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2019年美国大学生数学建模竞赛（ICM）F题特等奖论文.pdf

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    Introduction
    Background
    Statement of the Problem
    Assumptions and Notations
    Assumptions
    Notation of Parameters
    Mathematical Model
    Digital Currency System
    The Money Market
    Basic Theory
    Individual Level - Micro Level
    National Level - Macro Level
    The Domestic Product Market
    The Foreign Exchange Market
    Analysis of Our Model
    Choice of Exchange Rate Institution
    Fixed Exchange Rate Institution
    Balance of International Payments
    Policy Effects
    Shocks to the Economy
    Internal and External Disequilibrium Adjustment
    Floating Exchange Rate Institution
    Balance of International Payments
    Policy Effects
    Impact effect under floating exchange rate system
    Real Cases Analysis
    Analysis of Fixed Exchange Institution – Sweden
    Analysis of Floating Exchange Institution – Israel
    Evaluations of the Model
    Sensitivity Analysis
    Test for Robustness
    Conclusions
    Strengths and Weaknesses
    Strengths
    Weaknesses
    Conclusion
    Appendices
    Appendix Real Data Result
    For office use only T1 T2 T3 T4 For office use only F1 F2 F3 F4 Team Control Number 1904381 Problem Chosen F 2019 MCM/ICM Summary Sheet Digital Currency System is Coming Summary In recent years, digital currency has developed rapidly and become an in- dispensable part of the modern economic system. We construct a brand-new dynamic macroeconomic operation system, taking digital currency into account. Our model is based on three perspectives: individual choice, national economy, international economy. We first consider the individual behavior choice deci- sion, which is because the aggregate demand of all micro-individuals selections determine the domestic economic operation state, describing as the product mar- ket equilibrium and money market equilibrium. We use models to analyze three short-term economic as well as a long-term economic performance objective, un- der different exchange rate systems, economic shocks and policies. We first build a virtual country whose central bank issues the new digital currency. The impact of digital currency on the current banking system and macroeconomic system is determined by the economic interaction with a par- ticular country in the real world. We discuss two different exchange rate systems that a country will choose. Using theoretical analysis and R, we find that either systems would achieve in- ternal and external equilibrium and realize economic objects if proper policies are implemented when digital currency is introduced. The innovation of our model is that we introduce the parameters of interna- tional capital flows (ICF) to the model. We analyze the impact of ICF on economy of the country both theoretically and empirically. It is inspiring that we find that decentralized digital currency can not only run the economy more efficiently by removing barriers to currency flows, but also enhance the world’s welfare by the ultra-high liquidity of digital currency. To ensure the new banking and the macroeconomic systems involving digital currency can operate with stability and effectiveness we propose all countries accepting digital currencies co-create a United Nations-affiliated organization– the World Digital Money Bank (WDCB) and develop a series of regulations. Keywords: digital currency;fixed exchange rate ;float exchange rate
    Team # 1904381 1 Introduction 1.1 Background Page 1 of 27 Considering the application of the digital currency, there lies both advantages and disadvantages. Its digital forms can enable instantaneous transactions and borderless transfer-of-ownership, which improves the efficiency of the markets and constructs a more convenient form of financial exchange. But in the mean- time, while not technically money, their value is tied to real-world currencies which from the very beginning has placed them in a very precarious situation. Lack of regulation around these currencies and their anonymity also bring risk to both citizens and economic analysts. Therefore, modeling the working mechanism of financial system adding the new type of currency is necessary. This model would enable us to have a clear overview of the market model and help make policies with regard to both regu- lations of the currency market and stablity of the system. 1.2 Statement of the Problem • Task1: Construct a model in which digital currency is taken into consider- ation, and the model should adequately elucidate this financial system. • Task2: Identify the viability and effects of a global decentralized digital financial market and develop a kind of general digital currency. • Task3: Identify major factors which will limit or facilitate its growth, access, security, and stability at both the individual, national, and global levels. • Task4: How will the countries modify their current banking and monetary models according to their different needs and their willingness to work with this new financial marketplace? And what are the consequences of these modifications? • Task5: Include the mechanisms for oversight of the global digital currency built above. • Task6: Analyze the long-term effects of this system on the current banking industry; the local, regional, and world economy; and international rela- tions between countries. • Task7: What will happen if the countries abandon their own currency and only use digital currency?
    Team # 1904381 Page 2 of 27 2 Assumptions and Notations 2.1 Assumptions In order to better quantify our model, we may relax some of the assumptions. • Consumption and savings occur in individuals, while production and in- vestment occur in the enterprise sector. Individuals and enterprises have to pay taxes. • Consumers are rational and pursue maximum utility. • The government has two kinds of behavior: government purchase expen- diture and tax revenue. • Depreciation and undistributed profits are zero. GDP, NDP, NI, PI and DPI are equal. • The government aims to achieve economic growth, adequate employment, price stability and internal and external equilibrium under balance of inter- national payments. 2.2 Notation of Parameters We make a list of parameters involving in the model as shown in Tabel 1. 3 Mathematical Model In order to construct a model that sufficiently represent the brand-new finan- cial system, we should take the willingness of countries to accept digital currency into account and primarily consider two different paths of developing digital currency, which laid the foundation of our mathematical model. Path I Central banks in all countries refuse to accept digital currency. In this case, the system will be the same as today’s, which rarely influence the tradi- tional currency market. Path II NOT ALL central banks refuse to accept digital currency. When a central bank of a country with certain international influence that regard digital currency as traditional currency, the central bank will be willing to pay a certain amount of traditional currency to purchase digital currency owned by the public, and will sell digital currency to potential demanders to earn a certain amount of traditional currency. From the analysis above, it can be seen that digital currency will directly enter the existing open macroeconomic model.
