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	0.000526	0	1.419
lnL	0.349844	0	2771.801
lnK	0.400084	0	3062.901
lnM	0.24974	0	2254.922
lnLlnL	-0.020116	0	-845.572
lnKlnK	-0.021153	0	-960.213
lnMlnM	-0.017214	0	-793.585
lnLlnK	0.024105	0	732.848
lnLlnM	0.016224	0	467.705
lnKlnM	0.018205	0	554.351

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	15.32	16.04	13.12	8.94	9.68	7.66
2	16	16	18.8	16.57	12.2	7.54	10	8.96
3	17	17	19.08	20.87	10.7	6.24	6.42	8.8
4	18	18	21.79	19.05	12.78	6.56	8.78	8.78
5	19	19	19.02	22.45	14.67	9.04	8.5	8.76
6	20	20	19.8	25.24	14.24	9.48	8.14	9.98
7	21	21	19	27.17	16.26	8.58	6.44	7.36
8	22	22	20.78	25.36	19.07	7.72	6.98	6.1
9	23	23	21.89	30.49	16.06	9.2	7.12	9.46
10	24	24	23.32	28.12	19.53	8.64	7.92	7.6
11	25	25	27.06	28.65	18.22	6.42	6.86	7.3
12	26	26	30.12	29.75	17.4	6.9	8	9.4
13	27	27	25.99	29.11	25.28	8.24	8.24	6.08
14	28	28	28.13	32.63	21.97	9.4	9.02	8.98
15	29	29	34.15	29.67	22.45	7.28	9.82	8.52
16	30	30	31.95	32.52	24.28	7.38	8.26	7.28
17	31	31	28.85	40.81	22.53	9	6.84	8.6
18	32	32	34.13	38.31	22.29	8.16	8.14	9.62
19	33	33	31.91	42.14	23.82	8.06	6.64	8.12
20	34	34	34.88	37.98	27.63	8.26	8.54	7.76
21	35	35	36.45	39.85	27.1	9.58	9.86	9.7
22	36	36	32.96	39.87	34.68	9.96	9.1	6.7
23	37	37	39.89	38.21	31.72	8.02	9.64	7.5
24	38	38	38.99	41.1	32.44	9.18	9.86	8.14
25	39	39	38.49	48.53	28.44	8.16	7.1	8.32
26	40	40	38.4	51.87	28.5	9.72	7.8	9.86
27	15	15	15.96	12.68	18.14	7.42	11.12	4.56
28	16	16	19.71	13.03	16.79	7.12	13.24	6.14
29	17	17	21.7	12.64	19.84	5.72	12.34	4.54
30	18	18	21.09	15.55	18.34	8.72	14.26	7.34
31	19	19	27.42	14.42	18.17	6.16	15	7.14
32	20	20	28.73	15.97	17.69	6	13.68	7.58
33	21	21	24.72	19.34	19.15	7.92	12.08	7.64
34	22	22	26.38	17.69	24.47	6.56	12	5.12
35	23	23	35.29	17.53	20.18	5.54	14.42	7.64
36	24	24	29.97	20.72	22.46	6.38	11.26	6.4
37	25	25	36.69	16.27	30.48	5.02	14.94	4.46
38	26	26	33.73	18.1	33.3	6.04	14.36	4.38
39	27	27	31.18	23.05	28.6	6.88	11.22	5.44
40	28	28	42.39	21.14	25.38	5.62	14.56	7.34
41	29	29	40.36	22.26	28.54	6.5	14.96	6.98
42	30	30	41.81	24.92	25.89	5.32	11.18	6.68
43	31	31	34.02	29.78	29.07	8.26	11.04	7.1
44	32	32	42.44	25.37	31.8	6.4	13.4	6.42
45	33	33	40.02	27.25	34.56	6.88	12.36	5.84
46	34	34	37.93	29.09	37.65	7.16	11.18	5.16
47	35	34.99	35.46	29.85	44.74	8.76	12.26	4.76
48	36	36	44.03	28.91	39.06	8	15	6.58
