econometric military production function

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Econometric Military Production FunctionBy Oleg Nekrassovski

The present paper aims to show the mathematical details behind an example of an econometric military production function, given in Hildebrandt (1999).

The measure of effectiveness of the interdiction campaign Commando Hunt V was determined to be described by the following multivariable function (Hildebrandt, 1999):

Y = X11.31 X2

0.57 X30.33 X4

0.28 X5-0.85 (I)

where

Y =IP-TP (reduction in throughput)X1 = gunship team sortiesX2=fighter attack sorties against trucks and storage areasX3=fighter attack sorties against the lines of communicationX4=fighter attack sorties in close air support roleX5 = southbound sensor-detected truck movements

If the dollar costs are operationally fixed, or the per sortie cost of each of the 3 fighter-attack sorties (X2, X3, X4) is the same, then the reduction in throughput (Y) will be greatest when the marginal productivities of the 3 fighter-attack sorties are equal (Hildebrandt, 1999). In other words, when

MPX2 = MPX3 = MPX4,

which is the same as

∂Y∂ X2

= ∂Y∂ X3

= ∂Y∂ X 4

(II)

Satisfying Eq.(II) using Eq.(I):

Y = X11.31 X2

0.57 X30.33 X4

0.28 X5-0.85

∂Y∂ X2

=¿0.57 X11.31 X2

-0.43 X30.33 X4

0.28 X5-0.85

∂Y∂ X3

=¿0.33 X11.31 X2

0.57 X3-0.67 X4

0.28 X5-0.85


∂Y∂ X4

=¿0.28 X11.31 X2

0.57 X30.33 X4

-0.72 X5-0.85

∂Y∂ X2

= ∂Y∂ X3

0.57 X11.31 X2

-0.43 X30.33 X4

0.28 X5-0.85 = 0.33 X1

1.31 X20.57 X3

-0.67 X40.28 X5

-0.85

0.57 X2-0.43 X3

0.33 = 0.33 X20.57 X3

-0.67

0.57 X30.33/X3

-0.67 = 0.33 X20.57/X2

-0.43

0.57 X3 = 0.33 X2

57 X3 = 33 X2

∂Y∂ X3

= ∂Y∂ X4

0.33 X11.31 X2

0.57 X3-0.67 X4

0.28 X5-0.85 = 0.28 X1

1.31 X20.57 X3

0.33 X4-0.72 X5

-0.85

0.33 X3-0.67 X4

0.28 = 0.28 X30.33 X4

-0.72

0.33 X40.28/X4

-0.72 = 0.28 X30.33/X3

-0.67

0.33 X4 = 0.28 X3

33 X4 = 28 X3

∂Y∂ X2

= ∂Y∂ X4

0.57 X11.31 X2

-0.43 X30.33 X4

0.28 X5-0.85 = 0.28 X1

1.31 X20.57 X3

0.33 X4-0.72 X5

-0.85

0.57 X2-0.43 X4

0.28 = 0.28 X20.57 X4

-0.72

0.57 X40.28/ X4

-0.72 = 0.28 X20.57/ X2

-0.43

0.57 X4 = 0.28 X2


57 X4 = 28 X2

In order to express, the determined proportions of the three types of fighter-attack sorties, as percentages of the total number of fighter-attack sorties, so as to make them more useful, let X2 + X3 + X4 = 1 = 100%. Then,

X2 = 57/28 X4 and X3 = 33/28 X4

57/28 X4 + 33/28 X4 + X4 = 1

118/28 X4 = 1

X4 = 28/118 = 0.237 = 23.7%

X3 = 33/57 X2 and X4 = 28/57 X2

X2 + 33/57 X2 + 28/57 X2 = 1

118/57 X2 = 1

X2 = 57/118 = 0.483 = 48.3%

X2 = 57/33 X3 and X4 = 28/33 X3

57/33 X3 + X3 + 28/33 X3 = 1

118/33 X3 = 1

X3 = 33/118 = 0.280 = 28.0%

Hence, the optimal allocation of the three types of fighter-attack sorties, according to each type, is:

Fighter-Attack Sorties Type Optimal Sorties Flown (%)X2 48.3X3 28.0X4 23.7


When fighter- attack sorties are allocated according to these percentages, the function that describes the reduction in throughput (Eq. (I)) can be simplified as follows:

Let XF = X2 + X3 + X4. If the fighter-attack sorties are allocated optimally, then

X2 = 0.483 XF

X3 = 0.28 XF

X4 = 0.237 XF.

So,

X20.57 X3

0.33 X40.28 = (0.483 XF)0.57 (0.28 XF)0.33 (0.237 XF)0.28

= (0.483)0.57 (0.28)0.33 (0.237)0.28 XF0.57+0.33+0.28

= 0.29 XF1.18

Hence,

Y = 0.29 X11.31 XF

1.18 X5-0.85 (III)

Holding X5 constant at the average weekly level, Eq. (III) can be used to construct two-dimensional isoquants. By specifying a value for the reduction in throughput (Y = IP-TP), the isoquants can be easily traced by varying the values of X1 and XF (Hildebrandt, 1999).

References

Hildebrandt, G. G. (1999). “The military production function.” Defence and Peace Economics, 10, 247-272.

econometric military production function

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