     Egwald Economics: Microeconomics

Production Functions

by

Elmer G. Wiens

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E. Comparing the Translog and Diewert Production Functions

To obtain estimates of the Translog 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 Translog 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 Diewert 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.

lnA = 0 aL = 0.348648 aK = 0.400012 aM = 0.25134
bLL = -0.105433 bKK = -0.122478 bMM = -0.091295
bLK = 0.136617 bLM = 0.07425 bKM = 0.10834

aL + aK + aM = 1
-2*bLL = 0.210866 =~ 0.210867 = bLK + bLM
-2*bKK = 0.244956 =~ 0.244957 = bLK + bKLM
-2*bMM = 0.18259 =~ 0.18259 = bLM + bKM

R2 = 1.0       all |t-values| >> 2

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

Translog Long Run Cost Data
Constant Returns to Scale
Elasticity of Substitution = .5
qest q LK Mtotal cost ave. costmarg. costsLKsLMsKM
2020 22.6817.79 20.71514.19 25.7125.710.50.50.5
2222 24.9419.56 22.78565.61 25.7125.710.50.50.5
2424 27.2121.34 24.85617.03 25.7125.710.50.50.5
2626 29.4823.12 26.92668.45 25.7125.710.50.50.5
2828 31.7524.9 28.99719.86 25.7125.710.50.50.5
3030 34.0126.68 31.06771.28 25.7125.710.50.50.5
3232 36.2828.46 33.13822.7 25.7125.710.50.50.5
3434 38.5530.24 35.2874.12 25.7125.710.50.50.5
3636 40.8232.01 37.27925.54 25.7125.710.50.50.5
3838 43.0933.79 39.34976.96 25.7125.710.50.50.5
4040 45.3535.57 41.411028.38 25.7125.710.50.50.5

Short Run: Capital Fixed.

If we set capital at the least cost level for q = 30, then K = 26.678769826985

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

Translog 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 17.8426.68 16.32569.65 28.48 15.730.610.56
2222 20.4826.68 18.73602.58 27.39 17.230.590.55
2424 23.3826.68 21.37638.67 26.61 18.940.570.54
2625.99 26.5726.68 24.28678.43 26.09 20.890.550.53
2828 30.126.68 27.49722.46 25.8 23.10.530.51
3030 34.0126.68 31.05771.24 25.71 25.70.50.5
3232 38.3726.68 35.02825.57 25.8 28.710.470.49
3433.99 43.2526.68 39.46886.32 26.07 32.280.440.47
3636 48.7926.68 44.49955.29 26.54 36.540.410.46
3837.99 55.0526.68 50.181033.27 27.19 41.730.360.44
4039.99 62.2626.68 56.731123.01 28.08 48.180.320.42

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.

lnA = 0.000348 aL = 0.347346 aK = 0.398656 aM = 0.253994
bLL = 0.062107 bKK = 0.055418 bMM = 0.047515
bLK = -0.070009 bLM = -0.054203 bKM = -0.040827

aL + aK + aM = 1
-2*bLL = -0.124214 =~ -0.124212 = bLK + bLM
-2*bKK = -0.110836 =~ -0.110836 = bLK + bKLM
-2*bMM = -0.09503 =~ -0.09503 = bLM + bKM

R2 = 1.0

Translog Long Run Cost Data
Constant Returns to Scale
Elasticity of Substitution = 2.0
qest q LK Mtotal cost ave. costmarg. costsLKsLMsKM
2020 30.9111.7 21.47497.29 24.8624.862.111.922.06
2222 3412.87 23.61547.02 24.8624.862.111.922.06
2424 37.114.04 25.76596.75 24.8624.862.111.922.06
2626 40.1915.21 27.91646.48 24.8624.862.111.922.06
2828 43.2816.38 30.05696.2 24.8624.862.111.922.06
3030 46.3717.55 32.2745.93 24.8624.862.111.922.06
3232 49.4618.72 34.35795.66 24.8624.862.111.922.06
3434 52.5519.89 36.49845.39 24.8624.862.111.922.06
3636 55.6421.06 38.64895.12 24.8624.862.111.922.06
3838 58.7322.23 40.79944.85 24.8624.862.111.922.06
4040 61.8323.4 42.93994.58 24.8624.862.111.922.06

Short Run: Capital Fixed.

If we set capital at the least cost level for q = 30, then K = 17.549858968771

Translog 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
2020 24.9217.55 17.64508.4 25.42 22.372.382.2
2221.99 28.9817.55 20.42553.49 25.16 23.052.252.13
2424.01 33.1417.55 23.4600.51 25.02 23.572.142.08
2626 37.7117.55 25.99648.07 24.93 24.062.062.04
2827.99 41.7217.55 29.35696.31 24.87 24.531.982
3030 46.0417.55 32.61746.07 24.87 24.861.921.97
3232.01 50.5317.55 35.73796.21 24.88 25.181.861.94
3434 55.0717.55 38.82846.61 24.9 25.51.821.92
3636 59.6817.55 41.99897.86 24.94 25.761.771.9
3838 64.3817.55 45.13949.6 24.99 25.991.731.88
4040.01 69.1217.55 48.351002.08 25.05 26.161.691.86

The short run average cost curve is (approx.) tangent to the long run average cost curve, at q = 30. Copyright © Elmer G. Wiens:   Egwald Web Services All Rights Reserved.    Inquiries 