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	-8.4E-5	0	-0.194
lnL	0.350176	0	2474.863
lnK	0.400186	0	2689.401
lnM	0.249665	0	2040.461
lnLlnL	-0.020238	0	-814.591
lnKlnK	-0.021125	0	-866.628
lnMlnM	-0.017156	0	-762.19
lnLlnK	0.024157	0	608.268
lnLlnM	0.016317	0	430.692
lnKlnM	0.018047	0	496.376

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	17.08	15.83	11.57	8	10	9.04
2	16	16	17.4	18.3	11.62	8.3	8.94	9.54
3	17	17	17.16	19.25	13.84	9.24	9.22	8.5
4	18	18	16.22	22.82	14.43	8.42	6.44	6.9
5	19	19	16.92	24.18	15.4	9.8	7.36	7.82
6	20	20	17.53	24.24	17.84	9.66	7.54	6.76
7	21	21	20.03	25	17.12	9.2	8.1	7.9
8	22	22	23.21	23.96	17.91	7.74	8.52	7.5
9	23	23	21.35	27.75	19.07	8.46	7.1	6.9
10	24	24	21.87	33.51	16.49	9.02	6.24	8.98
11	25	25	26.54	24.9	23.17	8.02	9.88	6.72
12	26	26	26.83	31.35	18.72	7.88	7.5	8.6
13	27	27	26.92	35.51	17.94	8.22	6.78	9.46
14	28	28	24.02	39.9	20.28	9.98	6.28	8.7
15	29	29	27.57	32.22	26.36	9.88	9.4	7.44
16	30	30	37.65	31.15	20.93	6.02	8.6	8.58
17	31	31	27.51	43.93	21.62	9.68	6.38	9.18
18	32	32	37.74	34.12	23.23	7.04	9.06	8.9
19	33	33	36.33	37.93	23.42	7.2	7.82	8.62
20	34	34	39.72	35.72	25.54	7.26	9.4	8.72
21	35	35	33.44	42.7	27.44	9.94	8.52	8.96
22	36	36	37.15	39.76	29.54	8.68	9.16	8.12
23	37	37	38.17	47.44	24.39	6.96	6.16	8.42
24	38	38	40.15	39.16	33.59	7.24	8.52	6.38
25	39	39	43.31	43.3	28.81	7.54	8.62	8.7
26	40	40	42.03	45.5	30.64	7.82	8.14	8.1
27	15	15	19.41	10.16	20.24	6	14.68	4.08
28	16	16	23.33	13.02	13.47	5.02	11.4	6.84
29	17	17	22.28	13.57	16.98	6.12	12.54	6.02
30	18	18	24.68	12.71	20.85	5.74	14.32	5
31	19	19	23.36	15.06	20.92	6.1	11.68	4.96
32	20	20	19.7	17.19	26.29	8.26	11.08	4.2
33	21	21	23.91	17.75	23.07	7.68	12.46	5.72
34	22	22	27.39	17.65	23.34	5.82	11.16	5.02
35	23	23	24.93	18.45	29.62	8.92	14.52	5.2
36	24	24	31.16	18.43	25.89	6.82	14.46	6.06
37	25	25	27.54	19.7	32.45	7.2	12.2	4.24
38	26	26	38.82	19.31	24.65	5.58	14.5	6.8
39	27	27	39.6	19.49	27.5	5.16	13.58	5.66
40	28	27.99	28.94	21.95	40.26	8.38	13.26	4.06
41	29	29	33.42	23.89	32.72	7.32	12.42	5.36
42	30	30	34.88	27.16	28.61	8.54	13.1	7.7
43	31	31	37.83	24.69	34.2	6.24	11.78	5.02
44	32	32	35.99	26.59	36.82	8.86	14.46	6.16
45	33	33	36.81	28.54	35.91	8.16	12.58	6
46	34	34	38.95	27.03	41.11	7.16	12.58	4.8
47	35	35	45.12	25.48	41.8	5.86	13.12	4.58
48	36	36	57.39	27.3	30.35	5.2	14.24	7.86
