The interplay between economic growth and military spending has been extensively studied, but a comprehensive quantitative analysis of the interaction between military expenditure and currency value remains limited. This research utilizes the concepts of hard power (coercive power through military or economic means) and soft power (non-coercive influence through cultural or political means) to explore this relationship. The study aims to determine whether a positive correlation exists between the growth rates of military hard power and monetary soft power. The significance of this research lies in understanding the potential impact of military spending on a country's currency and economic stability, offering valuable insights for policymakers and economic analysts. While existing literature examines the relationship between military spending and economic growth, this study uniquely focuses on the interaction between military hard power (measured by military expenditure per GDP) and monetary soft power (measured as the ratio of exchange rate growth to broad money growth). The research posits that excessive broad money issuance may depreciate a currency, and aims to investigate the role of military spending in counteracting this effect.

Previous research has explored the multifaceted relationship between economic growth and military expenditure. Some studies suggest that economic growth facilitates military development and spending (Beckley, 2010), while others posit that military expenditure can stimulate economic growth (Dunne & Smith, 2020; Santamaría et al., 2021; Su et al., 2020) or have little to no effect (Manamperi, 2016). The impact of monetary policy on exchange rates and economic development has also been studied (Pham, 2019; Gala, 2008). However, the causal relationship between hard power and soft power, particularly focusing on monetary soft power, remains an under-explored area. Existing studies largely focus on the relationship between international trade and exchange rates (Kang & Dagli, 2018), neglecting the potential influence of military spending on currency value. This study aims to bridge this gap by analyzing the direct causal link between military expenditure and monetary soft power.

The study employs a novel methodology combining mathematical and statistical techniques to analyze the causal relationship between military hard power (MHP) and monetary soft power (MSP). MHP is defined as the growth rate of military expenditure per GDP, while MSP is defined as the ratio of the annual growth of the exchange rate (in terms of US dollars) to the annual monetary growth. Data from 1961 to 2019, sourced from The World Bank, was used. The data includes time series for broad money, exchange rate, and military expenditure per GDP for available countries. The data handling process involved creating samples of countries with available data for each year, calculating MHP and MSP values, and categorizing them using k-means clustering (k=5 for MHP, k=4 for MSP). The core of the methodology centers on evaluating causality through the identification of optimal empirical relations and comparing them to absolute causal relations (benchmark). The study utilizes a formalization of causal relations based on total linear relations, employing several key definitions: probabilistic absolute causal relation (Definition 1.2), causal product (Definition 1.3), and causal correlation coefficient (CCC, Definition 1.4). Optimal empirical relations are identified by maximizing the difference in CCC values between different pairs of cause-and-effect. The Hausdorff distance between the sets of optimal empirical relations and the set of absolute causal relations serves as a measure of the strength and consistency of the causal link. This distance measures the deviation from a perfect linear relationship, thereby indicating the degree of causality. Time series of these distances are analyzed to identify the dynamic relationship between MHP and MSP over time. The detailed algorithms are outlined in the supplementary appendix.

The study presents its findings through a series of graphs visualizing the causal correlation coefficients and Hausdorff distances. Figures 1 and 2 show the maximal difference between optimal causal relations for MHP to MSP (MaxCCCYX) and MSP to MHP (MaxCCCXY), respectively. These figures indicate a relatively consistent and observable causality between the two variables. Figures 3 and 4 illustrate the Hausdorff distances (average and minimal) between optimal empirical and absolute causal relations for both directions (MHP to MSP and MSP to MHP). These figures demonstrate a distinct separation between the two directions, implying a clear unidirectional causality. The asymmetry between the Hausdorff distances in Figures 3 and 4, and further visualized in Figure 5, clearly indicates a strong unidirectional causality from MHP to MSP. The consistent and substantial gap between the distances strongly suggests a one-way causal relationship from MHP to MSP. Specifically, the analysis of the maximal causal correlation coefficients (Figs. 1 and 2) shows a consistent difference between the relations from MHP to MSP and from MSP to MHP, indicating an asymmetry in the relationship. The analysis of Hausdorff distances between optimal empirical and absolute causal relations (Figs. 3, 4, and 5) reveals a persistent and substantial gap, supporting the unidirectional causal effect from MHP to MSP. These findings indicate that military hard power has a consistent and stronger influence on monetary soft power compared to the vice versa. The overall findings show that while the specific values of the correlation coefficients and Hausdorff distances fluctuate somewhat over time, the core relationship of a unidirectional causal effect from MHP to MSP remains consistently observed and statistically significant.

The results strongly support the hypothesis that military hard power bolsters monetary soft power, but not the other way around. The significant and persistent asymmetry observed in the causal correlation coefficients and Hausdorff distances clearly indicates a unidirectional causal relationship. This suggests that appropriate levels of military spending can positively influence a country's currency value and overall economic stability. This finding challenges traditional perspectives that view military spending as purely a burden on the economy. The methodology employed provides a robust framework for analyzing the causal relationships between complex variables, avoiding limitations of traditional statistical approaches by utilizing mathematical tools that offer more intuitive interpretations. The findings contribute valuable insights to the fields of international relations, political economy, and economic policy. This understanding has significant implications for policymakers in allocating resources and formulating economic and military strategies.

This study presents a novel approach to analyzing the causal relationship between military hard power and monetary soft power using data-driven probabilistic methods. The findings strongly indicate a unidirectional causal link where military hard power positively influences monetary soft power. This suggests that appropriate military spending can enhance currency stability. Future research could explore the incorporation of additional variables into this framework, such as the number of nuclear warheads or interest rates. Furthermore, incorporating fuzzy logic into the analysis could provide a more nuanced understanding of the causality. Combining this approach with other predictive modeling techniques would allow for improved quantitative estimations of the causal relationships.

The study relies on optimal surjective relations for inferences, potentially neglecting other potential causal relationships. The choice of k-means clustering parameters (number of clusters) could also introduce some degree of arbitrariness. Additionally, the focus on linearity might overlook non-linear relationships that may exist between the variables. While the study uses a comprehensive dataset, the availability of data for certain countries might affect the generalizability of the findings. Finally, other indicators of hard and soft power could potentially provide a more thorough analysis.