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	8.3E-5	0	0.237
lnL	0.350126	0	3076.861
lnK	0.399935	0	3269.832
lnM	0.249902	0	2384.95
lnLlnL	-0.020208	0	-951.414
lnKlnK	-0.021056	0	-993.626
lnMlnM	-0.0172	0	-822.752
lnLlnK	0.024101	0	702.516
lnLlnM	0.016325	0	539.867
lnKlnM	0.018041	0	557.439

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	14.97	18.84	10.64	8.9	7.76	9.5
2	16	16	14.81	19.27	13.36	9.3	7.8	7.5
3	17	17	16.66	19.07	14.6	7.98	7.78	6.66
4	18	18	17.29	20.92	15.06	9.02	8.24	7.58
5	19	19	17.47	24.91	14.13	8.66	6.52	7.94
6	20	20	22.39	21.66	15.16	6.16	7.32	6.96
7	21	21	21.78	22.43	18	8.66	9.56	7.74
8	22	22	20.54	26.58	18.06	9.46	7.98	7.86
9	23	23	22.64	31.16	14.9	8.56	6.72	10
10	24	24	29.07	23.75	18.85	6.36	9.22	7.56
11	25	25	23.79	29.93	20.28	9.56	8.34	8.24
12	26	26	26.41	28	22.65	9.3	9.92	7.96
13	27	27	24.42	34.53	21.27	9.02	6.86	7.58
14	28	28	28.94	29.14	25.12	7.94	9	6.7
15	29	29	30.47	32.04	23.21	8.58	9.24	8.44
16	30	30	30.22	35.03	23.39	8.1	7.78	7.82
17	31	31	32.08	35.39	24.1	6.5	6.62	6.5
18	32	32	32.47	40.58	21.89	7.78	6.84	8.84
19	33	33	36.51	33.13	28.56	6.9	8.84	6.58
20	34	34	42.18	34.33	25.1	6.66	9.7	8.76
21	35	35	34.28	44.82	24.74	9.14	7.62	9.58
22	36	36	38.41	41.02	26.95	6.22	6.58	6.74
23	37	37	33.94	48.52	27.57	8.42	6.32	7.68
24	38	38	37.09	47.59	27.86	9.62	8.2	9.62
25	39	39	37.68	43.27	34.76	7.64	7.42	6
26	40	40	46.04	48.8	24.61	6.26	6.68	9.34
27	15	15	22.36	10.96	14.65	5.6	14.8	6.58
28	16	16	19.33	14.2	14.96	6.92	11.36	6.68
29	17	17	17.98	13.79	22.24	8.16	12.74	4.54
30	18	18	18.01	16.65	20.42	8.86	11.1	5.46
31	19	19	22.1	16.54	19.31	8.36	13.44	7
32	20	20	23.81	17.22	20.06	6.98	11.68	6.1
33	21	21	24.09	16.36	26.23	7.86	14.16	5.08
34	22	22	26.26	18.39	23.07	6.78	11.78	5.64
35	23	23	25.66	17.17	32.25	7.84	14.38	4.28
36	24	24	27.17	20.68	25.73	8.06	12.7	6.14
37	25	25	32.18	20.21	25.03	6.5	12.84	6.24
38	26	26	28.15	24.82	25.08	8.68	11.5	7.1
39	27	27	29.69	25.12	26.57	8.18	11.38	6.66
40	28	28	36.88	22.75	26.97	5.6	11.3	5.78
41	29	29	33.92	27.95	24.83	7.76	11.14	8
42	30	30	35.79	24.78	32.12	6.46	11.38	5.24
43	31	31	47.25	22.95	28.78	5.5	14.7	7.04
44	32	32	41.06	26.54	30.84	5.76	11	5.76
45	33	33	44.82	27.07	30.04	5.84	12.08	6.68
46	34	34	42.14	23.67	46.45	6.5	14.64	4.14
47	35	35	39.43	29.32	39.63	8.62	13.96	6.12
48	36	36	42.65	31.39	35.59	8.26	13.54	7.3
