Mathematics in Economics

Abstract

Economics is the study of how resources are used as well as an analysis of the decisions made in allocating resources and distributing goods and services. Mathematics is the language of numbers and symbols that can be used to logically solve problems and precisely describe size, quantity and other concepts. Some complex problems could not be described and complicated problems could not be acted upon without the language of mathematics and its support of logical processes to solve problems. Mathematical modeling is used in economic analysis to study existing economic relationships and it helps economists study what-if scenarios to see what might happen to the economy if a certain action is applied. Economic concepts and relationships can be measured in mathematical indexes, formulas and graphs. Several areas of mathematics can be utilized in economic analysis, including linear algebra, calculus, and geometry. Since economic concepts can be complex, it is important to use care in representing data and relationships in isolation. Results can be misinterpreted based on the representation of data.

Overview

Economics. Economics is "the study of how people choose to use resources which can include land, labor, money, equipment, taxes and investments" ("What is economics?", n.d.). The broad field of economics has many descriptions each trying to discover a way to make clear what the field covers. De Rooy (1995, xiii) states that economics is about people and "…the things they do." Decisions have to be made in life and people are constantly making decisions about resources at the individual, company and government levels. Gottheil (2007, p.1) describes economics as "an important branch of the social sciences" that is all about resources. This discussion of resources surrounds the fact that resources are limited and people have needs and wants. Scarcity is a term used to explain these limited resources. Typically, resources have to be distributed to people in the form of goods and services. Gottheil notes that resources are limited but the wants of people are not. The three main problems which Gottheil ascribes to economic study are:

• The problems related to scarce resources and unlimited wants of humans;
• The problem of making choices about allocation of resources to produce goods and services;
• The problem of distributing completed goods and services to people.

Gottheil points to distribution of goods and services as a way to understand the economy. Investorwords.com (n.d.) defines economy as "Activities related to the production and distribution of goods and services in a particular geographic region."

Mathematics. Large (2006, p. 2) describes the field of mathematics as "the study of the relationship between size, shape and quantity, using numbers and symbols." These numbers and symbols convey meaning in a clear, concise and consistent way. Mathematics is a way to explain, explore, decipher and analyze complex concepts that otherwise might be resistant to synthesis. Math can also demonstrate whether or not a theory is true. The logical process of problem solving using mathematics can give precise answers to complicated, compound and multi-faceted problems.

Mathematics in Economics. Mathematics is helpful in economics because it can help quantify or provide measurement and meaning to economic concepts. Mathematics also plays a large role in the area of economic analysis. Economics uses modeling to describe certain states of being and to analyze economic scenarios. Modeling suggests what will happen if certain actions are taken. Simulation of real world situations is possible with economic analysis and modeling and would not be possible without mathematics. Barnett, Ziegler & Byleen (2008, p. 210) describe mathematical modeling as "the process of using mathematics to solve real-world problems." The authors further break mathematical modeling into three steps:

• Constructing the model. This step provides the basis for understanding the problem or question being asked. Once an answer is given to question asked by the model, it can be used to apply the information learned to a real-world problem.
• Solving the mathematical model.
• Interpreting the solution to the mathematical model in the context of the real-world problem that needs to be solved.

These steps are repeated over and over again as long as information is needed to solve the real-world problem. Gottheil notes that modeling in economics helps "to understand cause-and-effect relationships in the economy." Lim (2001, p.2) calls mathematics "a useful tool for the analysis of all economic problems" and says it is "practically indispensable for many." Lim lists the following mathematical concepts as important to economics:

• Calculus;
• Linear algebra;
• Differential equations.

