Topics in Modern Macroeconomics

Transcription

1 Topics in Modern Macroeconomics Michael Bar July 4, 20 San Francisco State University, department of economics.

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3 Contents Introduction. The Scope of Macroeconomics Models in Economics and Science What is a model? Why Models? Models are not realistic and are not supposed to be Modern Macroeconomics Business Cycles 5 2. Introduction The Classical Model The description of the model Important remarks about models in general Working with the classical model Real business cycle doctrine Unemployment 7 3. Labor Market De nitions The Search Model Preferences Job O er Acceptance Distribution of wage o ers Equilibrium Experiments with the Search Model Saving and Investment Saving and Investment Equation Saving and Investment in the U.S Intertemporal Choice Model (Saving Theory) The Model Optimal Choice Changes in income Changes in the real interest rate Changes in taxes and Ricardian equivalence Two-Period Model of Investment iii

4 iv CONTENTS 4.4. Optimal investment decision Changes in interest rate Changes in technology Solving for optimal investment Capital Market Decline in government de cit (S G ") Increase future productivity at home (A 2 ") Summary Appendix: Firm With Unlimited Life Span Economic Growth Introduction The Solow Model Description of the model Working with the model Endogenous Growth Model Description of the model Working with the model Economic Policy and Growth Evidence Appendix Money and Prices What is Money? The Demand for Money Quantity Theory of Money Money in the Utility Function Money Supply Example of Money Creation Illustration of the Money Multiplier Phillips Curve 8 7. Introduction Pillips Curve The impact of the Phillips curve on monetary policy Expectations-Augmented Phillips Curve (Edmund Phelps) The impact of the expectations-augmented Phillips curve on monetary policy Rational Expectation (Robert Lucas) Numerical example Credibility of Monetary Policy (Finn E. Kydland, Edward C. Prescott) Appendix: Estimating the Expectations-Augmented Phillips Curve

5 CONTENTS v 8 International Macroeconomics Balance of Payments Exchange Rates Using the exchange rates Law of One Price and Purchasing Power Parity (PPP) Predicting future trends in exchange rates Fixed vs. oating exchange rate Review Questions Math Review 05 0 Micro Review Consumer s Choice Budget Constraint Indi erence Curves Optimal Choice: graphical illustration Optimal Choice: mathematical treatment Examples Invariance of utility functions Income and substitution e ects Producer s Choice Firm s pro t maximization problem Factor shares Appendix Transitivity assumption The slope of indi erence curves Demand with Cobb-Douglas Preferences and n goods Rates of Change 27. Measuring Rates of change Discrete time variables Continuous time variables Rate of change of a product and ratio Examples Logarithmic scale

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7 Chapter Introduction Traditionally, economics is divided into two broad elds: (i) Microeconomics, and (ii) Macroeconomics. Microeconomics is the study of individual behavior of consumers and rms. A microeconomist might study questions like what a ects the price of individual good, or what causes changes in output of particular rm or industry? Macroeconomics is the study of aggregate behavior of consumers and rms. A macroeconomist might study questions like what a ects the aggregate price level" in the economy, or what causes changes in the aggregate output level?. The Scope of Macroeconomics The two main areas of macroeconomics are: (i) business cycles and (ii) growth, as described in the next diagram.

8 2 CHAPTER. INTRODUCTION Each of these two areas has many sub elds. For example, some business cycle economists study the evolution of real GDP and unemployment rate, while others focus on money and prices. Growth economists study the long-term growth trend of an economy. Economics of growth also has several sub elds..2 Models in Economics and Science.2. What is a model? A model is a simpli ed version of the real object that we study. Examples models include: (i) a map in geography, (ii) a rat in neuroscience, and (iii) supply and demand model in economics. Model is another word for theory. In any model, we distinguish between exogenous variables - determined outside of the model, and endogenous variables - determined within the model. In any model, the endogenous variables are determined by the exogenous variables. Models generate prediction about the endogenous variables for di erent values of exogenous variables. For example, in the model of supply and demand the endogenous variables are price (P) and quantity traded (Q), and exogenous variables are those that determine the location of supply and demand curves (such as income, prices of related goods, tastes of consumers, prices of inputs, technology of rms, etc.). The model generates a prediction about P and Q for any set of exogenous variables. The model s prediction about P and Q is competitive equilibrium..2.2 Why Models? Models can explain some features of the real world. Models don t tell us what the world looks like. Instead, they tell us what we can expect to happen in the world if the world was like the model. For example, the supply and demand diagram doesn t look anything like the markets in the real world. The diagram does not show the identities of the buyers and sellers, their feelings and emotions, their physical appearance. The supply and demand diagram only captures two features of real markets: () buyers typically want to buy less when the price goes up, and (2) sellers want to produce more when the price goes up. It turns out that the supply and demand diagram is very useful for explaining why prices di er across goods and why there are changes in prices over time. After testing the predictions of the model against the data we conclude that indeed the two features of buyers and sellers that we included in the model were important. Models can be used to perform controlled experiments. In the real world many things change at the same time; the technology changes, government policies change, etc. In the model we can perform controlled experiments of changing one thing at a time (ceteris paribus). This is impossible to do with actual economies..2.3 Models are not realistic and are not supposed to be When the object of study is very complicated, we need models that will highlight some important features of the object and leave out many other features. For example, when we

9 .3. MODERN MACROECONOMICS 3 study the economy of an entire country with millions of people, thousands of markets and rms, it is di cult for us to understand the behavior of the economy by just looking at it. Moreover, if we don t have any models to work with, we don t even know what data should be collected about the object of our study. For example, the model of supply and demand tells us that we don t need to collect data of all the names of buyers and sellers in a market in order to understand how it works..3 Modern Macroeconomics The common feature of modern macroeconomics, regardless of its eld, is micro-foundations. This means that in macroeconomic models we want to see the choices of individuals and rms. The common elements of modern macroeconomics are: (i) consumers, who maximize utility subject to budget constraints, and (ii) rms, who maximize pro t. The old approach to macroeconomics was to make assumptions about how individuals behave, while in modern macroeconomics economists make assumptions about individuals objectives, and then derive their behavior. The advantage of micro-foundations approach is that we get more economic intuition and see more clearly the choices that individuals make. For example, in the old (Keynessian) macroeconomic model, the consumers were represented by a demand for consumption function: C = C 0 + MP C(Y T ). According to this model, the average consumer starts with some consumption level C 0, and increases consumption by M P C for each dollar increase in the after-tax (disposable) income. In modern macroeconomics, consumers choose optimal consumption, when facing time and budget constraints: max U (C; l) C;l;L S {z } utility s:t: [Budget constraint] : C [wl S + ] ( t) [Time constraint] : L S + l = h The above says that consumers derive utility from consumption and leisure, and therefore they wish to maximize their utility (objective function). Consumers however are facing constraints and cannot choose arbitrary levels of consumption and leisure. Spending on consumption cannot exceed the income from labor (wl S ) and non labor ( - pro t), net of taxes (t - is the tax rate). Time is also constrained, so that leisure l and worktime L S must equal to the time endowment. Notice that this optimization problem must be solved in order to obtain the demand for consumption. This is the di erence from the old macroeconomics, which assumes a particular demand for consumption. The next section illustrates the modern approach to macroeconomics, with the the classical model, applied to the study of real business cycles.

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11 Chapter 2 Business Cycles 2. Introduction Recall from the introduction that the output per capita in the U.S. is growing steady, but there are uctuations about the trend. These uctuations are called business cycles. Figure 2. shows the ln of real GNP per capita in the U.S. in the last century, together with a linear trend. The linear trend ts the data pretty well, which means that the original variable, GNP per capita, was growing at constant rate. Figure 2.: Ln of real GNP per capita in U.S. 4 ln(real GNP per capita) Year The study of the long run growth trend belongs to the eld of economic growth. In these notes we focus on the uctuations of the output around the trend. Subtracting the growth trend from the time series in gure 2., results in a series of deviations from trend, displayed in gure 2.2. The series of deviations from trend is called detrended real output, or the cyclical part of the real output. The questions that we want to ask in these notes are:. What causes business cycles? 5

12 6 CHAPTER 2. BUSINESS CYCLES Figure 2.2: Cyclical part of GNP per capita in U.S. 0.3 Deviations from trend Year 2. Can the government smooth out the business cycles? 3. Should the government smooth out the business cycles? In order to answer these questions, economists use models. We will see that di erent models give di erent answers to those questions. 2.2 The Classical Model 2.2. The description of the model The model consists of a representative consumer, representative rm, and a government. The consumer receives income from supplying his labor and from dividends from the rm he owns. The consumer chooses his consumption and time allocation between labor and leisure. The rm is owned by the consumer, it owns a xed amount of capital, and it chooses the optimal amount of labor to maximize pro ts. The government consumption is exogenous to the model. The government balances its budget by collecting taxes at the amount of expenditures. The formal description of the model economy:. Consumer: max ln C + ( C;l s:t: ) ln l C = [w(h l) + ] ( t) where C is consumption, l is leisure, w is real wage, h is time endowment (say 00 hours per week), is the pro ts or dividends from the rm, t is the at tax rate. Thus,

13 2.2. THE CLASSICAL MODEL 7 the time spent working (labor supply) is L S = h l 2. Firm: max = AK L D L D wl D where A is productivity parameter (called Total Factor Productivity, TFP), K is the capital stock, and L D is labor employed by the rm. The productivity parameter re- ects the idea that with technological improvement (A ") more output can be produced with the same inputs. The total output in the economy is thus Y = AK L. 3. Government: collects taxes on all income at the rate of t, and spends them on government consumption. The government budget is G = t (wl + ) 4. De nition: Competitive equilibrium consists of (w; C; G; L; l; ; Y ) such that (a) Given w, the values of (C; l) solve the consumer s problem, (b) Given w, the value of L solves the rm s problem, (c) Markets are cleared: i. L D = L S = L (labor market), ii. C + G = Y ( nal goods market) Important remarks about models in general This section is a philosophical discussion of our approach in general. It is essential to read it in order to understand the material of this entire course, and many other courses that you are taking. You should come back and read this again after you have practiced working with the classical model.. The competitive equilibrium is the model s prediction about the endogenous variables. Endogenous variables are determined inside the model, i.e. the variables which the model is trying to explain. The exogenous variables are those that are determined outside of the model. For example, in the model of a market the endogenous variables are price and quantity traded, and exogenous variables are those that determine the location of supply and demand curves (such as income, prices of related goods, etc.). In the classical model the exogenous variables are: (A; t; K), and the endogenous variables are: (w; C; G; L; l; ; Y ). 2. Causality: what causes what? In any model, the exogenous variables are causing the endogenous variables. For example, we can change A (the technology level) and observe the changes in real wage, employment, consumption, output, etc. All the endogenous variables are caused by the exogenous variables. If we don t change any

14 8 CHAPTER 2. BUSINESS CYCLES of the exogenous variables, no change in the endogenous variables can occur. Thus, in this model we cannot say that output causes employment, since both output and employment are endogenous variables, and cannot change unless we change some of the exogenous. Be very careful about making statements of causality in the real world. 3. Why models? (a) Models can explain some features of the real world. Models don t tell us what the world looks like. Instead, they tell us what we can expect to happen in the world if the world was like the model. For example, the supply and demand diagram doesn t look anything like the markets in the real world. The diagram does not show the identities of the buyers and sellers, their feelings and emotions, their physical appearance. The supply and demand diagram only captures two features of real markets: () buyers typically want to buy less when the price goes up, and (2) sellers want to produce more when the price goes up. It turns out the supply and demand diagram is very useful in explaining why prices di er across goods and why there are changes in prices. After testing the predictions of the model with the data we conclude that indeed the two features of buyers and sellers that we included in the model were important. (b) Models can be used to perform controlled experiments. In the real world many things change at the same time; the technology changes, government policies change, etc. In the model we can perform controlled experiments of changing one thing at a time. This is impossible to do with actual economies. 4. Models are not realistic and are not supposed to be. When the object of study is very complicated, we need models that will highlight some important features of the object and leave out many other features. For example, when we study the economy of an entire country with millions of people, thousands of markets and rms, it is di cult for us to understand the behavior of the economy by just looking at it. Moreover, if we don t have any models to work with, we don t even know what data should be collected about the object of our study. For example, the model of supply and demand tells us that we don t need to collect data of all the names of buyers and sellers of the market in order to understand how it works Working with the classical model The de nition of competitive equilibrium is instructive about how the model should be solved. The de nition suggests the following steps: () solve the consumer s problem to get the labor supply, (2) solve the rm s problem to get the labor demand, and (3) use the market clearing conditions to nd the real wage, the equilibrium employment, and the rest of the endogenous variables. Mathematical solution Step : solving the consumer problem

15 2.2. THE CLASSICAL MODEL 9 The consumer s problem can be written as max ln C + ( C;l s:t: ) ln l C + w ( t) l = (wh + ) ( t) This is a standard consumer choice problem with two goods: C and l, the prices of the goods are and w ( t) respectively, and the consumer s income is (wh + ) ( t). We know already how to solve a consumer choice problem with Cobb-Douglas preferences. Thus, the demand is C = (wh + ) ( t) (wh + ) ( t) l = ( ) w ( t) = ( ) h + w and the labor supply is L S = h ( ) h + w (2.) Observe that consumption is increasing in w and, and decreases in t. The labor supply is increasing in w, decreasing in and does not depend on taxes. The intuition why the labor supply is decreasing in goes as follows. The dividend income is non labor income, so when it goes up the consumer does not need to work as much. Figure 2.3 shows the graph of the labor supply curve. i.e. how much labor the consumer wants to supply at any given wage, holding everything else xed. This means that changes in w are re ected by movements along the curve, while changes in will shift the entire curve. Figure 2.3: Labor supply curve Real wage (w) Labor Supply Curve Labor (N) Ls Step 2: Solving the rm s problem

16 0 CHAPTER 2. BUSINESS CYCLES The rm s problem is The rst order condition max = AK L L D D wl = ( ) AK L D w = D ( ) AK L D = w (2.2) which tells us that the rm maximizes pro t when it equates the marginal product of labor the the real wage. Equation (2.2) thus gives us the labor demand of the rm. We can solve for L D explicitly from equation (2.2) to get ( L D = ) AK = w Observe that this curve is decreasing in w. Figure 2.4 shows the labor demand curve, i.e. how much labor the rm wants to employ at any given wage, holding everything else constant. Thus, changes in w are re ected by movements along the curve while changes in A will shift the entire curve.the pro t is therefore given by Figure 2.4: Labor demand curve Real wage (w) Labor Deamand curve Labor (N) Ld = AK L D ( ) AK L D L D = AK L D (2.3) Step 3: equilibrium in the labor market Letting L S = L D = L and substituting equations (2.2) and (2.3) into equation (2.) gives L = h ( ) h + AK L ( ) AK L

17 2.2. THE CLASSICAL MODEL Solving for equilibrium L: L = h ( ) h + ( ) L ( ) L = h ( ) h ( ) L ( ) h = L + ( ) L ( ) h = L + + ( ) h = L h = L Equilibrium employment: L = ( ) h Once we found the equilibrium employment L, all the other endogenous variables can be found in terms of L. Equilibrium leisure: l = h L To solve for equilibrium wage, use equation (2.2): w = ( ) AK L Equilibrium output: Y = AK L Equilibrium pro t, using equation (2.3): = AK L To nd equilibrium consumption we use the budget constraint: Equilibrium government expenditures: C = [w L + ] ( t) C = ( ) AK L + AK L D ( t) C = ( t) Y G = Y C = ty

