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	0.000206	0	0.433
lnL	0.349954	0	2205.87
lnK	0.400351	0	2540.484
lnM	0.249522	0	1773.4
lnLlnL	-0.020117	0	-741.531
lnKlnK	-0.021143	0	-842.887
lnMlnM	-0.017193	0	-653.604
lnLlnK	0.024024	0	613.754
lnLlnM	0.016279	0	395.175
lnKlnM	0.018181	0	446.169

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	14.33	15.73	14.84	9.22	9.44	6.32
2	16	16	16.98	18.06	12.24	9.28	9.86	9.74
3	17	17	17.16	19.92	13.14	9.28	8.9	9.08
4	18	18	18.66	20.95	13.56	8	7.98	8.32
5	19	19	23.05	17.9	16.06	6.5	10	7.1
6	20	20	19.36	24.55	15.28	7.8	6.74	7.36
7	21	21	21.92	26.59	13.88	6.92	6.3	8.46
8	22	22	23.95	25.28	15.85	7.7	8.26	8.94
9	23	23	21.44	25.43	21.66	8.64	8.08	6.1
10	24	24	23.23	30.89	17.1	9.74	7.96	9.98
11	25	25	24.23	28.53	21.25	8.42	7.94	7.02
12	26	26	26.41	31.97	18.57	6.9	6.3	7.46
13	27	27	28.68	32.48	18.78	6.38	6.3	7.5
14	28	28	28.38	34.57	19.94	8.54	7.74	9.24
15	29	29	28.96	34.54	22.19	8.72	8.1	8.52
16	30	30	25.58	41.99	22.48	9.72	6.2	8.08
17	31	31	30.8	31.96	29.79	8.48	9.28	6.3
18	32	32	35.22	33.42	26.23	7.88	9.58	7.96
19	33	33	31.29	38.64	27.79	7.38	6.58	6.06
20	34	34	34.85	39.11	26.48	9.02	9	8.9
21	35	35	36.78	40.25	26.38	8.62	8.86	9.1
22	36	36	34.45	37.28	36.23	9.18	9.56	6.18
23	37	37	38.82	49.15	22.7	7.3	6.32	9.8
24	38	38	46.36	39.17	27.78	6.56	9.14	8.56
25	39	39	39.41	44.48	31.35	9.24	9.16	8.64
26	40	40	41.88	44.98	31.32	7.46	7.84	7.5
27	15	15	21.61	12	13.17	5.82	13.28	7.44
28	16	16	21.8	12.1	16.6	5.06	11.56	4.98
29	17	17	24.81	13.11	15.6	5.42	13.12	6.68
30	18	18	22.71	14.08	19.58	6.16	12.36	5.24
31	19	19	24.97	13.01	24.6	5.64	13.88	4.1
32	20	20	26.56	15.38	20.88	6.9	15	6.54
33	21	21	27.04	14.26	28.43	6	14.56	4.04
34	22	22	24.23	19.57	23.25	8.48	12.46	6.36
35	23	23	33.7	16.49	23.74	5.6	14.84	6.04
36	24	24	33.89	19.13	21.77	5.84	13.08	7.02
37	25	25	30.68	21.7	23.73	6.52	11.2	6.3
38	26	26	37.02	20.62	23.53	5.52	12.56	6.72
39	27	27	33.29	22.39	27.46	7.12	12.98	6.38
40	28	28	34.45	25	25.32	6.84	11.4	7.02
41	29	29	37.92	21.37	33.24	5.18	11.62	4.32
42	30	30	38.47	23.99	30.74	7.42	14.78	6.9
43	31	31	37.58	25.2	33.35	6.18	11.3	5.08
44	32	32	39.1	26.29	33.45	7.34	13.38	6.3
45	33	33	42.84	27.2	31.63	6.8	13.26	6.94
46	34	34	38.09	28.17	39.53	8.54	13.92	5.84
47	35	35	47.03	26.42	37.11	6.42	14.46	6.06
48	36	36	52.13	26.27	36.6	5.54	14.18	6