Micro- & Macroeconomics. There are sub-branches of economics. Two important ones are microeconomics and macroeconomics. Each branch covers different types of decisions. Microeconomics is the study of economic relationships and decisions of the individual. Macroeconomics looks at the economy as a whole. Economists utilize models to analyze both micro and macroeconomics using supply and demand as the foundation (Baumol & Blinder, 2001). "Supply and demand are the most fundamental tools of economic analysis" (McAfee, p. 14). Macroeconomics looks at measures that affect the whole economy including inflation, unemployment, Gross Domestic Product and prices. Aggregate demand and supply are macroeconomic measures that provide "a way of illustrating macroeconomic relationships and the effects of government policy changes" (Riley, 2006). Every part of the economy has an interrelationship with other parts of the economy. As an example, if the population increases, the demand for certain goods will increase. Similarly, as goods become obsolete, demand will decrease. Various economic measures are represented as mathematical variables. Mathematics is a way to track variables under consideration and to model changes in variables.

Mathematics in Practice. There are many ways in which math in economics can be used. For example, Cooper (2009) discusses the importance of an increase in housing starts and draws a direct relationship between the decline in homebuilding and the sluggish and declining financial markets. At the current rate of decline, housing starts will fall to zero by November of 2009. In an additional example, Magner (2000, para. 2.) discusses Markus Mobius, a Harvard economics professor, who used his skills in mathematics and economics to study social problems. In his thesis, Mobius "used economic theory to explore the formation of ghettos." Mobius found that economics was inconsistent since it relied on human behavior, but purported that mathematics can compensate for these changes and variations. Mobius' strength in math helped him to become an excellent economist. Successful economists are those able to master economic concepts and mathematics at the same time (Magner).

Math's Limits. Scrubbing the Skies (2009) looks at a traditional problem of costs in attempting to reduce carbon emissions from global warming. Interestingly enough, part of the economic problem in this case is the politics involved in getting action on carbon dioxide reductions. Politics is inextricably related to economics because economics is tied to behavior and politics influences behavior. Math helps us measure, monitor and predict behaviors but cannot help to completely overcome political issues. Many industries and powerful companies may be reluctant to make costly changes to comply with tougher carbon emission regulations and there may be little 'political will' to force anyone's hand. Powerful companies can afford lobbyists to forestall change efforts and have the wherewithal to persuade politicians to adopt a position since many will need campaign contributions to remain in office. McAfee (p.7) calls the controlling interest of politics in government "political economy" or "the study of allocation by politics."

Economic Literacy. Many people and groups care about economics including governments and economists. However, economics affects individuals as well. De Rooy introduces a concept called economic literacy where an individual understands the economic environment and how the environment affects the individual. Some people may be frightened away from economic data because of mathematics. Economically literate people can interpret how economic metrics and news affects them and what economic relationships will create a successful environment for them as individuals. As a result, economically literate people will be interested in when conditions signal a recession or when prices rise along with unemployment, for example. Individuals might care about inflation because purchasing power declines with rising inflation. All of these economic concepts are represented in mathematical terms.

Economic Concepts & Measures. Mathematics can help in visualizing and quantifying economic concepts. Formulas and graphs can be used to describe and display such concepts. Mathematics is also a way to deal with uncertainty in a problem. Many economic terms can be represented mathematically. The terms are used to describe values and behaviors concerning supply and demand, the U.S. economy, producer and consumer theory, imperfections in the market and strategic behavior. The national economy is very complex so it is impossible for one single number or measure to accurately represent it whether as snapshot in time or over a period of time (De Rooy).

Mathematics in Price & Cost Measurements. Mathematics can be use to show relative size or whether something is high or low or large or small. A basic economics concept related to scarcity of resources is price or cost. Price is "the exchange of goods and services for money" (McAfee, 2006, p.7). However, the true cost or opportunity cost is based on what must be given up or what cannot be purchased because funds are allocated elsewhere. McAfee (p.9) calls opportunity cost "the value of the best foregone alternative." Price-related economic measures include the consumer price index (CPI) and personal consumption expenditures (PCE). Both CPI and PCE are indexes. An index is "a number or ratio (a value on a scale of measurement) derived from a series of observed facts; and can reveal relative changes as a function of time" (WordNet). Baumol and Blinder (p. 522) define index as a number that "expresses the cost of a market basket of goods relative to its cost in some base period." The CPI is weighted over several years while the PCE compares the current period to the preceding period. CPI and PCE play "important roles… in guiding economic policy" (McCully, Moyer, & Stewart, p. 26). De Rooy (p. 100) says prices "are at the center of a market economy because they make things happen." Price levels and changes directly impact the economy.

