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Chapter 1. Economic Models| Chapter 2.Mathematics for Microeconomics
This part contains two chapters. Chapter 1 examines the general philosophy of how economists build models of economic behavior. Chapter 2 then reviews some of the mathematical tools used in the construction of these models. The mathematical tools from Chapter 2 will be used throughout the remainder of this book.
CHAPTER ONE Economic Models
The main goal of this book is to introduce you to the most important models that econo- mists use to explain the behavior of consumers, firms, and markets. These models are central to the study of all areas of economics. Therefore, it is essential to understand both the need for such models and the basic framework used to develop them. The goal of this chapter is to begin this process by outlining some of the conceptual issues that determine the ways in which economists study practically every question that interests them.
Theoretical Models
A modern economy is a complicated entity. Thousands of firms engage in producing mil- lions of different goods. Many millions of people work in all sorts of occupations and make decisions about which of these goods to buy. Let’s use peanuts as an example. Pea- nuts must be harvested at the right time and shipped to processors who turn them into peanut butter, peanut oil, peanut brittle, and numerous other peanut delicacies. These processors, in turn, must make certain that their products arrive at thousands of retail outlets in the proper quantities to meet demand.
Because it would be impossible to describe the features of even these peanut markets in complete detail, economists have chosen to abstract from the complexities of the real world and develop rather simple models that capture the ‘‘essentials.’’ Just as a road map is helpful even though it does not record every house or every store, economic models of, say, the market for peanuts are also useful even though they do not record every minute feature of the peanut economy. In this book we will study the most widely used economic models. We will see that, even though these models often make heroic abstractions from the complexities of the real world, they nonetheless capture essential features that are common to all economic activities.
The use of models is widespread in the physical and social sciences. In physics, the notion of a ‘‘perfect’’ vacuum or an ‘‘ideal’’ gas is an abstraction that permits scientists to study real-world phenomena in simplified settings. In chemistry, the idea of an atom or a molecule is actually a simplified model of the structure of matter. Architects use mock-up models to plan buildings. Television repairers refer to wiring diagrams to locate prob- lems. Economists’ models perform similar functions. They provide simplified portraits of the way individuals make decisions, the way firms behave, and the way in which these two groups interact to establish markets.
Verification of Economic Models
Of course, not all models prove to be ‘‘good.’’ For example, the earth-centered model of planetary motion devised by Ptolemy was eventually discarded because it proved incapa- ble of accurately explaining how the planets move around the sun. An important purpose of scientific investigation is to sort out the ‘‘bad’’ models from the ‘‘good.’’ Two general methods have been used for verifying economic models: (1) a direct approach, which seeks to establish the validity of the basic assumptions on which a model is based; and (2) an indirect approach, which attempts to confirm validity by showing that a simplified model correctly predicts real-world events. To illustrate the basic differences between the two approaches, let’s briefly examine a model that we will use extensively in later chapters of this book—the model of a firm that seeks to maximize profits.
The profit-maximization model
The model of a firm seeking to maximize profits is obviously a simplification of reality. It ignores the personal motivations of the firm’s managers and does not consider conflicts among them. It assumes that profits are the only relevant goal of the firm; other possible goals, such as obtaining power or prestige, are treated as unimportant. The model also assumes that the firm has sufficient information about its costs and the nature of the market to which it sells to discover its profit-maximizing options. Most real-world firms, of course, do not have this information readily available. Yet such shortcomings in the model are not necessarily serious. No model can exactly describe reality. The real ques- tion is whether this simple model has any claim to being a good one.
Testing assumptions
One test of the model of a profit-maximizing firm investigates its basic assumption: Do firms really seek maximum profits? Some economists have examined this question by sending questionnaires to executives, asking them to specify the goals they pursue. The results of such studies have been varied. Businesspeople often mention goals other than profits or claim they only do ‘‘the best they can’’ to increase profits given their limited in- formation. On the other hand, most respondents also mention a strong ‘‘interest’’ in prof- its and express the view that profit maximization is an appropriate goal. Therefore, testing the profit-maximizing model by testing its assumptions has provided inconclusive results.
Testing predictions
Some economists, most notably Milton Friedman, deny that a model can be tested by inquiring into the ‘‘reality’’ of its assumptions.1 (1See M. Friedman, Essays in Positive Economics (Chicago: University of Chicago Press, 1953), chap. 1. For an alternative view stressing the importance of using ‘‘realistic’’ assumptions, see H. A. Simon, ‘‘Rational Decision Making in Business Organiza- tions,’’ American Economic Review 69, no. 4 (September 1979): 493–513.) They argue that all theoretical models are based on ‘‘unrealistic’’ assumptions; the very nature of theorizing demands that we make certain abstractions. These economists conclude that the only way to determine the valid- ity of a model is to see whether it is capable of predicting and explaining real-world events. The ultimate test of an economic model comes when it is confronted with data from the economy itself.
Friedman provides an important illustration of that principle. He asks what kind of theory one should use to explain the shots expert pool players will make. He argues that the laws of velocity, momentum, and angles from theoretical physics would be a suitable model. Pool players shoot shots as if they follow these laws. But most players asked whether they precisely understand the physical principles behind the game of pool will undoubtedly answer that they do not. Nonetheless, Friedman argues, the physical laws provide accurate predictions and therefore should be accepted as appropriate theoretical models of how experts play pool.
Thus, a test of the profit-maximization model would be provided by predicting the behavior of real-world firms by assuming that these firms behave as if they were maxi- mizing profits. (See Example 1.1 later in this chapter.) If these predictions are reasonably in accord with reality, we may accept the profit-maximization hypothesis. However, we would reject the model if real-world data seem inconsistent with it. Hence the ultimate test of any theory is its ability to predict real-world events.
