I have long had a "thinking like an economist" lecture in the can. But I very rarely give it. It seems to me that it is important stuff—that people really should know it before they begin studying economics, because it would make studying economics much easier. But it also seems to me—usually—that it is pointless to give it at the start of a course to newBs: they just won't understand it. And it also seems to me—usually—that it is also pointless to give it to students at the end of their college years: they either understand it already, or it is too late.

By continuity that would seem to imply that there is an optimal point in the college curriculum to teach this stuff. But is that true?

What do you think?

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## How to Think Like an Economist

key: https://www.icloud.com/keynote/0Lvx3IfBcDQWga2Y3ERk72Zkg

Every new subject requires new patterns of thought; every intellectual discipline calls for new ways of thinking about the world. After all, that is what makes it a *discipline*: a discipline that allows people to think about a subject in some new way. Economics is no exception.

In a way, learning an intellectual discipline like economics is similar to learning a new language or being initiated into a club. Economists’ way of thinking allows us to see the economy more sharply and clearly than we could in other ways. (Of course, it can also cause us to miss certain relationships that are hard to quantify or hard to think of as purchases and sales; that is why economics is not the only social science, and we need sociologists, political scientists, historians, psychologists, and anthropologists as well.) In this chapter we will survey the intellectual landmarks of economists’ system of thought, in order to help you orient yourself in the mental landscape of economics.

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### Economics: What Kind of Discipline Is It?

If you are coming to economics from a background in the natural sciences, you probably expect economics to be something like a natural science, only less so: You probably think that to the extent that it works, it works more or less like chemistry, though it does not work as well. It does not work as well because economic theories are unsettled and poorly described. It does not work as well because economists’ predictions are often wrong.

If you are coming to economics from a background in the humanities, you probably see it as a combination of two centuries out-of-date psychology and moral philosophy, coupled with obscure and often wrong—yet somehow authoritative in some way—mathematical manipulations.

If you hold either of these opinions, you are half-right.

While economics is not a *natural* science, it is a science—a social science. Its subject is not electrons or elements but human beings: people and how they behave. This makes it something more and something more difficult than it would be if it were just like a natural science but underdeveloped and badly done. The fact that economics' subject matter is people has important consequences, that make economics easier than a natural science in some ways, harder than a natural science in other ways, and in yet other ways just plain different. Moreover, economics is an inherently *quantitative* social science. And it has developed as an *abstract* social science.

While economics is not a humanity, it is humanistic. Its subject matter is made up not of quarks or molecules or animals but of people. And to understand people you have to get inside their heads: understand their hopes, fears, desires, reasoning, plans, expectations, and actions. Thus one of the principal intellectual moves in economics is one that is totally absent from the natural sciences: it is for you to imagine yourself in the place of the people you are studying. Thus economics often turns into an exercise in introspective psychology.

**A Social Science**: The principal things to remember that flow because economics is a *social* and not a *natural* science are:

Because economics is a social science, debates within economics last a lot longer and are much less likely to end in a clear consensus than are debates in the natural sciences. The major reason is that different people have different views of what makes a free, a good, a just, or a well-ordered society. They look for an economy that harmonizes with their vision of what a society should be. They ignore or explain away facts that turn out to be inconvenient for their particular political views. People are, after all, only human.

Economists*try*to approach the objectivity that characterizes most work in the natural sciences. After all, what is is, and what is not is not. Even if wishful thinking or predispositions contaminate the results of a single study, later studies can correct the error. But economists never approach the unanimity with which physicists embraced the theory of relativity, chemists embraced the oxygen theory of combustion, and biologists rejected the Lamarckian inheritance of acquired characteristics. Biology departments do not have Lamarckians. Chemistry departments do not have phlogistonists. But economics departments do have a wide variety of points of view and schools of thought.That economics is about people means that economists cannot ethically undertake large-scale experiments. Economists cannot set up special situations in which potential sources of disturbance are reduced to a minimum, then observe what happens, and generalize from the results of the experiment (where sources of disturbance are absent) to what happens in the world (where sources of disturbance are common). Thus the experimental method, the driver of rapid progress in many of the natural sciences, is lacking in economics. This flaw makes economics harder to analyze, and it makes economists’ conclusions much more tentative and subject to dispute, than is the case with natural sciences.

That economics studies people means that the subjects economists study have minds of their own. They observe what is going on around them, plan for the future, and take steps to avoid future consequences that they foresee and fear will be unpleasant. At times they simply do what they want, just because they feel like doing it. Thus in economists’ analyses the present often depends not just on the past but on the future as well—or on what people expect the future to be. Box 3.1 presents one example of this: how people’s expectations of the future and particularly their fear that there might be a depression contributed to the coming of the Great Depression of the 1930s.