Econometrics. Economics has grown because of the ability to use statistical models on 50+ years of data to "test the accuracy of economic models." Econometrics is the name used for the sub-branch of economics using "statistical methods to analyze economic data" (Gottheil, p. 1). An application of statistics can be seen in the measurement of inflation ("the overall general upward price movement of goods and services in an economy"). A few of the indexes that are used by the Bureau of Labor Statistics (2008) to look at different components of inflation are:

• Consumer Price Index: "Produces monthly data on changes in the prices paid by urban consumers for a representative basket of goods and services." Two categories of consumer are measured, urban consumers, and urban wage earners and clerical workers.
• Producer Price Indexes: Measures "changes in the selling prices received by domestic producers of goods and services."
• Import and Export Prices: Measures non-military trade to and from the United States.
• Employment Cost Trends: Changes in employment and compensation costs over time.

Gross Domestic Product. The U.S. Department of Commerce has a Bureau of Economic Analysis (BEA). The BEA tracks major economic trends and makes available current information about the U.S. economy. One of the measures tracked by the BEA is GDP or Gross Domestic Product. GDP is the "broadest measure of the economic activity that goes on in the nation during the year" (De Rooy, p. 3). GDP is also described as the total market value of final goods and services produced in a year. Final goods are those that reach the end user and are not components in other products. Mathematically, GDP reports are made in percentages and typically describe either an increase or decrease in GDP. Reports will also indicate where the changes are coming from, for example, a decrease due to a decline in software.

Demographic Information. Many of the statistics that reflect information about the United States and its economy have to do with people. Tracked demographic information includes number of people, ethnicity, gender, education, place of habitation, household type, earnings and spending. Statistics regarding production are also gathered. McAfee (p. 62) gives an example of the mathematical formula for production:

The formula shows how income is divided based on what we spend, save or give to the government.

Economic Analysis

Government Analysis. There are many reasons why a country or government might want to track and analyze information related to the national or regional economy. Analysis can determine the general health of the economy and highlight trends of growth or decline. Economic analysis is "analysis that is undertaken using economic values, reflecting the values that society would be willing to pay for a good or service" (European Union funds, n.d.). Economies are driven by the production of goods and services.

The Food and Agriculture Organization of the United Nations describes economic analysis as:

"a way of looking at the worth of a development from the viewpoint of the whole economy, i.e. the general public interest or public good. These costs are adjusted to account for any distortions, such as subsidies, taxes and transfer payments."

It is the responsibility of government to work for the common good. Governments analyze the economy in order to determine how fiscal policy might impact the growth and health of the economy.

Individual & Company Analysis. Individuals and companies can engage in economic analysis as well. McAfee gives the example of a company, British Petroleum, engaging in economic analysis by using a demand model which estimates the demand of oil refineries and individual gasoline consumers. The anticipated demand from both refineries and customers impact the decisions British Petroleum makes. "Stock market analysts use economic models" to predict stock prices and company profits while governments use modeling to forecast budget deficits. Changes in "laws, rules and other government intervention in markets" can be suggested by economic analysis (McAfee, p. 8).

Mathematical Graphing. Mathematical graphing can be used in economic analysis to represent economic concepts and measures. Baumol and Blinder (p. 18) note "Economic graphs are invaluable because they can display a large quantity of data quickly and because they facilitate data interpretation and analysis." The authors continue to note that it would not be possible to take a lot of information and immediately see the relationships in the data without graphs. But, they also warn against graphs that are poorly constructed which can lead to confusion and misinterpretation. Baumol and Blinder (p.18) use an example of a misleading graph of exploding unemployment between 1990 and 1992 possibly indicating a collapse of the U.S. economy that showed only partial information. Baumol and Blinder caution against the overexposure of a single metric or isolated statistics in this way, something they decry as common in print media.

