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	-2.5E-5	0	-0.069
lnL	0.350082	0	2766.25
lnK	0.400023	0	3289.937
lnM	0.249926	0	2375.441
lnLlnL	-0.020126	0	-810.924
lnKlnK	-0.021064	0	-1019.938
lnMlnM	-0.017151	0	-795.454
lnLlnK	0.024067	0	749.162
lnLlnM	0.016202	0	454.309
lnKlnM	0.018064	0	566.297

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	17.08	17.39	10.07	6.54	7.32	8.7
2	16	16	18.21	18.41	10.86	7.48	8.44	9.82
3	17	17	17.15	20.03	13.05	6.7	6.38	6.6
4	18	18	20.83	16.65	16.7	6.52	9.7	6.04
5	19	19	17.72	21.98	16.68	9.52	8.44	7.3
6	20	20	21	23.48	14.63	8.18	8.2	8.94
7	21	21	25.04	19.35	18.81	6.1	9.44	6.1
8	22	22	25.27	21.96	18.27	6.1	8.22	6.38
9	23	23	21.73	26.39	20.08	7.96	7.24	6.24
10	24	24	23.63	30.6	16.95	8.66	7.3	9.14
11	25	25	22.99	30.53	20.62	8.7	7.12	7.06
12	26	26	30.42	28.58	18.23	7.22	8.88	9.42
13	27	27	29.29	30.03	20.51	8.56	9.5	9.3
14	28	28	33.82	28.2	21.48	6.5	9.2	7.92
15	29	29	32.01	31.91	21.88	7.48	8.58	8.36
16	30	30	30.45	33.26	25.02	9.2	9.48	8.28
17	31	31	36.98	32.18	23.09	6.82	9.18	8.48
18	32	32	34.24	36.3	24.04	8.68	9.26	9.4
19	33	33	42.15	35.4	21.46	6	8.42	9.48
20	34	34	37.84	44.12	20	6.52	6.22	9.86
21	35	35	37.71	41.14	24.72	7.6	7.84	8.92
22	36	36	37.32	44.75	24.65	8.32	7.68	9.68
23	37	37	38.73	40.75	29.91	6.26	6.74	6.06
24	38	38	40.76	39.27	32.77	6.8	8.12	6.28
25	39	39	35.62	54.4	26.76	9.36	6.5	9.36
26	40	40	38.37	51.76	28.63	9.78	7.86	9.86
27	15	15	17.94	11.36	18.56	6.06	11.86	4.16
28	16	16	18.18	12.67	19.7	6.74	11.78	4.38
29	17	17	22.75	12.36	19.33	5.48	12.84	4.74
30	18	18	24.6	14.24	17.25	6.66	14.48	7.22
31	19	19	26.89	14.52	18.42	5.56	13.12	6.2
32	20	20	28.3	16.55	17.03	5.9	12.68	7.66
33	21	21	22.18	16.2	30.1	8.22	13.6	4.1
34	22	22	25.52	18.68	23.37	7.98	13.16	6.32
35	23	23	30.12	16.79	26.75	6.38	14.5	5.24
36	24	24	28.28	18.68	28.92	6.9	12.84	4.8
37	25	25	26.06	20.71	32.22	7.8	11.68	4.34
38	26	26	28.23	21.32	32.15	8.68	13.8	5.32
39	27	27	27.94	23.08	33.34	8.72	12.48	5.06
40	28	28	34.96	25.38	24.24	7.06	11.76	7.76
41	29	29	32.51	27.1	27.61	8.18	11.58	7.08
42	30	30	38.46	22.56	34.13	5.4	11.56	4.44
43	31	31	36.6	26.89	31.04	8.12	13.34	7.04
44	32	32	42.71	26.27	29.78	5.66	11.46	6.18
45	33	33	34.57	27.13	42.81	7.38	11.22	4.1
46	34	34	44.04	26.6	35.64	6.2	12.82	5.68
47	35	35	47.47	28.63	32.06	6.46	13.38	7.32
48	36	36	50.99	29.27	31.5	5.72	12.56	7.2
