Egwald Economics: Microeconomics

Production Functions

by

Elmer G. Wiens

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

F. Comparing the Translog and Diewert Production Functions

To obtain the estimates of the Diewert production function, I used the CES production function to generate a sequence of observations relating the CES least cost factor inputs to factor prices and levels of output.

Estimating the Diewert production function using multiple regression yielded the specified coefficient estimates for sigma = .5 and 2.

For values of sigma significantly greater than 1, the Diewert production function provides better estimates of the Allen elasticites than the Translog production function. See the Translog production function.

1. Elasticity of substitution < < 1

elasticity of scale = 1.0, and

elasticity of substitution = .5

and alpha = .35, beta = .4, and gamma = .25.

nu = 1
aLL = -0.4669 aKK = -0.5905 aMM = -0.4703
bLK = 1.0794 bLM = 0.5468 bKM = 0.9017

aLL + aKK + aMM + bLK + bLM + bKM = 1

R2 = 1.0       all |t-values| >> 2

The firm buys its inputs at the prices:     wL = 7   wK = 13   wM = 6

Diewert Long Run Cost Data
Constant Returns to Scale
Elasticity of Substitution = .5
qest q LK Mtotal cost ave. costmarg. costsLKsLMsKM
2020 22.6817.78 20.72514.14 25.7125.71 0.510.50.5
2222 24.9419.56 22.79565.56 25.7125.71 0.510.50.5
2424 27.2121.33 24.86616.97 25.7125.71 0.510.50.5
2626 29.4823.11 26.93668.38 25.7125.71 0.510.50.5
2828 31.7524.89 29719.8 25.7125.71 0.510.50.5
3030 34.0126.67 31.07771.21 25.7125.71 0.510.50.5
3232 36.2828.45 33.15822.63 25.7125.71 0.510.50.5
3434 38.5530.22 35.22874.04 25.7125.71 0.510.50.5
3636 40.8232 37.29925.46 25.7125.71 0.510.50.5
3838 43.0833.78 39.36976.87 25.7125.71 0.510.50.5
4040 45.3535.56 41.431028.28 25.7125.71 0.510.50.5

Short Run: Capital Fixed.

If we set capital at the least cost level for q = 30, then K = 26.667327433218

See the discussion of the short-run elasticity of substitution, sLM, on the translog production function page.

Diewert Short Run Cost Data
Constant Returns to Scale
Elasticity of Substitution = .5
qest q LK Mtotal cost ave. costmarg. cost3 factor
sLM
2 factor
sLM
2020 17.8826.67 16.44570.49 28.52 15.41 0.850.72
2222 20.4926.67 18.8602.91 27.4 17.04 0.790.68
2423.99 23.3626.67 21.41638.69 26.61 18.84 0.730.63
2625.99 26.5526.67 24.31678.36 26.09 20.86 0.660.59
2828 30.0926.67 27.52722.38 25.8 23.11 0.580.55
3030 34.0126.67 31.07771.2 25.71 25.71 0.50.51
3232 38.3826.67 35.02825.45 25.8 28.66 0.420.47
3433.99 43.2526.67 39.43886.01 26.06 32.12 0.330.43
3636 48.7626.67 44.38954.32 26.51 36.21 0.240.39
3837.99 54.9526.67 50.011031.39 27.14 41.19 0.150.35
4039.98 62.0726.67 56.41119.57 27.99 47.39 0.050.31

We get a U-shaped, short run average cost curve, with capital fixed.

The short run average cost curve is (approx.) tangent to the long run average cost curve, at q = 30.

2. Elasticity of substitution > > 1.

elasticity of scale = 1.0, and

elasticity of substitution = 2.0.

and alpha = .35, beta = .4, and gamma = .25.

nu = 1
aLL = 0.1225 aKK = 0.16 aMM = 0.0625
bLK = 0.28 bLM = 0.175 bKM = 0.2

aLL + aKK + aMM + bLK + bLM + bKM = 1

R2 = 1.0

Diewert Long Run Cost Data
Constant Returns to Scale
Elasticity of Substitution = 2.0
qest q LK Mtotal cost ave. costmarg. costsLKsLMsKM
2020 30.911.7 21.46497.21 24.8624.86 222
2222 33.9912.87 23.61546.93 24.8624.86 222
2424 37.0814.04 25.75596.65 24.8624.86 222
2626 40.1715.21 27.9646.37 24.8624.86 222
2828 43.2616.38 30.04696.1 24.8624.86 222
3030 46.3517.55 32.19745.82 24.8624.86 222
3232 49.4418.72 34.34795.54 24.8624.86 222
3434 52.5319.89 36.48845.26 24.8624.86 222
3636 55.6221.06 38.63894.98 24.8624.86 222
3838 58.7122.24 40.77944.7 24.8624.86 222
4040 61.823.41 42.92994.42 24.8624.86 222

Short Run: Capital Fixed.

If we set capital at the least cost level for q = 30, then K = 17.554007079253

Diewert Short Run Cost Data
Constant Returns to Scale
Elasticity of Substitution = 2.0
qest q LK Mtotal cost ave. costmarg. cost3 factor
sLM
2 factor
sLM
2019.99 24.9717.55 17.52508.13 25.41 22.42 22
2221.99 29.0417.55 20.34553.53 25.16 23.04 22
2424.01 33.1817.55 23.34600.5 25.02 23.55 22
2625.99 37.6917.55 25.96647.78 24.91 24.06 22
2828 41.7417.55 29.36696.52 24.88 24.48 22
3030.01 46.0417.55 32.58745.96 24.87 24.86 22
3232 50.4517.55 35.74795.78 24.87 25.23 22
3434 55.0317.55 38.86846.57 24.9 25.54 22
3636.01 59.7317.55 41.99898.25 24.95 25.79 22
3838.01 64.3717.55 45.22950.16 25 26.06 22
4040 68.9817.55 48.541002.26 25.06 26.34 22

The short run average cost curve is (approx.) tangent to the long run average cost curve, at q = 30.