This third wrinkle makes economics in some sense very hard. Natural scientists can always assume the arrow of causality points from the past to the future. In economics people’s expectations of the future mean that the arrow of causality often points the other way, from the (anticipated) future back to the present, so that the effect of future (expected) economic events and processes can be to "cause" things in the past—which then cause the future. And things can get very weird indeed.

Here's an Example of Things Getting Weird Indeed: In the United States the fall of 1929 saw the biggest stock market crash ever. It was the start of a prolonged period of decline that carried the values of stocks—shares of ownership of and thus both claims on the profits of and authority over the operations of America's companies—down by 1932 to less than one fifth of their 1929 peak:

The crash changed what Americans expected about the future of the economy, and the shifts in spending caused by their changed expectations played a key role in causing the greatest eco- nomic depression in American history, the Great Depression. On October 29, 1929, the price of shares traded on the New York Stock Exchange suffered their largest one-day percentage drop in history. Stock values bounced back a bit initially, but by the end of the week they were down by more than a quarter. Gloom fell over Wall Street. Many rich people had lost a lot of money.

At that time stock ownership was confined to the rich. Middle-class Americans owned little stock. Nonetheless, the crash affected nearly everyone's perceptions of the economy: Bad times were coming.

Because people expected the economic future to be dimmer, many cut back on spending, especially on big-ticket consumer durables. The 1920s had been the first decade in which consumer credit was widely available to finance purchases of cars, refrigerators, stoves, and washing machines. With the economic future uncertain, spending on consumer durables collapsed. It made sense to borrow to buy a consumer durable only if you were confident that you could make the payments and pay off the loan. If you thought the economic future might be bad, you had a powerful incentive to avoid debt. And in the short run the easiest way to avoid debt is to refrain from purchasing large consumer durables on credit.

You can probably guess what happened in the months after the crash. Most people simply stopped buying big-ticket items like cars and furniture. This massive drop in demand reduced new orders for goods. The drop in output generated layoffs in many industries. Even though most people’s incomes had not yet changed, their expectations of their future income had.

Moreover, the drop in demand greatly reduced the profits of businesses. stocks had had a high value in 1929 because the future looked bright: profits were expected to be high, and so it was worth paying high prices for shares of stock that got you the high future dividends that were expected were going to be paid out of those high future profits. But with stock prices low causing depressed demand those high future profits were no longer visible.

The drop in demand produced by this shift in expectations helped bring on what people feared; it put America on the path to the Great Depression. The Great Depression happened in large part because people expected something bad to happen. Without that pessimistic shift in expectations triggered by the crash of 1929, there would have been no Great Depression. And the fact that the Great Depression did arrive validated the lowered valuations of the American stock market that followed the crash of 1929.

Many economists talked—and some still talk—about how the high stock market values of 1928-1929 were "irrational", how there had to be a crash, and how the Great Depression was the result not of the shift of expectations that followed the crash—a shift could have been avoided if the crash had been avoided—but rather of the fact that stock market values were elevated in 1928-1929. The British economist John Maynard Keynes argued strongly against this point of view. "While some part of the investment", he said:

which was going on... was doubtless ill judged and unfruitful, there can, I think, be no doubt that the world was enormously enriched by the constructions of the quinquennium from 1925 to 1929; its wealth increased in these five years by as much as in any other ten or twenty years of its history.... Some austere and puritanical souls regard [the current Great Depression] both as an inevitable and a desirable nemesis on so much overexpansion, as they call it; a nemesis on man’s speculative spirit. It would, they feel, be a victory for the mammon of unrighteousness if so much prosperity was not subsequently balanced by universal bankruptcy. We need, they say, what they politely call a ‘prolonged liquidation’ to put us right. The liquidation, they tell us, is not yet complete. But in time it will be. And when sufficient time has elapsed for the completion of the liquidation, all will be well with us again...

They were, Keynes judged, completely wrong:

Doubtless, as was inevitable in a period of such rapid changes, the rate of growth of some individual commodities could not always be in just the appropriate relation to that of others. But, on the whole, I see little sign of any serious want of balance such as is alleged by some authorities. The rates of growth [of different sectors]…seem to me, looking back, to have been in as good a balance as one could have expected them to be. A few more quinquennia of equal activity might, indeed, have brought us near to the economic Eldorado where all our reasonable economic needs would be satisfied.

It seems an extraordinary imbecility that this wonderful outburst of productive energy [over 1925–29] should be the prelude to impoverishment and depression.... I find the explanation of the current business losses, of the reduction in output, and of the unemployment which necessarily ensues on this not in the high level of investment which was proceeding up to the spring of 1929, but in the subsequent cessation of this investment. I see no hope of a recovery except in a revival of the high level of investment. And I do not understand how universal bankruptcy can do any good or bring us nearer to prosperity...

