Economics and Operations Research

Frederic H. Murphy

Temple University, Philadelphia

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Introduction

To understand the relationship between economics and operations research, we need to understand some of the history of both fields. Operations research was developed prior to and during World War II with the pragmatic goal of improving military operations through the use of mathematics. The founders of the field of operations research came from diverse backgrounds, including physics, mathematics, engineering and economics. Operations research as a field has tried to maintain its multidisciplinary character.

The textbooks on the subject contain a common set of techniques: stochastic modeling, simulation, optimization, and game theory. Operations research emphasizes certain application areas such as operations management and encompasses some, such as inventory management. Mainly, operations research provides tools to analyze the operations of and assist in decision making in organizations.

Economics as a subject has been explored and developed for centuries. The field used to be called political economy, reflecting its public policy orientation, which carries through to today. The subject areas of economics can be defined broadly as follows: macroeconomics, the study of economic aggregates and microeconomics, the study of economic agents, such as firms, and the market structures within which these agents operate to optimize their utility or profits. Examples of markets include monopolies, oligopolies, and perfect competition. Economists also develop tools such as the statistical techniques of econometrics, which are used for estimating the parameters of economic models. Despite strong connections between microeconomics and operations research, little overlap exists between macroeconomics and operations research.

While some early writers such as Adam Smith and Karl Marx were very influential as social philosophers, professional economists did not play an important role in day-to-day discussions of specific public policies and programs until well into the 20th century. For example, the first tax cut based on macroeconomic theory and intended to stimulate the U.S. economy was implemented in the Eisenhower administration in the late 1950s, and regulatory reform based on microeconomic theory started to make headway in the 1970s, decades after the basics of the theory of workably competitive markets was understood. That is, economics, as we now know it, was developing contemporaneously with operations research and its broad impact in decision making is a post-World War II phenomenon as with operations research.

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The common history

The rapid growth of mathematical economics and operations research during and after World War II stemmed from the same root: the application of mathematics to build and understand models that only approximate the reality being studied. Given this common starting point and the interest of operations researchers in finding the most economic solutions, the overlap in the fields has to be significant. The connections were most prominent in the early days of operations research and they involved the areas of optimization, inventory theory, and game theory.

Hitchcock (1941), a physicist, and Koopmans (1951), an economist, independently developed the first useful optimization model, the transportation model. Kantorovitch (1939), a mathematician in the Russian central planning agency, developed several linear programming (LP) models for production and distribution including the transshipment model. Stigler (1944), an economist, developed the diet/feed mix model. Dantzig (1951-1), (1951-2), and (1963), at the time, a mathematician in the US Air Force, invented the first generic linear programs and the simplex algorithm for solving them. The simplex algorithm has survived for 50 years as the primary method for solving linear programs. The collection of papers, Koopmans (1951), defined the beginning of the subject of optimization, game theory and the relationship between the two. It also devoted a substantial amount of space to generalizing the input-output model of an economy. Dantzig (1963) pointed to the work of Leontief in input-output models of the US economy as an important beginning for his ideas. The contributing authors to Koopmans' book were a mix of economists and mathematicians. Another important early book on linear programming, Dorfman, Samuelson and Solow (1958), was written by economists. Indeed, Dorfman (1953) wrote the intuitive description of linear programming models that we see in all of the textbooks today. Current texts on microeconomics continue to include chapters on optimization and game theory.

Many of the first articles on optimization appeared in such journals as Econometrica (see the references in Dantzig, 1963). Charnes and Cooper, the developers of many of the first linear programming models, also published in the economics journals (see, e.g., Charnes, Cooper and Mellon, 1952). Agricultural economists were quick to develop the feed-mix model for farmers. Mathematical programming has become a mainstay for agricultural economists (see Hazell, 1986).

In the early days of inventory theory, the links between economists and operations researchers were equally strong. This area involved using such optimization techniques as dynamic programming and traditional, calculus-based methods to find optimal inventory policies (Arrow, Karlin and Scarf, 1958; Whitin, 1957). However, the development of the field moved very quickly into the hands of operations researchers because the issues in inventory analysis evolved into the implementation of inventory systems and situation-specific models, away from the more broadly-based economic considerations.

Game theory was developed by von Neumann to study issues of conflict and cooperation at a theoretical level. The RAND Corporation became an early center for the development of game theory right after World War II, in good part to understand geopolitical and military strategy. The link between game theory and optimization was understood from very early on (see Dantzig, 1951–2; Gale, Kuhn and Tucker, 1951).

