Macroeconomics Theory through Applications Chapter 16 Macroeconomics Toolkit

Chapter 16 Toolkit In this chapter , we present the key tools used in the part of this textbook . This toolkit serves two main functions . Because these tools appear in multiple chapters , the toolkit serves as a reference . When using a tool in one chapter , you can refer back to the toolkit to find a more concise description of the tool as well as links to other parts of the book where the tool is used . You can use the toolkit as a study guide . Once you have worked through the material in the chapters , you can review the tools using this toolkit . The chart below shows the main uses of each tool in green , and the secondary uses are in orange . URL books 613

The Labor Market The labor market is the market in which labor services are traded . Individual labor supply comes from the choices of individuals or households about how to allocate their time . As the real wage ( the nominal wage divided by the price level ) increases , households supply more hours to the market , and more households decide to participate in the labor market . Thus the quantity of labor supplied increases . The labor supply curve of a household is shifted by changes in wealth . A wealthier household supplies less labor at a given real wage . Labor demand comes from firms . As the real wage increases , the marginal cost of hiring more labor increases , so each demands fewer hours of labor is , a labor URL books 614

demand curve is downward sloping . The labor demand curve of a firm is shifted by changes in productivity . If labor becomes more productive , then the labor demand curve of a shifts rightward the quantity of labor demanded is higher at a given real wage . The labor market equilibrium is shown in Figure Labor Market Equilibrium . The real wage and the equilibrium quantity of labor traded are determined by the intersection of labor supply and labor demand . At the equilibrium real wage , the quantity of labor supplied equals the quantity of labor demanded . Figure Labor Real wage Labor supply Equilibrium real wage Labor demand Hours worked Key Insights . Labor supply and labor demand depend on the real wage . Labor supply is upward sloping as the real wage increases , households supply more hours to the market . Labor demand is downward sloping as the real wage increases , demand fewer hours of work . A market equilibrium is a real wage and a quantity of hours such that the quantity demanded equals the quantity supplied . URL books ( 615

The Main Uses of This Tool . Chapter The Interconnected Economy . Chapter Globalization and Competitiveness . Chapter Jobs in the . Chapter 11 Big and Small . Chapter 12 Income Taxes . Chapter 14 Balancing the Budget Choices over Time Individuals make decisions that unfold over time . Because individuals choose how to spend income earned over many periods on consumption goods over many periods , they sometimes wish to save or borrow rather than spend all their income in every period . Figure Choices over Time shows examples of these choices over a horizon . The individual earns income this year and next . The combinations of consumption that are affordable and that exhaust all of an individual income are shown on the budget line , which in this case is called an budget constraint . The slope of the budget line is equal to ( real interest rate ) which is equivalent to the real interest factor . The slope is the amount of consumption that can be obtained tomorrow by giving up a unit of consumption today . The preferred point is also indicated it is the combination of consumption this year and consumption next year that the individual prefers to all the points on the budget line . The individual in part ( a ) of Figure Choices over Time is consuming less this year than she is earning she is saving . Next year she can use her savings to consume more than her income . The individual in part ( of Figure Choices over Time is consuming more this year than he is earning he is borrowing . Next year , his consumption will be less than his income because he must repay the amount borrowed this year . When the real interest rate increases , individuals will borrow less and ( usually ) save more ( the effect of interest rate changes on saving is unclear as a matter of theory because income effects and substitution effects act in opposite directions ) Thus individual loan supply slopes upward . Of course , individuals live for many periods and make frequent decisions on consumption and saving . The lifetime budget constraint is obtained using the idea of discounted present value discounted present Value of lifetime income discounted present Value of lifetime consumption . The left side is a measure of all the disposable income the individual will receive over his lifetime ( disposable means after taking into account taxes paid to the government and transfers received from the government ) The right side calculates the value of consumption of all goods and services over an individual lifetime . URL books 516

Key Insights . Over a lifetime , an individual discounted present value of consumption will equal the discounted present value of income . Individuals can borrow or lend to obtain their preferred consumption bundle over their lifetimes . The price of borrowing is the real interest rate . Figure over Time ( mu sax ' um nu . mum , lu . um yea I Iu ' am . mu Ihn um you The Main Use of This Tool . Chapter 13 social Security Discounted Present Value Discounted present value is a technique used to add dollar amounts over time . We need this technique because a dollar today has a different value from a dollar in the future . The discounted present value this year of that you will receive next year is as follows nominal interest nominal interest rate If the nominal interest rate is 10 percent , then the nominal interest factor is , so next year is worth this year . As the interest rate increases , the discounted present value decreases . More generally , we can compute the value of an asset this year from the following formula flow from asset this of asset next year nominal interest factor ' value this year The depends on the asset . For a bond , the is a coupon payment . For a stock , the is a dividend payment . For a fruit tree , the is the yield of a crop . URL books 617

If an asset ( such as a bond ) yields a payment next year of 10 and has a price next year of 90 , then the benefit from asset price of the asset next year is 100 . The value of the asset this year is then If the nominal interest rate is 20 percent , then the value of the asset is . We discount nominal using a nominal interest factor . We discount real ( that is , already corrected for ) using a real interest factor , which is equal to ( real interest rate ) Key Insights . If the interest rate is positive , then the discounted present value is less than the direct sum of . If the interest rate increases , the discounted present value will decrease . More Formally Denote the dividend on an asset in period as . Define as the cumulative effect of interest rates up to period For example , Then the value of an asset that yields dollars in every year up to year is given by , If the interest rate is constant ( equal to ) then the one period interest factor is , and . The discounted present value tool is illustrated in Table Discounted Present Value with Different Interest Rates . The number of years ( is set equal to . The table gives the value of the dividends in each year and computes the discounted present values for two different interest rates . For this example , the annual interest rates are constant over time . Table Discounted Present Value with Different Interest Rates Year Dividend Discounted Discounted Present ( Present Value Value with with ( 100 100 100 100 90 120 400 Discounted present value The Main Uses of This Tool URL books 518

. Chapter Jobs in the . Chapter 10 Understanding the Fed . Chapter 13 Social Security . Chapter 14 Balancing the Budget . Chapter 15 The Global Financial Crisis The Credit ( Loan ) Market ( Macro ) Consider a simple example of a loan . Imagine you go to your bank to inquire about a loan of , to be repaid in one year time . A loan is a contract that specifies three things . The amount being borrowed ( in this example , The date ( at which repayment must be made ( in this example , one year from now ) The amount that must be repaid What determines the amount of the repayment ?

The a supplier of credit , and the a demander of credit . We use the terms credit and loans interchangeably . The higher the repayment amount , the more attractive this loan contract will look to the bank . Conversely , the lower the repayment amount , the more attractive this contract is to you . If there are lots of banks that are willing to supply such loans , and lots of people like you who demand such loans , then we can draw supply and demand curves in the credit ( loan ) market . The equilibrium price of this loan is the interest rate at which supply equals demand . In , we look at not only individual markets like this but also the credit ( loan ) market for an entire economy . This market brings together suppliers of loans , such as households that are saving , and of loans , such as businesses and households that need to borrow . The real interest rate is the price that brings demand and supply into balance . The supply of loans in the domestic loans market comes from three different sources . The private saving of households and . The saving of governments ( in the case of a government surplus ) The saving of foreigners ( when there is a of capital into the domestic economy ) Households will generally respond to an increase in the real interest rate by reducing current consumption relative to future consumption . Households that are saving will save more households that are borrowing will borrow less . Higher interest rates also encourage foreigners to send funds to the domestic economy . Government saving or borrowing is little affected by interest rates . The demand for loans comes from three different sources . The borrowing of households and firms to purchases , such as housing , durable goods , and investment goods . The borrowing of governments ( in the case of a government deficit ) URL books 619

. The borrowing of foreigners ( when there is a of capital from the domestic economy ) As the real interest rate increases , investment and durable goods spending decrease . For firms , a high interest rate represents a high cost of funding investment expenditures . This is an application of discounted present value and is evident if a borrows to purchase capital . It is also true if it uses internal funds ( retained earnings ) to investment because the firm could always put those funds into an asset instead . For households , higher interest rates likewise make it more costly to borrow to purchase housing and durable goods . The demand for credit decreases as the interest rate rises . When it is expensive to borrow , households and will borrow less . Equilibrium in the market for loans is shown in Figure The Credit Market . On the horizontal axis is the total quantity of loans in equilibrium . The demand curve for loans is downward sloping , whereas the supply curve has a positive slope . Loan market equilibrium occurs at the real interest rate where the quantity of loans supplied equals the quantity of loans demanded . At this equilibrium real interest rate , lenders lend as much as they wish , and borrowers can borrow as much as they wish . Equilibrium in the aggregate credit market is what ensures the balance of into and out of the financial sector in the circular diagram . Key Insights . As the real interest rate increases , more loans are supplied , and fewer loans are demanded . Adjustment of the real interest rate ensures that , in the circular diagram , the into the financial sector equal the from the sector . The Main Uses of This Tool . Chapter The Interconnected Economy . Chapter Money A User Guide . Chapter 10 Understanding the Fed . Chapter 11 Big and Small . Chapter 14 Balancing the Budget URL books

