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	-0.000424	0	-1.078
lnL	0.35009	0	2587.627
lnK	0.400517	0	2874.841
lnM	0.249638	0	2098.145
lnLlnL	-0.020196	0	-799.486
lnKlnK	-0.021164	0	-834.625
lnMlnM	-0.017152	0	-736.416
lnLlnK	0.024122	0	657.967
lnLlnM	0.016291	0	437.08
lnKlnM	0.018064	0	539.149

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	14.45	17.35	12.59	7.64	7.04	6.42
2	16	16	18.72	16.38	12.47	7.36	9.84	8.48
3	17	17	18.58	18.76	12.94	8.74	9.88	9.56
4	18	18	16.04	24.76	13.02	9.42	6.46	8.6
5	19	19	17.28	22.9	16.23	9.36	7.68	7.2
6	20	20	22.83	24.23	12.56	6.42	6.84	9.26
7	21	21	19.17	25.01	18.16	9.14	7.64	6.96
8	22	22	25.42	21.24	19.1	7.08	10	7.08
9	23	23	24.11	26.65	17.19	7.58	7.7	8.06
10	24	24	25.16	27.36	18.38	8.92	9.24	9.22
11	25	25	21.97	29.46	23.19	9.64	7.8	6.46
12	26	26	24.45	32.72	19.9	9.74	7.9	8.86
13	27	27	25.92	31.99	21.97	7.24	6.46	6.28
14	28	28	30.62	28.78	23.73	7.24	8.9	6.98
15	29	29	32.36	31.37	22.13	7.34	8.7	8.2
16	30	30	32.81	34.24	21.7	8.28	9	9.62
17	31	31	28.62	36.23	27.17	9.56	8.28	7.26
18	32	32	35.1	32.39	27.65	6.64	8.34	6.28
19	33	33	31.85	45.28	21.55	8.26	6.24	9.34
20	34	34	32.82	42.38	25.47	8.58	7.26	8.26
21	35	35	33.86	42.48	27.19	7.7	6.74	7.12
22	36	36	35.16	42.83	28.45	7.66	6.94	7.02
23	37	37	37.53	48.12	24.43	8.04	6.86	9.52
24	38	38	33.96	45.7	33.37	8.86	7.14	6.46
25	39	39	41.39	42.33	31.64	7.96	8.86	7.8
26	40	40	32.37	51.93	36.01	9.52	6.24	6
27	15	15	18.39	12.31	15.65	7.2	13.2	6.22
28	16	16	22.74	13.17	13.66	5.9	12.82	7.68
29	17	17	20.91	14.94	15.77	7.98	13.54	7.94
30	18	18	23.09	13.22	21.32	6.4	14.1	5.02
31	19	19	23.89	15.3	19.77	5.76	11.12	5.14
32	20	20	28.48	14.5	21.02	5.76	14.56	5.88
33	21	21	25.56	14.61	29.45	6.78	14.96	4.1
34	22	22	25.82	18.08	24.3	7.04	12.24	5.4
35	23	23	30.81	16.79	25.93	5.28	12.32	4.62
36	24	24	26.75	19.51	29.02	8.04	13.32	5.22
37	25	25	29.41	22.66	23.41	8.5	13.2	7.94
38	26	26	33.93	22.05	23.62	7.04	13.36	7.7
39	27	27	38.64	21.26	24.55	5.78	13.34	7.04
40	28	27.99	27.64	22.7	40.63	8.9	12.82	4.04
41	29	29	36.45	23.77	29.29	7.64	14.44	7.06
42	30	30	37.73	26.63	26.58	7.12	12.26	7.68
43	31	31	34.8	26.52	34.03	7.5	11.8	5.5
44	32	32	39.88	27.44	30.35	8.06	14.3	7.94
45	33	33	34.61	28.31	39.72	8	11.58	4.86
46	34	34	40.58	26.99	38.9	7.54	13.92	5.66
47	35	35	51.32	26.16	33.62	5.6	14.14	6.58
48	36	36	37.69	32.29	40.3	8.36	11.46	5.52
