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	-2.6E-5	0	-0.063
lnL	0.350234	0	2527.073
lnK	0.399918	0	2776.511
lnM	0.249876	0	2093.652
lnLlnL	-0.020203	0	-843.109
lnKlnK	-0.021076	0	-902.621
lnMlnM	-0.017221	0	-794
lnLlnK	0.024087	0	665.558
lnLlnM	0.016299	0	506.079
lnKlnM	0.018106	0	513.736

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	17.64	14.88	12.19	6.16	8.6	6.8
2	16	16	16.32	19.37	11.63	7.86	7.34	8.36
3	17	17	15.44	18.33	17.28	9.94	9.28	6.22
4	18	18	16.08	21.77	15.68	8.7	6.96	6.4
5	19	19	17.63	23.74	14.96	8.84	7.12	7.66
6	20	20	21.37	24.74	13.27	7.88	7.58	9.86
7	21	21	20.17	24.19	17.82	9.34	8.62	7.72
8	22	22	22.97	24.99	17.03	9.06	9.38	9.2
9	23	23	26.18	25.39	16.59	7.68	9.1	9.38
10	24	24	22.48	28.36	20.28	9.82	8.54	7.92
11	25	25	24.65	29.19	20.05	8.54	8	7.78
12	26	26	25.53	31.27	20.06	7.02	6.32	6.66
13	27	27	29.09	29.59	21.15	8.66	9.7	9
14	28	28	32.16	30.13	20.74	6.92	8.54	8.28
15	29	29	30.31	31.52	23.96	8.56	9.34	8.06
16	30	30	26.89	41.16	21.61	9.96	6.9	9.2
17	31	31	31.75	35.63	24.19	9.9	9.88	9.74
18	32	32	36.1	36.25	22.47	6.7	7.62	8.36
19	33	33	31.72	39.28	26.62	7.36	6.54	6.46
20	34	34	33.99	40.42	26.06	6.78	6.32	6.62
21	35	35	37.29	40.9	25.3	8.8	9.02	9.92
22	36	36	34.16	44.34	28.09	9.3	7.82	8.36
23	37	37	37.29	42.19	29.85	8.62	8.52	8
24	38	38	43.82	44.11	25.05	6.12	6.94	8.44
25	39	39	41.96	47.24	26.41	6.82	6.78	8.4
26	40	40	35.94	52.37	30.74	8.34	6.12	7.16
27	15	15	20.45	11.87	14.42	5.16	11.18	5.56
28	16	16	16.89	13.39	19.89	8.86	13.3	5.22
29	17	17	18.87	16.39	15.6	8.44	11.38	7.54
30	18	18	29.62	13.07	15.66	5	14.96	7.56
31	19	19	22.86	16.54	18.44	8.18	13.68	7.52
32	20	20	28.65	15.13	19.42	6.06	14.68	6.84
33	21	21	26.29	17.08	21.6	7.8	14.8	7.02
34	22	22	28.07	15.11	29.55	5.98	14.16	4.02
35	23	23	29.61	20.1	20.27	6.9	12.44	7.7
36	24	24	28.45	18.47	29.25	6.42	12.2	4.44
37	25	24.99	25.62	19.72	36	8.94	13.9	4.28
38	26	26	33.52	18.26	33.06	5.62	13.12	4.08
39	27	27	30.28	25.36	25.48	8.82	12.42	7.72
40	28	28	36.4	20.7	32.17	5.18	11.5	4.28
41	29	29	32.46	23.08	36.16	7.44	12.7	4.68
42	30	30	38.66	21.81	35.93	5.96	13.36	4.64
43	31	31	44.32	25.04	27.08	5.82	13.02	7.42
44	32	32	43.9	27.53	26.63	5.6	11.08	7.2
45	33	33	42.73	26.35	33.45	5.52	11.14	5.26
46	34	34	45.7	25.87	35.54	5.52	12.32	5.3
47	35	35	43.4	30.61	32.37	7.08	12.2	7.14
48	36	36	45.55	25.64	45.84	5.84	13.12	4.14
