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	-0.000362	0	-0.893
lnL	0.350152	0	2551.825
lnK	0.400185	0	2669.267
lnM	0.24988	0	2125.771
lnLlnL	-0.020191	0	-759.354
lnKlnK	-0.021076	0	-877.576
lnMlnM	-0.017154	0	-809.421
lnLlnK	0.0241	0	622.733
lnLlnM	0.016282	0	459.334
lnKlnM	0.018006	0	497.189

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	13.29	20.91	10.73	9.76	6.54	8.96
2	16	16	18.31	18.78	10.48	7.1	7.88	9.78
3	17	17	16.75	21.01	12.56	8.34	7.3	8.36
4	18	18	19.44	21.32	12.53	6.38	6.54	7.64
5	19	19	20.71	21.94	13.57	7.66	8.18	9
6	20	20	17.82	26.25	15.49	9	6.52	7.58
7	21	21	23.97	25.58	13.08	6.84	7.24	9.96
8	22	22	21.19	29.18	15.11	8.54	6.7	9.08
9	23	23	25.44	23.63	19.2	7.14	8.9	7.1
10	24	24	23.9	24.88	22.79	9.08	9.9	6.86
11	25	25	23.64	30.29	20.08	8.48	7.24	7.34
12	26	26	27	28.85	21	8.06	8.52	7.74
13	27	27	31.18	27.59	21.48	6.88	9.08	7.62
14	28	28	28.34	33.51	20.9	9.12	8.56	9.32
15	29	29	30.56	33.75	21.39	7.14	7.26	7.76
16	30	30	32.38	35.18	21.21	7.66	7.94	9
17	31	31	26.83	39.54	26.05	9.14	6.62	6.76
18	32	32	35.82	37.1	21.95	7.68	8.42	9.76
19	33	33	32.83	35.54	29.58	9.24	9.62	7.46
20	34	34	39.82	40.08	21.49	6.26	7.1	9.24
21	35	35	35.45	37.07	31.4	8.3	9	6.84
22	36	36	35.92	39.1	31.72	9.36	9.68	7.74
23	37	37	38.78	45.47	25.39	8.42	7.98	9.9
24	38	38	45.92	38.67	28.68	6.92	9.68	8.6
25	39	39	49.67	37.92	29.5	6.28	9.86	8.28
26	40	40	42.45	45.2	30.54	8.44	8.96	8.88
27	15	15	18.76	12.82	14.24	6.62	11.84	6.54
28	16	16	19.52	14.23	14.73	7.78	12.9	7.74
29	17	17	22.31	13.27	17.59	5.82	12.26	5.5
30	18	18	19.82	14.29	23.08	7.5	12.6	4.48
31	19	19	25.74	12.89	23.98	5.64	14.54	4.38
32	20	20	19.3	17.03	27.55	8.64	11.44	4.06
33	21	21	32.75	14.6	21.04	5.04	14.9	6.06
34	22	22	27.17	18.72	21.39	7.44	13.18	7.04
35	23	23	29.23	17.4	26.21	5.54	11.66	4.5
36	24	24	29.02	18.57	28.17	6.76	13.06	5
37	25	25	29.19	22.72	23.56	8.42	12.92	7.74
38	26	26	30.54	21.11	29.28	6.9	12.18	5.18
39	27	27	33.25	23.06	26.19	6.68	11.74	6.32
40	28	28	35.39	24.32	25.53	7.36	13.08	7.72
41	29	29	32.7	27.27	27.11	8.58	12.14	7.64
42	30	30	32.78	27.05	31.35	8.74	12.52	6.58
43	31	31	31.11	27.96	36.53	9	11.66	5.32
44	32	32	37	23.45	44.03	7.42	14.5	4.32
45	33	33	36.55	28.87	35.57	8.46	12.76	6.24
46	34	34	35.81	28.44	42.47	7.46	11.18	4.36
47	35	35	48.68	27.61	32.94	5.8	12.92	6.56
48	36	36	38.37	29.35	46.2	7.56	11.84	4.34