49	37	37	46.84	32.59	32.93	7.4	12.96	8
50	38	38	53.57	25.59	46.05	5.18	14.12	4.42
51	39	39	56.07	30.19	36.27	5.92	14.02	7.06
52	40	40	44.8	31.17	51.71	7.16	12.54	4.32
53	15	15	25.17	15.39	7.52	4.18	8.52	12.36
54	16	16	17.65	23.32	8.13	6.7	5.52	11.92
55	17	17	21.27	21.12	9.2	7.46	8.6	14.28
56	18	18	24.78	21.59	9.13	5.88	7.9	13.6
57	19	19	24.13	25.17	9.26	6.64	7.22	14.64
58	20	20	29.78	25.34	8.61	4.14	5.72	12.74
59	21	21	24.32	27.18	11.8	7.5	7.52	12.54
60	22	22	26.17	30.91	10.71	7.3	6.86	14.92
61	23	23	27.83	35.12	9.85	6.12	5.32	14.84
62	24	24	32.68	31.27	11.02	5.5	6.62	14.12
63	25	25	29.48	35.31	12.21	5.54	5.12	11.16
64	26	26	32.07	35.48	12.6	6.86	6.96	14.7
65	27	27	37.83	29.96	14.94	5.68	8.54	12.1
66	28	28	42.94	31.06	13.95	5.26	8.8	14.1
67	29	29	39.71	37.75	13.25	4.98	6.04	12.94
68	30	30	33.08	48.31	13.39	6.94	5.08	14.36
69	31	31	31.8	44.18	17.76	7.96	6.18	11.28
70	32	32	40.34	41.47	16.2	6.96	7.7	14.54
71	33	33	52.02	35.36	16.72	4.2	7.56	11.4
72	34	34	54.04	35.94	17.4	4.7	8.68	12.74
73	35	35	37.49	52.43	17.78	6.96	5.36	11.96
74	36	36	46.23	47.05	17.63	6.54	7.32	14.52
75	37	37	50.9	43.67	19.2	5.86	8.02	13.18
76	38	38	49.5	44.73	21.12	6.96	8.96	13.54
77	39	39	57.23	44.79	19.49	5.26	8.02	13.34
78	40	40	46.59	51.73	22.35	7.62	7.7	12.92
79	15	15	11.22	18.54	16.33	12.1	7.66	5.56
80	16	16	11.53	19.98	18.14	12.6	7.54	5.28
81	17	17	13.5	22.94	14.83	11.72	7.18	7.5
82	18	18	14.71	24.47	14.92	11.84	7.44	8.32
83	19	19	13.85	30.61	14.5	12.86	5.78	8.7
84	20	20	14.19	27.66	19.82	13.82	7.2	6.66
85	21	21	16.37	31.88	15.87	11.12	5.8	8.24
86	22	22	15.23	34.88	18.49	13.22	5.7	7.52
87	23	23	16.68	29.53	24.71	13.6	7.94	6.12
88	24	24	17.37	36.19	20.33	11.08	5.34	6.58
89	25	25	17.55	39.19	20.93	14.36	6.38	8.34
90	26	26	16.68	43.76	22.43	14.04	5.16	7.08
91	27	27	22.78	35.96	22.06	11.92	7.96	8.84
92	28	28	20.46	37.19	28.26	12.16	6.88	5.94
93	29	29	21.33	44.7	23.28	13.5	6.46	8.7
94	30	30	18.26	57.21	23.49	13.48	4.02	7.16
95	31	31	18.77	56.37	26.13	13.02	4.08	6.3
96	32	32	23.09	54.54	22.91	12.12	5.04	8.74
97	33	33	22.44	53.03	27.94	12.9	5.36	7.12
98	34	34	21.35	54.15	32.81	11.98	4.58	5.16
99	35	35	23.72	58.77	27.98	13.64	5.36	8.02
100	36	36	26.02	53.76	31	14.18	6.9	8.24
101	37	37	24.89	54.14	36.49	14.54	6.66	6.62
102	38	38	25.81	56.64	35.94	12.44	5.64	6.02
103	39	39.01	22.93	61.29	42.46	14.8	5.32	5.12