49	37	37	48.34	25.87	46.56	5.52	13.16	4.12
50	38	38	48.22	30.92	38.36	5.84	11.26	5.46
51	39	39	48.93	34.89	34.25	6.9	11.74	7.5
52	40	40	59	29.57	38.84	5	12.88	5.84
53	15	15	18.06	21.13	7.17	5.66	5.38	11.98
54	16	16	21.29	18.88	8.64	5.88	7.74	12.14
55	17	17	22.51	19.57	9.56	6.62	8.92	12.94
56	18	18	27.41	19.58	9.3	4.84	8.22	12.34
57	19	19	25.84	22.55	9.91	6.44	8.64	14.2
58	20	20	28.05	25.18	9.3	5.04	6.54	13.2
59	21	21	31.32	24.44	10.12	4.98	7.62	13.44
60	22	22	33.86	26.44	9.79	4.16	6.36	12.8
61	23	23	24.08	34.22	12.12	6.88	5.2	11.02
62	24	24	38.94	25.58	11.86	4.28	8.02	12.38
63	25	25	30.68	30.81	14.01	7.14	8.12	12.82
64	26	26	27.55	41.51	12.31	7.88	5.56	14.52
65	27	27	30.34	40.69	12.83	7.02	5.68	13.8
66	28	28	31.38	40.6	14.05	7.32	6.18	13.46
67	29	29	34.83	34.53	17.52	7.26	8.38	11.64
68	30	30	39.49	42.37	12.88	5	5.26	13.34
69	31	31	48.05	34.61	15.13	4.58	7.7	12.74
70	32	32	39.68	43.4	15.54	6.34	6.52	13.64
71	33	33	38.5	49.23	15.2	6.66	5.7	14.2
72	34	34	45.79	44.88	15.54	5.64	6.6	14.36
73	35	35	43.07	46.93	17.42	6.68	6.9	13.84
74	36	36	41.58	47.06	20.03	6.58	6.5	11.1
75	37	37	47.56	49.48	17.55	5.72	6.24	13.2
76	38	38	40.99	50.59	22.44	7.98	7.12	11.58
77	39	39	50.45	48.55	20.23	6.84	8.18	14.32
78	40	40	46.54	58.02	19.18	5.92	5.22	12
79	15	15	9.33	28.21	11.6	13.96	4.34	7.72
80	16	16	12.71	22.9	12.81	11.76	6.72	8.32
81	17	17	12	22.88	17.69	14.58	7.8	6.6
82	18	18	11.53	35.52	12.51	13.68	4.16	8.88
83	19	19	13.66	27.91	16.89	14.52	7.16	8.08
84	20	20	14.54	33.25	14.67	12.6	5.44	8.9
85	21	21	13.9	33.38	18.78	13.88	5.66	6.96
86	22	22	15.91	32.83	18.94	13.54	6.6	7.88
87	23	23	16.48	36.42	18.43	13.7	6.16	8.58
88	24	24	19.72	31.7	20.6	11.32	7.4	7.68
89	25	25	18.96	33.84	23.2	13.56	7.84	7.64
90	26	26	15.87	41.54	26.06	14.6	5.38	5.82
91	27	27	19.57	37.47	25.81	13.38	7.12	6.9
92	28	28	16.66	43.97	29.96	14.8	5.4	5.3
93	29	29	21.5	41.02	26.06	12.54	6.7	7.14
94	30	30	22.33	42.51	26.72	12.24	6.56	7.08
95	31	31	20.3	53.51	25.06	12.7	4.64	7.08
96	32	32	23.08	45.81	29.41	11.92	6.08	6.4
97	33	33	25.01	51.03	25.27	11.42	5.64	8.06
98	34	34	19.08	58.88	34.47	14.82	4.5	5.28
99	35	35	24.95	48.19	34.68	11.92	6.28	5.78
100	36	36	21.34	63.01	33.06	14.76	4.72	6.3
101	37	37	25.12	54.4	35.72	13.68	6.3	6.46
102	38	38	24.59	60.79	34.83	13.96	5.5	6.62
103	39	39	26.66	51.56	43.8	14.26	7.5	5.68