49	37	37	42.67	26.74	52.41	7.24	14.34	4.06
50	38	37.99	40.11	29.45	54.09	8.2	13.48	4.12
51	39	39	42.58	38.91	34.69	8.8	11.18	8
52	40	40	44.47	36.59	39.86	7.68	11.04	6.24
53	15	15	16.81	21.29	7.74	6.38	5.52	11.34
54	16	16	19.87	19.26	9.13	6.46	7.66	11.52
55	17	17	25.98	19.37	8.2	5.24	8.46	14.54
56	18	18	26.91	21.07	8.57	5	7.62	13.72
57	19	19	23.24	27.43	8.64	6.36	5.98	14.56
58	20	20	26.52	25.43	9.79	5.48	6.58	12.64
59	21	21	28.97	26.72	9.8	5.3	6.66	13.54
60	22	22	25.34	29.76	11.72	7.4	7	13.1
61	23	23	34.65	25.27	11.86	4.84	8.02	12.2
62	24	24	32.2	31.67	11.02	5.68	6.62	14.32
63	25	25	41.44	26.45	12.18	4.52	8.76	13.64
64	26	26	28.14	38.56	13.21	7.58	5.98	13.18
65	27	27	37.87	37.44	11.05	4.54	5.26	13.82
66	28	28	32.08	43.79	12.4	6.36	5.04	13.9
67	29	28.99	50.7	29.71	13.92	4.04	8.66	13.2
68	30	30	33.12	45.71	14.36	7.08	5.54	13.52
69	31	31	40.17	38.46	16.12	6.8	8.18	14.22
70	32	32	37.86	42.03	17.18	7.48	7.56	13.54
71	33	33	41.18	39.65	18.78	6.98	8.34	12.56
72	34	34	36.45	49.39	17.97	7.98	6.38	13.1
73	35	35	36.43	52.66	18.31	7.64	5.66	12.26
74	36	35.99	61.97	36.51	17.84	4.16	8.86	12.86
75	37	37.01	51.23	53.59	14.53	4.76	5.16	14.98
76	38	38	50.73	51.41	16.99	5.28	5.94	13.66
77	39	39	44.34	53.65	20.63	6.92	6.32	12.16
78	40	40	49.09	53.98	19.8	6.32	6.46	13.14
79	15	15	10.89	18.83	16.67	11.6	6.96	5.02
80	16	16	11.61	23.04	14.44	11.36	5.8	6.28
81	17	17	10.51	29.07	15.15	13.44	4.64	6.24
82	18	18	13.37	22.21	19.87	12.5	7.86	5.6
83	19	19	12.44	32.53	15.54	11.5	4.24	6.32
84	20	20	12.44	39.45	14.5	14.56	4.28	8.68
85	21	21	14.6	36.52	15.41	13.28	5.16	8.9
86	22	22	14.7	37.57	17.51	13.14	4.98	7.64
87	23	23	14.92	36.68	21.11	11.5	4.56	5.46
88	24	24	14.96	44.67	18.77	14.26	4.5	7.8
89	25	25	16.6	41.86	20.66	14.6	5.62	8.06
90	26	26	20.02	41.24	18.79	11.3	5.52	8.7
91	27	27	21.35	38.67	21.73	11.58	6.58	8.1
92	28	28	17.06	43.19	29.67	14.88	5.7	5.54
93	29	29	21.82	37.37	29.35	12.82	7.78	6.46
94	30	30	17.54	51.48	28.95	13.48	4.34	5.34
95	31	31	22.7	49.74	23.66	12.2	5.54	8.3
96	32	32	19.64	51.55	31.45	13.4	4.92	5.5
97	33	33	21.48	60.36	24.82	13.28	4.5	8
98	34	34	21.92	57.74	28.77	11.76	4.3	6.1
99	35	35	22.9	54.57	32.76	11.52	4.74	5.4
100	36	36	23.74	66.55	26.2	13.24	4.5	8.42
101	37	37	27.93	53.54	31.33	13.62	7.24	8.5
102	38	38	23.31	60.72	37.81	13.6	5.04	5.5
103	39	38.99	25.29	74.42	27.85	13.46	4.32	8.58
