Macro-micro Economic System Simulation B.S.S. Onggo1, K. Kusano2, and T. Sato2 1) Management Science Department, Lancaster University Management School Lancaster, LA1 4YX, United Kingdom 2) The Earth Simulator Center, Japan Agency for Marine-Earth Science and Technology, 3173-25 Showa-machi, Kanazawa, Yokohama, 236-0001, Japan Emails: [email protected], [email protected], tetsuya@ jamstec.go.jp

Abstract The compute power required in the simulation of mutually cooperating systems in the physical sciences is huge. Therefore, it is common to simulate one system and treat others as parameters or environmental conditions to the simulated system. Macro-Micro Interlocked (MMI) simulation framework has been proposed to simulate such cooperating systems. In this paper, we study the feasibility of applying this framework on the simulation of macro-micro economic model. This simulation allows us to observe the system’s behaviour at the macro and micro levels concurrently. Policy changes at the macroeconomic level may affect the behaviour of the entities at the microeconomic level. Similarly, the behaviour changes at the microeconomic level may affect the macroeconomic system. We implement an MMI simulation library suitable for execution on a cluster of PCs. At this early stage, the experiment shows a promising result which provides us with a foundation to experiment with a larger model.

1. Introduction Based on his observation on a number of physical system phenomena, Sato [12] proposed a simulation framework called the Macro-Micro Interlocked (MMI) simulation to model the interlocking behaviour of two (and possibly more) subsystems, namely macro subsystem and micro subsystem. The main objective of this paper is to study the possibility of applying the MMI framework to simulate similar systems in the social sciences. A macro-micro economic system is chosen as a case study. The result may open a possibility to simulate the mutually cooperating macro-

micro economic systems concurrently. This simulation allows us to observe the effect of changes in the individual behaviour at the microeconomic system on the macroeconomic system. Similarly, we can study the effect of policy changes at the macroeconomic system on individuals at the microeconomic system. The other objective of this paper is to design an MMI simulation library to allow the end-users to use the simulator easily. The remainder of this paper is organized as follows. The previous works on MMI simulation are discussed in section 2. We present the MMI simulation framework in section 3. Section 4 shows how macromicro economic simulation could fit into the MMI simulation framework. We present the design and implementation of MMI simulation library in section 5. The experiment results are presented in section 6. Our concluding remark is in section 7.

2. Related Works The Earth Simulation Center has been very active in applying computer simulation to study the atmospheric system, oceanic system, and solid-earth system, as well as other fields such as: energy, life science and material science. For example, Nakajima proposed an efficient parallel iterative method with selective blocking preconditioning to enhance the performance of a 3D geophysical simulation [8]. The Large Atmospheric Computation on the Earth Simulator (LACES) project, a joint-work with the Canadian meteorological services and academic institutions, produced a high resolution simulation of the 1998’s hurricane Earl [4]. Most studies at the Earth Simulation Center have focused on systems that are governed by the same physical law [12]. In other words, other systems outside the simulated system are treated as parameters or environmental conditions to the system. This raises a

new challenge, namely, would it be possible to simulate a number of mutually cooperating systems concurrently? No doubt, this would require compute power that may even exceed the capacity of the Earth Simulator. Sato proposed a framework called the Macro-Micro Interlocked (MMI) simulation that may provide an answer to this challenge [12]. The idea of MMI simulation framework comes from his observation of a number of physical system phenomena such as solar flares and galactic jets. If we could divide each of these systems into two cooperating subsystems, namely, macro system and micro system, we could observe that changes in the macro system might affect some entities at the micro system depending on their sensitivity towards the changes at the macro system. Subsequently, if the accumulation of changes in the affected entities at the micro systems exceeds a certain level, it may affect the macro system which may further affect the micro system (this time could be on different entities). Therefore, it is not necessary to simulate all entities at the micro system at the same time. In parallel simulation terms, this concept attempts to reduce the number of logical processes. To prove this concept, Watanabe et al. developed an MMI Aurora simulation [17]. The simulation models the macroscopic magnetosphere and ionosphere processes in the scale of 100,000km, and their interactions with the microscopic behaviour of electrons and ions in the scale of 10cm. The magnetohydrodynamic (MHD) simulation [16] was used to simulate the macroscopic interaction between the magnetosphere and ionosphere; and particle in cell with the open boundary condition [2] was used for the microscopic processes. Other applications include the MMI Rainfall formation [12].

