A Game-Theoretic Analysis of the Coexistence and Competition Between Hard and Fiat Money

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Comments and Suggestions

1. Introduction:

   • Examples for Improvement:

     1. Simplify Technical Terms: The phrase "utility exponents for hard money and fiat money" is overly technical for the introduction.
        • Improvement: It could be replaced it with "preferences of the player and bank for holding different types of money" to make it more accessible.
     2. Highlight Real-World Implications: The introduction states, "This article contributes to understanding the competition and coexistence between hard and fiat money."
        • Improvement: You could expand this to include specific examples, e.g., "such as Bitcoin's role in modern economies or the impact of CBDC adoption on inflation."
     3. State Main Findings Clearly: The conclusion in the introduction is vague, e.g., "The findings have implications for regulators and central banks."
        • Improvement: You could specify, e.g., "The findings indicate that increasing interest rates for fiat money can lead to inflationary pressures, while higher hard money rates stabilize inflation but may disadvantage consumers."

2. Materials and Methods:

   • Add Software Information: The methodology lacks mention of tools or software.
     • Improvement: You could include, e.g., "Simulations were conducted using MATLAB, and the code is available upon request."
   • Explain Assumptions: Assumptions like the one-period game model are not fully justified.
     • Improvement: You could add a rationale, e.g., "The one-period assumption simplifies the analysis and allows for a focus on strategic interactions without temporal complexity."
   • Provide Visual Aids: The dense mathematical explanations could use diagrams.
     • Improvement: You could add a conceptual flowchart summarizing the relationship between variables, such as the player’s utility function and the bank’s strategic choices.

3. Results:

   • Improve Figure Labels: Figures like Figure 1 lack self-explanatory labels.
     • Improvement: Replace "n (number of players)" with "Number of players (n): Effect on Inflation and Utility."
   • Interpret Practical Implications: Numerical results, such as "inflation increases when fiat money is printed," lack broader discussion.
     • Improvement: You could add, e.g., "This result suggests that central banks should carefully monitor the impact of money printing to avoid destabilizing inflation."
   • Highlight Key Takeaways: The results section does not emphasize the most important insights.
     • Improvement: Use a summary table to present key findings, such as the relationship between interest rates and inflation.

4. Discussion:

   • Simplify technical language to ensure accessibility for readers from related fields.
   • Expand on policy implications, such as recommendations for central banks managing fiat money printing or withdrawal.
   • Discuss broader macroeconomic impacts, such as the implications of digital currencies on global trade.

5. Conclusions:

   • Emphasize the broader relevance of the findings for regulators, financial institutions, and individuals.
   • Suggest specific areas for future research, such as extending the model to multi-period settings.
   • Conclude with a strong, impactful takeaway to leave a lasting impression.

Comments on the Quality of English Language

Suggestions about language

1. Introduction

Examples for Improvement:

1. Rephrase Jargon-Heavy Sentences: The sentence "Hard money has fixed supply regardless of whether it is physical or digital" is repetitive and verbose.
   • Improvement: Simplify to "Hard money has a fixed supply, whether physical or digital."
2. Simplify Sentence Structure: "Central banks research, explore, and test CBDCs to develop efficient competitive fiat payment systems."
   • Improvement: Use clearer phrasing, e.g., "Central banks explore CBDCs to enhance fiat payment systems."
3. Clarify Phrasing of Research Questions: The sentence "How do transaction costs impact the player's consumption, assets holdings, borrowing, and utilities?" can be clearer.
   • Improvement: Change to "What impact do transaction costs have on consumption, asset holdings, borrowing, and utility?"

2. Materials and Methods

Examples for Improvement:

1. Use Clearer Language for Equations: The description "Utility is additive and not multiplicative" is unclear for non-specialists.
   • Improvement: Rephrase to "The utility function assumes independent contributions from each component, summing their effects rather than combining them multiplicatively."
2. Avoid Overloading Sentences: The sentence "The bank’s utility function reflects both kinds of banks" is ambiguous.
   • Improvement: Specify, e.g., "The bank’s utility function incorporates features of both commercial and central banks."
3. Clarify Parameter Explanations: The sentence "The parameter γ represents the fraction of helicopter money assigned to the player" could be more intuitive.
   • Improvement: Use, e.g., "The parameter γ defines the share of newly printed money allocated to the player in the model."

3. Results

Examples for Improvement:

1. Explain Results in Layman Terms: The phrase "Inverse U-shaped utility curve" is too technical.
   • Improvement: Use, e.g., "Utility initially increases with interest rates but declines after a certain point."
2. Simplify Descriptions of Findings: The phrase "The player decreases its hard money loan due to the bank’s negative money printing strategy" is unclear.
   • Improvement: Rephrase to "The player borrows less hard money as the bank reduces the money supply."
3. Clarify Graph Interpretation: The sentence "The inflation rate becomes negative as hard money interest exceeds 4.61%" lacks context.
   • Improvement: Add, e.g., "A negative inflation rate suggests deflation, reflecting reduced fiat money in circulation."