18 2 CHAPTER 2. BUSINESS CYCLES Summary of equilibrium: L ( ) h = l = h L w = ( ) AK L Y = AK L = AK L C = Y ( t) G = ty As you can see, an increase in productivity A, causes an increase in equilibrium output, equilibrium real wage, equilibrium consumption, equilibrium government consumption, and equilibrium pro t. Equilibrium employment does not depend on the level of technology, even though the real wage went up. The e ect of higher K is similar because A and K always appear together in the equations. An increase in the tax rate a ects only the distribution of the total output between the private sector and the government sector. If t = 30% for example, then the government consumes 30% of the total output, while the private consumers get to consume the rest 70%. Graphical analysis The classical model can be analyzed graphically with only two diagrams, the labor market and the production function, as shown in gure 2.5. These graphs correspond to the following equations: Production function : Y = AK L Labor supply curve : L S = h ( ) h +, where = AK L w ( Labor demand curve : L D = ) AK w = It is important to repeat here that labor supply curve is increasing in w. On the other hand, if increases, this leads to a shift of the entire supply curve to the left. The labor demand curve is decreasing in w. If A or K increase, the entire labor demand curve will shift to the right. Now we use this graphical framework in order to perform 3 experiments with the model:. An increase in productivity (A "). Figure 2.6 shows the e ects of an increase in A in the classical model. As A ", there is an increase in labor demand (shift of the labor demand curve to the right) and a decrease in labor supply (shift of the supply curve to the left) and an increase in production function. The e ect of the increase in labor demand on employment is an increase in employment, while the e ect of a decrease in labor supply on employment is a decrease in employment. Thus, without solving the model with

19 2.2. THE CLASSICAL MODEL 3 Figure 2.5: Classical model: graphical illustration Y Production function Y * w L * L L S Labor market w * L D L * L particular functional forms we cannot tell what is the e ect of A " on equilibrium employment. In the pervious section however we solved the model with Cobb-Douglas technology and preferences and found that the equilibrium employment does not change as A ". In other words, the e ect on employment of a decline in labor demand and of an increase in labor supply cancel each other. Both e ects however increase the equilibrium real wage. 2. An increase in K. The e ect of an increase in K is the same as the e ect of an increase in A. Notice that A and K always appear together as AK. 3. An increase in the tax rate (t "). Neither the labor demand nor the labor supply depend on the tax rate, hence nothing will change in the labor market. The production function does not depend on the tax rate as well, and therefore none of the curves in gure 2.5 will shift. As we have seen before, the only e ect that an increase in the tax rate has on the economy is the increase in the government share of the total output.

20 4 CHAPTER 2. BUSINESS CYCLES Figure 2.6: An increase in productivity (A ") Y Y 2 Production function Y w L L L S Labor market W 2 W L D L L Answering the questions Now we are ready to answer the questions we posed in the beginning of these notes, within the framework of the classical model.. What causes business cycles? The exogenous variables in this model are (A; t; K). As we have seen before, a positive shock to productivity (A ") increases the equilibrium output while a negative shock to productivity (A #) decreases it. We can think of shocks to productivity as agglomeration of many factors such as innovations, shocks to oil prices, weather, political events, etc., that change the amount produced with the same inputs. So this model suggests that business cycles might be a result of productivity shocks. As we have seen before, changes in t do not a ect the equilibrium output. How about K? It is possible that a hurricane, or a terrorist attack would destroy part of the nation s capital and cause a decline in output. It is harder to think of how the stock of capital can experience a sudden increase. In any case, when we look at the data on capital stock, it looks very For a more detailed discussion about productivity shocks see the next section.

21 2.2. THE CLASSICAL MODEL 5 smooth and does not exhibit uctuations that can potentially be the cause of business cycles. 2. Can the government smooth out the business cycles? We have seen before that changes in the tax rate in this economy do not a ect the equilibrium output. Changes in the tax rate only a ect the fraction of total output that is consumed by the government. Recall that C = ( t) Y G = ty So the answer to the question is, NO, the government cannot smooth the business cycles (in this model). 3. Should the government smooth out the business cycles? It shouldn t because it can t (in this model) Real business cycle doctrine Real business cycle theory suggests that the main source of business cycles is shocks to productivity. The real business cycle school is led by Edward Prescott and Finn Kydland. They were awarded a Nobel Prize in Economics in economics in 2004 "for their contributions to dynamic macroeconomics: the time consistency of economic policy and the driving forces behind business cycles". Finn Kydland and Edward Prescott developed a methodology that allows them to answer the following quantitative equation: "how much of the uctuations in output around a trend can be accounted for by random shocks to productivity?". Their answer was 2/3. Kydland and Prescott used a model that is a more complex version of the classical model (their model is called "the Neoclassical Growth Model"). But the idea can be illustrated with the classical model. Step : Choose functional forms for utility and production function. In the data, although the real wage went up in the last decades, the average worktime did not change. Notice that in our model with Cobb-Douglas utility function, we get the same result, i.e. in equilibrium the worktime is constant and does not depend on the real wage. In the data, the labor share of total output is roughly constant over time. The Cobb- Douglas production function delivers this property. Recall that the capital share is and the labor share is, and these are constant. Step 2: Choose the parameter values for the utility and production functions. In the data the labor share is about 2=3 of the total output. Thus, set = =3 so that ( = 2=3). In the real world people have approximately 00 hours per week that they can allocate between labor and leisure activity (24 hours per day, minus 8 hours of sleep and 2 hours of maintenance such as bathroom, eating, resting). In the data the average worktime is 40 hours per week, so using our equilibrium equation for employment we can nd as

22 6 CHAPTER 2. BUSINESS CYCLES follows: L ( ) h = 40 = :4 = :4 0:4 3 = 2 3 :2 0:4 = 2a :2 = 2:4a = 0:5 Thus, = 0:5, = 3. Step 3: Estimate the shocks to productivity We assume that aggregate output is produced with Y = AK L (2.4) We have data on real GDP (Y ), on capital (K) and labor employed (L). This means that we can nd Afrom the above equation as a residual. Because of this procedure A is called the Solow Residual, since we nd it as the residual that would equate the left hand side and the right hand side of equation (2.4). Step 4: Model simulation Having found the time series of A we can simulate the model and generate time series of consumption and output. We have seen that an increase in A causes an increase in output in the classical model and a decline in A will cause a decline in output. It turns out that the time series of Y generated by the model is very similar to the data in gure 2.2. In fact, the variance of the output generated by the model is about 2/3 of the actual variance of the real GDP/capita in the data. This means that random shocks to productivity can explain most, but not all the variation in real GDP/capita over the business cycles.

23 Chapter 3 Unemployment In the classical model the labor market is cleared by assumption. This means that all people looking for a job were able to nd a job. In practice, there are unemployed people. Even though in the U.S. unemployment rate is about 5% and does not represent a huge problem; there are countries in Europe whose unemployment rate is in the double digits. All policymakers agree on the importance of the objective to keep unemployment low. High unemployment stands in the way of achieving full productive capacity and increases inequality. In this chapter we focus on the labor market, and in particular, on the determinants of unemployment rate. We begin with key concepts related to the labor market, and later present the search model of unemployment. 3. Labor Market De nitions Civilian noninstitutional population - Included are persons 6 years of age and older residing in the 50 States and the District of Columbia who are not inmates of institutions (for example, penal and mental facilities, homes for the aged), and who are not on active duty in the Armed Forces. Unemployed persons - Persons aged 6 years and older who had no employment during the reference week, were available for work, except for temporary illness, and had made speci c e orts to nd employment sometime during the 4-week period ending with the reference week. Persons who were waiting to be recalled to a job from which they had been laid o need not have been looking for work to be classi ed as unemployed. Employed persons - Persons 6 years and over in the civilian noninstitutional population who, during the reference week, (a) did any work at all (at least hour) as paid employees; worked in their own business, profession, or on their own farm, or worked 5 hours or more as unpaid workers in an enterprise operated by a member of the family; and (b) all those who were not working but who had jobs or businesses from which they were temporarily absent because of vacation, illness, bad weather, childcare problems, maternity or paternity leave, labor-management dispute, job training, or other family or personal reasons, whether or not they were paid for the time o or 7

24 8 CHAPTER 3. UNEMPLOYMENT were seeking other jobs. Each employed person is counted only once, even if he or she holds more than one job. Excluded are persons whose only activity consisted of work around their own house (painting, repairing, or own home housework) or volunteer work for religious, charitable, and other organizations. Labor force - The labor force includes all persons classi ed as employed or unemployed. Not in the labor force - Includes persons aged 6 years and older in the civilian noninstitutional population who are neither employed nor unemployed. The next diagram illustrates the breakdown of the population into di erent categories. Population = labor force z } { employed+unemployed+ not in labor force {z } + The next table shows the data for the U.S., January 2006 (in thousands). 8 < Total Population Civilian Noninstitutional Population Labor Force 504 Employed Unemployed 7040 Not in the Labor Force : age < 6 in the military institutionalized The most important indicators of the labor market are: () Unemployment Rate, and (2) Labor Force Participation Rate. Unemployment Rate = #Unemployed #Labor Force #Labor Force Labor Force Participation Rate = #Civilian Noninstitutional Population Based on the above data, Unemployment Rate = = 4:7% Labor Force Participation = = 66% Labor force participation in the U.S. is approximately 70% for men and 60% for women. In the last 50 years, participation rates for women more than doubled while for men, participation rate slightly declined. Some of the prominent hypotheses for why women increasingly entered the labor force include the closing of the gender wage gap, the declining price of home appliances, and the wide spread use of a contraceptive pill. All of these stories have

25 3.. LABOR MARKET DEFINITIONS 9 the feature of increasing the return to women of educating themselves and working relative to staying at home. Participation rate is a completely di erent thing from the unemployment rate. It measures the degree of willingness of people to work for paid wage. Unemployment rate measures the degree of di culty of nding a job. In these notes, we only focus on the unemployment rate. Some Determinants of the Unemployment Rate. Aggregate economic activity. High levels of output are associated with lower unemployment rates. In other words, unemployment is countercyclical, as the gure 7.2 shows. Figure 6.2 Deviations from Trend in the Unemployment Rate and Percentage Deviations from Trend in Real GDP for Figure 3.: Unemployment rate is countercyclical. 2. Demographic structure of the population. For example, younger workers tend to switch jobs more often, they have less to loose by getting red, etc. Hence, younger populations, all things equal, tend to have higher unemployment rates. For example, if during the 50 s there was a baby boom in the U.S., then 20 years later when the baby boom cohort enters the labor market, we expect the unemployment rate to increase. 3. Sectorial Shifts. For example, a shift away from manufacturing has displaced many workers. Finding a new job for these workers involves acquiring di erent skills. Hence, societies with a greater degree of restructuring tend to have higher unemployment rates. 4. Government policies. These include unemployment insurance programs as well as welfare, training programs and job matching services for the unemployed. The unemployment

26 20 CHAPTER 3. UNEMPLOYMENT insurance (UI) program in the U.S. is run by state governments. Typically, unemployed workers in the U.S. draw bene ts for 6 months and the replacement ratio (ratio of UI bene- ts to the wage the unemployed worker used to receive) is /2. Existence of this government policy a ects the behavior of both, employed and unemployed. The unemployment rate in the data exhibits both, uctuations at business cycle frequencies (determinant #) as well as longer run trends (determinants #2,3,4). 3.2 The Search Model This model will provide us with some simple insight into how the unemployment rate is determined. This model will also allows us to study how the unemployment rate can be a ected by government policy with respect to unemployment bene ts, labor income or unemployment income taxation as well as changes in informational technology Preferences There are many jobs with di erent real wages w. The only characteristics of a job that people care about is the wage that it pays. People are either employed or unemployed. This means that everybody is in the labor force. Let U represent the fraction of people that are unemployed. Then fraction U of people are employed. A fraction s of all the employed will be separated from their jobs at any given period. We call s the separation rate and assume that it is xed and the same for all jobs. The separation rate is exogenously given parameter and can be thought of as the probability that any employed worker will loose his job. A fraction p of all unemployed people get a job o er. Again, p is just a given parameter; people have no control over it. We can think of p as the probability that any unemployed worker will get a job o er. Let V e (w; s; t w ) denote the utility of being employed at wage w, with separation rate s and taxes on labor income t w. We assume that V e is increasing in w, but at a decreasing rate (which means that V e is concave in w). Also assume that V e is decreasing in separation rate s and in taxes on labor income t w. Thus V e w ; s; t w. For example, V e could be of the + following form V e (w; s; t w ) = ( s) p w ( t w ) If we plot V e as a function of w (keeping s and t w xed), the graph would look like the following

27 3.2. THE SEARCH MODEL 2 Ve ( w, s, tw) Utility of Employed, as a Function of w The notation s and t w means that the above graph was plotted for some xed values of s and t w. Changes in these values will shift the entire curve. In particular, an increase in either s or t w will shift the entire curve down, as shown in the next picture w Ve ( w, s, tw) Shift in the Utility as s " or t w ". In what follows, we will use the notation V e (w) to denote the utility of employed person for given and xed values of s and t w. Let V u (b; p; t b ) denote the utility of being unemployed, where b is the real unemployment bene t, p is the probability of getting a job o er, and t b is the tax on income from unemployment. We assume that V u b ; p ; t b, which means that V u is increasing in the + + unemployment bene ts b, increasing in the chances of getting an o er p, and decreasing in the taxes on unemployment bene t. For example, the function V u could be of the following form V u (b; p; t b ) = p p b ( t b ) Plotting the graph of V u against the real wage (for given values of b; p; t b ) looks like a horizontal curve since V u does not depend on w, as shown in the next picture w

28 22 CHAPTER 3. UNEMPLOYMENT Vu ( b, p, tb ) In what follows, we will use the notation V u as a shorthand for the utility of unemployed for given and xed values of b; p and t b Job O er Acceptance When an unemployed worker receives a job o er w he accepts it if V e (w) V u : The minimum wage o er which an unemployed worker accepts is an o er w such that V e (w ) = V u. We refer to this w as the reservation wage. For all job o ers w w ; V e (w) V u and therefore the unemployed will accept those job o ers. For all job o ers w < w ; V e (w) < V u and therefore the unemployed will reject those job o ers. The next graph illustrates the job o er acceptance decision. w (w) V e V u w* w The reservation wage w is crucial for determining the unemployment rate. Suppose that there are 000 job o ers, and 700 of them are w. Then we know that 70% of those receiving job o ers will accept them and become employed. Examples. Suppose that V e (w; s; t) = ( s) p w ( t w ), and V u (b; p; t b ) = 3p p b ( t b ). Find the reservation wage w.