3. Macro-Micro Interlocked Simulation MMI simulation framework is suitable for simulating two interrelated subsystems, namely, macro subsystem and micro subsystem as shown in Figure 1. It comprises a macro simulation, a micro simulation, and an interlocking mechanism. Conceptually, different simulation techniques such as discrete-event simulation, agent-based simulation, or numerical simulation can be used for each subsystem. The interlocking mechanism is implemented as a set of parameters. The macro simulation provides the parameters for the micro simulation and vice versa.

Macro simulation Interlock parameters Micro simulation Figure 1. MMI Simulation Framework Because of the interlocked relationship between the two subsystems, the macro simulation and micro simulation need to be calibrated to ensure that their simulation results are consistent before the MMI simulation is started.

4. Macro-Micro Economic Model Economics is commonly divided into two branches: microeconomics and macroeconomics. Microeconomics studies how individuals, such as: firms, households, and labour, make decisions and how the interaction among them affects markets. Macroeconomics, on the other hand, studies the behaviour of economy at the aggregate level. The two interrelated branches seem to be suitable for the MMI simulation framework. A change in the government policy at the macro level may affect individuals at the micro level. Similarly, a change in the economic behaviour of individuals at the micro level may affect the macro level. This motivates us to explore the possibility of applying the MMI simulation framework to simulate a macro-micro economic model. In this paper, a variant of Dornbusch model is chosen to simulate the movement of price level and spot exchange rate at the macro level. These two parameters are fed into the firms-labour simulation at the micro level. The firms-labour simulation gives the total real income that will be fed back into the macro level. The following subsections explain the Dornbusch model, firms-labour model, and their interlocked relationship in more detailed.

4.1. Dornbusch Model Dornbusch proposed a macroeconomic model comprising goods market, money market, and international asset market [5]. The goods market is used to model a country’s total expenditure and inflation rate. The money market is used to model the real money balances in the country. The international asset market is used to model the country’s interest rate

and exchange rate. A number of variations have since been proposed. In this study, we choose a variation of the Dornbusch model described in [11]. This model assumes a perfect foresight and capital mobility. The perfect foresight assumes that we can perfectly foresee the future movement of the exchange rate. The perfect capital mobility assumes that capital can move from one country to another without any restriction. The model is captured in the following equations. Note that all variables are in the natural logarithm except for the interest rates (r and r*). Goods market e = cy + g + h(s-p) dp/dt = a(e – y) Money market md = p + ky – ur md = ms = m

demonstrate that the framework is suitable for the simulation of a macro-micro economic model. Further, this model also fits into our future plan because it can be extended to represent a more complex relation among a number of countries, i.e., more dots at the macro level. The relaxation of the assumptions will make the system dynamic more complex. Different exchange rate regimes will further complicate the model.

0 < c < 1, h > 0 a>0

k > 0, u > 0 Figure 2. MMI Economic System

International asset market r = r* + dse/dt dse/dt = ds/dt

4.2. Firms-Labour Model

where e = total expenditure y = total real income g = government spending s = spot exchange rate p = domestic price level dp/dt = inflation rate md = demand for money ms = supply of money m = money balances r = domestic interest rate r* = foreign interest rate dse/dt= expected depreciation/appreciation ds/dt = change in spot exchange rate We can derive the following two ordinary differential equations from the above equations. dp/dt = a(g – (1 – c)y) – ahp + ahs ds/dt = (ky – m)/u – r* + p/u

(1) (2)

The two equations form a system dynamic that models the interaction between a country (shown as the black dot at the macro level in Figure 2) and an international market consisting of other countries (the grey dots at the macro level). For the purpose of this study, this macroeconomic model is suitable because it can be easily validated analytically. Hence, we can focus on the implementation of the MMI simulation to

At the micro level, we will use the agent-based simulation to simulate a number of sectors (such as manufacturing and agriculture) and labour in one country. Each sector comprises a number of firms. At this stage, we use a simplified version of the similar models used in Basu et al. [1], Sprigg et al. [13], and Tongeren [15].

