Chapter 31 Toolkit

31.1 Individual Demand

Individual demand refers to the demand for a good or a service by an individual (or a household). