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	-0.000333	0	-0.992
lnL	0.350347	0	3140.536
lnK	0.400155	0	3468.576
lnM	0.249683	0	2432.056
lnLlnL	-0.020208	0	-989.383
lnKlnK	-0.021065	0	-996.02
lnMlnM	-0.017197	0	-853.587
lnLlnK	0.024021	0	765.772
lnLlnM	0.016347	0	516.616
lnKlnM	0.018075	0	544.229

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	16.23	15.17	13.22	6.82	8.44	6.2
2	16	16	18.87	19.24	9.75	6	6.7	9.32
3	17	17	18.05	18.9	13.3	6.04	6.54	6.18
4	18	18	18.85	18.82	15.75	8.16	9.34	7.2
5	19	19	20.52	21.85	13.81	6.82	7.24	7.76
6	20	20	23.43	21.21	14.76	7.38	9.48	9.08
7	21	21	18.37	25.01	19.28	8.98	7.14	6.06
8	22	22	19.43	26.46	19.63	9.44	7.5	6.66
9	23	23	23.05	25.07	20.03	9.64	9.98	8.12
10	24	24	28.71	21.98	21.63	6.02	9.42	6
11	25	25	27.62	25.02	21.77	7.26	9.32	6.86
12	26	26	33.01	25.99	18.92	6.46	9.78	8.88
13	27	27	25.92	35.94	18.54	9.1	7.08	9.64
14	28	28	27.07	34.92	20.91	9.94	8.42	9.62
15	29	29	26.37	38.43	21.54	9.62	7.06	8.72
16	30	30	38.26	29.95	21.76	6.56	10	9.1
17	31	31	31.33	33.77	26.72	7.36	7.7	6.34
18	32	32	30.53	39.09	25.09	7.98	6.82	7.18
19	33	33	33.08	34.94	30.05	8.68	9.3	6.94
20	34	34	32.76	37.1	31.21	9.44	9.32	7.14
21	35	35	38.49	43.32	22.35	7.28	7.24	9.86
22	36	36	38.45	42.4	25.64	8.64	8.8	9.94
23	37	37	33.39	51.05	26.2	9.2	6.38	8.74
24	38	38	38.18	45.5	28.63	9.68	9	9.7
25	39	39	40.86	41.94	32.65	8.3	9.2	7.72
26	40	40	38.73	44.06	35.94	10	9.82	7.8
27	15	15	22.15	11.21	14.29	5.82	14.82	6.96
28	16	16	22.36	12.77	14.69	6.78	14.98	7.94
29	17	17	17.46	14.65	20.93	8.84	12.42	5.1
30	18	18	23.24	13.55	20.24	5.64	12.16	4.74
31	19	19	24.96	16.45	16.56	6.86	12.8	7.94
32	20	20	29.19	16.21	16.92	5.1	11.64	6.92
33	21	21	26.64	17.93	19.59	7.14	13	7.32
34	22	22	27.41	16.83	25.27	6.36	12.9	5
35	23	23	30.14	19.49	20.81	6.54	12.48	7.22
36	24	24	29.23	20.24	24.13	8.38	14.76	7.5
37	25	25	28.81	22.04	25.2	7.2	11.28	6.02
38	26	26	41.3	20.04	21.45	5	13.38	7.72
39	27	27	28.63	24.31	29.51	8.56	11.86	5.9
40	28	28	40.04	22.78	24.16	5.3	11.76	6.86
41	29	29	39.6	25.29	23.75	6	11.62	7.82
42	30	30	35.3	27.42	27.72	8.28	12.74	7.86
43	31	31	35.33	23.3	41.65	7	13.06	4.12
44	32	32	45.34	27.42	25.73	5.52	11.4	7.68
45	33	33	45.78	26.91	29.5	6.36	13.58	7.62
46	34	34	37.3	26.1	46.5	8.6	14.96	4.74
47	35	35	39.85	27	44.97	6.52	11.78	4.04
48	36	36	46.82	27.13	40	5.7	12.38	4.9