Production Plan Labour Market Production Goods Market End? Measurement Figure 3. Micro Level Simulation The main algorithm is shown in Figure 3. Each firm generates a production plan based on its historical sales data, price level, and spot exchange rate. At this stage, we focus on firms that produce goods only. The production plan states the number of workers needed to

meet the production targets. Based on the offered wage and the number of workers available at each sector, workers are assigned to each firm in the labour market. The workers join the work force if the wages offered by firms exceed their reservation wages. Next, production begins. The number of products produced by each firm depends on the size and productivity of its workers. The products are then sold in the goods market. The demand for goods comes from domestic consumption as well as from overseas (export). Finally, measurements such as total income can be calculated. At the production stage, firms in a sector may need goods from other sectors. The interdependency among sectors is reflected in the flow of incomes and expenditures among them. In the simulation, this flow of incomes and expenditures is represented using a simplified version of the widely-used Social Accounting Matrix [15]. Each cell in this matrix represents a credit (income) to the row account and a debit (expenditure) to the column account. This model is chosen because it provides the basic mechanisms of goods market and labour market. We plan to extend this model by incorporating results from research in the behavioural analysis of the microeconomic agents.

interaction of the parameters. The Dornbusch model gives the movement of the price level (equation 1) and the spot exchange rate (equation 2) which are fed into the firms-labour simulation at the micro level. First, the price level and the spot exchange rate determine the export and import values at the micro level (i.e. firms). This will determine the total demand which affects production and employment. The firms-labour simulation gives the total real income (y) which is the summation of the capital income and the labour income. The total real income will then be fed back into the Dornbusch model at the macro level. This is different from the original Dornbusch model where the total real income is treated as a constant. The total real income directly affects the change in the spot exchange rate and indirectly affects the inflation rate through consumption. This may cause subsequent changes at the macro level and the simulation cycle continues. Figure 4 also shows two parameters: government spending (g) and money balances (m) that can be used to control the macro system. The main challenge happens when a larger number of agents with more realistic behaviour are used at the micro level model. In this case, the performance of the simulator will be affected. E et al. summarized a number of possible solutions in the numerical simulation of physical systems which could be applied here [6]. If the micro level simulations that experience some perturbations can be isolated (it is referred to as the type A problem in the paper), then only these micro level simulations need to be simulated in parallel. The results are then aggregated and sent to the macro level

4.3. Macro-Micro Interactions After the macro model and the micro model are identified, the next step is to define their interactions. In this example, the interaction is done through three parameters: price level (p), spot exchange rate (s), and total real income (y). Figure 4 shows detailed

Micro

Macro

p Export - Import

Consumption

Inflation (equation 1)

g

Exchange rate (equation 2)

m

s

Firms - Labour

y

Figure 4: Macro-Micro Economic Model Interaction

simulation. If the macro level simulation must be derived from all micro level simulations (the type B problem) then a sample of micro level simulations will be run and the results will be adjusted to represent the population of micro level simulations.

simulation library, or a discrete-event simulation library. In this example, we use a numerical simulation library for the macro model (Dornbusch) and an agentbased simulation library for the micro model (firmslabour).

5. Design and Implementation

6. Experiments

The previous prototypes, such the Aurora simulation, are written in FORTRAN and are specially designed for the vector processors at the Earth Simulation Center. In this paper, we also aim to implement the MMI simulation on a cluster of PCs. Figure 5 shows the architecture of our proposed MMI simulation library.

We implement our simulator in C++ and MPI library is used for the interprocessor communication. The experiments are conducted on a cluster of Sun Fire X4100 servers connected via a gigabit Ethernet switch. Each node has two dual-core 2.4GHz Opteron CPUs and 8GB memory. In this section, we present the following: the parameters used in the simulation, the validation of our simulator, the experimental results and the performance of our simulator.