29 3.2. THE SEARCH MODEL 23 Thus Solution: the reservation wage solves ( s) p w ( t w ) = 3p p b ( t b ) w = 3p 2 b ( t b ) s ( t w ) 2. Explain how does w depend on the exogenous parameters b; p; s; t w ; t b and give some intuition for your results. Solution: The reservation wage, w, is increasing in b; p; s and t w. This makes intuitive sense. Higher unemployment bene t b means that the unemployed are more comfortable with being unemployed and it will take higher wage to induce them to accept the job o er. Higher p means that greater fraction of unemployed receive job o ers, so the chances of nding a higher paid job (everything else equal) is higher and therefore w is higher. Higher s means greater separation rate, or greater risk of loosing the job. Thus, the unemployed will demand higher wage to compensate for that risk. Finally, higher tax on labor t w lowers the net of tax real wage and hence it takes higher before tax wage to induce the unemployed to accept a job. Also observe that w is decreasing in the tax on unemployment bene t t b, which is also intuitive; higher tax on unemployment bene t lowers the net of tax unemployment bene t and hence lowers the utility from being unemployed. Thus, it will take lower wage to induce the unemployed to accept the job. 3. In the above example, suppose that all types of income are taxed at the same rate t, how does the reservation rate w depend on t? Solution: It doesn t, t cancels out w = 2 2 3p b ( t) 3p s ( t) = b s 4. Suppose that in the above example we have b = 5, p = 0:6, s = 0:, t w = t b = 0:3. Find the reservation wage w. Solution: w = = = 2 3p b ( t b ) s ( t w ) 2 3 0:6 5 ( 0:3) 0: ( 0:3) 2 :8 5 = 20 0: Distribution of wage o ers We need one more piece of information in order to nd the unemployment rate, namely the distribution of wage o ers. We assume that the distribution of job o ers is given by a

30 24 CHAPTER 3. UNEMPLOYMENT function H (w) which gives the probability that an o er is at least w. For example, suppose that H (w) = 00 w The next gure is the plot of this function. Pr(offer >= w) H(w) w Important! H (w) does not give the probability of receiving an o er of at least w. What it does tell us is that if some unemployed person received an o er, than H (w) is the probability that that o er is at least w. Examples. Suppose that the distribution of job o ers is as given above, and suppose that I am unemployed who received and o er. What is the probability that this o er is above 20? Solution: H (20) = 20 = 0: Suppose that I am an unemployed person and a fraction p = 0:6 receive job o ers. What is the probability that I get an o er of at least 20? Solution: p H (20) = 0:6 0:8 = 0:48 3. Explain the di erence between part and 2. Solution: In part it was already given that I received an o er, so H (w) tells us what is the probability that an o er that was received is at least w. In part 2 it is not known whether I will receive an o er or not. In fact, there is only 60% chance that I will. Thus, the chances that I will get an o er of at least 20 are 0:6 times what they are in part Equilibrium Now we have all the information we need in order to compute the law of motion of unemployment rate: U t+ = U t + s ( U t ) ph (w ) U t

31 3.2. THE SEARCH MODEL 25 The unemployment rate in the next period U t+ is equal to the sum of 3 elements. The rst is the current unemployment rate U t. The term s ( U t ) is the addition to the unemployment rate due to separation of currently employed from their jobs. Suppose that 90% of the labor force are currently employed ( U t = 0:9) and the separation rate is 0:, which means that 0% of the currently employed will loose their jobs. Thus the term s ( U t ) = 0: 0:9 = 0:09, i.e., the unemployment rate next period will increase by 0:9% due to some of the currently employed loosing their jobs. The last term on the right hand side represents the decline in unemployment rate due to some of the currently unemployed nding jobs. Suppose that currently there is 0% unemployment rate. Suppose that p = 0:6, which means that 60% of the currently unemployed will receive a job o er. What fraction of them will accept the o er? It is given by H (w ), which is the probability that an o er exceeds the reservation wage (remember that unemployed people accept an o er if it is at least as high as their reservation wage). Suppose that in our example w = 20, so H (w ) = 0:8. This means that 80% of the unemployed who received and o er will accept it. Thus, ph (w ) U t = 0:6 0:8 0: = 0:048%, which is the decline in the unemployment rate due to unemployed nding jobs. In this numerical example we see that the unemployment rate will increase from period t to period t + because the addition to unemployment rate from employed who loose their jobs is greater than the decline in unemployment rate from unemployed who nd jobs. Long-run equilibrium Rearranging the law of motion of unemployment rate gives U t+ = U t + s ( U t ) ph (w ) U t U t+ = U t + s su t ph (w ) U t U t+ = U t [ s ph (w )] + s If s ph (w ) < then the law of motion has a steady state, as shown in the next gure Law of motion of U_t U_t U_t U_t+ 45_deg We can see that starting from any unemployment rate, the economy will converge to a steady state U such that U t = U t+ = U for all t. To nd the steady state we solve U = U + s ( U) ph (w ) U s ( U) = ph (w ) U U = s ph (w ) + s Once we nd the reservation wage w ; we can plot ph (w ) U as a function of U; it is just a linear function of U with slope ph (w ) : The intersection of this line with s ( U) (also

32 26 CHAPTER 3. UNEMPLOYMENT a linear function of U with slope state) equilibrium level of U : s and intercept s) determines the long-run (or steady s ph ( w*) U U* s( U ) U In our numerical example U = s ph (w ) + s = 0: 0:6 0:8 + 0: = 0: Intuitively, the term s ( U) represents the " ow in" to the unemployment as a result of employed people separating from their jobs, while the term ph (w ) U represents the " ow out" of the unemployment as a result of unemployed accepting job o ers Experiments with the Search Model An increase in unemployment insurance bene t (b ") As b ", the value of being unemployed shifts up, hence, the reservation wage increases (the unemployed get more picky about job o ers). As a result, fewer of those who are o ered jobs (the same job o ers are made) accept their o er, i.e., H (w ) # : Hence, U s = goes ph(w )+s up. Notice that in this experiment, s and p remained the same while H (w ) went down. As the denominator became smaller, the fraction became larger. Intuitively, the ow out of the unemployed is reduced since fewer job o ers are accepted. The gures below illustrate all of the steps we mentioned.

33 3.2. THE SEARCH MODEL 27 * 2 * w w ) ( ) ( * 2 * w H w H * 2 * U U * 2 * w w w H (w) s U w ),, ( + e t w s w V ),, ( + + u t b p b V U w ph *) ( ) ( U s U t i l i t y

34 28 CHAPTER 3. UNEMPLOYMENT An increase in probability of getting a job o er (p ") This increase can be a result of an improvement in information technology that facilitates a job search, or government policy that is successful at increasing the chances of the unemployed of nding a job, such as retraining programs. As p goes up, the value of being unemployed increases, driving the reservation wage up (again the unemployed become more picky). As a result, a lower fraction of unemployed with job o ers actually accept their job, that is, H (w ) #. Let s consider the equilibrium unemployment rate U s =. What happens to this fraction? s remained unchanged. ph(w )+s p went up but H (w ) went down. It is unclear whether the product ph (w ) increased or decreased. Thus, U t i l i t y V ( w, s, t e w + V ( b, p, u t b + + ) ) H (w) * * w w2 w * H ( w ) * H ( w ) 2 * * w w2 w s? ph ( w*) U * * U U 2 s( U ) U

35 3.2. THE SEARCH MODEL 29 An increase in labor income taxes (t w ") An increase in labor income taxes leads to a fall in the value of being employed for any given wage. As V e (w) shifts down, the reservation wage goes up and fewer of those unemployed with job o ers actually accept their o er, that is, H (w ) goes down. So, the equilibrium unemployment rate U = s ph(w )+s increases. Thus, U t i l i t y V ( w, s, t e w + V ( b, p, u t b + + ) ) H (w) * * w w2 w * H ( w ) * H ( w ) 2 * * w w2 w s ph ( w*) U * U U * 2 s( U ) U

36 30 CHAPTER 3. UNEMPLOYMENT An increase in separation rate (s ") This increase can be a result of moving from a command economy (operated by government) to a market economy where the jobs are less secured and there is a higher risk of being separated from the job (since people are employed based on their skill and not based on their connections to the ruling party). An increase in the separation rate decreases the utility from being employed, so the value of being employed goes down and the reservation wage goes up w ". As a result, the probability of getting o ers that are accepted goes down H (w ) #. Thus, the ow out of unemployment is reduced (ph (w ) goes down). At the same time the ow in to the unemployment goes up s ( U) ". Thus, U t i l i t y V ( w, s, t e w + V ( b, p, u t b + + ) ) H (w) * * w w2 w * H ( w ) * H ( w ) 2 * * w w2 w s 2 s ph ( w*) U * * U U 2 s( U ) U

37 3.2. THE SEARCH MODEL 3 Numerical Examples Suppose that V e (w; s; t) = ( s) p w ( t w ), and V u (b; p; t b ) = 3p p b ( t b ), b = 5, p = 0:6, s = 0:, t w = t b = 0:3, H (w) = w. 00. Find the steady state unemployment rate. Solution: Step : nd the reservation wage w Thus ( s) p w ( t w ) = 3p p b ( t b ) w = = = 2 3p b ( t b ) s ( t w ) 2 3 0:6 5 ( 0:3) 0: ( 0:3) 2 :8 5 = 20 0:9 Step 2: Find the fraction of o ers that are at least w ( nding H (w )) H (20) = 0:0 20 = 0:8 Step 3: Find the steady state unemployment rate (U ) s ( U) = ph (w ) U U = s ph (w ) + s = 0: 0:6 0:8 + 0: = 0: Suppose that p =, so that everybody receives an o er. Find the steady state unemployment rate. Solution: Step : nd the reservation wage w Thus ( s) p w ( t w ) = 3p p b ( t b ) w = = = 2 3p b ( t b ) s ( t w ) ( 0:3) 0: ( 0:3) = 55: :9 Step 2: Find the fraction of o ers that are at least w ( nding H (w )) H (20) = 0:0 55: = 0:

38 32 CHAPTER 3. UNEMPLOYMENT Step 3: Find the steady state unemployment rate (U ) s ( U) = ph (w ) U U = s ph (w ) + s = 0: 0:6 0: : = 0: So that the unemployment rate increased.

39 Chapter 4 Saving and Investment The process of economic growth depends, among other things, on the ability of rms to expand their productive capacity through investment in additional equipment. Firms can nance their investment from retained earnings (also called undistributed pro ts or business saving) or borrow funds from households who save. In this chapter we discuss the relationship between saving and investment in the macroeconomy, and present a theory of saving and investment. We will examine what factors a ect investment decisions by the rms and saving decisions by the households, and how those decisions are a ected by government policies. Before we start the formal discussion of saving and investment, we need to introduce two general concepts. A stock variable is a magnitude measured at a point in time (say at the end of the year), and a ow variable is a variable measured over a given time interval (say over the year). For example, the stock of capital in the U.S. on December 3 st 2005, is a stock variable. Investment that took place in 2005 is a ow variable. As another example, the saving during 2005 is a ow variable and the savings at the end of 2005 (the value of all the balances of saving accounts) is a stock variable. The ow variables determine the value of the stock variable at the end 4. Saving and Investment Equation In any economy there exists an accounting identity that relates saving and investment. In this section we derive this relationship, called the saving and investment equation. The GDP is given by GDP = C + G + I + NX (4.) where C is personal consumption expenditure, G is government consumption expenditure, I is gross domestic investment, and NX = X IM is net exports (exports minus imports). We de ne disposable income as Y D = GDP + T R T (4.2) where T R are transfer payments by the government (such as unemployment insurance bene ts, social security, medicare,...). and T is taxes. We de ne the private saving as S P = Y D C (4.3) 33

40 34 CHAPTER 4. SAVING AND INVESTMENT That is, the private saving is the disposable income that is not consumed. Similarly, the government saving is S G = T T R G (4.4) which is the government income that is not spent on government consumption or transfer payments. Government de cit is then Now add T R and subtruct T from equation (4.) Def = S G = G + T R T (4.5) GDP + T R T = C + G + T R T + I + NX Now using the de nition of disposable income in equation (4.2) Y D = C + Def + I + NX Finally, using the de nition of the government de cit in equation (4.5) gives the saving and investment equation S = S P + S G = I + NX (4.6) The left hand side of (4.6) is the gross national saving S, which is the sum of private and government saving. On the right hand side we have the gross domestic investment I and net exports NX, which is also known as Net Foreign Investment. In a closed economy we have S P + S G = I that is, in a closed economy the total saving is equal to total investment. In an open economy, on the other hand, part of the domestic saving can fund investment abroad, if S > I. It is also possible in an open economy that the domestic saving are insu cient to fund all of the domestic investment, if S < I, and in this case part of the domestic investment is funded by foreigners. If NX > 0, then the economy is exporting more goods and services than what it imports, i.e. the country is experiencing a trade surplus. This means that the domestic economy is accumulating foreign assets, since the rest of the world has to borrow from the domestic economy. If on the other hand, NX < 0, this means that the economy is importing more goods and services than its exports to the rest of the world, i.e. the country is experiencing trade de cit (trade de cit is de ned as N X). In this case the domestic economy has to borrow from the rest of the world and the rest of the world is accumulating domestic assets. Thus, in the open economy, total saving is equal to the domestic investment plus net foreign investment, as equation (4.6) states What can we learn from the saving and investment equation (4.6)?. The domestic investment can be nanced by domestic saving and by foreign saving. Rewriting (4.6) gives I = S P + S G NX The term NX represents the investment of the rest of the world in the domestic economy. For example, if I = 20, S P + S G = 5 and NX = 5, then I {z} 20 = S P + S G {z } 5 NX {z} 5

41 4.2. SAVING AND INVESTMENT IN THE U.S. 35 which means that part of the domestic investment is nanced by domestic saving (5) and part is nanced by foreign saving (5). If on the other hand we have I = 20, S P + S G = 25 and NX = 5, then I {z} 20 = S P + S G {z } 25 NX {z} 5 which means that the domestic saving nances not only the domestic investment, but also nances some of the investment in the rest of the world. 2. The government can nance its de cit in two ways: () borrowing from domestic residents or (2) borrow from the rest of the world. Rewriting equation (4.6) gives S P + S G = I + NX Def = S P I NX Thus, we can see that an increase in government de cit has to be associated with either increase in private saving, or decrease in gross domestic investment or an increase in the trade de cit (borrowing from abroad). 4.2 Saving and Investment in the U.S. Lets take a look at the behavior of total investment in the U.S. and how it was nanced. Figure (4.) shows the total investment in the U.S. as a fraction of GDP.We see that the Gross Domestic Investment as a fraction of GDP (I/GDP) Fraction of of GDP Time Figure 4.: Gross Domestic Investment as a fraction of GDP. total investment since 929 tends to uctuate around 20% of GDP. In other words, the investment rate in the U.S. is about 20% of GDP, and this rate does not change much over time. From equation (4.6) we know that domestic investment can be nanced by domestic saving (private and government), or carried out by foreigners. That is, I = S P + S G NX.