49	37	36.99	38.81	28.45	53.88	8.32	13.7	4.04
50	38	38	42.96	30.23	46.76	6.62	11.44	4.28
51	39	39	42.91	30.29	52.03	7.34	12.64	4.18
52	40	40	54	29.44	44.01	6.36	14.84	5.78
53	15	15	16.61	21.57	7.73	6.76	5.68	11.88
54	16	16	21.1	19.23	8.51	7	8.92	14.54
55	17	17	20.24	22.68	8.86	5.88	5.88	11.1
56	18	18	23.78	23.11	8.73	6.38	7.54	14.82
57	19	19	25.11	22.19	10.49	6.66	8.8	13.28
58	20	20	22.55	26.64	11.17	7.94	7.46	12.96
59	21	21	28.27	25.58	10.7	5.18	6.66	11.6
60	22	22	22.75	33.71	11.33	7.2	5.18	11.68
61	23	23	33.85	27.5	10.84	5.18	7.56	14.12
62	24	24	32.57	29.67	11.87	4.86	6.2	11.38
63	25	25	30.67	33.54	12.48	6.24	6.42	12.84
64	26	26	37.57	33.79	11.19	4.48	5.8	13.3
65	27	27	31.5	34.9	15.07	7.56	7.66	12.86
66	28	28	42.5	30.78	14.3	5.34	8.92	13.74
67	29	29	38.94	41.62	11.92	4.94	5.22	14.2
68	30	30	42.33	38.88	13.32	4.26	5.38	11.86
69	31	31	46.26	35.15	15.47	4.7	7.42	12.18
70	32	32	42	37.38	17.82	6.24	8.18	12.22
71	33	33	41.38	49.04	14.04	5.62	5.26	14.32
72	34	34	36.45	53.05	16.35	7.62	5.6	13.98
73	35	35	41.18	47.36	18.16	6.54	6.34	12.24
74	36	36	46.79	47.82	17.01	5.44	6.06	12.78
75	37	37	54.97	41.31	18.95	4.84	7.74	12.1
76	38	38	46.96	49.44	19.6	6.6	7.1	13.18
77	39	39	53.6	51.5	17.41	5.16	6.18	13.84
78	40	40	63.43	44.88	18.89	4.1	7.04	12.18
79	15	15	9.74	23.51	14.09	13.96	5.66	6.46
80	16	16	10.87	27.16	12.54	14.44	5.62	8.72
81	17	17	12.52	27.29	12.86	12.08	5.52	8.36
82	18	18	14.35	24.68	15.25	12.78	7.72	8.5
83	19	19	11.99	29.99	18.43	13.74	5.34	5.92
84	20	20	13.34	30.44	18.83	12.98	5.62	6.18
85	21	21	14.99	31.57	18.23	11.14	5.3	6.32
86	22	22	15.87	32.25	19.53	12.94	6.42	7.24
87	23	23	16.83	31.47	22.12	12.9	7.06	6.68
88	24	24	18.38	31.7	22.79	11.86	7.14	6.58
89	25	25	16.01	38.04	25.05	13.42	5.54	5.66
90	26	26	15.88	44.32	23.7	11.88	4.06	5.3
91	27	27	16.72	41.92	27.64	12.8	4.96	5.06
92	28	28	22.5	38.88	23.04	12.72	7.64	8.84
93	29	29	19.39	50.54	22.44	14.1	5.22	8.48
94	30	30	18.47	51.76	26.56	13.82	4.7	6.44
95	31	31	20.23	50.56	27.29	13.06	5.08	6.56
96	32	32	20.49	43.89	37.56	14.28	6.66	5
97	33	33	24.79	42.16	33.92	11.54	7.06	5.7
98	34	34	21.31	50.98	35.98	14.26	5.84	5.5
99	35	35	22.94	53.77	33.38	12.54	5.26	5.76
100	36	35.99	24.15	66.41	25.64	13.46	4.68	8.96
101	37	37	23.48	61.65	32.92	11.88	4.36	5.7
102	38	38	22.64	62.23	38.12	13.8	4.8	5.34
103	39	39	26.89	66	30.08	13.64	5.42	8.54
