NATO ascendant
A model of the levels of Western support needed for a Ukrainian victory by level of Russian mobilisation, under a Cobb-Douglas production function
Since the invasion of Ukraine by Russia began in February 2022, the support of NATO members has been vital to Ukraine’s continuing ability to fight the war. This post will provide a very macro and abstract model of the war to try to estimate the implied levels of Western support necessary at the maximum possible level of Russian mobilisation in order to achieve given objectives, and thus respresents an upper bound on necessary expenditure to achieve those objectives. A graphical representation of the model can be found here for readers wishing to vary the parameter estimates.
A Cobb-Douglas production, shown above, takes productivity, A, labour, L, and capital, K, as inputs, and outputs income, Y. In this case, labour is the number of individuals currently serving under arms and capital is the sum of military expenditure in this year and all prior (somewhat depreciated) expenditures. The theoretical capital capacity of Ukraine is far greater, as the combined GDP of NATO is $32.4 trillion in comparison to Russia’s $1.78 trillion, but this advantage is lower than first appeared as absent the recruitment of foreign mercenaries (which may be undesirable for political reasons) the Ukrainian labour component is fixed while Russia has considerable capacity to expand labour allocated to the war through conscription due to the large divergence in population between the two states, although whether Russia has the political ability to mobilise such a fraction of its population is another question.
Since independence in 1991, Ukraine has spent the equivalent of $70 billion on its military. However, this steeply overestimates the capital that was accessible to Ukraine upon the outbreak of the war due to not accounting for the Ukrainian army consisting of humans rather than entirely robots and capital depreciation over time.1 Assuming Ukrainian capital and labour expenditure proportions are identical to those of the British military accounts, personnel costs account for about 41% of the budget (modelled as 0.3 to 0.5 normal) and depreciation is (less £700 million to account for impairment expenditure) roughly 28% annually.23 However, not the entirety of Russian capital could be deployed, due to Russia’s possessing borders with the Baltic States, Poland and Finland: this effect will be modelled as 0.6 to 0.85 normal. 4
Russia does not publish breakdowns of its military budget by service, and although only 20% of active duty Russian army personnel were directly deployed in Ukraine before the mobilisation three months ago a considerably greater proportion of total capital would have been allocated to the war. The US spends 49% of the military budget on the services which Russia would have been able to deploy at a large scale during the war (army and air force), but as Russia is primarily a land power this ratio would be expected to be rather higher: however, Russia has regions (Kaliningrad, St Petersburg and shortly the entire Finnish border) that border NATO, so would not be able to deploy all land and air force based capital. Additionally, as naval, air force and nuclear services all have higher capital-to-labour ratios than land forces, they will all likely occupy a lower proportion of the military budgets of countries with lower GDPs per capita: this effect would probably be larger with Ukraine than Russia due to Ukrainian pre-war GDP per capita being $4800 per capita compared to Russia’s $12000. This will be modelled as 70-90% of Ukrainian military spending being on land and air forces and 50-75% of Russian military spending being on land and air forces, with 80-100% of Ukrainian land and air forces being deployed and 65-85% of Russian relevant capital being deployed, all distributed normally.
Lastly, the value of A can in this case be interpreted as the relative institutional quality of the Russian and Ukrainian armies. To avoid needing to compute the exact path of Ukrainian familiarity with NATO technology and other effects, the opening of the war provides some data, although of potentially limited use due to the substantial expansion in both Russian and Ukrainian forces since then. A complicating factor is that warfare success likely depends on both the absolute level of capital and willingness to expend capital - tanks matter in relation to their number rather than casualty figures, while tank ammunition matters only insofar as it is in fact expended. However, without access to data on the exact rates of ammunition consumption of both powers such considerations will be neglected. At the beginning of the invasion, Russia had about 180,000 regular military and 34,000 Donbas militia, while Ukraine fielded around 200,000 regular military and 100,000 paramilitary troops. Including paramilitaries, weighting Ukrainian (0.6 to 0.85) more highly than Russian (0.2 to 0.5) due to the much higher performance of Ukrainian paramilitaries than the forces of the LPR and DPR, thus provides a relative estimate for Aa, with a as the ratio of Russo-Ukrainian military power in the opening days of the war and A as the ratio of Ukrainian-Russian efficiency, of 7.9 (90% 3.2 to 17). a is also chosen with very high uncertainty, but is modelled as lognormal 1.2 to 3.7. This gives an implied value of A of 5.6 (90% 1.1 to 21, implying that the Ukrainian military utilised the resources it had access to 5.6 times more efficiently than the Russian military did.
Given the value A, three separate groups of estimates need to be made: possible maximum Russian mobilisation, the state of both sides’ militaries in 2023 and the required ratios the Ukrainians would need to achieve various possible goals. For the first, Russia has a GDP of $1.78 trillion, and during WW2 the military of the Soviet Union peaked at 7 million in size, with the USSR’s population then around a third higher than the current Russian population. During WW2, Britain’s defence spending peaked at 52% of GDP and America’s defence spending peaked at 43%: both these population and GDP figures likely represent upper bounds on plausible expenditure, but as defence spending occasionally exceeded 15% of GDP even in 18th century Britain a 90% confidence interval of 20% to 40% will be used for defence spending as a proportion of GDP and a military size of 3 million to 5 million.
