The economic theory of alliances developed by Olson and Zeckhauser (1966) touches upon a number of issues, notably those related to burden sharing and interaction between members. In its original form, the concept of an alliance is taken to represent security and, along with it, defence spending as a pure public good.¹ In this context, several public finance issues arise, such as non-rivalry and non-excludability, which, in their turn, lead to considering problems like unequal burden sharing and, consequently, free-riding possibilities, to which the literature refers as the “exploitation hypothesis” (Olson and Zeckhauser 1966; Sandler and Hartley 1995, p. 23). Once this is the alliance environment, one can then argue that defence expenditure for each ally, as well as in the overall context, is sub-optimal in terms of Pareto optimisation, pointing instead to the direction of a Nash equilibrium.²

In technical terms, following George and Sandler (2018, 2021, 2022), the welfare function of each ally can be expressed as depending on the consumption of a non-defence good cⁱ, the collective defence good

and the external threat T. Therefore, the welfare function takes the following form: Ui=Uici,Q,T. This function is maximized subject to each member’s budget constraint, expressed as Ii=aci+pqi, where a and p are the per unit prices of c and q, respectively. In this context, the aggregate defence level in all allies except the ith is called a spill to ally i and is given as:

Sandler (1977) extended the analysis by proposing a joint product model, for which the defence expenditure of an ally can produce either public output or private output. Sandler and Hartley (1995, pp. 35–36) distinguish between the demand functions of a pure public good and the joint product model by using the concept of what they call “full income,” which incorporates the sum of both income and spill-ins enjoyed by an alliance member. Thus, in the case of a joint product model, the income constraint, including the spill-in benefits p Q˜i is added to both sides of the pure public good model constraint and leads to: Fi=Ii+pQ˜i=aci+pQi and in this case, the corresponding utility function to maximise will be: Ui=Uici,Qi,Q˜i,T subject to the above income constraint and bearing in mind that: qi=Qi−Q˜i. The maximisation exercise will lead to the first-order condition that represents ally i’s defence demand function as follows:

where Qⁱ is the total defence output of the alliance.

A basic assumption of the standard economic theory of alliance (Sandler and Hartley, 1995, p.22) is that ally i receives spill-ins, derived from the defence expenditures of its N-1 allies, as follows:

Where q^k represents the defence expenditures of the N-1 allies, but k≠i. ωk denotes the weight applied to the k-th ally’s defence spending, qk.

However, in the context of an alliance, ally i has to meet certain obligations, which imply that some of the defence part of each member’s defence expenditure is devoted to infrastructure and other operating expenditures to support the alliance’s needs. To incorporate the spills that stem from the obligations of each ally, we expand the standard economic theory of alliance by focusing on the concept of the spill-outs of ally i, representing some of its defence expenditure and directed to the remaining N-1 allies, as follows:

where q^t represents the defence expenditure of ally i, and δk is the fraction of the military spending of ally i, provided to the k-th ally as spill-out, but k≠i. We note that the value of δk is not necessarily equal to the value of ωk, as the transmission mechanism of the spills received (spill-ins) may be different than the transmission mechanism of the defence expenditure incurred (spill-outs). Therefore, the net spills of ally i are:

Substituting Equations (1) and (2) into Equation (3), we get the net spills constraint:

We interpret the net spills as the difference between the spills provided by ally i to other allies and the spills received by ally i from other allies. In other words, the net spills show the net contribution of each member to the common security good produced by the alliance. A positive value of net spills from ally i to other members of the alliance implies that the ally is a net provider of security spills and the allies are net receivers in their relationship with ally i, while the opposite holds for negative values of net spills.

We assume a political realism approach, according to which each country gains utility from spill-ins and loses utility from spill-outs. The political realism assumption argues that the international system consists of states that aim to overpower other rival states and thus dominate in the international power hierarchy (Hobbes, 1946, p.82; Morgenthau, 1985, p.12). In that sense, power is a tool for a state to serve its national interest and overcome the consequences of security dilemma (Hertz, 1951), as this is rooted in the anarchy of the state system (Sørensen et al., 2022, pp. 67–102). In a political realism environment, it is straightforward to assume that even in a military alliance, each state seeks to serve its own interest, thus determining its behaviour in the alliance by considering not only the possible spill-ins but the spill-outs too, as expressed by Equation (4). Extreme negative values of net spills may be interpreted as possible free-riding, given that the ally concerned receives more security spills than it provides to the alliance.