Importance of empirical analysis
The primary concern of this book is the construction of theoretical models. But the goal of such models is always to learn something about the real world. Although the inclusion of a lengthy set of applied examples would needlessly expand an already bulky book,2 the Extensions included at the end of many chapters are intended to provide a transition between the theory presented here and the ways that theory is applied in empirical studies.
General Features of Economic Models
The number of economic models in current use is, of course, large. Specific assumptions used and the degree of detail provided vary greatly depending on the problem being addressed. The models used to explain the overall level of economic activity in the United States, for example, must be considerably more aggregated and complex than those that seek to interpret the pricing of Arizona strawberries. Despite this variety, practically all economic models incorporate three common elements: (1) the ceteris pari- bus (other things the same) assumption; (2) the supposition that economic decision- makers seek to optimize something; and (3) a careful distinction between ‘‘positive’’ and ‘‘normative’’ questions. Because we will encounter these elements throughout this book, it may be helpful at the outset to describe the philosophy behind each of them.
The ceteris paribus assumption
As in most sciences, models used in economics attempt to portray relatively simple rela- tionships. A model of the market for wheat, for example, might seek to explain wheat prices with a small number of quantifiable variables, such as wages of farmworkers, rain- fall, and consumer incomes. This parsimony in model specification permits the study of wheat pricing in a simplified setting in which it is possible to understand how the specific forces operate. Although any researcher will recognize that many ‘‘outside’’ forces (e.g., presence of wheat diseases, changes in the prices of fertilizers or of tractors, or shifts in consumer attitudes about eating bread) affect the price of wheat, these other forces are held constant in the construction of the model. It is important to recognize that econo- mists are not assuming that other factors do not affect wheat prices; rather, such other variables are assumed to be unchanged during the period of study. In this way, the effect of only a few forces can be studied in a simplified setting. Such ceteris paribus (other things equal) assumptions are used in all economic modeling.
Use of the ceteris paribus assumption does pose some difficulties for the verification of economic models from real-world data. In other sciences, the problems may not be so severe because of the ability to conduct controlled experiments. For example, a physicist who wishes to test a model of the force of gravity probably would not do so by dropping objects from the Empire State Building. Experiments conducted in that way would be subject to too many extraneous forces (e.g., wind currents, particles in the air, variations in temperature) to permit a precise test of the theory. Rather, the physicist would conduct experiments in a laboratory, using a partial vacuum in which most other forces could be controlled or eliminated. In this way, the theory could be verified in a simple setting, without considering all the other forces that affect falling bodies in the real world.
With a few notable exceptions, economists have not been able to conduct controlled experiments to test their models. Instead, they have been forced to rely on various statisti- cal methods to control for other forces when testing their theories. Although these statis- tical methods are as valid in principle as the controlled experiment methods used by other scientists, in practice they raise a number of thorny issues. For that reason, the limi- tations and precise meaning of the ceteris paribus assumption in economics are subject to greater controversy than in the laboratory sciences.
Structure of Economic Models
Most of the economic models you will encounter in this book will have a mathematical structure. They will highlight the relationships between factors that affect the decisions of households and firms and the results of those decisions. Economists tend to use different names for these two types of factors (or, in mathematical terms, variables). Variables that are outside of a decision-maker’s control are called exogenous variables. Such variables are inputs into economic models. For example, in consumer theory we will usually treat individuals as price-takers. The prices of goods are determined outside of our models of consumer behavior, and we wish to study how consumers adjust to them. The results of such decisions (e.g., the quantities of each good that a consumer buys) are endogenous variables. These variables are determined within our models. This distinction is pictured schematically in Figure 1.1. Although the actual models developed by economists may be complicated, they all have this basic structure. A good way to start studying a particular model is to identify precisely how it fits into this framework.
This distinction between exogenous and endogenous variables will become clearer as we explore a variety of economic models. Keeping straight which variables are determined outside a particular model and which variables are determined within a model can be con- fusing; therefore, we will try to remind you about this as we go along. The distinction between exogenous and endogenous variables is also helpful in understanding the way in which the ceteris paribus assumption is incorporated into economic models. In most cases we will want to study how the results of our models change when one of the exogenous variables changes. It is possible, even likely, that the change in such a single variable will change all the results calculated from the model. For example, as we will see, it is likely that the change in the price of a single good will cause an individual to change the quantities of practically every good he or she buys. Examining all such responses is precisely why econo- mists build models. The ceteris paribus assumption is enforced by changing only one exog- enous variable, holding all others constant. If we wish to study the effects of a change in the price of gasoline on a household’s purchases, we change that price in our model, but we do not change the prices of other goods (and in some cases we do not change the individual’s income either). Holding the other prices constant is what is meant by studying the ceteris paribus effect of an increase in the price of gasoline.
FIGURE 1.1 Structure of a Typical Microeconomic Model
Values for exogenous variables are inputs into most economic models. Model outputs (results) are values for the endogenous variables.
EXOGENOUS VARIABLES
Households: Prices of goods Firms: Prices of inputs and output
ECONOMIC MODEL
Households: Utility maximization Firms: Profit maximization
ENDOGENOUS VARIABLES Households: Quantities bought Firms: Output produced, inputs hired
Optimization assumptions
Many economic models start from the assumption that the economic actors being studied are rationally pursuing some goal. We briefly discussed such an assumption when investi- gating the notion of firms maximizing profits. Example 1.1 shows how that model can be used to make testable predictions. Other examples we will encounter in this book include consumers maximizing their own well-being (utility), firms minimizing costs, and gov- ernment regulators attempting to maximize public welfare. Although, as we will show, all these assumptions are unrealistic, and all have won widespread acceptance as good start- ing places for developing economic models. There seem to be two reasons for this accep- tance. First, the optimization assumptions are useful for generating precise, solvable models, primarily because such models can draw on a variety of mathematical techniques suitable for optimization problems. Many of these techniques, together with the logic behind them, are reviewed in Chapter 2. A second reason for the popularity of optimiza- tion models concerns their apparent empirical validity. As some of our Extensions show, such models seem to be fairly good at explaining reality. In all, then, optimization models have come to occupy a prominent position in modern economic theory.