42 36 CHAPTER 4. SAVING AND INVESTMENT 0.3 Private Saving as a fraction of GDP 0.25 Fraction of of GDP Time Figure 4.2: Private Saving as a fraction of GDP. Lets examine the funding sources of investment. Figure (4.2) shows the private saving as a fraction of GDP. Notice that in the last data point (in 2004) the private investment is about 5% of GDP. Thus, the private saving is about 75% of the domestic investment. The rest, according to the identity (4.6) has to come from government saving or from foreign investment. Figure (4.3) shows the government saving as a fraction of GDP.Notice that recently the government is 0.08 Government Saving as a fraction of GDP Fraction of of GDP Time Figure 4.3: Government saving as a fraction of GDP. running a budget de cit, and in 2004 the de cit was about 2% of GDP. Thus, the government does not "help" to nance the domestic investment. The rest of the funding for the domestic investment has to come from abroad. Figure (4.4) shows the net exports as a fraction of GDP.We can see that in the last 30 years the U.S. is experiencing trade de cit. This means that foreigners accumulate U.S. assets. In

43 4.3. INTERTEMPORAL CHOICE MODEL (SAVING THEORY). 37 Fraction of of GDP Net Exports as a fraction of GDP (NX/GDP) Time Figure 4.4: Net Exports as a fraction of GDP. particular, in 2004 the trade de cit was about 6% of GDP, which corresponds to foreign investment in the U.S. 4.3 Intertemporal Choice Model (Saving Theory). In the rst section of these notes we showed the relationship between saving and investment in the economy, called the saving and investment equation: S = I + NX Our next goal is to investigate the determinants of saving and investment. In this section we build a model in which consumers make explicit decisions about consumption and saving. Before presenting the model, let s take a moment to think about what factors might a ect the saving decision of households. We point out three factors that might be important determinates of saving. Why do people save? Saving is a process of giving up current consumption in order to increase the future consumption. Therefore, we expect that our saving decisions would depend on our current and future income. Typically, individuals who work and expect a decrease in their income when they retire, tend to save some of their current income for retirement. In contrast, other individuals who expect an increase in their future income, tend to borrow (have negative saving). For example, many students take student loans while they are studying, and plan to repay the loan when they graduate and earn higher income. Therefore, current and future income, are among the most important factors that a ect the saving decisions. Another important factor that a ects the saving decision is the interest rate. If an individual gives up some of his current consumption and decides to save, he will be able to increase his future consumption. But the question is by how much? The real interest rate gives the answer to that question. If you walk into a bank and open a savings account, the

44 38 CHAPTER 4. SAVING AND INVESTMENT interest rate that the bank will o er you is a nominal interest rate. The nominal interest rate tells you how many extra dollars you will get in the next period when you save one dollar today. For example, if the annual nominal interest is 0%, this means that when you deposit $ today, you will receive your $ back, plus $0: interest. What consumers care about though is not how much money they got, but how much consumption they can buy with their money. Suppose that as before, the nominal interest rate is 0%. In addition, suppose that there is some consumption good, say burger, which currently costs $. What you really care about is how many extra burgers you will get next year when you give up one burger this year. Suppose that the in ation rate is 5%, so that the price of a burger next year is $:05. Now, when you give up one burger today, and save the $ in the savings account, you will receive $: in the next year, and with this money you can buy :=:05 :048 burgers. Thus, when you give up one burger today, you get 0:048 extra burgers in the future. We therefore say that the real interest rate is 0:048 or 4:8%. Formally, the real interest rate is the extra amount of consumption that one gets in the future when he gives up one unit of current consumption. In contrast, the nominal interest rate is the extra dollars that one gets in the future when he gives up one dollar today. Let i be the nominal interest rate, r be the real interest rate, and be the in ation rate. Then the relationship between the nominal interest rate and the real interest rate is given by: + i + = + r If the nominal interest rate and the in ation rates are small, then we can derive an approximation formula to the above. Taking ln from both sides gives ln ( + i) ln ( + ) = ln ( + r) If i; r and are small, the above is approximately r i Thus, the real interest rate is approximately equal to the nominal interest rate minus the in ation rate. In the burger example, the nominal interest rate 0% and the in ation rate is 5%, and then the real interest rate is approximately 5%. Recall that the exact real interest rate was 4:8%, which is close to 5%. To summarize this discussion, since real interest rate determines how much extra future consumption we expect to get when we save one unit of current consumption, then we suspect that the real interest rate would be one of the pivotal factors that a ect our saving decisions. The third factor that determines saving is our preferences. An individual who values current consumption a lot and does not value future consumption much (someone who lives the day ), will tend to save little or even borrow. On the other hand, someone who values future consumption or consumption of his children and grand children a lot will tend to save more. The three factors that a ect the saving decision are therefore, preferences, current and future income, and the real interest rate The Model Consumers: There are N identical consumers that live for two periods ( and 2) and derive utility from consumption c and c 2 in the two periods: U (c ; c 2 ). Consumers receive income

45 4.3. INTERTEMPORAL CHOICE MODEL (SAVING THEORY). 39 y and y 2 in the two periods and pay a lump sum tax t and t 2 to the government. The consumers decide how much to consume in each period and how much to save in the rst period. We denote the saving in the rst period by s. Consumers can borrow and lend at real interest rate r, which is assumed exogenously given. Thus the budget constraints in the two periods are: The consumers problem is therefore [BC ] : c + s = y t [BC 2 ] : c 2 = y 2 t 2 + ( + r) s max U (c ; c 2 ) c ;c 2 ;s s:t: [BC ] : c + s = y t [BC 2 ] : c 2 = y 2 t 2 + ( + r) s Government: The government collects tax revenues T = N t and T 2 = N t 2 in the two periods and spends G and G 2 in the two periods. The government can borrow and lend at real interest rate r with the constraint that the present value of spending = present value of taxes G + G 2 + r = T + T 2 + r This means that if the government runs a de cit in the rst period, it must borrow the amount of the de cit and pay that amount with the second period s surplus. And if the government has a surplus, it can save the surplus at interest r and be able to a ord a de cit in the second period. To see this, rearrange the above condition (G T ) ( + r) = T 2 G 2 Suppose that the interest rate is r = 5% and in the rst period the government runs a de cit of 00, thus G T = 00. The above condition means that in the second period the government must have a surplus of 05 to pay the debt, i.e. T 2 G 2 = 05. Now that we have completed the description of the model we would like to analyze the impact on consumers of the following changes:. Changes in income: y and y 2 2. Changes in the real interest rate: r 3. Changes in government taxes: T and T 2 To answer the above questions we need to solve the consumers problem. It is convenient to derive the lifetime budget constraint of the consumer. Substitute s from the second period budget constraint into the rst period s budget constraint. It is easy to do when you divide both sides of BC 2 by + r to get BC 2 : c 2 + r = y 2 + r t 2 + r + s

46 40 CHAPTER 4. SAVING AND INVESTMENT Now add the two budget constraints and get the lifetime budget constraint: c + c 2 + r {z } PV of lifetime consumption = y t + y 2 t 2 + r {z } we =PV of lifetime wealth Thus, the left hand side is the present value of lifetime consumption, and the right hand side is the present value of lifetime net of taxes income, which we call the lifetime wealth (we). The consumers problem can then be rewritten as c + c 2 + r max U (c ; c 2 ) c ;c 2 s:t: = y t + y 2 t 2 + r Interpretation: Recall the consumer choice model of choosing the optimal amounts of two goods x and y, with budget constraint p x x+p y y = I (from the micro foundations appendix). In that model the slope of the budget constraint (in absolute value) is p x =p y, which is the relative price of good x in terms of good y. Notice that the two period model is very similar to the general model of consumer choice, where the two goods are current consumption and future consumption (c and c 2 ). We can write the budget constraint concisely as c + c 2 + r = we which is similar to p x x + p y y = I The price of c is, and the price of c 2 is, thus the slope of the budget constraint is (+r), +r and it represents the relative price of current consumption in terms of future consumption. Indeed, consider the cost of increasing current consumption by unit. If the consumer saved that unit, then he would have enjoyed an increase of + r units in the future consumption. Hence the cost of current consumption in terms of future consumption is + r. Thus the consumer can choose the optimal bundle of two goods (c and c 2 ), given his preferences and given the prices of the two goods. The next gure shows the graph of the lifetime budget

47 4.3. INTERTEMPORAL CHOICE MODEL (SAVING THEORY). 4 constraint. c 2 we ( + r) y2 t 2 E Slope = ( + r) y t we c With free borrowing and lending, it is feasible for this consumer to consume all his wealth in the rst period and nothing in the second: (c = we; c 2 = 0). Similarly, it is feasible for this consumer not to consume anything in the rst period and consume all his wealth in the second period: (c = 0; c 2 = we ( + r)). Also notice that it is feasible for the consumer to consume in each period the income (net of taxes) received in that period: (c = y t ; c 2 = y 2 t 2 ). This bundle is denoted by E is the consumer s endowment. If the consumer was not allowed to borrow or lend, then he would be forced to consume his endowment, i.e., his net of taxes income in each period. Because the consumers are free to borrow and lend at real interest rate r, they can chose other points on the budget constraint. If the consumer chooses a point above E on the lifetime budget constraint, then he is a lender (his current consumption is less then current income, so he is saving a positive amount). If the consumer chooses a point below E on the lifetime budget constraint, then he is a borrower (his current consumption is greater than his current income, so he has negative saving).

48 42 CHAPTER 4. SAVING AND INVESTMENT Optimal Choice The next gure shows the optimal choice for a consumer who is a lender. c 2 we ( + r) * c 2 A y2 t 2 E s * >0 ( + r) * c y t we c At the optimal bundle (point A) we have the usual condition that the marginal rate of substitution between and is equal the relative price. That is U (c ; c 2 ) U 2 (c ; c 2 ) = + r The left hand side is the (absolute value of) the slope of the indi erence curves and the right hand side is the (absolute value of) the slope of the budget constraint. This should look very familiar to you and similar to the optimality condition U x (x; y) U y (x; y) = p x p y

49 4.3. INTERTEMPORAL CHOICE MODEL (SAVING THEORY). 43 Notice that in the case of a lender, the saving is positive. The next gure shows the optimal choice for a consumer who is a borrower. c 2 we ( + r) y2 t 2 E * c 2 A s * <0 ( + r) y t * c we c Notice that the saving is negative for a borrower Changes in income In this section we want to analyze the impact of changes in y and y 2 on the consumer s choice (c ; c 2; s ). The consumer s lifetime budget constraint is An increase in current income (y ") c + c 2 + r = y t + y 2 t 2 + r We see from the budget constraint that an increase in will shift the budget constraint to the right. If we assume both goods (current consumption and future consumption) are normal, the consumer will increase the consumption in both periods. In order to increase the consumption in the second period the consumer must increase his saving. Thus, an increase in the current income will increase the current consumption by less than the change in the current income. We call this result consumption smoothing. To summarize: y "=) c "; c 2 "; s "; c < y

50 44 CHAPTER 4. SAVING AND INVESTMENT An increase in future income We see from the budget constraint that an increase in will shift the budget constraint to the right. Given that both goods (current consumption and future consumption) are normal, the consumer will increase the consumption in both periods. In order to increase the consumption in the rst period the consumer must decrease his saving. Thus, an increase in the future income will increase the future consumption by less than the change in the future income. To summarize: y 2 "=) c "; c 2 "; s #; c 2 < y 2 An increase in current and future income. The budget constraint will shift to the right and again consumption in both periods will go up. It is unclear however what will happen to the saving. The impact on saving depends on the relative magnitudes of the changes in y and y 2. To summarize: y "; y 2 "=) c "; c 2 "; s? Temporary vs. Permanent changes in income The main point of the above experiments was to show that if the increase in income happens only in one period, then the consumer will increase his consumption in that period by less than the change in that period s income. This is called consumption smoothing. Is there evidence in the data of consumption smoothing? The next gure shows the percentage deviation from trend of real consumption per capita and real GDP per capita in the U.S. What we can see from the next gure is that consumption is smoother than GDP Detrended GDP, Consumption 0.04 % Deviation det_gdp det_c I 953 II 959 III 965 IV 972 I 978 II 984 III 990 IV 997 I 2003 II Time

51 4.3. INTERTEMPORAL CHOICE MODEL (SAVING THEORY) Changes in the real interest rate Suppose that. What is the e ect of the increase in real interest rate on the budget constraint? First of all, the slope of the budget constraint will increase in absolute value (it will become more steep). Also notice that whatever the interest rate, since the incomes and taxes did not change, the new budget constraint has to pass through the endowment point E. Regardless of the interest rate it is always feasible to consume in each period that period s income. The next graph shows the e ect of on the budget constraint. c 2 we ( + r) A y2 t 2 E B y t we c The dashed line is the new budget constraint, after r ". An increase in the real interest rate has two e ects on the consumer. On the one hand the relative price of current consumption in terms of future consumption ( + r) has gone up. As a result, the consumer would like to substitute (the now more expensive) current consumption with the (the now cheaper) future consumption. This is called the substitution e ect the change in consumption that results from the change in the relative prices. As a result of the substitution e ect the consumer will reduce current consumption and increase future consumption. Since current income did not change, the saving must increase. Thus, as a result of the substitution e ect we have:. But there is another e ect. Notice that if the consumer was lender before the change (for example chose the point A), then after the change he can still a ord the original bundle, and even bundles that contain more of both goods than A. In other words, his purchasing power increased. We call this increase in purchasing power a positive income e ect. For a borrower (one who consumed at point B for example) the opposite happened. After the change he

52 46 CHAPTER 4. SAVING AND INVESTMENT can no longer a ord the previous bundle. We call this decrease in the purchasing power a negative income e ect. Our de nition of income e ect is not precise, but it is intuitive and su cient for the purpose at hand. So for us a positive income e ect occurs if at the new prices, even if you take some of the consumer s income he can still a ord the old bundle. We say that a negative income e ect occurs if at the new prices the old bundle is not a ordable. Assuming that both goods are normal implies that the positive income e ect will cause for the lender and for the borrower. The next table summarizes the results: Lender Borrower Substitution e ect c #; c 2 " c #; c 2 " Income e ect c "; c 2 " c #; c 2 # Total e ect c?; c 2 " c #; c 2? Changes in taxes and Ricardian equivalence We have already analyzed the impact on consumers of the changes in incomes and. Notice that our analysis also covers the changes in and since the budget is So an increase in is just like a decrease in and an increase in is just like a decrease in. There is however an important result in public nance the Ricardian equivalence theorem. Theorem (Ricardian equivalence): If the present value of government spending remains unchanged, then changes in the taxes do not a ect the households optimal consumption choice ( ). Proof: The government budget constraint is Thus, We see that any changes in and must be such that the present value of taxes that the consumer has to pay remain constant. This means that the consumer s budget constraint remains unchanged,, since the last term on the right hand side (the present value of taxes) is unchanged. This implies that the optimal choice of consumption ( ) for each consumer will remain unchanged. Notice however that the saving decision of consumers will change, and the aggregate saving can change as well. To see that recall that. If the government changes the taxes ( and ), then since will not change, then the saving must change by the amount of the change in, but in the opposite direction. More formally, So if the government reduces the taxes in the rst period for each consumer by unit, then the consumers will increase their saving exactly by unit. Discussion: The Ricardian equivalence theorem is a useful starting point for thinking about the e ects of government de cit. Governments can nance their de cit by taxing people or by borrowing (i.e. issuing debt). However, the government must eventually pay the debt by raising taxes in the future. The choice is therefore between "tax now" and "tax later". Our simple framework suggests that from the consumers point of view there is no di erence