6.1. Parameters

Figure 5. Software Architecture The MMI simulation library consists of a simulation engine and two interfaces: MMI model and MMI parameter. The macro model and micro model must implement the MMI model interface which has two sets of interlocked parameters, namely InParams and OutParams. The interlocked parameters implement the MMI parameters interface. The current simulation engine uses simple time-step synchronization. The macro level simulation runs on a coarser step than the micro level simulation, for example, one simulation time unit at the macro level is a quarter and the one at the micro level is one week. Synchronization is done every time the simulation time at all micro level simulations reach one simulation time unit used in the macro level simulation. In this architecture, the user needs to develop a macro model, a micro model, and their interaction. The models may use any existing simulation libraries which can be a numerical simulation library, an agent-based

Before the MMI simulation is run, we set both the price level and the spot exchange rate to be 100. Economists often arbitrarily set index numbers, such as the price level (p) and the spot exchange rate (s) at particular time, which do not have any economic interpretation. However, their rates of change have a clear economic interpretation, i.e., inflation rate (dp/dt) and depreciation rate (ds/dt), respectively. With these values and other parameters, we run the firms-labour (micro level) simulation. We simulate 11 sectors, and the number of firms within each sector varies from 45 to 9410 (on average 2400 firms per sector). The production capacity for each sector varies from 514m units per quarter to 14b units per quarter, which are distributed to each firm based on a uniform distribution. The fixed production costs are generated as a function of capacity. The total number of workers is 30m. At this stage, we assume workers are provided to each sector and they cannot switch to other sectors. Further, we assume workers within a sector have the same productivity level. This simulation gives the total real income which is used to calibrate the Dornbusch (macro level) simulation. The calibration gives us the following parameters. c = 0.94 a = 0.1 m = 17.2

g = 1.8 k = 0.1 r* = 4

h = 8.22 u = 21.43

6.2. Validation We validate the macro simulation (Dornbusch model) analytically based on the equilibrium theory. At

p = s + g / h – (1 – c) y / h p = m – ky + ur*

(3) (4)

From equation 3 and 4, we have the equilibrium point ( s , p ) = (100, 100). Figure 6 shows the phase plane of this system. The two equilibrium lines intersect at the equilibrium point and create four quadrants. The vector of forces in each quadrant is also shown. Shone provides a detailed explanation on how these vectors are derived [11]. This shows, for example, any point in quadrant I will be directed towards either quadrant II or quadrant IV, depending on its initial position. p

make the comparison to historical data inappropriate even if such data exist. As a matter of fact, validation is still one of the main obstacles that hinder the widespread use of simulation in the social sciences. Therefore, at this initial stage, we are more concerned about the correctness of basic mechanisms such as: the effect of changes in the price level and spot exchange rate on: export and import, offered wage (hence the unemployment rate), consumption, and total real income. Preliminary tests show no contradictory results. 100.0 99.0 98.0 p

the equilibrium, dp/dt = ds/dt = 0. From equation 1 and 2, we can derive:

97.0 96.0

dp/dt = 0

95.0

I

II

94.0 97.5

s, p

98.0

98.5

99.0

99.5

100.5

100.0

s

ds/dt = 0

Figure 8. Simulation Result – Quadrant III

III IV

6.3. Simulation Result s Figure 6. Phase Plane The macro simulation produces consistent results. Two of them are shown in Figure 7 and 8. In Figure 7, the simulation starts at (100, 105) in quadrant I. After 1000 time steps, the forces move it to a new position in quadrant II. Similarly, in Figure 8, the starting point is (100, 95) in quadrant III, after 1000 time steps, it moves to a position in quadrant IV.

The MMI simulation allows us to observe the behaviour of the system at the macro and micro level as shown in Figure 9. At the macro level, to control the price level, the central bank reduces the money supply (m) when the price level drops for more than 5 points and increases the money supply when the price level soars for more than 5 points. The changes in the price level and spot exchange rate affect the real incomes of firms and labour at the micro level. Subsequently, the total real income affects the price level and spot exchange rate, and so on.

