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	-0.000254	0	-0.567
lnL	0.350195	0	2332.307
lnK	0.400163	0	2400.482
lnM	0.249787	0	1879.048
lnLlnL	-0.020177	0	-798.643
lnKlnK	-0.021133	0	-696.579
lnMlnM	-0.01715	0	-734.574
lnLlnK	0.024132	0	581.253
lnLlnM	0.016212	0	438.875
lnKlnM	0.018097	0	448.062

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	16.44	15.98	12	8.06	9.52	8.34
2	16	16	17.08	15.74	15	7.28	9.16	6.06
3	17	17	16.81	23.49	10.65	7.86	6.06	9.6
4	18	18	21.83	18.72	13.08	6.88	9.42	8.98
5	19	19	17.6	22.03	16.79	8.16	7.16	6.16
6	20	20	20.05	23.04	16	8.8	8.54	8.2
7	21	21	20.18	23.34	18.81	9.2	8.86	7.14
8	22	22	23.68	23.99	17.4	8.28	9.32	8.5
9	23	23	25.25	25.12	17.68	8.52	9.8	9.26
10	24	24	21.75	33.11	16.9	9.7	6.76	9.32
11	25	25	29.58	24.39	20.69	6.64	9.52	7.22
12	26	26	28.75	31.69	16.85	6.26	6.38	8.38
13	27	27	23.41	33.39	23.71	9.38	7.06	6.6
14	28	28	23.6	36.7	23.46	9.62	6.54	6.92
15	29	29	24.22	38.84	23.84	9.18	6.02	6.68
16	30	30	30.21	37.65	21.05	7.12	6.28	7.78
17	31	31	36.69	30.04	25.92	6.68	9.66	7.18
18	32	32	35.89	38.91	20.44	6.64	6.9	9.2
19	33	33	33.5	34.71	29.85	7.48	8.2	6.12
20	34	34	28.06	45.67	28.29	9.84	6.34	6.96
21	35	35	33.88	45.31	24.73	8.08	6.56	8.36
22	36	36	35.5	40.51	30.53	7.34	7.18	6.26
23	37	37	34.42	45.11	30.11	8.42	7	7.04
24	38	38	39.97	49.49	23.89	7.38	6.56	9.66
25	39	39	34.22	51.26	30.77	8.5	6.04	6.88
26	40	40	42.17	44.76	31.27	6.4	6.82	6.5
27	15	15	16.95	13.54	14.93	8.68	12.92	7.2
28	16	16	17.34	14.62	16.54	8.42	11.76	6.36
29	17	17	19.6	14.16	18.81	8.64	14.48	6.48
30	18	18	28.77	13.58	15.25	5.24	14.48	7.9
31	19	19	25.69	13.79	21.36	6.06	14.4	5.38
32	20	20	22.47	19	18.49	8.1	11.28	7.28
33	21	21	23.01	17.44	25.07	8.64	13.68	5.58
34	22	22	29.96	18.95	18.44	5.82	11.4	7.36
35	23	23	27.32	18.22	26.59	6.4	11.78	4.72
36	24	24	31.83	20.3	21.44	5.68	11.02	6.46
37	25	25	26.04	20.15	33.83	7.96	12.3	4.18
38	26	26	29.73	20.4	32.24	6.9	12.28	4.48
39	27	27	33.47	19.71	33.9	5.94	12.66	4.18
40	28	28	27.37	24.83	35.26	8.6	11.02	4.56
41	29	29	37.72	24.14	27.28	7.38	14.26	7.72
42	30	30	36.13	24.92	31.41	7.1	12.56	5.98
43	31	31	31.21	26.39	40.09	8.72	12.14	4.64
44	32	32	36.06	24.44	42.45	8.04	14.52	4.74
45	33	33	45.37	24.65	34.55	6.36	14.9	6.26
46	34	34.01	49.19	22.31	41.64	5.04	14.6	4.38
47	35	35	38.59	30	39.28	8.12	12.48	5.68
48	36	36	46.81	26.67	41.21	6.7	14.84	5.56