EXAMPLE 1.1 Profit Maximization
The profit-maximization hypothesis provides a good illustration of how optimization assumptions can be used to generate empirically testable propositions about economic behavior. Suppose that a firm can sell all the output that it wishes at a price of p per unit and that the total costs of production, C, depend on the amount produced, q. Then profits are given by
profits 1⁄4 p 1⁄4 pq " CðqÞ: (1:1)
8 Part 1: Introduction
Maximization of profits consists of finding that value of q which maximizes the profit expression in Equation 1.1. This is a simple problem in calculus. Differentiation of Equation 1.1 and setting that derivative equal to 0 give the following first-order condition for a maximum:
dp1⁄4p"C0ðqÞ1⁄40 or p1⁄4C0ðqÞ: (1:2) dq
In words, the profit-maximizing output level (q%) is found by selecting that output level for which price is equal to marginal cost, C 0ðqÞ. This result should be familiar to you from your introductory economics course. Notice that in this derivation the price for the firm’s output is treated as a constant because the firm is a price-taker. That is, price is an exogenous variable in this model.
Equation 1.2 is only the first-order condition for a maximum. Taking account of the second- order condition can help us to derive a testable implication of this model. The second-order condition for a maximum is that at q% it must be the case that
d2p 1⁄4 "C00ðqÞ < 0 or C00ðq%Þ > 0: (1:3) dq2
That is, marginal cost must be increasing at q% for this to be a true point of maximum profits. Our model can now be used to ‘‘predict’’ how a firm will react to a change in price. To do so, we differentiate Equation 1.2 with respect to price (p), assuming that the firm continues to
choose a profit-maximizing level of q: d1⁄2p"C0ðq%Þ1⁄40'1⁄41"C00ðq%Þ(dq% 1⁄40:
(1:4)
(1:5)
Rearranging terms a bit gives
dp dp
dq% 1⁄4 1 dp C00ðq%Þ
> 0:
Here the final inequality again reflects the fact that marginal cost must be increasing at q% if this point is to be a true maximum. This then is one of the testable propositions of the profit- maximization hypothesis—if other things do not change, a price-taking firm should respond to an increase in price by increasing output. On the other hand, if firms respond to increases in price by reducing output, there must be something wrong with our model.
Although this is a simple model, it reflects the way we will proceed throughout much of this book. Specifically, the fact that the primary implication of the model is derived by calculus, and consists of showing what sign a derivative should have, is the kind of result we will see many times. Notice that in this model there is only one endogenous variable—q, the quantity the firm chooses to produce. There is also only one exogenous variable—p, the price of the product, which the firm takes as a given. Our model makes a specific prediction about how changes in this exogenous variable affect the firm’s output choice.
QUERY: In general terms, how would the implications of this model be changed if the price a firm obtains for its output were a function of how much it sold? That is, how would the model work if the price-taking assumption were abandoned?
Positive-normative distinction
A final feature of most economic models is the attempt to differentiate carefully between ‘‘positive’’ and ‘‘normative’’ questions. Thus far we have been concerned primarily with positive economic theories. Such theories take the real world as an object to be studied, attempting to explain those economic phenomena that are observed. Positive economics seeks to determine how resources are in fact allocated in an economy. A somewhat differ- ent use of economic theory is normative analysis, taking a definite stance about what should be done. Under the heading of normative analysis, economists have a great deal to
say about how resources should be allocated. For example, an economist engaged in posi- tive analysis might investigate how prices are determined in the U.S. health-care econ- omy. The economist also might want to measure the costs and benefits of devoting even more resources to health care by, for example, offering government-subsidized health in- surance. But when he or she specifically advocates that such an insurance plan should be adopted, the analysis becomes normative.
Some economists believe that the only proper economic analysis is positive analysis. Drawing an analogy with the physical sciences, they argue that ‘‘scientific’’ economics should concern itself only with the description (and possibly prediction) of real-world economic events. To take political positions and to plead for special interests are consid- ered to be outside the competence of an economist acting as such. Of course, an econo- mist, like any other citizen, is free to express his or her views on political matters. But when doing so he or she is acting as a citizen, not an economist. For other economists, however, the positive-normative distinction seems artificial. They believe that the study of economics necessarily involves the researchers’ own views about ethics, morality, and fairness. According to these economists, searching for scientific ‘‘objectivity’’ in such cir- cumstances is hopeless. Despite some ambiguity, this book tries to adopt a positivist tone, leaving normative concerns for you to decide for yourself.
Development of the Economic Theory of Value
Because economic activity has been a central feature of all societies, it is surprising that these activities were not studied in any detail until fairly recently. For the most part, eco- nomic phenomena were treated as a basic aspect of human behavior that was not suffi- ciently interesting to deserve specific attention. It is, of course, true that individuals have always studied economic activities with a view toward making some kind of personal gain. Roman traders were not above making profits on their transactions. But investiga- tions into the basic nature of these activities did not begin in any depth until the eight- eenth century.3 Because this book is about economic theory as it stands today, rather than the history of economic thought, our discussion of the evolution of economic theory will be brief. Only one area of economic study will be examined in its historical setting: the theory of value.
