# 1 Macroeconomics Modeling The Behavior Of Aggregate Variables

## Transcript Of 1 Macroeconomics Modeling The Behavior Of Aggregate Variables

Economics 314 Coursebook, 2010

Jeffrey Parker

1 MACROECONOMICS: MODELING THE BEHAVIOR OF AGGREGATE VARIABLES

Chapter 1 Contents

A. Topics and Tools ............................................................................. 2 B. Methods and Objectives of Macroeconomic Analysis................................ 2

What macroeconomists do.............................................................................................3 C. Models in Macroeconomics: Variables and Equations .............................. 4

Economic variables.......................................................................................................5 Economic equations......................................................................................................6 Analyzing the elements of a familiar microeconomic model ..............................................8 The importance of the exogeneity assumptions ..............................................................10 Static and dynamic models..........................................................................................11 Deterministic and stochastic models .............................................................................12 D. Mathematics in Macroeconomics ....................................................... 13 E. Forests and Trees: The Relationship between Macroeconomic and Microeconomic Models ........................................................................ 14 The complementary roles of microeconomics and macroeconomics ..................................14 Objectives of macroeconomic modeling..........................................................................17 F. Social Welfare and Macro Variables as Policy Targets ............................. 18 Utility and the social objective function.........................................................................18 Real income as an indicator of welfare..........................................................................20 The welfare effects of inflation ......................................................................................21 Real interest rates and real wage rates: Measures of welfare?...........................................22 G. Measuring Key Macroeconomic Variables ............................................ 23 Measuring real and nominal income and output ...........................................................23 Output = expenditure = income...................................................................................24 GDP and welfare .......................................................................................................26 Measuring prices and inflation.....................................................................................27 Employment and unemployment statistics ....................................................................29 International comparability of macroeconomic data ......................................................30 H. Works Referenced in Text................................................................ 31

A. Topics and Tools

In this chapter, we sketch the basic outlines of macroeconomic analysis: What do we study in macroeconomics and how do we go about it? This chapter is less about specific topics and analytical tools than it is a survey of the topics and tools that we’ll use throughout the course. It contains details about the macroeconomic variables we will study and about how macro models attempt to describe theories about the relationships among them.

The students in this class likely have widely varying exposure to macroeconomics, from a brief sampling of a few macro concepts to a full course at the intermediate level. Those who have a more extensive background may omit sections of this chapter with which they are very familiar.

B. Methods and Objectives of Macroeconomic Analysis

Macroeconomics is the study of the relationships among aggregate economic variables. We are interested in such questions as: Why do some economies grow faster than others? Why are economies subject to recessions and booms? What determines the rates of price and wage inflation?

These kinds of questions often involve issues of macroeconomic policy: What are the effects of government budget deficits? How does Federal Reserve monetary policy affect the economy? What can policymakers do to increase economic growth? Can monetary or fiscal policies help stabilize business-cycle fluctuations?

An important but elusive goal of macroeconomics is accurate forecasting of economic conditions. Although economic forecasting may well be a billion-dollar-a-year industry, forecasters are far better at predicting each other’s forecasts than at anticipating economic fluctuations with any degree of precision. Experience suggests that in most cases the pundits who predicted the last recession correctly are not very likely to get the next one right.

The failure to achieve finely tuned forecasts is not surprising. Modern economies are huge entities of unfathomable complexity. The behavior of the U.S. economy depends on the decisions of more than 100 million households about how much to buy, how much to work, and how to use their resources. Equally important are millions of business firms—from small to gigantic—deciding how much to produce, what prices to charge, how many workers to hire, and how much plant and equipment to buy and use.

1 – 2

Economists attempt to bypass this complexity with simple, abstract models of the macroeconomy. A good model is simple enough to make transparent the key mechanisms that make it work yet capable of replicating salient aspects of the economic activity it attempts to describe. By understanding how the simple model operates we hope to gain flashes of insight about the workings of the incomprehensibly vast macroeconomy itself.

Of course, no simple model can pretend to describe all aspects of an economy, to apply to all macroeconomies, or to be relevant to any macroeconomy under all possible conditions. We must be very careful in applying models: use the insights they provide about the economy but don’t stretch them too far. Simple models are made from simplistic assumptions. For some applications, those assumptions may be innocuous; for others they are likely to be misleading.

What macroeconomists do

Macroeconomics consists broadly of two modes of analysis. Theoretical macroeconomics creates models that purport to describe important aspects of macroeconomic behavior and demonstrates their properties by solving the mathematical systems that describe them. Empirical macroeconomics uses aggregate (or sometimes disaggregated) data to test the conclusions of the theoretical models.