The social welfare of ally i is given by the following function:

In Equation (5), let Qi=qi+Q˜ri−Q˜pi, where Qⁱ is the common defence good for each ally i, considering its own military spending and the net spills. Further, the following common economic assumptions hold:

Substituting the net spills constraint of Equation (1) and the budget constraint into Equation (5), we get ally’s i maximization problem, with respect to its defence spending as follows:

The first-order conditions of the above equation give:

z denotes the fraction of the military spending of ally i that has not been devoted to support the alliance’s needs.

Rearranging the above First-Order Conditions (FOC) gives:

The interpretation of Equation (8) suggests that the lower the z and/or the higher the p, the more expensive Qⁱ, the amount of defence good consumed by ally i. However, z represents ally’s i remaining fraction of defence spending after spill-outs have been subtracted. Therefore, according to Equation (8), a lower value for z (indicating a higher value of spill-outs) increases the cost of defence. Consequently spill-outs increase the defence costs for ally i, thus playing the same role as the price p of the defence.

Further, applying the total differential in Equation (7), we get:

which indicates free-riding, namely when the other allies, except for ally i, increase spending, then ally i decreases its defence spending. Therefore, the larger the spill-ins from the other allies to ally i, the higher the incentive for free-riding, as in George and Sandler (2021). Consequently, introducing the notion of spill-outs in our model confirms the free-riding incentive.

In the analysis, we employ yearly data concerning military spending on equipment as a percentage of the total military expenditure for seven NATO members (the United States, the United Kingdom, the Netherlands, Germany, Italy, Greece, and Türkiye) over the period of 1971–2022. The data was obtained from NATO’s database,³ and the above countries were selected on the basis of the availability of data due to their long-standing membership of the alliance. Table 1 provides summary statistics for the variables. The statistical properties of the series indicate a non-normal distribution, since the kurtosis is lower than 3 in all cases (platykurtic distribution), except for the United States and Türkiye, where we can see a kurtosis close to 3. The values for skewness also provide evidence in favour of a non-normal distribution, as in most cases it is either higher than 0.5 or lower than –0.5. In addition, the Jarque–Bera (JB) normality test rejects the null hypothesis of normal distribution of the time-series, except for the cases of Greece and Türkiye. When it comes to unit-root, we can see that according to the augmented Dickey–Fuller (ADF) test, we cannot reject the null hypothesis of a unit-root, with the exception of Türkiye, implying that the variables are non-stationary in levels, which is a strong sign of mean aversion behavior that justifies the use of non-linear econometric techniques.

Furthermore, the quantile-mean covariance (QC) normality test (Bera et al., 2016) results, as shown in Table 2, rejects the null hypothesis of normality overall, thus indicating the asymmetric behavior of all series distribution.

The analysis of the defence spending data revealed the existence of possible asymmetric features, as in Palaios and Papapetrou (2023). The latter indicates the need for applying econometric techniques that allow us a more comprehensive description of the conditional distribution than the ordinary mean approach, thus offering a more robust econometric technique methodology in the presence of conditional heterogeneity and departures from the Gaussian conditions. Therefore, to test the empirical implications and the validity of our theoretical model, we estimate the spillover effects among the selected NATO allies by initially applying the static and dynamic connectedness -analysis developed by Diebold and Yilmaz (2009, 2012, 2014), and thereafter the quantile connectedness methodology developed by Ando et al. (2022), which allows an in-depth analysis of spill-ins, spill-outs and net spills across the whole distribution of our time series.

In this section, we examine the evolution of the spills produced by the common defence good of the alliance. As a measurement for that purpose, we use the TSI evaluated at various quantiles. We interpret the TSI as the total spills available, stemming from the military expenditure of the alliance’s members that consist the common good of defence. An increase (decrease) in the level of the TCI is an evidence that the available security spills increase (decrease). Consequently, the index indicates the overall ability of the alliance to provide its members with a higher level of security during periods of crisis. In this context, it is straightforward to interpret an increase (decrease) in the TSI as an increase (decrease) in the cohesion and effectiveness of the alliance, as more (less) spills are the result of a higher (lower) level of security provision from the alliance to its members.

