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	0.000155	0	0.39
lnL	0.349996	0	2666.167
lnK	0.400171	0	3019.094
lnM	0.24973	0	2264.521
lnLlnL	-0.020174	0	-868.513
lnKlnK	-0.021122	0	-898.552
lnMlnM	-0.017204	0	-836.135
lnLlnK	0.024095	0	641.37
lnLlnM	0.016307	0	492.3
lnKlnM	0.018115	0	566.409

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	15.5	15.75	13.28	8.24	9.24	7.06
2	16	16	16.77	17.37	13.2	7.88	8.64	7.46
3	17	17	15.41	24.34	11.37	9.38	6.26	9.58
4	18	18	19.68	20.65	12.93	7.26	7.84	8.5
5	19	19	20.71	23.32	12.42	6.78	6.74	8.84
6	20	20	22.02	21.13	16.1	6.82	8.18	7.04
7	21	21	20.17	27.91	14.44	7.82	6.1	8.28
8	22	22	22.57	25.24	17.17	9.76	9.78	9.62
9	23	23	24.81	24.23	19.12	7.42	8.72	7.2
10	24	24	26.58	23.63	21.37	7.3	9.58	6.74
11	25	25	22.94	31.17	20.05	8.46	6.74	7.08
12	26	26	30.87	27.5	18.93	6.08	7.96	7.72
13	27	27	32.44	28.36	19.56	6.44	8.62	8.34
14	28	28	29.11	35.91	18.26	8.04	7.18	9.94
15	29	29	31.95	35.62	18.68	7.06	7.1	9.48
16	30	30	33.65	35.24	20.14	6.06	6.56	7.92
17	31	31	32.59	33.61	25.52	8.42	9.28	8.02
18	32	32	37.99	32.76	24.48	6.16	8.38	7.38
19	33	33	33.92	39.18	24.43	7.9	7.62	8.3
20	34	34	32.18	38.1	30.71	8.56	8.02	6.46
21	35	35	35.84	38.63	29.05	8.2	8.58	7.5
22	36	36	41.63	38.08	27.13	7.58	9.62	8.96
23	37	37	40.78	38.96	29.9	6.22	7.5	6.4
24	38	38	37.03	43.9	31.47	9.94	9.3	8.6
25	39	39	34.92	49.81	31.22	9.94	7.48	8.1
26	40	40	39.89	54.51	25.28	7.86	6.22	9.6
27	15	15	17.4	12.95	15.5	8.06	13.04	6.6
28	16	16	18.33	12.51	19.9	6.72	12.04	4.36
29	17	17	20.67	13.45	19.07	7.64	14.48	6
30	18	18	19.75	15.47	20.25	7.38	11.24	5.12
31	19	19	21.08	17.13	19.45	8.94	13.04	7.02
32	20	20	25.07	16.62	19.82	6.54	12.12	6.16
33	21	21	26.04	14.69	28.4	6.48	14.52	4.18
34	22	22	32.06	14.73	25.74	5.04	14.38	4.66
35	23	23	25.64	19.31	26.33	7.02	11.2	4.86
36	24	24	30.2	20.95	21.84	6.52	11.46	6.82
37	25	25	39.25	18.62	22.13	5.08	13.96	7.12
38	26	26	30.23	23.71	24.51	8.4	12.78	7.68
39	27	27	31.12	19.99	36.64	6.82	13.12	4.02
40	28	28	36.16	24.05	25.26	7.16	13.22	7.8
41	29	29	40.44	24.62	24.15	6.06	12.42	7.94
42	30	30	40.06	25.89	25.73	5.82	11.12	7
43	31	31	47.43	23.06	28.43	5.52	14.74	7.2
44	32	32	43.42	22.52	37.8	5.78	14.3	4.86
45	33	33	37.87	28.5	34.59	8.28	13.22	6.58
46	34	34	44.15	26.45	35.88	5.88	12.28	5.36
47	35	35	42.64	28.64	36.98	6.18	11.28	5.22
48	36	36	46.98	30.46	32.84	7.02	13.36	7.64