49	37	37	50.69	29.89	34.15	6.42	13.66	7.3
50	38	38	43.84	29.01	48.78	8	14.86	5.04
51	39	39	41.92	32.77	46.93	8.92	13.62	5.58
52	40	40	53.38	31.72	39.4	6.64	14	6.78
53	15	15	22.13	18.68	6.67	4.76	6.64	13.94
54	16	16	19.82	23.79	6.9	5.66	5.22	14
55	17	17	21.61	19.47	10.12	6.72	8.68	11.72
56	18	18	20.21	24.08	10.05	7.74	7.2	12.58
57	19	19	28.8	23.02	8.5	4.8	7.14	14.4
58	20	20	34.32	21.12	9.4	4.24	8.58	13.9
59	21	21	29.14	22.4	12.44	5.7	8.88	11.08
60	22	22	35.98	25.76	9.47	4.1	6.94	14.08
61	23	23	31.09	27.63	11.89	5.18	6.8	11.46
62	24	24	29.77	29.17	13.53	7.62	8.92	13.76
63	25	25	41.32	30.01	10.29	4	6.66	14.66
64	26	26	30.19	37.15	12.74	6.88	6.16	13.56
65	27	27	34.31	35.52	13.27	5.98	6.56	13.06
66	28	28	40.19	29.81	15.98	5.22	8.48	11.04
67	29	29	36.93	40.31	13.21	5.8	5.98	13.88
68	30	30	39.05	35.57	16.52	6.32	8.06	12.42
69	31	31	31.62	47.64	16.17	7.2	5.08	11.32
70	32	32	36.91	40.63	18.57	7.66	7.82	12.28
71	33	33	38.49	50.34	14.77	6	5	13.22
72	34	34	42.4	39.35	20.41	6.94	8.66	11.72
73	35	35	52.09	38.56	18.23	5	8.14	12.28
74	36	36	48.54	40.91	20.17	6.14	8.58	12.32
75	37	37	40.45	51.79	20.06	7.56	6.46	12.32
76	38	38	46.32	50.05	19.6	7.42	7.74	14.58
77	39	39	46.06	51.46	20.84	6.42	6.44	11.66
78	40	40	52.24	46.09	22.84	6.6	8.74	12.48
79	15	15	11.41	19.68	14.54	13	7.82	6.98
80	16	16	12.47	24.31	12.08	11.52	6	8.54
81	17	17	11.74	27.04	14.28	14.52	6.22	8.24
82	18	18	10.67	28.33	19.31	14.58	5.28	5.18
83	19	19	14.53	28.83	14.76	12.78	6.52	8.96
84	20	20	13.35	27.89	21.46	14.2	6.82	5.8
85	21	21	13.9	39.72	14.75	12.22	4.06	8.14
86	22	22	15.38	38.5	15.88	13.08	5.08	9
87	23	23	17.09	38.07	16.46	11.36	5.06	8.48
88	24	24	16.86	34.19	23.09	14.68	7.3	7.24
89	25	25	17.22	39.75	21.08	12.08	5.16	6.8
90	26	26	19.78	39.61	20.22	12.24	6.18	8.52
91	27	27	19.18	42.37	22.19	13.56	6.1	8.16
92	28	28	18.67	47.36	22.68	13.24	5.06	7.52
93	29	29	22.6	41.25	24.06	12.54	7.06	8.32
94	30	30	20.93	41.16	30.85	14.54	7.5	6.58
95	31	31	22.15	52.13	22.93	11.4	4.76	7.82
96	32	32	18.13	57.22	30.52	14	4.14	5.42
97	33	33	22.29	46.04	34.74	14.58	7.1	6.18
98	34	34	26.49	45.96	30.43	13.12	7.84	7.96
99	35	35	23.66	62.47	25.8	12.34	4.5	7.96
100	36	36	23.28	68.46	25.94	13.2	4.24	8.3
101	37	37	23.53	66.81	29.29	13.26	4.44	7.32
102	38	38	27.01	54.82	35.26	13.56	6.74	7.08
103	39	39	24.65	56.21	43.25	14.54	6.3	5.36
