Model Structure of Agent-Based Artificial Economic System Responsible for Reproducing Fundamental Economic Behavior of Goods Market

Abstract

Agent-based modeling (ABM) features its capability not only to deal with the heterogeneity of agents but also to elucidate the causal mechanism of social phenomena. The latter can be done by clarifying the model structure required to reproduce the phenomenon through systematic computer experiments. This article presents some examples of such studies that uncovered the causal mechanism of the goods market’s fundamental economic behaviors, including price equilibrium, business cycles, the effect of tax cuts in both income and corporate taxes. The condition of the validity of ABM and the causal mechanism of business cycles are also discussed.

Appendix: Overview, Design Concepts, and Details Protocol

This appendix describes the model in terms of the Overview, Design Concepts, and Details (ODD) protocol by Grim et al. (2006).

1.1 Purpose

The purpose of this model is to experimentally elucidate the underline mechanism of the complex macroeconomic phenomena. The model also aims to clarify the conditions under which the model structures reproduce these phenomena in an agent-based artificial economic system where macroeconomic indicators emerge as a result of agents’ actions and interactions. In the present study, the purpose of this model focuses on elucidating the model structure to reproduce the positive influence of corporate tax reduction on GDP and to obtain a clearer understanding of the mechanism behind this effect.

1.2 Entities, State Variables, and Scales

The entities included in this model are agents, goods, and markets, which are the minimum requirements of a macro economy of a nation. Agents include the following: consumers, comprising workers, and executives of private sector firms and public workers; producers, comprising retailers, raw-material makers, and an equipment maker; a bank; and a government. Goods include consumption goods for any agents, raw-material goods for retailers, and equipment as capital goods for retailers and raw-material makers. The market is divided into a consumption goods market and a raw-material goods market. We assume that capital goods transactions take place directly between equipment makers and buyers. Figure 4.19 shows the relationship between these entities, including the flows of goods, labor, and funds among the agents.

Fig. 4.19

Outline of the entities and their relationships

The entities included in this model and their characteristics are described in Table 4.4.

Table. 4.4 Entities included in the model and their characteristics

State variables are divided into those for agents and those for other entities. Each agent belongs to a different category, according to its behavior, such as agents in general, buyers, enterprises, and producers. State variables for agents are divided into those for an agent’s behavioral category and those peculiar to each type of agent. State variables for each behavioral category are described in Table 4.5, and the state variables peculiar to each type of agent are described in Table 4.6. The state variables of other types of entities are described in Table 4.7.

Table. 4.5 State variables for agents’ behavioral category

Table. 4.6 State variables peculiar to each type of agent

Table. 4.7 State variables of other types of entities

Tables 4.5, 4.6, and 4.7 present the characteristics of the state variables: the initial settings, differences among agents, and how values change with a change in time step. The initial settings are the values assigned to the state variables of the objects when the objects are created. The difference among agents shows whether the values are the same or different among the agents. The change in time step indicates whether the values are time dependent.

1.3 Process Overview and Scheduling

The present model consists of three submodels: a fund circulation submodel, a price equilibrium submodel, and an investment submodel. The fund circulation submodel constitutes the fundamental structure of the model in which the latter two submodels are implemented. The model consists of three processes: initialization, where the objects of the entities are created and initialized; the sequence of seven actions performed by agents during each time step; and the calculation of the average GDP and other statistical data of macroeconomic indicators. The seven steps comprise the actions at the beginning of every time step, the production of raw materials, the production of consumption goods, purchasing of consumption goods, payment of wages, actions for investment, and the actions at the end of every time step. The pseudocode that describes this process is given in Fig. 4.20, and the sequential events conducted by each type of entity during each time step are described in Fig. 4.21.