49	37	36.99	40.75	27.2	54.37	8.06	14.82	4.1
50	38	38	46.48	28.03	47.71	6.38	13.22	4.42
51	39	39	47.36	29.39	47.65	6.12	12.26	4.34
52	40	40	58.61	32.27	33.92	5.62	12.96	7.64
53	15	15	19.8	20.28	6.8	5.74	6.38	14.42
54	16	16	18.93	21.35	8.4	7.28	7.22	13.52
55	17	17	26.33	17.78	9.1	4.94	8.96	12.32
56	18	18	22.48	25.81	8.06	5.62	5.46	13.42
57	19	19	26.68	24.1	8.73	4.52	5.82	12.02
58	20	20	25.71	24.29	10.81	6.68	8.16	13.22
59	21	21	25.55	28.17	10.59	7.44	7.58	14.98
60	22	22	24.5	31.92	11.11	7.5	6.28	13.58
61	23	23	28.28	30.91	11.42	6.12	6.3	12.7
62	24	24	32.38	27.3	13.42	5.9	8.24	11.88
63	25	25	39.14	28.25	11.86	4.4	7.38	12.8
64	26	26	29.32	37.82	12.9	7.98	6.76	14.98
65	27	27	34.82	31.34	15.49	6.48	8.38	12
66	28	28	45.32	34.3	11.48	4.06	6.44	14.58
67	29	29	38.94	41.9	11.82	4.94	5.18	14.34
68	30	30	31.97	47.19	14.38	7.94	5.74	14.52
69	31	31	36.93	41.02	16.34	7.98	8.06	14.88
70	32	32	39.41	39.64	17.73	6.52	7.4	11.92
71	33	33	46.72	41.86	15.02	4.06	5.28	11.02
72	34	34	42.82	38.94	20.47	6.84	8.74	11.64
73	35	35	54.15	37.09	18.41	4.72	8.42	12
74	36	36	51.65	45.9	16.02	4.6	6.04	13.02
75	37	37	50.63	48.07	16.96	5.12	6.22	13.24
76	38	37.99	61.86	40.63	18.62	4.44	8.32	13.02
77	39	39	41.94	59.15	19.39	7.5	5.72	13.28
78	40	40	45.42	61.34	18.36	6.58	5.28	13.64
79	15	15	9.78	24.69	13.05	14	5.38	7.12
80	16	16	10.94	28.59	11.59	11.32	4.18	7.56
81	17	17	11.9	22.98	17.79	13.66	7.2	6.08
82	18	18	12.24	31.76	13.36	12.36	4.6	7.96
83	19	19	12.97	33.87	13.84	12.18	4.5	8.06
84	20	20	15.1	25.08	20.99	12.46	7.84	6.04
85	21	21	14.58	32.98	17.82	11.84	5.18	6.68
86	22	22	16.68	32.04	18.34	13.28	7.04	8.48
87	23	23	13.76	35.11	25.48	14.86	5.64	5.14
88	24	24	17.22	33.25	23.32	12.52	6.6	6.26
89	25	25	17	36.66	24.17	12.96	6	6.12
90	26	26	21.3	34.31	22.45	11.38	7.42	7.64
91	27	27	18.48	36.48	29.27	12.7	6.52	5.28
92	28	28	21.37	42.92	21.49	12.2	6.14	8.66
93	29	29	18.51	44.48	28.86	14.78	6.02	6.26
94	30	30	20.71	46.66	26.03	14.88	6.54	8.12
95	31	31	22.9	50.14	23.1	12.28	5.58	8.68
96	32	32	20.2	54.81	27.57	13.08	4.62	6.48
97	33	33	23.72	53.45	25.51	11.56	5.08	7.58
98	34	34	22.89	56.74	27.67	11.3	4.44	6.46
99	35	35	24.78	49.56	33.6	12.62	6.38	6.3
100	36	36	30.82	47.64	29.11	11.1	7.6	8.48
101	37	37	23.11	65.03	31.24	12.18	4.12	6.1
102	38	38	23.62	57.29	40.4	14.74	5.94	5.6
103	39	39	26.13	56.78	38.98	14.52	6.66	6.48