49	37	37	51.86	28.78	35.28	5.34	12.2	6
50	38	38	41.73	29.3	51.49	8.46	14.66	4.72
51	39	39	47.28	32.11	41.07	8.28	14.92	6.98
52	40	40	43.9	31.37	52.64	7.8	13.24	4.5
53	15	15	21.74	17.68	7.34	5.16	7.52	13.22
54	16	16	15.84	24.81	8.56	7.92	5.34	11.68
55	17	17	19.25	25.23	8.17	6.4	5.32	12.52
56	18	18	26.47	20.66	8.98	5.36	8.2	13.66
57	19	19	24.09	24.64	9.55	6.74	7.5	14.3
58	20	20	21.67	29.37	10.28	7.28	5.82	12.5
59	21	21	24.83	27.28	11.45	6.86	7.02	12.18
60	22	22	30.78	28.5	9.87	5.26	6.58	14.32
61	23	23	30.42	26.36	13.03	5.96	8.06	11.54
62	24	24	40.16	25.72	11.37	4.04	7.8	12.74
63	25	25	27.64	35.77	12.98	6.8	5.74	11.82
64	26	26	41.23	29.1	12.32	4.46	7.68	13.2
65	27	27	32.1	38.72	12.8	6.96	6.38	14.66
66	28	28.01	42.82	37.6	10.84	4.04	5.38	14.52
67	29	28.99	51.14	29.58	13.88	4.08	8.88	13.52
68	30	30	39.81	35.42	16.24	5.84	7.66	11.98
69	31	31	41.45	37.15	16.29	6.6	8.58	14.14
70	32	32	50.31	33.75	16.63	4.76	8.7	12.5
71	33	33	34.04	52.33	16.29	7.46	5.14	12.68
72	34	34	45.92	38.27	19.27	6.06	8.58	12.02
73	35	35	48.39	46.76	15.28	4.32	5.14	11.98
74	36	36	48.21	40.59	20.57	5.86	8.2	11.4
75	37	37	44.12	43.25	23.15	7.54	8.82	11.5
76	38	38	55.43	44.09	18.85	5.44	8.14	13.82
77	39	39	50.59	50.56	19.08	6.4	7.32	14.4
78	40	40	55.33	46.63	20.98	5.18	7.24	11.58
79	15	15	10.52	19.95	16.06	13.86	7.46	6.02
80	16	16	10.97	23.43	15.31	12.6	5.9	6.08
81	17	17	12.7	25.82	13.62	12.46	6.18	8.2
82	18	18	12.45	29.15	14.7	12.76	5.36	7.5
83	19	19	11.77	35.61	14.84	13.64	4.24	7.42
84	20	20	15.51	25.66	19.48	11.42	7.22	6.24
85	21	21	11.76	36.07	21.62	14.32	4.38	5
86	22	22	15.94	29.51	22.13	13.18	7.3	6.4
87	23	23	17.75	35.45	17.24	12.12	6.14	8.96
88	24	24	18.96	32.26	21.24	11.84	7.24	7.4
89	25	25	18.78	35.96	21.5	13	6.92	7.92
90	26	26	16.43	37.64	28.65	14.76	6.36	5.48
91	27	27	19.87	40.18	22.77	12.98	6.48	7.9
92	28	27.99	17.93	54.53	19.81	13.6	4.2	8.64
93	29	29	19.86	41.07	29.23	14.56	7.08	6.6
94	30	30	19.29	50	26.16	14.06	5.24	7.02
95	31	31	23.34	37.16	35.07	11.98	7.92	5.3
96	32	32	20.72	53.21	27.72	11.04	4.16	5.6
97	33	33	27.07	45.01	27.08	12.16	7.64	8.68
98	34	34	23.59	45.84	36.24	11.74	6.14	5.06
99	35	35	21.2	56.32	35.14	13.7	4.96	5.4
100	36	36	28.13	53.54	27.91	11.68	6.26	8.42
101	37	37	24.86	49.12	42.39	13.22	6.78	5.04
102	38	38	26.52	63.5	29.32	12.76	5.22	8.1
103	39	39	29.35	52.61	36.79	11.58	6.66	6.34
