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	0.000641	0	1.811
lnL	0.350095	0	2971.89
lnK	0.399634	0	3141.165
lnM	0.249894	0	2471.418
lnLlnL	-0.020193	0	-892.358
lnKlnK	-0.021115	0	-947.462
lnMlnM	-0.017183	0	-877.1
lnLlnK	0.024204	0	680.702
lnLlnM	0.016189	0	504.663
lnKlnM	0.018152	0	555.269

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	16.73	15.64	12.1	6.58	8.14	6.88
2	16	16	16.27	16.62	14.71	7.66	8.54	6.16
3	17	17	21.21	17.92	11.68	6.2	8.64	8.94
4	18	18	16.94	21.96	14.4	9.9	8.34	8.56
5	19	19	18.56	21.25	16.48	7.96	7.76	6.54
6	20	20	25.05	21.07	13.69	6.08	8.52	8.84
7	21	21	24.9	20.67	17.09	6.24	8.88	6.94
8	22	22	22.44	22.9	20.09	8.6	9.6	7
9	23	23	25.28	27.95	15.09	6.56	6.66	8.6
10	24	24	22.63	31.05	17.58	8.28	6.52	7.96
11	25	25	24.47	29.71	19.73	9.74	8.86	8.96
12	26	26	24.65	29.11	23.46	8.32	7.82	6.3
13	27	27	30.31	28.66	21.03	6.52	7.96	7.16
14	28	28	27.16	37.78	18.6	8.18	6.34	9.12
15	29	29	28.54	37.18	20.33	8.36	7	8.9
16	30	30	29.89	34.49	24.28	8.2	7.92	7.48
17	31	31	33.86	37.04	21.01	6.34	6.52	7.94
18	32	32	35.56	32.03	27.66	6.5	8.4	6.24
19	33	33	35.17	38	24.37	7.8	8.14	8.58
20	34	34	33.55	34.23	34.26	8.78	9.8	6.12
21	35	35	38.19	39.49	25.83	8.5	9.34	9.62
22	36	36	36.23	40.28	29.96	8.82	8.9	7.88
23	37	37	33.39	49.37	27.49	9.62	6.94	8.64
24	38	38	36.13	41.79	35.11	9.18	8.84	6.78
25	39	39	36.91	51.92	27.28	8.34	6.38	8.5
26	40	40	48.93	38.55	32.3	6.08	9.2	7.08
27	15	15	15.24	13.45	17.52	8.84	11.7	5.36
28	16	16	18.84	12.82	18.37	8.1	14.56	5.96
29	17	17	19.61	14.82	17.44	7.78	12.36	6.38
30	18	18	25.49	12.29	21.17	5.22	14.08	4.64
31	19	19	20.3	14.34	27.82	8.64	14.86	4.26
32	20	20	24.52	17.24	19.23	8.12	14.04	7.72
33	21	21	23.8	16.23	27.07	6.58	11.8	4.04
34	22	22	27.83	16.47	25.7	5.56	11.78	4.36
35	23	23	26	20.83	22.78	8.94	13.26	7.46
36	24	24	30.84	18.94	25.06	7.26	14.72	6.62
37	25	25	29.69	18.64	32.1	6.66	13.16	4.34
38	26	26	36.87	20.76	23.38	6.26	14.06	7.64
39	27	27	33.96	22.37	26.77	7.24	13.52	6.84
40	28	28	36.05	21.17	31.37	6.68	14.28	5.62
41	29	29	32.01	24.76	32.73	7.76	12	5.4
42	30	30	35.35	21.86	40.6	7.02	14.12	4.26
43	31	31	35.62	26.33	33.35	8.5	13.86	6.56
44	32	32	43.33	26.67	28.5	6.14	12.42	7.18
45	33	33	36.75	27.28	38.85	8.22	13.34	5.5
46	34	34	37.44	32.12	32.6	8.9	12.18	7.48
47	35	35	48.34	27.02	34.46	6.62	15	7.04
48	36	36	50.31	28.91	32.69	5.48	12.02	6.5