105.0

103.0

Index

p

104.0

102.0 101.0 100.0 99.9

100.0

100.1

100.2

100.3

100.4

100.5

100.6

100.7

s

Figure 7. Simulation Result – Quadrant I At the micro level, the validation is very difficult without adequate historical data. Furthermore, at this stage, many simplifications have been made which

104 102 100 98 96 94 92 90 88 86 84 82

£40 £35 £30

Trillions

106.0

£25

p

£20

s

£15

y

£10 £5 £0 Timestep

Figure 9. Movement of Price Level (p), Spot Exchange Rate (s) , and Total Real Income (y)

Billions

£3,700

8 7 6

Speed-up

Figure 10 shows another result at the micro level, i.e., the movement of capital income and labour income over time. Figure 11 shows the distribution of income among different sectors at the end of the simulation.

5 4 3 2

£3,600

1

£3,500

0

£3,400 labours

£3,300

capital

£3,200 £3,100

2

4

8

Processors

Figure 12. Simulation Performance

£3,000 £2,900

7. Conclusions and Future Works

£2,800 Timestep

Figure 10. Capital Income and Labour Income

11

12

10

3 4 5

9

6 8 7

Figure 11. Income Distribution Among Sectors

6.4. Performance Although our research is still at an early stage, the early performance indicator is important for us to analyze the feasibility of using our simulator to simulate a larger macro-micro economic model. For this performance study, we simulate eight countries with the same characteristics. In terms of economic relation, these countries form a ring-like network, i.e., each country is economically influenced by and influences its two neighbouring countries. Figure 12 shows the speed-up obtained by executing the simulator on 2, 4, and 8 processors. This initial study shows that our simulator can achieve good speedup. This is partly due to the many simplifications in the micro model which reduces the amount of data to be passed across processors. For example, we do not consider the direct relationships between firms from different countries, workers do not move from country to country, and so on. Therefore, it is reasonable to expect the performance to drop if a more realistic model is attempted.

This paper shows our work in analysing the feasibility of extending the application of the MMI simulation framework to social science, specifically in the simulation of a macro-micro economic model. This study has provided us with a foundation to continue with our research. There are three future directions to be pursued. First, on the modelling side, we will attempt to simulate a larger and more realistic macro-micro economic model. A larger number of micro level simulations will affect the scalability of the overall MMI simulation which leads us into the second research direction. Second, on the technology side, we need to address the performance issues such as the potential bottleneck at the macro level simulation. The current time-step synchronization may not cope with larger and more realistic models. In the future, we plan to use the parallel discrete-event simulation (PDES) kernel. Recently, highly scalable PDES kernels have been reported. Park et al. reported a good scalability result on their conservative PDES kernel running on a cluster of 128 processors [9]. Riley et al. applied federated simulation techniques to create large-scale parallel simulations of computer networks [10]. They simulated a network size of 2 million nodes using 128 processors. Chen and Szymanski ran a large-scale simulation of a synthetic benchmark called PHOLD on 1033 processors [3]. Third, on the usability aspect, we need to address how to make the simulation transparent to the potential users who are typically trained as social scientists or professionals. This will complement the works of other researchers in promoting the application of parallel simulation in wider real world problems such as: the work by Tang et al. who demonstrated the initial study in applying the parallel discrete-event simulation on a

plasma physics application [14] and the work by Lobb et al. in simulating a neuron model [7].

8. Acknowledgement This research is supported by the Daiwa AngloJapanese Foundation grant no: 5913/6090 and the Lancaster University Management School 2006 pump priming research fund.

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“Optimistic Parallel Discrete Event Simulations of Physical Systems Using Reverse Computation”, Proceedings of the 19th Workshop on Principles of Advanced and Distributed Simulation, 2005, pp. 26-35. [15] Tongeren, F.W, Microsimulation Modelling of the Corporate Firms: Exploring Micro-Macro Economic Relations. Springer, 1995. [16] K. Watanabe, M. Ashour-Abdalla, and T. Sato, “A Numerical Model of the Magnetosphere-Ionosphere Coupling, Preliminary Results”, Journal of Geophysics Research, 91 (A6), 1986, pp. 6973-6978. [17] K. Watanabe, T. Sugiyama, A. Kageyama, K. Kusano and N. Ohno, “Development of Micro-Macro Interaction Simulation Algorithm”, Annual Report of the Earth Simulator Center, April 2004 - March 2005.