The position of both sides’ militaries currently depends on the rate of extant capital depreciation, the current size of the Ukrainian army and the capital transfer that has occurred to date. As financial backing pledged thus far by the European Union alone is only slightly less than the entire Ukrainian military budget scheduled for 2023, Ukrainian military expenditure will be assumed to effectively be a subset of capital transfers from NATO (which is an assumption with no impact if Ukrainian labour is fixed, which it is believed to be for reasons mentioned above). The Ukrainian army currently stands at 700,000 and has received just over $40 billion in external military funding. For extant capital depreciation rates, Russia in 2021 had over 10,000 tanks in storage, of which 1500 have been confirmed captured or destroyed: however, as Russia likely deployed the more modern tanks first and many remaining tanks have been cannibalised for spare parts, and the 1500 figure only refers to losses photographically confirmed, the actual depreciation rate will be much higher. As Russia appears to be deploying tanks mostly seen in museums elsewhere, very high depreciation rates for usable capital seem reasonable and a lognormal estimate of 0.4 to 0.7 will be used.
A final estimate needed is the Solow ratios required for various objectives: these are modelled here as a stalemate at current front lines, a return to 2014-2022 boundaries or restoration of pre-2014 boundaries, respectively. Due to a lack of previous study of this approach to modelling warfare, these estimates, as those of relative effectiveness were above, are necessarily rather arbitrary: quantitative work on this model would be extremely helpful. However, the rough intuition is: defence is possible even if a military is rather weaker than its opponent, so a ratio between 0.3 and 0.8 normally distributed would be sufficient: a return to pre-war lines would require a much higher ratio, as it would require taking territory currently occupied by Russia, so a ratio of 1.4 to 3.9 lognormally distributed would be required: and a full return to pre-2014 frontiers including the reintegration of Crimea would be rather more difficult and would require a ratio between 2.2 and 4.5 distributed lognormally.
Table 1 shows the results. As the annual NATO military budget is well in excess of $1 trillion, this implies that even in the case of complete Russian mobilisation NATO could likely still provide sufficient funding for a Ukrainian victory and would certainly be able to provide sufficient funding to allow a stalemate at Russian front lines. Russian victory, absent the use of nuclear weapons or similar, is thus extremely unlikely assuming NATO will continue financing Ukraine.
However, this model also allows us to investigate a different question. Ukraine retains complete victory as a goal, which would maximise the incentive against future state-on-state aggression worldwide and minimise the increase in the incentive to possess nuclear weapons, while many of its NATO allies are concerned about the potential for nuclear escalation during any battle for Crimea due to current opinion polls indicating strong support within Russia for defending the peninsula. What probabilities does this model give existing Western aid of victory or a draw for Ukraine? The model predicts that if Russia were to double both the number of soldiers allocated to Ukraine and the military budget existing Western transfers would have been sufficient to ensure the probability of Ukrainian victory would be at around 11% while the probability of at least a stalemate at current front lines would be 95%.5 This provides some evidence towards this being the strategy that NATO states are employing: the much slower delivery of offensive weapons such as tanks and APCs compared to more defensive infantry equipment and missile systems provides qualitative evidence.
What principle shortcomings are there of this model? A key problem is that military production, unlike industrial production, uses single-use inputs - ammunition - to create output, but this output cannot be sold to create new inputs.6 This means that actual combat power is probably a function of capital stock, rate of ammunition expenditure, and manpower, which would be a useful development of the model here. Another issue is that the Solow ratios and effectiveness ratios are all parameters chosen with very low confidence, due to a lack of previous studies on them. Finally, implicitly the model has assumed that A - relative Ukrainian to Russian efficiency of converting inputs into combat power - has stayed fixed during the war, while, in reality, the number of trained officers, quality of weaponry and level of corruption may have varied considerably as the war progressed.
This post thus implies that Russian victory in Ukraine is effectively impossible given Western willingness to continue financing Ukraine merely out of current defence budgets - which given that Ukrainian military spending per soldier remains much below Western levels remains considerably cheaper than financing comparable NATO militaries. However, the model lacks empirical verification elsewhere and does not incorporate ammunition expenditure: key shortcomings which will be rectified in a future post.
It also underestimates it slightly due to not accounting for leftover Soviet capital: however, under depreciation rates implied by the British government’s accounts (discussed in more detail after) maintaining such capital requires such high expenditure as to functionally be equivalent to worthless 30 years after the collapse of the USSR.
This was calculated by dividing depreciation by the sum of depreciation, other equipment support costs, capital spending and inventory consumption.
The limitations of the software used means that uncertainty in this will not be modelled, and the uncertainty in the model is hence underestimated.
This and all subsequent “x to y” statements refer to the 90% confidence interval for that value.
The ammunition issue mentioned below likely renders this a considerable overestimate of both probabilities.
In the ancient world, in contrast, war very much could be self-financing in this manner.