49	37	37	48.93	25.3	47.66	6	14.9	4.42
50	38	38	48.95	27.07	47.18	6.06	13.9	4.52
51	39	39	46.05	31.37	44.3	7.92	14.22	5.92
52	40	40	46.02	34.71	41.47	8.1	12.9	6.54
53	15	15	17.58	20.77	7.58	6.24	5.86	11.98
54	16	16	17.8	23.86	7.81	6.82	5.52	12.84
55	17	17	20.94	24.59	7.65	5.54	5.24	12.94
56	18	18	28.2	21.32	8	4.76	7.56	14.96
57	19	19	21.17	24.26	11.41	7.62	7.42	11.26
58	20	20	20.2	33.01	9.62	7.92	5.08	13.54
59	21	21	29.72	24.91	10.47	5.12	7.2	12.48
60	22	22	30.85	25.39	11.51	5.54	7.96	12.62
61	23	23	25.8	32.3	12.03	7.5	6.58	13.14
62	24	24	37.31	27.78	11.11	4.12	6.66	12.24
63	25	25	41.47	27.38	11.6	4.06	7.56	12.98
64	26	26	36.28	33.53	11.77	4.72	5.92	12.68
65	27	27	37.83	32.09	13.58	5.32	7.38	12.68
66	28	28	32.92	40.29	13.4	6.68	6.02	13.74
67	29	29	38.33	34.93	15.34	6.76	8.62	14.18
68	30	30.01	41.51	43.5	11.77	4.66	5.04	14.66
69	31	31	36.4	43.5	15.36	7.1	6.58	14
70	32	32	46.16	38.87	15.11	4.86	6.8	12.92
71	33	33	46.53	38.31	17.02	5.18	7.44	12.08
72	34	34	51.42	44.39	13.83	4.18	5.68	14
73	35	35	35.65	52.96	18.67	7.86	5.64	12.02
74	36	36	42.27	48.02	19.09	6.12	6.02	11.14
75	37	37	59.09	42.39	16.85	4.12	6.96	12.88
76	38	38	57.76	46.83	16.58	4.32	6.32	13.4
77	39	39	56.79	43.43	20.53	4.94	7.74	11.68
78	40	40	44.92	50.78	24	7.58	7.5	11.32
79	15	15	11.59	22.38	11.76	12.34	6.5	8.66
80	16	16	11.48	27.29	11.53	12.36	5.1	8.78
81	17	17	11.69	25.05	16.07	11.2	5.22	5.5
82	18	18	12.93	23.51	19.1	11.74	6.64	5.3
83	19	19	12.07	28.29	19.9	13.16	5.52	5.22
84	20	20	12.15	32.98	19.29	14.28	5.04	5.92
85	21	21	12.9	36.41	18.53	13.76	4.64	6.42
86	22	22	15.49	34.59	18.26	14.14	6.28	8.32
87	23	23	14.65	39.44	19.53	14.92	5.32	7.6
88	24	24	17.22	29.96	27.4	13.06	7.78	5.4
89	25	25	17.78	34.74	24.52	11.2	5.82	5.48
90	26	26	15.84	45.79	22.72	14.34	4.7	6.7
91	27	27	18.63	40.86	24.39	14.42	6.54	7.5
92	28	28	21.36	33.64	30.96	11.44	7.66	5.28
93	29	28.99	19.1	56.97	19.48	12.72	4.02	8.88
94	30	30	17.67	53.43	27.14	13.32	4.14	5.74
95	31	31	19.67	60.14	22.33	13.42	4.12	8.26
96	32	32	22.05	44.39	32.98	11.96	6	5.32
97	33	33	21.96	44.6	37.33	14.74	7.32	5.64
98	34	34	25	47.12	31.88	14.76	8	7.92
99	35	35	24.84	48.01	35.11	11.78	6.2	5.6
100	36	36	23.46	55.07	34.85	14.9	6.24	6.68
101	37	37	25.17	64.36	28	13.36	5.06	8.42
102	38	38	30.11	51.76	32.82	12.58	7.6	8.12
103	39	39	24.89	60.08	38.57	13.08	5.3	5.58