Fig. 4.20

Pseudocode of the model

Fig. 4.21

The sequential events in each time step for each type of entity

1.4 Design Concepts

1.4.1 Basic Principles

The general concept underlying the model design of ABM is that the behavior of an artificial economic system can mimic the real-world behavior if the model structure and the structure of the real systems have a homomorphic relationship. This relationship is considered to be fulfilled when the structural factors of the modeled system are essentially the same as those of the real system with respect to the relevant macroscopic economic phenomenon. Therefore, ABM can be useful in describing the mechanism of a macroeconomic phenomenon by performing controlled experiments in which only one factor of interest varies at a time, while holding other factors constant. In this way, ABM clarifies the structural conditions necessary for the model to reproduce the macroeconomic phenomenon being studied.

1.4.2 Emergence

The modeled artificial systems should include heterogeneous and autonomous agents. Their behavioral rules might be similar, but the values of their state variables should be different. Therefore, the heterogeneous agents behave differently and interact with each other. Macroscopic phenomena emerge from these actions and interactions, which affect the microscopic behavior of the agents, resulting in a micro–macro link in the dynamics of the systems. In this way, artificial economic systems can behave as complex systems.

1.4.3 Adaptation

Retailers and raw-material makers adjust the price and number of products they supply to the market by gauging the demand in the market. To do so, they observe the number of stock items that remain unsold at the end of each time step. These producers also use the market demand to adjust their production capacity and number of employees. In this way, the artificial economic systems in this study possess an internal adaptation mechanism.

1.4.4 Objectives

Consumers and producers hold their own objective functions, such as maximizing utility or profits.

1.4.5 Prediction

Retailers and raw-material makers predict the total sales of their goods based on the sales figures from the ten most recent periods. Based on this prediction, they decide on the amount of production in the next time step so that the probability of goods being out of stock is less than 5%. When they decide to invest in equipment, they first predict the financial benefit of the investment by estimating the increase in profit gained from a one-unit increase of equipment and the subsequent increase in production capacity.

1.4.6 Sensing

Retailers and raw-material makers gauge the market demand by observing the amount of goods still in stock at the end of each time step. They also calculate the optimal number of employees based on the profit of the current term, as well as the potential financial benefit of increasing or decreasing the number of employees and, therefore, their production capacity.

1.4.7 Interaction

The price equilibrium is loosely attained by the interaction between agents’ purchasing actions and producers’ actions when adjusting their production levels and product prices. In addition, the circulation of funds and the emergence of various macroeconomic indicators, such as GDP, are the result of the actions of agents and their interactions. The investment behavior is also a result of the interaction between buyers and producers.

1.4.8 Stochasticity

Various state variables are randomly defined at the start of the simulation or during the simulation. Typical examples are agents’ initial funds, state variables that distinguish agents (e.g., product classes), production capacities, utility weights, and a consumer’s workplace. These are defined using random numbers with a uniform distribution.

The order of agents’ actions for the same type of agent is also defined by shuffling the set of agent id numbers using a uniform random number at every time step.

1.4.9 Observation

At the end of each time step, each agent settles its account using the double-entry bookkeeping method. An input-output table for the artificial system is defined by summing the calculated data for all agents. The macroeconomic indicators, such as GDP, tax revenue, total funds for each type of agent, total salaries paid by producers, and the total number of investments, are calculated based on the input-output table and other account data of the agents. In addition, statistical data, such as the total number of goods produced or bought during each time step, the average price of products, and the amount of funds circulated between the bank and other types of agents, are also calculated at the end of each time step. The average values of these data for the overall simulation can also be obtained and used for various types of analysis.

1.5 Initialization

All state variables of the agents are initialized when the agent objects are created. These initial values are described in Tables 4.8 and 4.9. An agent id is sequentially assigned for each agent, but this number is only used to distinguish agents. Each agent object is initially assigned randomly to one of the types of agents. Public employee’s salaries are calculated in each fiscal period so that they are equal to the average income of private employees.

Table. 4.8 Initialization of state variables for agent’s behavioral category

Table. 4.9 Initialization of state variables peculiar to each type of agents

1.6 Input Data

No data from the real system is used as input data for the simulation.

1.7 Submodels

1.7.1 Funds Circulation Submodel

This submodel constitutes the fundamental structure of the model, the outline of which is described in the pseudocode presented in Fig. 4.20. The basic principles of the circulation of funds and additional behavioral rules are presented below.