49	37	37	47.23	30.95	35.4	7.42	13.94	7.44
50	38	38	46.73	32.16	37.49	6.44	11.42	5.96
51	39	39	50.53	32.07	37.62	7.34	14.32	7.42
52	40	40	55.77	32.75	35.3	5.38	11.5	6.58
53	15	15	19.39	18.4	7.95	5.48	6.66	11.18
54	16	16	18.53	20.27	9.27	7	7.2	11.3
55	17	17	21.54	24.2	7.56	5.7	5.68	13.96
56	18	18	26.04	18.89	10.39	5.3	8.84	11.16
57	19	19	20.66	27.1	10.11	6.72	5.58	11.12
58	20	20	28.15	22.43	10.85	6	8.96	13.16
59	21	21	25.37	25.84	12.02	7.28	8.14	12.52
60	22	22	27.43	24.99	13.56	6.76	8.62	11.06
61	23	23	34.35	30.78	9.18	4.2	5.46	14.16
62	24	24	26.5	34.75	12.29	7.22	6	12.74
63	25	25	29.37	33.79	13.02	7.72	7.48	14.36
64	26	26	30.78	33.5	14.31	7.02	7.26	12.34
65	27	27	31.21	40.45	12.51	6.34	5.34	13.28
66	28	28	40.51	32.36	14.1	6.02	8.96	14.88
67	29	29	39.2	37	13.81	6	7.34	14.62
68	30	30	37.68	35.43	17.34	7.26	8.92	12.92
69	31	31	44	37.62	14.93	5.08	6.98	12.94
70	32	32	42.88	41.68	14.98	5.06	5.98	12.46
71	33	33	43.79	37.72	18.7	5.74	7.82	11.16
72	34	34	45.52	42.52	16.83	5.54	6.86	12.76
73	35	35	57.29	41.44	14.83	4.22	7.06	14.78
74	36	36	57	40.08	17.21	4.88	8.44	14.26
75	37	37.01	52.11	50.71	15.3	4.78	5.64	14.44
76	38	38	39.92	60.28	18.33	7.9	5.56	14.1
77	39	39	59.95	45.32	18.18	4.52	7.18	13.14
78	40	40	47.03	52.68	21.55	7.88	7.88	14.1
79	15	15	10.52	22.14	13.73	12.6	6	6.58
80	16	16	11.06	20.18	19.04	13.32	7.5	5.02
81	17	17	11.23	27.83	14.62	14.56	5.72	7.62
82	18	18	12.82	31.56	12.63	11.62	4.6	8.44
83	19	19	11.65	31.53	17.9	14.62	5.18	6.3
84	20	20	14.71	31.1	15.85	13.3	6.3	8.7
85	21	21	14.13	31.6	19.86	12.5	5.54	5.98
86	22	22	15.01	35.41	18.49	11.1	4.62	6.2
87	23	23	17.02	34.5	19.02	13.78	6.86	8.64
88	24	24	15.79	43.86	17.8	13.16	4.52	8.16
89	25	25	19.07	33.01	23.88	11.38	6.82	6.24
90	26	26	20	31.23	28.4	11.38	7.7	5.38
91	27	27	19.36	46.76	19.09	11.46	4.64	8.32
92	28	28	22.5	41.95	20.66	11.4	6.26	9
93	29	29	20.78	43.38	25.22	14.56	7	8.28
94	30	30	20.49	40.5	32.67	12.8	6.56	5.28
95	31	31	22.41	47.19	25.96	14.88	7.08	8.94
96	32	32	21.78	51.15	27.25	12.76	5.34	7
97	33	33	25.37	41.55	33.52	12.16	7.78	6.26
98	34	34	22.75	44.82	39.64	13.56	6.98	5.04
99	35	35	25.21	55.45	27.83	12.52	5.66	7.96
100	36	36	21.42	71.26	27.71	14.9	4.14	7.86
101	37	37	23.37	64.15	31.32	11.54	4.02	5.84
102	38	38	24.25	63.82	33.14	11.2	4.1	5.54
103	39	39	22.73	65.24	39.25	14.22	4.7	5.34