49	37	37	43.33	33.61	34.75	8.62	13.28	7.98
50	38	38	50.72	25.49	50.01	5.7	14.64	4.14
51	39	39	42.31	37.29	37.44	8.9	11.8	7.34
52	40	40	53.61	33.66	35.55	5.94	11.74	6.88
53	15	15	23.23	16.95	7.21	4.48	7.42	12.68
54	16	16	21.93	19.62	7.91	4.76	6.2	11.28
55	17	17	21.53	20.67	9.34	7.24	8.68	13.82
56	18	18	25.42	21.7	8.8	5.84	8.04	14.54
57	19	19	21.1	26.37	10.22	7.44	6.54	12.46
58	20	20	30.04	25.59	8.41	4.68	6.46	14.94
59	21	21	24.42	28.72	10.9	7.92	7.48	14.62
60	22	22	24.29	33.17	10.68	7.88	6.24	14.8
61	23	23	37.9	22.82	12.36	4.18	8.68	11.16
62	24	24	25.29	35.23	12.77	7.96	6.16	12.7
63	25	25	36.54	31.47	11.08	4.74	6.46	13.78
64	26	26	37.41	28.49	14.2	5.5	8.66	12.28
65	27	27	37.91	30.97	14.23	5.96	8.64	13.48
66	28	28	34.45	36.12	14.69	5.7	6.16	11.1
67	29	29	35.93	38.11	14.71	6.94	7.4	14.18
68	30	30	37.49	39.74	14.9	6.28	6.7	13.28
69	31	31	41.84	33.99	18.26	5.88	8.58	11.14
70	32	32	38.09	45.94	15.13	6.02	5.52	12.74
71	33	33	39.85	42.31	17.86	6.74	7.18	12.38
72	34	34	42.13	42.57	18.42	6.2	7	11.72
73	35	35	45.53	41.33	19.4	5.84	7.48	11.38
74	36	36	54.25	40.11	18.19	4.6	7.5	11.88
75	37	37	42.03	52.84	18.63	7.56	6.6	14.06
76	38	38	56.37	46.69	17.11	4.22	6.02	12.26
77	39	39	52.79	46.7	20.22	6.12	8.08	13.52
78	40	40	51.63	46.64	22.78	6.32	8.14	11.82
79	15	15	11.04	22.97	12.14	13.06	6.3	8.34
80	16	16	10.83	27.39	12.47	14.18	5.44	8.58
81	17	17	11.99	28.73	12.72	11.4	4.66	7.6
82	18	18	14.77	23.41	15.84	12	7.98	7.9
83	19	19	12.04	36.86	13.71	13.9	4.26	8.52
84	20	20	12.64	35.68	16.24	13.76	4.64	7.32
85	21	21	13.98	31.17	20.58	12	5.34	5.44
86	22	22	18.28	30.09	17.67	12.08	7.68	8.98
87	23	23	15.07	35.08	22.19	11.26	4.76	5.1
88	24	24	17.82	32.85	22.6	12.18	6.78	6.58
89	25	25	17.08	42.84	19.2	14.04	5.44	8.74
90	26	26	18.58	38.94	22.65	13.08	6.26	7.4
91	27	27	17.57	48.23	21	12.92	4.5	7.48
92	28	28	21.74	35.22	28.07	12.1	7.84	6.4
93	29	29	18.58	50.29	24.03	14.4	5.1	7.6
94	30	30	18.6	52.09	26.06	14.7	5	7.06
95	31	31	21.14	47.92	27.63	12.24	5.34	6.38
96	32	32	21.76	46.43	31.45	11.1	5.2	5.14
97	33	33	26.44	44.75	28.25	12.32	7.58	8.14
98	34	34	26.26	48.17	28.73	13.54	7.58	8.7
99	35	35	21.62	65.83	27.34	13.62	4.2	7.38
100	36	36	23.11	65.56	27.79	13.84	4.64	7.96
101	37	37	27.99	50.47	34.08	13.06	7.46	7.4
102	38	38	24.49	55.34	40.27	14.98	6.56	5.96
103	39	39	29.32	51.44	38.12	11.02	6.5	5.78