104	40	40	24.28	72.79	33.57	14.84	4.66	7.24
105	15	15	18.17	14.72	11.92	9.26	13.56	10.86
106	16	16	19.96	16.05	11.85	9.48	14	12.5
107	17	17	22.14	16.1	13.01	8	13.3	10.68
108	18	18	21.06	16.69	16.38	10.1	15.18	9.7
109	19	19	26.27	17.51	14.06	8.44	15.54	12.58
110	20	20	25.75	17.91	16.95	7.98	13.98	9.32
111	21	21	30.6	19.34	14.58	7.08	13.88	12.1
112	22	22	29.97	18.92	18.48	8.18	16.06	10.32
113	23	23	25.85	23.59	18.86	10.92	13.9	11.3
114	24	24	34.51	21.72	17.39	8	15.76	12.8
115	25	25	34.12	23.78	17.91	8.08	14.12	12.32
116	26	26	38.8	21.18	21.21	7.1	16.54	10.32
117	27	27	34.87	24.26	22.68	8.22	14.4	9.74
118	28	28	36.08	24.2	25.09	8.62	15.76	9.44
119	29	29	39.43	25.63	23.4	7.94	15.06	10.48
120	30	30	34.77	29.69	24.95	10.1	13.9	10.66
121	31	31	34.12	30.09	28.48	10.28	13.62	9.08
122	32	32	49.69	28.14	22.06	7	15.62	13
123	33	33	39	31.98	27.63	10.06	14.52	10.78
124	34	34	43.27	33.4	25.33	8.44	13.08	11.32
125	35	35	43.76	31.94	29.92	8.88	14.7	9.92
126	36	36	45.44	33.78	29.1	9.48	15.36	11.44
127	37	37	52.63	31.03	30.62	7.52	16	10.16
128	38	38	49.94	34.29	30.99	8.48	15.08	10.62
129	39	39	54.26	33.13	32.54	7.12	14.54	9.28
130	40	40	52.64	34.36	35.22	8.22	15.52	9.42
131	15	15	14.22	22.33	8.96	12.5	8.4	15.36
132	16	16	16.9	23.21	8.63	9.66	7.6	15.22
133	17	17	20.16	21.32	9.69	9.96	10.66	16.84
134	18	18	17.6	26.48	10.5	11.12	7.86	14.58
135	19	19	17.91	27.3	12.01	12.3	8.56	14.06
136	20	20	21.48	28.37	10.89	10	8.24	15.88
137	21	21	20.05	29.8	13.27	12.72	9.12	14.76
138	22	22	20.77	31.56	13.9	12.74	8.9	14.6
139	23	23	25.04	29.35	14.24	11.16	10.58	15.48
140	24	24	22.96	33.62	15.4	12.36	9.02	14.12
141	25	25	24.6	32.57	17.13	12.22	10.04	13.36
142	26	26	25.35	37.79	15.48	12.82	9.16	16.36
143	27	27	28.08	35.47	17.02	10.3	8.94	13.26
144	28	28	29.05	38.47	16.63	10.08	8.28	13.88
145	29	29	30.15	40.66	16.7	10.3	8.28	14.74
146	30	30	32.57	42.69	16.05	9.84	8.18	16.16
147	31	31	26.64	48.68	19.57	12.84	7.22	13.18
148	32	32	31.43	50.11	17.07	11.06	7.3	16.2
149	33	33	38.94	40.85	19.29	9.2	9.94	15.02
150	34	34	40.1	38.86	22.27	9.14	10.84	13.04
151	35	35	31.08	52.95	22.27	12.94	7.9	13.68
152	36	36	35.96	46.7	24.34	12.9	10.84	14.58
153	37	37	37.7	50.9	22.45	11.06	8.88	14.54
154	38	38	38.92	53.18	22.37	9.6	7.6	13.16
155	39	39	41.13	53.53	22.72	11.62	9.74	16.68
156	40	40	40.84	60.11	21.47	10.92	7.92	16.62
