Econometric Military Production Function
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The document describes a mathematical model of an econometric military production function that measures the effectiveness of an interdiction campaign. The model defines variables like fighter attack sorties and gunship sorties that impact the reduction in enemy throughput. The model shows that reducing throughput is maximized when the marginal productivities of different fighter attack sorties are equal. An optimal allocation of 48.3% for X2 sorties, 28% for X3, and 23.7% for X4 is determined. The production function can then be simplified to show reduction in throughput as a function of these optimal allocations.
Econometric Military Production Function
- 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.