A new strand of macroeconomic literature often begins with observation of a phenomenon that is not satisfactorily explained by existing theories. To consider a simple historical example: Why is per-capita income in the United States much higher than in India? Consideration of the unexplained phenomenon stimulates the development of one or more simple theoretical models attempting to explain in very simple terms the mechanism that led to the phenomenon. For example, the Solow growth model could be used to argue that the income difference between the U.S. and India is due to higher saving rates (and therefore greater capital accumulation) in the United States.

Once a theoretical model has been proposed, empirical scholars swarm over it like an army of ants, testing in many ways whether the theory’s implications match actual macroeconomic observations. The answer is invariably ambiguous: there are so many different (but equally valid) ways of testing any model that most new theories will be supported by some pieces of evidence and refuted by others. For example, we may find that differences in capital accumulation are an important determinant of the U.S./India income differential, but that they are not large enough to explain the entire gap. Or we may find that capital accumulation can adequately explain the gap between incomes in the U.S. and France in 1948, but not between the U.S. and Zimbabwe in 2009.

The results of these empirical tests give us clues about the ways in which the model succeeds in providing insight into macroeconomic behavior and the ways in

1 – 3

which it falls short. This then leads to extensions and modifications of the original theory, attempting to reconcile it with those empirical facts that conflicted with the original model’s conclusions. These revised theories will then be subjected to empirical scrutiny until, perhaps, a consensus emerges about what (if anything) the model can teach us about the economy. Given the inevitable conflicting evidence, there is always room for disagreement about a model’s relevance, which often leads to spicy debates among macroeconomists.

Because the empirical testing of major macroeconomic theories consists of scores of individual studies that often reach conflicting conclusions, the student of empirical macroeconomics faces a daunting challenge. Reading individual empirical studies is essential to appreciate the methods used to test hypotheses. However, it is impossible to understand the breadth of evidence on a widely studied question without reading a half-dozen or more separate papers. The approach that we will take in this class will be to look in detail at a few key studies, then to survey the broader literature to get an idea of the degree of consensus or disagreement in results. Several coursebook chapters are devoted to discussion of empirical evidence using this format.

C. Models in Macroeconomics: Variables and Equations

Introducing models As noted above, the macroeconomy is a system of tremendous complexity. Hun-

dreds of millions of individual people and firms make daily decisions about working, producing, investing, buying, and selling. These decisions are affected by countless factors (many of them unobservable and some unmeasurable) including abilities, preferences, incomes, prices, laws, technology, and the weather. To understand a modern macroeconomy in all its detail is clearly beyond the power of the human mind, even when aided by powerful computers.

As do other scientists, economists try to understand important features of the macroeconomy by building models to answer particular questions. Because they must be simple enough to work with, all models necessarily omit most of the details of the interactions among economic agents. A good model for any particular question is one that captures the interactions that bear most importantly on that question, while omitting those that are less relevant. Since different questions address different aspects of the economy, a model that is good for analyzing one question may be very poor for looking at others. For example, a model in which the labor market is perfectly competitive might provide a reasonable framework for looking at long-run wage behavior, but because it assumes full employment it would not yield useful in-

1 – 4

sights about movements in the unemployment rate. There is no universally correct model of the macroeconomy, only models that have proved to be useful in answering specific sets of questions.

Some scientific models have tangible representations, such as a globe in geography or a physical molecular model in chemistry. Economic models do not have such a physical representation. Rather they are abstract mathematical models composed of variables linked together by equations.

Economic variables

The variables of economic models are measurable magnitudes that are outcomes of or inputs to economic decisions. Familiar examples include the number of hours worked, the amount of income earned, the amount of milk purchased, the price of a pound of kumquats, or the interest rate on a loan. Many of these variables can be observed at different levels of aggregation. For example, an income variable could be the income of one person or household, or the aggregate income of all of the people in a city, state, or nation. Price variables can be specific to one commodity such as a 2009 Ford Focus with air-conditioning and automatic transmission, or an index of the prices of many commodities such as an automobile-price index or a price index for all consumer goods purchased in the United States.

In macroeconomics, we shall most often be interested in the behavior of economy-wide aggregates, including gross domestic product, economy-wide price indexes, and total employment. However, because decisions in market economies are made by individual households and firms, most of the theories underlying our models must be built at the microeconomic level then aggregated to form a macroeconomic model. This aggregation is usually accomplished by making extreme simplifying assumptions, such as assuming that all consumers are alike or that differences among individual consumer goods are irrelevant.

The purpose of an economic model is to describe how the values of some of its variables are determined, and especially how they are affected by changes in the values of other variables. The variables whose determination is described by the model are called endogenous variables. Variables whose values are assumed to be determined outside the model are exogenous variables. Because they are determined outside, exogenous variables are assumed to be unaffected by changes in other variables in the model. For example, the price and production of corn would be endogenous variables in a national model of agricultural markets, while variables measuring the weather would be exogenous. (It is reasonable to suppose that the other variables of the corn market do not affect the weather.) In macroeconomic models, aggregate output, the general price level, interest rates, and the unemployment rate are usually endogenous variables. Variables set by government policymakers and those determined in other countries are often assumed to be exogenous.