Operations research techniques and operations researchers have contributed significantly to economics. Once Samuelson (1952) recognized the connection between mathematical programming and economic equilibrium models, mathematical programming became an important tool for economic analysis. In fact, the GAMS modeling language was developed by operations researchers at the World Bank for the purpose of solving computable general equilibrium models for evaluating national development plans (Brooke, Kendrick and Meeraus, 1993). An economist, Gustafson (1958), used dynamic programming, an operations research tool, to develop the first grain storage models to protect against famine. One of the most prominent microeconomic policy-analysis models of the 1970s, the Project Independence Evaluation System (PIES), was built by a team of operations researchers and economists led by William Hogan (1975), an operations researcher, who went on to organize the International Association of Energy Economists.

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The different perspectives of Economics and Operations Research

Economics and operations research are distinct fields because the economists and operations researchers have different interests. Economists are primarily interested in qualitative analysis for policymaking and operations researchers are more interested in assisting decision making within the firm and have a strong computational orientation. For example, oil companies use the results of their mathematical programming models for operating their refineries. Even when economists are interested in numbers, they are looking to measure the impact of the sum of individual decisions rather than determining the decisions. This distinction between the fields is not absolute. Econometricians are interested in computational issues and the theoretical properties of their estimation methods. Scarf (1973) has developed algorithms for computing economic equilibria. The function of corporate planning and public policy studies produced by operations researchers is to provide insight rather than specific numbers. This is done through constructing multiple scenarios, examining alternative policies, and analyzing the sensitivity of the results to the underlying parameters.

The different perspectives can be seen in the study of inventories. For the past few decades operations researchers and computer scientists have been implementing inventory systems, while the economists have been focusing on the effect of inventories in the business cycle rather than inventory policies per se. The recent popularity of scientific inventory management in corporations and the desire to reduce inventories to free up capital and gain operational flexibility with just-in-time manufacturing has led to a significant decline in the inventory-to-sales ratio. That is, inventories turn over more quickly and companies are able to adapt to fluctuations in demand more rapidly with less draconian changes in production levels.

Inventory changes are known accelerators of business cycles. See Forrester (1961) for an illustration of this at the firm level. The smaller aggregate inventories are, relative to GNP, the less effect they have on business cycles. Economists measure this drop at the national level and factor this secular change into their macroeconomic models to explain the resultant dampening of business cycles. For example, the recession in the early 1990s was slow in coming and going but also shallow relative to past recessions because of the cumulative impact of individual improvements of inventory systems and production management.

The different views of production functions taken by the two fields further illustrates the distinctions between the fields. When economists estimate production functions, they typically posit a differentiable functional form, gather data and estimate the parameters of the function using regression techniques. They do this to estimate output prices and understand the rates of substitution of inputs as a function of input prices. They are not looking inside at the production process. Instead they are looking at market consequences. For example, from the rates of substitution, one can derive a demand curve for an input given the prices of the other inputs.

The operations research tool of data envelopment analysis (Charnes, Cooper, and Rhodes, 1978) estimates production functions using an alternative approach with different assumptions and goals. In data envelopment analysis the goal is to identify which decision-making units are efficient and which are not. That is, data envelopment analysis is a bench-marking tool for finding the best production practices with the ultimate goal of improving production processes. Unlike the econometric assumption that errors are in the data, data envelopment analysis assumes the data is error free and differences among decision-making units are due to different resource mixes and managerial effectiveness. The production function is the inputs and outputs of the decision-making units as activities in a linear program. A linear program is solved for each decision-making unit to see if it is on the efficient frontier. If it is, it represents best practices, given its mix of resources and products. If not, it is a candidate for improvement.

For every differentiator between operations research and economics, we can find an exception. We have noted that economists have focused on how agents interact in a framework and draw conclusions about the effect of changes in the framework, while operations researchers have been more interested in aiding the agents to make decisions. An exception to this is the study of traffic equilibria where the agents are travelers on a network of roads. The defining paper of this subject appeared in a civil engineering journal. Wardrop (1952) stated a set of equilibrium conditions based on trip times that are directly related to the equilibrium conditions for spatial economic equilibria based on cost. Although economists contributed to the early literature (e.g., Beckmann, McGuire and Winsten, 1956, established the relationship with economic equilibria), the bulk of the literature is in transportation journals with an operations research connection. See Nagurney (1993) for a description of different types of equilibrium models.

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Where economists and operations researchers have common interests

The two fields overlap in several areas. We mention four: public policy analysis, finance, game theory and decision analysis. The convergence of the fields in policy analysis comes about because politicians want quantitative analyses of programs. Economic models have a lot to say about how economic agents behave and operations researchers have the computational skills and modeling expertise to implement the economic theories and solve for the economic impacts of policy alternatives.