1.7.1.1 Basic Principles of Fund Circulation

Consumers work for one of the other agents, receive wages, buy consumption goods produced by retailers, and pay income tax to the government. Retailers produce consumption goods using raw material goods supplied by raw-material makers, where minimum units of supply chain processes are implemented in the model. The behavioral rules for the strategies of consumer purchasing and producers’ production are described in the price-equilibrium submodel. Retailers and raw-material makers invest in equipment when doing so will increase their profit. The investment strategies are described in investment submodel.

The government levies income tax and corporation tax, pays wages to public employees, and conducts public expenditure, comprising market purchasing as an extreme case of efficient public spending and firm subsidies as an extreme case of inefficient public spending.

In this way, funds circulate among agents in the artificial economic systems as a result of agents’ actions and interactions.

1.7.1.2 Related Behavioral Rules

• 1. Agents’ behavioral rules for determining their consumption budget.

  Every agent, other than the bank, determines a consumption budget at the beginning of each time step. The definitions of the budget are different each type of agent.

  For the consumer agent:

  $$ {E}_{b}^{t}=a+{bI}^{t}+{r}_{\mathrm{wd}}^{t}{D}^t $$

  where E _b ^t: Consumer’s consumption budget; a: Basic consumption; b: Marginal propensity to consume; I ^t: after-tax income; r _wd: Withdrawal ratio; D ^t: Bank deposit.

  For the producer agent: Purchasing ratio multiplied by the amount of internal funds.

  For the government agent:

  $$ {E}_b^t={E}_{\mathrm{all}\_b}^t-{\mathrm{wage}}_G^t $$

  where ${E}_b^t$: Total public expenditure budget; ${E}_{\mathrm{all}\_b}^t$: Total amount of tax revenue; ${\mathrm{wage}}_G^t$: Total salaries paid to public employees.

  The budgets for market purchasing and for firm subsidies are defined as the ratio of the respective amount of public expenditure to the total budget.

• 2. Payment of salaries.

  • 2-1 Salaries paid by enterprises.

    Each enterprise agent pays a fixed salary, a bonus, and executive compensation.

    The total amount paid as salaries depends on both the before-tax profit and accumulated profit, as given below:

    $$ {E}_w^t=\left\{\begin{array}{l@{\qquad}l@{\qquad}l}{W}_f&\mathrm{if}&{\pi}^{t-1}<0\\ {}{W}_f+{W}_b^{t-1}&\mathrm{if}&{\pi}^{t-1}>0\kern0.24em \mathrm{and}\kern0.24em \mathrm{AC}<0\\ {}{W}_f+{W}_b^{t-1}+{\mathrm{EC}}^{t-1}& \mathrm{if}&{\pi}^{t-1}>0\kern0.24em \mathrm{and}\kern0.24em \mathrm{AC}>0\end{array}\right. $$

    where E ^t _w: Total salary amount; W _f: Fixed salary; W _b: Bonus; EC: Executive compensation; AC: Accumulated profits.

    The total amount of salaries paid to workers or to executives is given below.

    $$ {W}_C^t={W}_f+{W}_b^{t-1}/\mathrm{ne}\operatorname{}\mathrm{for}\kern0.34em \mathrm{workers}\vspace*{-6pt} $$
    $$ {W}_C^t={W}_f+{W}_b^{t-1}/\mathrm{ne}+{\mathrm{EC}}^{t-1}\operatorname{}\mathrm{for}\kern0.17em \mathrm{executives} $$

    where W ^t _c: Total salaries paid to workers or to executives; ne: The number of employees.

  • 2-2 Salaries paid by the government

    The government pays fixed salaries to public workers based on the previously determined budget for wages.

• 3. Agents’ behavioral rules for settling accounts at the end of each time step

  • 3-1 The rules for consumers

    Consumers define the amount of income tax to be paid based on their income and remember this as the amount of unpaid tax:

    $$ {\mathrm{Tax}}_i={W}_C^t\;r{}_{i\_\mathrm{tax}} $$

    where Tax_i: The amount of income tax; r _{i_tax}: The income tax rate.