Early economic thoughts on value
The theory of value, not surprisingly, concerns the determinants of the ‘‘value’’ of a com- modity. This subject is at the center of modern microeconomic theory and is closely intertwined with the fundamental economic problem of allocating scarce resources to al- ternative uses. The logical place to start is with a definition of the word ‘‘value.’’ Unfortu- nately, the meaning of this term has not been consistent throughout the development of the subject. Today we regard value as being synonymous with the price of a commodity.4 Earlier philosopher-economists, however, made a distinction between the market price of a commodity and its value. The term value was then thought of as being, in some sense, synonymous with ‘‘importance,’’ ‘‘essentiality,’’ or (at times) ‘‘godliness.’’ Because ‘‘price’’ and ‘‘value’’ were separate concepts, they could differ, and most early economic
3For a detailed treatment of early economic thought, see the classic work by J. A. Schumpeter, History of Economic Analysis (New York: Oxford University Press, 1954), pt. II, chaps. 1–3.
4This is not completely true when ‘‘externalities’’ are involved, and a distinction must be made between private and social value (see Chapter 19).
Chapter 1: Economic Models 9
10 Part 1: Introduction
discussions centered on these divergences. For example, St. Thomas Aquinas believed value to be divinely determined. Because prices were set by humans, it was possible for the price of a commodity to differ from its value. A person accused of charging a price in excess of a good’s value was guilty of charging an ‘‘unjust’’ price. St. Thomas believed that, in most cases, the ‘‘just’’ rate of interest was zero. Any lender who demanded a pay- ment for the use of money was charging an unjust price and could be—and sometimes was—prosecuted by church officials.
The founding of modern economics
During the latter part of the eighteenth century, philosophers began to take a more scien- tific approach to economic questions. The 1776 publication of The Wealth of Nations by Adam Smith (1723–1790) is generally considered the beginning of modern economics. In his vast, all-encompassing work, Smith laid the foundation for thinking about market forces in an ordered and systematic way. Still, Smith and his immediate successors, such as David Ricardo (1772–1823), continued to distinguish between value and price. To Smith, for example, the value of a commodity meant its ‘‘value in use,’’ whereas the price represented its ‘‘value in exchange.’’ The distinction between these two concepts was illus- trated by the famous water–diamond paradox. Water, which obviously has great value in use, has little value in exchange (it has a low price); diamonds are of little practical use but have a great value in exchange. The paradox with which early economists struggled derives from the observation that some useful items have low prices whereas certain non- essential items have high prices.
Labor theory of exchange value
Neither Smith nor Ricardo ever satisfactorily resolved the water–diamond paradox. The concept of value in use was left for philosophers to debate, while economists turned their attention to explaining the determinants of value in exchange (i.e., to explaining relative prices). One obvious possible explanation is that exchange values of goods are determined by what it costs to produce them. Costs of production are primarily influenced by labor costs—at least this was so in the time of Smith and Ricardo—and therefore it was a short step to embrace a labor theory of value. For example, to paraphrase an example from Smith, if catching a deer takes twice the number of labor hours as catching a beaver, then one deer should exchange for two beavers. In other words, the price of a deer should be twice that of a beaver. Similarly, diamonds are relatively costly because their production requires substantial labor input, whereas water is freely available.
To students with even a passing knowledge of what we now call the law of supply and demand, Smith’s and Ricardo’s explanation must seem incomplete. Did they not recog- nize the effects of demand on price? The answer to this question is both yes and no. They did observe periods of rapidly rising and falling relative prices and attributed such changes to demand shifts. However, they regarded these changes as abnormalities that produced only a temporary divergence of market price from labor value. Because they had not really developed a theory of value in use, they were unwilling to assign demand any more than a transient role in determining relative prices. Rather, long-run exchange values were assumed to be determined solely by labor costs of production.
The marginalist revolution
Between 1850 and 1880, economists became increasingly aware that to construct an adequate alternative to the labor theory of value, they had to devise a theory of value in use. During the 1870s, several economists discovered that it is not the total usefulness of a commodity that helps to determine its exchange value, but rather the usefulness of the last unit consumed. For example, water is certainly useful—it is necessary for all life.
Chapter 1: Economic Models 11
Marshall theorized that demand and supply interact to determine the equilibrium price (p%) and the quantity (q%) that will be traded in the market. He concluded that it is not possible to say that either demand or supply alone determines price or therefore that either costs or usefulness to buyers alone determines exchange value.
FIGURE 1.2
The Marshallian Supply–Demand Cross
Price
D
p*
S
D
q* Quantity per period
S
However, because water is relatively plentiful, consuming one more pint (ceteris paribus) has a relatively low value to people. These ‘‘marginalists’’ redefined the concept of value in use from an idea of overall usefulness to one of marginal, or incremental, usefulness— the usefulness of an additional unit of a commodity. The concept of the demand for an incremental unit of output was now contrasted with Smith’s and Ricardo’s analysis of production costs to derive a comprehensive picture of price determination.5
Marshallian supply–demand synthesis
The clearest statement of these marginal principles was presented by the English econo- mist Alfred Marshall (1842–1924) in his Principles of Economics, published in 1890. Mar- shall showed that demand and supply simultaneously operate to determine price. As Marshall noted, just as you cannot tell which blade of a scissors does the cutting, so too you cannot say that either demand or supply alone determines price. That analysis is illustrated by the famous Marshallian cross shown in Figure 1.2. In the diagram the quan- tity of a good purchased per period is shown on the horizontal axis, and its price appears on the vertical axis. The curve DD represents the quantity of the good demanded per pe- riod at each possible price. The curve is negatively sloped to reflect the marginalist princi- ple that as quantity increases, people are willing to pay less for the last unit purchased. It is the value of this last unit that sets the price for all units purchased. The curve SS shows how (marginal) production costs increase as more output is produced. This reflects the increasing cost of producing one more unit as total output expands. In other words, the upward slope of the SS curve reflects increasing marginal costs, just as the downward slope of the DD curve reflects decreasing marginal value. The two curves intersect at p%, q%. This is an equilibrium point—both buyers and sellers are content with the quantity being traded and the price at which it is traded. If one of the curves should shift, the equi- librium point would shift to a new location. Thus, price and quantity are simultaneously determined by the joint operation of supply and demand.