157	15	15	11.85	19.75	13.68	15.54	9.74	9.38
158	16	16	12.76	18.9	16.99	16.86	12.14	8.6
159	17	17	14.4	19.83	16.88	15.12	11.86	8.96
160	18	18	13.16	24.48	17.51	16.82	9.26	8.58
161	19	19	17.61	22.02	16.77	13.56	11.92	10.26
162	20	20	15.44	25.47	19.83	14.28	9.06	7.6
163	21	21	18.22	25.06	19.45	16.42	12.9	10.86
164	22	22	17.05	27.88	21.87	14.64	9.38	7.8
165	23	23	19.28	25.67	24.84	14.9	12.16	7.9
166	24	24	19.38	30.29	22.62	16.9	11.42	10.06
167	25	25	19.15	30.95	26.2	16.56	10.76	8.18
168	26	26	20.65	31.71	26.45	14.46	9.98	7.72
169	27	27	24.2	31.58	24.63	13.26	11.08	9.28
170	28	28	24.06	32.68	27.21	16.28	12.98	10.06
171	29	29	24	37.14	25.8	14.48	9.9	9.5
172	30	30	23.4	34.52	34.35	16.98	12.28	7.72
173	31	31	27.92	36.59	27.7	13.76	11.44	9.92
174	32	32	25.55	43.03	27.86	16.06	9.94	10.36
175	33	33	28.99	35.21	35.78	13.2	12	7.36
176	34	34	29.29	44.75	27.46	13.08	9.08	10.08
177	35	35	27.24	40	40.64	16.36	11.9	7.3
178	36	36	25.64	47.48	38.14	16.62	9.2	7.44
179	37	37	29.25	48.59	33.85	16.76	10.54	10.08
180	38	38	32.15	49.06	32.38	15.16	10.54	10.74
181	39	39	32.57	46.82	37.79	13.84	10.32	8.3
182	40	40	37.33	44.31	37.5	13.68	12.78	9.72
183	15	15	14.9	18.08	11.36	11.84	10.78	11.64
184	16	16	15.2	18.24	14	13.06	12.04	10.28
185	17	17	18.92	19.03	12.37	10.78	12.24	12.7
186	18	18	19.34	21.08	12.83	10.86	11.22	12.58
187	19	19	19.69	21.78	14.65	10.5	10.66	10.62
188	20	20	19.99	21.7	17.61	13.22	13.72	10.96
189	21	21	22.25	23.26	16.56	10.66	11.56	10.78
190	22	22	20.87	26.13	18.13	12.2	10.7	10.28
191	23	23	21.77	25.44	21.2	13.6	12.94	10.02
192	24	24	26.9	24.81	19.51	10.8	13.58	11.26
193	25	25	25.73	29.75	18.41	11.96	11.52	12.66
194	26	26	27.64	31.51	17.87	10.58	10.36	12.62
195	27	27	24.47	35.42	20.45	13.9	10.28	12.26
196	28	28	29.74	30.08	23.05	10.5	11.84	10.12
197	29	29	30.13	33.7	21.84	12.6	12.62	13.14
198	30	30	29.47	38.79	20.82	12.5	10.34	13.44
199	31	31	35.11	31.15	25.96	10.46	13.76	10.66
200	32	32	36.23	35.61	22.97	10.94	12.76	13.36
201	33	33	37.47	33.45	27.17	10.4	13.58	10.84
202	34	34	34.37	40.14	25.95	13.78	13.12	13.7
203	35	35	32.54	46.34	25.27	13.66	10.3	13.14
204	36	36	35.94	45.59	25.21	11.62	10.04	12.6
205	37	37	34.21	44.18	31.33	12.98	10.98	10.28
206	38	38	33.75	46.37	32.93	13.68	10.76	10.06
207	39	39	42.81	41.09	31.66	11.62	13.94	11.84
208	40	40	39.17	46.29	32.82	13.76	12.92	12.1