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).