49	37	37	44.81	30.53	38.87	8.1	14.54	6.84
50	38	38	50.96	29.05	39.54	5.9	13.06	5.68
51	39	39	49.52	31.18	40.51	7.34	14.46	6.64
52	40	40	54.32	32.84	36.38	6.36	13.14	7.28
53	15	15	18.83	18.96	7.9	6.86	7.78	13.62
54	16	16	19.2	20.56	8.69	6.24	6.58	11.32
55	17	17	20.68	20.2	10.13	7.56	8.88	12.5
56	18	18	27.62	21.34	8.19	4.56	7.06	13.62
57	19	19	23.65	24.14	10.03	7.64	8.52	14.96
58	20	20	23.64	26.55	10.59	7.5	7.48	13.78
59	21	21	26.74	28.78	9.75	5.82	6.1	13.62
60	22	22	26.4	25.66	13.7	7.16	8.46	11.06
61	23	23	27.17	31.33	11.77	7.16	6.92	13.68
62	24	24	27.24	36	11.34	7.36	6.06	14.74
63	25	25	26.92	35.01	13.81	7.94	6.66	12.44
64	26	26	37.6	29.66	13.34	4.74	7.16	11.46
65	27	27	35.17	36.07	12.63	4.74	5.26	11.3
66	28	28	37.84	34.36	14.07	5.28	6.76	12.08
67	29	29	46.38	36.81	11.5	4	6	14.74
68	30	30	34.71	38.29	17.21	7.78	7.92	12.68
69	31	31	32.53	42.36	18.29	7.88	6.6	11.08
70	32	32	47.49	36.76	15.78	5.4	8.34	14.1
71	33	33	46.87	41.5	15.14	5.46	7.2	14.74
72	34	34	49.58	36.12	19.1	5.04	8.36	11.06
73	35	35	41.34	50.37	16.65	6.58	5.96	13.7
74	36	36	45.52	44.99	19.09	7.18	8.32	14.26
75	37	37	39.92	52.94	19.8	8	6.56	13.04
76	38	38	50.71	51.87	16.8	5.32	5.92	13.94
77	39	39	41.92	57.87	19.97	7.62	5.96	13.02
78	40	40	52.55	45.44	23.14	6.4	8.68	12
79	15	15	9.51	25.48	13.01	14.46	5.18	7.14
80	16	16	12.05	23.38	13.4	11.3	5.92	7.12
81	17	17	10.38	27.22	16.99	13.5	4.96	5.4
82	18	18	12.47	24.28	19.2	14.98	7.82	6.44
83	19	19	12.74	34.56	13.81	13.36	4.72	8.68
84	20	20	13.76	28.84	19.49	14.46	6.92	6.86
85	21	21	14.92	34.96	15.87	12.1	5.08	8.04
86	22	22	14.98	28.48	25.69	13.68	7.34	5.18
87	23	23	15.02	37.48	20.25	11.34	4.42	5.7
88	24	24	15.32	35.67	25.02	14.76	6.24	5.92
89	25	25	19.66	35.37	20.64	11.56	6.62	7.8
90	26	26	17.44	39.98	23.93	13.98	6.02	6.88
91	27	27	21.45	35.33	24.65	12.4	7.88	7.52
92	28	28	20.06	48.82	19.63	12.26	4.92	8.98
93	29	29	19.1	46.89	25.49	14.2	5.64	7.22
94	30	30	23.22	44.31	23.8	11.04	5.9	7.66
95	31	31	20.04	58.62	22.52	12.56	4.06	7.82
96	32	32	22.74	50.9	25.79	12.6	5.58	7.76
97	33	33	24.54	46.54	29.65	13.08	7.04	7.48
98	34	34	27.21	49.5	26.28	11.96	6.76	8.9
99	35	35	22.89	59.93	28.65	12.76	4.7	7
100	36	36	24.21	62.56	27.73	11.92	4.46	7.26
101	37	37	26.24	58.42	30.22	13.06	5.82	7.9
102	38	38	24.6	61.81	33.99	12.78	4.94	6.24
103	39	39	26.76	64.99	30.94	13.82	5.56	8.32