I Ira ! Loan ny ol Figure The ( Correcting for If you have some data expressed in nominal terms ( for example , in dollars ) and you want to convert them to real terms , you should use the following four steps . Select your . In most cases , the Consumer Price Index ( is the best to use . You can data on the ( for the United States ) at the Bureau of Labor Statistics website ( Select your base year . Find the value of the index in that base year . For all years ( including the base year ) divide the value of the index in that year by the value in the base year . The value for the base year is . For each year , divide the value in the nominal data series by the number you calculated in step . This gives you the value in base year Table Correcting Nominal Sales for shows an example . We have data on the for three years , as listed in the second column . The price index is created using the year 2000 as a base year , following steps . Sales measured in millions of dollars are given in the fourth column . To correct for , we divide sales in each year by the value of the price index for that year . The results are shown in the fifth column . Because there was each year ( the price index is increasing over time ) real sales do not increase as rapidly as nominal sales . Table Correcting Nominal Sales for Year Price Index Sales Real Sales ( Millions of ( 2000 Base ) Millions ) Year 2000 Dollars ) 2000 URL books 621

2001 2002 Source Bureau of Labor Statistics for the Consumer Price Index This calculation uses the , which is an example of a price index . To see how a price index like the is constructed , consider Table Constructing a Price Index , which shows a very simple economy with three goods , music downloads , and meals . The prices and quantities purchased in the economy in 2012 and 2013 are summarized in the table . Table Constructing a Price Index Year Music Meals Cost of Price Downloads 2013 Basket Price ( Quantity Price ( Quantity Price ( Quantity Price ( 2012 20 10 50 25 425 2013 22 12 60 26 442 To construct a price index , you must choose a fixed basket of goods . For example , we could use the goods purchased in 2013 ( 12 , 60 downloads , and meals ) This fixed basket is then priced in different years . To construct the cost of the 2013 basket at 2013 prices , the product of the price and the quantity purchased for each good in 2013 is added together . The basket costs 442 . Then we calculate the cost of the 2013 basket at 2012 prices that is , we use the prices of each good in 2012 and the quantities purchased in 2013 . The sum is 425 . The price index is constructed using 2012 as a base year . The Value of the price index for 2013 is the cost of the basket in 2013 divided by its cost in the base year ( 2012 ) When the price index is based on a bundle of goods that represents total output in an economy , it is called the price level . The and gross domestic product ( are examples of measures of the price level ( they differ in terms of exactly which goods are included in the bundle ) The growth rate of the price level ( its percentage change from one year to the next ) is called the rate . We also correct interest rates for . The interest rates you typically see quoted are in nominal terms they tell you how many dollars you will have to repay for each dollar you borrow . This is called a nominal interest rate . The real interest rate tells you how much you will get next year , in terms of goods and services , if you give up a unit of goods and services this year . To correct interest rates for , we use the Fisher equation real interest rate nominal interest rate rate . For more details , see Section The Fisher Equation Nominal and Real Interest Rates on the Fisher equation . Key Insights . Divide nominal values by the price index to create real values . URL books 622

. Create the price index by calculating the cost of buying a basket in different years . The Main Uses of This Tool . Chapter The State of the Economy . Chapter The Interconnected Economy . Chapter Globalization and Competitiveness . Chapter Money A User Guide . Chapter 11 Big and Small . Chapter 12 Income Taxes Supply and Demand The framework is the most fundamental framework in economics . It explains both the price of a good or a service and the quantity produced and purchased . The market supply curve comes from adding together the individual supply curves of firms in a particular market . A competitive , taking prices as given , will produce at a level such that price marginal cost . Marginal cost usually increases as a produces more output . Thus an increase in the price of a product creates an incentive for firms to produce is , the supply curve of a is upward sloping . The market supply curve slopes upward as well if the price increases , all firms in a market will produce more output , and some new firms may also enter the market . A supply curve shifts if there are changes in input prices or the state of technology . The market supply curve is shifted by changes in input prices and changes in technology that affect a significant number of the in a market . The market demand curve comes from adding together the individual demand curves of all households in a particular market . Households , taking the prices of all goods and services as given , distribute their income in a manner that makes them as well off as possible . This means that they choose a combination of goods and services preferred to any other combination of goods and services they can afford . They choose each good or service such that price marginal valuation . Marginal valuation usually decreases as a household consumes more of a product . If the price of a good or a service decreases , a household will substitute away from other goods and services and toward the product that has become is , the demand curve of a household is downward sloping . The market demand curve slopes downward as well if the price decreases , all households will demand more . The household demand curve shifts if there are changes in income , prices of other goods and services , or tastes . The market demand curve is shifted by changes in these factors that are common across a number of households . URL books 523

A market equilibrium is a price and a quantity such that the quantity supplied equals the quantity demanded at the equilibrium price ( Figure Market Equilibrium ) Because market supply is upward sloping and market demand is downward sloping , there is a unique equilibrium price . We say we have a competitive market if the following are true . The product being sold is homogeneous . There are many households , each taking the price as given . There are many , each taking the price as given . A competitive market is typically characterized by an absence of barriers to entry , so new firms can readily enter the market if it is profitable , and existing firms can easily leave the market if it is not . Key Insights . Market supply is upward sloping as the price increases , all firms will supply more . Market demand is downward sloping as the price increases , all households will demand less . A market equilibrium is a price and a quantity such that the quantity demanded equals the quantity supplied . Figure ( I ) URL books 624

Price of chocolate supply demand curve Figure Market Equilibrium shows equilibrium in the market for chocolate bars . The equilibrium price is determined at the intersection of the market supply and market demand curves . More Formally If we let denote the price , the quantity demanded , and I the level of income , then the market demand curve is given by , where a , and are constants . By the law of demand , For a normal good , the quantity demanded increases with income . If we let denote the quantity supplied and the level of technology , the market supply curve is given by , where , and are constants . Because the supply curve slopes upward , Because the quantity supplied increases when technology improves , In equilibrium , the quantity supplied equals the quantity demanded . Set and set in both equations . The market clearing price ( and quantity ( are as follows I ( URL books 625

and . The Main Uses of This Tool . Chapter The Interconnected Economy . Chapter Money A User Guide Comparative Advantage Comparative advantage explains why individuals and countries trade with each other . Trade is at the heart of modern economies individuals specialize in production and generalize in consumption . To consume many goods while producing relatively few , individuals must sell what they produce in exchange for the output of others . Countries likewise specialize in certain goods and services and import others . By so doing , they obtain gains from trade . Table Hours of Labor Required shows the productivity of two different countries in the production of two different goods . It shows the number of labor hours required to produce two and two countries and Mexico . From these data , Mexico has an absolute advantage in the production of both goods . Workers in Mexico are more productive at producing both tomatoes and beer in comparison to workers in . Table Hours of Labor Required ( Beer ( Kilogram ) Liter ) Mexico In , the opportunity cost of kilogram of tomatoes is liters of beer . To produce an extra kilogram of tomatoes in , hours of labor time must be taken away from beer production hours of labor time is the equivalent of liters of beer . In Mexico , the opportunity cost of kilogram of tomatoes is liter of beer . Thus the opportunity cost of producing tomatoes is lower in Mexico than in . This means that Mexico has a comparative advantage in the production of tomatoes . By a similar logic , has a comparative advantage in the production of beer . and Mexico can have higher levels of consumption of both beer and tomatoes if they trade rather than produce in isolation each country should specialize ( either partially or completely ) in the good in which it has a comparative advantage . It is never efficient to have both countries produce both goods . Key Insights . Comparative advantage helps predict the patterns of trade between individuals or countries . A country has a comparative advantage in the production of a good if the opportunity cost of producing that good is lower in that country . URL books 626

. Even if one country has an absolute advantage in all goods , it will still gain from trading with another country . Although this example is cast in terms of countries , the same logic is also used to explain production patterns between two individuals . The Main Use of This Tool . Chapter Jobs in the Comparative Statics Comparative statics is a tool used to predict the effects of exogenous variables on market outcomes . Exogenous variables shift either the market demand curve ( for example , news about the health effects of consuming a product ) or the market supply curve ( for example , weather effects on a crop ) By market outcomes , we mean the equilibrium price and the equilibrium quantity in a market . Comparative statics is a comparison of the market equilibrium before and after a change in an exogenous variable . A comparative statics exercise consists of a sequence of five steps . Begin at an equilibrium point where the quantity supplied equals the quantity demanded . Based on a description of the event , determine whether the change in the exogenous variable shifts the market supply curve or the market demand curve . Determine the direction of this shift . After shifting the curve , the new equilibrium point . Compare the new and old equilibrium points to predict how the exogenous event affects the market . Figure A Shift in the Demand Curve and Figure A Shift in the Supply Curve show comparative statics in action in the market for Curtis replica shirts and the market for beer . In Figure A Shift in the Demand Curve , the market demand curve has shifted leftward . The consequence is that the equilibrium price and the equilibrium quantity both decrease . The demand curve shifts along a fixed supply curve . In Figure A Shift in the Supply Curve , the market supply curve has shifted leftward . The consequence is that the equilibrium price increases and the equilibrium quantity decreases . The supply curve shifts along a demand curve . Key Insights . Comparative statics is used to determine the market outcome when the market supply and demand curves are shifting . Comparative statics is a comparison of equilibrium points . If the market demand curve shifts , then the new and old equilibrium points lie on a market supply curve . If the market supply curve shifts , then the new and old equilibrium points lie on a fixed market demand curve . URL books 627