104	40	40	24.7	65.7	37.83	13.22	4.78	5.72
105	15	15	20.29	13.82	11.41	7.4	13.28	10.4
106	16	16	19.93	15.25	12.84	10.5	16.44	12.58
107	17	17	20.74	15.64	14.81	9.74	15.52	10.34
108	18	18	24.28	15.69	14.99	8.18	15.62	10.3
109	19	19	25.09	17.1	15.47	7.5	13.46	9.46
110	20	20	23.18	18.13	19.12	10.96	16.72	9.82
111	21	21	26.03	18.07	19.95	9.5	16.68	9.28
112	22	22	26.1	20.69	19.22	10.2	15.32	10.44
113	23	23	27.2	20.89	21.32	10.7	16.68	10.18
114	24	24	30.09	22.04	20.24	8.48	13.98	9.66
115	25	25	32.95	22.83	19.93	8.74	15.38	11.28
116	26	26	31.42	26.11	20.03	10.6	15.06	12.86
117	27	27	31.53	25.43	24.03	10.8	15.9	10.62
118	28	28	37.1	26.66	20.79	8.46	14.26	11.94
119	29	29	32.71	29.53	23.93	10.72	13.82	11.06
120	30	30	37.5	27.11	26.05	9.94	16.64	10.9
121	31	31	42.39	26.64	25.95	8.38	16.54	10.66
122	32	32	43.04	28.74	25.52	7.78	14.3	10.28
123	33	33	46.51	27.22	28.39	7.66	16.44	9.78
124	34	34	39.23	35.12	26.64	10.14	13.2	11.42
125	35	35	43.68	32.53	29.13	9.7	15.68	11.16
126	36	36	48.23	33.11	27.81	8.6	15.3	11.74
127	37	37	46.29	34.28	30.87	10.36	16.86	11.92
128	38	38	52.72	31.97	32.3	7.9	16.26	10.04
129	39	39	52.35	37.09	28.54	8.6	14.74	12.54
130	40	40	62.61	34.03	28.74	7.08	16.58	12.64
131	15	15	16.06	20.82	8.45	11.02	9.28	16.76
132	16	16	15.76	22.23	10	11.86	9.04	14.46
133	17	17	16.86	24.87	9.81	10.2	7.38	13.78
134	18	18	17.29	28.38	9.79	11.04	7.04	15.4
135	19	19	19.83	24.93	11.94	11.34	9.9	14.72
136	20	20	18.89	28.06	13.02	12.6	9.04	13.94
137	21	21	20.59	30.63	12.35	12.4	8.88	16.16
138	22	22	25.09	28.51	12.58	9.66	9.5	15.54
139	23	23	26.39	28.46	13.93	9.64	10.08	14.6
140	24	24	25.88	32.84	13.66	10.28	8.88	15.58
141	25	25	22.63	37.92	15.46	11.82	7.36	13.22
142	26	26	25.85	35.86	16.22	12.4	9.64	15.32
143	27	27	27.72	36.04	16.92	10.7	8.98	13.66
144	28	28	32.46	34.92	16.55	9.86	10.34	15.56
145	29	29	31.86	37.07	17.71	10.58	10.12	15.08
146	30	30	34.65	37.11	18.02	9.98	10.52	15.38
147	31	31	35.13	37.13	19.97	10.2	10.92	14.16
148	32	32	34.35	44.96	17.66	9.32	7.76	14.56
149	33	33	35.89	43.73	19.42	9.12	8.26	13.42
150	34	34	35.88	49.28	18.38	9.1	7.16	14.28
151	35	35	37.57	49.26	19.27	9.7	8.06	15.2
152	36	36	37.19	52.44	19.85	11.3	8.62	16.9
153	37	37	35.18	58.44	20.33	11.16	7.02	15.2
154	38	38	42.06	51.32	21.31	10.04	9.08	15.96
155	39	39	38.93	51.68	25.6	12.6	10.32	14.74
156	40	40	42.56	51.87	24.94	11.66	10.56	15.62
