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	-0.000264	0	-0.76
lnL	0.350253	0	2846.91
lnK	0.40017	0	3383.056
lnM	0.249728	0	2535.515
lnLlnL	-0.020214	0	-826.031
lnKlnK	-0.021083	0	-1005.141
lnMlnM	-0.017181	0	-933.12
lnLlnK	0.024071	0	706.883
lnLlnM	0.016327	0	518.79
lnKlnM	0.018056	0	593.789

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	16.43	14.58	13.84	7.48	9.84	6.54
2	16	16	18.84	16.65	12.07	7.48	9.88	9.02
3	17	17	15.18	21.28	14.07	9.06	6.96	7.08
4	18	18	19.65	18.73	15	6.4	7.74	6.28
5	19	19	18.77	21.87	15.54	9.26	8.84	8.26
6	20	20	23.4	20.4	15.68	7.22	9.7	8.26
7	21	21	21.62	21.36	19.63	7.52	8.72	6.02
8	22	22	21.03	29.01	15.39	8.94	7	9.22
9	23	23	22.3	26.76	18.97	9.26	8.54	8
10	24	24	26.61	24.45	20.24	7	8.84	6.9
11	25	25	25.95	26.85	21.24	7.94	8.72	7.18
12	26	26	30.54	30.27	16.67	6.84	7.9	9.96
13	27	27	29.24	28.29	22.51	7.76	9.22	7.54
14	28	28	31.52	32.86	18.73	6.8	7.4	8.96
15	29	29	22.96	39.37	25.19	9.9	6	6.34
16	30	30	37.03	33.05	19.59	6.26	8.18	9.46
17	31	31	35.08	39.1	18.59	6.5	6.54	9.8
18	32	32	34.38	32.89	27.77	6.86	8.26	6.3
19	33	33	30.97	35	32.87	9.7	9.6	6.46
20	34	34	43.66	36.64	21.82	6.18	8.68	9.98
21	35	35	36.47	36.4	31.09	8.54	9.78	7.36
22	36	36	43.5	39.59	24.21	6.14	7.84	8.74
23	37	37	36.76	40.74	32.11	9.36	9.48	7.84
24	38	38	34.02	42.89	36.68	9.24	8.04	6.04
25	39	39	35.67	46.73	33.35	8.2	6.82	6.34
26	40	40	36.87	55.65	27.22	9.8	6.9	10
27	15	15	18.68	10.58	19.89	6.48	14.46	4.3
28	16	16	21.64	13.27	14.39	6.36	12.92	7.34
29	17	17	22.75	12.93	17.9	5.68	12.62	5.38
30	18	18	20.2	13.87	23.66	7.62	13.56	4.52
31	19	19	27.43	14.18	18.67	5.5	13.66	6.18
32	20	20	24.89	15.47	22.57	5.94	11.88	4.76
33	21	21	30.35	16.5	18.92	6.34	14.84	7.9
34	22	22	25.74	18.85	22.76	7.56	12.46	6.24
35	23	23	29.99	19.11	21.63	6.48	12.58	6.8
36	24	24	23.98	20.61	30.94	8.2	11.2	4.34
37	25	25	28.05	24.02	22.75	8.56	11.74	7.82
38	26	26	33.03	18.86	31.88	5.72	12.64	4.26
39	27	27	30.2	23.03	29.95	8.32	13.08	6
40	28	28	32	21.49	36.11	7.36	13.44	4.56
41	29	29	39.55	22.75	28.27	5.66	12.4	6
42	30	30	30.41	26.53	36.01	8.3	11.14	4.86
43	31	31	33.41	24.44	41.5	8.42	13.9	4.66
44	32	32	38.83	27.02	32.28	8.36	14.64	7.42
45	33	33	43.96	23.12	40.28	5.76	14.02	4.56
46	34	34	39.77	28.38	36.75	7.4	12.58	5.8
47	35	35	40.12	30.71	35.8	8.18	12.8	6.68
48	36	36	46.88	30.36	33.11	7.16	13.64	7.7