104	40	40	27.71	66.01	31.74	11.76	4.84	7.16
105	15	15	21.7	12.89	11.69	7.72	16.28	11.42
106	16	16	22.07	13.37	13.85	7.66	15.78	9.46
107	17	17	23.98	15.67	12.26	7.34	13.84	11.54
108	18	18	22.38	17.16	14.47	8.68	13.56	10.36
109	19	19	26.83	17.14	14.15	8.52	16.5	12.92
110	20	20	24.32	17.95	18.21	10.08	16.46	10.12
111	21	21	24.63	20.14	18.05	9.5	13.76	9.78
112	22	22	26.05	19.19	21.75	10.26	16.8	9.06
113	23	23	32.49	20.42	17.55	7.68	15.16	11.32
114	24	24	31.67	21.75	19.33	9.1	16.18	11.62
115	25	25	34.88	23.8	17.41	7.96	14.26	12.88
116	26	26	32.49	25.12	20.35	10.24	15.84	12.68
117	27	27	37.05	25.07	19.94	7.78	14.08	11.52
118	28	28	33.68	28.28	21.51	10.26	14.4	12.42
119	29	29	33.9	30.1	22.18	10.74	14.12	12.64
120	30	30	37.77	28.24	24.2	10.38	16.7	12.52
121	31	31	39.2	27.9	26.69	8.62	14.7	9.68
122	32	32	37.9	32.5	24.86	10.4	14.24	12.2
123	33	33	42.8	30.25	26.7	8.68	14.92	10.8
124	34	34	37.22	33.1	31.33	10.38	13.62	9.08
125	35	35	39.74	35.29	29.07	10.06	13.22	10.38
126	36	36	46.32	32.02	30.87	8.06	14.22	9.28
127	37	37	51.05	32.79	29.17	8.42	16.2	11.62
128	38	38	55.18	33.91	27.8	7.4	15	11.84
129	39	39	56.76	32.01	32.46	7.5	16.82	10.34
130	40	40	54.02	36.13	31.45	8.46	15.52	11.42
131	15	15	15.13	22.49	8.2	10.1	7.24	14.84
132	16	16	16.9	22.91	8.78	9.04	7.22	13.94
133	17	17	15.88	27.37	9.31	11.82	7.12	15.82
134	18	18	20.79	23.13	10.26	9.84	9.92	16.14
135	19	19	21.44	23.12	12.03	10	10.46	14.1
136	20	20	20.41	30.18	10.68	9.9	7.14	15.14
137	21	21	22.84	26.9	12.95	10.48	9.88	14.6
138	22	22	22.95	29.4	13.49	12.48	10.66	16.66
139	23	23	24.59	30.97	13.52	11.32	9.86	16.34
140	24	24	26.06	32.75	13.59	11.04	9.64	16.96
141	25	25	24.38	37.69	14.15	10.72	7.34	14.52
142	26	26	23.81	38.7	16.25	12.52	8.08	14.02
143	27	27	27.3	41.33	14.31	10.66	7.48	16.28
144	28	28	27	42.31	16.03	11.22	7.56	14.8
145	29	29	26.82	44.32	17.25	12.2	7.72	14.64
146	30	30	30.61	40.15	18.88	12.86	10.68	16.22
147	31	31	29.46	47.37	17.82	10.74	7.02	13.86
148	32	32	32.97	40.22	21.74	11.7	10.58	13.64
149	33	33	36.7	47.67	16.8	9	7.56	16.12
150	34	34	40.36	41.77	19.93	9.84	10.8	16.12
151	35	35	36.37	50.29	19.51	10.66	8.32	15.84
152	36	36	33.74	53.49	21.88	12.16	8.08	14.46
153	37	37	45.08	43.76	22.17	9.16	10.84	15.08
154	38	38	41.31	54.32	20.17	10.24	8.48	17
155	39	39	41.83	58.26	19.85	9.82	7.6	16.86
156	40	40	39.53	58.82	23.03	11.06	7.92	14.92