104	40	40	24.02	67.81	37.71	12.52	4.22	5.26
105	15	15	18.2	15.44	11.06	10.04	13.92	12.88
106	16	16	20.11	14.97	13.06	9.72	15.72	11.54
107	17	17	20.41	17.42	12.8	9.54	13.14	11.8
108	18	18	24.57	16.68	13.43	8.5	15.32	12.36
109	19	19	23.09	17.07	17.28	9.56	15.58	9.6
110	20	20	28.01	17.41	15.91	7.96	15.92	11.06
111	21	21	28.16	19.14	16.44	8.62	15.52	11.6
112	22	22	31.85	18.24	18.13	7.62	16.78	10.56
113	23	23	28.55	21.12	19.65	8.26	13.46	9.16
114	24	24	28.77	23.55	19.35	9.64	13.94	10.98
115	25	25	33.69	22.86	19.33	8.04	14.5	11.04
116	26	26	36.79	24.44	18.14	7.3	13.5	11.98
117	27	27	29.19	27.6	23.43	10.68	13.04	9.88
118	28	28	36.42	27.9	19.86	8.34	13.04	12.16
119	29	29	40.33	25.27	23.25	8.58	17	11.72
120	30	30	35.59	26.67	28.68	10.38	16.66	9.56
121	31	31	39.48	27.17	27.58	9.2	16.32	10.02
122	32	32	41.74	28.38	27.08	7.86	14.14	9.34
123	33	33	42.8	28.58	29.22	8.38	15.4	9.38
124	34	34	47.69	32.53	23.34	7.6	13.62	12.58
125	35	35	44.27	30.74	31.33	9.58	16.82	10.28
126	36	36	51.92	29.87	29.77	7.38	16.16	10.14
127	37	37	49.89	35.21	26.9	8.34	14.36	12.32
128	38	38	54.4	35.86	25.97	7.44	13.88	12.68
129	39	39	55.17	36.33	27.59	7.72	14.42	12.46
130	40	40	51.03	36.72	33	9.34	15.72	11.14
131	15	15	15.84	21.49	8.23	10.02	8	15.46
132	16	16	15.47	24	9.22	10.94	7.46	14.36
133	17	17	17.64	24.34	9.54	11.5	9	16.94
134	18	18	18.23	25.38	10.64	11.54	8.94	15.54
135	19	19	19.85	23.99	12.59	11.46	10.48	13.98
136	20	20	24.16	24.93	11.25	9.26	10.2	16.26
137	21	21	24.12	28.05	11.42	9.72	9.3	16.74
138	22	22	22.62	30.39	13.12	11.96	9.66	16.22
139	23	23	26.37	27.27	14.81	9.32	10.24	13.12
140	24	24	24.29	32.52	15	12.48	10.12	15.72
141	25	25	25.13	35.91	14.54	10.28	7.72	13.98
142	26	26	27.18	34.64	15.96	11.2	9.62	14.96
143	27	27	28.19	33.22	18.59	11.38	10.72	13.26
144	28	28	28.03	42.17	15.36	10.5	7.42	15.22
145	29	29	32.26	35.61	18.46	10.16	10.34	14
146	30	30	29.1	41.93	18.97	12.96	9.64	15.32
147	31	31	33.73	37.66	20.61	10.82	10.86	13.8
148	32	32	34.81	39.18	21.05	10.66	10.6	13.76
149	33	33	39.47	41.24	18.72	9.14	9.92	15.7
150	34	34	31.77	49.44	21.37	12.66	8.6	14.42
151	35	35	38.85	47.97	19.17	9.62	8.58	15.78
152	36	36	40.02	44.6	22.64	10	10.06	13.96
153	37	37	36.23	48.33	25.42	12.48	10.16	13.52
154	38	38	37.84	47.51	27.2	12.4	10.84	13.06
155	39	39	43.1	48.86	24.32	10	9.86	14
156	40	40	37.24	64.72	21.86	12.54	7.48	16.76
