I think that I need to include an additional element into this model which will reduce the need for such massive discrepancies of munitions required by the equation: introducing defense as a function of munitions invested per soldier.

Guaporense wrote:So how we could determine the value of munitions? Well, the ultimate function of weapons is to inflict casualties on the enemy (gains in territory occur when the enemy forces are either destroyed or retreat due to superior enemy casualty infliction capabilities in proportion to the forces engaged).

Using a conventional Cobb-Douglas casualty production function, where the volume of casualties is determined by the equation:

Enemy casualties = **A** x **M**^{z} x **P**^{1-z} (1)

Which can be interpreted as the equation determining the output of the military sector of the economy given by the number of casualties it inflicts on the enemy.

Where:

**A** = fighting power parameter (total factor productivity of the armed force, i.e. efficiency)

**M** = munitions

**P** = personnel

z = parameter between 0 and 1 that yields the relative importance between munitions and personnel, in civilian sectors, human resources are usually around 60% while capital (i.e. munitions in the military sector) are 40%. However, workers have human capital, which is not taken into account here (we abstract human capital away from personnel and allocate it to parameter A). So I would give 50% / 50% of the relative contribution of personnel and material capital (munitions) to the output of an armed force. That's it, I give z a value of 0.5, that's the only assumption I make in this post.

Note that if we double both M and P we will double our casualty infliction capability. That makes sense, doesn't it? I mean, an army twice the size with twice everything should inflict twice the casualties. One thing that I don't consider in this model would be the size of the enemy forces, that's it, the density of enemies along the front. If there are few men to man the frontlines, artillery barrages, for instance, would tend to produce fewer casualties than if manpower is plenty.

Anyway, my objective is to determine M(german)/M(soviets)given that we know P, A and the number of casualties. These are the statistics for Germany and the USSR on force strength and the number of casualties:

German force strength:

2,500,000 - March 1942

2,550,000 - May 1942

2,600,000 - June 1942

2,600,000 - July 1942

2,500,000 - August 1942

2,490,000 - October 1942

Average: 2,540,000

German casualties:

1,080,950

Soviet force strength:

1st quarter 1942 - 4,186,000

2nd quarter 1942 - 5,060,300

3rd quarter 1942 - 5,664,600

4th quarter 1942 - 6,343,600

Average: 5,313,600

That means that Germany had 47.8% of the personnel strength of the Soviet Union in the Eastern front in 1942.

Soviet casualties:

7,369,278

However, Soviet casualties include besides killed, wounded and missing also sick and frostbitten, while German casualty figures don't. I don't have 1942 sick and frostbitten numbers, but during the whole war they were 12.4% of all Soviet losses. So cutting down 12.4% of the Soviet numbers we reach:

6,455,488 casualties.

That means that Germany inflicted 597% of the casualties using 47.8% of the personnel. The main reason was that Germany was much richer and hence could afford to invest a greater amount of physical capital on their soldiers.

The additional element I think needs inclusion is "armor". Defined as the capability of resisting attack, in other words, reducing the opponents casualty infliction. We defined as

**B** = basic casualties, which is the value determined in equation (1).

**C** = inflicted casualties would follow a simple equation:

**C** =

**B** x 1/(

**M**/

**P**)^{z} (2)

Where

**M** is the value of munitions of the enemy and

**P** is the manpower/personnel. That's it: the number of casualties inflicted is the number of basic casualties divided by a function of munitions per capita. Meaning that better equipped armies will be able to resist attacks suffering fewer casualties and also, that armies with the same amounts of munitions invested, if they have fewer soldiers they will tend to suffer fewer casualties since each soldier would be better equipped.

Where better equipped means the use of tanks, aircraft, fortifications, armored infantry transports, helmets and other things that make harder to kill or wound the enemy. Using eq. 2 with 1 will help to increase the overall effectiveness of munitions in determining casualties.

Applying this model to the Eastern front in 1942 we have adding equations 1 and 2:

**C** =

**A** x

**M**^{z} x

**P**^{1-z} x 1/(

**M(enemy)**/

**P(enemy)**)^{z}

A is equal to 250%,

Soviet casualties / German casualties = 2.5 x (

**M(German)**/

**M(Soviet)**)^{z} x

**P(German)/P(Soviet)**^{1-z} x [(

**P(Soviet)**/

**M(Soviet)**)/(

**P(German)**/

**M(German)**)]^{z}

Using z = .5,

Soviet casualties / German casualties = 2.5 x (

**M(German)**/

**M(Soviet)**)^{1/2} x (2.54/5.314)^{1/2} x [(5.314/

**M(Soviet)**)/(2.54/

**M(German)**)]^{1/2}

----->

Soviet casualties / German casualties = 2.5 x (

**M(German)**/

**M(Soviet)**)^{1/2} x (0.478)^{1/2} x [(2.092 x

**M(German)**/

**M(Soviet)**)]^{1/2}

----->

Soviet casualties / German casualties = 2.5 x 1.446 x (

**M(German)**/

**M(Soviet)**)^{1/2} x (0.478)^{1/2} x [

**M(German)**/

**M(Soviet)**)]^{1/2}

----->

6,455,488 / 1,080,950 = 3.615 x (

**M(German)**/

**M(Soviet)**) x 0.691

----->

5.97 = 2.49 x (

**M(German)**/

**M(Soviet)**)

So, with this modification, where if I increase the munitions per soldier by 100% it become 41% harder to kill/wound the soldiers, we reach a very reasonable estimate that the Germans invested 239.7% of the value of munitions the Soviets invested.

And this implies that each German soldier had 5.014 more munitions invested in him in 1942 than each Soviet soldier in 1942 on a per head basis. A reasonable discrepancy that fits with the long run difference in economic development between the two countries, where Germany's GDP in terms of munitions goods (which are tradeable goods and thus reflected in Market Exchange Rate GDP and not on Maddison's GDP numbers) is usually around 5 times Russia's.

"In tactics, as in strategy, superiority in numbers is the most common element of victory." - Carl von Clausewitz