    Team # 1904381 Page 3 of 27 Parameters Table 1: List of Parameters and Notations Descriptions Y C I G T X M ¯m y i m1 m2 u α, β L1(Y ) L2(i) ¯Ms ¯Ms1 ¯Ms2 P k, h CA F A N X N F AX AM Pf e iw γ, n, σ, φ GDP of a country Consumption of a country Investment Government purchase Tax Exports Imports Total currency demand of individuals, a constant in a short period Individual income level Market interest rate level The amount of money held in traditional currency The amount of money held in digital currency Individual utility level Parameters. Depending on individual micro-traits Money demand caused by transactions motives and precautionary motives Money demand caused by speculative motive is a function with regards to interest rate Total money supply in the form of digital currency Total money supply in the form of traditional currency Total money supply in the form of digital currency Price level measured by inflation rate Constants The balance of current account The balance of financial account Net export Net capital inflow Capital outflow Capital inflow Price level of foreign countries, measured by inflation rates of foreign countries Exchange rate of national currency under direct price method Interest rates of foreign countries Parameters
    Team # 1904381 Page 4 of 27 In the following sections, we will mainly focus on path two. Our idea is to discuss the details of our mathematical model from three different perspectives in macroeconomics: 1) the domestic product market 2) the money market in- cluding individual level and nationanl level 3) the foreign exchange market. In addition, we would observe how every single model changes with the introduc- tion of digital currency and analyze the working mechanism of all these three general models. 3.1 Digital Currency System The figure above demonstrates us the development of international mone- tary system. It is evident that this sytem will be changed with the introduction of brand-new type of currency–digital currency. Therefore, we name this new system as Digital Currency System. • Build a virtual country Regardless of all the existing digital currencies, we assume a mysterious orga- nization issues a brand-new digital currency, which is initially held by a minority. Also, we envision there is a virtual country. The central bank of this country is the very mysterious organization that issues the digital currency, the currency in circulation is digital currency and the residents of this country are the individuals holding the currency. We assume the total amount of digital money issued by the central bank of this virtual country as a constant. Each unit of electronic money can be infinitely divided. The unit price of digital money measured in real-world currency de- pends on the economic interactions between the virtual country and the coun- tries in the real world. Denote the virtual country as A. In the following analysis, we suppose a particular country in the real world will be economically associated with virtual countries through existing curren- cies and digital currencies. And this international economic relationship can be characterized by FE curve. Starting with the FE curve, we will in turn analyze the IS curve and the LM curve of this particular country which has economically associated with the virtual country.
    Team # 1904381 Page 5 of 27 3.2 The Money Market 3.2.1 Basic Theory • Money Supply Money supply refers to the process in which the economic entity creates the money supply and puts it into circulation. That is, the behavior and process of the central bank and commercial banks investing, expanding or contracting the money in circulation. The money supply refers to the sum of the currency held by enterprises, in- dividuals and foreign countries outside the banking system and other deposit currencies that are freely available at any time. • Money Demand Money demand refers to the amount of money the public is willing to hold after comprehensively weighing the benefits and costs of various assets. • Liquidity Preference Theory Keynes’s study of money demand is based on a study of the motives of de- mand for economic agents. Keynes believed that peopleâ ˘A ´Zs demand for money is determined by three motives: 1. The transactions motive: in order to make daily transactions, people must hold money 2. The precautionary motive: people prefer to have liquidity to cope with unexpected situations. 3. Speculative motive: due to the uncertainty of the future interest rate, peo- ple will adjust the asset structure and demand more money to hold in order to avoid capital loss or increase capital gains. 3.2.2 Individual Level - Micro Level Modern new classical macroeconomics tend to focus on the determination of money demand from the perspective of dynamic optimization. Based on the new classical currency model, we interpret currency as the amount of treasure that people are willing to retain under the premise of invariant income. Hence we propose the following model to describe the economic behavious at the micro level. • Individual Currency Choice
    Team # 1904381 Page 6 of 27 Denote ¯m ¯m(y, i). According to our model, we have ¯m = m1 + m2 • Individual Money Utility Function Indifferent Curve: u = mα 1 mβ 2 where α, β are parameters. They depend on individual micro-traits, such as risk preferences, liquidity preferences, etc. Assumptions: 1. Individual preference is normal. 2. Monotonicity. 3. Con- vexity. In the next section, we try to solve this equation through both mathematical calculation and explicit figures. max m1,m2 u = mα 1 mβ 2 subject to ¯m = m1 + m2 We assume that both u and ¯m have continuous first partial derivatives. To find the maxium of this function, we introduce a new variable λ as the Lagrange multiplier. L(m1, m2, λ) = mα 2 − λ(m1 + m2 − ¯m) 1 mβ We take the partial derivatives with regard to m1, m2 and λ.  ∂L ∂m1 ∂L ∂m2 ∂L ∂λ = αmα−1 1 mβ 1 mβ−1 2 − λ = 0 2 − λ = 0 = βmα = ¯m − m1 − m2 = 0 =⇒  m1 = m2 = α α + β β α + β ¯m ¯m (1)
    Team # 1904381 Page 7 of 27 (a) individual choice between currency (b) The points that are not on the curve Figure 1: The slope 3.2.3 National Level - Macro Level 1. Total Demand for Money: M = ¯m = m1 + m2 2. Total Supply for Money: Ms = ¯Ms1 + ¯Ms2 3. Balance: L(Y, i) = L1(Y ) + L2(i) = kY − hi M = L(Y, i)  M = ¯Ms1+ ¯Ms2 P =⇒ i = − 1 h Ms P + k h Y , where k, h : are parameters. Figure 1 explains the economic meaning of points that are not on the curve. As noted in the figure 1, the points above the LM curve indicate insufficient money demand of the country while those below the curve represent excessive demand for money.
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