    A part of consumers’ income, including unpaid tax, is kept on hand as cash and deposited in the bank, as given below:

    $$ \mathrm{deposit}=\left(1-b\right)\left({W}_C^t\left(1-{r}_{i\_{\mathrm{tax}}}\right)\right)-a $$

  • 3-2 The rules for producers

    Producers define their profit based on total sales and total expenses:

    $$ {\Pr}_p^t={S}^t-\left({W}_f+\sum {\mathrm{co}}^t+{\operatorname{int}}^t+{\mathrm{dep}}^t\right) $$

    where Pr_p: The profit before bonus; S: Total sales; Σco: Total expenses for raw materials; int: Interest to be paid; dep: Depreciation expenses.

    Producers define the amount to be paid as bonuses to employees based on the profit before bonuses, as given below. They remember this as the amount of unpaid bonuses:

    $$ {W}_b^t={\Pr}_p^t{r}_{\mathrm{bonus}} $$

    where r _bonus: The ratio of bonus.

    Based on this value, they define their before-tax profit as given below:

    $$ {\Pr}_{a\_{\mathrm{tax}}}^t={\mathit{\Pr}}_p^t\left( 1-{r}_{\mathrm{bonus}}\right) $$

    where Pr^t _{a_tax}: The before-tax profit.

    Then, they calculate the amount of corporation tax to be paid and remember this as the amount of unpaid tax:

    $$ {\mathrm{Tax}}_c={\Pr}_p^t\left( 1-{r}_{\mathrm{bonus}}\right){r}_{c\_{\mathrm{tax}}} $$

    where Tax_c: The amount of corporation tax; r _{c_tax}: The rate of corporation tax.

    Based on this value, they define their after-tax profit and executive compensation, and remember this as unpaid executive compensation:

    $$ {\mathrm{EC}}^t={\Pr}_p^t\left( 1-{r}_{\mathrm{bonus}}\right)\left( 1-{r}_{c\_{\mathrm{tax}}}\right){r}_{\mathrm{exec}} $$

    where r _ecex: The ratio of executive compensation.

    Extracting the executive compensation from their after-tax profit enables producers to define their accumulated profit, as given below:

    $$ {\mathrm{Ac}}^t={\mathrm{Ac}}^{t-1}+{\Pr}_p^t\left(1-{r}_{\mathrm{bonus}}\right)\left(1-{r}_{c\_{\mathrm{tax}}}\right)\left(1-{r}_{\mathrm{ecex}}\right) $$

  • 3-3 The rules for the government.

    The government defines the total amount of tax revenue and expenses, and passes the resultant money on to the next period.

• 4. Others

  • 4-1 The rules for dismissal

    At the end of each time step, the retailer fires one of its employees if its dismissal flag reaches a critical value. The employee to be fired is selected at random and is assigned to the producer with the largest accumulated profit.

  • 4-2 The rules for stopping production and for bankruptcy

    At the end of each time step, the producer stops production of a certain class of product if its flag reaches a critical value. When a producer stops all its product classes, it then goes bankrupt, and a new producer object is created with new initial variables.

1.7.2 Price Equilibrium Submodel

The present model mimics the price equilibrium in the market according to the following two principles.

1.7.2.1 Lowest-Price-Oriented Purchasing Strategy by Buyers

All buyers purchase consumption goods within the limits of their consumption budget. If there are products within the same product class, but with different prices, they will select the cheapest of them. The consumption goods bought are indexed by buyer’s id and are removed from the market and moved to the buyer.

In addition, consumers purchase products to maximize their utility within the limit of their consumption budget.

$$ \max u=\sum \limits_i{w}_i{x}_i^{\alpha}\operatorname{}\operatorname{}\mathrm{s}.\mathrm{t}.\sum \limits_i{p}_i^t{x}_i\le {E}_b^t $$

where w _i: The weight of utility for each product of class i; x _i: The number of products to purchase; p _i: The price of a product; α: An exponent of x _i.