53 4.3. INTERTEMPORAL CHOICE MODEL (SAVING THEORY). 47 between tax now or tax later. What matters for the optimal choice of consumption is the present value of the lifetime taxes,, but not the speci c magnitudes of and. It is useful to list the assumptions of our two-period model, which are responsible for the Ricardian equivalence theorem.. We assume that all the households are taxed equally. In the real world it could be that the tax burden is not shared equally, so that tax policies will have an e ect on the distribution of wealth in the economy. 2. In the model, the same people who receive a tax cut are the ones who have to pay the government debt in the future. In the real world the government can postpone the tax increase until long in the future, when consumers who received the tax cut are either retired or dead. In this case, the government tax policy will involve intergenerational transfer. 3. In the model the taxes were lump-sum. If we change this assumption and let the taxes be a fraction of income and also tax the interest rate earnings in the second period, then the timing of taxes will matter for the optimal choice of the household. To see this, notice that if the government taxes the interest earnings in the second period, then the household s net-of-taxes saving in the second period is. This means that the slope of the budget constraint is now, so if the government changes the timing of the taxes, the household s budget constraint will not remain una ected. 4. Finally, in the model consumers can borrow or lend at the same interest rate as much as they please. If we relax these assumptions, the timing of taxes would a ect the optimal consumption choices. A simple example can illustrate why. Suppose that some consumer is not allowed to borrow, so he is forced to consume his endowment. Changes in taxes will change the consumer s endowment, and therefore the consumption in both periods will change. Appendix: Solving the consumer choice problem with Cobb-Douglas preferences Suppose that the utility function has the form, where. This is a version of Cobb-Douglas preferences. The coe cient is a weight on the utility from consumption in the second period. The greater is, the more patient is the consumer. The consumer s problem that we need to solve is therefore This is a standard problem, similar to the ones we have solved before. The Lagrange function is First order conditions: Thus the familiar condition of MRS = slope of the budget constraint is, or Plugging this into the budget constraint gives the demand for The demand for is therefore Thus, the consumer spends a fraction of his lifetime income on rst period s consumption and a fraction of his lifetime income on the second period s consumption. It makes intuitive sense that the higher the relative weight on a particular good in the utility, the greater is the demand for that good. Substituting the expression of we in the demand gives the demand for and : The saving is then Notice that higher interest rate implies lower consumption in the rst period, and higher saving and consumption in the second period. This makes intuitive sense, because is the relative price of current consumption in terms of future consumption. If interest rate goes

54 48 CHAPTER 4. SAVING AND INVESTMENT up, the consumer will substitute future consumption for current consumption and therefore will save more in the rst period. Also, the higher the current net-of-tax income is, the greater is the saving, and the higher is the second period net-of-tax income, the lower is the saving. This makes intuitive sense. If the current income is relatively high, we would like to save more for the future, and if the future income is relatively high, we would like to save less for the future. Finally notice that an increase in by unit will increase by less than unit (by to be precise). The saving will increase by. Thus we see the consumption smoothing result here. 4.4 Two-Period Model of Investment. There is one rm that can produce output in two periods according to Y = A K L Y 2 = A 2 K 2L 2 where A, A 2 are productivity parameters (TFP - Total Factor Productivity), K, K 2 are the levels of physical capital in the two periods, and L, L 2 are labor inputs (number of workers employed by the rm). 2. The rm owns the capital stock in each period, and consumers own the rm. That is, the rm belongs to the shareholders, who are entitled to the stream of pro ts from the rm. 3. The capital stock evolves according to K 2 = ( ) K + I (4.7) where is depreciation rate of capital and I is investment in capital in the rst period. 4. The capital in the rst period, K, is exogenously given in the model, while the second period capital is a result of the rm s investment decision. 5. The rm decides on the labor demand in each period (L ; L 2 ) and on the magnitude of the investment in the rst period, I, and thus implicitly chooses K The distributed pro t (dividends) in each period is given by = Y w L I 2 = Y 2 + ( ) K 2 w 2 L 2 Notice that since the economy lasts for two periods, the rm can sell the nondepreciated capital stock, so the revenue in the second period is Y 2 + ( ) K 2.

55 4.4. TWO-PERIOD MODEL OF INVESTMENT Optimal investment decision Recall that the rm is owned by the consumer and the lifetime value of the rm to the consumer is given by V = r where r is the real interest rate. Thus, the rm s problem is to choose L ; L 2 ; I; K 2 to maximize the present value of the stream of dividends max V L ;L 2 ;I;K 2 = A KL w L I + A 2K2L 2 + ( ) K 2 + r w 2 L 2 s:t: K 2 = ( ) K + I Notice that choosing particular value of I essentially pins down K 2, we can let the rm directly choose K 2. Substitute the constraint into the objective and obtain the following rm s problem: max V = A K L ;L 2 ;K 2 L w L K 2 + ( ) K + A 2K2L 2 + ( ) K 2 w 2 L 2 + r The rst order conditions with respect to L and L 2 = 0, ) ( ) A L = = 0, ) ( ) A 2 2L 2 2 = w 2 This means that in each period the rm wants to hire labor up to the point where the marginal product of labor equals the wage. The rst order condition with respect to K 2 = + A 2K 2 L 2 + = 2 + r The interpretation is as follows. Increasing next period s capital by unit costs unit of current dividends. The bene t in the next period comes from two sources: () the revenue in the next period will increase by the marginal product of capital, A 2 K2 L2, and (2) the nondepreciated capital can be sold in the next period and thus we have units of aditional revenue. The bene t from investment is collected in the second period, so it is discounted by + r to obtain its present value. Thus, the optimal level of K 2 (and therefore of I) is determined by the condition + A 2K2 L 2 + = 0 + r A 2 K2 L2 = r (4.8) which means that the real interest rate is equal to the marginal product of capital net of depreciation. This condition makes intuitive sense. In equilibrium the return in the nancial

56 50 CHAPTER 4. SAVING AND INVESTMENT market must be equal to the return in the capital market. If we invests unit in the physical capital, the net return will be the marginal product of capital net of depreciation. If one invests in the nancial market, the net return is r. In equilibrium there should be no arbitrage opportunities, and thus the rates of return must be equalized. It should make intuitive sense that an increase in r will decrease the investment in physical capital since the return in the nancial market becomes relatively higher Changes in interest rate The next gure illustrates the e ect of an increase in r on the optimal investment decision. MP K r' + δ r + δ θa θ θ 2K2 L2 K 2 ' K 2 K 2 The downward slopping curve is the marginal product of capital MP K. The optimal K 2 is obtained at the point where the marginal product of capital is equal to r +. Higher real interest rate leads to lower K 2 and thus lower investment (since I = K 2 ( ) K ). The intuition is as follows. The real interest rate represents the opportunity cost of investment in physical capital. Higher real interest rate means that the return in the nancial market is higher, which makes the investment in physical capital less attractive. As shown in the gure, when interest rate goes up from r to r 0, the optimal future capital falls from K 2 to K Changes in technology The next gure illustrates the e ect of an increase in A on the optimal investment decision.

57 4.4. TWO-PERIOD MODEL OF INVESTMENT 5 MP K r + δ θ θ 2 2 θ A 2 ' K L θa θ θ 2K2 L2 K 2 K 2 ' K 2 Notice that the marginal product curve shifts upward, so that for any given level of K 2 its marginal product increases. The optimal level of K 2 (and also of investment) will therefore increase. The net return on investment in physical capital is the marginal product of capital (net of depreciation), hence it is intuitive that when the marginal product of capital goes up, the investment in physical capital should go up Solving for optimal investment Solving equation (4.8) gives the optimal future capital K 2 as a function of real interest rate. A 2 K2 L 2 = r + A 2 L2 = K2 r + A2 L 2 K 2 = r + Notice that from the above equation we clearly see that optimal K 2 is decreasing in interest rate r and increasing in the productivity level A. Now substitute this into equation (4.7) to nd the optimal investment A2 L2 I = r + ( ) K (4.9) Observe that the optimal investment is decreasing in r and decreasing in current capital (K ). The intuition for the last observation is simple: if today s capital stock is big, we don t need to invest as much in order to attain the optimal level of future capital. The next gure show the investment demand curve as a function of real interest rate.

58 52 CHAPTER 4. SAVING AND INVESTMENT r I (r) I The price of investing in physical capital is the real interest rate - the opportunity cost of investment. Changes in the real interest rate is re ected in a movement along the same demand curve, but the curve itself will not shift. Changes in parameters other than r will shift the entire curve. For example, higher expected productivity (A 2 ") will shift the entire demand curve to the right. With higher future productivity the rm would like to invest more at any given real interest rate. The next gure shows the impact of (A 2 ") on the demand for investment. r I r; A ) ( 2 I ( r; A' 2 ) I In the above gure A 0 2 > A 2 results in shift to the right of the demand for investment. It is clearly seen from equation (4.9) that an increase in future productivity should increase investment. 4.5 Capital Market Now we are ready to put together the supply of saving with the demand for investment. We start with closed economy rst. The next gure shows the capital market. For simplicity, we draw linear supply and demand curves.

59 4.5. CAPITAL MARKET 53 r S(r) * r I (r) I *, S * I,S The supply of saving curve is assumed upward slopping. Recall from our analysis of the saving decisions of household, we concluded that a lender will not necessarily increase his saving when the real interest rate goes up. Nevertheless, we assume that the substitution e ect is stronger than the income e ect, which ensures that the total saving of households is increasing in interest rate. In a closed economy, national saving and domestic investment are equal. The above graph shows that the real interest rate and the amount of saving and investment is determined in equilibrium, at the intersection of supply and demand. In an open economy, national saving can di er from domestic investment. For example, if the world interest rate is below r then the national saving will not be enough to nance the domestic investment, and the di erence has to be borrowed. As we discussed before, borrowing from the rest of the world is the negative of trade de cit. The next gure illustrates an economy with S < I. r S(r) * r NX I (r) S * I * I,S In this economy (as in the U.S. currently) the national saving falls short of the domestic investment, and the economy borrows the amount of N X (the trade de cit). What a ects the trade de cit according to our theory? It is obvious from the graph that national saving and domestic investment together determine the size of the trade de cit. In the next sections we will work with this model, to analyze the impact of di erent events on the saving, investment and the trade de cit in the economy.

60 54 CHAPTER 4. SAVING AND INVESTMENT 4.5. Decline in government de cit (S G ") The total saving curve will shift to the right, as shown in the next gure. r S(r) * r NX I(r) S * S ** I * I,S For simplicity we assume that this change will not a ect the world interest rate. We see that an increase in government saving leads to an increase the national saving and lowers the trade de cit. As bigger fraction of the domestic investment is funded by national saving, there is less borrowing from the rest of the world Increase future productivity at home (A 2 ") The demand for investment curve will shift to the right, as demonstrated in the next gure. r S(r) * r NX I (r) S * I * I ** I,S If initially the national saving were not su cient to fund the domestic investment, then after an increase in domestic investment the shortage is greater. The borrowing from the rest of the world is increasing, meaning higher trade de cit. The domestic investment increases, where all of the increase is funded by foreigners.

61 4.6. SUMMARY Summary. We showed that saving and investment in any economy are related. This relationship is called the "Saving and Investment Equation". 2. We presented a theory of saving in a two period model of intertemporal choice. The model allows us to study the impact real interest rate, current and future income, and government tax policies on consumption and saving behavior. The model delivers an intuitive result, that people tend to smooth consumption. Another important result is Richardian Equivalence Theorem, which says that under certain conditions changes in government taxes do not a ect the consumption and saving decisions of households. 3. We presented a theory of Investment decision by rms. The demand for investment is decreasing in real interest rate and increasing in future productivity. 4. Putting together the saving and investment theories, allows us to analyze the capital market. We demonstrated how real interest rate, the level of saving, investment and trade de cit are determined in the capital market. 4.7 Appendix: Firm With Unlimited Life Span In these notes we analyzed the investment decision of rms that live for two periods. Now we show that the same condition for optimal investment holds when the rm lives unlimited number of periods. Suppose that the production function is more general than we used above, namely Y t = F (K t ; L t ; t) The present value of the stream of pro ts is V = X t=0 t ( + r) t = X t=0 F (K t ; L t ; t) w t L t I t ( + r) t The rm s problem is max fk t+ ;L tg t=0 = X t=0 s:t: F (K t ; L t ; t) w t L t I t ( + r) t K t+ = ( ) K t + I t Substituting the constraint into the objective, gives max = fk t+ ;L tg t=0 X t=0 F (K t ; L t ; t) w t L t K t+ + ( ) K t ( + r) t

62 56 CHAPTER 4. SAVING AND INVESTMENT F.O.C. with respect to K t+ is ( + r) t + F (K t+ ; L t+ ; t) + ( + r) t+ = 0 F (K t+ ; L t+ ; t) + = + r F (K t+ ; L t+ ; t) = r + Thus, as before, the rm will invest up to the point where the marginal product of capital F (K t+ ; L t+ ; t) is equal to the sum of the real interest rate and depreciation. Intuitively, the cost of investment is forgone interest rate and depreciation, and that has to be balanced by the return to investment - marginal product of capital.

63 Chapter 5 Economic Growth 5. Introduction The most important macroeconomic observation in the world is the huge di erences in output (and income) per capita across countries. More than 60% of the world population is at least 7 times poorer than the average American. In the poorest countries the GDP per capita is at least 40 times smaller than that in the U.S. What accounts for such big di erences? We will not attempt to answer this question here, but rather focus on one important source of cross country di erences - the di erence in capital per worker. The next gure shows the scattergram of capital per worker and GDP/capita in a large sample of countries. We see that there is a strong positive correlation between capital per worker and GDP/capita across countries. In other words, countries with high GDP/capita tend to have higher capital per worker. 57

64 58 CHAPTER 5. ECONOMIC GROWTH But why some countries have much less capital per worker than others? To answer this question recall that capital is created through investment in new capital. The next graph shows the average investment over v.s. physical capital per worker in Observe that, not surprisingly, there is a strong positive correlation between the investment rate and physical capital per worker. These observations motivated the Solow growth model. Robert Solow received a Nobel Prize in Economics in 987 "for his contributions to the theory of economic growth". Here is part of the press release which describes Solow s contribution: "The study of the factors which permit production growth and increased welfare has been a central feature in economic research for many years. Robert M. Solow s prize recognizes his exceptional contributions in this area. It is eminently reasonable to imagine that increased per capita production in a country may be the result of more machines and more factories (a greater stock of real capital). But this increased production may also be due to improved machines and more e cient production methods (which may be termed technical development). In addition, better education and training, and improved methods of organizing production may also give rise to increased productivity. The discovery of fresh natural resources, or improvements in a country s position on the world market, may also lead to higher standards of living. Solow has created a theoretical framework which can be used in discussing the factors which lie behind economic growth in both quantitative and theoretical terms. This framework can also be exploited to measure empirically the contributions made by various production factors in economic growth."

65 5.2. THE SOLOW MODEL The Solow Model 5.2. Description of the model Output is produced according to Y t = A t Kt Lt, 0 < <. Capital evolves according to K t+ = K t ( and I t is aggregate investment. ) + I t, where is the depreciation rate People save a fraction s of their income. This fraction is exogenous. Thus, the total saving and total investment in this (closed) economy is S t = I t = sy t The population of workers grows at a constant rate of n, which is exogenous in this model. Thus, L t+ = ( + n) L t. We neglect the di erences between population and population of workers for the sake of simplicity, and use the terms "output per worker" and "output per capita" Interchangeably Working with the model Now we derive the predictions of the model. The output per worker is: y t = Y t = A tkt L t Kt = A t = A t kt L t L t The law of motion of capital per worker is L t K t+ = K t ( ) + I t L t+ L t+ L t+ k t+ = K t ( ) L t ( + n) + sy t L t ( + n) k t+ = k t ( ) + n + sa tkt (5.) + n Equation (5.) describes the law of motion of physical capital per worker. If A t is xed at some level A, then the law of motion can be illustrated graphically, as in gure (5.). With xed productivity it can be shown that the capital per worker converges to a steady state level, such that k t+ = k t = k ss 8t The steady state level of capital per worker can be seen in the graph at the intersection of the law of motion equation with the 45 0 line. It can be shown that starting from any level of capital per worker, it converges to the steady state level k ss. Thus, the prediction of the Solow model is that with xed A, the capital per worker will converge to k ss. We call a variable endogenous if it is determined within the model and exhogenous if it is determined outside the model. For example, in the model of a market (supply and demand diagram), the price and quantity traded of the good are endogenous variables, while other variables that determine the location of the supply and demand curve, such as income and prices of other goods, are assumed exogenous.