McAfee states that economic analysis serves two purposes: Positive analysis and normative analysis. Positive analysis evaluates existing economic relationships (Gottheil). McAfee (p. 8) calls positive analysis the "scientific understanding of how allocation of… scarce resources" is decided. Normative analysis is "the study of what ought to be in the economy" (Gottheil, p. 1). A central figure or star in economic analysis modeling is homo economicus — representing people acting in their own self interest. An assumption when examining the behavior of people such as consumers is that they always make decisions based on what is in their own self interest. In an economy, everyone demonstrates "maximizing behavior" as in companies wanting to maximize prices and consumers wanting to maximize value (McAfee).

Viewpoint

A Look at the Current Economic Crisis. Economic analysis is a "Systematic approach to determining the optimum use of scarce resources…" Mathematics can support this process but cannot achieve an ideal result. Nor can mathematics overcome the influence of politics on an issue. A good example of this is the economic stimulus efforts in the United States and other countries attempting to resolve the global economic crisis. There are many disagreements as to what the appropriate action should be for government, financial institutions and other private sector companies. Coy (2009) examined the varying views of economists on whether or not government intervention helps, hurts or is necessary. Although everyone may have access to the same information, it may not be interpreted the same way. "Is the economy in a dangerous downward spiral, or is this a painful but ultimately healthy adjustment leading to a sustainable growth path?" (Coy, para. 1). Misdiagnosing the situation could be a significant problem just as overreacting might increase the problem. Mathematics can help quantify the problem but it can only be used as one tool in the decision making process. Similarly, the economic problems have trickled down to the individual who may be facing foreclosure on a home or job loss. Economic analysis can describe the situation for the individual and model alternatives, but it cannot assure the success of any particular path.

In early 2008, the federal government passed the Economic Stimulus Act of 2008 which provided tax rebates in an effort to stimulate the economy. Experts are divided on the benefit of the Act. Later in 2008, the U.S. Senate approved a government rescue plan called the Emergency Economic Stabilization Act which has a provision called TARP — Troubled Asset Relief Program. The economy had suffered due to an overabundance of risky mortgage loans. TARP is considered to be a tool the government can use to repair the economy through support for financial institutions. TARP allows the federal government to purchase up to $700 billion in troubled assets as relief to financial institutions. In 2009, a second bailout or stimulus bill was passed. The American Recovery and Reinvestment Act of 2009 is a $789 billion compilation of tax cuts, additional federal spending and bolstering of social programs. Stimulus packages are controversial because the added spending will increase the national debt since the government is borrowing or creating money that does not exist. Stimulus packages are also controversial because they usually contain provisions that some see as unnecessary but were politically necessary to get votes to pass the bill. Mandel (2009) suggests that education and health have to be the focus of any economic stimulus because he views them as the most needed items that provide the best results. Mandel criticizes prior efforts to give money directly to people or banks because he does not feel that has stimulated the economy in the way funding for health and education would. McGregor (2009) reports on the companies grabbing stimulus work opportunities as private sector ones dry up. The downside is being under government control and rules. Davis (2009) also warns against accepting government support which could lead to a loss of autonomy in a business.

While money does not solve every problem, it can help. Garber (2009) reports on the Department of Energy's Energy Information Administration (EIA) struggling because of funding. EIA is falling short because of an inability to provide up to date and accurate statistics about energy production, usage and availability. In addition to stimulus money allocated for energy, EIA is seeking an increase in federal budget allocations.