5Ricardo had earlier provided an important first step in marginal analysis in his discussion of rent. Ricardo theorized that as the production of corn increased, land of inferior quality would be used and this would cause the price of corn to increase. In his argument Ricardo recognized that it is the marginal cost—the cost of producing an additional unit—that is relevant to pricing. Notice that Ricardo implicitly held other inputs constant when discussing decreasing land productivity; that is, he used one ver- sion of the ceteris paribus assumption.
12 Part 1: Introduction
EXAMPLE 1.2 Supply–Demand Equilibrium
Although graphical presentations are adequate for some purposes, economists often use algebraic representations of their models both to clarify their arguments and to make them more precise. As an elementary example, suppose we wished to study the market for peanuts and, based on the statistical analysis of historical data, concluded that the quantity of peanuts demanded each week (q, measured in bushels) depended on the price of peanuts (p, measured in dollars per bushel) according to the equation:
quantity demanded 1⁄4 qD 1⁄4 1,000 " 100p: (1:6)
Because this equation for qD contains only the single independent variable p, we are implicitly holding constant all other factors that might affect the demand for peanuts. Equation 1.6 indicates that, if other things do not change, at a price of $5 per bushel people will demand 500 bushels of peanuts, whereas at a price of $4 per bushel they will demand 600 bushels. The negative coefficient for p in Equation 1.6 reflects the marginalist principle that a lower price will cause people to buy more peanuts.
To complete this simple model of pricing, suppose that the quantity of peanuts supplied also depends on price:
quantity supplied 1⁄4 qS 1⁄4 "125 þ 125p: (1:7)
Here the positive coefficient of price also reflects the marginal principle that a higher price will call forth increased supply—primarily because (as we saw in Example 1.1) it permits firms to incur higher marginal costs of production without incurring losses on the additional units produced.
Equilibrium price determination. Therefore, Equations 1.6 and 1.7 reflect our model of price determination in the market for peanuts. An equilibrium price can be found by setting quantity demanded equal to quantity supplied:
or or thus,
qD 1⁄4 qS
1,000 " 100p 1⁄4 "125 þ 125p
225p 1⁄4 1,125
p% 1⁄4 5:
(1:8)
(1:9)
(1:10)
(1:11)
At a price of $5 per bushel, this market is in equilibrium: At this price people want to purchase 500 bushels, and that is exactly what peanut producers are willing to supply. This equilibrium is pictured graphically as the intersection of D and S in Figure 1.3.
A more general model. To illustrate how this supply–demand model might be used, let’s adopt a more general notation. Suppose now that the demand and supply functions are given by
qD 1⁄4 aþbp and qS 1⁄4 cþdp (1:12)
where a and c are constants that can be used to shift the demand and supply curves, respec- tively, and b (<0) and d (>0) represent demanders’ and suppliers’ reactions to price. Equilib- rium in this market requires
Thus, equilibrium price is given by6
qD 1⁄4 qS or a þ bp 1⁄4 c þ dp:
p% 1⁄4 a " c : d"b
(1:13) (1:14)
6Equation 1.14 is sometimes called the ‘‘reduced form’’ for the supply–demand structural model of Equations 1.12 and 1.13. It shows that the equilibrium value for the endogenous variable p ultimately depends only on the exogenous factors in the model (a and c) and on the behavioral parameters b and d. A similar equation can be calculated for equilibrium quantity.
Chapter 1: Economic Models 13
Notice that in our previous example a 1⁄4 1,000, b 1⁄4 "100, c 1⁄4 "125, and d 1⁄4 125; therefore,
p% 1⁄4 1,000 þ 125 1⁄4 1,125 1⁄4 5: (1:15)
125 þ 100 225
With this more general formulation, however, we can pose questions about how the equilibrium price might change if either the demand or supply curve shifted. For example, differentiation of Equation 1.14 shows that
dp%1⁄4 1 >0; da d"b
dp%1⁄4 "1 <0: dc d"b
(1:16)
That is, an increase in demand (an increase in a) increases equilibrium price, whereas an increase in supply (an increase in c) reduces price. This is exactly what a graphical analysis of supply and demand curves would show. For example, Figure 1.3 shows that when the constant term, a, in the demand equation increases to 1,450, equilibrium price increases to p% 1⁄4 7 [1⁄4 (1,450 þ 125)/225].
QUERY: How might you use Equation 1.16 to ‘‘predict’’ how each unit increase in the exogenous constant a affects the endogenous variable p%? Does this equation correctly predict the increase in p% when the constant a increases from 1,000 to 1,450?
FIGURE 1.31Changing Supply–Demand Equilibria
The initial supply–demand equilibrium is illustrated by the intersection of D and S (p% 1⁄4 5, q% 1⁄4 500). When demand shifts to qD 0 1⁄4 1; 450 " 100p ðdenoted as D 0 Þ, the equilibrium shifts to p% 1⁄4 7, q% 1⁄4 750.
Price ($)
14.5 D′ 10 D
7 5
S
S
0 500 750
D D′ 1,000 1,450
Quantity per period (bushels)
14 Part 1: Introduction
Paradox resolved
Marshall’s model resolves the water–diamond paradox. Prices reflect both the marginal evaluation that demanders place on goods and the marginal costs of producing the goods. Viewed in this way, there is no paradox. Water is low in price because it has both a low marginal value and a low marginal cost of production. On the other hand, diamonds are high in price because they have both a high marginal value (because people are willing to pay quite a bit for one more) and a high marginal cost of production. This basic model of supply and demand lies behind much of the analysis presented in this book.