In Keynes's view—and in mine, and in the view of the greater and wiser part of economists who have taken a serious look at the origins of the Great Depression—you can say that American stock prices at the end of 1929 and through the fall of 1929 were in a "bubble": so high that it is hard to see how rational and reasonable people could have thought it would pay to buy stocks and then simply hold them and collect the dividends that would flow from them, even in a continued boom. But the fall of stock prices below their start-of-1928 values could itself be rationalized only as a reaction to a forthcoming depression.

Thus there is a sense in which the fall in the stock market from 1930-1932 is both an effect of the gathering shadow of the forthcoming Great Depression

anda cause of that Great Depression. And there is a sense in which the Great Depression was both an effect of the collapse into pessimism produced by and measured by the erosion of the stock market from 1929-1932anda cause of a large part of that erosion.As I said: when expectations of the future matter for what happens in the present, things get weird. They get weird indeed.

Okay. Now back from this example to the main thread of the argument...

**A Quantitative Social Science**: In spite of the political complications, the nonexperimental nature, and the peculiar problems of cause and effect in economics, the discipline remains a quantitative science. Most of the relationships that economists study come quantified. Thus economics makes heavy use of arithmetic and algebra, while political science, sociology, and most of history do not. Economics makes heavy use of arithmetic to measure economic variables of interest. Moreover, economists use mathematical models to relate these variables.

**An Abstract Social Science**: The American economy is complex: 130 million workers, 10 million firms, and 90 million households buying and selling $24 trillion worth of goods and services per year. Economists must simplify it. To understand this complex phenomenon, they restrict their attention to a few behavioral relationships—cause-and-effect links between economic quantities—and a handful of equilibrium conditions—conditions that must be satisfied for economic activity to be stable and for supply and demand to be in balance. They attempt to capture these behavioral relationships and equilibrium conditions in simple algebraic equations and geometric diagrams. Then they try to apply their equations and graphs to the real world, while hoping that their simplifications have not made the model a distorted and faulty guide to how the real-world economy works.

Among economists, the process of reducing the complexity and variation of the real-world economy to a handful of equations is known as “building a model.” Using models to understand what is going on in the complex real-world economy has been a fruitful intellectual strategy. But model building tends to focus on the variables and relationships that fit easily into the algebraic model. It overlooks other factors.

Economics might have developed as a descriptive science, like sociology or political science. If so, courses in economics would concentrate on economic institutions and practices and the institutional structure of the economy as a whole. But it has not; it has instead become a more abstract science that emphasizes general principles applicable to a variety of situations. Thus a large part of economics involves a particular set of tools: a unique way of thinking about the world that is closely linked with the analytical tools economists use and that is couched in a particular technical language and a particular set of data. While one can get a lot out of sociology and political science courses without learning to think like a sociologist or a political scientist (because of their focus on institutional description), it is not possible to get much out of an economics course without learning to think like an economist.

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### The Rhetoric of Economics

**Analogies and Metaphors**: Surrounding the models economists build a special rhetoric: a set of analogies and metaphors that economists use to help laypeople grasp the functioning of the macroeconomy. Metaphors and analogies are the basic stuff of human thought. To understand something we do not know, we will often compare it to something we do know. Economics is no exception. In economics, curves “shift” and money has a “velocity.”

When the central bank raises interest rates and throws people out of work, economists say that this “pushes the economy down the Phillips curve.” Conversely, when the central bank lowers interest rates and the economy booms, economists say that this “pushes the economy up the Phillips curve”—as if the economy were a dot on a diagram drawn on a piece of paper, constrained to move along a particular curve called the Phillips curve, and monetary policy really did push the dot up and to the left:

That is, of course, not what happened. There was no dot pushed up and to the left. What happened is that more people got jobs and went to work, leaving fewer people unemployed; with a smaller pool of unemployed workers to draw on, businesses wanting to hire began raising wages faster than they had in the past; and to cover the costs of paying those more rapidly rising wages, they began to raise prices at a more rapid rate as well.

As a student you should be conscious of (and a little critical of) the rhetoric of economics for two reasons. First, if you don’t understand the metaphors, much of economics may simply be incomprehensible. All will become clear, or at least clearer, if you take care to remain very conscious of the metaphors.

Much of the rhetoric of economics can be reduced to four dominant elements:

The use of the word “market” to describe intricate and decentralized processes of exchange, as if all the workers and all the jobs in the economy really were being matched in a single open-air market.

The idea of equilibrium: that economic processes tend to move the economy into some sort of balance and to keep the economy at this point of balance.

The use of graphs and diagrams as an alternative to equations and arithmetic in expressing economic relationships. Economists identify equations with geometric curves, situations of equilibrium with points where curves cross, and changes in the economic environment or in economic policy with shifts in the positions of particular curves.

