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	0.000436	0	1.01
lnL	0.350132	0	2553.626
lnK	0.400081	0	2584.692
lnM	0.249479	0	2029.392
lnLlnL	-0.020149	0	-827.708
lnKlnK	-0.021173	0	-851.877
lnMlnM	-0.017116	0	-733.585
lnLlnK	0.024153	0	590.742
lnLlnM	0.016144	0	462.226
lnKlnM	0.018193	0	484.383

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	13.2	20.32	11.29	9.16	6.3	7.86
2	16	16	17.65	18.51	11.22	7.9	8.54	9.62
3	17	17	16.42	18.82	15.21	9.8	9.54	7.66
4	18	18	20.82	17.26	15.76	6.92	9.86	6.86
5	19	19	21.12	19.98	15.22	7.18	8.76	7.54
6	20	20	18.74	26.33	14.4	9.32	7.14	9.08
7	21	21	24.9	24.41	13.34	6.58	7.7	9.8
8	22	22	21.85	24.03	19.33	8.34	8.52	6.88
9	23	23	26.32	24.07	17.86	7.42	9.42	8.36
10	24	24	29.07	24.02	18.52	6.08	8.7	7.38
11	25	25	31.27	24.92	18.62	6.62	9.88	8.7
12	26	26	24.91	30.41	21.62	8.32	7.52	7.02
13	27	27	31.19	25.65	24.06	6.52	9.38	6.32
14	28	28	25.38	39.1	19.38	9.6	6.6	9.42
15	29	29	34.94	29.15	22.39	6.42	9.08	7.74
16	30	30	28.85	41.33	19.57	8.44	6.32	9.52
17	31	31	40.5	30.33	22.49	6.04	9.7	8.62
18	32	32	36.18	32.75	26.11	6.18	7.94	6.48
19	33	33	26.58	43.63	29.1	9.56	6.1	6.14
20	34	34	33.27	36.15	31.8	8.76	9.08	6.6
21	35	35	38.56	41.54	23.68	7.26	7.6	9.2
22	36	36	33.24	44.41	29.09	8.76	7.12	7.32
23	37	37	40.56	42.7	26.24	6.86	7.38	8.18
24	38	38	39.05	42.1	31.21	8.24	8.62	7.66
25	39	39	32.89	50.74	33.01	9.56	6.56	6.8
26	40	40	38.39	48.43	31.53	9.04	7.86	8.14
27	15	15	19.69	12.67	13.62	6.48	12.44	7.14
28	16	16	16.24	14.41	18.59	8.96	11.78	5.46
29	17	17	16.95	15.05	20.86	8.58	11.28	4.8
30	18	18	25.78	12.17	21.2	5.16	14.26	4.64
31	19	19	25.26	15.27	18.39	6.34	13.1	6.58
32	20	20	24.89	17.1	19.11	7.92	14.08	7.72
33	21	21	25.73	17.6	21.16	7.12	12.72	6.4
34	22	22	28.72	17.64	21.91	5.56	11.28	5.46
35	23	23	31.74	15.13	29.9	5.22	14.26	4
36	24	24	30.63	20.28	22.6	7.4	13.74	7.56
37	25	25	31.27	17.71	32.69	6.58	14.68	4.46
38	26	26	29.15	22.98	27.09	8.4	12.7	6.54
39	27	27	34.5	23.18	24.74	6.4	11.68	6.76
40	28	28	33.32	24.41	27.51	8.36	13.78	7.48
41	29	29	31.15	23.23	37.95	8.76	14.14	4.96
42	30	30	37.24	25.62	28.8	7.72	13.7	7.46
43	31	31	41.57	23.58	32.54	6.36	14.16	6.06
44	32	32	38.77	25.14	36.5	6.78	12.9	5.2
45	33	33	42.76	25.76	34.73	6.16	12.78	5.62
46	34	34	35.67	29.39	40.37	8.68	12.46	5.36
47	35	35	50.7	24.13	39.23	5.26	14.4	5.08
48	36	36	44.99	30.91	33.97	6.74	11.98	6.7