General equilibrium models
Although the Marshallian model is an extremely useful and versatile tool, it is a partial equilibrium model, looking at only one market at a time. For some questions, this nar- rowing of perspective gives valuable insights and analytical simplicity. For other, broader questions, such a narrow viewpoint may prevent the discovery of important relationships among markets. To answer more general questions we must have a model of the whole economy that suitably mirrors the connections among various markets and economic agents. The French economist Leon Walras (1831–1910), building on a long Continental tradition in such analysis, created the basis for modern investigations into those broad questions. His method of representing the economy by a large number of simultaneous equations forms the basis for understanding the interrelationships implicit in general equilibrium analysis. Walras recognized that one cannot talk about a single market in iso- lation; what is needed is a model that permits the effects of a change in one market to be followed through other markets.
For example, suppose that the demand for peanuts were to increase. This would cause the price of peanuts to increase. Marshallian analysis would seek to understand the size of this increase by looking at conditions of supply and demand in the peanut market. General equilibrium analysis would look not only at that market but also at repercussions in other markets. An increase in the price of peanuts would increase costs for peanut but- ter makers, which would, in turn, affect the supply curve for peanut butter. Similarly, the increasing price of peanuts might mean higher land prices for peanut farmers, which would affect the demand curves for all products that they buy. The demand curves for automobiles, furniture, and trips to Europe would all shift out, and that might create additional incomes for the providers of those products. Consequently, the effects of the initial increase in demand for peanuts eventually would spread throughout the economy. General equilibrium analysis attempts to develop models that permit us to examine such effects in a simplified setting. Several models of this type are described in Chapter 13.
Production possibility frontier
Here we briefly introduce some general equilibrium ideas by using another graph you should remember from introductory economics—the production possibility frontier. This graph shows the various amounts of two goods that an economy can produce using its available resources during some period (say, one week). Because the production possibil- ity frontier shows two goods, rather than the single good in Marshall’s model, it is used as a basic building block for general equilibrium models.
Figure 1.4 shows the production possibility frontier for two goods: food and clothing. The graph illustrates the supply of these goods by showing the combinations that can be produced with this economy’s resources. For example, 10 pounds of food and 3 units of clothing could be produced, or 4 pounds of food and 12 units of clothing. Many other combinations of food and clothing could also be produced. The production possibility frontier shows all of them. Combinations of food and clothing outside the frontier cannot
Chapter 1: Economic Models 15
The production possibility frontier shows the different combinations of two goods that can be produced from a certain amount of scarce resources. It also shows the opportunity cost of producing more of one good as the amount of the other good that cannot then be produced. The opportunity cost at two different levels of clothing production can be seen by comparing points A and B.
FIGURE 1.4
Production Possibility Frontier
Quantity of food per week
A
Opportunity cost of clothing = 12 pound of food
Opportunity cost of clothing = 2 pounds of food
B
10 9.5
4 2
0 34
1213 Quantity of clothing per week
be produced because not enough resources are available. The production possibility fron- tier reminds us of the basic economic fact that resources are scarce—there are not enough resources available to produce all we might want of every good.
This scarcity means that we must choose how much of each good to produce. Figure 1.4 makes clear that each choice has its costs. For example, if this economy produces 10 pounds of food and 3 units of clothing at point A, producing 1 more unit of clothing would ‘‘cost’’ 1⁄2 pound of food—increasing the output of clothing by 1 unit means the production of food would have to decrease by 1⁄2 pound. Thus, the opportunity cost of 1 unit of clothing at point A is 1⁄2 pound of food. On the other hand, if the economy ini- tially produces 4 pounds of food and 12 units of clothing at point B, it would cost 2 pounds of food to produce 1 more unit of clothing. The opportunity cost of 1 more unit of clothing at point B has increased to 2 pounds of food. Because more units of clothing are produced at point B than at point A, both Ricardo’s and Marshall’s ideas of increasing incremental costs suggest that the opportunity cost of an additional unit of clothing will be higher at point B than at point A. This effect is shown by Figure 1.4.
The production possibility frontier provides two general equilibrium insights that are not clear in Marshall’s supply and demand model of a single market. First, the graph shows that producing more of one good means producing less of another good because resources are scarce. Economists often (perhaps too often!) use the expression ‘‘there is no such thing as a free lunch’’ to explain that every economic action has opportunity costs. Second, the production possibility frontier shows that opportunity costs depend on how much of each good is produced. The frontier is like a supply curve for two goods: It
16 Part 1: Introduction
shows the opportunity cost of producing more of one good as the decrease in the amount of the second good. Therefore, the production possibility frontier is a particularly useful tool for studying several markets at the same time.
EXAMPLE 1.3 The Production Possibility Frontier and Economic Inefficiency
General equilibrium models are good tools for evaluating the efficiency of various economic arrangements. As we will see in Chapter 13, such models have been used to assess a wide variety of policies such as trade agreements, tax structures, and environmental regulations. In this simple example, we explore the idea of efficiency in its most elementary form.
Suppose that an economy produces two goods, x and y, using labor as the only input. The production function for good x is x 1⁄4 l 0:5 (where l is the quantity of labor used in x
x x 0:5
production), and the production function for good y is y 1⁄4 2ly . Total labor available is
constrained by lx þ ly * 200. Construction of the production possibility frontier in this economy is extremely simple:
lx þly 1⁄4x2 þ0:25y2 *200 (1:17)
where the equality holds exactly if the economy is to be producing as much as possible (which, after all, is why it is called a ‘‘frontier’’). Equation 1.17 shows that the frontier here has the shape of a quarter ellipse—its concavity derives from the diminishing returns exhibited by each pro- duction function.