That economics is done by building

*models*—abstract algebraic structures that serves as a map between what we know about the structure of the economy—resources, capabilities, preferences, technologies, markets and institutions—and what economic outcomes might be. The purpose of these models is to be used as frameworks to make calculations of likely and possible economic outcomes—and to evaluate how effects might be different if the structure of the economy or the policies of the government were different.

**Markets**: Economists often speak as if all economic activity took place in great open-air marketplaces like those of medieval merchant cities. Contracts between workers and bosses are made in the “labor market.” All the borrowing of money from and the depositing of money into banks take place in the “money market.” Supply and demand balance in the “goods market.” Indeed, in the market squares of preindustrial trading cities you could survey the buyers and sellers and form a good idea of what was being sold and for how much.

In using the open-air markets of centuries past as a metaphor for the complex processes of matching and exchange that take place in today’s modern industrial economy, economists are assuming that information travels fast enough and buyers and sellers are well informed enough for prevailing prices and quantities to be as they would be if we actually could walk around the perimeter of a marketplace and examine all buyers and sellers in an hour. In most cases this bet that a decentralized matching process is like an open-air market will be a good intellectual bet to make. But sometimes (for example, in situations of so-called structural unemployment) it may not be.

**Equilibrium**: Economists spend most of their time searching for the state of equilibrium—a point or points of balance at which some economic quantity is neither rising nor falling. The dominant metaphor is that of an old-fashioned scale whose two pans are in balance. The search for equilibrium is an attempt to simplify the problem of understanding how the economy will behave. Economists’ questions are much easier to analyze if we can identify “points of rest” where the pressures for economic quantities to rise and fall are evenly balanced. Once the potential points of rest have been identified, economists can figure out how fast economic forces will push the economy to those points of equilibrium. The search for points of equilibrium, followed by an analysis of the speed of adjustment to equilibrium, is the most common way of proceeding in any economic analysis.

Do not, however, forget that this pattern of thought is merely an aid to understanding economic theories and principles. It is not the theories and principles themselves. The theories and principles, in turn, are just aids to understanding the reality; they are not themselves the reality.

**Graphs and Equations**: In the seventeenth century, the French philosopher and mathematician René Descartes spent much of his life demonstrating that graphs and equations are two different representations of the same reality. Specifically, an algebraic equation relating two variables can also be represented as a curve drawn on a graph. Each of the variables in the equation can be thought of as one of the axes of the graph. The set of points whose x-axis value is the first variable and whose y-axis value is the second—that is, the set of points for which the equation holds—makes up a line or curve on the graph. That line or curve is the equation. Thus the solution to a set of two equations is the point on the graph where the two curves that represent the equations intersect. Moreover, you can just as easily move back in the other direction, by thinking of a curve in terms of the equation that generates it. Today economists make very extensive use of these ideas from Descartes’ Analytic Geometry.

Just after the end of World War II Professor Paul Samuelson of MIT discovered that many of his students were much more comfortable manipulating diagrams than solving algebraic equations. With diagrams, they could see what was going on in a hypothetical economy. Thinking of how a particular curve would shift was often easier than thinking of the consequences of changing the value of the constant term in an equation.

When economists translate their algebraic equations into analytical geometric diagrams, they do things that may annoy you. Economists (like mathematicians) think of a “line” as a special kind of “curve” (and it is: it is a curve with zero curvature). So you may find words—in this book, in your lecture, or in your section—referring to a “Phillips curve,” but when you look at the accompanying diagram, you see that it is a straight line. Do not let this bother you. Economists use the word “curve” to preserve a little generality.

If you find analytic geometry easy and intuitive, then Samuelson’s intellectual innovation will make economics more accessible to you. Behavioral relationships become curves that shift about on a graph. Conditions of economic equilibrium become dots where curves describing two behavioral relationships cross (and thus both behavioral relationships are satisfied). Changes in the state of the economy become movements of a dot. Understanding economic theories and arguments becomes as simple as moving lines and curves around on a graph and looking for the place where the correct two curves intersect. And solving systems of equations becomes easy, as does changing the presuppositions of the problem and noting the results.

If you are not comfortable with analytic geometry, then you need to find other tools to help you think like an economist. Remember that the graphs are merely tools to aid your understanding. If they don’t, then you need to concentrate on understanding and manipulating the algebra or understanding and using the verbal descriptions of a problem. Use whatever method feels most comfortable. Grab hold of what makes most sense to you, and recognize that all three approaches are ways of reaching the same conclusions.

**Model Building**: This element of the rhetoric of economics is important enough that it deserves to be hoisted to its own proper section—in fact, sections:

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### Model Building: Assumptions

There. That's better...

