.79	13.35	11.48	6.38	8.26
82	18	18	14.28	25	15.09	12.28	7.26	8.22
83	19	19	12.78	30.7	16.2	12.66	5.16	6.84
84	20	20	13.13	37.33	14.45	12.38	4.14	7.9
85	21	21	16.46	29.91	17.25	11.62	6.58	7.86
86	22	22	13.39	36.91	20.63	14.02	4.86	6.02
87	23	23	16.54	31.91	22.23	14.86	7.84	7.5
88	24	24	17.09	37.03	20.13	12.08	5.56	7.12
89	25	25	16.4	49.07	16.92	12.9	4.06	8.88
90	26	26	20.72	36.48	21.32	13	7.64	8.98
91	27	27	18.12	43.38	23.29	12.9	5.28	6.86
92	28	28	17.8	49.82	22.65	12.16	4.14	6.54
93	29	29	24.71	37.91	24.03	11.38	7.86	8.4
94	30	30	21.34	37.04	35.31	12.66	7.56	5
95	31	31	19.96	55.12	24.63	12.66	4.38	7.06
96	32	32	20	47.53	34.49	14.78	6.1	5.56
97	33	33	22.48	54.72	26.64	11.86	4.76	6.94
98	34	34	25.27	46.98	31.51	12.92	7.12	7.12
99	35	35	24.99	50.69	32.08	14.76	7.34	7.86
100	36	36	23.92	61.91	28.62	14.14	5.28	8.18
101	37	37	26.41	51.82	35.67	14	7.24	7.02
102	38	38	29.32	51.75	34.09	11.2	6.56	6.7
103	39	39	27.45	65.8	29.32	12.92	5.28	8.54
104	40	40	31.44	48.08	42.26	11.22	7.78	5.66
105	15	15	20.47	14.02	11.05	8.16	14.56	12.04
106	16	16	23.22	15.39	10.45	7.04	13.06	12.86
107	17	17	22.72	14.51	14.82	8.76	16.96	10.34
108	18	18	24.88	15.88	14.26	8.42	16.32	11.58
109	19	19	25.96	16.25	16.05	8.24	16.34	10.36
110	20	20	30.73	17.19	14.46	7.28	16.48	12.62
111	21	21	24.87	20.37	17.51	9.84	14.22	10.62
112	22	22	27.71	20.79	17.64	10.48	16.8	12.74
113	23	23	27.37	23.11	18.05	10.38	14.48	12.1
114	24	24	29.32	23.56	18.87	9.54	14.1	11.44
115	25	25	31.41	24.38	19.15	8.56	13.18	10.94
116	26	26	33.55	22.37	23.44	9.2	16.94	10.02
117	27	27	36.95	26.01	18.92	7.86	13.58	12.34
118	28	28	37.57	26.23	20.99	8.06	14.06	11.42
119	29	29	36.36	27.26	23.58	8.58	13.76	10.2
120	30	30	33.86	29.5	26.1	10	13.44	9.7
121	31	31	39.16	28.23	26.24	8.18	13.74	9.36
122	32	32	45.06	28.49	24.41	8.4	16.46	12.34
123	33	33	47.19	28.28	26.21	7.78	16.24	11.1
124	34	34	42.36	31.64	28.3	8.38	13.5	9.62
125	35	35	40.29	34.73	29.26	10.86	14.78	11.3
126	36	36	50.32	31.38	28.66	7.58	15.1	10.5
127	37	37	43.33	38.17	28.48	10.22	13.56	11.96
128	38	38	45.71	36.15	32	9.72	14.64	10.56
129	39	39	47.97	39.19	29.33	9.7	14.06	12.36
130	40	40	53.76	38.46	28.75	8.46	14.34	12.62
131	15	15	16.95	19.8	8.46	10.06	9.58	16.28
132	16	16	16.12	23.78	8.86	11.78	8.52	17
133	17	17	17.69	21.09	11.61	11.28	10.48	13.22
134	18	18	16.95	28.04	10.2	12.82	8.1	16.64
135	19	19	16.95	31.29	10.71	12.78	7.1	15.66
136	20	20	21.69	24.3	13.35	10.34	10.34	13.08
137	21	21	21.37	29.13	12.62	12.4	9.84	16.46
138	22	22	25.24	29.82	11.74	9.58	9	16.84
139	23	23	26.59	29.78	12.95	9.6	9.6	15.98
140	24	24	23.67	32.47	15.54	12.72	10.02	14.9
141	25	25	24.81	34.76	15.46	12.78	9.82	15.92
142	26	26	26.42	38.08	14.53	10.52	7.82	15.18
143	27	27	29.07	36.84	15.45	10.2	8.82	15.32
144	28	28	29.67	41.26	14.73	10.06	7.8	16.38
145	29	29	28.36	41.54	17.54	11.82	8.62	14.86