1.7.2.2 Stock-Control-Oriented Production Strategy by Sellers

• 1. The behavioral rules used by producers to determine the price of their products.

  The price of a product is defined according to the number of products in stock and the amount bought in the market.

  $$ {p^t}_i=\left\{\begin{array}{l@{\qquad}l@{\qquad}l}\left(1+{\gamma}_i\right){p^{t-1}}_i&\mathrm{if}&{s}_i^{t-1}=0\\ {}\left(1-{\gamma}_d\right){p^{t-1}}_i&\mathrm{if}&{s}_i^{t-1}>0\kern0.24em \mathrm{and}\kern0.24em {p^{t-1}}_i<{p}_{\mathrm{avei}}^{t-1}\end{array}\right. $$

  where γ _i: The ratio of a price increasing; γ _d: The ratio of a price decreasing; s ^t−1 _i: The amount of goods in stock at the end of previous period; p ^t−1 _avei: The average price of goods bought in the market in the previous period.

• 2. The strategy for amount to be produced (the production plan).

  • (a) The number of products to be produced in a given period is defined so that the probability of goods being out of stock is 5%:

    $$ {q}_{si}^t\operatorname{}={q}_{\mu i}^t+1.65{q}_{\sigma i}^t $$

    where q ^t _si: Target number of goods in stock; q ^t _μi: Average sales during the past ten periods; q ^t _σi: Sigma of total sales during the past ten periods

  • (b) Producers decide on the number of products to produce according to the number of products in stock, adjusting their target as shown below.

    $$ {\displaystyle \begin{array}{l}{q^t}_i=\left\{\begin{array}{l@{\qquad}l@{\qquad}l}{q_s^t}_i\left(1+\varepsilon \right)&\mathrm{if}&{s}_i^{t-1}=0\\ {}{q_s^t}_i\left(1-\varepsilon \right)-{s}_i^{t-1}&\mathrm{if}&{s}_i^{t-1}>0\end{array}\right.\\ {}\qquad\quad\mathrm{If}\kern0.36em {q^t}_i>{Y}_i\left(K,L\right)\kern0.48em {q^t}_i={Y}_i\left(K,L\right)\end{array}} $$

    where q ^t _i: The amount of production; ε: The ratio of changing amount of production; Y _i(K, L) = A _i K ^α L ^1-α: Production capacity; K: The number of units of equipment for production; L: The number of employees; A: A proportionality constant.

1.7.3 Investment Submodel

1.7.3.1 Producers’ Behavioral Rules for Investment Decisions

The retailer or raw-material maker decides to invest when the three conditions listed below are fulfilled. Once the agent decides to invest, it becomes a candidate for investment and is included in the list of candidates owned by an equipment maker.

Conditions for investment:

• 1. The investment flag number exceeds a critical value for investment.

• 2. The financial benefit from an increase in one unit of equipment is positive, as given below.

  $$ \Delta {\pi}_K=\underset{i}{\max}\left[\left({p}_i^t-{c}_i^t\right)\left\{{Y}_i\left(K+1,L\right)-{Y}_i\left(K,L\right)\right\}-\left({r}_0+1/N\right)F\right]>0 $$

  where p _i: The price of goods of product of class i; c: The variable cost per unit product; r ₀: The borrowing interest rate; F: The borrowed money required to buy one unit of equipment; N: The repayment period.

• 3. The accumulated profit at the end of current term is greater than half the necessary funds for investment.

1.7.3.2 The Behavioral Rules for Equipment Makers

The equipment makers randomly select one of the candidates for investment and sell that agent a unit of equipment. If there is more than one candidate, the equipment maker continues to produce and sell until the number of equipment units reaches the equipment maker’s production capacity.

1.7.3.3 Producers’ Behavioral Rules for Financing and Buying Equipment

The selected retailer or raw-material maker purchases one unit of equipment. Before purchasing, the agent finances half the necessary funds using internal funds from accumulated profits and the other half from the bank. After purchasing, the retailer or raw-material maker renews its production capacity by increasing the number of units of equipment by one in the Cobb–Douglas-type equation.