104	40	40	25.66	58.87	42.01	13.9	5.98	5.56
105	15	15	18	14.93	11.81	10.9	15.52	12.78
106	16	16	18.77	16.17	12.69	10.6	14.44	12
107	17	17	22.35	15.59	13.51	8.42	14.7	10.88
108	18	18	23.56	17.19	13.52	8.2	13.58	11.26
109	19	19	23.71	18.29	14.97	9	13.96	11.04
110	20	20	26.47	19.59	14.25	8.34	13.58	12.34
111	21	21	25.04	19.37	18.78	10.78	16.66	10.8
112	22	22	29.34	20.04	17.35	8.92	15.96	11.82
113	23	23	32.89	21.33	16.16	7.32	13.92	12.06
114	24	24	28.37	24.87	18.13	9.92	13.24	12
115	25	25	32.84	21.14	22.64	8.64	16.58	9.56
116	26	26	30.05	27.71	19.39	10.8	13.58	12.92
117	27	27	34.38	26.15	20.55	9.84	15.52	12.88
118	28	28	35.68	26.89	21.56	9.6	15.3	12.4
119	29	29	38.49	26.26	23.23	7.9	14.16	10.22
120	30	30	34.67	32.3	22.03	10.58	13.14	12.88
121	31	31	43.8	26.31	25.39	8.02	16.7	10.88
122	32	32	44.31	28.21	25.33	8.6	16.72	11.86
123	33	33	40.39	30.75	28.07	9.56	15.06	10.48
124	34	34	39.64	31.91	30.48	9.64	14.22	9.38
125	35	35	45.39	31.51	29.15	8.28	14.54	9.96
126	36	36	42.65	31.97	34.52	10	16.04	9.16
127	37	37	45.25	34.9	30.91	9.14	14.18	10.22
128	38	38	47.63	36.89	29.4	10.16	15.68	12.8
129	39	39	52.87	36.4	29.01	7.74	13.72	11.2
130	40	40	47.63	37.73	34.6	8.94	13.44	9.3
131	15	15	14.52	22.27	8.75	11.34	7.84	14.7
132	16	16	17.84	21.2	9.12	9.84	9.18	15.48
133	17	17	18.38	22.04	10.39	10.94	10.1	15.3
134	18	18	20.23	22.34	11.13	9.4	9.56	13.56
135	19	19	19.4	27.71	10.61	10.78	8.1	15.66
136	20	20	21.52	25.12	12.86	10.6	10.1	13.88
137	21	21	23.98	26.1	12.71	10.44	10.8	15.74
138	22	22	20.08	33.1	13.6	12	7.62	13.56
139	23	23	25.22	33.28	11.89	9.36	7.72	16.2
140	24	24	24.56	32.07	15.08	12.72	10.62	16.12
141	25	25	27.49	31.58	15.52	10.26	9.96	14.36
142	26	26	27.26	36.24	14.94	11.72	9.58	16.98
143	27	27	25.38	41.59	15.56	11.7	7.48	14.86
144	28	28	29.37	36.8	17.44	11.18	9.8	14.74
145	29	29	32.49	38.45	16.43	9.24	8.66	14.72
146	30	30	31.65	40.19	18.06	12.1	10.44	16.72
147	31	31	30.28	43.46	19.35	12.1	9.04	14.64
148	32	32	29.15	52.88	17.49	12.8	7.26	16.68
149	33	33	34.72	42.45	21.11	10.82	9.76	13.88
150	34	34	31.62	51.82	20.16	11.36	7.26	13.78
151	35	35	38.48	47.05	19.92	10.4	9.38	16.12
152	36	36	32.48	53.53	22.97	12.76	8.1	13.7
153	37	37	38.97	48.83	22.8	12.3	10.78	16.5
154	38	38	38.05	54.26	22.39	11	8.28	14.66
155	39	39	41.56	49.44	25.06	11.72	10.92	15.18
156	40	40	43.51	50.12	25.45	9.92	9.6	13.32
