1. Interdisciplinary, unpaid research opportunities are available. Various academic specialties
are required. If interested, email me at dr.freedom@hotmail.ca.
Econometric Military Production Function
By 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 = X1
1.31 X2
0.57 X3
0.33 X4
0.28 X5
-0.85 (I)
where
Y =IP-TP (reduction in throughput)
X1 = gunship team sorties
X2=fighter attack sorties against trucks and storage areas
X3=fighter attack sorties against the lines of communication
X4=fighter attack sorties in close air support role
X5 = southbound sensor-detected truck movements
If the dollarcostsare operationallyfixed,orthe persortie cost of each of the 3 fighter-attacksorties(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
𝜕X4
(II)
Satisfying Eq.(II) using Eq.(I):
Y = X1
1.31 X2
0.57 X3
0.33 X4
0.28 X5
-0.85
𝜕Y
𝜕X2
= 0.57 X1
1.31 X2
-0.43 X3
0.33 X4
0.28 X5
-0.85
𝜕Y
𝜕X3
= 0.33 X1
1.31 X2
0.57 X3
-0.67 X4
0.28 X5
-0.85

2. Interdisciplinary, unpaid research opportunities are available. Various academic specialties
are required. If interested, email me at dr.freedom@hotmail.ca.
𝜕Y
𝜕X4
= 0.28 X1
1.31 X2
0.57 X3
0.33 X4
-0.72 X5
-0.85
𝜕Y
𝜕X2
=
𝜕Y
𝜕X3
0.57 X1
1.31 X2
-0.43 X3
0.33 X4
0.28 X5
-0.85 = 0.33 X1
1.31 X2
0.57 X3
-0.67 X4
0.28 X5
-0.85
0.57 X2
-0.43 X3
0.33 = 0.33 X2
0.57 X3
-0.67
0.57 X3
0.33/X3
-0.67 = 0.33 X2
0.57/X2
-0.43
0.57 X3 = 0.33 X2
57 X3 = 33 X2
𝜕Y
𝜕X3
=
∂Y
𝜕X4
0.33 X1
1.31 X2
0.57 X3
-0.67 X4
0.28 X5
-0.85 = 0.28 X1
1.31 X2
0.57 X3
0.33 X4
-0.72 X5
-0.85
0.33 X3
-0.67 X4
0.28 = 0.28 X3
0.33 X4
-0.72
0.33 X4
0.28/X4
-0.72 = 0.28 X3
0.33/X3
-0.67
0.33 X4 = 0.28 X3
33 X4 = 28 X3
𝜕Y
𝜕X2
=
∂Y
𝜕X4
0.57 X1
1.31 X2
-0.43 X3
0.33 X4
0.28 X5
-0.85 = 0.28 X1
1.31 X2
0.57 X3
0.33 X4
-0.72 X5
-0.85
0.57 X2
-0.43 X4
0.28 = 0.28 X2
0.57 X4
-0.72
0.57 X4
0.28/ X4
-0.72 = 0.28 X2
0.57/ X2
-0.43
0.57 X4 = 0.28 X2

3. Interdisciplinary, unpaid research opportunities are available. Various academic specialties
are required. If interested, email me at dr.freedom@hotmail.ca.
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 48.3
X3 28.0
X4 23.7

4. Interdisciplinary, unpaid research opportunities are available. Various academic specialties
are required. If interested, email me at dr.freedom@hotmail.ca.
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,
X2
0.57 X3
0.33 X4
0.28 = (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 XF
0.57+0.33+0.28
= 0.29 XF
1.18
Hence,
Y = 0.29 X1
1.31 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.