104	40	40	25.56	73.86	30.51	14.76	4.84	8.56
105	15	15	18.22	13.54	13.56	10.3	16.7	10.42
106	16	16	21.44	14.97	12.03	8.34	14.54	11.76
107	17	17	24.64	16.01	11.48	7.06	13.4	12.38
108	18	18	24.47	15.77	14.73	7.98	15.3	10.36
109	19	19	25.93	17.17	14.73	8.88	16.48	12.34
110	20	20	29.07	17.18	15.51	7.66	16.26	11.46
111	21	21	25.39	20.49	16.89	9.1	13.38	10.5
112	22	22	25.99	22.31	17.18	10.6	14.5	12.32
113	23	23	25.75	22.64	20.19	10.56	14.04	10.04
114	24	24	27.09	24.59	19.6	10.96	14.04	11.46
115	25	25	35.78	21.54	19.68	7.8	16.2	11.26
116	26	26	30.14	25.92	21.38	9.98	13.62	10.68
117	27	27	40.74	23.13	20.07	7.5	16.68	12.32
118	28	28	40.41	23.65	22.49	7.66	16.44	10.9
119	29	29	36.12	29.68	20.87	9.5	13.68	12.94
120	30	30	33.42	30.59	25.11	10.3	13.06	10.3
121	31	31	37.68	28.68	26.92	9.48	14.94	10.06
122	32	32	42.87	30.37	23.55	8.62	14.78	12.46
123	33	33	41.58	33	24.23	8.96	13.44	12.08
124	34	34	37.6	33.86	29.81	10.78	13.94	10.12
125	35	35	44.35	34.2	26.44	8.96	13.9	11.76
126	36	36	43.77	32.63	32.28	10.12	16.34	10.34
127	37	37	46.04	33.93	31.58	10.28	16.82	11.44
128	38	38	52.89	35.52	27.28	7.8	14.24	12.14
129	39	39	43.41	38.93	33.78	10.84	14.08	10.4
130	40	40	45.5	41.36	31.87	10.26	13.12	11.14
131	15	15	16.77	19.77	8.59	10.26	9.66	16.1
132	16	16	16.65	23.28	8.76	10.12	7.8	15.4
133	17	17	16.46	25.87	9.59	11.02	7.4	14.86
134	18	18	18.4	27.71	9.34	10.6	7.48	16.8
135	19	19	17.87	29.47	10.84	11.6	7.36	14.92
136	20	20	21.41	28.39	10.92	10.76	8.82	16.96
137	21	21	24.13	27.52	11.71	9.5	9.3	15.88
138	22	22	21.71	32.22	12.76	10.22	7.34	13.64
139	23	23	26.76	29.72	12.88	9.94	10.04	16.78
140	24	24	25.49	32.55	14.09	10.22	8.76	14.66
141	25	25	24.88	33.61	16.14	12.86	10.32	15.28
142	26	26	30.19	30.98	16.36	9.18	10.18	13.48
143	27	27	25.35	42	15.39	11.38	7.18	14.62
144	28	28	31	35.93	16.85	10.7	10.28	15.66
145	29	29	33.46	34.94	18.11	9.26	10.06	13.62
146	30	30	34.97	38.1	17.17	9.06	9.36	14.94
147	31	31	32.61	41.24	18.92	11.88	10.3	16.1
148	32	32	36.11	42.63	17.85	10	9.4	16.36
149	33	33	32.76	48	19.17	10.12	7.38	13.58
150	34	34	36.56	48.47	18.36	10.56	8.66	16.96
151	35	35	33.04	48.54	23.18	12.22	8.88	13.24
152	36	36	34.31	54.59	20.83	10.88	7.2	13.98
153	37	37	38.23	50.93	22.04	11.16	9.1	15.24
154	38	38	44.58	46.24	22.91	9.7	10.62	15.16
155	39	39	37.58	51.7	26.8	12.66	9.94	13.46
156	40	40	39.56	63	20.99	10.68	7.06	16.08