104	40	40	30.69	58.59	32.52	12.62	6.74	8.42
105	15	15	17.01	15.65	11.85	10.36	13.06	11.32
106	16	16	19.04	16.09	12.55	9.36	13.04	10.92
107	17	17	21.51	14.57	15.81	9.22	16.66	9.46
108	18	18	22.83	16.72	14.69	9.62	15.86	11.54
109	19	19	21.96	18.54	16.22	10.74	14.98	10.96
110	20	20	23.71	19.3	16.8	9.26	13.48	9.92
111	21	21	27.29	19.69	16.36	8.82	14.8	11.5
112	22	22	27.64	21.13	17.25	10.08	15.8	12.54
113	23	23	35.58	19.19	17.27	7.12	16.82	11.9
114	24	24	28.07	23.46	20.1	10.94	15.44	11.58
115	25	25	35.64	23.01	17.84	7.06	13.5	11.38
116	26	26	36.66	22.47	20.74	7.88	16.02	11
117	27	27	32.62	25.49	22.89	9.78	14.94	10.6
118	28	28	34.31	26.58	23.09	9.54	14.72	10.86
119	29	29	39.86	25.19	23.71	8.66	16.98	11.4
120	30	30	38.57	26.63	25.83	9	15.9	10.3
121	31	31	38.22	31.85	22.47	9.28	13.14	12.38
122	32	32	45.85	27.51	25.24	7.56	15.76	10.9
123	33	33	43.36	27.46	30.65	8.62	16.86	9.26
124	34	34	47.14	31.61	24.73	8.52	15.58	13
125	35	35	47.98	28.6	31.67	7.78	16.34	9.06
126	36	36	50.44	32.51	27.03	7.42	14.22	11.04
127	37	37	54.59	32.1	27.72	7.14	15.24	11.32
128	38	38	51.17	34.94	29.17	8.96	16.04	12.4
129	39	39	45.39	37.7	33.46	10.72	15.24	10.96
130	40	40	48.67	39.71	31.08	9.78	14.2	11.84
131	15	15	13.53	23.25	9.05	12.54	7.58	14.38
132	16	16	18.36	19.77	9.7	10.46	10.96	15.82
133	17	17	16.82	24.77	9.9	11.54	8.36	15.38
134	18	18	18.01	26.04	10.43	12.48	9.24	16.94
135	19	19	19.81	25.84	11.38	10.72	8.96	14.7
136	20	20	22.6	25.79	11.65	10.26	10.04	15.98
137	21	21	23.33	27.82	12.04	10.72	9.96	16.68
138	22	22	26.41	28.48	11.83	9.22	9.64	16.94
139	23	23	27.5	28.33	13.32	9.46	10.44	15.86
140	24	24	21.36	37.85	14.39	12.38	7.22	14.08
141	25	25	24.99	35.98	14.6	11.34	8.44	15.24
142	26	26	28.28	31.88	17.07	10.96	10.88	14.18
143	27	27	27.54	37.35	16.24	12.42	9.92	16.52
144	28	28	25.72	44.41	15.99	12.64	7.6	15.8
145	29	29	27.75	46.62	15.44	11.44	7.1	16.28
146	30	30	30.87	40.93	18.18	10.9	8.94	14.52
147	31	31	32.46	42.1	18.5	10.36	8.72	14.34
148	32	32	36.14	42.67	17.81	9	8.46	14.78
149	33	33	36.39	48.2	16.73	9.28	7.62	16.54
150	34	34	39.47	42.14	20.25	9.64	10.2	15.1
151	35	35	34.47	51.01	20.49	12.74	9.18	16.78
152	36	36	36.29	49.95	21.89	12.82	10.06	16.6
153	37	37	39.88	45.37	24.57	11.04	10.84	13.94
154	38	38	43.72	46.61	23.22	10	10.6	15.04
155	39	39	40.17	52	24.38	12.26	10.34	15.76
156	40	40	43.1	52.28	24.27	12.08	11	16.96