1 – 5

The “final product” of a macroeconomic modeling exercise is a characterization of the relationship between the exogenous and endogenous variables under the assumptions of the model. The exogenous variables are assumed to affect the endogenous variables but not vice versa, so this relationship has a causal interpretation: a change in an exogenous variable causes a change in the endogenous variables. This usually takes the form of statements such as “a one-unit increase in exogenous variable X (holding all the other exogenous variables constant) in period t would lead to an increase in endogenous variable Y of 1.6 units.”

The example above would be appropriate to a static model in which time does not enter in an important way. However, all important models in modern macroeconomics are dynamic models, where instead of examining changes in the levels of variables we consider changes in their time paths. In a dynamic model, the final product is a statement more like “a one-unit permanent increase in X starting in period t (holding the paths of all other exogenous variables constant) would increase in endogenous variable Y by 1.6 units in period t, 2.2 units in period t + 1, …, and 3.4 units in the steady state of the new growth path.”

In order to arrive at this final product of our model, we must solve the model’s equations. In simple models, we can use algebra to calculate a solution; more complex models can only be solved by numerical simulation methods. The process of solving economic models is discussed below.

Economic equations

A model's assumptions about individual and market behavior are represented by its structural equations. Each equation expresses a relationship among some of the model’s variables. For example, a demand equation might express the economic assumption that the quantity of a good demanded is related in a given way to its price, the prices of related goods, and aggregate income.

Endogenous variables are ones whose behavior is described by the model. Mathematically, their values are determined by the equations of the model. A model’s “solution” consists of a set of mathematical equations that express each of the endogenous variables as a function solely of exogenous variables. This is an extremely important definition: you will be called upon to solve macro models in many homework sets during the course. You have not solved the model until you have an equation for each important endogenous variable that does not involve any other endogenous variables. These equations are often called reduced-form equations. Each reduced-form equation tells how the equilibrium value of one endogenous variable depends on the values of the set of exogenous variables.

1 – 6

One of the most common mistakes that students make in solving macroeconomic models is to leave endogenous variables on the right-hand side of an alleged solution.

Structural Model

Exogenous variables Endogenous variables

Equations

X1

Y1

Y1 = F1(Y2, X1, X2)

X2

Y2

Y2 = F2(Y1, X2, X3)

X3

Reduced-Form Model

Exogenous variables Endogenous variables Equations

X1

Y1

Y1 = G1(X1, X2, X3)

X2

Y2

Y2 = G2(X1, X2, X3)

X3

Figure 1. Schematic representation of structural and reduced-form questions

The model is not solved until you have expressed the endogenous variable solely as a function of exogenous variables.

Figure 1 shows the relationship between the structural and reduced-form equations for a simple model with three exogenous variables X1, X2, and X3, and two endogenous variables Y1 and Y2. The arrows in the schematic in the left of the diagram show directions of causal effects. The equations in the boxes at right represent these relationships using functional notation.

Examining the structural equations in the top section we see that our economic theory says that X1 and X2 affect Y1, but X3 does not. Similarly, X2 and X3 affect Y2,

1 – 7

but X1 does not. Each of the endogenous variables affects the other, as shown by the vertical arrows joining them.

Solving this model for its reduced-form equations consists of expressing each of the endogenous variables as a separate function of only the exogenous variables. Thus, in the reduced form we solve out the dependence of Y1 and Y2 on each other. The reduced-form equations (the G functions) depend only on the X variables. However, note that although X3 has no direct effect on Y1 in the structural model, changes in X3 usually will have an effect on Y1 through its effect on Y2 and the effect of Y2 on Y1. Therefore all of the exogenous variables generally appear in the reduced-form equations for all of the endogenous variables, although in special cases the effect may be absent.

Two aspects of Figure 1 are worth stressing: • There are never any arrows pointing toward exogenous variables and there are no equations determining the X variables. This is what exogeneity means. • There are generally arrows from some endogenous variables to others in the structural model, but there can never be arrows pointing from endogenous variables in the reduced form. Y variables can appear in the functions of the structural equations but not the reduced-form equations. This is what solving the model to get the reduced form means.

Analyzing the elements of a familiar microeconomic model

A simple example from introductory microeconomics may help clarify the nature of exogenous and endogenous variables, equations, and solving for the reduced form. Suppose that the demand for corn is assumed to be

cc = α0 + α1 pc + α2 y,

(1)

where cc is consumption of corn, pc is the price of corn, and y is aggregate income. Equation (1) expresses the economic assumption that the quantity of corn demanded is a linear function of the two variables appearing on the right-hand side (and no others). In order to draw conclusions from the model, we usually add assumptions about the signs of some of the coefficients in the equation. In equation (1) we might assume that α1 < 0, which reflects a downward-sloping demand curve, and that α2 > 0, which says that corn is a normal good: demand rises when income goes up.