146	30	30	31.67	39.94	18.21	10.16	8.84	13.92
147	31	31	32.67	41.82	18.52	9.92	8.48	13.82
148	32	32	31.98	45.95	18.76	11.5	8.58	15.38
149	33	33	36.61	46.28	17.53	9.36	8.12	15.9
150	34	34	35.09	49.51	18.78	9.84	7.5	14.66
151	35	35	32.97	51.73	21.29	11.18	7.52	13.36
152	36	36	39.87	46.54	21.43	10.56	10.06	15.66
153	37	37	38.76	53.09	20.46	10.06	7.94	15.24
154	38	38	42.21	50.83	21.49	10.3	9.46	16.28
155	39	39	41.13	49.65	25.24	11.16	10.22	14.16
156	40	40	42.04	55.99	22.78	11.18	9.12	16.42
157	15	15	11.3	18.85	15.75	16.26	10.18	7.86
158	16	16	12.45	22.76	13.3	16.62	9.34	10.98
159	17	17	13.78	19.96	17.81	16.24	12	8.58
160	18	18	15.04	20.69	18.65	14.82	11.64	8.22
161	19	19	16.38	24.64	15.68	13.78	9.74	10.36
162	20	20	16.58	25.61	17.75	16.14	11.06	10.64
163	21	21	17.55	23.95	22.03	13.24	10.5	7.24
164	22	22	17.69	27.42	21.26	16.06	10.96	9.24
165	23	23	19.02	29.85	20.09	14.36	9.66	9.62
166	24	24	20.18	30.99	20.63	13.48	9.3	9.38
167	25	25	20.32	28.71	27.02	13.98	10.64	7.14
168	26	26	23.46	30.95	22.88	13.7	11.3	10.08
169	27	27	22.28	36.29	22.45	14.6	9.4	10.34
170	28	28	21.38	37.7	25.93	16.3	9.56	9.28
171	29	29	24.48	32.31	31.1	13.58	11.2	7.32
172	30	30	25.6	35.3	29.07	16.06	12.58	9.88
173	31	31	24.66	37.33	32.09	13.4	9.4	7.02
174	32	32	27.49	36.72	31.94	13.06	10.62	7.82
175	33	33	28.31	39.32	31.14	14.94	11.6	9.54
176	34	34	30.57	41.18	29.29	13.34	10.74	10.02
177	35	35	28.5	48.51	28.36	14.92	9.12	10.72
178	36	36	28.91	45.14	34.53	15.46	10.46	8.96
179	37	37	30.47	44.37	36.66	13.4	9.84	7.7
180	38	38	32.01	44.53	37.75	15.64	12.12	9.2
181	39	39	31.58	46.48	39.98	16.3	11.82	8.82
182	40	40	33.23	48.46	38.52	14.46	10.6	8.68
183	15	15	16.4	17.62	10.4	11.14	11.7	13.6
184	16	16	17.06	18.4	11.83	10.58	11.06	11.62
185	17	17	16.86	20	13.37	11.7	10.94	10.98
186	18	18	18.8	20.74	13.63	13.16	13.4	13.72
187	19	19	19.25	21.23	15.69	12	12.22	10.9
188	20	20	20.6	25.49	13.32	11.62	10.34	13.86
189	21	21	24.84	21.84	15.76	10.26	13.64	12.52
190	22	22	23.02	24.69	17.29	13	13.68	13
191	23	23	23.43	27.31	17.23	13.2	12.6	13.54
192	24	24	23.16	28.08	19.77	11.9	10.84	10.24
193	25	25	26.71	26.18	21.24	11.4	13.34	10.66
194	26	26	25.44	30.96	20.46	11.44	10.38	10.56
195	27	27	31.08	30.98	18.14	10.18	11.68	13.7
196	28	28	29.48	32.02	21.22	10.84	11.24	11.4
197	29	29	28.89	33.61	23.2	12.42	11.88	11.48
198	30	30	28.64	34.83	25.36	13.9	12.62	11.46
199	31	31	35.56	35.32	21.14	10.28	11.84	13.54
200	32	32	32.36	37.75	24.44	12.44	11.86	12.36
201	33	33	37.27	37.52	23.03	10.6	12.02	13.34
202	34	34	33.37	37.88	29.47	12.92	12.72	10.68
203	35	35	32.99	44	26.74	13.78	11.22	12.6
204	36	36	35.48	38.62	32.87	13.36	13.82	10.44
205	37	37	39.15	40.43	29.84	11.94	13.14	11.74
206	38	38	39.12	46.91	26.52	11.68	10.78	13.18
207	39	39	43.93	43.71	27.89	10.96	12.6	13.36
208	40	40	40.19	47.25	30.75	12.28	11.6	12.02