66 60 CHAPTER 5. ECONOMIC GROWTH Figure 5.: Law of motion of physical capital per worker. k_t Law of motion of capital per worker k_t k_t+ 45_deg Finding the steady state Using the law of motion and the de nition of the steady state k t+ = k t ( ) + n + sak t + n k ( ) k = + n + sak + n k ( + n) = k ( ) + sak k (n + ) = sak (5.2) The intuition behind the last equation is as follows. The left hand side shows the decline in capital per worker due to depreciation and growth in the number of workers. The right hand side is the investment per worker, i.e. the increase in capital per worker. At the steady state the decline in capital per worker due to depreciation and growth in the labor force must be o set by the increase in capital per worker due to investment. The steady state capital per worker is k ss = The steady state output per worker is sa n + sa y ss = Akss = A = A n + The steady state consumption per worker is s n + (5.3) (5.4) c ss = ( s) Ak ss (5.5)

67 5.2. THE SOLOW MODEL 6 The predictions of the model Observe that k ss is increasing in the saving (=investment) rate and productivity, and decreasing in the population growth rate and depreciation. This result is fairly intuitive. First, higher saving rate means that investment per worker is higher, and thus capital per worker should be higher. Higher productivity means that investment per worker is higher simply because more output is produced per worker. This also leads to higher capital per worker in the steady state. Equation (5.2) can help with developing the intuition. Think of the right hand side of (5.2) as the " ow in" the stock of capital per worker, i.e. the investment per worker. Higher s or higher A increase the " ow in", and result in higher stock of capital per worker. Now from equation (5.3) we can see that steady state capital per worker is decreasing in depreciation and the growth rate of population. Higher depreciation means that the " ow out" of the stock of capital is higher, so the stock of capital is lower. Similarly, higher growth rate of the workers population also reduces the capital available per worker, and in a sense works just like depreciation. Observe that n and appear together in equations (5.3) and (5.4). Similarly, the steady state output per worker y ss is increasing in the saving rate and productivity, and decreasing in the population growth rate and depreciation, just like k ss. This is just because output per worker is increasing function of the capital per worker. The Solow model therefore predicts that countries with higher investment rates, should on average, enjoy higher standard of living. This prediction is consistent with the data, as can be seen from gure (5.2). Figure 5.2: Investment rate and GDP per capita. The Solow model also predicts that countries with higher growth rate of population, should on average enjoy lower standard of living. This prediction is also consistent with the

68 62 CHAPTER 5. ECONOMIC GROWTH data. According to the Solow model, higher saving rate leads to higher output per worker. Does this mean that the policy recommendation implied by the model is to save as much as possible? The answer to this question is absolutely no. Suppose that at the extreme the consumers save all their income. If this happens, the consumers will not consume anything, and just starve and die. The next section discusses the optimal saving rate, i.e. the saving rate that maximizes the steady state consumption per capita. Optimal saving rate Notice that although higher saving rate leads to higher steady state level of capital per worker and output per worker, it does not necessary lead to higher consumption per worker. Observe from equation (5.5) that on the one hand higher s leads to higher income per worker, but on the other hand higher saving rate means that a smaller fraction of that income is consumed. We can nd the optimal saving rate, i.e. the saving rate that maximizes the steady state consumption per worker. This saving rate is called the golden rule saving rate. c ss = ( s) Akss = Akss (n + ) k ss First order condition: max k ss c ss = Ak ss (n + ) k ss Ak GR = n + k GR = Now comparing this with the steady state capital k ss = A n + sa n + implies that s GR = 5.3 Endogenous Growth Model Recall that in the Solow growth model, when the TFP (= A) is xed, capital per worker, output per worker and consumption per worker, all converge to a steady state level: k ss = sa n + y ss = Ak ss c ss = ( s) y ss

69 5.3. ENDOGENOUS GROWTH MODEL 63 Higher TFP means higher levels of the endogenous variables (k ss ; y ss ; c ss ), higher saving rate (s) results in higher k ss and y ss, but not necessarily higher c ss and higher n results in lower (k ss ; y ss ; c ss ). In the Solow growth model, sustained growth in output per worker is possible only if A is growing all the time (perpetual growth). It is important to realize that the Solow growth model does not o er an explanation for why growth occurs (i.e. why A grows). In the Solow model, country will grow faster than country 2 because the A of country grows faster than that in country 2. The question is what is A and what causes it to grow? This is the question that we address in this section. In addition to the TFP, we observed that di erences in saving rate in the Solow model create di erences in the steady state output per worker. In the Solow model the saving rate is assumed to be xed and exogenous. Is this a good assumption? People decide how much of their current income they wish to consume and how much to save based on many factors. Among those factors one can list the interest rate on saving, expected future income, etc. Chapter 4 studies the consumption and saving decision of households in more depth. Now we modify the Solow model slightly such that the TFP and its growth depend on other parameters in the model. In particular, the TFP re ects the level of human capital in the economy. Human capital is an abstract term which is typically associated with the skills and education of workers in the economy. The resulting model will be called Endogenous Growth Model. The human capital in this model depends on the fraction of time that people spend on human capital accumulation (education), and the e ciency of human capital accumulation. It is debatable whether the name Endogenous is justi ed here, since the factors that determine the TFP are themselves exogenous. Having said that, we will follow the convention of naming models in which the growth rate of endogenous variables depends on some other parameters of the model, as Endogenous Growth Models. The main results of the model are:. The greater the fraction of time spent on human capital accumulation, the greater is the growth rate of endogenous variables. 2. The greater is the e ciency of human capital accumulation, the greater is the growth rate of endogenous variables Description of the model Consumers: Each consumer saves a fraction s of his income (and thus consumes a fraction s of his income). Each consumer is endowed with unit of time, with fraction u spent on work, and a fraction u is spent on human capital accumulation. Technology: Output is produced using Capital and labor. Output at time t is given by: Y t = K t (uh t L t ) where K t is capital L t is the population of workers, and h t is the level of human capital per worker. Thus, the labor input, t uh t L t, is not just the number of bodies, but depends on the time worked and the quality of workers (h t ). The above production function can be written as Y t = (uh t ) Kt Lt = A t Kt L t

70 64 CHAPTER 5. ECONOMIC GROWTH This production technology is the same as in the Solow model, except now we have a story about the formation of the TFP, i.e., the TFP depends on the stock of human capital in the economy: A t = (uh t ). Population: Population of workers grows at constant and exogenous rate n. Thus, the population of workers evolves according to L t+ = ( + n) L t Physical Capital: Physical capital stock evolves according to K t+ = ( ) K t + I t where is depreciation rate and I t is aggregate investment. Human Capital: The human capital per worker evolves according to h t+ = b ( u) h t where ( u) is the fraction of time the people spend on human capital accumulation and the parameter b re ects the e ciency of human capital accumulation. We assume that b > 0, otherwise the human capital becomes negative. Notice that in order to enable growth in human capital, we must have b ( u) > Working with the model We start by deriving the growth rate of TFP. A t+ A t = (uh t+) (uh t ) = ht+ h t = [b ( u)] It is important to realize that in this model the endogenous variables will not converge to a steady state since A t is not xed. It is possible to formally prove that in this model, all the endogenous per capita variables (k t ; y t ; c t ) will be growing at a constant rate in the long run. We will postpone the formal proof for the appendix to this chapter, and here we only derive the long run growth rate of endogenous variables. As a side comment, a situation in which all the endogenous variables are growing at constant, but not necessarily the same, rate is called Balanced Growth. We rst nd the long run growth rate of capital per worker. The law of motion of capital per worker is the same as in the Solow model Dividing both sides by k t gives k t+ = k t ( ) + n + sa tk t + n k t+ k t = ( ) + n + sa tk t + n

71 5.3. ENDOGENOUS GROWTH MODEL 65 If capital per worker grows at constant rate, then the left hand side must be constant. This implies that the second term on the right hand side must be constant as well, and in particular, A t kt must be constant. Thus, A t+ kt+ = A t kt [b ( u)] kt+ = k t kt+ = [b ( u)] k t k t+ k t = b ( u) We proved above that if capital per worker grows at constant rate, then this rate must be k t+ k t = b ( u) Next, we nd the long run growth rate of output per worker. y t+ y t = A t+k t+ A t k t = [b ( u)] [b ( u)] = b ( u) Thus, output per worker gros at the same rate as physical capital per worker. Another way to prove the above is to use tha fact that A t kt is constant, wich is the ratio of output to capital A t k t = A tk t k t = y t k t If the output/capital ratio is constant, this means that the output must grow at the same rate as capital. Finally, it should be clear that consumption per worker must grow at the same constant rate as output per worker, since consumption is proportional to output: c t = ( s) y t Thus, in the endogenous growth model, all the per worker variables grow (in the long run) at a constant rate of b ( u). Summary: A t+ = [b ( u)] A t k t+ = y t+ = c t+ = b ( u) k t y t c t

72 66 CHAPTER 5. ECONOMIC GROWTH Economic Policy and Growth In this endogenous growth model, the higher the fraction of time that is allocated to human capital accumulation ( u), the higher will be the long run growth rate of the endogenous variables (A t ; k t ; y t ; c t ). Similarly, the higher is the e ciency of human capital accumulation (b), the higher is the long run growth rate of the endogenous variables (A t ; k t ; y t ; c t ). Two questions regarding government policy arise immediately: () can the government a ect b and ( u)? and (2) Should the government try to a ect b and ( u)? Many believe that the answer to the rst question is yes. The government can increase b by implementing better incentives for performance in the school system, or by changing the mix of private and public schools. The government can increase ( u) by subsidizing education. The answer to the second question is not clear. Notice that Y t = (uh t ) Kt Lt. We see right away that increasing ( u) means that u will drop and the current output will drop. Thus, the short run e ect is a drop in (A t ; k t ; y t ; c t ). True, in the long run these variables will grow faster, but this does not come without cost. What about increasing b? In this framework we do not model the cost of increasing b, but we can be sure that nothing comes without cost in the real world, including reforming the educational system Evidence The next gure suggests that there is a positive correlation between educational attainment and growth rate of real GDP. To me it seems that an inverse U curve can better t the data than a straight line with positive slope. This could mean that there is a level of educational attainment that maximizes growth.

73 5.3. ENDOGENOUS GROWTH MODEL 67 The next graph shows a clear positive trend between educational attainment and output per worker. The two graphs thus support the prediction of the model that higher ( u) is associated with faster growth of output per worker Appendix We now prove that in this model all the endogenous per capita variables, (k t ; y t ; c t ), grow at constant rate in the long run. The production function is Y t = K t where t = uh t L t is the labor input. Notice that the growth rate of this quantity is: t+ t = uh t+l t+ uh t L t = b ( u) ( + n) We de ne the variables in e ciency units, i.e mormalized by the total labor input, as follows: k t = K t t ; y t = Y t t ; c t = C t t Next we show that all the variables converge to a steady state. We derive the law of motion of k t : K t+ = ( ) K t + sk t t K t+ t+ = k t+ = ( ) K t + b ( u) ( + n) t b ( ( ) b ( u) ( + n) k t + b ( t skt t u) ( + n) t s u) ( + n) (k t )

74 68 CHAPTER 5. ECONOMIC GROWTH The law of motion of kt has the same shape as the one in the Solow model, and therefore in the long run kt converges to some constant level kss. Similarly, is is easy to see that yt and c t converge to a steady state in the long run. Then, if kt is constant, we have: k t+ k t = K t+= t+ = K t+=uh t+ L t+ = k t+ h t = K t = t K t =uh t L t k t h t+ ) k t+ k t = h t+ h t = b ( u) Similarly, we show that the growth rate of y t and c t is also constant in the long run.

75 Chapter 6 Money and Prices 6. What is Money? We de ne money as the medium of exchange in the economy, i.e. a commodity or nancial asset that is generally acceptable in exchange for goods and services. Currency which consists of coins and bank notes are a medium of exchange. Checking accounts can also be used as a medium of exchange, since a consumer can write a check in exchange for goods. Travelers checks are another example of money. There are other assets where it is not clear if they should be considered as money, for example saving accounts. A consumer can withdraw from a saving account and pay for goods, but the main purpose saving accounts is to store value not to serve as a medium of exchange. Are credit cards considered a form of money? The answer is no. If you buy goods from a supermarket using a credit card, the credit card company will pay the shopkeeper today and you will have an obligation to pay the credit card company when your credit card bill comes in. This obligation to the credit card company does not represent money. The money part of the transaction between you and the credit card company only comes into play when you pay your bill. There are several de nitions money aggregates:. M - Measure of the U.S. money stock that consists of currency held by the public, travelers checks, demand deposits and other checkable deposits including NOW (negotiable order of withdrawal) and ATS (automatic transfer service) account balances and share draft account balances at credit unions. 2. M2 - Measure of the U.S. money stock that consists of M, certain overnight repurchase agreements and certain overnight Eurodollars, savings deposits (including money market deposit accounts), time deposits in amounts of less than $00,000 and balances in money market mutual funds (other than those restricted to institutional investors). 3. M3 - Measure of the U.S. money stock that consists of M2, time deposits of $00,000 or more at all depository institutions, term repurchase agreements in amounts of $00,000 Take a look at $ bill, or any other U.S. paper money. It is called "Federal Reserve Note". 69

76 70 CHAPTER 6. MONEY AND PRICES or more, certain term Eurodollars and balances in money market mutual funds restricted to institutional investors. The most important monetary aggregate is M and is often referred to as the money supply in the economy. 6.2 The Demand for Money There are many theories that try to explain how the quantity of money that households want to hold is determined. Most of the modern theories are quite complicated and cannot be presented in this class. We will describe two theories here: () Quantity Theory of Money 2, and (2) Money in the Utility Function Quantity Theory of Money According to this theory, households want to hold money in proportion to the dollar value of goods produced in the economy. Let Y t be the real GDP at time t and let P t be the price level (GDP de ator). Thus, the nominal GDP is P t Y t. The demand for money according to this theory is given by M D t = k t P t Y t (6.) where Mt D is the demand for money and k t is the propensity to hold money. Typically, k t < since each dollar can be used more than once every year, so if the households spend P t Y t during year t, they need to keep only a fraction of their planned spending as money. In 2005 k t = 0:, which means that in 2005 households held money at the amount of % of the GDP. In other words, each dollar circulated 9 times during 2005 (=0: = 9). We de ne the velocity of money as the average number of times a piece of money circulates during the year. The velocity is denoted by V t and de ned as Thus, equation (6.) can be written as V t = P ty t : Mt D M D t V t = P t Y t (6.2) Notice that V t = =k t. In equilibrium, the money demand Mt D is equal to the money supply, and denoted by Mt and the above can be written as M S t M t V t = P t Y t (6.3) Equation (6.3) is called the quantity equation. It is important to realize that the quantity equation always holds for every economy in the world. This is simply because we de ne the velocity to be such that the above equation holds. 2 Sometimes called the Classical Theory.