The second equation of our model is a supply curve describing production of corn, which we denote by qc. We could assume that the supply curve for corn can be represented by

qc = β0 + β1 pc + β2 R,

(2)

1 – 8

where R is rainfall in major corn-producing states during the growing season. The additional assumption β1 > 0 means that the supply curve slopes upward, while β2 > 0 implies that production increases with more rainfall.

We complete our model with an assumption about how consumption is related to production. The simplest assumption is that they are equal, which is an assumption of market clearing:

qc = cc

(3)

More sophisticated models could allow for stocks of corn inventories to absorb differences between production and consumption, but we will keep things simple in this example.

Equations (1), (2), and (3) express the assumptions of the model about demand, supply, and market clearing. We must also specify which variables are to be considered endogenous and which are assumed exogenous. In order for the model to have a single, unique solution, there must usually be the same number of equations as endogenous variables. The three variables that would typically be assumed endogenous in the corn-market model would be cc , pc , and qc .

This leaves R and y as exogenous variables. Exogeneity assumptions are critical to the specification of a model. Is it reasonable to assume that income and rainfall are exogenous? Can we assume that a change in any of the other variables of the model (endogenous or exogenous) would not affect income or rainfall? It seems safe to assume that rainfall is exogenous because it is unlikely that changes in corn production, corn consumption, prices, or incomes would affect the weather in Nebraska. The exogeneity of income is a little less clear cut, but since the corn market is only a very small part of the economy, it is unlikely that aggregate GDP would be affected very much by changes to the model’s other variables. Thus, our exogeneity assumptions seem reasonable here. (Later in this section we will consider adding another variable for which exogeneity is more questionable.)

The purpose of a model is to examine how changes in the variables are related. In particular, since the model is supposed to represent the process by which the endogenous variables are determined, we are interested in knowing how each endogenous variable would be affected by a change in one or more of the exogenous variables. We do this by solving the model’s equations to find reduced-form expressions for each endogenous variable as a function only of exogenous variables. In this example, we seek equations representing corn consumption, production, and price as a function only of rainfall and income.

In this simple model, we can find a reduced-form equation for pc with simple algebra. First, we use equation (3) to set the right-hand sides of equations (1) and (2)

1 – 9

equal: α0 + α1 pc + α2 y = β0 + β1 pc + β2 R. Isolating the two pc terms on the same side and dividing yields

p = α0 − β0 − β2 R + α2 y.

(4)

c β −α β −α β −α

1

1

1

1

1

1

Equation (4) is a reduced-form equation for pc because the only other variables appearing in the equation are the exogenous variables R and y.

Since (4) is a linear reduced form for pc , the coefficients in front of the R and y terms measure the effect of a one-unit increase in R or y on pc .1 Thus, a one-unit increase in R causes pc to change by –β2 /(β1 – α1). Earlier, we assumed that α1 < 0 and β1 > 0, so the denominator consists of a negative number subtracted from a positive number and is surely positive. We also assumed that β2 > 0, so the numerator is positive until the negative sign in front makes the whole expression negative. Since we can establish that the coefficient measuring Δpc /ΔR is negative, we have shown that under the assumptions of our model, an increase in rainfall will lower the equilibrium price of corn (by shifting the supply curve outward). A similar analysis can be used to show that Δpc /Δy is positive, so we conclude that an increase in income raises the price of corn.

Equation (4) tells us how pc is affected by R and y, but what about the effects of R and y on c and q? Since c = q at all times, Δc/ΔR and Δc/Δy are the same as Δq/ΔR and Δq/Δy. The effects of the exogenous variables on pc can be found by substituting equation (4) into either equation (1) or equation (2) and simplifying. If you do this, you will find that, as expected, either an increase in rainfall or an increase in income raises the quantity produced and consumed.

The importance of the exogeneity assumptions

Choosing which variables are to be endogenous and which are to be exogenous is an important part of the specification of a model. If a variable is (incorrectly) assumed to be exogenous when it is actually influenced in important ways by other

1

Two comments are important here. First, if this were not a reduced-form equation, then it would not be correct to use the coefficients of the exogenous variables to measure their effects on the endogenous variables. This is because when an exogenous variable changes, all of the endogenous variables in the model usually change. In (4), a change in R does not change y, thus the only change in pc is the effect measured by the coefficient on R. If another endogenous variable were on the right-hand side, then it would also be changing and pc would change by the sum of the two effects. Second, only when the model is linear can the effect of the exogenous variable on the endogenous variable be read off directly as a coefficient. In nonlinear models, it is necessary to differentiate the reduced-form equation with respect to an exogenous variable to evaluate the effect.