157	15	15	12.1	18.79	14.32	14.34	9.76	8.4
158	16	16	14.31	18.8	14.55	15.1	12.52	10.58
159	17	17	14.64	18.1	19.03	14.6	13	7.66
160	18	18	16.22	22	15.25	13.82	11.04	10.62
161	19	19	16.73	21.05	19.33	13.64	11.9	8.22
162	20	20	17.07	22.87	20.27	14.84	12.02	8.66
163	21	21	18.66	23.23	21.14	14.72	13	9.08
164	22	22	17.89	24.05	25.69	15.34	12.38	7.16
165	23	23	19.11	28.06	21.91	14.8	10.76	9
166	24	24	20.05	28.17	24.07	13.26	10.16	7.64
167	25	25	21.34	28.57	25.35	13.1	10.62	7.64
168	26	26	22.06	32.59	23.07	15.7	11.34	10.64
169	27	27	23.71	33.52	23.15	14.62	11.12	10.74
170	28	28	20.2	36.69	29.39	16.84	9.54	7.74
171	29	29	24.05	38.71	24.2	14.98	9.78	10.62
172	30	30	23.99	35.07	32.29	16.8	12.28	8.46
173	31	31	26.89	36.53	29.29	15.76	12.56	10.18
174	32	32	24.52	41.61	31.09	16.4	10.06	8.86
175	33	33	26.31	42.71	30.49	14.02	9.06	8.42
176	34	34	31.1	37.71	32.72	13.94	12.7	9.38
177	35	35	30.94	40.66	32.89	13.9	11.52	9.24
178	36	36	31.5	43.43	32.4	13.66	10.7	9.44
179	37	37	30.22	50.14	30.84	15.08	9.5	10.52
180	38	38	32.11	42.36	40.65	13.82	11.4	7.48
181	39	39	32.46	46.71	38.13	14.18	10.56	8.38
182	40	40	37.43	43.7	38.17	13.44	12.8	9.38
183	15	15	15.15	16.48	12.77	12.36	12.8	10.8
184	16	16	15.12	20.2	12.1	12.5	10.16	11.6
185	17	17	14.95	21.41	14.24	13.72	10.28	10.38
186	18	18	20.04	18.49	14.91	10.32	12.96	10.44
187	19	19	21.36	21.65	13.29	11.02	12.4	13.76
188	20	20	21.24	23.1	14.77	11.98	12.4	13.12
189	21	21	20.45	26.13	15.59	13.96	11.96	13.72
190	22	22	21.84	26.07	17.1	13.88	12.88	13.22
191	23	23	22.98	24.66	20.64	12.34	12.98	10
192	24	24	26.23	27.87	16.94	11.2	11.92	13.38
193	25	25	23.52	30.71	19.81	12.24	10.22	10.7
194	26	26	25.17	30.61	21.11	12.18	11.06	10.7
195	27	27	24.41	34.68	21.15	13.86	10.48	11.72
196	28	28	28.34	30.9	23.61	11.88	12.26	10.52
197	29	29	27.66	36.53	21.74	13.98	11.52	13.26
198	30	30	29.7	37.63	21.54	11.84	10.24	12.34
199	31	31	29.9	37.72	24.09	12.88	11.2	11.86
200	32	32	37.21	34.69	23.08	10.36	12.86	12.98
201	33	33	35.46	39.46	22.8	10.16	10.24	12.2
202	34	34	32.99	39.24	28.36	13.78	12.84	11.76
203	35	35	36.33	36.98	30.49	12.1	13.54	10.62
204	36	36	39.46	40.07	26.94	11.42	12.82	12.78
205	37	37	40.81	40.33	28.34	10.32	11.96	11.32
206	38	38	35.98	47.2	29.38	13.68	11.36	12.4
207	39	39	37.71	49.8	28.15	12.84	10.58	12.94
208	40	40	44.05	43.95	30.34	11.92	13.66	13.2