**Simplification**: Simplification is the essence of model building. Economists use simple models for two reasons. First, no one really understands excessively complicated models, and a model is of little use if economists cannot understand the logic behind its prediction. Second, the predictions generated by simple models are nearly as good as the ones generated by more complex models. While the economic models used by the Federal Reserve or the Congressional Budget Office are more complicated than the models presented in this textbook, in essence they are cousins of the models used here.
You may have heard that economics is more of an art than a science. This means that the rules for effective and useful model building—for omitting unnecessary detail and complexity while retaining the necessary and important relationships—are nowhere written down. In this important respect, economists tend to learn by doing or by example. But there are fundamental steps that almost every successful construction of a macroeconomic model follows. They include the use of representative agents, a focus on opportunity costs in understanding agents’ decisions, and careful attention to the effect of people’s expectations on events.

**Representative Agents**: One simplification that economist often make—especially macroeconomists, but also, albeit much more rarely, microeconomists—is that all participants in the economy are the same or, rather, that the differences between businesses and workers do not matter much for the issues macroeconomists study. Thus macroeconomists analyze a situation by examining the decision making of a single representative agent—be it a business, a worker, or a saver. They then generalize to the economy as a whole from what would be the rational decisions of that single representative agent.

The use of representative agents makes pieces of economics simpler, yet it also makes some questions very hard to analyze. Consider unemployment: The key concept is that some workers have jobs but others do not. If one has adopted the simplifying assumption of a single representative worker, how can one worker represent both those who are employed and those who are not?

The assumption of a representative agent is also useless when the relative distribution of income and wealth among people in the economy is important. Most of the time our judgments about social welfare are influenced by distribution. Consider an economy in which everyone works equally hard, but 1 million lucky people receive $1,000,000 a year in income and 99 million receive $10,000 a year in income. Now consider an economy in which all 100 million people work equally hard and each receives $80,000 a year in income. Almost all of us would think that the second was a better—a happier and a fairer—economy, even though total income in the first economy amounts to $10.9 trillion and total income in the second economy amounts to only $8 trillion. As noted above, every discipline sees some things clearly and some things fuzzily. Distribution and its impact on social welfare are an area that economics has trouble bringing into focus.

**Opportunity Cost**: Perhaps the most fundamental principle of economics is that there is always a choice. But making a choice excludes the alternatives. If you keep your wealth in the form of easily spendable cash, you pass up the chance to keep it earning interest in the form of bonds. If you keep your wealth in the form of interest-earning bonds, you pass up the capability of immediately spending it on something that suddenly strikes your fancy. If you spend on consumption goods, you pass up the opportunity to save. Economists use the term opportunity cost to refer to the value of the best alternative that you forgo in making any particular choice.

At the root of every behavioral relationship is somebody’s decision. In analyzing such decisions, economists always think about the decision maker’s opportunity costs. What else could the decision maker do? What opportunities and choices does the decision maker foreclose by taking one particular course of action? Many students make economics a lot harder than it has to be by not remembering that this opportunity-cost way of thinking is at the heart of every behavioral relationship in an economic model.

**Expectations**: Many times the opportunity cost of taking some action today is not an alternative use of the same resources today but a forgone opportunity to save one’s resources for the future. A worker trying to decide whether to quit a job and search for another will be thinking about future wages after a successful search. A consumer trying to decide whether to spend or save will be thinking about what interest rate savings will earn in the future.

No one, however, knows the future. At best people can form rational and reasonable expectations of what the future might be. Hence nearly every behavioral relationship in macroeconomic models depends on expectations of the future. Expectations formation is a central, perhaps the central, piece of macroeconomics. Indeed, every economic model must account for the amount of time people can spend thinking about the future, the information they have available, and the rules of thumb they use to turn information into expectations.

Economists tend to consider three types of expectations:

Static expectations, in which decision makers simply don’t think about the future.

Adaptive expectations, in which decision makers assume that the future is going to be like the recent past.

Rational expectations, in which decision makers spend as much time as they can thinking about the future, and know as much (or more) about the structure and behavior of the economy as the model builder does.

The behavior of an economic model will differ profoundly depending on what kind of expectations economists build into the model.

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### Model Building: "Solving" Models

**Behavioral Relationships and Equilibrium Conditions**: When economists are trying to analyze the implications of how people act—say, how the overall level of production would change with a larger capital stock—they almost always write an equation that represents a behavioral relationship. This behavioral relationship states how the effect (the total level of production) is related to the cause (the available capital stock). The economist usually draws a diagram to help visualize the relationship and writes it down verbally: “A larger capital stock means that the average employee will have more machines and equipment to work with, and this will increase total production per worker. But increases in capital will probably be subject to diminishing returns to scale, so the gain in production from increasing the capital stock per worker from $40,000 to $80,000 will be less than the gain in production from increasing the capital stock per worker from $0 to $40,000.”