104	40	40	28.7	56.68	37.61	12.24	6.28	6.36
105	15	15	21.15	13.61	11.09	7.96	15.28	12.16
106	16	16	22.14	14.53	12.1	8.36	15.68	12.16
107	17	17	25.47	14.15	13.34	7.26	16.56	11.1
108	18	18	20.79	18.69	13.96	10.9	14.12	12.44
109	19	19	24.86	17.75	14.77	7.98	13.56	10.52
110	20	20	28.4	18.36	14.4	7.24	13.82	11.5
111	21	21	27.15	19.37	16.9	8.72	14.82	10.88
112	22	22	30.65	18.68	18.33	7.4	15.14	9.68
113	23	23	33.12	19.21	18.86	7.58	16.44	10.5
114	24	24	31.25	22.51	18.65	7.9	13.28	10.36
115	25	25	36.52	22.93	17.41	7.46	14.74	12.74
116	26	26	34.05	24.75	19.6	8.32	13.84	11.38
117	27	27	36.86	24.29	21.07	9.02	16.84	12.44
118	28	28	37.76	26.74	20.24	8.7	14.92	12.94
119	29	29	38.02	26.56	23.18	8.7	15.16	11.12
120	30	30	33.83	29.16	26.62	10.14	13.8	9.6
121	31	31	38.62	29.92	24.39	9.54	14.72	11.7
122	32	32	43.56	29.35	24.33	8.82	16.04	12.5
123	33	33	44.61	29.89	25.8	8.72	15.96	11.86
124	34	34	43.16	30.82	28.8	9.22	15.66	10.6
125	35	35	51.47	32.8	23.4	7.2	13.98	13
126	36	36	52.44	30.62	28.28	7.88	16.96	11.64
127	37	37	45.84	33.25	32.81	8.54	14.24	9.04
128	38	38	48.53	38.79	26.6	8.82	13.12	12.78
129	39	39	44.93	37.78	33.81	9.72	13.62	9.7
130	40	40	56.47	34.56	31.88	8.26	16.82	11.56
131	15	15	13.72	24.03	8.5	12.66	7.48	15.88
132	16	16	16.22	22.07	9.74	10.56	8.4	13.74
133	17	17	15.77	26.1	10.02	11.08	7	13.5
134	18	18	20.66	24.45	9.58	9.6	9	16.94
135	19	19	17.27	31.14	10.53	12.92	7.38	16.52
136	20	20	20.14	30.07	10.91	11.44	8.16	16.8
137	21	21	22.77	28.09	12.24	9.52	8.5	14.12
138	22	22	21.03	29.71	14.88	12.9	9.82	13.84
139	23	23	25.35	31.78	12.57	10	8.76	16.3
140	24	24	26.43	33.87	12.75	9.28	7.92	15.62
141	25	25	26.92	34.02	14.36	10.44	9.06	15.62
142	26	26	26.92	33.12	17.21	11.48	10.28	13.88
143	27	27	29.28	36	15.81	10.4	9.32	15.34
144	28	28	31.16	38.47	15.24	9.62	8.58	15.94
145	29	29	33.95	37.75	15.96	9.3	9.38	16.14
146	30	30	29.26	39.64	20.38	12.88	10.3	14.08
147	31	31	30.6	42.76	19.52	11.96	9.22	14.5
148	32	32	32.48	43.06	20.12	12.8	10.5	16.06
149	33	33	34.02	47.34	18.63	10.74	8.32	15.58
150	34	34	34.6	43.46	22.91	12.22	10.68	14.18
151	35	35	33.61	51.54	20.87	11.78	8.14	14.74
152	36	36	38.82	46.65	22.08	11.1	10.22	15.4
153	37	37	38.41	51.36	21.65	11.28	9.16	15.82
154	38	38	36.83	54.44	23.24	11.86	8.56	14.56
155	39	39	38.27	53.08	25.21	11.7	9.1	13.66
156	40	40	37.03	63.83	22.43	12.12	7.3	15.62