104	40	40	24.27	72.02	34.09	14.16	4.5	6.78
105	15	15	17.13	13.56	14.69	10.64	16	9.1
106	16	16	18.79	15.91	12.98	10	13.9	11.04
107	17	17	25.38	15.45	11.69	7.2	14.76	12.8
108	18	18	22.05	17.78	13.97	10.38	15.28	12.68
109	19	19	27.95	16.44	14.35	7.82	16.68	12.24
110	20	20	28.49	16.74	16.57	7.8	16.66	10.54
111	21	21	25.36	19.95	17.63	9.58	14.52	10.5
112	22	22	26.63	21.26	17.93	10.62	15.82	12.08
113	23	23	30.8	19.54	20.15	8.04	15.7	9.46
114	24	24	29.23	21.42	22	9.92	16.34	9.9
115	25	25	34.32	21.18	21.3	7.54	15.2	9.44
116	26	26	31.91	23.75	22.73	9.88	15.98	10.52
117	27	27	31.57	26.46	22.54	10.38	14.6	11.02
118	28	28	35.99	26.12	22.31	8.82	14.7	11.06
119	29	29	40.08	26.37	21.92	8.48	15.86	12.32
120	30	30	36.48	26.93	27.31	9.44	15.42	9.48
121	31	31	39	30.89	22.94	9.06	13.62	12.08
122	32	32	37.64	31.06	26.9	10.96	15.7	11.62
123	33	33	41.12	32.64	25	9.48	14.22	12.16
124	34	34	41.02	30.32	31.61	10.14	16.54	9.84
125	35	35	40.77	33.64	30.27	10.02	14.36	10.16
126	36	36	49.46	35.04	24.71	7.76	13.3	12.54
127	37	37	42.43	33.78	35.47	10.84	16.2	9.56
128	38	38	54.63	34.14	27.85	7.58	15.06	11.96
129	39	39	49.08	35.91	32.59	9.72	16.04	11.24
130	40	40	56.62	34.72	31.55	8.12	16.5	11.54
131	15	15	14.9	21.59	8.83	12.7	9.38	16.78
132	16	16	16.54	21.31	9.96	12.92	10.96	16.76
133	17	17	15.26	26.46	10.26	12	7.18	13.68
134	18	18	18.2	22.59	12.56	11.98	10.62	13.24
135	19	19	19.47	26.86	11.02	11.98	9.38	16.72
136	20	20	19.96	30.59	10.79	11.1	7.68	16.36
137	21	21	23.96	26.48	12.48	10.02	10.18	15.42
138	22	22	24.9	30.08	11.8	9.64	8.82	16.58
139	23	23	26.75	26.6	15.08	9.44	10.86	13.24
140	24	24	25.04	32.16	14.66	9.96	8.48	13.36
141	25	25	24.51	35.02	15.54	11.88	8.92	14.5
142	26	26	25.42	35.04	17.12	12.66	9.92	14.4
143	27	27	32.37	32.22	16.29	9.14	10.5	14.64
144	28	28	29.36	40.66	15.22	9.5	7.4	14.7
145	29	29	31.07	36.23	18.89	11.32	10.8	14.52
146	30	30	33.2	36.29	19.64	10.22	10.52	13.54
147	31	31	31.88	42.49	18.68	12.66	10.32	16.96
148	32	32	34.62	42.34	18.99	11.4	10.28	16.5
149	33	33	32.05	47.49	20.01	10.84	7.8	13.48
150	34	34	37.5	43.3	20.77	9.74	9.4	13.94
151	35	35	31.9	53.48	21.23	12.92	8.04	14.9
152	36	36	39.66	48.72	20.24	10.7	9.6	16.86
153	37	37	32.58	57.63	22.88	12.36	7.22	13.38
154	38	38	37.8	54.56	22.42	10.16	7.54	13.42
155	39	39	35.63	57.71	24.64	12.5	8.1	13.78
156	40	40	42.02	57.39	22.05	9.9	7.84	15.1