49	37	37	45.23	32.66	34.37	7.48	12.54	7.38
50	38	38	51.06	28.31	41.2	5.22	11.94	4.8
51	39	39	47.15	32.31	40.8	6.26	11.16	5.3
52	40	40	54	29.46	43.96	6.22	14.5	5.66
53	15	15	17.47	21.01	7.52	7.26	6.68	13.98
54	16	16	19.26	21.95	7.93	6.88	6.74	13.96
55	17	17	19.17	27.08	7.49	6.7	5.1	14.46
56	18	18	23.6	21.95	9.45	5.56	6.92	11.66
57	19	19	26.49	20.47	11.04	5.62	8.7	11.24
58	20	20	23.54	30.45	8.88	6.42	5.42	14.44
59	21	21	24.06	26.53	12.37	7.5	7.64	11.72
60	22	22	35.56	26.16	9.4	4.28	7.02	14.62
61	23	23	35.11	26.47	10.95	5.12	8.16	14.4
62	24	24	32.83	33.9	9.86	4.58	5.04	13.46
63	25	25	27.97	36.66	12.38	7.58	6.3	14.12
64	26	26	31.75	35.05	12.96	5.72	5.82	11.72
65	27	27	30.64	32.27	17.41	7.98	8.58	11.08
66	28	28	42.65	31.92	13.54	5.04	8.1	13.88
67	29	29	35.52	38.51	14.71	6.64	6.9	13.38
68	30	30	45.45	34.88	14.21	5.14	8.02	14.42
69	31	31	38.81	41.77	15.02	5.8	6.08	12.66
70	32	32	49.18	33.71	17.1	4.5	8.02	11.14
71	33	33	43.64	45.76	14.43	4.7	5.08	12.34
72	34	34	46.2	41.17	17.28	4.92	6.44	11.18
73	35	35	46.69	43.88	17.33	5.14	6.32	11.78
74	36	36	47.37	43.92	18.81	6.1	7.62	12.92
75	37	37	49.66	47.06	17.84	5.98	7.28	14.24
76	38	38	54.36	49.51	16.49	4.28	5.46	12.44
77	39	39	59.9	44.62	18.59	5.26	8.5	14.88
78	40	40	58.76	44.66	20.76	5.22	8.24	12.68
79	15	15	11.17	20.59	14	13.08	7.28	7.16
80	16	16	10.99	25.85	13.24	13.98	5.84	8.02
81	17	17	13.96	23.58	13.6	11.22	6.92	8.26
82	18	18	14.8	24.6	14.68	11.42	7.18	8.24
83	19	19	13.98	27.96	16.29	12.74	6.44	7.6
84	20	20	14.49	32.48	15.23	11.4	5.04	7.68
85	21	21	15.5	30.47	18.29	14.22	7.34	8.36
86	22	22	14.09	36.33	19.56	13.76	5.16	6.68
87	23	23	15.31	36.52	20.44	13.14	5.4	6.68
88	24	24	17.15	36.98	20.07	14.82	6.86	8.8
89	25	25	18.53	39.43	19.2	13.02	6.12	8.92
90	26	26	19.44	34.85	25.02	12.66	7.28	6.72
91	27	27	15.93	47.61	24.7	12.94	4.08	5.52
92	28	28	20.74	38.46	26.31	12.48	6.9	6.74
93	29	29	19.99	36.73	34.39	13.78	7.7	5.2
94	30	30	18.03	48.15	30.59	13.56	4.88	5.2
95	31	31	23.62	43.92	26.72	12.56	6.92	7.76
96	32	32	24.01	44.16	29.32	11.9	6.64	6.72
97	33	33	23.83	47.46	30.07	14.84	7.54	8.06
98	34	34	24.11	52.67	28.54	13.08	5.96	7.66
99	35	35	25.42	49.46	32.44	14.36	7.5	7.7
100	36	36	26.46	51.32	32.37	12.74	6.68	7.18
101	37	37	27.06	60.16	27.76	12.32	5.5	8.54
102	38	38	23.86	72.23	28.57	14.88	4.62	8.6
103	39	39	28.32	56.31	35.04	13.24	6.74	7.36