Let's drop down into another example:

A Behavioral Relationship: The Production Function: One of the key behavioral relationships in macroeconomics is the production function, which specifies the relationship between the economy’s productive resources. The production function relates:

- The economy’s capital-labor ratio (how many machines, tools, and structures are available to the average worker), written K/L (K for capital and L for labor).
- The level of technology or efficiency of the labor force, written E.
- The level of real GDP per worker, written Y/L (Y for real GDP or total output and L for the number of workers in the economy).
An economist could write the production function in this general, abstract form:

This formula states that the level of output per worker is some function, F, of K/L and E; that is, the level of output per worker depends in a systematic and predictable way on the capital-labor ratio and the efficiency of labor in the current year. But this abstract form does not specify the particular form of this systematic and predictable relationship.

Alternatively, an economist could write the production function in a particular algebraic form, for example, the Cobb-Douglas form which is convenient to use:

This equation states: “Take the capital-labor ratio, raise it to the exponential power α, and multiply the result by the efficiency of labor E raised to the exponential power (1–α). The result is the level of output per worker that the economy can produce.” While this equation is more particular than the abstract form Y/L = F(K/L, E), it remains flexible: The parameter α could have any of a wide range of different values, and the efficiency of labor E could have any value. This Cobb-Douglas algebraic form still stands for a whole family of possible production functions. The values chosen for E and α will tell us exactly which production function is the real one and thus what the behavioral relationship is between the economy’s resources and its output. Once we know the values of E and α, we can calculate what the output per worker will be for every possible value of capital per worker.

Why do economists choose to write the production function in this particular algebraic form? Ease of use is the key reason. This form of the production function makes a lot of calculations much simpler and more straightforward than other forms.

That is worth explaining enough for us to digress down another level:

Tools: Working with Exponents: What is the point behind the use of the Cobb-Douglas production function, with its exponents?

Recall that exponents greater than 1 are a means of repeated multiplication. Thus 2

^{1}is 2 multiplied by itself once, that is, just 2; 2^{2}is 2 times itself twice, that is, 2 × 2 = 4; 2^{4}is 2 times itself four times, that is, 2 × 2 × 2 x 2 = 16.Recall also that once one has decided that exponents are repeated multiplication, it is natural to decide that an exponent of 1/2 is a way of taking a square root. In fact, it becomes natural to take all fractions between 1 and zero as ways of taking various roots: 2

^{1/3}is the cube root of 2, 2^{2/5}is the fifth root of two-squared, and so forth.Whenever the Cobb-Douglas production function is used, the parameters that are the exponents will be between zero and 1.

In fact, in most applications of this production function the parameter α will be something like 1⁄2. Raising the capital-labor ratio to the α power is something like taking the square root of the capital-labor ratio. So the production function will state that output per worker is proportional to the square root of the capital-labor ratio. This function (a) is easy to calculate or look up for particular cases, (b) is one with which we have a lot of experience, (c) is an increasing function (so it fits the intuitive requirement that more capital is useful), and (d) is a function with diminishing returns—the higher the capital stock, the less valuable is the next investment in expanding the capital stock still further. Thus a lot of features that economists would like a sensible behavioral relationship between the capital-labor ratio and output per worker to have are already built into the Cobb-Douglas production function.

Moreover, there is an additional advantage to using the Cobb-Douglas production function. It makes calculating growth rates easy. Recall from Chapter 2 the rule of thumb for calculating the growth rate of a quantity raised to a power: The propor- tional change of a quantity raised to a power is equal to the proportional change in the quantity times the power to which it is raised.

Output per worker is proportional to the capital-output ratio raised to the power α. Thus if nothing else is changing and if we know the growth rate of the capital- labor ratio, we can immediately calculate the growth rate of output per worker. If α is 1⁄2 and if the capital-labor ratio is growing at 4 percent per year, then output per worker is growing at:

Now back to the main narrative thread...

**Equilibrium Conditions**: In addition to analyzing behavioral relationships, economists consider equilibrium conditions—conditions that must be true if the economy is to be in balance. If an equilibrium condition does not hold, then the state of the economy must be changing rapidly, moving toward a state of affairs in which the equilibrium condition does hold. In microeconomics the principal equilibrium condition is that supply must equal demand. If it does not, then buyers who find themselves short are frantically raising their bids (and prices are rising) or sellers who find themselves with excess inventory are frantically trying to shed it (and prices are falling). Only if supply equals demand can the price in a market be stable. In macroeconomics, the supply-must-equal-demand equilibrium condition is the most important, but there are other important equilibrium conditions too.