157	15	15	13.86	16.66	14.2	13.86	12.76	9.62
158	16	16	14.49	17.88	15.43	13.62	12.16	9.04
159	17	17	15.39	20.48	14.62	13.1	10.7	9.94
160	18	18	13.92	22.38	18.48	14.5	9.48	7.42
161	19	19	14.49	23.22	20.45	17	11.16	8.1
162	20	20	14.87	24.58	22.16	16.02	10.14	7.16
163	21	21	17.11	27.05	18.94	15.78	10.52	10
164	22	22	19.88	26.05	19.48	14.38	11.96	10.52
165	23	23	19.74	28.83	20.08	15	10.98	10.5
166	24	24	19.67	27.77	25.33	14.18	10.8	7.52
167	25	25	21.17	29.51	24.37	13.94	10.78	8.44
168	26	26	20.42	31.83	26.73	16.26	11.02	8.46
169	27	27	23.47	29.11	29.25	13.28	11.78	7.32
170	28	28	25.48	30.26	28.29	13.84	12.92	8.74
171	29	29	22.28	35.48	30.83	14.4	9.52	7.02
172	30	30	26.69	37.8	24.72	13.02	9.88	10.18
173	31	31	26.47	35.87	30.81	16.14	12.9	9.64
174	32	32	30.05	36.94	27.91	13.14	11.78	10.24
175	33	33	25.86	41.11	33.13	15.7	10.4	8.38
176	34	34	28.36	38.7	35.87	13.62	10.8	7.38
177	35	35	25.62	43.76	38.65	16.62	10.12	7.32
178	36	36	29.29	44.36	34.8	15.74	11.04	9.18
179	37	37	29.04	49.23	33.53	16.38	10.06	9.88
180	38	38	32.3	42.31	40.39	14.28	11.88	7.84
181	39	39	35.9	44.71	35.35	14.66	12.94	10.66
182	40	40	30.42	51.02	40.46	16.72	10.4	8.54
183	15	15	17.07	16.74	10.65	10.72	12.54	13.34
184	16	16	17.25	17.92	12.13	11.82	12.92	12.78
185	17	17	17.4	18.44	14.49	11.36	12.12	10.06
186	18	18	18.18	21.71	13.33	12.18	11.3	12.54
187	19	19	19.69	21.23	15.22	10.9	11.4	10.54
188	20	20	19.26	21.76	18.45	13.9	13.76	10.44
189	21	21	19.5	28.4	14.74	14	10.28	13.9
190	22	22	24.39	25.05	15.69	10.98	12.16	13.18
191	23	23	23.3	27.28	17.37	11.08	10.52	11.18
192	24	24	26.72	26.6	17.71	11.82	13.58	13.7
193	25	25	25.75	28.04	20.08	13.04	13.48	12.48
194	26	26	25.86	27.96	23.36	12.72	13.26	10.24
195	27	27	31.32	28.79	20.01	10.38	13.1	12.56
196	28	28	31.12	29.44	22.42	11.08	13.52	11.64
197	29	29	28.92	34.12	22.65	10.82	10.18	10.3
198	30	30	30.69	37.5	20.73	12.14	10.96	13.76
199	31	31	33.63	33.91	24.14	12.02	13.6	12.68
200	32	32	35.65	35.62	23.45	10.28	11.76	12.02
201	33	33	30.6	40.86	26.36	13.7	11.14	11.66
202	34	34	35.37	38.78	26.29	10.9	11.18	11.04
203	35	35	38.93	38.45	26.21	11.92	13.82	13.56
204	36	36	35.49	40.9	30.1	13.98	13.52	12.12
205	37	37	35.89	45.52	28.07	12.52	10.82	11.94
206	38	38	37.97	42.64	31.79	13.86	13.82	12.2
207	39	39	42.49	43.84	28.99	11.82	13.02	13.24
208	40	40	37.68	46.39	34.49	13.12	11.74	10.4