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	-0.000401	0	-1.101
lnL	0.35018	0	2928.533
lnK	0.400113	0	3044.18
lnM	0.249971	0	2294.187
lnLlnL	-0.020143	0	-930.988
lnKlnK	-0.021109	0	-827.082
lnMlnM	-0.017192	0	-777.224
lnLlnK	0.024077	0	687.794
lnLlnM	0.016204	0	453.307
lnKlnM	0.018119	0	561.749

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	18.23	15.25	11.26	6.88	9.7	8.66
2	16	16	17.31	17.24	12.79	7.14	8.2	7.28
3	17	17	19.88	17.58	13.07	6.12	8.08	7.16
4	18	18	18.66	21.45	13.1	7.28	7.06	7.88
5	19	19	22.66	20.11	13.74	6.92	9.1	8.9
6	20	20	18.91	25.55	14.86	9	7.22	8.54
7	21	21	21.71	25.16	15.22	7.6	7.3	8.24
8	22	22	20.23	24.53	20.84	9.68	8.82	6.68
9	23	23	23.93	27.65	16.45	7.8	7.52	8.66
10	24	24	21.8	30.95	18.57	9.62	7.28	8.3
11	25	25	24.07	28.77	21.17	8.96	8.3	7.44
12	26	26	25.54	31.73	19.63	6.82	6.04	6.64
13	27	27	25.81	35.45	19.01	9.2	7.24	9.42
14	28	28	26.74	30.71	25.82	9	8.74	6.7
15	29	29	29.39	30.2	26.7	7.68	8.5	6.14
16	30	30	35.89	32.23	21.14	6.46	8.38	8.6
17	31	31	29.61	37.69	24.44	7.46	6.42	6.68
18	32	32	33.69	34.63	26.36	6.4	7.08	6.1
19	33	33	42.02	33.32	23.57	6.14	9.22	8.66
20	34	34	39.16	34.08	27.96	6.48	8.72	6.88
21	35	35	32.4	43.4	27.96	9.18	7.44	7.8
22	36	36	44.97	32.82	30.86	6.04	10	6.72
23	37	37	45.13	38.4	26.79	6.7	9.26	8.84
24	38	38	36.76	50.76	25.7	7.7	6.02	8.38
25	39	39	37.51	47.43	30.45	9.64	8.36	8.8
26	40	40	35.79	52.44	30.85	9.38	6.84	7.98
27	15	15	18.13	13.1	14.4	6.9	11.56	6.46
28	16	16	21.99	12.88	14.8	6.06	13	6.9
29	17	17	20.52	14.19	17.6	7.2	12.7	6.16
30	18	18	22.26	16.59	15.36	7.06	11.4	7.8
31	19	19	20.17	16.89	21.18	8.72	12.28	5.88
32	20	20	23.7	15.53	24.01	6.14	11.54	4.32
33	21	21	25.95	17.86	20.43	7.42	13.16	7.02
34	22	22	23.6	16.72	31.65	8.62	14.78	4.36
35	23	23	35.75	17.91	19.17	5.38	13.86	8
36	24	24	31.75	18.59	24.88	6.6	14.16	6.28
37	25	25	28.97	22.39	24.37	8.84	13.68	7.74
38	26	26	36.68	19.15	26.93	5.9	14.48	6.06
39	27	27	30.33	20.49	36.41	8.02	14.54	4.62
40	28	28	36.48	24.29	24.59	6.02	11.1	6.84
41	29	29	30.5	25.17	34.09	8.46	12.12	5.3
42	30	30	38.84	25.14	28.09	7.46	14.22	7.8
43	31	31	37.16	26.42	31.3	7.3	12.46	6.38
44	32	32	41.69	25.18	32.97	6.16	12.74	5.8
45	33	33	41.36	24	41.06	5.94	12.88	4.28
46	34	34	40.4	31.94	29.69	7.58	11.42	7.78
47	35	35	42.8	26.96	40.76	6.9	13.58	5.22
48	36	36	48.89	26.79	38.56	6.46	14.98	6.1