157	15	15	13.64	16.46	14.81	13.6	12.46	8.82
158	16	16	14.58	17.37	16.01	13.72	12.76	8.78
159	17	17	13.81	20.88	16.55	14.52	10.2	8.38
160	18	18	14.54	22.29	17.45	14.4	9.96	8.3
161	19	19	14.18	25.64	18.12	15.84	9.02	8.48
162	20	20	15.33	24.79	20.9	15.68	10.18	7.78
163	21	21	18.43	22.82	22.14	13.8	12.26	7.94
164	22	22	18.55	27.11	20.22	16.3	11.92	10.52
165	23	23	19.24	26.01	24.41	15.74	12.62	8.5
166	24	24	19.11	28.68	25.1	16.28	11.54	8.44
167	25	25	19.52	32.36	23.79	16.68	10.52	9.44
168	26	26	20.11	30.06	29.93	16.54	11.78	7.4
169	27	27	20.99	29.83	33.17	16.88	12.76	7.04
170	28	28	25.1	32.54	25.8	15.3	12.88	10.58
171	29	29	25.83	31.55	29.88	13.86	12.52	8.34
172	30	30	24.56	36.5	29.31	15.48	11.1	8.98
173	31	31	26.63	37.61	28.4	16.34	12.44	10.82
174	32	32	25.98	40.75	29.51	16.46	11.08	10.12
175	33	33	25.68	39.71	35.32	15.4	10.54	7.56
176	34	34	29.55	39.04	33.34	14.62	12.04	9.06
177	35	35	29.21	45.15	30.44	14.08	9.64	9.58
178	36	36	31.87	43.11	32.24	13.48	10.8	9.5
179	37	37	31.05	43.75	36.45	16.4	12.52	9.7
180	38	38	30.92	44.67	39.52	15.62	11.58	8.36
181	39	39	34.06	46.45	35.91	13.26	10.52	8.9
182	40	40	37.44	44.59	36.97	13.82	12.86	10.02
183	15	15	16.15	17.89	10.38	10.4	10.54	12.5
184	16	16	17.47	18.24	11.61	10.22	11.1	11.8
185	17	17	16.9	19.31	14.06	12.24	11.96	10.86
186	18	18	17.36	22.61	13.35	13.3	11.14	12.94
187	19	19	17.46	24.47	14.5	13.48	10.36	11.98
188	20	20	21.75	24.78	12.94	10.36	10.16	13.64
189	21	21	19.45	28.68	14.58	13.96	10.1	14
190	22	22	22.59	24.39	18.06	10.84	11.32	10.08
191	23	23	20.27	29.19	18.99	13.84	10.3	10.68
192	24	24	25.4	27.92	17.62	10.76	11	11.82
193	25	25	23.92	29.36	20.71	11.98	10.76	10.14
194	26	26	27.14	29.32	20.36	12.86	13.42	12.88
195	27	27	26.23	33.08	20.57	13.86	12.06	13.18
196	28	28	31.58	30.54	20.81	11.66	13.86	13.6
197	29	29	30.14	33.77	21.77	10.06	10.06	10.54
198	30	30	29.31	34.87	24.54	12.86	11.98	11.32
199	31	31	30.23	38.25	23.26	13.8	11.96	13.42
200	32	32	31.54	37.41	25.63	11.38	10.64	10.38
201	33	33	35.56	37.15	24.84	10.04	10.9	10.94
202	34	34	33.97	38.16	28.43	12.4	12.36	10.92
203	35	35	36.39	41.86	25.25	11.64	11.28	12.78
204	36	36	39.19	37.42	30.17	11.22	13.54	10.9
205	37	37	36.77	38.97	34.38	13.04	13.92	10.08
206	38	38	40.83	42.07	29.43	10.86	11.98	11.4
207	39	39	45.33	43.16	27.29	10.54	12.76	13.68
208	40	40	42.58	47.01	28.71	10.48	10.66	11.9