Price of shins Market supply curve Old equilibrium price New Market price demand curve Quantity of shins New Old equilibrium equilibrium quantity quantity Figure Shift in the Demand Curve Figure A Shift in the Supply Curve URL books 111 628 Price of beer supply curve Equilibrium price demand curve Quantity of beer Equilibrium quantity The Main Uses of This Tool . Chapter The Interconnected Economy Chapter The Great Depression . Chapter 11 Big and Small Nash Equilibrium A Nash equilibrium is used to predict the outcome of a game . By a game , we mean the interaction of a few individuals , called players . Each player chooses an action and receives a that depends on the actions chosen by everyone in the game . A Nash equilibrium is an action for each player that two conditions . The action yields the highest payoff for that player given her predictions about the other players actions . The player predictions of others actions are correct . Thus a Nash equilibrium has two dimensions . Players make decisions that are in their own , and players make accurate predictions about the actions of others . Consider the games in Table Prisoners Dilemma , Table Dictator Game , Table Ultimatum Game , and Table Coordination Game . The numbers in the tables give URL books 629

the payoff to each player from the actions that can be taken , with the payoff of the row player listed first . Table Prisoners Dilemma Left Right Up , 10 Down 10 , Table Dictator Game Number of 100 , dollars ( Table Ultimatum Game Accept Reject Number of dollars ( 100 , Table Coordination Game URL books Left Right UP , Down , Prisoners dilemma . The row player chooses between the action labeled Up and the one labeled Down . The column player chooses between the action labeled the one labeled Right . For example , if row chooses Up and column , then the row player has a payoff of , and the column player has a payoff of 10 . If the row player predicts that the column player will choose Left , then the row player should choose Down ( that is , down for the row player is her best response to left by the column player ) From the column player perspective , if he predicts that the row player will choose Up , then the column player should choose Right . The Nash equilibrium occurs when the row player chooses Down and the column player chooses Right . Our two conditions for a Nash equilibrium of making optimal choices and predictions being right both hold . Social dilemma . This is a version of the prisoners dilemma in which there are a large number of players , all of whom face the same payoffs . Dictator game . The row player is called the dictator . She is given 100 and is asked to choose how many dollars ( to give to the column player . Then the game ends . Because the column player does not move in this game , the dictator game is simple to analyze if the dictator is interested in maximizing her payoff , she should offer nothing ( Ultimatum game . This is like the dictator game except there is a second stage . In the first stage , the row player is given 100 and told to choose how much to give to the column player . In the second stage , the column player accepts or rejects the offer . If the column player rejects the offer , neither player receives any money . The best choice of the row player is then to offer a penny ( the smallest amount of money there is ) The best choice of the column player is to accept . This is the Nash equilibrium . 630

Coordination game . The coordination game has two Nash equilibria . If the column player plays Left , then the row player plays Up if the row player plays Up , then the column player plays Left . This is an equilibrium . But is also a Nash equilibrium . Both players prefer , but it is possible to get stuck in a bad equilibrium . Key Insights . A Nash equilibrium is used to predict the outcome of games . In real life , payoffs may be more complicated than these games suggest . Players may be motivated by fairness or spite . More Formally We describe a game with three players ( but the idea straightforwardly to situations with any number of players . Each player chooses a strategy ( Suppose ( is the payoff to player if ( is the list of strategies chosen by the players ( and similarly for players and ) We put an asterisk ( to denote the best strategy chosen by a player . Then a list of strategies ( is a Nash equilibrium if the following statements are true ( In words , the first condition says that , given that players and are choosing their best strategies ( then player can do no better than to choose strategy . If a similar condition holds for every player , then we have a Nash equilibrium . The Main Uses of This Tool Chapter The Great Depression . Chapter 11 Big and Small . Chapter 15 The Global Financial Crisis Foreign Exchange Market A foreign exchange market is where one currency is traded for another . There is a demand for each currency and a supply of each currency . In these markets , one currency is bought using another . The price of one currency in terms of another ( for example , how many dollars it costs to buy one Mexican peso ) is called the exchange rate . Foreign currencies are demanded by domestic households , and governments that wish to purchase goods , services , or assets in the currency of another economy . For example , if a US auto importer wants to buy a German car , the importer must URL books 631

buy euros . The law of demand holds as the price of a foreign currency increases , the quantity of that currency demanded will decrease . Foreign currencies are supplied by foreign households , firms , and governments that wish to purchase goods , services , or financial assets in the domestic currency . For example , if a Canadian bank wants to buy a US government bond , the bank must sell Canadian dollars . As the price of a foreign currency increases , the quantity supplied of that currency increases . Exchange rates are determined just like other the interaction of supply and demand . At the equilibrium exchange rate , the supply and demand for a currency are equal . Shifts in the supply or the demand for a currency lead to changes in the exchange rate . Because one currency is exchanged for another in a foreign exchange market , the demand for one currency entails the supply of another . Thus the dollar market for euros ( where the price is dollars per euro and the quantity is euros ) is the mirror image of the euro market for dollars ( where the price is euros per dollar and the quantity is dollars ) To be concrete , consider the demand for and the supply of euros . The supply of euros comes from the following . European households and firms that wish to buy goods and services from countries that do not have the euro as their currency . European investors who wish to buy assets ( government debt , stocks , bonds , etc . that are in currencies other than the euro The demand for euros comes from the following . Households and in countries that wish to buy goods and services from Europe . Investors in countries that wish to buy assets ( government debt , stocks , bonds , etc . that are in euros Figure The Foreign Exchange Market shows the dollar market for euros . On the horizontal axis is the quantity of euros traded . On the vertical axis is the price in terms of dollars . The intersection of the supply and demand curves determines the equilibrium exchange rate . URL books 632

Exchange NI ?

Supply of euros Equilibrium exchange rate Demand for euros Quantity Equilibrium of euros traded Figure The Foreign Market The foreign exchange market can be used as a basis for comparative statics exercises . We can study how changes in an economy affect the exchange rate . For example , suppose there is an increase in the level of economic activity in the United States . This leads to an increase in the demand for European goods and services . To make these purchases , US households and will demand more euros . This causes an outward shift in the demand curve and an increase in the dollar price of euros . When the dollar price of a euro increases , we say that the dollar has relative to the euro . From the perspective of the euro , the depreciation of the dollar represents an appreciation of the euro . URL books ' 633

Key Insight . As the exchange rate increases ( so a currency becomes more valuable ) a greater quantity of the currency is supplied to the market and a smaller quantity is demanded . The Main Uses of This Tool . Chapter The Interconnected Economy . Chapter The Great Depression . Chapter Money A User Guide Chapter 10 Understanding the Fed Growth Rates If some Variable ( for example , the number of gallons of gasoline sold in a week ) changes from to , then we can define the change in that variable as Ax . But there are with this simple . The number that we calculate will change , depending on the units in which we measure . If we measure in millions of gallons , will be a much smaller number than if we measure in gallons . If we in liters rather than gallons ( as it is measured in most countries ) it would be a bigger number . So the number we calculate depends on the units we choose . To avoid these problems , we look at percentage changes and express the change as a fraction of the individual value . In what follows , we use the notation Ax to mean the percentage change in and it as follows Ax ( A percentage change equal to means that gasoline consumption increased by 10 percent . Why ?

Because 10 percent means 10 per hundred , so 10 percent . Very often in economics , we are interested in changes that take place over time . Thus we might want to compare gross domestic product ( between 2012 and 2013 . Suppose we know that in the United States in 2012 was 14 trillion and that in 2013 was trillion . Using the letter denote measured in trillions , we write and . If we want to talk about at different points in time without specifying a particular year , we use the notation . We express the change in a variable over time in the form of a growth rate , which is just an example of a percentage change . Thus the growth rate of in 2013 is calculated as follows ( 14 ) The growth rate equals percent . In general , we write ( Occasionally , we use the gross growth rate , which simply equals the growth rate . So , for example , the gross growth rate of equals , or . There are some useful rules that describe the behavior of percentage changes and growth rates . The Product Rule . Suppose we have three variables , and , and suppose . URL books 634

Then Ax Ay Az . In other words , the growth rate of a product of two variables equals the sum of the growth rates of the individual variables . The Quotient Rule . Now suppose we rearrange our original equation by dividing both sides by to obtain If we take the product rule and subtract Az from both sides , we get the following Ay Ax Az . The Power Rule . There is one more rule of growth rates that we make use of in some advanced topics , such as growth accounting . Suppose that a . Then Ay a ( Ax ) For example , if , then the growth rate of is twice the growth rate of . If , then the growth rate of is half the growth rate of ( remembering that a square root is the same as a power of ) More Formally Growth rates compound over time if the growth rate of a variable is constant , then the change in the variable increases over time . For example , suppose in 2020 is , and it grows at 10 percent per year . Then in 2021 , is ( an increase of ) but in 2022 , is ( an increase of ) If this compounding takes place every instant , then we say that we have exponential growth . Formally , we write exponential growth using the number If the value of Yat time equals and if at the constant rate ( where is an or per year growth rate ) then at time ( measured in years ) A version of this formula can also be used to calculate the average growth rate of a variable if we know its value at two different times . We can write the formula as URL books 635

, which also means ( where ( is the natural logarithm . You do not need to know exactly what this means you can simply calculate a logarithm using a scientific calculator or a spreadsheet . Dividing by get the average growth rate ( For example , suppose in 2020 is and in 2030 is . Then . Using a calculator , we can find ( Dividing by 10 ( since the two dates are 10 years apart ) we get an average growth rate of , or percent per year . The Main Uses of This Tool . Chapter The State of the Economy . Chapter Globalization and Competitiveness . Chapter Global Prosperity and Global Poverty . Chapter The Great Depression . Chapter Money A User Guide Chapter 11 Big and Small Mean and Variance To start our presentation of descriptive statistics , we construct a data set using a spreadsheet program . The idea is to simulate the of a coin . Although you might think it would be easier just to a coin , doing this on a spreadsheet gives you a full range of tools embedded in that program . To generate the data set , we drew 10 random numbers using the spreadsheet program . In the program we used , the function was called RAND and this generated the choice of a number between zero and one . Those choices are listed in the second column of Table . The third column creates the two events of heads and tails that we normally associate with a coin . To generate this last column , we adopted a rule if the random number was less than , we termed this a tail and assigned a to the draw otherwise we termed it a head and assigned a to the draw . The choice of as the cutoff for heads the fact that we are considering the of a fair coin in which each side has the same probability . Table Draw Heads ( or Number Tails ( URL books 535

10 Keep in mind that the realization of the random number in draw i is independent of the realizations of the random numbers in both past and future draws . Whether a coin comes up heads or tails on any particular does not depend on other outcomes . There are many ways to summarize the information contained in a sample of data . Even before you start to compute some complicated statistics , having a way to present the data is important . One possibility is a bar graph in which the fraction of observations of each outcome is easily shown . Alternatively , a pie chart is often used to display this fraction . Both the pie chart and the bar diagram are commonly found in spreadsheet programs . Economists and statisticians often want to describe data in terms of numbers rather than figures . We use the data from the table to and illustrate two statistics that are commonly used in economics discussions . The first is the mean ( or average ) and is a measure of central tendency . Before you read any further , ask , What do you think the average ought to be from the coin exercise ?