49	37	37	39.26	34.96	37.33	8.84	11.58	6.7
50	38	38	45.16	33.85	36.12	7.58	12.16	7.04
51	39	39	48.13	33.74	36.95	6.36	11.04	6.2
52	40	39.99	40.18	31.43	59.79	8.94	13.64	4
53	15	15	15.88	23.87	7.14	7.18	5.08	13.14
54	16	16	17.38	22.51	8.68	7.14	6.02	11.54
55	17	17	26.95	17.95	8.74	4.1	7.56	11.02
56	18	18	20.8	21.07	11.69	7.98	8.98	11.22
57	19	19	32.83	20.66	8.51	4.16	8.2	14.54
58	20	20	22.68	30.34	9.33	7	5.68	14.22
59	21	21	32.81	22.47	10.77	4.26	7.6	11.28
60	22	22	28.9	27.39	11.2	5.06	6.16	11.02
61	23	23	26.35	34.34	10.82	6.62	5.54	13.48
62	24	24	39.47	26.9	10.9	4.38	7.86	14.22
63	25	25	29.11	36.02	12.08	7.42	6.6	14.92
64	26	26	34.63	32.04	13.2	6.42	8.04	14.26
65	27	27	43.3	31.59	11.89	4.48	7.42	14.64
66	28	28	36.89	32.96	15.35	6.76	8.82	13.54
67	29	29	38.47	33.23	16.38	6.48	8.8	12.64
68	30	30	43.36	39.03	12.89	4.84	6.26	14.4
69	31	31	41.17	41.77	14.01	5.02	5.64	12.74
70	32	32	34.19	48.87	15.9	7.62	5.72	13.4
71	33	33	35.65	49.28	16.65	7.58	5.92	13.26
72	34	34	35.9	49.95	18.04	7.2	5.58	11.56
73	35	35	56.24	39.23	16.3	4.88	8.52	14.96
74	36	36	41.19	47	20.3	7.38	7.22	12.12
75	37	37.01	56.81	48.51	14.71	4.04	5.56	14.14
76	38	38.01	48.02	57.23	15.75	5.4	5.02	14.32
77	39	38.99	67.13	40.56	18.66	4.14	8.56	13.34
78	40	39.99	67.42	41.99	19.34	4	7.98	12.42
79	15	15	10.59	21.1	14.61	12.76	6.48	6.24
80	16	16	11.36	23.14	14.82	11.6	5.74	6.06
81	17	17	11.06	25.62	16.87	12.74	5.42	5.54
82	18	18	12.84	24.22	18.47	13.96	7.56	6.5
83	19	19	13.67	31.6	14.13	11.96	5.1	8.22
84	20	20	12.62	35.76	16.23	13.7	4.6	7.28
85	21	21	14.47	27.69	23.41	12.92	6.88	5.24
86	22	22	14.36	35.57	19.59	11.9	4.68	5.9
87	23	23	15.53	38.87	18.32	12.62	4.9	7.42
88	24	24	16.31	35.51	22.93	12.5	5.72	5.98
89	25	25	20.98	34.15	19.83	11.58	7.46	8.84
90	26	26	20.18	38.61	20.4	11.94	6.36	8.42
91	27	27	17.38	44.25	24.07	13.92	5.3	6.78
92	28	28	19.32	44.3	23.71	12.68	5.46	7.12
93	29	29	20.75	44.06	24.72	11.8	5.56	6.86
94	30	30	24.55	41.35	24.33	11.44	7.08	8.26
95	31	31	22.05	43.31	30.14	14.56	7.52	7.2
96	32	32	21.88	52.07	26.39	14.8	6.1	8.48
97	33	33	25.36	49	26.24	11.28	5.94	7.74
98	34	34	21.52	61.8	26.87	13.2	4.36	7.26
99	35	35	23.75	58.95	27.8	11.32	4.44	6.72
100	36	36	23.98	47.95	41.58	13.8	6.98	5.16
101	37	37	23.31	53.7	40.81	14.34	6.14	5.3
102	38	38	25.37	61.53	32.67	11.46	4.62	6.08
103	39	39	24.75	65.06	34.62	15	5.5	7.22