49	37	37	52.69	29.45	33.27	6.18	14	7.58
50	38	38	56.3	32.29	29.26	5.16	11.34	7.96
51	39	39	50.51	35.1	32.54	6.66	11.68	7.98
52	40	40	51.44	28.74	49.04	6.14	13.92	4.64
53	15	15	20.29	19.18	7.12	5.34	6.52	13.08
54	16	16	19.59	21.25	8.12	7.34	7.62	14.78
55	17	17	24.09	19.1	9.12	5.02	7.54	11.24
56	18	18	26.84	21.51	8.36	4.14	6.14	11.66
57	19	19	21.36	27.51	9.52	7.92	6.72	14.64
58	20	20	26.39	26.55	9.29	4.74	5.38	11.56
59	21	21	34.58	21.4	10.87	4.34	8.72	12.1
60	22	22	29.03	31.47	9.27	5.1	5.3	13.96
61	23	23	27.21	29.35	12.85	7.04	7.36	12.16
62	24	24	26.99	34.66	12.05	7.94	6.76	14.64
63	25	25	40.49	26.29	12.61	4.38	8.32	12.34
64	26	26	29.45	37.86	12.81	7.62	6.48	14.5
65	27	27	36.7	32.54	13.81	5.24	6.9	11.82
66	28	28	27.92	45.81	13.84	7.96	5.08	12.98
67	29	29	41.44	32.61	15.39	5.44	8.24	12.46
68	30	30	33.75	42.58	15.41	6.28	5.46	11.28
69	31	31	38.66	43.43	14.32	5.92	5.9	13.6
70	32	32	38.03	44.42	15.85	6.26	5.96	12.52
71	33	33	46.15	37.86	17.47	6.1	8.8	13.66
72	34	34	40.08	47.05	17.08	7.48	7.08	14.58
73	35	35	44.92	42.64	18.88	7.26	8.82	14.38
74	36	36	39.85	48.75	20.11	6.9	6.22	11.02
75	37	37	45.19	54.03	16.6	5.68	5.26	13.18
76	38	38	58.82	43.37	18.01	4.28	7	12.3
77	39	39	61.8	44.03	18.27	4.78	8.14	14.32
78	40	40	47.66	51.67	21.78	7.12	7.4	12.78
79	15	15	9.65	25.99	12.37	14.82	5.28	7.9
80	16	16	10.48	24.45	15.41	12.52	5.28	5.68
81	17	17	11.07	24.62	17.87	12.54	5.6	5.1
82	18	18	12.66	31.2	13.07	11.28	4.46	7.76
83	19	19	11.94	33.71	15.69	14.24	4.8	7.38
84	20	20	15.42	28.34	16.93	12.28	6.86	7.86
85	21	21	14.07	33.28	18.52	13.54	5.62	7
86	22	22	16.49	29.88	20.67	13.84	7.86	7.58
87	23	23	15.97	38.01	18.15	14.22	5.86	8.74
88	24	24	16.64	36.9	21.04	13.18	5.9	7.14
89	25	25	19.08	37.04	20.14	12.68	6.64	8.5
90	26	26	18.06	38.09	24.39	12.84	6.1	6.44
91	27	27	19.88	40.08	22.84	12.5	6.26	7.58
92	28	28	21	39.55	24.79	13.34	7.24	7.84
93	29	29	19.92	42.63	27.52	12.12	5.66	5.92
94	30	30	21.69	45.91	24.92	14.8	7	8.98
95	31	31	19.21	53.23	27.36	14.22	4.9	6.7
96	32	32	23.78	43.23	30.7	11.42	6.46	6.04
97	33	33	21.9	60.45	24.1	12.54	4.34	8
98	34	34	25.48	48.22	29.96	12.64	6.82	7.46
99	35	35	21.45	57.31	33.63	13.4	4.82	5.64
100	36	36	25.98	49.99	34.58	14.78	7.82	7.54
101	37	37	24.08	60.49	32.57	13.86	5.36	6.94
102	38	38	25.02	57.16	37.13	14.52	6.28	6.52
103	39	39	25.81	62.14	34.77	13.68	5.56	6.88