104	40	40	29.6	56.49	36.13	14.34	7.66	8.1
105	15	15	19.34	12.58	14.1	8.8	16.68	9.12
106	16	16	18.02	15.95	13.67	10.9	14.38	10.78
107	17	17	21.05	16.14	13.83	9.26	14.46	10.84
108	18	18	25.25	16.9	12.71	7.18	13.16	11.5
109	19	19	24.54	17.96	14.73	9.08	14.98	11.82
110	20	20	23.62	19.97	16	10.8	15.04	12.2
111	21	21	23.09	20.48	19.18	10.42	13.72	9.26
112	22	22	24.67	22.9	17.69	10.66	13.3	11.26
113	23	23	31.38	18.68	21.15	8.08	17	9.18
114	24	24	30.53	21.74	20.29	9.52	16.22	11
115	25	25	30.17	25.58	18.74	9.9	13.74	12.38
116	26	26	39.04	21.58	20.43	7.24	16.62	11.08
117	27	27	36.22	24.97	20.64	8.28	14.66	11.46
118	28	28	40.97	22.68	23.65	7.26	16.64	9.9
119	29	29	43.8	24.93	21.42	7.42	16.46	12.3
120	30	30	41.25	28.19	21.7	8.52	15.24	12.96
121	31	31	43.2	25.57	27.06	7.46	15.8	9.24
122	32	32	39.07	29.08	28.42	9.26	14.98	9.62
123	33	33	42.45	32.86	23.77	8.96	13.84	12.66
124	34	34	45.1	28.78	30.33	8.46	16.4	9.64
125	35	35	38.47	37.15	28.04	11	13.1	11.4
126	36	36	43.63	34.55	29.61	10.88	16.36	12.26
127	37	37	44.36	34.58	32.2	9.7	14.86	10.1
128	38	38	55.2	33.85	27.86	7.04	14.3	11.24
129	39	39	59.15	33.48	28.71	7.14	15.94	11.94
130	40	40	54.86	38.91	27.54	7.84	13.42	12.6
131	15	15	14.69	23.07	8.22	10.86	7.3	15.36
132	16	16	17.3	20.1	10.22	11.32	10.84	15.02
133	17	17	16.43	24.63	10.27	11.86	8.42	14.72
134	18	18	17.97	25.73	10.63	10.12	7.58	13.4
135	19	19	17.65	28.92	11.31	12.42	7.94	14.98
136	20	20	23.04	25.5	11.56	9.8	9.94	15.76
137	21	21	23.98	28.31	11.35	9.7	9.12	16.7
138	22	22	20.25	32.08	14.04	12.54	8.34	13.78
139	23	23	20.98	35.97	13.5	12.8	7.76	15.36
140	24	24	24.22	30.47	16.51	12.06	10.52	13.52
141	25	25	29.08	30.23	15.38	9.58	10.46	14.48
142	26	26	27.16	33.24	16.93	11.54	10.4	14.38
143	27	27	24.94	43.85	14.84	12.92	7.6	17
144	28	28	27.99	41.88	15.53	10.54	7.5	15.06
145	29	29	30.02	42.47	15.83	10.5	7.98	15.92
146	30	30	28.29	46.83	16.92	12.54	7.92	16.4
147	31	31	29.56	47.79	17.53	11.64	7.56	15.38
148	32	32	35.32	40.06	20.01	11.12	10.96	15.5
149	33	33	32.45	44.02	21.89	13	10.38	14.76
150	34	34	36.99	43.16	21.23	10.68	10.18	14.66
151	35	35	32.9	56.88	18.79	12.2	7.32	16.84
152	36	36	36.84	51.53	20.57	11.74	9.04	16.64
153	37	37	39.57	47.53	23.22	10.68	9.84	14.28
154	38	38	40.19	47.21	25.37	11.44	10.82	14.04
155	39	39	45.24	51.81	21.11	9.34	9.1	16.36
156	40	40	40.34	58.15	22.8	9.66	7.18	13.5