104	40	40	23.71	74.63	33.57	14.16	4.2	6.72
105	15	15	18.8	14.24	12.02	10.7	16.96	12.94
106	16	16	20.92	13.37	14.85	8.42	16.3	9
107	17	17	21.61	16.78	12.59	9.54	14.68	12.86
108	18	18	20.9	17.33	15.59	9.66	13.76	9.74
109	19	19	22.91	17.49	16.8	9.62	15.1	9.9
110	20	20	24.9	19.95	14.98	9.98	14.8	12.96
111	21	21	26.06	18.61	19	9.9	16.82	10.26
112	22	22	29.69	19.51	17.82	7.86	14.72	10.24
113	23	23	27.16	22.42	19.11	9.54	13.66	10.3
114	24	24	28.55	25.21	17.62	10.28	13.6	12.96
115	25	25	37.53	20.88	19.46	7.06	16.08	10.92
116	26	26	31.94	26.34	19.34	9.78	14.02	12.6
117	27	27	34.96	22.81	24.95	8.98	16.96	9.54
118	28	28	33.05	27.88	22.52	10.66	14.88	11.96
119	29	29	34.32	26.32	26.88	10.86	16.96	10.34
120	30	30	41.08	26.45	24.08	8.56	16.42	11.46
121	31	31	37.1	29.9	25.73	10.94	16.12	12.02
122	32	32	38.8	29.85	27.52	10	15.56	10.7
123	33	33	40.25	32.89	25.4	10.1	14.64	12.4
124	34	34	40.28	34.93	25.95	10.18	13.76	12.2
125	35	35	43.58	33.17	28.35	9.42	14.84	11.16
126	36	36	44.93	33.7	29.64	9.18	14.72	10.7
127	37	37	50.42	31.5	31.6	8.16	16.22	10.1
128	38	38	47.37	35.17	31.88	8.84	14.34	10.06
129	39	39	49.3	35.48	33.02	8.58	14.44	9.82
130	40	40	49.14	37.51	33.53	10.52	16.52	11.78
131	15	15	14.76	21.68	8.89	10.78	7.84	13.98
132	16	16	18	19.94	9.83	9.28	9.4	13.5
133	17	17	17.66	23.72	9.87	10.6	8.56	15
134	18	18	20.55	23.22	10.34	9.64	9.54	15.44
135	19	19	19.99	26.91	10.64	10.1	8.14	15.16
136	20	20	22.26	26.57	11.39	9.48	8.8	14.9
137	21	21	22.42	27.86	12.62	11.48	10.16	16.12
138	22	22	22.67	28.72	14.16	11.86	10.26	14.74
139	23	23	27.01	29.35	12.96	9.34	9.68	15.82
140	24	24	21.65	37.11	14.52	11.68	7.08	13.34
141	25	25	25.87	31.62	16.73	11.7	10.56	13.96
142	26	26	28.15	34.44	15.39	10.76	9.7	15.64
143	27	27	29.1	36.03	15.91	11.7	10.4	17
144	28	28	27.5	39.51	17.19	13	9.7	16.14
145	29	29	30.34	37.88	18.28	10.86	9.56	14.08
146	30	30	28.53	41.95	19.45	13	9.44	14.58
147	31	31	31.43	40.97	20.02	12.98	10.86	15.76
148	32	32	32.58	42.87	20.16	12.4	10.26	15.58
149	33	33	38.54	42.65	18.4	9.76	9.9	16.64
150	34	34	37.74	42.72	21	9.96	9.84	14.18
151	35	35	34.43	52.65	19.65	11.48	7.96	15.86
152	36	36	35.31	56.6	19.13	11.1	7.28	16.3
153	37	37	33.38	58.39	21.78	12.3	7.28	14.52
154	38	38	34.81	61.3	21.29	11.92	7	15.18
155	39	39	38.78	56.91	22.51	10.66	7.76	14.44
156	40	40	46.45	51.02	22.87	10.18	10.42	16.74