104	40	40	27.81	66.58	31.2	13.98	5.72	8.72
105	15	15	20.25	14.71	10.4	8.16	13.58	12.76
106	16	16	21.2	15.62	11.44	7.94	13	11.72
107	17	17	23.64	14.14	14.68	7.96	16.66	9.96
108	18	18	21.87	17.62	14.31	10.08	14.86	11.86
109	19	19	22.98	18.01	15.99	10.56	16.08	11.56
110	20	20	23.83	17.49	19.51	9.96	16.38	9
111	21	21	31.14	17.91	16.06	7.44	16.3	11.58
112	22	22	27.67	21.2	17.14	10.26	16.04	12.88
113	23	23	28.12	22.15	18.61	8.68	13.14	10.08
114	24	24	30.67	21.57	20.42	8.78	15.18	10.12
115	25	25	30.73	21.83	23.45	9.9	16.92	9.72
116	26	26	29.4	25.21	23.07	10.82	14.82	10.28
117	27	27	31.24	23.95	26.79	10.68	16.68	9.14
118	28	28	37.14	24.44	23.78	9.08	16.98	10.96
119	29	29	35.08	27.02	25.05	10.08	15.66	10.7
120	30	30	41.32	26.36	24.03	8.26	16.02	11.16
121	31	31	39.45	29.54	24.21	8.78	14.1	11.14
122	32	32	34.94	31	29.81	10.64	14	9.16
123	33	33	41.46	29.62	28.79	9.06	15.38	9.94
124	34	34	42.13	33.47	26.13	9.8	14.68	12.28
125	35	35	47.59	32.72	25.87	8.64	15.34	12.64
126	36	36	51.26	30.21	29.72	7.64	16.26	10.36
127	37	37	44.71	34.91	31.39	9.64	14.74	10.44
128	38	38	47.3	34.76	32.55	10.28	16.88	11.4
129	39	39	56.46	33.76	30.03	7.78	16.28	11.68
130	40	40	45.63	40.37	32.95	10.94	14.44	11.46
131	15	15	16.03	19.63	9.18	11.82	10.64	16.26
132	16	16	18.04	20.8	9.24	9.46	9.14	14.84
133	17	17	17.09	23.19	10.62	11.28	9	14.1
134	18	18	20.76	23.38	10.12	9.44	9.38	15.7
135	19	19	21.85	23.55	11.45	9.48	9.92	14.48
136	20	20	20.96	29.16	10.82	10.4	8.06	16.18
137	21	21	24.55	25.75	12.59	9.7	10.48	15.2
138	22	22	25.92	26.98	13.05	9.98	10.88	15.98
139	23	23	23.37	33.41	13	9.86	7.4	14.04
140	24	24	23.2	33.96	14.99	12.66	9.24	15.12
141	25	25	26.32	34.76	14.34	10.22	8.42	14.92
142	26	26	25.69	36.45	15.99	12.02	9.1	15
143	27	27	24.95	38.54	17.66	12.14	8.32	13.02
144	28	28	31.14	39.81	14.56	9.72	8.32	16.98
145	29	29	33.28	35.93	17.52	10.36	10.82	15.74
146	30	30	28.28	43.8	18.53	12.94	8.84	15.2
147	31	31	31.35	38.87	21.65	12.08	10.72	13.34
148	32	32	34.14	45.96	17.28	10.18	8.2	16.2
149	33	33	30.01	48.74	21.03	12.48	8.06	13.54
150	34	34	32.33	55.58	17.85	11.82	7.14	16.98
151	35	35	35.06	45.6	23.43	11.78	9.88	13.52
152	36	36	36.76	49.26	21.94	11.46	9.28	15.02
153	37	37	43.28	42.87	24.01	9.24	10.68	13.2
154	38	38	39.2	56.54	20.4	10.96	8.14	16.88
155	39	39	40.16	52.46	24.09	12.2	10.18	15.9
156	40	40	37.54	57.3	25.51	12.32	8.56	13.86