77 6.2. THE DEMAND FOR MONEY 7 The quantity theory is silent about what determines the velocity V t and real GDP Y t, but nevertheless this equation is useful for relating money, in ation and real GDP. Divide equation (6.3) at time t + by the same equation at time t: + ^M + ^V M t+ V t+ = P t+y t+ M t V t P t Y t = + ^P + ^Y (6.4) where "hat" above a variable denotes its growth rate, for example ^M = (M t+ M t ) =M t. Taking logs of equation (6.4) gives ln + ^M + ln + ^V = ln + ^P + ln + ^Y For small growth rates the above is approximately Suppose that velocity is constant, i.e. ^V = 0. Then we have Rearranging ^M + ^V = ^P + ^Y (6.5) ^M = ^P + ^Y ^P = ^M ^Y (6.6) The growth rate of the price level is in ation = ^P. Equation (6.6) tells us that if velocity is constant, then the in ation rate in the economy is approximately equal to the growth rate of money supply minus the growth rate of the real GDP. For example, suppose that during 2005 the money supply increased by 4% and the growth rate of real GDP was.5%, then the in ation rate must be 2.5%, if velocity did not change during The next gure shows the velocity in the U.S. since 959. Velocity Velocity of M in the U.S Years Notice that velocity has increased during the time period in question by a factor of 2.5. During the decade of the velocity increased from 6 to 9, which is 50% increase. This means that each piece of money circulates more times than it used to in the past. In other words, people economize on money holdings. In the next section we attempt to develop a theory of velocity.

78 72 CHAPTER 6. MONEY AND PRICES Money in the Utility Function Consumers derive utility from consumption C and real balances M=P. We can think of the later as liquidity services, or the purchasing power of the nominal money holdings. The opportunity cost of holding money is the nominal interest rate i. The consumer s income is equal to the nominal GDP (P Y ). The consumer s problem is therefore max ln C + ( C;M s:t: P C + im = P Y Now divide the budget constraint by the price level M ) ln P max ln C + ( ) ln (M) ( ) ln P C;M s:t: C + i M P = Y Notice that the last term in the utility function can be dropped because utility is invariant with respect to monotone transformations. Hence, the consumer s problem is max ln C + ( C;M s:t: ) ln (M) C + i M P = Y We know that when preferences are of the Cobb-Douglas form, the demand is C = Y M D = ( ) P Y i The demand for money that we derived, is increasing GDP and decreasing in nominal interest rate - which represents the cost of holding money. Thus, according to this model the velocity is V = i This result makes intuitive sense: as interest rate goes up, the opportunity cost of holding money goes up and the households economize on money holdings. As a result, each piece of money is used more times during the year. The next gure shows the graphs of velocity and federal funds rate since 959.

79 6.3. MONEY SUPPLY 73 Federal Funds Rate Federal Funds rate (i) and Velocity (V) Year i V Poly. (V) Poly. (i) Velocity The dashed lines are polynomial trends (degree 6). As we see from the above gure, until early 80 s the trends of velocity and interest rates move in the same direction, as the model predicts. After 98 however, the trends move in the opposite directions. The only way that our model can reconcile this observation is by increasing. Recall that is the weight of consumption in the utility function while is the weight on real balances in the utility function. Decline in means that liquidity services provided by the real balances are not as valuable as before. Our model cannot o er any explanation why this might happen though. One might conjecture that the sharp increase in velocity in the last decade (50% increase) has something to do with innovations in the banking and payment system. These include ATM machines, electronic transfers, etc. Anything that allows faster payments and smaller average holdings of money will increase the velocity. For example, if households receive income every two weeks instead of every month, the average money holding will be lower and velocity higher. 6.3 Money Supply In this section we explain how the Federal Reserve System 3 (FED for short) and the commercial banks create money. We de ne the Monetary Base (MB) as all the coins and paper money that is created by the FED. The Monetary Base is held partly as reserves (R) of the commercial banks and partly as currency in the hands of public (CU). Thus, MB = R + CU The money supply M consists of currency and checking deposits, which is the M money aggregate de ned above. Thus M = CU + D The commercial banks are required to keep certain minimum percentage of deposits (D) as liquid reserves of cash and currency. This fraction is called the required reserve ratio and 3 To learn about the history, the structure and the activities of the Federal Reserve System, visit

80 74 CHAPTER 6. MONEY AND PRICES denoted by rd, where rd = R D Suppose that consumers want to hold certain amount of cash, as a fraction of their deposits. Let this fraction (currency/deposits ratio) be cd, that is cd = CU D Under these assumptions, M, D, R, and CU are all proportional to the monetary base. M MB = D + CU R + CU = + cd rd + cd The last equality is obtained by dividing the numerator and the denominator by D. Similarly, D MB = D R + CU = rd + cd R MB = R R + CU = rd rd + cd CU MB = CU R + CU = cd rd + cd Thus M = D = R = CU = + cd MB rd + cd MB rd + cd rd MB rd + cd cd MB rd + cd The magnitude mm = rd+cd +cd is called the money multiplier, and it gives us the change in the money supply that results from $ change in the monetary base. Now we are ready to illustrate how the FED, together with the commercial banks, a ects the money supply. The FED has several policy instruments, the most popular of which is open market operations. An open market operation is purchasing and selling government bonds. Every purchase of bonds by the FED increases the monetary base (injects money into the economy). Because of the partial reserve requirements, the commercial banks can lend some of the money received from the FED and thereby generating additional money. The FED can also alter the reserve requirement ratio, but it rarely does so. We will demonstrate the two modes of operation in the next example.

81 6.3. MONEY SUPPLY Example of Money Creation Suppose the currency/deposit ratio that the public wants is 20% and the reserve requirement ratio is 0%. The following is a consolidated balance sheet of the commercial banks. Balance sheet of the commercial banks Assets Liabilities R = 0 D = 00 B G = 25 L = where R is the reserves, B G government bonds, L is loans, and D is deposits. Notice that the balance sheet must always be balanced. Also observe that the banks conform to the reserve requirement and indeed R=D = 0:.. Find the monetary base in this economy. MB = CU + R CU = cd D = 0:2 00 = 20 R = 0 Thus MB = = Find the money supply in this economy M = CU + D = = Find the money multiplier in this economy. + cd mm = = rd + cd + 0:2 = 4 0: + 0:2 4. Now suppose that the FED performs an open market operation and buys government bonds at the amount of 5. Find the new monetary base, the money supply and describe the new balance sheet of the commercial banks. MB = = 35 + cd + 0:2 M = MB = 5 = 20 rd + cd 0: + 0:2 D = MB = 5 = 6 2 rd + cd 0: + 0:2 3 rd 0: R = MB = 5 = 2 rd + cd 0: + 0:2 3 cd 0:2 CU = MB = 5 = 3 rd + cd 0: + 0:2 3

82 76 CHAPTER 6. MONEY AND PRICES Thus M = = 40 D = = 62 3 R = = 2 3 CU = = 23 3 Balance sheet of the commercial banks Assets Liabilities R = 2 D = B G = 20 L = The loans are simply set to balance the balance sheet. Notice that when the FED increased the monetary base by 5, the money supply increased by 20. This illustrates the fact that the FED does not have direct control over the money supply, but rather the commercial banks together with the FED create the money supply. 5. Suppose that instead of the open market operation, the FED sets the required reserve ratio to 5%. Find the new monetary base, money multiplier, the money supply and describe the new balance sheet of the commercial banks. The monetary base does not change, because the FED did not buy or sell any asset. MB = 30 mm = + cd rd + cd = + 0:2 0:05 + 0:2 = 4:8 + cd + 0:2 M = MB = 30 = 44 rd + cd 0:05 + 0:2 D = MB = 30 = 20 rd + cd 0:05 + 0:2 rd 0:05 R = MB = 30 = 6 rd + cd 0:05 + 0:2 cd 0:2 CU = MB = 30 = 24 rd + cd 0:05 + 0:2 Balance sheet of the commercial banks Assets Liabilities R = 6 D = 20 B G = 25 L =

83 6.4. ILLUSTRATION OF THE MONEY MULTIPLIER 77 Just to check that we did not make an error, lets verify that the consumers hold currency/deposits at the right ratio, and also that the banks hold reserves/deposits at the right ratio: CU D = = 0:2 R D = 6 20 = 0:05 As was mentioned before, the FED does not use the second type of policy (changing the required reserve ratio) frequently. One of the goals of the FED is to maintain a stable banking system and if the required reserve ratio changes, the banks have to adjust their loans and deposits in a complicated way. 6.4 Illustration of the Money Multiplier In this section we show in detail how an open market operation a ects the balance sheet of the commercial banks and the money supply. The following steps illustrate the working of the money multiplier when the FED buys bonds at the amount of x from the commercial banks. Balance Sheet of Banks 0. CU = 20 Assets Liabilities 3. +cd x +cd 0. R = 0 0. D = 00 ( rd)x 4. +cd (+cd) 2. +x 3. + x +cd 5. +cd ( rd)2 x ( rd)x 2. x 4. + (+cd) 3 (+cd) rd x +cd 5. + ( rd)2 x (+cd) 3 ( rd)x 4. +rd (+cd) rd ( rd)2 x (+cd) B G = 25. x.. 0. L = x ( rd)x cd 4. + ( rd)2 x (+cd) ( rd)3 x (+cd) 3..

84 78 CHAPTER 6. MONEY AND PRICES Each step is numbered. Step "0" is the initial balance sheet. It is important to make sure that the balance sheet is balanced after each step, i.e. assets are equal to liabilities. In step the FED buys bonds at the amount of x from the commercial banks. As a result, the bonds decreased by x and the reserves increased by x. Now after step, the commercial banks have too much reserves and they can lend the extra reserves to households. As a result of step 2 the reserves decreased by x while loans increased by x. In step 3, the loans at the amount of x are distributed between currency and deposits such that CU=D = cd and CU + D = x. Also, the commercial banks keep a fraction rd of the new deposits in reserves and lend the rest (a fraction rd of the new deposits). Observe that all the changes in step 3 in the balance sheet keep it balanced. The same is repeated in step 4, that is the extra loans are distributed between CU and D according to the currency/deposits ratio, and a fraction rd of the new deposits is kept in reserves while the rest is used for extra loans. This process continues inde nitely. In order to follow the above process more easily, color each step in a di erent color and make sure that the impact of each step on the balance sheet keeps it balanced. Suppose that the above steps continue forever. We can use the rule of summation of an in nite geometric series X q t = (0 < q < ) q to get the following results: D = = t=0 x + cd x + cd X t rd = + cd! = t=0 +cd +rd +cd cd CU = cd D = x rd + cd + cd M = CU + D = rd + cd rd R = rd D = x rd + cd x + cd rd + cd x rd +cd Thus, we derived all the multipliers in section 3 as the limit when t! of the sequence of loans and deposits generated by the open market operation. We can also compute all the magnitudes above after T steps. This is more realistic because in the real world the money circulates only limited number of times per year. In particular, the money velocity in 2005 is 9. Thus, we would like to compute the summations of the rst 9 steps. This can be done using the rule summation of a nite geometric series x! TX q t = t=0 qt + q

85 6.4. ILLUSTRATION OF THE MONEY MULTIPLIER 79 This gives D = = x + cd x + cd TX t rd = x + cd + cd! t=0 rd T + +cd +cd +rd +cd rd T + +cd rd +cd = rd +cd rd + cd T +! x! CU = cd D = cd rd T + +cd rd + cd M = CU + D = ( + cd) R = rd D = rd rd +cd rd + cd! x rd T + +cd T + rd + cd! For example, if the FED buys bonds at the amount of 5, then after 9 rounds of loans and deposits we have! D = CU = cd rd +cd rd + cd M = ( + cd) R = rd 9 5 rd 9 +cd rd + cd rd +cd! 5 rd 9 +cd rd + cd rd + cd! This completes the illustration of money creation by the FED and the commercial banks. 9 5! x 5! x

86 80 CHAPTER 6. MONEY AND PRICES

87 Chapter 7 Phillips Curve 7. Introduction In ation and unemployment are both considered undesirable for an economy. Why is in ation bad? High in ation leads to decline in the purchasing power of nominal assets, such as money and wages. It is also argued that in ation brings with it a lot of uncertainty about future prices since not all the prices tend to rise at the same rate. Therefore rms are having hard time planning its future production and how the particular prices of its inputs and production evolve relative to other prices. Fortunately for the U.S., we did not experience very high in ation rates, as can be seen in Figure 7..Some countries were not as Inflation Rate % Inflation Rate in U.S. Jan 47 Jan 5 Jan 55 Jan 59 Jan 63 Jan 67 Jan 7 Jan 75 Jan 79 Jan 83 Jan 87 Jan 9 Jan 95 Jan 99 Jan 03 Time Figure 7.: In ation in the U.S. 8

88 82 CHAPTER 7. PHILLIPS CURVE fortunate however. For example, Austria in had 0,000% annual in ation rate, and Argentina during had annual in ation rate of 20,000%. Can you imagine living in an economy where prices change by the minute. High unemployment is also considered undesirable for a number of reasons. The most important reason is the waste of resources, since unemployed workers do not produce output. There is also emotional cost for those who loose their job or stay unemployed for long periods of time. Figure 7.2 shows the unemployment rate in the U.S. since 948.Compared 2 Unemployment Rate in U.S. Unemployment Rate % Jan 47 Jan 5 Jan 55 Jan 59 Jan 63 Jan 67 Jan 7 Jan 75 Jan 79 Jan 83 Jan 87 Jan 9 Jan 95 Jan 99 Jan 03 Time Figure 7.2: Unemployment rate in the U.S. to other European countries during the same period, the unemployment rate in the U.S. is considered quite low. So both in ation and unemployment are undesirable. The search model of unemployment shed some light on the factors that determine the unemployment rate (e.g. separation rate, government policies about unemployment insurance bene ts, etc.). We also mentioned cyclical unemployment, which varies with the business cycle. So far however we did not suggest that In ation and Unemployment are related in any way. 7.2 Pillips Curve The New Zealand-born economist A.W. Phillips, in his 958 paper "The relationship between unemployment and the rate of change of money wages in the UK " published in Economica, observed an inverse relationship between money wage changes and unemployment in the British economy over the period examined. He concluded that government "demand" policies can move the economy along the curve, and thereby changing the level

89 7.2. PILLIPS CURVE 83 of unemployment in the economy. Today, what is known as "The Phillips Curve" is a graph that shows the relationship between in ation and unemployment over a particular period of time. The negative relationship between in ation and unemployment was observed in other countries during di erent periods. Figure 7.3 shows the Phillips Curve in the U.S. during the 60 s. We can see the during the 60 s there a signi cant negative relationship between in ation and unemployment was observed in the U.S. 7 Philips Curve ( ) y =.082x R 2 = Inflation Rate (annual %) Unemployment Rate (in %) Figure 7.3: Phillips curve in the U.S., 960 s During other periods in the U.S., the negative relationship between in ation and unemployment rate seem to weaken and sometimes even a positive relationship was observed. The next gures show the Phillips curve in the U.S. for the 70 s, 80 s, , and

90 84 CHAPTER 7. PHILLIPS CURVE 4 Philips Curve ( ) y = 0.245x R 2 = Inflation Rate (annual %) Unemployment Rate (in %) 4 Philips Curve ( ) y = 0.245x R 2 = Inflation Rate (annual %) Unemployment Rate (in %)

91 7.2. PILLIPS CURVE 85 4 Philips Curve ( ) y = x R 2 = Inflation Rate (annual %) Unemployment Rate (in %) 7 Philips Curve ( ) y = 0.205x R 2 = 0.03 Inflation Rate (annual %) Unemployment Rate (in %)

92 86 CHAPTER 7. PHILLIPS CURVE Inflation Rate (annual %) Philips Curve ( ) y = x R 2 = Unemployment Rate (in %) Observe that the negative relationship between in ation and unemployment that was observed during the 60 s breaks down during the subsequent decades. Moreover, for the entire period of (last gure), the observed relationship between in ation and unemployment is negative The impact of the Phillips curve on monetary policy Phillips article in 958 and subsequent empirical work on the Phillips curve convinced many economists and policy makers that there exists a stable trade-o between in ation and unemployment. Some economists modi ed the standard Keynesian model so that it would predict a negative trade-o between in ation and unemployment. Some of you may have seen in your principle class the so called AD-AS (aggregate demand, aggregate supply) model, illustrated in the next gure.