Let's give an example of one that isn't supply-equals-demand: here is a balanced-growth equilibrium condition:

Balanced Growth Equilibrium—The Steady-State Capital-Output Ratio: An important equilibrium condition that is not of the supply-must-equal-demand form is the equilibrium condition for balanced growth, which plays a big part in the study of long-run economic growth. This equilibrium condition relates the following economic variables and parameters:

- The net share of total net income in the economy that is saved and invested, written s.
- The proportional rate of growth of the labor force, written n.
- The proportional rate of growth of the efficiency of the labor force, written g.
- The economy's stock of produced means of production—of capital, written K.
- The economy's level of production and output, written Y.
- The ratio of the capital stock to the rate at which output is produced, written κ.
- The particular value of that capital-output ratio κ* at which it is neither rising nor falling over time.
Growth will be balanced if, and only if, the ratio of the economy’s stock of capital K to its level of output Y is constant. This equilibrium condition holds if, and only if, the capital-to-output ratio K/Y is equal to:

The capital-output ratio must be equal to the economy’s net savings rate s divided by the sum of the population growth rate n and the labor efficiency growth rate g.

If the capital-output ratio is lower than this value, it will grow because net investment will be high relative to the capital stock. If the capital-output ratio is higher than this value, it will shrink because net investment will be low relative to the capital stock. In either case, the capital-output ratio will converge to its balanced-growth equilibrium level over time.

Thus this balanced-growth capital-output-ratio equation satisfies the two requirements for being an equilibrium condition. If the economy does not satisfy the equilibrium condition, it will be heading toward it. If the economy satisfies the equilibrium condition, it will remain in the same place.

Now back to the main thread...

**Solving Models**: As we have seen, economists solve models by combining behavioral relationships with equilibrium conditions. If the equilibrium conditions are not satisfied, then the economy cannot be stable. Someone’s expectations must turn out to be false, or someone’s plans for what to buy and sell must be unsatisfied. If the behavioral relationships are not satisfied, the relationships are not behavioral. The behavioral relationships must be satisfied. Economists hope that the simplified behavioral relationships of models are a good enough match to actual behavior. And they hope that the actual economy moves to equilibrium rapidly enough that the only situations they need to consider are those in which the equilibrium conditions hold.

Let's consider three views of how one can put an economic model together: of how to combine the behavioral relationship of the production function with the balanced-growth equilibrium condition to determine the value of steady-state output per worker in the economy. We can use arithmetic. We can use algebra. And we can use analytical geometry—graphical techniques—Descartes’ idea that anything that can be done with equations can also be done with curves on a graph:

*Arithmetic*: We start out knowing the structure of the economy in terms of the coefficients of the behavioral relationships: how people act in response to the incentives they face. Using simple arithmetic, we can then combine the production function (a behavioral relationship) with the balanced-growth equilibrium condition to calculate the economy’s equilibrium level of output per worker.

Suppose the efficiency of labor E is $10,000 a year, the diminishing-returns-to-investment coefficient α is 1⁄2, the net savings rate s is 10 percent of total output, and the population growth rate n and the labor efficiency growth rate g are each 1 percent per year. In that case the balanced- growth capital-output ratio K/Y will be

If K/Y = 5, then K = 5 × Y and K/L = 5 × Y/L. Thus, in this economy, the equilibrium capital stock per worker will be five times output per worker:

Given the current efficiency of labor, the production function is

What is the equilibrium? In equilibrium, both the behavioral relationship and the equilibrium condition hold. To solve these two equations together, substitute one into the other:

In its balanced-growth equilibrium, in the current year, the level of output per worker is $50,000 a year, and the balanced-growth level of the capital stock per worker is $250,000.

*Algebra*: We have now seen how to use arithmetic to calculate the economy’s steady-state growth equilibrium level of output per worker for particular coefficient value for α and the particular value of the efficiency-of-labor variable E (E = $10,000 a year, and α = 1⁄2).

But if we wanted to find the answer for a different value of the variable E, and for another coefficients—for a different value of what we will call the parameter α—we would have to do the whole process all over again.

We can save a lot of work in the long run if we are willing to use algebra instead of arithmetic.

Once again, start with the behavioral relationship:

This equation states the influence of the capital-labor ratio K/L on output per worker Y/L. But our equilibrium condition doesn’t address the capital-labor ratio K/L; it is phrased in terms of the capital-output ratio K/Y. So we need to rewrite the equation using the fact that K/L = K/Y × Y/L—that is, the capital-labor ratio is equal to the capital-output ratio times output per worker. We proceed as follows:

Dividing both sides of this equation by (Y/L)^{α, we obtain:}

Raising both sides to the ^{1}/_{(1 – α)} power produces an equation that we can work with:

There is an important general principle here: If a behavioral relationship and an equilibrium condition refer to different variables, the first step is to rewrite one or the other so that both refer to the same thing.