1 – 10

Jeffrey Parker

1 MACROECONOMICS: MODELING THE BEHAVIOR OF AGGREGATE VARIABLES

Chapter 1 Contents

A. Topics and Tools ............................................................................. 2 B. Methods and Objectives of Macroeconomic Analysis................................ 2

What macroeconomists do.............................................................................................3 C. Models in Macroeconomics: Variables and Equations .............................. 4

Economic variables.......................................................................................................5 Economic equations......................................................................................................6 Analyzing the elements of a familiar microeconomic model ..............................................8 The importance of the exogeneity assumptions ..............................................................10 Static and dynamic models..........................................................................................11 Deterministic and stochastic models .............................................................................12 D. Mathematics in Macroeconomics ....................................................... 13 E. Forests and Trees: The Relationship between Macroeconomic and Microeconomic Models ........................................................................ 14 The complementary roles of microeconomics and macroeconomics ..................................14 Objectives of macroeconomic modeling..........................................................................17 F. Social Welfare and Macro Variables as Policy Targets ............................. 18 Utility and the social objective function.........................................................................18 Real income as an indicator of welfare..........................................................................20 The welfare effects of inflation ......................................................................................21 Real interest rates and real wage rates: Measures of welfare?...........................................22 G. Measuring Key Macroeconomic Variables ............................................ 23 Measuring real and nominal income and output ...........................................................23 Output = expenditure = income...................................................................................24 GDP and welfare .......................................................................................................26 Measuring prices and inflation.....................................................................................27 Employment and unemployment statistics ....................................................................29 International comparability of macroeconomic data ......................................................30 H. Works Referenced in Text................................................................ 31

A. Topics and Tools

In this chapter, we sketch the basic outlines of macroeconomic analysis: What do we study in macroeconomics and how do we go about it? This chapter is less about specific topics and analytical tools than it is a survey of the topics and tools that we’ll use throughout the course. It contains details about the macroeconomic variables we will study and about how macro models attempt to describe theories about the relationships among them.

The students in this class likely have widely varying exposure to macroeconomics, from a brief sampling of a few macro concepts to a full course at the intermediate level. Those who have a more extensive background may omit sections of this chapter with which they are very familiar.

B. Methods and Objectives of Macroeconomic Analysis

Macroeconomics is the study of the relationships among aggregate economic variables. We are interested in such questions as: Why do some economies grow faster than others? Why are economies subject to recessions and booms? What determines the rates of price and wage inflation?

These kinds of questions often involve issues of macroeconomic policy: What are the effects of government budget deficits? How does Federal Reserve monetary policy affect the economy? What can policymakers do to increase economic growth? Can monetary or fiscal policies help stabilize business-cycle fluctuations?

An important but elusive goal of macroeconomics is accurate forecasting of economic conditions. Although economic forecasting may well be a billion-dollar-a-year industry, forecasters are far better at predicting each other’s forecasts than at anticipating economic fluctuations with any degree of precision. Experience suggests that in most cases the pundits who predicted the last recession correctly are not very likely to get the next one right.

The failure to achieve finely tuned forecasts is not surprising. Modern economies are huge entities of unfathomable complexity. The behavior of the U.S. economy depends on the decisions of more than 100 million households about how much to buy, how much to work, and how to use their resources. Equally important are millions of business firms—from small to gigantic—deciding how much to produce, what prices to charge, how many workers to hire, and how much plant and equipment to buy and use.

1 – 2

Economists attempt to bypass this complexity with simple, abstract models of the macroeconomy. A good model is simple enough to make transparent the key mechanisms that make it work yet capable of replicating salient aspects of the economic activity it attempts to describe. By understanding how the simple model operates we hope to gain flashes of insight about the workings of the incomprehensibly vast macroeconomy itself.

Of course, no simple model can pretend to describe all aspects of an economy, to apply to all macroeconomies, or to be relevant to any macroeconomy under all possible conditions. We must be very careful in applying models: use the insights they provide about the economy but don’t stretch them too far. Simple models are made from simplistic assumptions. For some applications, those assumptions may be innocuous; for others they are likely to be misleading.

What macroeconomists do

Macroeconomics consists broadly of two modes of analysis. Theoretical macroeconomics creates models that purport to describe important aspects of macroeconomic behavior and demonstrates their properties by solving the mathematical systems that describe them. Empirical macroeconomics uses aggregate (or sometimes disaggregated) data to test the conclusions of the theoretical models.

A new strand of macroeconomic literature often begins with observation of a phenomenon that is not satisfactorily explained by existing theories. To consider a simple historical example: Why is per-capita income in the United States much higher than in India? Consideration of the unexplained phenomenon stimulates the development of one or more simple theoretical models attempting to explain in very simple terms the mechanism that led to the phenomenon. For example, the Solow growth model could be used to argue that the income difference between the U.S. and India is due to higher saving rates (and therefore greater capital accumulation) in the United States.