The current economic stimulus plan may be a plan that does not add up. The global economic crisis puts countries like the United States in an untenable position. Something has to be done. Since the economy has so many interrelated parts, impact in one part of the economy creates a ripple effect in another part of the economy. If people are unemployed, they will spend less, if they spend less, business will sell fewer items and have to lay off people, and the cycle continues. Many areas of the economy are showing decline and the United States government has a responsibility to not allow the economy to fail. Countries holding U.S. debt are requesting assurances that they will be paid while they deal with their own economic problems.

Mathematics can measure the debt and count the size of the stimulus package. It can help create impressive graphs that map economic relationships, revenues, output and spending. It can support arguments on both sides with economic analysis and models. However, mathematics cannot guarantee that the funds will reach the intended targets with the intended results.

Terms & Concepts

Aggregate demand: "Aggregate demand is the total amount that all consumers, business firms, and government agencies are willing to spend on final goods and services" (Baumol & Blinder, 2001, p. 529).

Allocation of resources: "Society's decisions on how to divide up scarce input resources among different outputs produced in the economy and among different firms or other organizations that produce those outputs" (Baumol & Blinder, p. 60).

Economic analysis: "Systematic approach to determining the optimum use of scarce resources, involving comparison of two or more alternatives in achieving a specific objective under the given assumptions and constraints" (Economic analysis, 2009).

Economic model: "A simplified, small-scale version of some aspect of the economy expressed in equations, graphs or words" (Baumol & Blinder, p.13).

Factors of production: (sometimes called inputs) "Broad categories including land, labor, capital, exhaustible natural resources, and entrepreneurship into which we divide the economy's different productive inputs" (Baumol & Blinder, p. 32).

Fiscal policy: "The government's fiscal policy is its plan for spending and taxation designed to steer aggregate demand in a certain direction" (Baumol & Blinder, 2001).

Final goods and services: "Goods and services purchased by the end user" (Baumol & Blinder, p. 485).

Gross domestic product (GDP): "The sum of the money values of all final goods and services produced in the domestic economy and sold on organized markets during a specific period of time, usually a year" (Baumol & Blinder, p. 482).

Inflation: "Refers to a sustained increase in the average level of prices" (Baumol & Blinder, p. 482).

Recession: "A period of time during which the total output of the economy falls" (Baumol & Blinder, p. 482).

Bibliography

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Suggested Reading

Aastveit, K. A., & Trovik, T. (2008). Estimating the output gab in real time: A factor model approach. Norges Bank: Working Papers, (23/24), 2-42. Retrieved March 31, 2009, from EBSCO online database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=35781670&site=ehost-live

Gotibovski, C., & Kahana, N. (2009). Second-degree price discrimination: A graphical and mathematical approach. Journal of Economic Education, 40, 68-79.

Hrygorkyv, V. S. (2009). Some approaches to modeling prices in an ecological-economic system. Cybernetics & Systems Analysis, 45, 1-7. Retrieved March 31, 2009, from EBSCO online database Academic Search Complete. http://search.ebscohost.com/login.aspx?direct=true&db=a9h&AN=36533237&site=ehost-live

Milkman, M., & Marjadi, R. (2018). An analysis of mathematics requirements and recommendations for admission to PhD programs in economics. American Economist, 63(1), 110-114. doi:10.1177/0569434517746389. Retrieved February 28, 2018, from EBSCO online database Business Source Ultimate. http://search.ebscohost.com/login.aspx?direct=true&db=bsu&AN=128071737&site=ehost-live&scope=site

Pippenger, J. (2008). Freely floating exchange rates do not systematically overshoot. Journal of International Finance & Economics, 8, 37-56. Retrieved March 31, 2009, from EBSCO online database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=35618438&site=ehost-live

Sohel Azad, A. S. M. (2009). Efficiency, cointegration and contagion in equity markets: Evidence from China, Japan and South Korea. Asian Economic Journal, 23, 93-118. Retrieved March 31, 2009, from EBSCO online database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=36892658&site=ehost-live