49	37	37	40.07	29.53	48.15	7.82	12.8	4.5
50	38	38	42.16	31.98	43.61	7.46	11.8	5.12
51	39	39	47.42	30.7	44.12	7.46	14.22	5.8
52	40	40	54.15	30.31	41.72	5.5	12.44	5.34
53	15	15	23.88	18.74	6.1	4	6.08	14.22
54	16	16	21.57	18.75	8.59	5.98	8.06	12.62
55	17	17	18.38	23.87	9.31	7.02	5.9	11.16
56	18	18	20.62	21.85	11.22	7.98	8.52	11.66
57	19	19	24.08	25.21	9.26	6.76	7.32	14.86
58	20	20	21.61	30.08	10	7.72	5.98	13.66
59	21	21	31.04	26.08	9.36	5.02	7.04	14.7
60	22	22	25.12	33.41	10.16	7.12	5.82	14.76
61	23	23	26.94	27.83	14.01	7.78	8.56	12
62	24	24	36.84	27.09	11.66	4.9	8.04	13.54
63	25	25	27.49	35.56	13.17	6.94	5.86	11.78
64	26	26	41.76	30.1	11.6	4.18	7.02	13.48
65	27	27	33.41	36.02	13.44	5.66	5.92	11.8
66	28	28	33.9	37.98	13.99	6.64	6.64	13.44
67	29	29	35.14	36.5	16.03	7.34	8.02	13.2
68	30	30	40.31	40.71	13.26	4.48	5.06	11.84
69	31	31	39.55	35.24	18.57	6.66	8.72	11.58
70	32	32	33.78	47.71	16.65	7.38	5.62	12.12
71	33	33	42.68	38.36	18.83	6.16	7.98	11.52
72	34	34	45.74	45.86	15.12	5.46	6.22	14.34
73	35	35	36.31	51.21	19.08	7.5	5.72	11.42
74	36	36	42.54	50.31	17.79	7.12	6.68	14.18
75	37	37	49.7	48.34	17.2	4.54	5.36	11.3
76	38	38	44.22	51.72	19.83	7.3	6.94	13.4
77	39	39	43.4	52.81	21.65	7.98	7.24	12.92
78	40	40	46.71	58.68	18.81	6.98	6.1	14.54
79	15	15	11.79	22.2	11.61	11.94	6.48	8.68
80	16	16	11.2	23.12	15.15	14.94	7.28	7.48
81	17	17	11.87	27.12	13.99	12.68	5.48	7.46
82	18	18	11.71	25	20.22	14.48	6.78	5.44
83	19	19	14.24	31.83	13.22	11.5	5.1	8.96
84	20	20	13.21	36.72	14.66	13.58	4.66	8.58
85	21	21	15.77	27.1	21.26	12.38	7.48	6.22
86	22	22	16.68	31.32	18.96	11.2	6.1	6.88
87	23	23	16.08	29.37	26.36	13.72	7.72	5.48
88	24	24	15.45	43.1	18.82	14.16	4.84	8.02
89	25	25	15.14	40.09	25.24	12.88	4.68	5.04
90	26	26	18.55	36.07	25.45	11.9	6.22	5.86
91	27	27	18.29	42.65	23.56	12.56	5.3	6.66
92	28	28	20.25	47.48	20.13	11.4	4.78	8.2
93	29	29	18.57	43.98	29.2	12.4	5.14	5.2
94	30	30	21.06	50.45	22.74	11.98	4.9	7.82
95	31	31	21.74	41.15	33.28	13.12	7.08	5.68
96	32	32	22.02	52.23	26.04	11.7	4.84	6.86
97	33	33	24.34	50.27	26.81	13.02	6.34	8.3
98	34	34	25.91	44.1	33.43	12.92	7.9	6.84
99	35	35	26.64	49.87	29.97	12.48	6.82	7.76
100	36	36	22.37	68.85	27.24	13.2	4.02	7.48
101	37	37	25.85	55.33	33.4	12.04	5.62	6.36
102	38	38	27.48	51.9	37.32	11.28	6.1	5.62
103	39	39	28.26	47.26	45.95	12.4	7.74	5