Once a theoretical model has been proposed, empirical scholars swarm over it like an army of ants, testing in many ways whether the theory’s implications match actual macroeconomic observations. The answer is invariably ambiguous: there are so many different (but equally valid) ways of testing any model that most new theories will be supported by some pieces of evidence and refuted by others. For example, we may find that differences in capital accumulation are an important determinant of the U.S./India income differential, but that they are not large enough to explain the entire gap. Or we may find that capital accumulation can adequately explain the gap between incomes in the U.S. and France in 1948, but not between the U.S. and Zimbabwe in 2009.

The results of these empirical tests give us clues about the ways in which the model succeeds in providing insight into macroeconomic behavior and the ways in

1 – 3

which it falls short. This then leads to extensions and modifications of the original theory, attempting to reconcile it with those empirical facts that conflicted with the original model’s conclusions. These revised theories will then be subjected to empirical scrutiny until, perhaps, a consensus emerges about what (if anything) the model can teach us about the economy. Given the inevitable conflicting evidence, there is always room for disagreement about a model’s relevance, which often leads to spicy debates among macroeconomists.

Because the empirical testing of major macroeconomic theories consists of scores of individual studies that often reach conflicting conclusions, the student of empirical macroeconomics faces a daunting challenge. Reading individual empirical studies is essential to appreciate the methods used to test hypotheses. However, it is impossible to understand the breadth of evidence on a widely studied question without reading a half-dozen or more separate papers. The approach that we will take in this class will be to look in detail at a few key studies, then to survey the broader literature to get an idea of the degree of consensus or disagreement in results. Several coursebook chapters are devoted to discussion of empirical evidence using this format.

C. Models in Macroeconomics: Variables and Equations

Introducing models As noted above, the macroeconomy is a system of tremendous complexity. Hun-

dreds of millions of individual people and firms make daily decisions about working, producing, investing, buying, and selling. These decisions are affected by countless factors (many of them unobservable and some unmeasurable) including abilities, preferences, incomes, prices, laws, technology, and the weather. To understand a modern macroeconomy in all its detail is clearly beyond the power of the human mind, even when aided by powerful computers.

As do other scientists, economists try to understand important features of the macroeconomy by building models to answer particular questions. Because they must be simple enough to work with, all models necessarily omit most of the details of the interactions among economic agents. A good model for any particular question is one that captures the interactions that bear most importantly on that question, while omitting those that are less relevant. Since different questions address different aspects of the economy, a model that is good for analyzing one question may be very poor for looking at others. For example, a model in which the labor market is perfectly competitive might provide a reasonable framework for looking at long-run wage behavior, but because it assumes full employment it would not yield useful in-

1 – 4

sights about movements in the unemployment rate. There is no universally correct model of the macroeconomy, only models that have proved to be useful in answering specific sets of questions.

Some scientific models have tangible representations, such as a globe in geography or a physical molecular model in chemistry. Economic models do not have such a physical representation. Rather they are abstract mathematical models composed of variables linked together by equations.

Economic variables

The variables of economic models are measurable magnitudes that are outcomes of or inputs to economic decisions. Familiar examples include the number of hours worked, the amount of income earned, the amount of milk purchased, the price of a pound of kumquats, or the interest rate on a loan. Many of these variables can be observed at different levels of aggregation. For example, an income variable could be the income of one person or household, or the aggregate income of all of the people in a city, state, or nation. Price variables can be specific to one commodity such as a 2009 Ford Focus with air-conditioning and automatic transmission, or an index of the prices of many commodities such as an automobile-price index or a price index for all consumer goods purchased in the United States.

In macroeconomics, we shall most often be interested in the behavior of economy-wide aggregates, including gross domestic product, economy-wide price indexes, and total employment. However, because decisions in market economies are made by individual households and firms, most of the theories underlying our models must be built at the microeconomic level then aggregated to form a macroeconomic model. This aggregation is usually accomplished by making extreme simplifying assumptions, such as assuming that all consumers are alike or that differences among individual consumer goods are irrelevant.

The purpose of an economic model is to describe how the values of some of its variables are determined, and especially how they are affected by changes in the values of other variables. The variables whose determination is described by the model are called endogenous variables. Variables whose values are assumed to be determined outside the model are exogenous variables. Because they are determined outside, exogenous variables are assumed to be unaffected by changes in other variables in the model. For example, the price and production of corn would be endogenous variables in a national model of agricultural markets, while variables measuring the weather would be exogenous. (It is reasonable to suppose that the other variables of the corn market do not affect the weather.) In macroeconomic models, aggregate output, the general price level, interest rates, and the unemployment rate are usually endogenous variables. Variables set by government policymakers and those determined in other countries are often assumed to be exogenous.