157	15	15	12.03	17.05	16.8	15.34	11.64	7.4
158	16	16	11.93	19.49	17.92	16.5	10.58	7.3
159	17	17	12.74	22.37	16.76	16.66	9.82	8.62
160	18	18	14.75	23.1	16.2	15.7	10.58	10.04
161	19	19	16.39	21.43	19.36	14.1	11.76	8.28
162	20	20	18.26	20.94	21.15	13.08	12.72	7.86
163	21	21	17.32	28.33	17.38	15.02	9.62	10.68
164	22	22	19.45	24.78	21.69	13.08	11.24	8.22
165	23	23	18.72	28.69	21.8	16.48	11.4	9.84
166	24	24	19.22	27.67	26.34	14.72	10.96	7.26
167	25	25	19.7	34.17	21.65	15.92	9.52	10.18
168	26	26	22.71	29.28	26.1	13.52	11.46	8.2
169	27	27	21.26	34.56	25.82	14.78	9.54	8.4
170	28	28	22.72	35.15	26.42	14.68	10.04	8.78
171	29	29	24.13	33	30.72	14.14	11.18	7.6
172	30	30	27.54	35.37	26.14	14.14	12.04	10.74
173	31	31	26.86	36.57	29.28	15.12	12.02	9.76
174	32	32	25.82	37.89	33.31	16.62	12.1	8.8
175	33	33	25.03	45.87	29.44	16.2	9.08	9.56
176	34	34	26	40.85	37.45	16.56	11.12	7.7
177	35	35	28.85	38.03	40.46	14.92	12.32	7.16
178	36	36	29.6	42.17	37.08	14.78	11.14	8.1
179	37	37	29.16	41.71	43.1	15.52	11.64	7
180	38	38	33.55	41.61	39.25	13.3	11.8	7.9
181	39	39	30.14	44.34	46.16	16.78	12.18	7.26
182	40	40	33.64	47.6	38.9	16.88	12.82	10.16
183	15	15	15.06	18.66	10.7	11.68	10.38	12.48
184	16	16	14.95	20.64	11.91	13.1	10.24	12.22
185	17	17	15.55	20.27	14.65	13.6	11.38	10.42
186	18	18	16.55	22.73	14.13	13.88	10.92	11.94
187	19	19	17.84	23.43	15.01	12.08	10.02	10.58
188	20	20	21.49	22.1	15.53	11.52	12.74	12.06
189	21	21	21.29	24.5	16.24	13.54	13.12	13.3
190	22	22	23.41	27.06	14.79	10.44	10.06	12.8
191	23	23	22.49	28.62	16.98	13.64	11.74	13.56
192	24	24	23.49	29.04	18.44	11.52	10.26	10.94
193	25	25	25.33	29.2	19.33	13.22	12.78	12.98
194	26	26	23.77	32.57	20.81	13.74	10.84	11.48
195	27	27	27.93	29.58	22.35	11.16	11.92	10.36
196	28	28	27.58	33.58	21.61	12.44	11.28	11.84
197	29	29	28.02	32.79	25.11	12.36	11.74	10.04
198	30	30	32.37	35.03	21.35	11.14	11.6	12.98
199	31	31	33.73	32.92	25.15	11.24	13.22	11.34
200	32	32	35.07	35.46	24.11	11.1	12.52	12.32
201	33	33	32.29	37.12	28.3	13.28	12.88	11.08
202	34	34	33.81	36.28	30.93	12.76	13.42	10.12
203	35	35	35.77	41.36	26.29	13.12	12.64	13.46
204	36	36	42.72	40.2	24.21	10.02	12.3	13.96
205	37	37	37.29	46.11	26.19	12.4	11.04	13.42
206	38	38	34.38	45.63	32.87	13.6	11.14	10.24
207	39	39	39.71	44.04	31.5	11.94	12.08	11.2
208	40	40	36.55	49.25	32.88	13.52	10.88	10.94