104	40	40	31.29	51.11	38.72	11.44	7.34	6.36
105	15	15	19.5	14.23	11.47	9.48	15.7	12.64
106	16	16	22.94	13.42	13.11	7.6	16.32	10.48
107	17	17	20.21	17.42	12.97	9.8	13.34	11.8
108	18	18	21.43	18.48	13.67	10.16	13.82	12.32
109	19	19	27.69	15.99	15.17	7.64	16.66	11.08
110	20	20	27.37	18.95	14.36	7.96	14.02	12.14
111	21	21	23.74	22.89	15.57	10.98	13.1	12.88
112	22	22	26.86	19.83	19.79	9.5	15.52	9.72
113	23	23	32.47	19.47	18.93	7.7	16.06	10.38
114	24	24	34.88	20.12	19.34	7.22	15.76	10.32
115	25	25	28.36	26.62	19.12	10.9	13.42	12.38
116	26	26	31.22	23.67	23.53	10.02	15.86	9.98
117	27	27	31.79	27.79	20.71	10.1	13.52	11.94
118	28	28	38.87	25.64	20.82	8.1	15.1	12.06
119	29	29	36.04	27.6	23.4	10.46	16.36	12.42
120	30	30	39.26	25.68	26.76	8.06	15.18	9.04
121	31	31	43.19	29.56	21.58	8.02	14.32	12.96
122	32	32	40.93	28.61	27.43	8.9	15.5	10.18
123	33	33	37.17	32.27	29.04	10.24	13.82	9.78
124	34	34	50.79	29.56	25	7.48	16.16	12.3
125	35	35	44.73	32.68	28.05	8.36	13.82	10.34
126	36	36	42.02	35.59	29.71	9.96	13.84	10.7
127	37	37	47.07	33.48	31.35	9.06	15.46	10.44
128	38	38	55.84	33.42	28.01	7.96	16.64	12.8
129	39	39	44.99	35.1	37.95	10.82	16.56	9.44
130	40	40	54.8	33.26	35.23	7.76	15.96	9.32
131	15	15	16.56	19.72	8.76	9.88	9.2	14.94
132	16	16	15.5	22.22	10.22	11.68	8.74	13.62
133	17	17	16.4	26.19	9.48	12.44	8.2	16.94
134	18	18	19.46	23.31	11.02	9.48	8.76	13.22
135	19	19	17.49	30.45	10.67	12.5	7.44	15.96
136	20	20	19.62	27.45	12.78	12.36	9.52	14.62
137	21	21	24.87	26.19	12.09	9.02	9.7	15.06
138	22	22	20.92	33.02	12.94	11.86	7.92	14.9
139	23	23	23.55	32.97	13.11	11.44	8.8	16.28
140	24	24	23.08	34.06	15.02	11.98	8.66	14.18
141	25	25	22.27	39.47	14.95	12.56	7.32	14.34
142	26	26	28.3	31.91	17.03	10.46	10.38	13.58
143	27	27	28.14	33.86	18.13	11.18	10.28	13.4
144	28	28	32.3	36.72	15.53	9.38	9.22	15.86
145	29	29	33.34	39.48	15.35	9.48	8.88	16.86
146	30	30	33.45	42.9	15.43	9.24	7.88	16.4
147	31	31	34.89	39.52	18.44	9.28	9.16	14.04
148	32	32	29.6	46.7	20.27	11.82	7.9	13.18
149	33	33	38.16	39.22	20.95	9.78	10.82	14.14
150	34	34	33.21	46.08	22.24	12.4	9.64	14.2
151	35	35	33.78	52.36	20.29	12.28	8.38	15.98
152	36	36	36.48	50.82	21.23	10.26	7.94	13.86
153	37	37	40.17	46.14	23.76	10.34	10.04	13.7
154	38	38	41.85	50.14	22.14	9.48	8.76	14.32
155	39	39	38.99	50.86	26.13	11.96	10	13.68
156	40	40	42.2	53.63	24.06	12.16	10.48	16.82