Author Response

See attachment

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Congratulations

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

Dear authors,

To enhance the paper " A Gametheoretic Analysis of the Coexistence and Competition Between Hard and Fiat Money", the following improvements could be considered:

While the literature section is comprehensive, the subsections sometimes overlap and lack a clear transition. Improving the structure can enhance readability. I suggest to introduce a brief overview paragraph at the beginning of Section 2, explaining the logic behind dividing the literature into seven categories; and conclude Section 2 with a summary that synthesizes how the reviewed literature connects to the current paper’s contributions.

Although the paper reviews various works, it could better emphasize the specific research gap that this paper aims to fill. Please highlight that most existing works on currency competition focus either on fiat vs. fiat currencies or cryptocurrencies vs. fiat, but few directly explore the interplay between hard money (e.g., gold, Bitcoin) and fiat money in a game-theoretic setting, and emphasize that the combination of player-bank dynamics and money printing/withdrawal strategies is underexplored in existing models.

Some areas, particularly on CBDCs and cryptocurrency adoption, have seen rapid development in the past few years. Please consider incorporating:

• Recent central bank reports (e.g., BIS 2023) on the evolution of CBDCs and their competitive implications.
• Empirical studies on countries adopting Bitcoin as legal tender (e.g., El Salvador post-2021) or implementing gold-backed digital currencies (e.g., Zimbabwe in 2023).
• Work on DeFi and algorithmic stablecoins, which further blur the lines between hard and fiat money (e.g., Terra collapse in 2022).

Regarding The Model Section, you have to Improve Intuition and Economic Interpretation. The section is heavily technical, and the economic intuition behind the equations could be better integrated. Please provide economic explanations alongside the derivations. For example, when defining the utility functions, explain why the exponents reflect preferences for consumption and asset holdings. Also, when introducing the transaction cost function, explain more clearly why the cost is modeled as a function of both the volume of transactions and money holdings, and how it aligns with real-world payment systems.

The utility functions and transaction costs are reasonable, but their forms are specific and could benefit from: abrief discussion on alternative functional forms (e.g., Cobb-Douglas, CES) and why the chosen additive and power-exponent structure is preferred and please clarify why the additive structure for utility is chosen, despite possible complementarities between consumption and money holdings.

The inflation rate is defined based on the money supply dynamics. It would be helpful to relate this formula (Equation 7) to standard monetary theories, such as the Quantity Theory of Money (MV = PY), to ground it in traditional monetary economics. Consider whether velocity of money or price-level expectations should play any role in the inflation formula, or at least acknowledge their absence as a simplification.

Consider adding a diagram to illustrate the flow of funds between the player and the bank. I recommend to show arrows representing loans, money printing, withdrawals, and holdings of different assets. This would help visualize the resource constraints (Equations 1 and 5) and clarify the simultaneous interaction of money flows and utility optimization.

Enhancing conceptual clarity alongside the mathematical rigor will broaden the paper’s accessibility to readers beyond mathematical economists, including policymakers and monetary theorists.

The section Illustrating and Interpreting the Model is dense with technical descriptions of Figure 1 panels. Consider breaking it into smaller subsections, such as:

• Impact of Interest Rates on Hard and Fiat Money
• Effect of Loans and Money Printing on Inflation
• Player vs. Bank Trade-offs
• Resource Allocation Dynamics

This structure will help readers navigate the interpretation of parameter variations more easily.

While the section provides detailed descriptions of parameter effects, the underlying intuition is sometimes lost. You could improve this by summarizing the economic logic after discussing each graph, and highlight the most surprising or counterintuitive results upfront, such as the paradox where higher hard money interest rates reduce player utility.

I recommend to condense or Summarize the Less Significant Panels, because the 14 less notable parameter variations in Appendix D could be further condensed into a summary table. Please focus  the main section on the 6-8 most impactful or non-intuitive parameter effects.

The Conclusion Section should start with a concise summary of the paper’s most important findings. Currently, the insights are somewhat scattered throughout the paper. Condense the core results into 3-4 key takeaways, such as:

• The bank’s control over fiat money supply grants it a strategic advantage over the player, enabling it to reduce inflation through money withdrawal, often at the expense of player welfare.
• Increasing interest rates on hard money can paradoxically lower player utility because it prompts the bank to reduce fiat money supply, leading to liquidity constraints for the player.
• Rising interest rates on fiat money benefit both the player and the bank, encouraging fiat money printing and increasing inflation, ultimately driving hard money out of circulation.
• The coexistence of hard and fiat money is fragile, as fiat money’s flexibility (printing and withdrawal) gives the bank long-term dominance in controlling inflation and shaping the economy.

A clear, structured, and policy-oriented conclusion that connects the theoretical findings to real-world monetary challenges will strengthen the paper’s impact on academic audiences.

Author Response

See attachment

Author Response File: Author Response.pdf

Round 2

Reviewer 3 Report

Comments and Suggestions for Authors

Dear Authors,

Thank you for your diligent efforts in addressing all the recommendations and suggestions provided during the review process. The revisions have significantly enhanced the quality and clarity of the paper, reflecting a thorough and thoughtful approach to the feedback received.

Given the substantial improvements, I am pleased to recommend the paper for publication. I appreciate your commitment to academic rigor and contribution to the field.