157	15	15	12.98	18	13.84	13.68	10.64	9.06
158	16	16	13.04	19.07	16.24	17	12.42	9.38
159	17	17	14.62	21.91	14.19	14.14	10.04	10.46
160	18	18	14.7	23.87	15.5	15.94	10.3	10.7
161	19	19	15.72	25.62	15.67	14.7	9.46	10.54
162	20	20	15.83	24.63	20.15	14.84	10.08	7.98
163	21	21	17.1	23.41	23.72	15.68	12.38	7.62
164	22	22	18.87	26.59	20.31	14.96	11.42	9.8
165	23	23	20.63	27.19	20.63	13.78	11.38	9.84
166	24	24	19.98	26.36	26.87	14.84	12.24	7.48
167	25	25	23.37	29.18	21.59	13.52	11.9	10.6
168	26	26	21.24	33.76	23.08	14.94	9.9	9.68
169	27	27	22.85	29.88	29.16	15	12.5	8.04
170	28	28	21.63	35.65	27.75	15.28	9.7	8.14
171	29	29	22.35	36.85	28.94	15.82	10.04	8.34
172	30	30	23.52	36.92	30.67	13.62	9.16	7.12
173	31	31	24.29	36.85	33.48	14.66	10.26	7.18
174	32	32	27.46	37.79	30.59	15.54	12.2	9.78
175	33	33	31.62	37.33	28.88	13.06	12.28	10.38
176	34	34	31.12	38.84	31.23	14.76	13	10.5
177	35	35	30.09	43.9	30.46	14.52	10.64	10.22
178	36	36	27.5	44.43	38.07	15.48	10.06	7.54
179	37	37	28.85	45.47	38.22	16.92	11.32	8.68
180	38	38	29.64	43.76	43.46	16.08	11.62	7.32
181	39	39	35.04	45.63	35.45	13.68	11.46	9.64
182	40	40	28.81	54.61	39.53	16.86	9.08	8.3
183	15	15	16.91	16.8	10.73	10.68	12.3	13.02
184	16	16	16.27	20.12	11.05	11.84	10.54	13.34
185	17	17	17.91	18.31	14.09	10.94	12.18	10.36
186	18	18	20.2	19.13	14.01	10.34	12.6	11.36
187	19	19	18.1	22.65	15.49	12.12	10.64	10.4
188	20	20	20.11	25.56	13.7	12.44	10.72	13.96
189	21	21	20.78	24.17	17.13	13.36	12.78	11.98
190	22	22	24.63	24.35	16.16	10.76	12.46	12.62
191	23	23	23.73	28.35	16.04	11.28	10.46	12.78
192	24	24	26.37	28.1	16.62	10.56	11.2	12.98
193	25	25	25.51	27.72	20.7	12.72	13.18	11.62
194	26	26	29.12	26.82	21.23	10.5	13.22	10.88
195	27	27	29.74	27.54	22.94	10.44	13.06	10.12
196	28	28	29.57	30.56	22.68	11.82	13	11.54
197	29	29	28.09	32.99	24.8	13.3	12.58	11
198	30	30	30.21	35.39	23.03	12.84	12.18	12.62
199	31	31	34.17	35.32	22.25	10.9	11.98	12.9
200	32	32	29.7	39.21	25.93	12.96	10.68	10.86
201	33	33	34.86	36.84	25.82	10.76	11.52	10.94
202	34	34	36.62	35.96	28.14	10.74	12.54	10.46
203	35	35	38.69	36.89	28.13	11.2	13.54	11.64
204	36	36	39.79	38.51	28.29	11.08	13.16	11.82
205	37	37	38.82	40.8	29.76	10.26	11.06	10.02
206	38	38	36.58	44.94	30.89	13.98	12.54	12.18
207	39	39	39.04	46.16	30.04	12.36	11.6	12.02
208	40	40	43.98	40.54	34.4	10.78	13.56	10.28