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	1.092
lnL	0.349991	0	2652.944
lnK	0.400124	0	2774.867
lnM	0.249599	0	2140.216
lnLlnL	-0.020154	0	-797.791
lnKlnK	-0.021166	0	-898.79
lnMlnM	-0.017131	0	-745.175
lnLlnK	0.024166	0	675.084
lnLlnM	0.016185	0	442.742
lnKlnM	0.018146	0	546.99

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	17.69	17.22	9.77	6.14	7.24	8.82
2	16	16	13.82	20.46	13.44	9.08	6.54	6.7
3	17	17	16.88	18.39	15.17	7.8	8.06	6.32
4	18	18	19.06	20	14.13	8.76	9.46	8.9
5	19	19	19.23	22.09	14.81	9.56	9.28	9.28
6	20	20	21.12	25.48	12.91	6.92	6.34	8.82
7	21	21	25.15	22.73	14.61	7.24	9.32	9.8
8	22	22	23.12	26.83	15.21	6.88	6.6	8.04
9	23	23	23.29	29.7	15.38	7.92	6.8	9.22
10	24	24	24.21	27.48	19.23	9.1	8.96	8.52
11	25	25	28.57	27.55	17.99	7.98	9.52	9.82
12	26	26	32.02	26.91	18.67	6.22	8.72	8.38
13	27	27	30.16	31.8	18.15	7.04	7.56	9.14
14	28	28	30.75	33.26	19.01	7.6	7.92	9.56
15	29	29	29.77	33.28	22.61	9.22	9.24	9.1
16	30	30	28.32	39.92	21.07	8.78	6.7	8.88
17	31	31	26.39	40.33	25.9	8.82	6.12	6.44
18	32	32	27.61	41.76	26.12	9.68	6.8	7.38
19	33	33	32.78	40.75	24.12	9.72	8.6	9.96
20	34	34	31.54	43.41	25.94	9.3	7.3	8.36
21	35	35	39.76	36.69	27.36	7.72	9.7	8.56
22	36	36	35.51	42.18	28.72	8.4	7.84	7.7
23	37	37	42.57	40.42	26.74	8.1	9.84	10
24	38	38	47.48	41.59	24.67	6.26	8.36	9.66
25	39	39	43.33	46.44	25.99	7.12	7.5	9.28
26	40	40	47.82	46.63	25.04	6.54	7.7	10
27	15	15	16.3	12.16	18.91	8.3	13.4	4.98
28	16	16	23.09	12.98	13.7	6	13.5	7.92
29	17	17	20.1	14.49	17.49	8.8	14.78	7.4
30	18	18	22.14	15.42	17.4	7.02	12.28	6.66
31	19	19	25.79	13.34	22.52	5.68	14.1	4.76
32	20	20	24.62	16.83	19.89	6.34	11.34	5.82
33	21	21	28.95	14.04	26.57	5.34	14.3	4.22
34	22	22	26.11	17.73	24.73	6.88	12.4	5.24
35	23	23	30.26	16.42	27.63	6.24	14.64	4.96
36	24	24	26.71	22.64	22.72	8.24	11.44	7.12
37	25	25	30.2	22.44	22.96	7.12	11.54	7.02
38	26	26	28.99	24.04	25.36	7.82	11.14	6.54
39	27	26.99	27	21.89	38.41	8.52	12.46	4.02
40	28	28	36.73	22.23	28.17	6.68	13.78	6.52
41	29	29	31.51	23.9	35.53	8.32	13.16	5.16
42	30	30	37.59	23.74	32.27	6.34	12.44	5.42
43	31	31	37.62	26.39	30.85	8.6	14.92	7.76
44	32	32	43.73	25.13	31.05	6.4	14.04	6.84
45	33	33	36.11	26.61	41.56	8.82	14.44	5.34
46	34	34	34.01	29.5	42.98	8.26	11.16	4.48
47	35	35	47.18	25.72	38.7	5.7	13.3	5.14
48	36	36	47.29	28.96	35.35	6.88	14	6.92
49	37	37	46.82	28.64	40.77	5.9	12.02	4.96
50	38	38	56.69	28.89	34.7	5.06	12.78	6.44
51	39	39	42.17	31.07	51.01	7.6	12.44	4.34
52	40	40	52.3	32.64	38.61	5.8	11.54	5.92
53	15	15	17.07	21.57	7.47	6.96	6.04	13.14
54	16	16	21.06	18.92	8.72	6.82	8.84	13.74
55	17	17	20.99	22.96	8.35	6.36	6.54	13.44
56	18	18	27.24	19.33	9.53	4.64	7.94	11.4
57	19	19	26.11	22.21	10	6.44	8.9	14.22
58	20	20	22.45	29.86	9.64	7.76	6.34	14.98
59	21	21	34.37	21.58	10.82	4.24	8.38	11.8
60	22	22	25.83	31.57	10.58	6.76	6.1	13.8
61	23	23	24.12	37.73	10.65	7.82	5.28	14.62
62	24	24	29.99	31.26	12.2	7.22	7.86	14.86
63	25	25	38.31	29.38	11.52	4.7	7.34	13.8
64	26	26	26.62	40.05	13.45	7.64	5.4	12.18
65	27	27	35.2	29.7	16.51	6.32	8.82	11
66	28	28	33.75	37.3	14.41	6.36	6.46	12.36
67	29	29	33.66	42.91	13.59	6.52	5.6	13.54
68	30	30	34.29	43.2	14.83	6.36	5.54	12.18
69	31	31	37.88	41.15	15.77	7.08	7.34	14.18
70	32	32.01	42.68	46.58	13.02	5.14	5.3	14.84
71	33	33	46.31	40.84	15.68	5.6	7.42	14.3
72	34	34	35.92	54.98	15.89	7.94	5.5	14.8
73	35	35	39.1	55.42	15.72	6.7	5.08	13.98
74	36	36	47.42	44.24	18.6	6.58	8.16	14.14
75	37	37	52.46	41.52	19.88	5.9	8.88	13.2
76	38	38	55.3	43.99	18.96	4.88	7.3	12.28
77	39	39	54.09	42.97	22.07	5.5	8.24	11.28
78	40	40	55.44	42.7	23.71	5.74	8.92	11.14
79	15	15	11.56	18.74	15.35	12.08	7.82	6.18
80	16	16	10.83	20.41	19.3	14.32	7.76	5.18
81	17	17	11.15	31.78	12.28	12.54	4.18	8
82	18	18	13.98	26.49	14.29	11.58	6.24	8.06
83	19	19	14.78	26.95	15.87	13.34	7.52	8.76
84	20	20	14.09	26.47	21.42	14.44	7.86	6.3
85	21	21	17.56	27.14	18.16	11.36	7.78	7.8
86	22	22	13.73	34.38	22.02	14.84	5.76	6.08
87	23	23	15.66	33.67	22.26	12.92	6	6.1
88	24	24	16.45	35.48	22.66	12.36	5.72	6.06
89	25	25	15.22	47.43	19.71	13.66	4.1	7.2
90	26	26	17.18	37.93	26.47	12.62	5.68	5.42
91	27	27	19.4	34.62	29.48	13.42	7.76	5.86
92	28	28	19.99	42.01	24.37	13.08	6.24	7.4
93	29	29	22.49	36.72	28.83	11.78	7.56	6.28
94	30	30	22.29	40.44	28.88	13.9	7.88	7.32
95	31	31	20.81	40.48	36.49	14.2	7.42	5.24
96	32	32	20.14	49.05	32.57	12.62	5.06	5.12
97	33	33	18.9	59.35	30.71	14.32	4.26	5.78
98	34	34	19.91	62.8	29.47	13.72	4.06	6.18
99	35	35	24.98	52.7	30.33	14.32	6.8	8.14
100	36	36	23.56	61.03	29.87	11.96	4.46	6.46
101	37	37	26.73	52.82	34.05	13.1	6.72	7.04
102	38	38	23.98	70.36	29.39	13.16	4.24	7.4
103	39	39	30.96	53.29	33.44	12.6	7.6	8.22