Now we can determine the balanced-growth equilibrium level of output per worker by substituting the equation for the balanced-growth equilibrium condition into our reworked equation for the production function:

Thus if the efficiency of labor variable E is $10,000 a year, the diminishing-returns-to- investment parameter α is 1⁄2, the savings rate parameter s is 10 percent of total output, and the population growth rate n and labor efficiency growth rate g are both 1 percent per year, we can calculate the equilibrium level of output per worker as:

Notice that this is the same answer we arrived at using arithmetic.

But now we have a general equation that can be used with any set of parameter values. To answer the question, “What would output per worker be if everything else was the same but the efficiency of labor was doubled?” we could simply substitute the new parameter values into our algebraic answer:

And we could immediately say, “Output per worker doubles.”

*Analytic Geometry*: To avoid using algebra to solve the problem in Box 3.7, or just to understand what our algebraic manipulations are telling us, we can turn to René Descartes’ tools and use analytic geometry. We can think about our behavioral relationship, the production function, as a curve on a graph, with output per worker on the vertical axis and capital per worker on the horizontal axis (see Figure 3.6). This curve always slopes upward: Increasing the amount of capital per worker increases the amount of output per worker. However, as capital per worker increases, the curve’s slope decreases: Diminishing returns to scale mean that each new increase in capital yields less additional output than the one before.

On the graph, our equilibrium condition—that the capital-output ratio K/Y be equal to the steady-state value of s/(n + g) in terms of the parame-ters of the economy—is simply a line with a constant slope, for which capital per worker is the appropriate constant multiple of output per worker. Diminishing re-turns to capital guarantee that eventually the slope of the production function curve will fall below the slope of the equilibrium-condition line. Thus the two curves must cross. The point where they cross is the equilibrium:

If we knew the coefficient values and had a steady hand, we could solve for the balanced-growth value of output per worker simply by finding the point on the graph where the curves cross, and reading off the x-axis and y-axis values.

Even if we don’t have a steady hand and don’t know the parameter values, the diagram is still worth drawing. Diagrams allow for qualitative if not quantitative predictions. For example, consider an increase in the equilibrium capital-output ratio. It will reduce the slope of the equilibrium-condition line. You can immediately draw an equilibrium-condition line with a lesser slope, and see that such a change pushes the equilibrium point up and to the right along the production function. Diagrams allow us to visualize the effect of a change in a way that equations do not.

**The Advantages of Algebra**: Model building is a powerful way of thinking, if the details that you omit are indeed unnecessary and if the factors emphasized are the most important factors. Algebraic equations are the best way to summarize cause-and-effect behavioral relationships in economics. Because so many economic concepts are easily quantified, arithmetic might seem a more natural choice, but arithmetic quickly reaches its limits. For example, to know what the level of output per worker would be for each of a great number of possible levels of the capital stock per worker and the efficiency of labor, you would need to carry around a huge table. Here is just a tiny part of what would be required in the way of the huge table:

It is much better to remember and to work not with a *table* but to move to *variables*, and to write down and work with a single equation with variables and coefficients, like this one:

Algebra, moreover, has another advantage. It allows us to think about the consequences of a host of different possible systematic relationships by replacing the fixed and known *coefficients* like $10,000 and 0.5 with unspecified and potentially varying *parameters*, in this case E and α:

Using algebra to analyze this single equation allows us to manipulate and analyze all at once, in shorthand form, all the systematic relationships corresponding to different parameter values and all the tables that they summarize.

Do you want to analyze a situation in which boosting capital per worker raises output per worker at almost the same rate indefinitely? You can do that with a value of α that is near 1. Do you want to analyze a situation in which boosting capital per worker beyond an initial minimal level does little to raise potential output per worker? You can do that with a value of α near zero. Do you want to analyze an intermediate case? You can do that, too, with an intermediate value of the parameter α that shows how quickly a diminishing marginal product to investment sets in. Do you want to analyze a productive economy, in which output per worker is high? Then pick a high value of the parameter E, which represents the efficiency of labor. Do you want to analyze a poor economy, close to subsistence levels, in which even mammoth amounts of capital per worker would not create an affluent society? Then pick a low value of the parameter E. A huge number of potentially different economies are all analyzed whenever you analyze this equation;

Here are some of them:

It is easy moving back to the specific when you want to consider a particular case with a particular set of parameter values. Just substitute the numerical values for that particular case ($10,000 and 0.5) for the abstract parameters (E and α).

It is possible to pass—even do well in—an economics course without feeling comfortable with all the algebra. In general economists through each topic three times: once in words, stating the logic of the argument and telling which quantities influence which others; once in algebra; and once in diagrams that represent the algebraic and verbal relationships. If you don’t understand a concept the first time it is presented, you have two more chances.

Recognize, however, that words, equations, and diagrams are simply three ways of presenting the same material. They should agree. So a discrepancy between your understanding of the algebra, the diagrams, and the words is a sign that something is wrong with your understanding.

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**Housekeeping:**