It is natural to say , since half the time the outcome will be a head and thus have a value of zero , whereas the remainder of the time the outcome will be a tail and thus have a value of one . Whether or not that guess holds can be checked by looking at Table and calculating the mean of the outcome . We let ki be the outcome of draw i . For example , from the table , and . Then the formula for the mean if there are draws is . Here means the sum of the ki outcomes . In words , the mean , denoted by , is calculated by adding together the draws and dividing by the number of draws ( In the table , 10 , and the sum of the draws of random numbers is about . Thus the mean of the 10 draws is about . We can also calculate the mean of the column , which is since heads came up times in our experiment . This calculation of the mean differs from the mean of the draws since the numbers in the two columns differ with the third column being a very discrete way to represent the information in the second column . A second commonly used statistic is a measure of dispersion of the data called the variance . The variance , denoted 02 , is calculated as ( ki ) From this formula , if all the draws were the same ( thus equal to the mean ) then the variance would be zero . As the draws spread out from the mean ( both above and below ) the variance increases . Since some observations are above the mean and others below , we square the difference between a single observation ( ki ) and the mean ( when calculating the variance . This means that values URL books 637

above and below the mean both contribute a positive amount to the variance . Squaring also means that values that are a long way away from the mean have a big effect on the variance . For the data given in the table , the mean of the 10 draws was given as . So to calculate the variance , we would subtract the mean from each draw , square the difference , and then add together the squared differences . This yields a variance of for this draw . A closely related concept is that of the standard deviation , which is the square root of the variance . For our example , the standard deviation is . The standard deviation is greater than the variance since the variance is less than . The Main Uses of This Tool Chapter Globalization and Competitiveness Chapter The Great Depression Chapter 11 Big and Small Correlation and Causality Correlation is a statistical measure describing how two variables move together . In contrast , causality ( or causation ) goes deeper into the relationship between two variables by looking for cause and effect . Correlation is a statistical property that summarizes the way in which two variables move either over time or across people ( firms , governments , The concept of correlation is quite natural to us , as we often take note of how two variables interrelate . If you think back to high school , you probably have a sense of how your classmates did in terms of two measures of performance grade point average ( GPA ) and the results on a standardized college entrance exam ( SAT or ACT ) It is likely that classmates with high also had high scores on the SAT or ACT exam . In this instance , we would say that the GPA and scores were positively correlated looking across your classmates , when a person GPA is higher than average , that person SAT or ACT score is likely to be higher than average as well . As another example , consider the relationship between a household income and its expenditures on housing . If you conducted a survey across households , it is likely that you would that richer households spend more on most goods and services , including housing . In this case , we would conclude that income and expenditures on housing are positively correlated . When economists look at data for a whole economy , they often focus on a measure of how much is produced , which we call real gross domestic product ( real ) and the fraction of workers without jobs , called the unemployment rate . Over long periods of time , when is above average ( the economy is doing well ) the unemployment rate is below average . In this case , and the unemployment rate are negatively correlated , as they tend to move in opposite directions . The fact that one variable is correlated with another does not inform us about whether one variable causes the other . Imagine yourself on an airplane in a relaxed mood , reading or listening to music . Suddenly , the pilot comes on the public address system and requests that URL books 638

you buckle your seat belts . Usually , such a request is followed by turbulence . This is a correlation the announcement by the pilot is positively correlated with air turbulence . The correlation is of course not perfect because sometimes you hit some bumps without warning , and sometimes the pilot announcement is not followed by turbulence . does not mean that we could solve the turbulence problem by turning off the public address system . The pilot announcement does not cause the turbulence . The turbulence is there whether the pilot announces it or not . In fact , the causality runs the other way . The turbulence causes the pilot announcement . We noted earlier that real and unemployment are negatively correlated . When real is below average , as it is during a recession , the unemployment rate is typically above average . But what is the causality here ?

If unemployment caused , we might be tempted to adopt a policy that makes unemployment illegal . For example , the government could fine firms if they lay off workers . This is not a good policy because we do not think that low unemployment causes high real . Neither do we necessarily think that high real causes low unemployment . Instead , based on economic theory , there are other that affect both real and unemployment . More Formally Suppose you have observations of two variables , and , where xi and yi are the values of these variables in observation i , The mean of , denoted , is the sum over the values of in the sample is divided by the scenario applies for . pix and We can also calculate the variance and standard deviations of and . The calculation for the variance of , denoted , is as follows 52 ( The standard deviation of is the square root of ( ix ) With these ingredients , the correlation of ( denoted corr ( is given by ! 12 ) The Main Uses of This Tool Chapter The Great Depression . Chapter 11 Big and Small Chapter 14 Balancing the Budget URL books 639

The Fisher Equation Nominal and Real Interest Rates When you borrow or lend , you normally do so in dollar terms . If you take out a loan , the loan is in dollars , and your promised payments are in dollars . These dollar must be corrected for to calculate the repayment in real terms . A similar point holds if you are a lender you need to calculate the interest you earn on saving by correcting for . The Fisher equation provides the link between nominal and real interest rates . To convert from nominal interest rates to real interest rates , we use the following formula real interest rate nominal interest rate rate . To the real interest rate , we take the nominal interest rate and subtract the rate . For example , if a loan has a 12 percent interest rate and the rate is percent , then the real return on that loan is percent . In calculating the real interest rate , we used the actual rate . This is appropriate when you wish to understand the real interest rate actually paid under a loan contract . But at the time a loan agreement is made , the rate that will occur in the future is not known with certainty . Instead , the borrower and lender use their expectations of future to determine the interest rate on a loan . From that perspective , we use the following formula contracted nominal interest rate real interest rate expected rate . We use the term contracted nominal interest rate to make clear that this is the rate set at the time of a loan agreement , not the realized real interest rate . Key Insight . To correct a nominal interest rate for , subtract the rate from the nominal interest rate . More Formally Imagine two individuals write a loan contract to borrow dollars at a nominal interest rate of i . This means that next year the amount to be repaid will be ( i ) This is a standard loan contract with a nominal interest rate of i . Now imagine that the individuals decided to write a loan contract to guarantee a constant real return ( in terms of goods not dollars ) denoted So the contract provides this year in return for being repaid ( enough dollars to buy ) units of real gross domestic product ( real ) next year . To repay this loan , the borrower gives the lender enough money to buy ( units of real for each unit of real that is lent . So if the rate is 91 , then the price level has risen to ( 21 ) so the repayment in dollars for a loan of dollars would be ( 31 ) URL books

Here ( 31 ) is one plus the rate . The rate is the percentage change in the price level from period to period . If a period is one year , then the price level next year is equal to the price this year multiplied by ( The Fisher equation says that these two contracts should be equivalent ( As an approximation , this equation implies i 31 . To see this , multiply out the side and subtract from each side to obtain ' 31 ' If and TI are small numbers , then is a very small number and can safely be ignored . For example , if and , then , and our approximation is about 99 percent accurate . The Main Uses of This Tool . Chapter The Interconnected Economy . Chapter Money A User Guide . Chapter 10 Understanding the Fed Chapter 11 Big and Small The Aggregate Production Function The aggregate production function describes how total real gross domestic product ( real ) in an economy depends on available inputs . Aggregate output ( real ) depends on the following . Physical , production facilities , and so forth that are used in production . number of hours that are worked in the entire economy . Human and education embodied in the workforce of the economy . scientific knowledge , and blueprints that describe the available production processes . Social general business , legal and cultural environment . The amount of natural resources available in an economy . Anything else that we have not yet included URL books

We group the inputs other than labor , physical , and human capital together , and call them technology . The aggregate production function has several key properties . First , output increases when there are increases in physical capital , labor , and natural resources . In other words , the marginal products of these inputs are all positive . Second , the increase in output from adding more inputs is lower when we have more of a factor . This is called diminishing marginal product . That is , The more capital we have , the less additional output we obtain from additional capital . The more labor we have , the less additional output we obtain from additional labor . The more natural resources we have , the less additional output we obtain from additional resources . In addition , increases in output can also come from increases in human capital , knowledge , and social infrastructure . In contrast to capital and labor , we do not assume that there are diminishing returns to human capital and technology . One reason is that we do not have a natural or an obvious measure for human capital , knowledge , or social infrastructure , whereas we do for labor and capital ( hours of work and hours of capital usage ) Figure shows the relationship between output and capital , holding fixed the level of other inputs . This shows two properties of the aggregate production function . As capital input is increased , output increases as well . But the change in output obtained by increasing the capital stock is lower when the capital stock is higher this is the diminishing marginal product of capital . Figure URL books 642