157	15	15	13.51	18.37	12.68	13.34	10.62	10.26
158	16	16	12.73	21.8	13.73	16.7	10.14	10.92
159	17	17	12.92	21.25	17.76	16.94	10.78	8.32
160	18	18	13.71	24.15	16.83	15.44	9.06	8.66
161	19	19	16.28	20.55	20.9	13.86	12.04	7.38
162	20	20	15.98	25.63	18.7	16.5	10.82	9.8
163	21	21	18.34	25.08	19.25	14.82	11.72	10
164	22	22	19.86	23.59	22.75	13.8	12.88	8.4
165	23	23	20.61	26.77	21.15	13.86	11.64	9.6
166	24	24	18.57	30.63	23.65	15.48	9.82	8.32
167	25	25	20.15	30.39	25.02	15.5	10.92	8.58
168	26	26	22.41	28.51	27.74	15.08	12.98	8.38
169	27	27	24.07	33.44	22.76	13.48	10.46	10.28
170	28	28	23.56	34.34	25.99	16.44	12.06	10.46
171	29	29	25.95	32.63	28.15	14.3	12.48	9.28
172	30	30	24.32	34.15	33	16.88	12.94	8.42
173	31	31	28.27	33.6	31.06	13.64	12.72	8.72
174	32	32	26.49	36.19	34.48	15.16	12	7.94
175	33	33	27.97	40.27	30.53	16.82	12.52	10.84
176	34	34	26.16	38.86	40.16	16.92	12.14	7.3
177	35	35	26.53	45.41	34.65	15.18	9.22	7.92
178	36	36	27.41	52.52	29.76	16.96	9.02	11
179	37	37	30.95	44.25	35.98	13.94	10.46	8.34
180	38	38	31.01	53.24	30.22	14.94	9.04	11
181	39	39	29.06	53.77	36.13	16.64	9.22	9.2
182	40	40	34.2	49.11	36.23	16.42	12.26	10.96
183	15	15	15.31	17.7	11.31	11.4	10.98	11.62
184	16	16	16.57	17.79	12.93	12.52	13.16	11.98
185	17	17	19.61	18.41	12.41	11.14	13.72	13.64
186	18	18	20.56	20.1	12.72	10.36	12.16	13.02
187	19	19	18.35	21.58	16.35	12.96	12.24	10.6
188	20	20	17.52	25.07	16.98	13.98	10.48	10.36
189	21	21	22.92	25.51	13.9	10.06	10.14	12.94
190	22	22	23.3	26.1	15.67	10.04	10.04	11.44
191	23	23	20.25	28.78	19.43	13.68	10.34	10.26
192	24	24	24.12	28.4	18.39	12.68	11.96	12.46
193	25	25	26.85	30.17	17.06	10.58	10.54	12.88
194	26	26	28.33	29.3	19.26	12.2	13.4	13.72
195	27	27	28.59	30.33	20.86	10.88	11.6	11.26
196	28	28	26.65	33.71	22.5	12.46	10.8	10.86
197	29	29	33.34	28.93	24.08	10.1	13.64	10.58
198	30	30	33.61	33.38	21.83	10.38	11.96	12.32
199	31	31	31.71	35.79	24.07	13.68	13.56	13.52
200	32	32	32.78	37	24.75	13.4	13.28	13.32
201	33	33	33.79	36.23	27.61	12.52	13.18	11.34
202	34	34	32.1	39.51	29.14	13.74	12.3	11
203	35	35	35.58	44.09	24.12	12.14	10.78	13.7
204	36	36	38.42	41.41	26.57	12.52	13.1	13.8
205	37	37	32.98	45.63	31.4	13.72	10.7	10.38
206	38	38	41.53	42.47	28.38	12.2	13.58	13.64
207	39	39	39.61	46.64	29.01	11.92	11.24	12.28
208	40	40	42.25	43.44	32.64	11.12	12.3	10.76