Opportunity cost. Assuming this economy is on the frontier, the opportunity cost of good y in terms of good x can be derived by solving for y as
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
y2 1⁄4800"4x2 or y1⁄4 800"4x2 1⁄41⁄2800"4x2'0:5
(1:18) (1:19)
And then differentiating this expression:
dy 1⁄4 0:51⁄2800 " 4x2'"0:5ð"8xÞ 1⁄4 "4x :
dx y
Suppose, for example, labor is equally allocated between the two goods. Then x 1⁄4 10, y 1⁄4 20, and dy/dx 1⁄4 "4(10)/20 1⁄4 "2. With this allocation of labor, each unit increase in x output would require a reduction in y of 2 units. This can be verified by considering a slightly different allocation, lx 1⁄4 101 and ly 1⁄4 99. Now production is x 1⁄4 10.05 and y 1⁄4 19.9. Moving to this alternative allocation would have
Dy1⁄4 ð19:9"20Þ 1⁄4"0:11⁄4"2, Dx ð10:05 " 10Þ 0:05
which is precisely what was derived from the calculus approach.
Concavity. Equation 1.19 clearly illustrates the concavity of the production possibility frontier. The slope of the frontier becomes steeper (more negative) as x output increases and y output decreases. For example, if labor is allocated so that lx 1⁄4 144 and ly 1⁄4 56, then outputs are x 1⁄4 12 and y + 15 and so dy/dx 1⁄4 "4(12)/15 1⁄4 "3.2. With expanded x production, the opportunity cost of one more unit of x increases from 2 to 3.2 units of y.
Inefficiency. If an economy operates inside its production possibility frontier, it is operating inefficiently. Moving outward to the frontier could increase the output of both goods. In this book we will explore many reasons for such inefficiency. These usually derive from a failure of some market to perform correctly. For the purposes of this illustration, let’s assume that the labor market in this economy does not work well and that 20 workers are permanently unemployed. Now the production possibility frontier becomes
x2 þ 0:25y2 1⁄4 180, (1:20)
Welfare economics
In addition to using economic models to examine positive questions about how the econ- omy operates, the tools used in general equilibrium analysis have also been applied to the study of normative questions about the welfare properties of various economic arrange- ments. Although such questions were a major focus of the great eighteenth- and nineteenth- century economists (e.g., Smith, Ricardo, Marx, and Marshall), perhaps the most significant advances in their study were made by the British economist Francis Y. Edgeworth (1848–1926) and the Italian economist Vilfredo Pareto (1848–1923) in the early years of the twentieth century. These economists helped to provide a precise definition for the concept of ‘‘economic efficiency’’ and to demonstrate the conditions under which markets will be able to achieve that goal. By clarifying the relationship between the allocation pricing of resources, they provided some support for the idea, first enunciated by Adam Smith, that properly functioning markets provide an ‘‘invisible hand’’ that helps allocate resources efficiently. Later sections of this book focus on some of these welfare issues.
Modern Developments
Research activity in economics expanded rapidly in the years following World War II. A major purpose of this book is to summarize much of this research. By illustrating how economists have tried to develop models to explain increasingly complex aspects of eco- nomic behavior, this book seeks to help you recognize some of the remaining unanswered questions.
The mathematical foundations of economic models
A major postwar development in microeconomic theory was the clarification and formal- ization of the basic assumptions that are made about individuals and firms. The first landmark in this development was the 1947 publication of Paul Samuelson’s Foundations of Economic Analysis, in which the author (the first American Nobel Prize winner in eco- nomics) laid out a number of models of optimizing behavior.7 Samuelson demonstrated the importance of basing behavioral models on well-specified mathematical postulates so that various optimization techniques from mathematics could be applied. The power of his approach made it inescapably clear that mathematics had become an integral part of modern economics. In Chapter 2 of this book we review some of the mathematical con- cepts most often used in microeconomics.
Chapter 1: Economic Models 17
and the output combinations we described previously are no longer feasible. For example, if x 1⁄4 10, then y output is now y + 17.9. The loss of approximately 2.1 units of y is a measure of the cost of the labor market inefficiency. Alternatively, if the labor supply of 180 were allocated evenly between the production of the two goods, then we would have x + 9.5 and y + 19, and the inefficiency would show up in both goods’ production—more of both goods could be pro- duced if the labor market inefficiency were resolved.
QUERY: How would the inefficiency cost of labor market imperfections be measured solely in terms of x production in this model? How would it be measured solely in terms of y production? What would you need to know to assign a single number to the efficiency cost of the imperfection when labor is equally allocated to the two goods?
7Paul A. Samuelson, Foundations of Economic Analysis (Cambridge, MA: Harvard University Press, 1947).
18
Part 1: Introduction
SUMMARY
New tools for studying markets
A second feature that has been incorporated into this book is the presentation of a num- ber of new tools for explaining market equilibria. These include techniques for describing pricing in single markets, such as increasingly sophisticated models of monopolistic pric- ing or models of the strategic relationships among firms that use game theory. They also include general equilibrium tools for simultaneously exploring relationships among many markets. As we shall see, all these new techniques help to provide a more complete and realistic picture of how markets operate.
The economics of uncertainty and information
A final major theoretical advance during the postwar period was the incorporation of uncertainty and imperfect information into economic models. Some of the basic assump- tions used to study behavior in uncertain situations were originally developed in the 1940s in connection with the theory of games. Later developments showed how these ideas could be used to explain why individuals tend to be adverse to risk and how they might gather information to reduce the uncertainties they face. In this book, problems of uncertainty and information enter the analysis on many occasions.
Computers and empirical analysis
One final aspect of the postwar development of microeconomics should be mentioned— the increasing use of computers to analyze economic data and build economic models. As computers have become able to handle larger amounts of information and carry out complex mathematical manipulations, economists’ ability to test their theories has dra- matically improved. Whereas previous generations had to be content with rudimentary tabular or graphical analyses of real-world data, today’s economists have available a wide variety of sophisticated techniques together with extensive microeconomic data with which to test their models. To examine these techniques and some of their limitations would be beyond the scope and purpose of this book. However, the Extensions at the end of most chapters are intended to help you start reading about some of these applications.