Figure 1 illustrates the variation of the TSI index evaluated at the mean and various quantiles along the conditional distribution. The quantile approach captures the impact of negative and positive political and geopolitical shocks in the international system of a different magnitude. As the spills primarily stem from the military expenditure of the allies, we interpret negative shocks (lower quantiles) as beneficial shocks (peace event) related to a lower level of military expenditure, and positive shocks (upper quantiles) as adverse shocks (crisis event) related to a higher level of military expenditure. Therefore, a period of geopolitical crisis is associated with a positive (adverse) shock that corresponds to the right-tail dependence of the conditional distribution, while the opposite holds for a peaceful event that is associated with a negative (beneficial) shock that corresponds to the left-tail dependence of the conditional distribution. The empirical evidence, presented in Figure 1, reveals an overall increasing trend of the TSI as we move from the lower to the upper quantiles. Specifically, in the case of a peaceful event, quantiles (τ = [0.10, 0.20, 0,30]), the values of the TSI are (62.2, 64.3, 69.0), while in the case of a normal geopolitical period, quantiles (τ = [0.40, 0.50, 0.60]), the values of the TSI are (73.4, 75.1, 75.4). Finally, in the upper tail of the distribution, which corresponds to a crisis event taking place, (τ = [0.70, 0.80, 0.90]), the values of the TSI are (76.2, 79.3, 81.8). The findings above align with the behavior of an effectively working alliance during crisis periods. The spills produced by the alliance for consumption by their members increase during these times. We interpret this as evidence that some selected members are motivated to contribute only during periods of crisis. This conclusion has been further reaffirmed since the onset of the war in Ukraine.

Further, we observe that the growth rate of the TSI, which is interpreted as the responsiveness of the alliance to exogenous geopolitical shocks, decreases at a higher rate after a beneficial shock (peaceful event), compared to the rate at which it increases after an adverse shock (crisis event). The above finding implies that despite the increasing trend of the TSI, as we move from the lower to the upper quantiles, there is space for improvement when it comes to the cohesion of the alliance during normal and peaceful periods.

In this section, we apply the connectedness methodology to empirically examine our model, detecting net spillover effects among NATO members included in our sample. We examine the net directional connectedness between the allies of our model. We interpret the net directional connectedness as the difference between the spills provided by ally i to others allies and the spills received by ally i from the other allies S⌣i,others=S⌣others←i−S⌣i←Others. Therefore, the net directional connectedness is a measurement of the net spill contribution of each member to the alliance. In other words, the net directional connectedness shows the net contribution of each member to the common security good produced by the alliance. A positive net directional connectedness from ally i to other members of the alliance S⌣i,others>0 implies that the ally is a net provider of security spills and the allies are net receivers in their relationship with ally i , while the opposite holds for negative values of net directional connectedness. Extreme negative values of net spills are a sign of possible free-riding as the corresponding country receives more security spills than it provides to the alliance. The same analysis applies when it comes to the net bidirectional connectedness which measures the spills received and provided between two specific members i, j of the alliance and S⌣i,j=S⌣j←i−S⌣i←jand S⌣j,i=S⌣i←j−S⌣j←i. A positive net bidirectional spills value from ally i to ally j S⌣i,j>0 or a negative spills value from ally j to ally iS⌣j,i<0 means that ally i is a net spill-over provider in its relationship with ally j and ally j a net spills receiver.

Table 3 shows the various connectedness measures estimated at the median (τ = 0.5). We observe that as expected, the United States is the largest security spills contributor of our system S⌣US←Others=101.1 and the largest net spills provider S⌣US,others=S⌣others←US−S⌣US←others=101.1−70.3=30.8. Further, we observe that the southeastern Mediterranean countries are net receivers of security spills. Specifically, the net spills transmission mechanism is from the allies to Türkiye S⌣TUR,others=S⌣others←TUR−S⌣TUR←Others=46.4−77.5=−31.1, from the allies to Italy S⌣ITA,others=S⌣others←IT−S⌣ITA←Others=53.2−80.6=−27.4 and from the allies to Greece S⌣GR,others=S⌣others←GR−S⌣GR←Others=64.1−70.1=−6.0. In contrast, the northern allies of NATO included in our empirical model, namely Germany S⌣DE,others=S⌣others←DE−S⌣DE←Others=89.0−77.7=11.3, the United Kingdom S⌣UK,others=S⌣others←UK−S⌣UK←Others=78.1−77.5=0.6, and the Netherlands S⌣NL,others=S⌣others←NL−S⌣NL←Others=93.9−72.0=21.9 are net providers of security spills. We can also see that the United States S⌣USA←Others=70.3 and Greece S⌣GR←Others=70.1 are the lowest spills receivers from other allies, while Italy is the largest spills receiver S⌣IT←Others=80.6.

Overall, during normal periods, the United States and the northern allies (the United Kingdom, the Netherlands, and Germany) are net providers of security spills, while the southeastern allies (Italy, Greece, and Türkiye) are net spills receivers. Therefore, our empirical results, in line with George and Sandler (2018, 2022) and the predictions of our theoretical model stated in Equation (5), support the possible free-riding behavior of some NATO allies, as they receive more spills than the spills they provide. It should be noted that a small negative value of net spills does not necessarily imply free-riding behavior, as it could be attributed to a disproportionately high level of spills provided by the dominant ally (United States). In contrast, an extreme negative value of net spills reinforces the indication of free-riding behavior. The empirically verified free-riding behavior during the normal period is alarming as it indicates insufficient military preparation, thus explaining the failure to deter Russia from invading Ukraine, in line with the findings of George and Sandler (2022).