157	15	15	14.06	17.08	13.4	13.7	12.46	10.36
158	16	16	12.64	19.4	16.56	13.7	9.46	7.12
159	17	17	13.84	21.25	16.07	14.32	9.88	8.58
160	18	18	15.41	20.13	18.81	14.16	11.82	8
161	19	19	15.39	21.47	21.18	16.1	12.44	7.9
162	20	20	17.02	23.36	19.68	16.28	12.82	9.8
163	21	21	17.67	22.72	23.71	15.2	12.92	7.68
164	22	22	17.07	27.28	22.58	15.8	10.4	8.12
165	23	23	18.38	27.32	24.16	13.86	9.94	7.18
166	24	24	18.12	32.74	22.17	15.9	9.06	8.96
167	25	25	20.34	29.08	26.44	15.06	11.3	7.9
168	26	26	21.44	32.71	23.89	15.96	11.1	10.04
169	27	27	24.42	30.5	25.67	13.48	11.86	9.08
170	28	28	23.38	36.14	24.33	13.32	9.12	9.08
171	29	29	22.8	40.1	24.78	16.46	9.68	10.66
172	30	30	26.62	34.42	28.59	14.68	12.4	9.64
173	31	31	24.62	35.58	34.68	16.38	12.14	7.82
174	32	32	27.04	37.21	32.04	13.2	10.36	7.72
175	33	33	26.54	42.03	30.84	16.44	10.94	9.84
176	34	34	29.7	39.3	32.77	15.06	12.38	9.58
177	35	35	28.41	42.42	34.84	16.52	11.78	9.28
178	36	36	32.25	40.09	35.46	13.9	12.3	8.88
179	37	37	30.14	42.09	40.46	15.76	12.16	7.96
180	38	38	33.22	45.29	34.89	14.92	11.84	10.06
181	39	39	31.44	46.65	40	13.42	9.64	7.22
182	40	40	34.85	43.05	43.29	13.26	11.82	7.34
183	15	15	15.09	17.92	11.32	13.94	13.02	13.96
184	16	16	16.81	19.7	10.92	11.66	11.06	13.84
185	17	17	15.5	20.31	14.67	13.76	11.44	10.48
186	18	18	19.95	19.88	13.43	10.16	11.66	11.56
187	19	19	18.08	24.11	14.14	12.6	10.26	12.02
188	20	20	21.55	22.93	14.65	11.28	11.98	12.68
189	21	21	19.85	24.72	17.63	12.98	11.46	10.66
190	22	22	22.04	23.56	19.7	12.44	13.14	10.14
191	23	23	23.24	28.12	16.68	12.66	11.56	13.36
192	24	24	26.11	26.86	17.99	12.12	13.4	13.42
193	25	25	23.97	29.78	20.21	12.74	11.28	11.12
194	26	26	30.2	26.53	20.58	10.52	14	11.8
195	27	27	26.73	33.26	19.9	13.22	11.68	13.36
196	28	28	31.58	29.18	22.3	10.3	12.92	11.08
197	29	29	28.84	34.42	22.44	13.38	12.42	12.84
198	30	30	32.16	33.19	23.34	10.08	11.1	10.5
199	31	31	33.12	37.16	21.51	10.6	10.58	12.58
200	32	32	30.34	41.62	23.08	13.48	10.62	13.28
201	33	33	31.96	41.26	24.51	12.48	10.56	12.18
202	34	34	36.11	40	24.43	12.06	12.22	13.64
203	35	35	36.42	42.37	24.78	12.44	11.9	13.98
204	36	36	32.93	44.89	28.99	12.8	10.16	10.62
205	37	37	36.66	45.36	27.43	13.38	11.9	13.44
206	38	38	39.16	43.75	29.34	13	13.04	13.04
207	39	39	46.34	41.76	27.87	10.76	13.9	13.98
208	40	40	44.54	43.27	30.61	11.42	13.5	12.68