Examples here include the previously mentioned activity at the World Bank. The close working relationships between economists and operations researchers have continued with the successor models to PIES, the Intermediate Future Forecasting System (Murphy, Conti, Sanders and Shaw, 1988) and the National Energy Modeling System (Energy Information Administration, 1994). See Murphy and Shaw (1995) for a history of the energy models at the Energy Information Administration.

A key feature of these kinds of policy models is that in some sectors they model the decisions using optimization by representing the technology choices directly in the model. The main reason for using optimization is that the models need to have representations for policies and technologies that affect more than input and output prices and quantities and there is no history to assess the resulting decisions for some sectors. Other reasons include the need to link more than one sector and the existence of a convoluted data history that muddies the econometric analysis for estimating such things as a production function for electric utilities. The optimization models are usually simplified versions of the planning models used by the industry with coefficients based on industry aggregates. They are treated as simulation models based on the result of Samuelson (1952) showing the connection between optimization and economic equilibrium models.

Markowitz (1954) proposed this approach almost 50 years ago as a way to model the whole economy, extending the input-output model to represent alternative production technologies. Henderson (1955) and Land (1956) successfully built models of coal markets using this approach. Only recently have databases, computers and algorithms progressed to the point where these ideas can be realized for economy-wide models.

Policy models almost always include econometric components as well. For example, the above-mentioned energy models include econometrically estimated demand curves along with optimization models for coal and electric utilities. In econometric models of production one measures the inputs and outputs to statistically estimate the parameters of a production function. The model makes no statement about the actual decisions made. Instead, it models the outcomes of the decisions made by the actors in the economic sector. Econometric approaches dominate optimization when there is too much heterogeneity among participants to specify the parameters of their decision environment, as in demand modeling or the behavior of producers when the industry has a large number of independent, small firms.

The finance literature is dominated by economic studies of financial markets and their efficiency. An example is the book A Random Walk Down Wall Street by Malkiel (1973). This book showed that movements in stock prices are a random walk, illustrating why stock pickers in general cannot beat the market. Also, Tobin's (1958) results on the relationship between risk and return were key to the development of decision models in finance.

Financial markets are not entirely efficient and the Black-Scholes (1973) model for pricing options created a whole new segment of the finance industry. Its basis is dynamic programming. The book by the economists Dixit and Pindyck (1994) emphasized the role of dynamic programming in properly valuing investments with uncertain returns.

Optimization models have come to play an important role in determining the mix of assets in a portfolio, the first one being the model by Markowitz (1952), which represented the beginning of computational finance. Because of the ability to solve far larger linear programs than in the past, stochastic programming models for building portfolios have made an important mark in the industry. For example, see Carino et al. (1994) for a description of the kind of operations research models used by the people known as “rocket scientists” in the financial press.

The interconnection between economics and operations research in game theory can be illustrated by the Averch-Johnson hypothesis (1962). This hypothesis states that regulated firms have a bias to overinvesting in capital rather than labor. They demonstrated their results by evaluating the Kuhn-Tucker conditions of an optimization model, where a firm with a monopoly maximizes profits subject to a rate-of-return constraint. With the deregulation of many industries, we now know that these firms were not only overcapitalized but the also had too much labor, in violation of the Averch-Johnson hypothesis.

The problem with the Averch-Johnson model was it presumed that the firm was a single entity and could optimize its behavior. However, one must not treat the firm as the atom. Instead, one must look at how the agents within the firm interact and look further into the nature of the behaviors of the agents who make up the firm. Figuring out the underlying incentives of the members of a firm and analyzing their behavior relative to the interests of stockholders is known as principal agent theory, an important area of microeconomics. For example, one could explain the behavior of regulated firms as follows: managers increase their importance by increasing the number of employees under them and buying labor peace by paying high wages to unionized employees.

Studying the behavior of economic agents and other individuals has a long tradition in economics and is the essence of game theory. Since little data exists for numerically evaluating game models, almost all studies examine the qualitative properties of the resulting games. Economists have focused mostly on markets (Shubik, 1959). Indeed, outside of von Neumann's early work on parlor games, the book by von Neumann and Morgenstern (1944) was the first major treatment of the subject and focused on economics. Operations researchers have studied other types of games such as war games and invented some of what are now the classics like the prisoner's dilemma game (Poundstone, 1992). The center for this work was RAND. An example of a strategic game that was studied was the stability of mutual assured destruction as a defense against nuclear war. Schelling (1980) presented an analysis of these strategic games. Both groups study the generic properties of games that abstract common situations. Shubik presented an interesting example of someone who does both strategic and economic games. As part of his examination of strategic issues he used the dollar auction game to describe games of escalation such as war and lawsuits (Poundstone, 1992).

Part of the reason for the common interest of economists and operations researchers in game theory is its universality in understanding conflict and cooperation. Political scientists and sociologists have become involved in game theory. The link between political science and games is direct through the games already mentioned and the use of game theory concepts in negotiation. Sociologists use games to understand social interactions. The prisoner's dilemma game has been used repeatedly to explain the behavior of individuals in social situations and social structures. Thus, the notions of game theory have moved beyond the disciplines in which they were developed and influence important areas of the social sciences.

Rational decision making encompasses game theory. Indeed, a still invaluable work on the subject that treated both together is the book by Luce and Raiffa (1957). What is a rational decision is subject to debate. To explore the subject, von Neumann and Morgenstern developed the concept of expected utility. Utility is a simple concept in many situations when the goal can be clearly stated as with maximizing profits. However, in real life we face many trade-offs. Examples include our willingness to bear risk, how we value income versus leisure, what we value in the products we consume and how we value the future over the present. In the decision-making literature, Keeney and Raiffa (1976) explored the issues associated with multi-attribute utility in decision making. The economists' notion of utility is central to the study of negotiation (Raiffa, 1982).

As in other areas, economists have not focused on making actual decisions except in the general properties that can be understood from the decision-making process as in Arrow (1951). In his seminal work on social choice he posited a set of axioms that define rationality and then shows how group interactions and voting processes lead to irrational decisions even though the original actors have rational utility functions. Another example of this is the economics literature on rational expectations. In its most basic form the question addressed in the context of macroeconomic models is: “How do the consequences of macroeconomic policy change when the participants in the economy have rational expectations about the effect of macroeconomic policies and adjust their decisions?” See Redman (1992) or Sargent (1993) for a discussion of this area of economics.

Operations research has come to dominate the subject so far as making actual decisions. For an example of a detailed decision analysis in a corporation see Borrison (1994). Some of the most important literature has come from psychologists trying to understand peoples' thought processes. The psychology literature is aimed squarely at the rational actor hypothesis of economics and finds it wanting. For a book that examines the approaches of all three disciplines see Bell, Raiffa, and Tversky (1988).

An operations researcher cannot be an effective modeler and analyst without a good understanding of the basics of economic theory. The simplex algorithm represents the behavior of a set of independent economic agents (activities) making decisions to act or not based on a set of incentives in the form of prices (dual variables). And if they choose to act based on profitability (reduced costs), they proceed until they reach a resource limit or drive a competitor out of business (the replaced activity reaches zero). By looking at a solution from the perspective of the simplex algorithm as an economic process, one gains the deeper insights into the story contained in the solution and takes one beyond just the value of the objective function and the level of activities.

The basic notions of substitutes and complements in production processes determines the character of optimization models. Network LP's are models of pure substitution. Whereas, a product-mix model consists of activities that have inputs that are pure complements. The vast majority of the constraints in an LP can be classified as supply, demand or material-balance constraints. Greenberg (1981) used the notions of substitutes and complements to gain deeper insights into LP models and their solutions.

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Summary

Economics and operations research have common roots. The fields often use the same tools, such as the Karush-Kuhn-Tucker conditions. In economics these conditions are used for marginal analysis, as with the search for institutional distortions of the marketplace in the Averch-Johnson hypothesis and for such uses as the derivation of cost functions from production functions. Operations researchers exploit these conditions to improve algorithms and use the actual duals and ranges for evaluating the stability of the model results, estimating the effects of uncertainty in the coefficients on the solution and determining the costs of constraints with an eye towards adding or reducing resources.

Typically, the fields use these tools differently for different purposes. This reflects the different professional goals of the individuals involved in these fields. Operations researchers focus on making specific decisions and economists study the consequences of different market structures and policies through an assumption of rational decision making. Both groups are interested in understanding rational decision making and the consequences of rational decisions. This can be seen in the different views of the firm. The traditional economic theory of the firm is really a theory of the interactions of firms or constituencies within the firm. Whereas, operations research models provide a theory of decision making within the firm and are an important component of a theory of the internals of the firm. The operations research models do not provide a complete theory of decision making in the firm because operations researchers, although commenting on conflicts in the firm, tend to not focus on the incentives and structures that create these conflicts. This is where agency theory fits in and one of the places where game theory links both fields.

The fields are now distinct because operations research takes an engineering perspective: the goal is to invent improved ways for making decisions, and in the process of doing this, inventing new models and algorithms as needed. Economics, instead, is a social science where the goal is to understand the existing world and study the consequences of policies that affect this world using the basic theme of exploring the consequences of rational self-interest. The two fields come together when there is the need to change the rules of the marketplace or when the marketplace creates opportunities to engineer new products that provide a profit as in policy analysis and finance. Both fields have their distinct niches, yet will always be connected by their tools and history.

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This text originally appeared in Encyclopedia of Operations Research and Management Science (ISBN 079237827x)

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