Egwald Economics: Microeconomics
Production Functions
by
Elmer G. Wiens
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Cobb-Douglas
| CES
| Generalized CES
| Translog
| Diewert
| Translog vs Diewert
| Diewert vs Translog
| Estimate Translog
| Estimate Diewert
| References and Linkss
H. Generate CES Data and Estimate a Diewert 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 Diewert production function is:
q^1/nu = aLL * L + aKK * K + aMM * M + bLK * L^1/2 * K^1/2 + bLM * L^1/2 * M^1/2 + bKM * K^1/2 * M^1/2
= f(L,K,M)
where L = labour, K = capital, M = materials and supplies, q = product, and nu = elasticity of scale parameter.
3. The coefficients of the Diewert production function vary with sigma and nu. The program will generate a set of 182 observations, and use ordinary least squares to estimate the coefficients. To estimate the coefficient of the elasticity of scale, nu - called nu1, of the Diewert production function, you must set it independently of nu (from the CES). This process mimics a nonlinear estimation procedure.
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 = 1 (Cobb-Douglas)
sigma < 1 → inputs complements; sigma > 1 → inputs substitutes
Coefficient Estimates
|
Variable
| Coefficient
|
std error
| t-ratio
|
L | -0.148053 | 0.001 | -183.569 | K | -0.18189 | 0 | -424.452 | M | -0.17997 | 0 | -380.444 | LK | 0.644377 | 0.001 | 555.523 | LM | 0.346619 | 0.001 | 322.232 | KM | 0.51906 | 0.001 | 465.723 |
R2 = 1 |
R2b = 1 |
# obs = 182 |
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 Diewert 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 | 18.44 | 12.21 | 15.8 | 7 | 13 | 6 |
2 | 16 | 16 | 19.67 | 13.02 | 16.85 | 7 | 13 | 6 |
3 | 17 | 17 | 20.9 | 13.84 | 17.9 | 7 | 13 | 6 |
4 | 18 | 18 | 22.13 | 14.65 | 18.96 | 7 | 13 | 6 |
5 | 19 | 19 | 23.36 | 15.46 | 20.01 | 7 | 13 | 6 |
6 | 20 | 20 | 24.59 | 16.28 | 21.06 | 7 | 13 | 6 |
7 | 21 | 21 | 25.82 | 17.09 | 22.11 | 7 | 13 | 6 |
8 | 22 | 22 | 27.05 | 17.9 | 23.17 | 7 | 13 | 6 |
9 | 23 | 23 | 28.28 | 18.72 | 24.22 | 7 | 13 | 6 |
10 | 24 | 24 | 29.51 | 19.53 | 25.27 | 7 | 13 | 6 |
11 | 25 | 25 | 30.74 | 20.35 | 26.33 | 7 | 13 | 6 |
12 | 26 | 26 | 31.97 | 21.16 | 27.38 | 7 | 13 | 6 |