This chapter provided background on how economists approach the study of the allocation of resources. Much of the material discussed here should be familiar to you from introductory economics. In many respects, the study of economics represents acquiring increasingly sophisticated tools for addressing the same basic problems. The purpose of this book (and, indeed, of most upper-level books on economics) is to provide you with more of these tools. As a starting place, this chapter reminded you of the following points:
• Economics is the study of how scarce resources are allocated among alternative uses. Economists seek to develop simple models to help understand that process. Many of these models have a mathematical basis because the use of mathematics offers a precise short- hand for stating the models and exploring their conse- quences.
• The most commonly used economic model is the supply–demand model first thoroughly developed by Alfred Marshall in the latter part of the nineteenth cen- tury. This model shows how observed prices can be taken to represent an equilibrium balancing of the pro- duction costs incurred by firms and the willingness of demanders to pay for those costs.
• Marshall’s model of equilibrium is only ‘‘partial’’—that is, it looks only at one market at a time. To look at many markets together requires an expanded set of general equilibrium tools.
• Testing the validity of an economic model is perhaps the most difficult task economists face. Occasionally, a model’s validity can be appraised by asking whether it is based on ‘‘reasonable’’ assumptions. More often, however, models are judged by how well they can explain economic events in the real world.
SUGGESTIONS FOR FURTHER READING
On Methodology
Blaug, Mark, and John Pencavel. The Methodology of Economics: Or How Economists Explain, 2nd ed. Cambridge: Cambridge University Press, 1992.
A revised and expanded version of a classic study on economic methodology. Ties the discussion to more general issues in the philosophy of science.
Boland, Lawrence E. ‘‘A Critique of Friedman’s Critics.’’ Journal of Economic Literature (June 1979): 503–22.
Good summary of criticisms of positive approaches to economics
and of the role of empirical verification of assumptions.
Friedman, Milton. ‘‘The Methodology of Positive Economics.’’ In Essays in Positive Economics, pp. 3–43. Chicago: University of Chicago Press, 1953.
Basic statement of Friedman’s positivist views.
Harrod, Roy F. ‘‘Scope and Method in Economics.’’ Economic Journal 48 (1938): 383–412.
Classic statement of appropriate role for economic modeling. Hausman, David M., and Michael S. McPherson. Economic Analysis, Moral Philosophy, and Public Policy, 2nd ed. Cambridge, UK: Cambridge University Press, 2006.
The authors stress their belief that consideration of issues in
moral philosophy can improve economic analysis.
McCloskey, Donald N. If You’re So Smart: The Narrative of Economic Expertise. Chicago: University of Chicago Press, 1990.
Discussion of McCloskey’s view that economic persuasion depends on rhetoric as much as on science. For an interchange on this topic, see also the articles in the Journal of Economic Literature, June 1995.
Sen, Amartya. On Ethics and Economics. Oxford, UK: Blackwell Reprints, 1989.
The author seeks to bridge the gap between economics and ethical studies. This is a reprint of a classic study on this topic.
Primary Sources on the History of Economics
Edgeworth, F. Y. Mathematical Psychics. London: Kegan Paul, 1881.
Initial investigations of welfare economics, including rudimentary
notions of economic efficiency and the contract curve.
Marshall, A. Principles of Economics, 8th ed. London: Macmillan & Co., 1920.
Complete summary of neoclassical view. A long-running,
popular text. Detailed mathematical appendix. Marx, K. Capital. New York: Modern Library, 1906.
Full development of labor theory of value. Discussion of ‘‘transformation problem’’ provides a (perhaps faulty) start for
general equilibrium analysis. Presents fundamental criticisms of
institution of private property.
Ricardo, D. Principles of Political Economy and Taxation. London: J. M. Dent & Sons, 1911.
Very analytical, tightly written work. Pioneer in developing careful analysis of policy questions, especially trade-related issues. Discusses first basic notions of marginalism.
Smith, A. The Wealth of Nations. New York: Modern Library, 1937.
First great economics classic. Long and detailed, but Smith had the first word on practically every economic matter. This edition has helpful marginal notes.
Walras, L. Elements of Pure Economics. Translated by W. Jaffe. Homewood, IL: Richard D. Irwin, 1954.
Beginnings of general equilibrium theory. Rather difficult reading.
Secondary Sources on the History of Economics
Backhouse, Roger E. The Ordinary Business of Life: The History of Economics from the Ancient World to the 21st Century. Princeton, NJ: Princeton University Press, 2002.
An iconoclastic history. Good (although brief) on the earliest economic ideas, but some blind spots on recent uses of mathematics and econometrics.
Blaug, Mark. Economic Theory in Retrospect, 5th ed. Cambridge, UK: Cambridge University Press, 1997.
Complete summary stressing analytical issues. Excellent
‘‘Readers’ Guides’’ to the classics in each chapter.
Heilbroner, Robert L. The Worldly Philosophers, 7th ed. New York: Simon & Schuster, 1999.
Fascinating, easy-to-read biographies of leading economists. Chapters on Utopian Socialists and Thorstein Veblen highly recommended.
Keynes, John M. Essays in Biography. New York: W. W. Norton, 1963.
Essays on many famous persons (Lloyd George, Winston Churchill, Leon Trotsky) and on several economists (Malthus, Marshall, Edgeworth, F. P. Ramsey, and Jevons). Shows the true gift of Keynes as a writer.
Schumpeter, J. A. History of Economic Analysis. New York: Oxford University Press, 1954.
Encyclopedic treatment. Covers all the famous and many not- so-famous economists. Also briefly summarizes concurrent developments in other branches of the social sciences.
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