157	15	15	12.53	18.64	13.79	14.02	10.04	8.94
158	16	16	13.17	20.59	14.24	13.32	9	8.68
159	17	17	15.34	19.71	15.57	14.78	12.58	10.38
160	18	18	16.53	20.06	17.09	13.8	12.56	9.48
161	19	19	15.19	24.39	17.7	16.96	11.1	10.12
162	20	20	16.38	23.29	20.92	16.02	12.1	8.58
163	21	21	16.5	24.14	23.82	15.14	11.06	7.02
164	22	22	16.67	30.19	20.03	16.26	9.24	9.36
165	23	23	17.67	26.15	27.47	16.98	12.24	7.22
166	24	24	20.1	29.45	22.41	14.32	10.44	9
167	25	25	23.19	29.6	21.34	13.48	11.56	10.62
168	26	26	22.75	32.67	22.02	13.26	9.9	9.84
169	27	27	20.49	31.73	31.14	16.54	11.3	7.22
170	28	28	23.46	36.89	23.47	13.56	9.1	9.68
171	29	29	25.84	30.13	32.15	13.54	12.92	7.48
172	30	30	24.46	32.65	35.16	15.02	12.22	7
173	31	31	29.4	34.04	28.8	13.2	12.7	9.66
174	32	32	28.13	35.05	33.25	13.74	12.12	8.06
175	33	33	28.07	36.38	35.6	14.22	11.98	7.68
176	34	34	29.33	40.18	32.22	14.04	11.08	8.98
177	35	35	26.07	44.6	36.55	15.04	9.14	7.22
178	36	36	26.94	49.92	32.87	16.52	9.14	9.34
179	37	37	30	44.45	37.39	14.62	10.52	8.06
180	38	38	33.05	46.13	34.17	13.86	10.7	9.52
181	39	39	29.67	51.49	37.41	16	9.56	8.7
182	40	40	30.48	51.41	39.87	15.86	9.8	8.26
183	15	15	17.11	15.65	11.75	10.54	13.38	11.72
184	16	16	16.35	19.66	11.36	11	10.12	12.06
185	17	17	18.57	18.48	13.25	10.2	11.72	10.84
186	18	18	21.94	18.97	12.74	10.06	13.64	13.62
187	19	19	17.46	22.85	16.06	13.86	11.54	10.92
188	20	20	21.55	24.14	13.58	10.62	10.62	13.06
189	21	21	20.9	25.17	16	12.74	11.7	12.46
190	22	22	19.4	27.91	18.17	13.56	10.1	10.46
191	23	23	25.18	25.41	17.43	11.46	12.96	12.62
192	24	24	22.61	31.14	17.52	12.78	10.02	12.32
193	25	25	26.05	28.76	19.05	10.5	10.68	10.84
194	26	26	29.54	28.92	18.59	11.3	13.24	13.92
195	27	27	27.44	29.06	23.52	12.36	13.2	10.58
196	28	28	26.57	35.29	21.12	13.64	11.16	12.76
197	29	29	32.64	30.12	23.28	11.1	13.94	11.8
198	30	30	32.06	37.5	19.6	10.86	10.32	13.84
199	31	31	30.42	34.36	27.07	12.4	12.28	10.16
200	32	32	35.92	34.18	24.69	11.44	13.86	12.7
201	33	33	35.64	36.9	25.01	11.72	12.86	12.7
202	34	34	33.24	41.38	25.94	13.7	12.1	13.1
203	35	35	35.48	41.09	26.83	10.48	10.08	10.4
204	36	36	39.98	38.94	27.65	10.18	12	11.22
205	37	37	35.08	46.97	27.66	13.42	10.88	12.68
206	38	38	42.35	38.93	31.55	10.16	12.82	10.26
207	39	39	45.89	41.24	28.74	10.14	13.14	12.56
208	40	40	44.06	42.14	32.33	11.36	13.68	11.68