13 | 27 | 27 | 33.2 | 21.97 | 28.43 | 7 | 13 | 6 |
14 | 28 | 28 | 34.43 | 22.79 | 29.49 | 7 | 13 | 6 |
15 | 29 | 29 | 35.66 | 23.6 | 30.54 | 7 | 13 | 6 |
16 | 30 | 30 | 36.89 | 24.42 | 31.59 | 7 | 13 | 6 |
17 | 31 | 31 | 38.12 | 25.23 | 32.65 | 7 | 13 | 6 |
18 | 32 | 32 | 39.35 | 26.04 | 33.7 | 7 | 13 | 6 |
19 | 33 | 33 | 40.58 | 26.86 | 34.75 | 7 | 13 | 6 |
20 | 34 | 34 | 41.81 | 27.67 | 35.8 | 7 | 13 | 6 |
21 | 35 | 35 | 43.04 | 28.48 | 36.86 | 7 | 13 | 6 |
22 | 36 | 36 | 44.27 | 29.3 | 37.91 | 7 | 13 | 6 |
23 | 37 | 37 | 45.5 | 30.11 | 38.96 | 7 | 13 | 6 |
24 | 38 | 38 | 46.73 | 30.93 | 40.02 | 7 | 13 | 6 |
25 | 39 | 39 | 47.95 | 31.74 | 41.07 | 7 | 13 | 6 |
26 | 40 | 40 | 49.18 | 32.55 | 42.12 | 7 | 13 | 6 |
27 | 15 | 15 | 16.64 | 13.21 | 15.96 | 8 | 12 | 6 |
28 | 16 | 16 | 17.75 | 14.09 | 17.03 | 8 | 12 | 6 |
29 | 17 | 17 | 18.86 | 14.97 | 18.09 | 8 | 12 | 6 |
30 | 18 | 18 | 19.97 | 15.85 | 19.16 | 8 | 12 | 6 |
31 | 19 | 19 | 21.08 | 16.73 | 20.22 | 8 | 12 | 6 |
32 | 20 | 20 | 22.19 | 17.61 | 21.29 | 8 | 12 | 6 |
33 | 21 | 21 | 23.3 | 18.49 | 22.35 | 8 | 12 | 6 |
34 | 22 | 22 | 24.41 | 19.37 | 23.42 | 8 | 12 | 6 |
35 | 23 | 23 | 25.52 | 20.25 | 24.48 | 8 | 12 | 6 |
36 | 24 | 24 | 26.63 | 21.13 | 25.54 | 8 | 12 | 6 |
37 | 25 | 25 | 27.73 | 22.01 | 26.61 | 8 | 12 | 6 |
38 | 26 | 26 | 28.84 | 22.89 | 27.67 | 8 | 12 | 6 |
39 | 27 | 27 | 29.95 | 23.77 | 28.74 | 8 | 12 | 6 |
40 | 28 | 28 | 31.06 | 24.65 | 29.8 | 8 | 12 | 6 |
41 | 29 | 29 | 32.17 | 25.53 | 30.87 | 8 | 12 | 6 |
42 | 30 | 30 | 33.28 | 26.41 | 31.93 | 8 | 12 | 6 |
43 | 31 | 31 | 34.39 | 27.29 | 32.99 | 8 | 12 | 6 |
44 | 32 | 32 | 35.5 | 28.17 | 34.06 | 8 | 12 | 6 |
45 | 33 | 33 | 36.61 | 29.05 | 35.12 | 8 | 12 | 6 |
46 | 34 | 34 | 37.72 | 29.94 | 36.19 | 8 | 12 | 6 |
47 | 35 | 35 | 38.83 | 30.82 | 37.25 | 8 | 12 | 6 |
48 | 36 | 36 | 39.94 | 31.7 | 38.32 | 8 | 12 | 6 |
49 | 37 | 37 | 41.05 | 32.58 | 39.38 | 8 | 12 | 6 |
50 | 38 | 38 | 42.16 | 33.46 | 40.44 | 8 | 12 | 6 |
51 | 39 | 39 | 43.27 | 34.34 | 41.51 | 8 | 12 | 6 |
52 | 40 | 40 | 44.38 | 35.22 | 42.57 | 8 | 12 | 6 |
53 | 15 | 15 | 20.8 | 12.08 | 13.71 | 6 | 13 | 7 |
54 | 16 | 16 | 22.19 | 12.88 | 14.62 | 6 | 13 | 7 |
55 | 17 | 17 | 23.57 | 13.69 | 15.53 | 6 | 13 | 7 |