157	15	15	12.72	16.6	16.15	13.42	11.22	7.24
158	16	16	12.09	20.24	16.55	14.82	9.24	7.32
159	17	17	15.23	19.41	16.1	14.5	12.46	9.7
160	18	18	15.28	21.09	17.69	15.7	12.28	9.44
161	19	19	17.69	21.31	17.53	13.16	12.08	9.5
162	20	20	18.21	21.67	20.11	13.94	12.98	8.86
163	21	21	18.27	23.74	21.07	14.5	12.18	8.76
164	22	22	17.99	29.1	18.97	16.24	10.54	10.9
165	23	23	17.49	28.39	24.47	14.88	9.62	7.16
166	24	24	18.8	27.34	27.73	16.72	12.3	7.56
167	25	25	21.57	31.72	21.27	15.04	10.92	10.92
168	26	26	23.32	32.3	21.63	13.76	10.72	10.74
169	27	27	22.35	34.76	23.83	15.04	10.22	9.96
170	28	28	23.83	35.39	24.44	13.24	9.5	9.18
171	29	29	27.55	32.52	26.03	13.36	12.56	10.2
172	30	30	24.73	39.03	26.22	15.3	10.22	10.2
173	31	31	27.07	36.1	29.54	15.54	12.66	10.02
174	32	32	27.09	35.67	34.15	13.64	11.28	7.42
175	33	33	25.19	44.64	30.37	16.36	9.54	9.38
176	34	34	29.29	37.86	35.43	14.08	11.9	8.04
177	35	35	28.41	40.39	37.59	14.32	10.82	7.36
178	36	36	30.56	41.44	36.37	14.02	11.2	8.16
179	37	37	31.09	48.37	31.25	14.48	9.84	10.28
180	38	38	33.15	43.25	37.58	15.48	12.94	9.54
181	39	39	37.12	43.95	34.64	13.32	12.48	10.32
182	40	40	29.86	51.24	41.32	15.76	9.54	7.68
183	15	15	13.85	18.58	12.05	12.44	10.06	10.46
184	16	16	17.84	17.98	11.55	10.46	11.84	12.46
185	17	17	16.21	19.9	14.21	13.46	12.08	11.22
186	18	18	17.97	22.41	12.91	11.48	10.12	12.1
187	19	19	18.86	23.58	13.8	12.04	10.58	12.42
188	20	20	19.33	25.59	14.4	13.8	11.34	13.94
189	21	21	20.93	24.42	16.7	11.08	10.56	10.32
190	22	22	23.73	24.68	16.62	10.98	11.98	11.92
191	23	23	23.61	27.53	16.85	12.2	11.64	12.96
192	24	24	24.93	25.08	21.24	11.9	13.5	10.26
193	25	25	26.94	28.9	18.09	10.38	10.92	11.84
194	26	26	27	29.09	20.74	11.6	12.14	11.3
195	27	27	31.75	28.07	20.43	10.22	13.5	12.26
196	28	28	30.18	31.67	20.91	12.04	13	13.24
197	29	29	30.54	33.63	21.53	11.84	12.08	12.76
198	30	30	29.59	35.19	23.89	13.52	12.6	12.42
199	31	31	35.13	30.65	26.61	10.36	13.9	10.26
200	32	32	35.68	33.28	25.95	10.74	13.32	11.16
201	33	33	35.58	39.38	22.78	11.1	11.26	13.4
202	34	34	35.02	36	29.83	12.24	13.54	10.56
203	35	35	34.8	37.71	31.38	12.52	13.02	10.1
204	36	36	38.11	40.51	27.75	10.68	11.36	11.08
205	37	37	37.85	44.16	27.35	10.68	10.18	11.18
206	38	38	43.8	40.54	28.38	11.04	13.82	13.14
207	39	39	38.25	51.23	26.53	12.56	10.18	13.8
208	40	40	41.83	42.28	34.47	11.66	13.16	10.46