# Welfare analysis for a military draft

How would the welfare analysis of a military draft look? My intuition tells me that it would look similar to a price ceiling below the equilibrium, with the exception that suppliers (civilians) are forced to supply the draft.

So we supply beyond P* x Q* and all the way out to the intersection of the demand curve and P-draft.

If we say that consumer surplus is the area above P-draft and below the demand curve, then how would we interpret the area that isn't our typical DWL? The area above P-draft, below the demand curve, and above the supply curve.

• To me there's a fundamental fallacy. Economics study human action, and particularly human decisions and utilisation of scarce resources. Here's there's no choice at all. Also, supply of soldiers will be a vertical line @ quantity = population capable to fight. – Commissar Vasili Karlovic Mar 21 '17 at 23:29
• @CommissarVasiliKarlovic I don't think this is a problem. People have to work. Otherwise they die of hunger. As such, you could think of a vertical supply of labour as a live or die issue. No choice there either. – luchonacho Apr 3 '17 at 20:05
• What is P? To whom is it paid? What is the demand? This is, what is "the marginal product of a soldier"? I think a welfare analysis of this sort might be better approached with a Cost-Benefit Analysis instead of a simple demand/supply, surplus study. – luchonacho Apr 3 '17 at 20:08

An obligatory draft is a tax collected in kind - productive time. Instead of producing in the private sector, citizens of the economy offer their services to the army.

Now, we should acknowledge that the existence of an army offers some desired public good (or at least that the society implementing the draft thinks so). Is this a utility-enhancing, or a production-enhancing public good?

Thompson, E. A. (1974). Taxation and national defense. Journal of political economy, 82(4), 755-782.

argued that it is safeguarding property rights of the local population against foreign aggressors, and so it is production-enhancing. Moreover it is subject to congestion: the larger the output/capital base of the economy, the larger military spending has to be to provide the same degree of security (one can find a mention of the above as well as a related model in Barro's book on growth but only on the 1st edition, p. 160).

Here, the tax to fund military expenses is collected in kind, so we could model this situation with the following tweaked Cobb-Douglas production function

$$Y = AK^a(N -L_a)^{1-a}\cdot (BL_a/Y)^\gamma \tag{1}$$

where $N$ is population/total available productive time, there is no labor-leisure choice, $L_a$ is the time spent in producing security services through the draft. The last term is the indirect contribution of military spending in output (by safeguarding property rights), and it is divided by output in order to reflect the congestion characteristic. It is treated as an externality from the private sector.

But the government/social planner can maximize output for any choices of the private sector regarding investment in physical capital, by maximizing the above production function, after taking into account the externality, i.e. by maximizing

$$Y = A^{1/(1+\gamma)}K^{a/(1+\gamma)}(N -L_a)^{(1-a)/(1+\gamma)}\cdot (BL_a)^{\gamma/(1+\gamma)} \tag{2}$$

with respect to $L_a$. We obtain

$$L_a^* = \frac {\gamma}{1-a+\gamma}N$$

namely, that the "size of the draft" in terms of productive time should be a constant proportion of the population.

The problem here is that the private sector looks at $(1)$ and "ignores" the last term. So it behaves as though the capital elasticity of output is $a$ while in reality is smaller, $a/(1+\gamma)$ as we see from $(2)$ (this is due to the existence of congestion in the public good).

This means that the private sector will tend to over-invest. In such cases a proportional tax on output is optimal from a welfare point of view because it would internalize the externality...

...and this means that it would be better, aside from possible non-economic issues, if instead of an obligatory draft the government applied a proportional tax on output in order to fund military expenses (i.e. have a professional, paid army). In this way it would both provide this desired public good, and internalize the externality, reducing the rate of capital investment down to the optimal level.

(Note: obviously the above is a very abstract approach. One could point out that armies need much more than just the labor of their soldiers, in terms of equipment, but even in terms of the subsistence expenses of the soldiers themselves. Then one could model the existence of a tax to fund these, and examine whether it can sustain the output maximizing level of Labor drafted and be at the optimal level as regards the target of internalizing the externality).

• Interesting, from OP I originally thought that such a question is beyond economic analysis. but as pointed out in the first sentence it is possible. Amazing. – EconJohn Apr 3 '17 at 21:12
• @EconJohn Thanks. Economics and economists are in the business of quantifying things that at first sight may appear not-quantifiable, and then subjecting them to quantitative or "quasi-quantitative" analysis. – Alecos Papadopoulos Apr 3 '17 at 21:18
• @AlecosPapadopoulos Great answer, but couldn't agreed less with your definition of economics. I'm not an Austrian Economist, but Hayek's 1974 remarks on the Pretense of Knowledge that economists presume is still a relevant point for today (btw, a lesson Modern Macro yet failed to learn). – luchonacho Apr 4 '17 at 18:34
• Oh look. Luchonacho disagrees with someone. – 123 Apr 7 '17 at 22:27