1 – 5

The “final product” of a macroeconomic modeling exercise is a characterization of the relationship between the exogenous and endogenous variables under the assumptions of the model. The exogenous variables are assumed to affect the endogenous variables but not vice versa, so this relationship has a causal interpretation: a change in an exogenous variable causes a change in the endogenous variables. This usually takes the form of statements such as “a one-unit increase in exogenous variable X (holding all the other exogenous variables constant) in period t would lead to an increase in endogenous variable Y of 1.6 units.”

The example above would be appropriate to a static model in which time does not enter in an important way. However, all important models in modern macroeconomics are dynamic models, where instead of examining changes in the levels of variables we consider changes in their time paths. In a dynamic model, the final product is a statement more like “a one-unit permanent increase in X starting in period t (holding the paths of all other exogenous variables constant) would increase in endogenous variable Y by 1.6 units in period t, 2.2 units in period t + 1, …, and 3.4 units in the steady state of the new growth path.”

In order to arrive at this final product of our model, we must solve the model’s equations. In simple models, we can use algebra to calculate a solution; more complex models can only be solved by numerical simulation methods. The process of solving economic models is discussed below.

Economic equations

A model's assumptions about individual and market behavior are represented by its structural equations. Each equation expresses a relationship among some of the model’s variables. For example, a demand equation might express the economic assumption that the quantity of a good demanded is related in a given way to its price, the prices of related goods, and aggregate income.

Endogenous variables are ones whose behavior is described by the model. Mathematically, their values are determined by the equations of the model. A model’s “solution” consists of a set of mathematical equations that express each of the endogenous variables as a function solely of exogenous variables. This is an extremely important definition: you will be called upon to solve macro models in many homework sets during the course. You have not solved the model until you have an equation for each important endogenous variable that does not involve any other endogenous variables. These equations are often called reduced-form equations. Each reduced-form equation tells how the equilibrium value of one endogenous variable depends on the values of the set of exogenous variables.

1 – 6

One of the most common mistakes that students make in solving macroeconomic models is to leave endogenous variables on the right-hand side of an alleged solution.

Structural Model

Exogenous variables Endogenous variables

Equations

X1

Y1

Y1 = F1(Y2, X1, X2)

X2

Y2

Y2 = F2(Y1, X2, X3)

X3

Reduced-Form Model

Exogenous variables Endogenous variables Equations

X1

Y1

Y1 = G1(X1, X2, X3)

X2

Y2

Y2 = G2(X1, X2, X3)

X3

Figure 1. Schematic representation of structural and reduced-form questions

The model is not solved until you have expressed the endogenous variable solely as a function of exogenous variables.

Figure 1 shows the relationship between the structural and reduced-form equations for a simple model with three exogenous variables X1, X2, and X3, and two endogenous variables Y1 and Y2. The arrows in the schematic in the left of the diagram show directions of causal effects. The equations in the boxes at right represent these relationships using functional notation.

Examining the structural equations in the top section we see that our economic theory says that X1 and X2 affect Y1, but X3 does not. Similarly, X2 and X3 affect Y2,

1 – 7

but X1 does not. Each of the endogenous variables affects the other, as shown by the vertical arrows joining them.

Solving this model for its reduced-form equations consists of expressing each of the endogenous variables as a separate function of only the exogenous variables. Thus, in the reduced form we solve out the dependence of Y1 and Y2 on each other. The reduced-form equations (the G functions) depend only on the X variables. However, note that although X3 has no direct effect on Y1 in the structural model, changes in X3 usually will have an effect on Y1 through its effect on Y2 and the effect of Y2 on Y1. Therefore all of the exogenous variables generally appear in the reduced-form equations for all of the endogenous variables, although in special cases the effect may be absent.

Two aspects of Figure 1 are worth stressing: • There are never any arrows pointing toward exogenous variables and there are no equations determining the X variables. This is what exogeneity means. • There are generally arrows from some endogenous variables to others in the structural model, but there can never be arrows pointing from endogenous variables in the reduced form. Y variables can appear in the functions of the structural equations but not the reduced-form equations. This is what solving the model to get the reduced form means.

Analyzing the elements of a familiar microeconomic model

A simple example from introductory microeconomics may help clarify the nature of exogenous and endogenous variables, equations, and solving for the reduced form. Suppose that the demand for corn is assumed to be

cc = α0 + α1 pc + α2 y,

(1)

where cc is consumption of corn, pc is the price of corn, and y is aggregate income. Equation (1) expresses the economic assumption that the quantity of corn demanded is a linear function of the two variables appearing on the right-hand side (and no others). In order to draw conclusions from the model, we usually add assumptions about the signs of some of the coefficients in the equation. In equation (1) we might assume that α1 < 0, which reflects a downward-sloping demand curve, and that α2 > 0, which says that corn is a normal good: demand rises when income goes up.