157	15	15	14.19	16.65	13.75	13.6	12.88	10.08
158	16	16	13.91	20.37	13.4	13.1	9.56	9.78
159	17	17	13.41	20.24	18.12	14.2	10	7.12
160	18	18	15.7	21.82	16.16	14.04	10.9	9.7
161	19	19	15.36	23.6	18.32	14.68	10.12	8.52
162	20	20	15.64	24.3	20.94	16.9	11.5	8.56
163	21	21	17.43	26.93	18.57	16.14	11.06	10.7
164	22	22	18.94	24.64	22.75	13.62	11.42	7.84
165	23	23	18.28	31.15	19.95	16.48	10.06	10.62
166	24	24	18.83	30.59	23.23	16.42	10.6	9.16
167	25	25	20.64	27.43	28.39	15.28	12.5	7.5
168	26	26	24.21	30.8	22.06	13.1	11.28	10.44
169	27	27	21.29	35.06	25.21	16.74	10.64	9.8
170	28	28	23.73	32.86	27.51	13.96	10.88	8.38
171	29	29	26.38	32.98	27.06	13.04	11.46	9.04
172	30	30	25.75	40.67	23.33	13.66	9.12	10.96
173	31	31	26.74	32.84	34.93	13.46	12.08	7.02
174	32	32	24.17	38.45	35.9	16.5	10.92	7.4
175	33	33	26.79	42.95	29.46	16.16	10.6	10.32
176	34	34	26.78	45.93	29.97	15.02	9.1	9.4
177	35	35	29.07	40.52	36.18	14.38	11.12	7.94
178	36	36	30.4	40.27	38.34	13.06	10.72	7.1
179	37	37	31.42	43.14	36.62	16.36	12.88	9.76
180	38	38	27.49	52.41	36.75	16.86	9.02	8.56
181	39	39	36.06	45.14	34.62	14.04	12.32	10.52
182	40	40	32.82	48.68	38.93	16.36	11.76	9.56
183	15	15	15.33	17.22	11.76	10.62	10.58	10.36
184	16	16	16.67	19.73	11.01	11.2	10.5	13.04
185	17	17	17.61	19.87	12.75	12.04	11.94	12.58
186	18	18	17.4	21.64	14.19	11.9	10.52	10.8
187	19	19	17.43	24.52	14.49	13.88	10.62	12.32
188	20	20	19.36	21.92	18.12	13.96	13.78	10.78
189	21	21	21.42	25.48	15.21	12.02	11.2	12.84
190	22	22	21.97	24.86	18.22	13.26	13.1	11.8
191	23	23	20.89	28.56	18.83	13.7	10.84	11.06
192	24	24	25.1	28.31	17.53	10.24	10.16	11.16
193	25	25	24.37	29.58	19.97	11.54	10.5	10.42
194	26	26	26.58	28.78	21.52	13.26	13.8	12.14
195	27	27	26.27	32.87	20.73	13.94	12.24	13.16
196	28	28	30.85	31.78	20.23	10.04	11.08	11.78
197	29	29	29.36	31.03	25.63	12.96	13.88	10.86
198	30	30	27.45	36.39	25.16	13.8	11.32	10.92
199	31	31	35.53	33.01	23.4	10.4	12.96	12.14
200	32	32	32.2	41.01	21.8	12.14	10.44	13.72
201	33	33	34.17	35.94	27.53	11.16	12.02	10.28
202	34	34	32.85	38.41	29.46	13.82	13.14	11.22
203	35	35	32.92	41.65	29.1	13.32	11.54	11
204	36	36	38.74	38.65	29.15	11.78	13.5	11.76
205	37	37	41.25	39.72	28.6	11.5	13.74	12.64
206	38	38	36.51	46.26	29.67	12	10.38	10.94
207	39	39	43.7	45.07	26.84	10.38	11.44	13.16
208	40	40	41.82	42.57	34.12	11.24	12.58	10.2