104	40	40	32.75	55.26	32.31	11.82	7.3	8.58
105	15	15	20.28	14.69	10.41	8.06	13.46	12.62
106	16	16	18.65	16.13	12.85	10.06	13.64	11.14
107	17	17	23.01	14.53	14.54	7.36	14.44	9.02
108	18	18	23.13	15.93	15.59	9.24	16.38	10.5
109	19	19	23.31	17.58	16.28	9.82	15.64	10.7
110	20	20	24.11	19.11	16.69	10.08	15.14	11.1
111	21	21	23.15	20.57	18.98	10.84	14.24	9.78
112	22	22	27.03	19.79	19.69	10.28	16.96	10.66
113	23	23	30.85	20.33	18.87	7.32	13.66	9.32
114	24	24	30.21	23.57	18.15	9.5	14.54	12.36
115	25	25	34.58	21.09	21.24	7.64	15.62	9.68
116	26	26	32.05	24.09	22.11	9.28	14.84	10.26
117	27	27	39.48	23.63	20.18	7.12	14.88	11.2
118	28	28	39.78	27.05	18.65	7.28	13.1	12.68
119	29	29	38.6	25.67	23.99	7.84	14.48	9.8
120	30	30	37.69	29.04	23.24	8.86	13.76	11.18
121	31	31	40.61	28.25	25	9.24	16.18	11.68
122	32	32	37.31	30.99	27.31	10.24	14.56	10.56
123	33	33	41.83	30.9	26.62	9.84	16.06	11.96
124	34	34	52.64	28.29	25.62	7.14	16.94	11.9
125	35	35	44.4	35.62	24.82	8.94	13.24	12.66
126	36	36	41.94	34.08	31.88	10.2	14.88	10.06
127	37	37	51.97	32.33	29.15	8.04	16.06	11.34
128	38	38	51.78	32.82	31.7	7.42	14.5	9.44
129	39	39	47.88	34.03	36.7	9.44	16.12	9.22
130	40	40	47.93	35.56	37.73	10.1	16.4	9.56
131	15	15	14.95	21.88	8.64	10	7.3	13.62
132	16	16	14.88	23.41	10.03	12.2	8.18	13.86
133	17	17	15.96	23.68	11.27	12.56	9.02	13.5
134	18	18	18.11	27.09	9.82	10.74	7.64	15.76
135	19	19	19.51	24.58	12.43	11.32	9.86	13.74
136	20	20	19.8	28.08	12.24	10.96	8.3	13.78
137	21	21	22.01	26.74	13.69	11	10	13.74
138	22	22	22.79	29.51	13.53	11.86	10	15.64
139	23	23	24.6	30.35	13.9	9.86	8.8	13.78
140	24	24	23.15	33	15.65	11.5	8.66	13.02
141	25	25	23.37	39.68	13.94	12.66	7.76	16.6
142	26	26	26.89	34.17	16.5	11.9	10.26	15.1
143	27	27	27.08	38.31	16.01	11.92	9.06	15.8
144	28	28	29.52	38.73	16.14	11.56	9.6	16.8
145	29	29	28.02	42.48	17.27	11.82	8.28	14.92
146	30	30	32.28	43.94	15.61	9.86	7.84	16.56
147	31	31	32.44	45.34	16.73	10.48	8.08	16.32
148	32	32	32.13	40.55	22.22	12.54	10.9	13.82
149	33	33	36.25	43.16	19.53	9.24	8.6	13.66
150	34	34	38.3	46.03	18.59	9.8	9.02	16.38
151	35	35	37.74	46.03	21.04	10.28	9.3	14.6
152	36	36	37.94	52.22	19.47	9.48	7.44	14.84
153	37	37	38.15	50.13	22.58	11.8	9.78	15.62
154	38	38	36.23	56.95	22.32	11.26	7.56	14.22
155	39	39	42.15	52.4	22.69	10.58	9.36	15.66
156	40	40	40.91	53.96	24.82	12.12	10	15.58