The second equation of our model is a supply curve describing production of corn, which we denote by qc. We could assume that the supply curve for corn can be represented by

qc = β0 + β1 pc + β2 R,

(2)

1 – 8

where R is rainfall in major corn-producing states during the growing season. The additional assumption β1 > 0 means that the supply curve slopes upward, while β2 > 0 implies that production increases with more rainfall.

We complete our model with an assumption about how consumption is related to production. The simplest assumption is that they are equal, which is an assumption of market clearing:

qc = cc

(3)

More sophisticated models could allow for stocks of corn inventories to absorb differences between production and consumption, but we will keep things simple in this example.

Equations (1), (2), and (3) express the assumptions of the model about demand, supply, and market clearing. We must also specify which variables are to be considered endogenous and which are assumed exogenous. In order for the model to have a single, unique solution, there must usually be the same number of equations as endogenous variables. The three variables that would typically be assumed endogenous in the corn-market model would be cc , pc , and qc .

This leaves R and y as exogenous variables. Exogeneity assumptions are critical to the specification of a model. Is it reasonable to assume that income and rainfall are exogenous? Can we assume that a change in any of the other variables of the model (endogenous or exogenous) would not affect income or rainfall? It seems safe to assume that rainfall is exogenous because it is unlikely that changes in corn production, corn consumption, prices, or incomes would affect the weather in Nebraska. The exogeneity of income is a little less clear cut, but since the corn market is only a very small part of the economy, it is unlikely that aggregate GDP would be affected very much by changes to the model’s other variables. Thus, our exogeneity assumptions seem reasonable here. (Later in this section we will consider adding another variable for which exogeneity is more questionable.)

The purpose of a model is to examine how changes in the variables are related. In particular, since the model is supposed to represent the process by which the endogenous variables are determined, we are interested in knowing how each endogenous variable would be affected by a change in one or more of the exogenous variables. We do this by solving the model’s equations to find reduced-form expressions for each endogenous variable as a function only of exogenous variables. In this example, we seek equations representing corn consumption, production, and price as a function only of rainfall and income.

In this simple model, we can find a reduced-form equation for pc with simple algebra. First, we use equation (3) to set the right-hand sides of equations (1) and (2)

1 – 9

equal: α0 + α1 pc + α2 y = β0 + β1 pc + β2 R. Isolating the two pc terms on the same side and dividing yields

p = α0 − β0 − β2 R + α2 y.

(4)

c β −α β −α β −α

1

1

1

1

1

1

Equation (4) is a reduced-form equation for pc because the only other variables appearing in the equation are the exogenous variables R and y.

Since (4) is a linear reduced form for pc , the coefficients in front of the R and y terms measure the effect of a one-unit increase in R or y on pc .1 Thus, a one-unit increase in R causes pc to change by –β2 /(β1 – α1). Earlier, we assumed that α1 < 0 and β1 > 0, so the denominator consists of a negative number subtracted from a positive number and is surely positive. We also assumed that β2 > 0, so the numerator is positive until the negative sign in front makes the whole expression negative. Since we can establish that the coefficient measuring Δpc /ΔR is negative, we have shown that under the assumptions of our model, an increase in rainfall will lower the equilibrium price of corn (by shifting the supply curve outward). A similar analysis can be used to show that Δpc /Δy is positive, so we conclude that an increase in income raises the price of corn.

Equation (4) tells us how pc is affected by R and y, but what about the effects of R and y on c and q? Since c = q at all times, Δc/ΔR and Δc/Δy are the same as Δq/ΔR and Δq/Δy. The effects of the exogenous variables on pc can be found by substituting equation (4) into either equation (1) or equation (2) and simplifying. If you do this, you will find that, as expected, either an increase in rainfall or an increase in income raises the quantity produced and consumed.

The importance of the exogeneity assumptions

Choosing which variables are to be endogenous and which are to be exogenous is an important part of the specification of a model. If a variable is (incorrectly) assumed to be exogenous when it is actually influenced in important ways by other

1

Two comments are important here. First, if this were not a reduced-form equation, then it would not be correct to use the coefficients of the exogenous variables to measure their effects on the endogenous variables. This is because when an exogenous variable changes, all of the endogenous variables in the model usually change. In (4), a change in R does not change y, thus the only change in pc is the effect measured by the coefficient on R. If another endogenous variable were on the right-hand side, then it would also be changing and pc would change by the sum of the two effects. Second, only when the model is linear can the effect of the exogenous variable on the endogenous variable be read off directly as a coefficient. In nonlinear models, it is necessary to differentiate the reduced-form equation with respect to an exogenous variable to evaluate the effect.

1 – 10