157	15	15	13.64	16.04	15.41	13.02	12.3	8.06
158	16	16	13.17	19.37	15.64	15.6	11.32	9.1
159	17	17	15.71	19.46	15.36	14.34	12.74	10.52
160	18	18	15.39	20.26	18.64	15.26	12.62	8.7
161	19	19	14.7	23.33	19.86	14.16	9.4	7.1
162	20	20	15.09	25.42	20.57	15.64	9.68	7.76
163	21	21	16.65	25.3	21.83	14.28	9.98	7.42
164	22	22	18.39	29.13	18.37	15.3	10.18	10.94
165	23	23	19.13	28.89	20.93	14.38	10.12	9.24
166	24	24	21.3	28.62	21.55	13.82	11.16	9.74
167	25	25	21.92	27.26	26.26	13.68	12.1	7.9
168	26	26	21.11	31.25	26.2	16.82	12.12	9.32
169	27	27	21.97	36.63	22.59	15.1	9.46	10.44
170	28	28	24.04	32.87	26.99	14.44	11.42	9
171	29	29	21.19	37.07	31.02	16.66	9.86	7.6
172	30	30	24.96	35.33	30.11	16.62	12.62	9.52
173	31	31	25.24	37.99	30.18	15.06	10.64	8.72
174	32	32	25.56	42.77	28.09	16.74	10.44	10.7
175	33	33	26.46	42.17	30.82	13.84	9.14	8.26
176	34	34	27.68	41.1	33.82	15.1	10.84	8.52
177	35	35	30.79	38.37	36.27	13.38	11.8	7.88
178	36	36	27.91	47.6	33.53	15.74	9.6	9.06
179	37	37	30.21	43.69	38.02	13.1	9.7	7.14
180	38	38	33.05	43.71	37.12	14.54	11.96	9.06
181	39	39	35.03	46.8	34.11	14.22	11.56	10.48
182	40	40	31.59	47.55	42.69	16.76	11.84	8.4
183	15	15	14.67	16.59	13.22	12.66	12.52	10.22
184	16	16	17.56	16.81	13.04	10.06	12.1	10.2
185	17	17	16.87	21.38	12.12	12.3	10.64	12.96
186	18	18	18.6	21.87	12.79	11.24	10.62	12.48
187	19	19	19.49	21.48	15.18	12.54	12.78	12.02
188	20	20	19.77	23.31	16.02	11.98	11.28	10.96
189	21	21	21	27.08	14.3	11.92	10.1	13.38
190	22	22	23.18	26.22	15.68	11.86	11.72	13.42
191	23	23	23.32	26.24	18.4	11.32	11.26	10.68
192	24	24	26.61	28.19	16.36	10.3	11	13.04
193	25	25	26.97	29.78	17.28	10.48	10.66	12.64
194	26	26	25.76	28.36	22.98	13.46	13.74	11
195	27	27	26.96	31.92	20.9	13.36	12.52	12.88
196	28	28	28.52	32.99	21.21	13.54	13.04	13.7
197	29	29	29.25	32.58	23.91	11.66	11.74	10.56
198	30	30	34.4	29.7	25.29	10.04	13.64	10.3
199	31	31	29.82	38.01	23.91	12.76	10.96	11.82
200	32	32	34.29	34.96	25.38	10.24	11.44	10.42
201	33	33	31.36	40.85	25.51	13.4	11.22	12.2
202	34	34	36.7	36.16	27.82	11.4	13.26	11.28
203	35	35	33.16	39.77	30.9	12.96	11.96	10.06
204	36	36	39.55	38.33	28.72	11.32	13.42	11.78
205	37	37	37.16	45.24	27.05	11.58	10.5	12.02
206	38	38	44.05	39.82	28.95	10.32	13.28	12.08
207	39	39	43.05	41.5	30.95	11.7	13.96	12.32
208	40	40	40.94	46.72	30.51	10.24	10.02	10.34