56 | 18 | 18 | 24.96 | 14.49 | 16.45 | 6 | 13 | 7 |
57 | 19 | 19 | 26.34 | 15.3 | 17.36 | 6 | 13 | 7 |
58 | 20 | 20 | 27.73 | 16.1 | 18.28 | 6 | 13 | 7 |
59 | 21 | 21 | 29.12 | 16.91 | 19.19 | 6 | 13 | 7 |
60 | 22 | 22 | 30.5 | 17.71 | 20.1 | 6 | 13 | 7 |
61 | 23 | 23 | 31.89 | 18.52 | 21.02 | 6 | 13 | 7 |
62 | 24 | 24 | 33.28 | 19.32 | 21.93 | 6 | 13 | 7 |
63 | 25 | 25 | 34.66 | 20.13 | 22.84 | 6 | 13 | 7 |
64 | 26 | 26 | 36.05 | 20.93 | 23.76 | 6 | 13 | 7 |
65 | 27 | 27 | 37.44 | 21.74 | 24.67 | 6 | 13 | 7 |
66 | 28 | 28 | 38.82 | 22.54 | 25.59 | 6 | 13 | 7 |
67 | 29 | 29 | 40.21 | 23.35 | 26.5 | 6 | 13 | 7 |
68 | 30 | 30 | 41.6 | 24.15 | 27.41 | 6 | 13 | 7 |
69 | 31 | 31 | 42.98 | 24.96 | 28.33 | 6 | 13 | 7 |
70 | 32 | 32 | 44.37 | 25.76 | 29.24 | 6 | 13 | 7 |
71 | 33 | 33 | 45.76 | 26.57 | 30.15 | 6 | 13 | 7 |
72 | 34 | 34 | 47.14 | 27.37 | 31.07 | 6 | 13 | 7 |
73 | 35 | 35 | 48.53 | 28.18 | 31.98 | 6 | 13 | 7 |
74 | 36 | 36 | 49.92 | 28.98 | 32.9 | 6 | 13 | 7 |
75 | 37 | 37 | 51.3 | 29.79 | 33.81 | 6 | 13 | 7 |
76 | 38 | 38 | 52.69 | 30.59 | 34.72 | 6 | 13 | 7 |
77 | 39 | 39 | 54.08 | 31.4 | 35.64 | 6 | 13 | 7 |
78 | 40 | 40 | 55.46 | 32.2 | 36.55 | 6 | 13 | 7 |
79 | 15 | 15 | 19.4 | 11.37 | 16.62 | 7 | 15 | 6 |
80 | 16 | 16 | 20.7 | 12.13 | 17.73 | 7 | 15 | 6 |
81 | 17 | 17 | 21.99 | 12.89 | 18.83 | 7 | 15 | 6 |
82 | 18 | 18 | 23.29 | 13.65 | 19.94 | 7 | 15 | 6 |
83 | 19 | 19 | 24.58 | 14.41 | 21.05 | 7 | 15 | 6 |
84 | 20 | 20 | 25.87 | 15.16 | 22.16 | 7 | 15 | 6 |
85 | 21 | 21 | 27.17 | 15.92 | 23.27 | 7 | 15 | 6 |
86 | 22 | 22 | 28.46 | 16.68 | 24.37 | 7 | 15 | 6 |
87 | 23 | 23 | 29.75 | 17.44 | 25.48 | 7 | 15 | 6 |
88 | 24 | 24 | 31.05 | 18.2 | 26.59 | 7 | 15 | 6 |
89 | 25 | 25 | 32.34 | 18.95 | 27.7 | 7 | 15 | 6 |
90 | 26 | 26 | 33.63 | 19.71 | 28.81 | 7 | 15 | 6 |
91 | 27 | 27 | 34.93 | 20.47 | 29.91 | 7 | 15 | 6 |
92 | 28 | 28 | 36.22 | 21.23 | 31.02 | 7 | 15 | 6 |
93 | 29 | 29 | 37.52 | 21.99 | 32.13 | 7 | 15 | 6 |
94 | 30 | 30 | 38.81 | 22.75 | 33.24 | 7 | 15 | 6 |
95 | 31 | 31 | 40.1 | 23.5 | 34.35 | 7 | 15 | 6 |
96 | 32 | 32 | 41.4 | 24.26 | 35.45 | 7 | 15 | 6 |
97 | 33 | 33 | 42.69 | 25.02 | 36.56 | 7 | 15 | 6 |
98 | 34 | 34 | 43.98 | 25.78 | 37.67 | 7 | 15 | 6 |