157	15	15	13.56	18.92	12.07	13.26	10.24	10.86
158	16	16	14.93	17.54	15.25	13.5	12.76	9.4
159	17	17	15.09	18.87	17.05	13.38	11.76	8.28
160	18	18	14.81	21.91	17.45	15.46	11.14	9.1
161	19	19	15.87	20.17	22.38	14.9	12.84	7.1
162	20	20	15.32	26.89	18.49	15.26	9	8.74
163	21	21	16.96	26.14	20.21	16.66	11.44	9.68
164	22	22	18.81	23.38	24.99	14.08	12.46	7.2
165	23	23	17.4	27	26.71	16.64	11.34	7.18
166	24	24	19.79	31.46	20.74	16.24	10.76	10.98
167	25	25	19	30.21	27.53	15.16	10.04	7
168	26	26	21.19	31.68	25.5	15.56	11.08	8.94
169	27	27	22.6	29.31	30.57	14.9	12.54	7.46
170	28	28	20.76	35.58	29.58	16.52	10.02	7.78
171	29	29	21.84	39.56	26.88	15.98	9.08	8.94
172	30	30	26.51	34.26	28.97	13.68	11.56	8.8
173	31	31	29.67	34.38	28.01	13.24	12.72	10.12
174	32	32	28.36	37.69	29.34	13.4	10.96	9.2
175	33	33	30.16	37.95	30.07	13.78	12.02	9.88
176	34	34	26.6	41.95	34.73	15.52	10.38	8.1
177	35	35	27.76	41.04	37.95	15.94	11.5	7.88
178	36	36	31.55	45.53	30.12	13.12	9.74	9.9
179	37	37	28.74	43.02	41.93	16.2	11.52	7.42
180	38	38	35.32	41.86	36.13	13.4	12.54	9.32
181	39	39	28.76	52.44	38.09	16.64	9.38	8.54
182	40	40	33.16	47.42	39.96	16.74	12.56	9.6
183	15	15	13.89	17.21	13.46	13.9	12.34	10.3
184	16	16	15.73	19.17	12.39	13.36	12.1	12.64
185	17	17	15.52	21.57	13.38	14	10.86	11.9
186	18	18	17.47	21.2	14.56	12.02	10.94	10.64
187	19	19	20.9	20.01	15.38	10.54	12.68	10.8
188	20	20	19.57	22.65	16.98	13.72	13.2	11.58
189	21	21	19.68	25.07	17.47	12.68	10.9	10.42
190	22	22	21.98	27.59	15.61	12.9	11.28	13.78
191	23	23	23.44	28.06	16.54	12.3	11.38	13.24
192	24	24	25.04	25.14	21.04	11.5	13.08	10.08
193	25	25	25.5	29.2	19.16	13.42	13.08	13.42
194	26	26	25.61	29.35	21.98	13.64	13.28	11.66
195	27	27	30.71	27.59	21.91	10.8	14	11.48
196	28	28	28.25	30.51	24.17	11.86	12.38	10.18
197	29	29	27.74	33.95	24.14	13.98	12.6	11.76
198	30	30	28.49	38.33	22.16	13.94	11.24	13.38
199	31	31	29.77	34.64	27.55	13.24	12.66	10.36
200	32	32	32.16	38.22	24.19	11.9	11.1	11.88
201	33	33	30.95	38.31	28.59	13.7	12.18	10.74
202	34	34	38.93	37.33	24.54	11.26	13.52	13.84
203	35	35	41.19	37.12	25.68	10.28	13.28	12.8
204	36	36	39.81	41.95	24.9	10.98	11.8	13.62
205	37	37	39.71	44.99	25.01	10.58	10.44	13.02
206	38	38	40.02	44.51	27.79	11.78	11.88	12.92
207	39	39	40.95	43.75	30.53	13.22	13.98	13.34
208	40	40	44.48	44.38	29.52	10.7	12.26	12.38