157	15	15	12.17	18.35	14.73	17	11.98	9.7
158	16	16	11.75	20.46	16.99	15.3	9.1	7.08
159	17	17	14.46	21.58	14.75	15.02	10.72	10.48
160	18	18	14.28	25.11	14.99	15.74	9.26	10.62
161	19	19	17.82	22.02	16.51	13.72	12.22	10.72
162	20	20	16.14	22.99	21.8	14.12	10.64	7.08
163	21	21	18.27	25.79	18.55	15.6	11.88	10.94
164	22	22	19.03	25.44	21.49	15.44	12.54	9.56
165	23	23	19.13	30.24	19.55	14.88	9.92	10.36
166	24	24	19.57	30.68	21.87	16.34	11	10.24
167	25	25	20.66	27.14	28.85	15.68	13	7.56
168	26	26	22.06	30.17	25.96	16.04	12.68	9.46
169	27	27	20.09	32.8	30.43	16.2	10.4	7.1
170	28	28	21.86	39.3	23.62	15.98	9.16	10.42
171	29	29	28.36	31.9	25.76	13	12.94	10.4
172	30	30	27.74	32.82	29.06	13.84	12.98	9.36
173	31	31	28.6	34.88	28.83	14.08	12.74	9.96
174	32	32	24.51	42.63	29.99	16.58	9.88	9.34
175	33	33	29.2	38.4	30.89	14.42	11.94	9.64
176	34	34	27.56	43.85	30.84	16.14	10.68	10.1
177	35	35	31.36	41.06	31.8	15.28	12.72	10.74
178	36	36	31.21	41.01	35.87	14.84	12.3	9
179	37	37	27.25	52.02	33.87	17	9.08	9.4
180	38	38	30.04	46.26	39.03	16.46	11.32	8.64
181	39	39	28.27	52.35	39.18	16.98	9.4	8.26
182	40	40	31.48	46.43	44.57	16.94	12.26	8.04
183	15	15	16	15.61	12.9	11	12.94	10.12
184	16	16	15.57	17.77	14.1	13.48	13.18	10.82
185	17	17	19.23	17.49	13.75	10	12.78	10.6
186	18	18	19.67	21.64	12.08	10.24	10.46	12.98
187	19	19	19.37	21	15.83	12.24	12.72	11.08
188	20	20	20.32	21.51	17.46	12.18	13.02	10.4
189	21	21	18.73	26.58	17.13	13.66	10.34	10.84
190	22	22	22.38	24.87	17.77	12.72	12.84	11.92
191	23	23	23.82	25.49	18.68	10.56	11.14	10.04
192	24	24	21.36	30.76	19.26	13.76	10.24	11.1
193	25	25	23.84	29.73	20.42	13.3	11.72	11.4
194	26	26	22.92	32.58	21.88	13.58	10.26	10.24
195	27	27	28.24	29.38	22.24	11.88	12.96	11.24
196	28	28	27.01	32.48	23.36	13.48	12.4	11.42
197	29	29	26.26	35.8	24.05	13.66	10.84	10.82
198	30	30	26.94	37.15	25.04	13.08	10.24	10.18
199	31	31	31.84	35.45	24.28	13.72	13.82	13.48
200	32	32	34.75	36.3	23.58	10.06	10.92	11.34
201	33	33	37.17	36.27	24.29	10.68	12.56	12.58
202	34	34	37.63	41.38	22.05	10.02	10.24	13.42
203	35	35	39.95	36.87	26.99	10.08	12.66	11.42
204	36	36	36.49	41.21	28.66	12.96	12.84	12.3
205	37	37	40.08	40.28	29.08	10.58	12.02	11.02
206	38	38	36.84	43.16	32.52	12.84	12.18	10.62
207	39	39	37.24	50.16	28.32	13.96	11.24	13.76
208	40	40	37.19	51.85	29.77	13.4	10.36	12.44