104	40	40	27.86	51.85	45.02	14.28	7.86	5.8
105	15	15	17.97	15.05	11.7	9.9	13.94	11.72
106	16	16	22.39	15.09	11.26	7.72	14.04	12.38
107	17	17	22.14	15.4	13.95	7.66	13.42	9.42
108	18	18	25.33	15.64	14.28	7.96	16.04	11.16
109	19	19	23.44	16.4	18.07	9.36	16.28	9.08
110	20	20	25.45	18.44	16.44	9.56	15.96	11.42
111	21	21	28.3	18.35	17.45	8.8	16.74	11.1
112	22	22	25.6	22.07	17.83	9.7	13.2	10.6
113	23	23	26.32	24.23	17.68	10.78	13.58	12.3
114	24	24	36.73	21.06	16.88	7.26	15.96	12.94
115	25	25	35.76	22.32	18.62	7.86	15.64	12.1
116	26	26	30.83	25.62	21.13	9.78	13.9	10.9
117	27	27	35.22	24.87	21.53	8.52	14.66	10.86
118	28	28	32.02	25.62	26.85	10.74	15.96	9.44
119	29	29	35.26	28.88	22.44	10.6	15.32	12.88
120	30	30	41.52	24.38	27.06	7.8	16.68	9.22
121	31	31	38.51	28.88	25.86	10.08	16.16	11.5
122	32	32	41.32	31.66	23.16	8.94	13.98	12.62
123	33	33	40.62	30.13	28.78	9.26	15.04	9.92
124	34	34	45.59	32.88	24.28	7.78	13.06	11.66
125	35	35	42.53	31.55	31.68	10.06	16.34	10.16
126	36	36	48.39	31.39	30.11	8.3	15.78	10.36
127	37	37	53.57	34.04	25.92	7.34	14.3	12.32
128	38	38	47.52	35.33	31.54	9.3	15.06	10.76
129	39	39	50.05	33.83	34.93	8.4	15.22	9.16
130	40	40	52.8	36.99	31.22	9.58	16.64	12.7
131	15	15	15.95	21.32	8.25	10.54	8.56	16.36
132	16	16	15.16	26.01	8.5	11.56	7	16.3
133	17	17	16.17	23.85	10.97	12.16	8.8	13.72
134	18	18	18.37	25.06	10.72	12.56	9.96	16.9
135	19	19	17.59	28.96	11.33	12.68	8.06	15.2
136	20	20	20.71	25.83	12.98	11.8	10.4	14.6
137	21	21	23.11	26.92	12.75	9.82	9.38	14.12
138	22	22	21.74	28.75	14.93	12.52	10.3	13.92
139	23	23	24.99	33.93	11.71	9	7.18	15.68
140	24	24	26.09	30.31	15.11	10	9.58	13.58
141	25	25	29.08	31.71	14.37	9.4	9.7	15.38
142	26	26	24.9	39.35	14.99	11.54	7.7	14.98
143	27	27	26.48	38.02	16.66	12.32	9.2	15.18
144	28	28	24.61	43.83	17.23	12.08	7	13.12
145	29	29	27.49	45	16.37	12.28	7.86	16.14
146	30	30	31.83	42.01	16.88	10.38	8.56	15.64
147	31	31	29.14	46.47	18.54	12.88	8.5	15.66
148	32	32	35.58	40.85	19.29	10.52	10.22	15.44
149	33	33	33.46	44.17	20.93	11.96	9.86	14.84
150	34	34	34.56	43.89	22.63	12.4	10.7	14.58
151	35	35	40.48	43.72	20.71	9.1	9.5	14.3
152	36	36	37.52	48.81	21.65	11.04	9.26	15.06
153	37	37	34.76	52.21	24.07	12.68	8.98	13.96
154	38	38	41.06	53.1	20.96	9.52	8.04	15
155	39	39	36.21	56.18	25.04	11.88	8.1	13.1
156	40	40	46.74	47.96	24.75	9	9.98	13.58