104	40	40	31.4	60.84	29.85	11.32	5.94	8.58
105	15	15	17.57	14.82	12.34	10.66	14.88	11.54
106	16	16	22.71	13.99	12.44	7.6	15.36	11.02
107	17	17	21.15	15.79	14.22	8.6	13.86	9.8
108	18	18	20.83	16.45	17.02	10.18	15.36	9.22
109	19	19	26.68	18.36	12.84	7.52	13.34	12.7
110	20	20	25	19.41	15.54	10.16	15.64	12.7
111	21	21	25.15	19.83	18	9.9	14.96	10.48
112	22	22	31.49	19.66	16.35	7.94	15.8	12.26
113	23	23	29.84	20.27	19.8	8.54	15.38	9.88
114	24	24	30.94	21.07	20.97	8.06	14.48	9.1
115	25	25	32.95	21.77	21.5	8.2	15.26	9.68
116	26	26	29.46	25.55	22.53	10.54	14.24	10.32
117	27	27	35.61	23.56	23.11	8.72	16.2	10.36
118	28	28	33.98	26.86	23.01	10.3	15.52	11.64
119	29	29	38	27.86	21.55	9.34	15.38	13
120	30	30	36.1	30.29	23.04	10.02	14.08	12.14
121	31	31	34.67	29.08	29.43	10.6	14.9	9.18
122	32	32	46.26	29.76	22.1	7.28	13.98	12.4
123	33	33	40.37	33.12	25.03	10.26	14.8	12.86
124	34	34	42.09	30.61	30.08	9.6	15.96	10.18
125	35	35	45.12	32.7	27.72	8.46	14.12	10.72
126	36	36	43.99	33.21	31.18	8.46	13.46	9.06
127	37	37	45.45	33.92	32.14	9.5	15.32	10.2
128	38	38	45.65	39.59	27.88	9.8	13.24	12.5
129	39	39	49.23	36.97	31.02	10.52	16.84	12.94
130	40	40	47.11	41.31	30.52	9.76	13.02	11.62
131	15	15	13.64	24.69	8.26	12.98	7.38	16.72
132	16	16	16.06	22.96	9.33	11.62	8.72	15.72
133	17	17	17.45	25.89	8.91	10.08	7.24	15.88
134	18	18	17.44	23.97	12.2	13	10.22	14.14
135	19	19	17.55	27.15	12.42	12.34	8.44	13.24
136	20	20	20.23	28.47	11.69	12.4	9.48	16.88
137	21	21	21.93	28.59	12.52	10.28	8.6	14.2
138	22	22	22.5	29.77	13.59	10.92	8.98	14.12
139	23	23	25.17	31.81	12.66	9.96	8.64	15.96
140	24	24	24.98	30.43	15.89	11.28	10.22	13.72
141	25	25	22.91	35.7	16.53	13	8.82	13.64
142	26	26	28.56	38	13.23	9.16	7.48	16.18
143	27	27	25.97	42.36	14.75	11.14	7.16	15.48
144	28	28	26.88	44.54	15.06	11.54	7.28	16.3
145	29	29	31.22	44.21	14.3	9.3	7.06	16.64
146	30	30	31.22	42.06	17.26	9.22	7.42	13.22
147	31	31	32.3	41.62	18.91	10.14	8.6	13.6
148	32	32	35.81	38.25	21.01	10.08	10.66	13.48
149	33	33	32.05	42.26	23.58	12.84	10.6	13.16
150	34	34	34.24	51.75	18.25	10.1	7.1	15.12
151	35	35	39.66	46.25	19.64	9.54	9.1	15.58
152	36	36	41.39	45.18	21.31	9.76	10.06	15.22
153	37	37	35.6	51.97	23.48	11.5	8.42	13.4
154	38	38	33.98	61.84	21.71	12.78	7.22	15.46
155	39	39	41.28	59.05	19.82	9.52	7.14	16.12
156	40	40	35.93	63.43	23.52	12.26	7.18	14.42