Physical capital In many applications , we want to understand how the aggregate production function responds to variations in the technology or other inputs . This is illustrated in Figure . An increase in , say , technology means that for a given level of the capital stock , more output is produced the production function shifts upward as technology increases . Further , as technology increases , the production function is steeper the increase in technology increases the marginal product of capital . URL books 643

Figure Technology , human . next year and workforce Output next year this year Technology . human capital . and workforce this year stock Capital stock Key Insight . The aggregate production function allows us to determine the output of an economy given inputs of capital , labor , human capital , and technology . More Formally Forms for the Production Function We can write the production function in mathematical form . We use represent real , to represent the physical capital stock , to represent labor , to represent human capital , andA to represent technology ( including natural resources ) If we want to speak about production completely generally , then we can write ( A ) Here ( means some function A lot of the time , economists work with a production function that has a specific mathematical form , yet is still reasonably simple Ka ( where a is just a number . This is called a production function . It turns out that this production function does a remarkably good job of summarizing aggregate URL books 644

production in the economy . In fact , we also know that we can describe production in actual economies quite well if we suppose that a . The Main Uses of This Tool Chapter Globalization and Competitiveness . Chapter Global Prosperity and Global Poverty Chapter The Great Depression The Circular Flow of Income The circular of income describes the of money among the main sectors of an economy . As individuals and buy and sell goods and services , money among the different sectors of an economy . The circular of income describes these of dollars ( pesos , euros , or whatever ) From a simple version of the circular , we learn a matter of gross domestic product ( income production spending . This relationship lies at the heart of analysis . There are two sides to every transaction . Corresponding to the of money in the circular , there are of goods and services among these sectors . For example , the wage income received by consumers is in return for labor services that from households to . The consumption spending of households is in return for the goods and services that from firms to households . A complete version of the circular is presented in Figure . Chapter The State of the Economy contains a discussion of a simpler version of the circular with only two sectors households and . Figure URL books 545

' Transfers . purchases ( Total income I production Disposable Income ( from plus . Imports , A , The complete circular has five sectors a household sector , a sector , a government sector , a foreign sector , and a financial sector . Different chapters of the book emphasize different pieces of the circular , and Figure shows us how everything fits together . In the following subsections , we look at the into and from each sector in turn . In each case , the balance of the into and from each sector underlies a useful economic relationship . The Firm Sector Figure includes the component of the circular associated with the into and from the firm sector of an economy . We know that the total of dollars from the sector measures the total value of production in an economy . The total of dollars into the firm sector equals total expenditures on . We therefore know that production consumption investment government purchases net exports . This equation is called the national income identity and is the most fundamental relationship in the national accounts . By consumption we mean total consumption expenditures by households on final goods and services . Investment refers to the purchase of goods and services that , in one way or another , help to produce more output in the future . Government purchases include all purchases of goods and services by the government . Net exports , which equal exports minus imports , measure the expenditure associated with the rest of the world . The Household Sector The household sector summarizes the behavior of private individuals in their roles as savers and suppliers of labor . The balance of into and from this sector is the basis of the household budget constraint . Households receive income from , in the form of wages and in the form of dividends resulting from their ownership of firms . The income URL books 646

that households have available to them after all taxes have been paid to the government and all transfers received is called disposable income . Households spend some of their disposable income and save the rest . In other words , disposable income consumption household savings . This is the household budget constraint . In Figure , this equation corresponds to the fact that the into and from the household sector must balance . The Government Sector The government sector summarizes the actions of all levels of government in an economy . Governments tax their citizens , pay transfers to them , and purchase goods from the sector of the economy . Governments also borrow from or lend to the sector . The amount that the government collects in taxes need not equal the amount that it pays out for government purchases and transfers . If the government spends more than it gathers in taxes , then it must borrow from the financial markets to make up the shortfall . The circular shows two into the government sector and two out . Since the into and from the government sector must balance , we know that government purchases transfers tax revenues government borrowing . Government borrowing is sometimes referred to as the government budget . This equation is the government budget constraint . Some of the in the circular can go in either direction . When the government is running a , there is a of dollars to the government sector from the markets . Alternatively , the government may run a surplus , meaning that its revenues from taxation are greater than its spending on purchases and transfers . In this case , the government is saving rather than borrowing , and there is a of dollars to the financial markets from the government sector . The Foreign Sector The circular includes a country dealings with the rest of the world . These include exports , imports , and borrowing from other countries . Exports are goods and services produced in one country and purchased by households , and governments of another country . Imports are goods and services purchased by households , and governments in one country but produced in another country . Net exports are exports minus imports . When net exports are positive , a country is running a trade surplus exports exceed imports . When net exports are negative , a country is running a trade deficit imports exceed exports . The third between countries is borrowing and lending . Governments , individuals , and in one country may borrow from or lend to another country . Net exports and borrowing are linked . If a country runs a trade deficit , it borrows from other countries to finance that . If we look at the into and from the foreign sector , we see that URL books 547

borrowing from other countries exports imports . Subtracting exports from both sides , we obtain borrowing from other countries imports exports trade . Whenever our economy runs a trade , we are borrowing from other countries . If our economy runs a trade surplus , then we are lending to other countries . This analysis has omitted one detail . When we lend to other countries , we acquire their assets , so each year we get income from those assets . When we borrow from other countries , they acquire our assets , so we pay them income on those assets . Those income are added to the trade to give the current account of the economy . It is the current account that must be matched by borrowing from or lending to other countries . A positive current account means that net exports plus net income from the rest of the world are positive . In this case , our economy is lending to the rest of the world and acquiring more assets . The Financial Sector The sector of an economy summarizes the behavior of banks and other financial institutions . The balance of into and from the financial sector tell us that investment is financed by national savings and borrowing from abroad . The financial sector is at the heart of the circular . The figure shows four into and from the sector . Households divide their income between consumption and savings . Thus any income that they receive today but wish to put aside for the future is sent to the financial markets . The household sector as a whole saves so , on net , there is a of dollars from the household sector into the markets . The of money from the sector into the firm sector provides the funds that are available to for investment purposes . The of dollars between the sector and the government sector the borrowing ( or lending ) of governments . The can go in either direction . When government expenditures exceed government revenues , the government must borrow from the private sector , and there is a of dollars from the sector to the government . This is the case of a government deficit . When the government revenues are greater than its expenditures , by contrast , there is a government surplus and a of dollars into the financial sector . The of dollars between the financial sector and the foreign sector can also go in either direction . An economy with positive net exports is lending to other countries there is a of money from an economy . An economy with negative net exports ( a trade ) is borrowing from other countries . The national savings of the economy is the savings carried out by the private and government sectors taken together . When the government is running a deficit , some of the savings of households and must be used to fund that , so there is less left over to finance investment . National savings is then equal to private savings minus the government is , private savings minus government borrowing national savings private savings government borrowing . URL books 548

If the government is running a surplus , then national savings private savings government surplus . National savings is therefore the amount that an economy as a whole saves . It is equal to what is left over after we subtract consumption and government spending from . To see this , notice that private savings government borrowing income taxes transfers consumption ( government purchases transfers taxes ) income consumption government purchases . This is the domestic money that is available for investment . If we are borrowing from other countries , there is another source of funds for investment . The into and from the sector must balance , so investment national savings borrowing from other countries . Conversely , if we are lending to other countries , then our national savings is divided between investment and lending to other countries national savings investment lending to other countries . The Main Uses of This Tool . Chapter The State of the Economy . Chapter The Interconnected Economy . Chapter Global Prosperity and Global Poverty . Chapter The Great Depression . Chapter Money A User Guide . Chapter 11 Big and Small . Chapter 12 Income Taxes . Chapter 14 Balancing the Budget . Chapter 15 The Global Financial Crisis Growth Accounting Growth accounting is a tool that tells us how changes in real gross domestic product ( real ) in an economy are due to changes in available capital , labor , human capital , and technology . Economists have shown that , under reasonably general circumstances , the change in output in an economy can be written as follows output growth rate a capital stock growth rate ( a ) labor hours growth rate ( a ) human capital growth rate technology growth rate . In this equation , a is just a number . For example , if a , the growth in output is as follows URL books 649

output growth rate ( capital stock growth rate ) labor hours growth rate ) human capital growth rate ) technology growth rate . Growth rates can be positive or negative , so we can use this equation to analyze decreases in as well as increases . This expression for the growth rate of output , by the way , is obtained by applying the rules of growth rates ( discussed in Section Growth Rates ) to the aggregate production function ( discussed in Section The Aggregate Production Function ) What can we measure in this expression ?

