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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

Cost Functions:   Cobb-Douglas Cost | Normalized Quadratic Cost | Translog Cost | Diewert Cost | Generalized CES-Translog Cost | Generalized CES-Diewert Cost | References and Links

Duality: Production / Cost Functions:   Cobb-Douglas Duality | CES Duality | Theory of Duality | Translog Duality - CES | Translog Duality - Generalized CES

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

CES Production Function Parameters
elasticity of scale parameter: nu
elasticity of substitution: sigma
alpha
beta
gamma
Diewert elasticity of scale: nu1
Coefficient Estimates
Variable Coefficient std error t-ratio
L-0.1480530.001-183.569
K-0.181890-424.452
M-0.179970-380.444
LK0.6443770.001555.523
LM0.3466190.001322.232
KM0.519060.001465.723
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 #qest qLKM 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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