99 | 35 | 35 | 45.28 | 26.54 | 38.78 | 7 | 15 | 6 |
100 | 36 | 36 | 46.57 | 27.29 | 39.89 | 7 | 15 | 6 |
101 | 37 | 37 | 47.87 | 28.05 | 40.99 | 7 | 15 | 6 |
102 | 38 | 38 | 49.16 | 28.81 | 42.1 | 7 | 15 | 6 |
103 | 39 | 39 | 50.45 | 29.57 | 43.21 | 7 | 15 | 6 |
104 | 40 | 40 | 51.75 | 30.33 | 44.32 | 7 | 15 | 6 |
105 | 15 | 15 | 20.13 | 12.51 | 13.5 | 7 | 14 | 8 |
106 | 16 | 16 | 21.47 | 13.34 | 14.4 | 7 | 14 | 8 |
107 | 17 | 17 | 22.81 | 14.18 | 15.3 | 7 | 14 | 8 |
108 | 18 | 18 | 24.16 | 15.01 | 16.2 | 7 | 14 | 8 |
109 | 19 | 19 | 25.5 | 15.85 | 17.1 | 7 | 14 | 8 |
110 | 20 | 20 | 26.84 | 16.68 | 18 | 7 | 14 | 8 |
111 | 21 | 21 | 28.18 | 17.51 | 18.9 | 7 | 14 | 8 |
112 | 22 | 22 | 29.52 | 18.35 | 19.8 | 7 | 14 | 8 |
113 | 23 | 23 | 30.87 | 19.18 | 20.7 | 7 | 14 | 8 |
114 | 24 | 24 | 32.21 | 20.02 | 21.6 | 7 | 14 | 8 |
115 | 25 | 25 | 33.55 | 20.85 | 22.5 | 7 | 14 | 8 |
116 | 26 | 26 | 34.89 | 21.68 | 23.4 | 7 | 14 | 8 |
117 | 27 | 27 | 36.24 | 22.52 | 24.3 | 7 | 14 | 8 |
118 | 28 | 28 | 37.58 | 23.35 | 25.2 | 7 | 14 | 8 |
119 | 29 | 29 | 38.92 | 24.19 | 26.1 | 7 | 14 | 8 |
120 | 30 | 30 | 40.26 | 25.02 | 27 | 7 | 14 | 8 |
121 | 31 | 31 | 41.6 | 25.86 | 27.9 | 7 | 14 | 8 |
122 | 32 | 32 | 42.95 | 26.69 | 28.8 | 7 | 14 | 8 |
123 | 33 | 33 | 44.29 | 27.52 | 29.7 | 7 | 14 | 8 |
124 | 34 | 34 | 45.63 | 28.36 | 30.6 | 7 | 14 | 8 |
125 | 35 | 35 | 46.97 | 29.19 | 31.5 | 7 | 14 | 8 |
126 | 36 | 36 | 48.31 | 30.03 | 32.4 | 7 | 14 | 8 |
127 | 37 | 37 | 49.66 | 30.86 | 33.3 | 7 | 14 | 8 |
128 | 38 | 38 | 51 | 31.69 | 34.2 | 7 | 14 | 8 |
129 | 39 | 39 | 52.34 | 32.53 | 35.1 | 7 | 14 | 8 |
130 | 40 | 40 | 53.68 | 33.36 | 36 | 7 | 14 | 8 |
131 | 15 | 15 | 15.63 | 14.76 | 14.54 | 9 | 11 | 7 |
132 | 16 | 16 | 16.67 | 15.75 | 15.51 | 9 | 11 | 7 |
133 | 17 | 17 | 17.71 | 16.73 | 16.47 | 9 | 11 | 7 |
134 | 18 | 18 | 18.75 | 17.71 | 17.44 | 9 | 11 | 7 |
135 | 19 | 19 | 19.8 | 18.7 | 18.41 | 9 | 11 | 7 |
136 | 20 | 20 | 20.84 | 19.68 | 19.38 | 9 | 11 | 7 |
137 | 21 | 21 | 21.88 | 20.67 | 20.35 | 9 | 11 | 7 |
138 | 22 | 22 | 22.92 | 21.65 | 21.32 | 9 | 11 | 7 |
139 | 23 | 23 | 23.96 | 22.63 | 22.29 | 9 | 11 | 7 |
140 | 24 | 24 | 25 | 23.62 | 23.26 | 9 | 11 | 7 |
141 | 25 | 25 | 26.05 | 24.6 | 24.23 | 9 | 11 | 7 |