We can measure the growth in output , the growth in the capital stock , and the growth in labor hours . Human capital is more difficult to measure , but we can use information on schooling , literacy rates , and so forth . We can not , however , measure the growth rate of technology . So we use the growth accounting equation to infer the growth in technology from the things we can measure . Rearranging the growth accounting equation , technology growth rate output growth rate ( a capital stock growth rate ) labor hours growth rate ( a ) human capital growth rate . So if we know the number a , we are can use measures of the growth in output , labor , capital stock , and human capital to solve for the technology growth rate . In fact , we do have a way of measuring a . The technical details are not important here , but a good measure of ( a ) is simply the total payments to labor in the economy ( that is , the total of wages and other compensation ) as a fraction of overall . For most economies , a is in the range of about to . Key Insight . The growth accounting tool allows us to determine the contributions of the various factors of economic growth . The Main Uses of This Tool Chapter Globalization and Competitiveness . Chapter Global Prosperity and Global Poverty Chapter The Great Depression The Growth Model The analysis in Chapter Global Prosperity and Global Poverty is ( implicitly ) based on a theory of economic growth known as the growth model . Here we present two formal versions of the mathematics of the model . The takes as its focus the capital accumulation equation and explains how the capital stock evolves in the economy . This version ignores the role of human capital and ignores the growth path of the economy . The second follows the exposition of the chapter and is based around the derivation of the balanced growth path . They are , however , simply two different ways of approaching the same problem . Presentation URL 650

There are three components of this presentation of the model technology , capital accumulation , and saving . The first component of the growth model is the specification of technology and comes from the aggregate production function . We express output per worker ( as a function of capital per worker ( and technology ( A ) A mathematical expression of this relationship is ) where ) means that output per worker depends on capital per worker . As in our presentation of production functions , output increases with technology . We assume ) has the properties that more capital leads to more output per capita at a diminishing rate . As an example , suppose . In this case the marginal product of capital is positive but diminishing . The second component is capital accumulation . If we let be the amount of capital per capita at the start of year , then we know that ( it . This expression shows how the capital stock changes over time . Here is the rate of physical depreciation so that between year and year , units of capital are lost from depreciation . But during year , there is investment ( it ) that yields new capital in the following year . The final component of the growth model is saving . In a closed economy , saving is the same as investment . Thus we link it in the accumulation equation to saving . Assume that saving per capita ( st ) is given by . Here is a constant between zero and one , so only a fraction of total output is saved . Using the fact that savings equals investment , along with the per capita production function , we can relate investment to the level of capital it sAf ( We can then write the equation for the evolution of the capital stock as follows ( sAf ( Once we have the function ( we can follow the evolution of the capital stock over time . Generally , the path of the capital stock over time has two important properties . Steady state . There is a particular level of the capital stock such that if the economy accumulates that amount of capital , it stays at that level of capital . We call this the steady state level of capital , denoted . URL books 651

. Stability . The economy will tend toward the per capita capital stock . To be more , the steady state level of capital solves the following equation ( sAf ( At the steady state , the amount of capital lost by depreciation is exactly offset by saving . This means that at the steady state , net investment is exactly zero . The property of stability means that if the current capital stock is below , the economy will accumulate capital so that . And if the current capital stock is above , the economy will capital so that . If two countries share the same technology ( A ) and the same production function ( then over time these two countries will eventually have the same stock of capital per worker . If there are differences in the technology or the production function , then there is no reason for the two countries to converge to the same level of capital stock per worker . Presentation In this presentation , we explain the path of the economy and prove some of the claims made in the text . The model takes as given ( exogenous ) the investment rate the depreciation rate and the growth rates of the workforce , human capital , and technology . The endogenous variables are output and physical capital stock . The notation for the presentation is given in Table Notation in the Growth A Model We use the notation to represent the growth rate of a variable that is , There are two key ingredients to the model the aggregate production function and the equation for capital accumulation . Table Notation in the Growth Model Variable Symbol Real gross domestic product Capital stock Human capital Workforce Technology A Investment rate Depreciation rate The Production Function The production function we use is the production function Equation Ka ( URL books 652

Growth Accounting If we apply the rules of growth rates to Equation , we get the following expression Equation ( a ) gA . Balanced Growth The condition for balanced growth is that . When we impose this condition on our equation for the growth rate of output ( Equation ) we get ( where the superscript indicates that we are considering the values of variables when the economy is on a balanced growth path . This equation to Equation ( gA . The growth in output on a path depends on the growth rates of the workforce , human capital , and technology . Using this , we can rewrite Equation as follows Equation . The actual growth rate in output is an average of the rate of output and the growth rate of the capital stock . Capital Accumulation The second piece of our model is the capital accumulation equation . The growth rate of the capital stock is given by Equation Divide the numerator and denominator of the first term by , remembering that i I . Equation i , The growth rate of the capital stock depends positively on the investment rate and negatively on the depreciation rate . It also depends negatively on the current ratio . URL books 653

The Ratio Now rearrange Equation to give the ratio of capital to gross domestic product ( given the depreciation rate , the investment rate , and the growth rate of the capital stock When the economy is on a balanced growth path , so . 36 We can also substitute in our expression for ( Equation ) to get an expression for the capital output ratio in terms of exogenous variables . as I ( Convergence The proof that economies will converge to the ratio of capital to is 45 relatively straightforward . We want to show that if I , then capital grows faster than output . If capital is growing faster than output , First , go back ag , Subtract both sides from the growth rate of capital Now compare the general expression for ratio of capital to with its balanced growth value . general expression and ( a balanced growth . 51 If , then it must be the case that , which implies ( from the previous equation ) that Output per Worker Growth If we want to examine the growth in output per worker rather than total output , we take the production function ( Equation ) and apply the rules of growth rates to that equation . a ) a ( a ) a ( a ) URL books 654

We then we divide by ( a ) to get ( and subtract from each side to obtain ( Finally , we note that With balanced growth , the first term is equal to zero , so Ex ( I ) Endogenous Investment Rate In this analysis , we made the assumption from the model that the investment rate is constant . The essential arguments that we have made still apply if the investment rate is higher when the marginal product of capital is higher . The argument for convergence becomes stronger because a low value of a higher marginal product of capital and thus a higher investment rate . This increases the growth rate of capital and causes an economy to converge more quickly to its path . Endogenous Growth Take the production function ( A . Now is constant and ( Al ( so a ( A ( A . The Main Uses of This Tool Chapter Globalization and Competitiveness . Chapter Global Prosperity and Global Poverty The Aggregate Expenditure Model The aggregate expenditure model relates the components of spending ( consumption , investment , government purchases , and net exports ) to the level of economic activity . In the short run , taking the price level as fixed , the level of spending predicted by the aggregate expenditure model determines the level of economic activity in an economy . An insight from the circular is that real gross domestic product ( real ) measures three things the production of firms , the income earned by households , and total spending on URL books 655

firms output . The aggregate expenditure model focuses on the relationships between production ( and planned spending planned spending consumption investment government purchases net exports . Planned spending depends on the level of production in an economy , for the following reasons . If households have higher income , they will increase their spending . This is captured by the consumption function . Firms are likely to decide that higher levels of if they are expected to that they should build up their capital stock and should thus increase their investment . Higher income means that domestic consumers are likely to spend more on imported goods . Since net exports equal exports minus imports , higher imports means lower net exports . The negative net export link is not large enough to overcome the other positive links , so we conclude that when income increases , so also does planned expenditure . We illustrate this in Figure Planned Spending in the Aggregate Expenditure Model where we suppose for simplicity that there is a linear relationship between spending and . The equation of the line is as follows spending autonomous spending marginal propensity to spend real . spending spending spending Marginal ) In Real spending Real ( Figure Planned Spending in Expenditure URL books 656

The intercept in Figure Planned Spending in the Aggregate Expenditure Model is called autonomous spending . It represents the amount of spending that there would be in an economy if income ( were zero . We expect that this will be positive for two reasons ( if a household finds its income is zero , it will still want to consume something , so it will either draw on its existing wealth ( past savings ) or borrow against future income and ( the government would spend money even if were zero . The slope of the line in Figure Planned Spending in the Aggregate Expenditure Model is given by the marginal propensity to spend . For the reasons that we have just explained , we expect that this is positive increases in income lead to increased spending . However , we expect the marginal propensity to spend to be less than one . The aggregate expenditure model is based on the two equations we have just discussed . We can solve the model either graphically or using algebra . The graphical approach relies on Figure . On the horizontal axis is the level of real . On the vertical axis is the level of spending as well as the level of . There are two lines shown . The is the line , which equates real on the horizontal axis with real on the vertical axis . The second line is the planned spending line . The intersection of the spending line with the line gives the equilibrium level of output . URL books 557

Figure Planned spending Planned spending Real Planned spend . pending Marginal pro to spend Real ( Autonomous spending Real ( income ) Equilibrium real More Formally We can also solve the model algebraically . Let us use denote the level of real and to denote planned expenditure . We represent the marginal propensity to spend by . The two equations of the model are as follows and . Here , is autonomous expenditure . We can solve the two equations to the values and are consistent with both equations . Substituting for in the first equation , we find that The equilibrium level of output is the product of two terms . The first ( called the multiplier . If , as seems reasonable , lies between zero and one , the multiplier is greater than one . The second term is the autonomous level of spending . URL books 658

Here is an example . Suppose that 100 , I 400 , 300 , and 200 , where is consumption , I is investment , is government purchases , and is net exports . First group the components of spending as follows ( 100 400 300 200 ) Adding together the first group of terms , we find autonomous spending 100 400 300 200 . Adding the coefficients on the income terms , we the marginal propensity to spend . Using , we calculate the multiplier . We then calculate real . The Main Uses of This Tool Chapter The Great Depression Chapter 10 Understanding the Fed Chapter 12 Income Taxes Chapter 15 The Global Financial Crisis Price Adjustment The price adjustment equation summarizes , at the level of an entire economy , all the decisions about prices that are made by managers throughout the economy . The price adjustment equation is as follows rate autonomous sensitivity output gap . URL books 659