To examine the dynamics of the system, we plot the time-varying evolution of the TSI evaluated at the conditional median (τ = 0.50) and the right-tail dependence (τ = 0.90) of the conditional distribution, which corresponds to positive shocks, namely to adverse political or geopolitical events that increase the tension between NATO and its adversaries (Figure 2). We see that the total spill effects (TSI) are of greater magnitude for adverse political or geopolitical shocks (τ = 0.90) (right-hand graph) than for normal periods. Specifically, our empirical results in the right-tail dependence (Figure 2B) are consistent with a higher magnitude of security spills after the Gulf War (1990–1991), the September 11^th, 2001 attacks in the United States, the War in Afghanistan (2001), and the Iraq War (2003). Further, we can see a gradual decrease of spills between 2010 and 2015, which can be explained by the gradual de-escalation of the war in Iraq and a decrease after 2015 because of the disengagement of the United States from the war in Afghanistan. The evolution of the TSI evaluated at the average connectedness exhibits similar patterns that are accompanied, however, by spills, as expected, of lower intensity. It should be noted that the increase in the spills that happened during the period of the Russian invasion of Ukraine is pronounced in the media quantile (τ = 0.5, left-hand side graph), rather than the right-tail dependence (τ = 0.9, right-hand graph), as expected. This finding is related to the fact that the reaction of the alliance to the Russian invasion of Ukraine is characterised by a low profile and an indirect engagement in order to avoid an escalation of conflict between NATO allies and Russia.

Figure 3 shows the evolution of dynamic spills evaluated at the right-tail dependence of the conditional distribution. Specifically, when it comes to the dynamic total directional spills from ally i to others (Figure 3A), we see that for the entire period depicted, the United States is the major spills provider to the rest of the allies at an increasing magnitude after the terrorist attacks of 9/11 up to 2016, when we observe that there was a relative shift in the United States’ stance concerning NATO. As a result, spills from the United States to other allies decreased thereafter. Our findings regarding the United States are also consistent with the general low-profile engagement of NATO allies in Russia’s invasion of Ukraine, as a low degree of spills from the United States during that period is evident. The same conclusions could be extracted by the evolution of the dynamic net total directional spills of the United States (Figure 3B). We also see a gradual decrease in both total directional spills from the United Kingdom to other allies and net total directional spills of the United Kingdom after the mid-2000s, which becomes more intense after Brexit, except for the period 2019–2021, which can be interpreted as spills stemming from the defence policy of the newly elected government, mostly because of the pressure on the United Kingdom to increase its defence spending to meet its NATO obligations. For the Netherlands, Germany, and Italy, we see mixed results, where periods of positive alternate with periods of negative net spills. For Greece, our results are consistent with a negative net spills effect after the 2008 economic crisis, which seems to change to a positive spills effect after 2020, because of the gradual restoration of macroeconomic and fiscal stability. Finally, we observe that Türkiye’s net spills contribution can be divided into two distinct periods, namely a period before and a period after the beginning of the 2010s, which could be interpreted as a gradual shift in the country’s defence policy.

Overall, our theoretical model incorporates not only the spill-ins but also the spill-outs, namely the spill cost that stems from the obligations of each ally. Our empirical results, in line with the predicted free-riding behaviour of our theoretical model, provide evidence in favour of free-riding of the northeastern member states due to the costly spill-outs added to the cost of acquiring defence equipment. Our findings suggest specifically that the North-European NATO member states included in our sample tend to be net contributors to the alliance, while the selected South-European member states tend to be net receivers. It also seems that the selected NATO member states are only motivated to contribute in crisis periods, a finding that may indicate insufficient military preparation and has been more evident in light of the war in Ukraine. The above findings are in line with the predictions of the seminal model of Olson and Zeckhauser (1966) and Plümper and Neumayer (2015) as well as with the more recent evidence provided by George and Sandler (2022). The latter concluded that the pattern of free-riding and lack of response to heightened Russian defence spending probably encouraged the invasion, as NATO appeared divided. On the other hand, our results are in contrast with those of Alley (2021) who finds that free-riding behavior based on economic weight is unusual in alliance politics, which may be due to limits on security as a public good or bargaining between alliance members. Finally, the above empirical analysis consistently interprets the major geopolitical events of the examined period.