142 | 26 | 26 | 27.09 | 25.59 | 25.2 | 9 | 11 | 7 |
143 | 27 | 27 | 28.13 | 26.57 | 26.17 | 9 | 11 | 7 |
144 | 28 | 28 | 29.17 | 27.55 | 27.14 | 9 | 11 | 7 |
145 | 29 | 29 | 30.21 | 28.54 | 28.1 | 9 | 11 | 7 |
146 | 30 | 30 | 31.26 | 29.52 | 29.07 | 9 | 11 | 7 |
147 | 31 | 31 | 32.3 | 30.51 | 30.04 | 9 | 11 | 7 |
148 | 32 | 32 | 33.34 | 31.49 | 31.01 | 9 | 11 | 7 |
149 | 33 | 33 | 34.38 | 32.47 | 31.98 | 9 | 11 | 7 |
150 | 34 | 34 | 35.42 | 33.46 | 32.95 | 9 | 11 | 7 |
151 | 35 | 35 | 36.47 | 34.44 | 33.92 | 9 | 11 | 7 |
152 | 36 | 36 | 37.51 | 35.43 | 34.89 | 9 | 11 | 7 |
153 | 37 | 37 | 38.55 | 36.41 | 35.86 | 9 | 11 | 7 |
154 | 38 | 38 | 39.59 | 37.39 | 36.83 | 9 | 11 | 7 |
155 | 39 | 39 | 40.63 | 38.38 | 37.8 | 9 | 11 | 7 |
156 | 40 | 40 | 41.67 | 39.36 | 38.76 | 9 | 11 | 7 |
157 | 15 | 15 | 18.69 | 10.95 | 18.69 | 7 | 15 | 5 |
158 | 16 | 16 | 19.93 | 11.68 | 19.93 | 7 | 15 | 5 |
159 | 17 | 17 | 21.18 | 12.41 | 21.18 | 7 | 15 | 5 |
160 | 18 | 18 | 22.43 | 13.14 | 22.43 | 7 | 15 | 5 |
161 | 19 | 19 | 23.67 | 13.87 | 23.67 | 7 | 15 | 5 |
162 | 20 | 20 | 24.92 | 14.6 | 24.92 | 7 | 15 | 5 |
163 | 21 | 21 | 26.16 | 15.33 | 26.16 | 7 | 15 | 5 |
164 | 22 | 22 | 27.41 | 16.06 | 27.41 | 7 | 15 | 5 |
165 | 23 | 23 | 28.65 | 16.79 | 28.65 | 7 | 15 | 5 |
166 | 24 | 24 | 29.9 | 17.52 | 29.9 | 7 | 15 | 5 |
167 | 25 | 25 | 31.15 | 18.25 | 31.15 | 7 | 15 | 5 |
168 | 26 | 26 | 32.39 | 18.98 | 32.39 | 7 | 15 | 5 |
169 | 27 | 27 | 33.64 | 19.71 | 33.64 | 7 | 15 | 5 |
170 | 28 | 28 | 34.88 | 20.44 | 34.88 | 7 | 15 | 5 |
171 | 29 | 29 | 36.13 | 21.17 | 36.13 | 7 | 15 | 5 |
172 | 30 | 30 | 37.38 | 21.9 | 37.38 | 7 | 15 | 5 |
173 | 31 | 31 | 38.62 | 22.63 | 38.62 | 7 | 15 | 5 |
174 | 32 | 32 | 39.87 | 23.36 | 39.87 | 7 | 15 | 5 |
175 | 33 | 33 | 41.11 | 24.09 | 41.11 | 7 | 15 | 5 |
176 | 34 | 34 | 42.36 | 24.83 | 42.36 | 7 | 15 | 5 |
177 | 35 | 35 | 43.6 | 25.56 | 43.6 | 7 | 15 | 5 |
178 | 36 | 36 | 44.85 | 26.29 | 44.85 | 7 | 15 | 5 |
179 | 37 | 37 | 46.1 | 27.02 | 46.1 | 7 | 15 | 5 |
180 | 38 | 38 | 47.34 | 27.75 | 47.34 | 7 | 15 | 5 |
181 | 39 | 39 | 48.59 | 28.48 | 48.59 | 7 | 15 | 5 |
182 | 40 | 40 | 49.83 | 29.21 | 49.83 | 7 | 15 | 5 |
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