157	15	15	11.57	19.56	14.38	14.9	9.18	8.24
158	16	16	12.58	20.03	15.87	16.16	10.68	8.78
159	17	17	13.37	22.2	15.81	16.72	10.52	9.8
160	18	18	13.39	23.11	18.62	16.42	9.88	7.96
161	19	19	15.91	24.84	16.13	15.46	10.46	10.86
162	20	20	17.83	23.36	18.42	14.02	11.66	9.64
163	21	21	19.19	23.79	19.58	13.7	12.16	9.56
164	22	22	16.01	29.01	22.57	16.94	9.62	8.08
165	23	23	19.96	24.77	25	14.12	12.52	7.74
166	24	24	19.97	27.28	25.46	16.06	12.72	8.62
167	25	25	20.65	33.01	21.31	14.56	9.58	10.02
168	26	26	21.04	33.21	23.98	16.56	11.06	10.14
169	27	27	21.44	36.68	23.34	14.94	9.08	9.66
170	28	28	20.5	39.01	26.21	16.94	9.08	9.06
171	29	29	24.48	36.04	26.25	13.1	9.5	8.62
172	30	30	28.65	34.97	25.19	13.14	11.88	10.92
173	31	31	25.31	36.46	32.06	13.2	9.82	7.14
174	32	32	25.54	35.83	37	16.5	12.66	7.62
175	33	33	27.88	42.98	27.82	15	10.3	10.74
176	34	34	27.44	42.66	32.34	13.38	9.1	7.88
177	35	35	31.18	41.76	31.25	14.88	12.06	10.6
178	36	36	29.73	49.26	29.14	14.74	9.3	10.78
179	37	37	27.68	45.88	40.06	16.52	10.42	7.64
180	38	38	33.2	43.34	37.37	13	10.86	8.08
181	39	39	33	41.98	44.04	13.96	12.02	7.1
182	40	40	34.68	47.65	37.18	15.92	12.52	10.48
183	15	15	16.42	17.45	10.54	10.88	11.58	13.1
184	16	16	15.26	20.12	12.03	12.24	10.1	11.56
185	17	17	18.5	18.17	13.65	11	12.84	11.24
186	18	18	18.94	19.59	14.71	11.56	12.7	11.12
187	19	19	18.75	22.67	14.75	11.42	10.44	10.82
188	20	20	19.75	22.32	17.15	13.8	13.66	11.64
189	21	21	19.88	25.63	16.66	11.92	10.1	10.48
190	22	22	21.56	26.61	16.88	12.64	11.28	12.04
191	23	23	20.83	28.44	19.02	13.94	11.04	11.08
192	24	24	24.65	29.02	17.31	12.38	11.68	13.4
193	25	25	28.45	26.96	18.69	11.22	13.66	13.14
194	26	26	26.33	30.98	19.53	13.12	12.38	13.32
195	27	27	25.99	34.52	19.58	13	10.64	12.96
196	28	28	29.43	32.6	20.72	10.6	10.74	11.44
197	29	29	35.23	30.71	20.47	10.2	13.7	13.8
198	30	30	29.28	34.64	24.81	12.56	11.78	10.9
199	31	31	32.51	35.11	23.96	10.44	10.9	10.68
200	32	32	36.46	31.89	26.93	10	13.38	10.2
201	33	33	33.02	38.72	25.77	11.82	11.2	11.3
202	34	34	38.45	36.72	25.56	10.84	13.08	12.52
203	35	35	38.73	40.07	24.83	10.82	11.88	13.04
204	36	36	38.46	41.23	26.71	11.34	11.94	12.44
205	37	37	39.7	43.68	26.11	11.2	11.44	13.1
206	38	38	39.17	43.56	29.52	11.48	11.58	11.44
207	39	39	40.4	45.96	28.9	11.12	10.92	11.78
208	40	40	46.79	41.61	30.45	10	13.12	11.84