93 7.2. PILLIPS CURVE 87 P AS P 2 P AD Q Q 2 Q In this model expansionary policies ( scal or monetary) lead to an outward of the AD curve, and equilibrium output goes up, but at a cost of higher price level. The obvious implication of higher output is lower unemployment. Equipped with this model and the empirical evidence about the negative relationship between in ation and unemployment, some economists and policy makers concluded that it is possible to exploit the Phillips curve. In other words, they believed that there is a stable relationship between in ation and unemployment rate, and the policy makers have a choice of lowering unemployment at the cost of higher in ation, or having higher unemployment rate but with lower in ation. It was often argued that as long as in ation is not too high, it does not pose much danger to the economy, so it is better to su er some in ation as long as this would lead to lower unemployment. Central banks of some countries (e.g. Israel) printed money with the hope that higher in ation would lower unemployment and thereby boost the economy. In most cases, if not in all of them, the in ationary policies that attempted to lower unemployment resulted in higher in ation but without lowering unemployment. In some countries these policies led to hyper in ation, and the collapse of the local currency, banking system and ultimately led to high unemployment. Why did in ationary policies fail so miserably? The key to answering this question is expectations, which is the topic of the next section.

94 88 CHAPTER 7. PHILLIPS CURVE 7.3 Expectations-Augmented Phillips Curve (Edmund Phelps) Edmund Phelps did most of his important work on In ation and Unemployment during the late 60 s, when the most compelling evidence that supported the downward slopping Phillips curve was becoming available (see gure 7.3). Contrary to the conventional wisdom that the downward slopping Phillips curve represents a stable and exploitable trade-o between in ation and unemployment, Phelps argued that this is not the case in the long-run. Phelps introduce an important notion of expected in ation, which plays a key role in his argument. The expectation-augmented Phillips curve is given by: or t = e t (u t u n ) (7.) t e t = (u t u n ) (7.2) where t is the actual in ation in period t, e t is the period t expectation of t (or formally, e t = E t ( t )), u t is the unemployment at time t and u n is some natural rate of unemployment or NAIRU (Non-Accelerating In ation Rate of Unemployment). According to the extended Keynesian model mentioned above, > 0. Thus, if the monetary policy is such that the actual in ation is equal to the expected in ation, then the unemployment in that period is by de nition equal to the natural rate. That is if t = e t then u t = u n. To motivate this formulation, think of workers and employers who set work contracts based on their expectations of future in ation. The contracts are set in such a way that when they perfectly anticipate the future in ation, the labor markets are cleared (the only unemployment is the natural one). If for some reason people made an error in predicting the future in ation, then the unemployment di ers from the natural rate. Suppose that the realized in ation is greater than the expected one, i.e. t > e t. In this case we can see that the realized unemployment rate will fall below the natural rate, i.e. u t < u n. We need to make some assumption about the way people form their expectations about future in ation. The most simple assumption that we can make is backward-looking expectations. In words, this means that the expected in ation at time t is the realized in ation at time t. Formally, backward-looking expectations mean that e t = t. Phelps argument that the there is no long-run trade-o between in ation and unemployment can be easily illustrated with Figure 7.4. Suppose that initially the Phillips curve is the curve labeled e = and the actual in ation is initially. This curve represents the Phillips curve for expected in ation of. Since the actual in ation is equal to the expected in ation, the actual unemployment rate is equal to the natural rate. Thus, at the initial point the economy has in ation rate of and unemployment rate of u n. Suppose now that the central bank decided to increase the in ation to 2. In the short run people still expect the in ation to be (as it was in the last period), so the economy moves to the point labeled Short run, with higher in ation and lower unemployment. In the following period however, people s expectations will adjust and they will expect in ation to be 2. The Phillips curve will shift up and the new Phillips curve is the one labeled e = 2. At the long-run equilibrium, the unemployment rate returns to the natural rate u n but the in ation is higher 2.Thus, we The Pillips curve shifts up because e is part of the intercept.

95 7.3. EXPECTATIONS-AUGMENTED PHILLIPS CURVE (EDMUND PHELPS) 89 π t P Short run Long run π 2 Initial point π u 2 u n e π = π 2 e π = π Figure 7.4: Phelps Expectations-Augmented Philips curve u t u see that even when we make the very simplistic assumption that expectations are backwardlooking, the result of this model is that there can be at best a short-run trade-o between in ation and unemployment, but not in the long run. As we will see below, with more reasonable assumptions of expectation formation, Phelps result becomes even more robust The impact of the expectations-augmented Phillips curve on monetary policy Phelps insight and his emphasis of the expected in ation revolutionized the way monetary policy is conducted. He switched the discussion from the permanent trade-o between in ation and unemployment to discussion about intertemporal trade-o (between lowering unemployment now but su ering from high in ation in the future). The theoretical underpinnings for the policy of in ation targeting, which many central banks have adopted since the early 990s, are to a large extent derived from the framework developed in Phelps 967 paper. He demonstrated that expansionary monetary policy can lower in ation only in the short run, in the best case scenario of backward looking expectations. We will see in what follows that if we make more reasonable assumptions about expectation formation, then in ationary monetary policy becomes even less e ective and can fail boosting the economy even in the short run.

96 90 CHAPTER 7. PHILLIPS CURVE 7.4 Rational Expectation (Robert Lucas) Rational Expectations theory assumes that people use all the information available to them to predict the future in ation. According to the backward-looking expectations, people s prediction of future in ation is current in ation. This is similar to predict tomorrow s weather to be exactly like today s weather. When meteorologists predict tomorrows weather, they use a host of information other than yesterdays weather. Similarly, when people form expectations about future in ation they use the media, the FED announcements and all sorts of other information that is available. What are the implications of the rational expectations assumption on monetary policy? Looking back at Figure 7.4, suppose that the public is able to predict that the FED is planning to increase the in ation. Then the public s expectations will immediately become e = 2 and the Phillips curve will shift upward at the same period when the in ation increases from to 2. Thus, with rational expectations, in ationary policy will not have an impact on unemployment even in the short run. The economy will jump from the initial point to the long run equilibrium with the same unemployment and higher in ation. Lucas argument strengthened Phelps argument and implied that even in the short-run, the central banks would not be able to boost the economy with in ationary monetary policies Numerical example Suppose that =, u n = 5, and e t = t (backward-looking expectations).. Suppose that the FED creates in ation of % until period 3 and then increases the in ation permanently to 2%. Show the time path of in ation, expected in ation and unemployment from period on. Rewriting the expectations-augmented Phillips curve in equation (7.) gives t = e t (u t u n ) t e t = u t u n u t = u n + t e t u t = u n + e t t The next table shows the time paths.

97 7.4. RATIONAL EXPECTATION (ROBERT LUCAS) 9 2. Now suppose that the FED creates in ation of % until period 3 and then increases the in ation permanently to 2%, but this time people have rational expectations and they anticipate perfectly the increase in in ation. Show the time path of in ation, expected in ation and unemployment from period on. 3. Now suppose that the FED creates in ation of % until period 3 and then increases the in ation permanently to 2%, but this time people have almost rational expectations so they are expecting that in period 3 the FED will increase the in ation to :9%. After period 3 they learn that the in ation is 2%. Show the time path of in ation, expected in ation and unemployment from period on.

98 92 CHAPTER 7. PHILLIPS CURVE Notice that when we make the assumption of backward-looking expectations, then in ationary policy is e ective in reducing unemployment only in the short run. If expectations are rational, then in ationary policy is not e ective in reducing unemployment even in the short run. The last example showed that if expectations are almost rational, then the monetary policy has some e ect in the short run, but this is getting smaller the better the publics prediction of in ation is. 7.5 Credibility of Monetary Policy (Finn E. Kydland, Edward C. Prescott) Kydland and Prescott illustrated another problem with monetary policy - time inconsistency. To illustrate their argument, consider again the expectations-augmented Phillips curve. Notice that if the public has expectations for low in ation, then the central bank can reduce the unemployment rate by creating in ation. In other words, there is an incentive for the central bank (or government) to promise a low in ation rate in the next period, and when the public sets the wage contracts according to the low in ationary expectations, there a temptation for the central bank to break the promise and create high in ation. If economic peacemakers lack the ability to commit in advance to a speci c decision rule, they will often not implement the most desirable policy later on. Kydland and Prescott s results o ered a common explanation for events that, until then, had been interpreted as separate policy failures, e.g., that economies become trapped in high in ation even though price stability is the stated objective of monetary policy. Their work established the foundations for an extensive research program on the credibility and political feasibility of economic policy. This research shifted the practical discussion of economic policy away from isolated policy measures towards the institutions of policymaking, a shift that has largely in uenced the reforms of central banks and the design of monetary policy in many countries over the last decade. Because of the time inconsistency problem, the recommended policy by Kydland and Prescott is "Rules Rather Than Discretion" 2. That is, they recommended that the central 2 Kydland, Finn E & Prescott, Edward C. 977 "Rules Rather Than Discretion: The Inconsistency of

99 7.6. APPENDIX: ESTIMATING THE EXPECTATIONS-AUGMENTED PHILLIPS CURVE93 banks should commit to a simple rule (for example in ation targeting at certain level) and not attempt to exercise discretionary policy. When the central bank commits to a certain rule by law, this commitment is credible, while a promise without commitment is not. In other words, they recommended that the central banks should tie their hands and not conduct monetary policy which responds to economic events. 7.6 Appendix: Estimating the Expectations-Augmented Phillips Curve We want to test whether the assumption that > 0 holds in the data. For simplicity we assume that expectations are backward-looking, e t = t, i.e. we assume that people s expectation about in ation at time t is the time t in ation rate. Also, we can assume for simplicity that u n is the average unemployment rate over the sample period, say 5%. Let t t t denote the di erence in in ation rate between period t and t. The statistical relationship that we want to estimate is then t = (u t u n ) + " t (7.3) Having obtained the estimate for the slope (^), we test whether ^ < 0. When we run this regression, we need to impose the restriction that the constant is zero. Remark. The speci cation in equation (7.3) is not the same as Rearranging equation (7.3) gives t = 0 + u t + " t (7.4) t = u n + u t + " t Since u n = 5 is given, then the rst speci cation in eq. (7.3) is equivalent to the second, eq. (7.4), if we impose a restriction on the second speci cation that the intercept is equal to the slope times u n. In other words, the speci cation that is implied by the theory requires estimating equation (7.3), which is the same as estimating the constrained relationship t = 0 + u t + " t s:t: 0 = u n Optimal Plans", The Journal of Political Economy.

100 94 CHAPTER 7. PHILLIPS CURVE

101 Chapter 8 International Macroeconomics 8. Balance of Payments Almost all the economies are open economies and have interactions in trade and nance with other countries. These interactions are documented in the balance of payment account which records the country s trade with other countries in goods, services and assets. The balance of payments consists of three accounts: the current account, the nancial account, and the capital account. The next table shows the balance of payments for the U.S. in All the values are in millions of dollars. The current account records the country s exports and imports as well as income from investment and unilateral transfers. Any payments received by U.S. residents are positive numbers and any payments made by the U.S. residents are negative numbers. For example, when the U.S. companies export goods or services, they receive payments, which are recorded 95

102 96 CHAPTER 8. INTERNATIONAL MACROECONOMICS as positive entries in line 2 above. Similarly, when a U.S. resident receives dividends from a company he owns in a foreign country, those payments are recorded as positive entries in line 3 above. Conversely, when U.S. residents import goods and services from other countries, these payments are recorded as negative entries in line 5 and payments to foreigners who earn income from the U.S. are negative entries in line 6. Finally, unilateral transfers include foreign aid, grants, gifts, donations, etc. The nancial account records purchases of assets that a country has made abroad and foreign purchases of assets in the country. These assets include physical capital as well as nancial assets, such as shares of stock and bonds. When investors in the U.S. buys foreign assets such as foreign government bonds, or when a U.S. rm builds a factory in another country, these payments are capital out ow from the U.S. and are recorded as negative entries in line. The capital in ow into the U.S. occurs when a foreign investor buys a bond issued by a U.S. company or government, or when a foreign rm builds a factory in the U.S. When rms build or buy facilities in foreign countries they engage in foreign direct investment, while purchases of nancial assets are called foreign portfolio investment. The capital account is the part of the balance of payments that records relatively minor transactions such as migrants transfers when they cross borders and also sales and purchases of assets that are neither produced nor nancial assets, such as copyrights, patents, trademarks or right to natural resource. The balance of payments is always balanced, up to statistical discrepancy. That is, the sum of the balance on current account and the nancial and capital accounts must be zero. Notice that in 2005 the U.S. spent $79.5 more on goods and services and transfers than it received. This means that foreigners have accumulated $79.5 during 2005, which they either invested in the U.S. as purchases of assets or not spent at all. In the later case the non-spent amount is added to foreign holding of dollars, which is a positive entry in the nancial account in line Exchange Rates The balance of payments that we have seen in the previous section showed the summary of transactions between the U.S. and other countries in millions of dollars. However, when the U.S. residents are engaged in trade of goods, services and assets with other countries, these transactions involve other currencies. The price of one currency in term of another currency is called the nominal exchange rate. For example, the exchange rate between the dollar ($) and the British pound ($) can be expressed as how many pounds are required to buy one dollar: e $ $ = 0:5$ $ The most confusing part about the exchange rates is that they can be expressed also in another way, for example how many dollars are needed in order to buy one pound: e $ $ = 2 $ $ The notation $=$ reads "pounds per dollar" and $=$ reads "dollars per pound". The next table shows a few selected exchange rates expressed in both ways. The middle column shows

103 8.2. EXCHANGE RATES 97 the price of the U.S. dollar in units of other currencies and the third column shows the price of the foreign currencies in U.S. dollars. Example. Q. Based on the above table, what is the exchange rate between the dollar and the euro? A. We can say that the price of one U.S. dollar is 0.76 euro or the price of one euro is.3 dollars. Using our notation, the two ways of describing the exchange rate between the dollar and the euro are e e $ = 0:76 e $ or e $ e = :3 $ e In what follows, to avoid confusion, when we talk about exchange rate between the dollar and other currency, we will express it as the price of the dollar in units of that currency. For example, e e is the exchange rate between euro and the dollar expressed in euros per dollar, $ and e U is the exchange rate between the yen and the dollar, expressed in yens per dollar. $ 8.2. Using the exchange rates The exchange rates are useful for converting prices in one currency into another. Suppose the price of a television in the U.S. is $200 and we want to convert this price into Japanese