157	15	15	13.09	17.59	14.15	15.4	12.44	10.04
158	16	16	12.76	18.68	17.32	14.8	10.8	7.38
159	17	17	15.15	18.96	16.84	13.44	11.8	8.48
160	18	18	15.83	21.76	16.03	13.84	10.88	9.74
161	19	19	15.3	22.61	19.68	15.96	11.52	8.48
162	20	20	15.95	27.51	16.87	15.02	9.04	10.04
163	21	21	17.16	23.94	22.77	16.36	12.64	8.38
164	22	22	16.19	29.59	21.56	16.08	9.04	8.2
165	23	23	18.91	30.08	20.02	16.46	10.9	11
166	24	24	19.11	31.43	21.82	15.74	10.02	9.62
167	25	25	20.47	28.63	26.83	14.36	11.06	7.46
168	26	26	22.14	31.88	23.72	16.58	12.34	10.92
169	27	27	20.93	36.51	24.31	16.16	9.6	9.68
170	28	28	23.69	33.23	27.11	16.18	12.42	9.86
171	29	29	21.37	39.44	27.87	16.88	9.38	8.82
172	30	30	26.9	36.02	26.29	13.82	11.2	10.14
173	31	31	28.35	35.38	28.54	14.08	12.4	9.98
174	32	32	27.1	39.19	29.48	15.18	11.24	9.82
175	33	33	29.09	40.55	28.6	14.82	11.46	10.8
176	34	34	25.67	48.4	29.48	16.94	9.18	10.28
177	35	35	28.05	46.84	30.53	16.06	10.04	10.38
178	36	36	30.41	37.9	42.23	14.42	12.72	7
179	37	37	31.82	46.57	32.02	14.3	10.44	10.14
180	38	38	34.62	44.23	34.13	13.82	11.84	10.04
181	39	39	37.03	43.83	34.9	13.08	12.26	10.02
182	40	40	34.2	47.57	38.03	13.26	10.28	8.36
183	15	15	15.9	17.29	11.14	10.82	11.2	11.74
184	16	16	14.67	20.37	12.46	13.42	10.42	11.62
185	17	17	18.13	18.78	13.35	12.4	13.6	12.7
186	18	18	16.71	21.38	15.28	13.98	11.96	11.1
187	19	19	20.8	20.75	14.66	10.54	12.08	11.36
188	20	20	20.03	23.72	15.35	13.6	12.74	13.28
189	21	21	19.48	28.05	15.02	13.98	10.4	13.56
190	22	22	20.58	27.77	16.88	13.44	10.8	12.12
191	23	23	22.44	28.09	17.5	13.56	11.9	12.98
192	24	24	25.66	28.84	16.57	10.48	10.44	12.52
193	25	25	27.25	28.19	18.49	12.24	13.44	13.8
194	26	26	24.72	29.44	22.95	13.24	12.32	10.32
195	27	27	29.15	29.44	21.26	12.36	13.96	12.8
196	28	28	28.85	34.76	19.36	11.44	10.5	13.06
197	29	29	28.95	32.39	24.47	12.52	12.54	10.9
198	30	30	30.96	35.53	22.17	12.76	12.4	13.5
199	31	31	31.71	32.95	27.31	12.12	13.24	10.32
200	32	32	32.01	36.53	26.04	12.98	12.7	11.82
201	33	33	31.17	40.16	26.38	13.16	11.16	11.44
202	34	34	36.22	38.08	26.18	11.64	12.54	12.18
203	35	35	33.66	43.09	26.85	12.94	11.06	12.06
204	36	36	35.61	40.47	30.45	11.74	11.54	10.08
205	37	37	38.54	40.45	30.45	12.82	13.84	12.08
206	38	38	40.54	40.13	31.93	11.48	13.28	10.86
207	39	39	37.83	47.91	29.66	13.38	11.58	12.72
208	40	40	39.03	46.64	32.61	11.72	10.86	10.34