The equation tells us that there are two reasons for rising prices . The is because the output gap is negative . The output gap is the difference between potential output and actual output output gap potential real gross domestic product ( real ) actual real . A positive gap means that the economy is in potential output . If the economy is in a boom , then the output gap is negative . The second reason for rising prices is that autonomous is positive . Autonomous refers to the rate that prevails in an economy when an economy is at potential output ( so the output gap is zero ) Looking at the second term of the price adjustment equation , we see that when real is greater than potential output , the output gap is negative , so there is upward pressure on prices in the economy . The rate will exceed autonomous . By contrast , when real is less than potential output , the output gap is negative , so there is downward pressure on prices . The rate will be below the autonomous rate . The sensitivity tells us how responsive the rate is to the output gap . The output gap matters because , as increases relative to potential output , labor and other inputs become scarcer . Firms are likely to see rising costs and increase their prices as a consequence . Even leaving this is , even when an economy is at potential firms are likely to increase their prices somewhat . For example , firms may anticipate that their suppliers or their competitors are likely to increase prices in the future . A natural response is to increase prices , so autonomous is Price Adjustment shows the price adjustment equation graphically . Figure Price URL books

rate Autonomous rate gap The Main Uses of This Tool . Chapter The Great Depression . Chapter 10 Understanding the Fed . Chapter 11 Big and Small . Chapter 12 Income Taxes Consumption and Saving The consumption function is a relationship between current disposable income and current consumption . It is intended as a simple description of household behavior that captures the idea of consumption smoothing . We typically suppose the consumption function is sloping but has a slope less than one . So as disposable income increases , consumption also increases but not as much . More specifically , we frequently assume that consumption is related to disposable income through the following relationship consumption autonomous consumption marginal propensity to consume disposable income . A consumption function of this form implies that individuals divide additional income between consumption and saving . We assume autonomous consumption is positive . Households consume something even if their income is zero . If a household has accumulated a lot of wealth in the past URL books 661

or if a household expects its future income to be larger , autonomous consumption will be larger . It captures both the past and the future . We assume that the marginal propensity to consume is positive . The marginal propensity to consume captures the present it tells us how changes in current income lead to changes in current consumption . Consumption increases as current income increases , and the larger the marginal propensity to consume , the more sensitive current spending is to current disposable income . The smaller the marginal propensity to consume , the stronger is the effect . We also assume that the marginal propensity to consume is less than one . This says that not all additional income is consumed . When a household receives more income , it consumes some and saves some . Figure The Consumption Function shows this relationship . Consumption Autonomous consumption Disposable Figure The Function More Formally In symbols , we write the consumption function as a relationship between consumption ( and disposable income ( where a and are constants . Here a represents autonomous consumption and is the marginal propensity to consume . We assume three things about a and a URL books ) 662

The first assumption means that even if disposable income is zero ( consumption will still be positive . The second assumption means that the marginal propensity to consume is positive . By the third assumption , the marginal propensity to consume is less that one . With , part of an extra dollar of disposable income is spent . What happens to the remainder of the increase in disposable income ?

Since consumption plus saving is equal to disposable income , the increase in disposable income not consumed is saved . More generally , this link between consumption and saving ( means that our model of consumption implies a model of saving as well . Using and we can solve for ( So is the level of autonomous saving and ( is the marginal propensity to save . We can also graph the savings function . The savings function has a negative intercept because when income is zero , the household will . The savings function has a positive slope because the marginal propensity to save is positive . Economists also often look at the average propensity to consume ( which measures how much income goes to consumption on average . It is calculated as follows . When disposable income increases , consumption also increases but by a smaller amount . This means that when disposable income increases , people consume a smaller fraction of their income the average propensity to consume decreases . Using our notation , we are saying that using a , so we can write . An increase in disposable income reduces the first term , which also reduces the . URL books 663

The Main Use of This Tool . Chapter 12 Income Taxes The Government Budget Constraint Like households , governments are subject to budget constraints . These can be viewed in two ways , either within a single year or across many years . The Government Budget Constraint In any given year , money into the government sector , primarily from the taxes that it imposes on individuals and corporations . We call these government revenues . Money out in the form of government purchases of goods and services and government transfers . The circular of income tells us that any difference between government purchases and transfers and government revenues represents a government . government revenues government purchases transfers tax revenues government purchases ( tax revenues transfers ) government purchases net taxes . Often , we it useful to group taxes and transfers together as net taxes and separate out government purchases , as in the last line of our . When are less than , then we say a government is running a surplus . In other words , a negative government is the same as a positive government surplus , and a negative government surplus is the same as a positive government . government surplus deficit . When a government runs a deficit , it must borrow from the markets . When a government runs a surplus , these funds into the markets and are available for firms to borrow . A government surplus is sometimes called government saving . Government Budget Constraint Tax and spending decisions at different dates are linked . Although governments can borrow or lend in a given year , a government total spending over time must be matched with revenues . When a government runs a , it typically borrows to finance it . It borrows by issuing more government debt ( government bonds ) To express the budget constraint , we introduce a measure of the deficit called the primary . The primary is the difference between government , excluding interest payments on the debt , and government revenues . surplus is minus the primary and is the difference between government revenues and government , excluding interest payments on the debt . URL books 564

The budget constraint says that if a government has some existing debt , it must run surpluses in the future so that it can ultimately pay off that debt . Specifically , it is the requirement that current debt outstanding discounted present Value of future primary surpluses . This condition means that the debt outstanding today must be offset by primary budget surpluses in the future . Because we are adding together in the future , we have to use the tool of discounted present value . If , for example , the current stock of debt is zero , then the budget constraint says that the discounted present value of future primary surpluses must equal zero . The stock of debt is linked directly to the government budget . As we noted earlier , when a government runs a budget deficit , it finances the deficit by issuing new debt . The deficit is a that is matched by a change in the stock of government debt change in government debt ( in given year ) deficit ( in given year ) The stock of debt in a given year is equal to the over the previous year plus the stock of debt from the start of the previous year . If there is a government surplus , then the change in the debt is a negative number , so the debt decreases . The total government debt is simply the accumulation of all the previous years . When a government borrows , it must pay interest on its debt . These interest payments are counted as part of the deficit ( they are included in transfers ) If a government wants to balance the budget , then government spending must actually be less than the amount government receives in the form of net taxes ( excluding interest ) This presentation of the tool neglects one detail . There is another way in which a government can fund its . As well as issuing government debt , it can print money . More precisely , then , every year , change in government debt change in money supply . Written this way , the equation tells us that the part of the that is not financed by printing money results in an increase in the government debt . More Formally We often denote government purchases of goods and services by and net tax revenues ( tax revenues minus transfers ) by The equation for tax revenues is as follows , where is the tax rate on income and Yis real gross domestic product ( real ) The is given as follows government deficit . From this equation , the depends on the following URL books 665

. Fiscal policy through the choices of and . The level of economic activity ( The Main Uses of This Tool . Chapter 11 Big and Small . Chapter 12 Income Taxes . Chapter 13 Social Security . Chapter 14 Balancing the Budget . Chapter 15 The Global Financial Crisis The Model of Consumption The model of consumption looks at the lifetime consumption and saving decisions of an individual . The choices made about consumption and saving depend on income earned over an individual entire lifetime . The model has two key components the lifetime budget constraint and individual choice given that constraint . Consider the saving decision of an individual who expects to work for a known number of years and be retired for a known number of years thereafter . Suppose his disposable income is the same in every working year , and he will also receive an annual retirement the same in every year . According to the model of consumption , the individual first calculates the discounted present value ( of lifetime income of lifetime income of income from working of retirement income . If the real interest rate is zero , then the calculation simply involves adding income across years . We assume the individual wants to consume at the same level in each period of life . This is called consumption smoothing . In the special case of a zero real interest rate , we have the following lifetime income More Formally Suppose an individual expects to work for a total of years and to be retired for years . Suppose his disposable income is equal to in every year , and he receives annual retirement income of . Then lifetime income , assuming a zero real interest rate , is given as follows lifetime income NY . If we suppose that he wants to have perfectly smooth consumption , equal to in each year , then his total lifetime consumption will be ( URL books 666

The lifetime budget constraint says that lifetime consumption equals lifetime income ( To obtain his consumption , we simply divide this equation by the number of years he is going to live ( NY Provided that income during working years is greater than income in retirement years , the individual will save during his working years and during retirement . If the real interest rate is not equal to zero , then the basic idea is the individual smooths consumption based on a lifetime budget the calculations are more complicated . Specifically , the lifetime budget constraint must be written in terms of the discounted present values of income and consumption . The Main Uses of This Tool Chapter The Great Depression . Chapter 13 Social Security . Chapter 14 Balancing the Budget Aggregate Supply and Aggregate Demand The aggregate supply and aggregate demand ( model is presented here . To understand the model , we need to explain both aggregate demand and aggregate supply and then the determination of prices and output . The aggregate demand curve tells us the level of expenditure in an economy for a given price level . It has a negative slope the demand for real gross domestic product ( real ) decreases when the price level increases . The downward sloping aggregate demand curve does not follow from the law of As the price level increases , all prices in an economy increase together . The substitution of expensive goods for cheap goods , which underlies the law of demand , does not occur in the aggregate economy . Instead , the downward sloping demand curve comes from other forces . First , as prices rise , the real value of nominal wealth falls , and this leads to a fall in household spending . Second , as prices rise today relative to future prices , households are induced to postpone consumption . Finally , a higher price level can lead to a higher interest rate through the response of monetary policy . All these factors together imply that higher prices lead to lower overall demand for real . Aggregate supply is equal to potential output at all prices . Potential output is determined by the available technology , physical capital , and labor force and is unaffected by the price level . Thus the aggregate supply curve is vertical . In contrast to a supply curve , as the price level increases , all prices in an economy increase . This includes the prices of inputs , such as labor , into the production process . Since no relative prices change when the price level URL books 667

increases , are not induced to change the quantity they supply . Thus aggregate supply is vertical . The determination of prices and output depends on the horizon the long run or the short run . In the long run , real equals potential , and real also equals aggregate expenditure . This means that , in the long run , the price level must be at the point where aggregate demand and aggregate supply meet . This is shown in Figure Aggregate Supply and Aggregate Demand in the Long Run . Figure Aggregate Supply and Aggregate Demand in the Long Run Price level price level A Real Potential output In the short run , output is determined by aggregate demand at the existing price level . Prices need not be at their equilibrium levels . If they are not , then output will not equal potential output . This is shown in Figure Aggregate Supply and Aggregate Demand in the Short Run . Figure Aggregate Supply and Aggregate Demand in the Short Run URL books . 668

level i A supply price level ( i output The price level is indicated on the vertical axis . The level of output is determined by aggregate demand at that price level . As prices are greater than the equilibrium level of prices , output is below potential output . The price level adjusts over time to its level , according to the equation . The Main Uses of This Tool We do not explicitly use this tool in our chapter presentations . However , the tool can be used to support the discussions in the following chapters . Chapter The Great Depression . Chapter 10 Understanding the Fed . Chapter 11 Big and Small . Chapter 12 Income Taxes The Model The model provides another way of looking at the determination of the level of run real gross domestic product ( real ) in the economy . Like the aggregate expenditure model , it takes the price level as fixed . But whereas that model takes the interest rate as , a change in the interest rate results in a change in autonomous model treats the interest rate as an endogenous variable . The basis of the model is an analysis of the money market and an analysis of the goods market , which together determine the equilibrium levels of interest rates and output in the economy , given prices . The model finds combinations of interest rates and output ( such that the money market is in equilibrium . This creates the curve . The model also combinations of interest rates and output such that the goods market is in equilibrium . This URL books 669

creates the IS curve . The equilibrium is the interest rate and output combination that is on both the IS and the curves . Curve The curve represents the combinations of the interest rate and income such that money supply and money demand are equal . The demand for money comes from households , and governments that use money as a means of exchange and a store of value . The law of demand holds as the interest rate increases , the quantity of money demanded decreases because the interest rate represents an opportunity cost of holding money . When interest rates are higher , in other words , money is less effective as a store of value . Money demand increases when output rises because money also serves as a medium of exchange . When output is larger , people have more income and so want to hold more money for their transactions . The supply of money is chosen by the monetary authority and is independent of the interest rate . Thus it is drawn as a vertical line . The equilibrium in the money market is shown in Figure Money Market Equilibrium . When the money supply is chosen by the monetary authority , the interest rate is the price that brings the market into equilibrium . Sometimes , in some countries , central banks target the money supply . Alternatively , central banks may choose to target the interest rate . This was the case we considered in Chapter 10 Understanding the Fed . Figure Money Market Equilibrium applies in either case if the monetary authority targets the interest rate , then the money market tells us what the level of the money supply must be . Figure ) URL books 670

Nominal rate Equilibrium rate Quantity of money To trace out the curve , we look at what happens to the interest rate when the level of output in the economy changes and the supply of money is held fixed . Figure A Change in Income shows the money market equilibrium at two different levels of real . At the higher level of income , money demand is shifted to the right the interest rate increases to ensure that money demand equals money supply . Thus the curve is upward sloping higher real is associated with higher interest rates . At each point along the curve , money supply equals money demand . We have not yet been about whether we are talking about nominal interest rates or real interest rates . In fact , it is the nominal interest rate that represents the opportunity cost of holding money . When we draw the curve , however , we put the real interest rate on the axis , as shown in Figure The Curve . The simplest way to think about this is to suppose that we are considering an economy where the rate is zero . In this case , by the Fisher equation , the nominal and real interest rates are the same . In a more complete analysis , we can incorporate by noting that changes in the rate will shift the curve . Changes in the money supply also shift the curve . URL books 671

Nominal rule New interest rate Old interest rate New money demand Old monty demand of money Figure Change in Income URL books 672 Real talc Real ( Figure The Curve IS Curve The IS curve relates the level of real and the real interest rate . It incorporates both the dependence of spending on the real interest rate and the fact that , in the short run , real equals spending . The IS curve is shown in Figure A Change in Income . We label the horizontal axis real since , in the short run , real is determined by aggregate spending . The IS curve is downward sloping as the real interest rate increases , the level of spending decreases . Figure The IS Curve URL books 99 673

Real interest rate IS tune Real In fact , we derived the IS curve in Chapter 10 Understanding the Fed . The dependence of spending on real interest rates comes partly from investment . As the real interest rate increases , spending by firms on new capital and spending by households on new housing decreases . Consumption also depends on the real interest rate spending by households on durable goods decreases as the real interest rate increases . The connection between spending and real comes from the aggregate expenditure model . Given a particular level of the interest rate , the aggregate expenditure model determines the level of real . Now suppose the interest rate increases . This reduces those components of spending that depend on the interest rate . In the aggregate expenditure framework , this is a reduction in autonomous spending . The equilibrium level of output decreases . Thus the IS curve slopes downwards higher interest rates are associated with lower real . Equilibrium Combining the discussion of the and the IS curves will generate equilibrium levels of interest rates and output . Note that both relationships are combinations of interest rates and output . Solving these two equations jointly determines the equilibrium . This is shown graphically in Figure . This just combines the curve from Figure The Curve and the IS curve from Figure The IS Curve . The crossing of these two curves is the combination of the interest rate and real , denoted ( such that both the money market and the goods market are in equilibrium . URL books 674

Figure Real rule Real Equilibrium in the . Comparative Statics Comparative statics results for this model illustrate how changes in exogenous factors the equilibrium levels of interest rates and output . For this model , there are two key exogenous factors the level of autonomous spending ( excluding any spending affected by interest rates ) and the real money supply . We can study how changes in these factors the equilibrium levels of output and interest rates both graphically and algebraically . Variations in the level of autonomous spending will lead to a shift in the IS curve , as shown in Figure A Shift in the IS Curve . If autonomous spending increases , then the IS curve shifts out . The output level of the economy will increase . Interest rates rise as we move along the curve , ensuring money market equilibrium . One source of variations in autonomous spending is policy . Autonomous spending includes government spending ( Thus an increase in leads to an increase in output and interest rates as shown in Figure A Shift in the IS Curve . URL books 675

Figure A Shift in the IS Curve Real rule New equilibrium Old equilibrium Old IS curve Real Old New equilibrium Variations in the real money supply shift the curve , as shown in Figure A Shift in the Curve . If the money supply decreases , then the curve shifts in . This leads to a higher real interest rate and lower output as the curve shifts along the IS curve . Figure A in the Curve URL books I 676

Real nun New Old equilibrium Real New Old equilibrium equilibrium More Formally We can represent the and IS curves algebraically . Curve Let ( represent real money demand at a level of real of a real interest rate of ( When we say real money demand , we mean that , as usual , we have by the price level . For simplicity , suppose that the rate is zero , so the real interest rate is the opportunity cost of holding money . Assume that real money demand takes a particular form ( In this equation , and are all positive constants . Real money demand is increasing in income and decreasing in the interest rate . Letting be the real stock of money in the economy , then money market equilibrium requires Given a level of real and the real stock of money , this equation can be used to solve for the interest rate such that money supply and money demand are equal . This is given by URL books 677

. From this equation we learn that an increase in the real stock of money lowers the interest rate , given the level of real . Further , an increase in the level of real increases the interest rate , given the stock of money . This is another way of saying that the curve is upward sloping . IS Curve Recall the two equations from the aggregate expenditure model and ( Here we have shown explicitly that the level of autonomous spending depends on the real interest rate We can solve the two equations to the values of and are consistent with both equations . We Given a level of the real interest rate , we solve for the level of autonomous spending ( using the dependence of consumption and investment on the real interest rate ) and then use this equation to the level of output . Here is an example . Suppose that 100 , I 400 , 300 , and 200 , where is consumption , I is investment , is government purchases , and is net exports . First group the components of spending as follows ( 100 200 ) Adding together the first group of terms , we find autonomous spending URL books 678

100 300 200 1000 . Adding the coefficients on the income terms , we find the marginal propensity to spend . Using , we calculate the multiplier . We then calculate real , given the real interest rate ( 1000 ) 2000 . Equilibrium Combining the discussion of the and the IS curves will generate equilibrium levels of interest rates and output . Note that both relationships are combinations of interest rates and output . Solving these two equations jointly determines the equilibrium . Algebraically , we have an equation for the curve ( And we have an equation for the IS curve mE ( where we let ( denote the multiplier . If we assume that the dependence of spending in the interest rate is linear , so that ( eo , then the equation for the IS curve is ( To solve the IS and curves simultaneously , we substitute from the IS curve into the curve to get ( Solving this for we get A . I I URL books 679

where both Ar and are constants , with Ar ( Lo ) and ( This equation gives us the equilibrium level of the real interest rate given the level of autonomous spending , summarized by , and the real stock of money , summarized by . To the equilibrium level of output , we substitute this equation for back into the equation for the IS curve . This gives us ( where and are constants , withA ( A ) and me equation gives us the equilibrium level of output given the level of autonomous spending , summarized by , and the real stock of money , summarized by . Algebraically , we can use the equations to determine the magnitude of the responses of interest rates and output to exogenous changes . An increase in the autonomous spending , will increase andA , implying that both the interest rate and output increase . An increase in the real money stock will reduce interest rates by and increase output by . A key part of monetary policy is the sensitivity of spending to the interest rate , given by . The more sensitive is spending to the interest rate , the larger ise and therefore the larger is . The Main Uses of This Too We do not explicitly use this tool in our chapter presentations . However , the tool can be used to support the discussions in the following chapters . Chapter Money A User Guide Chapter 10 Understanding the Fed Chapter 11 Big and Small Chapter 14 Balancing the Budget If we wanted to include in our analysis , we could write the real demand for money as ( 31 ) where is the rate . To see that Ay increases with requires a bit more algebra URL books 680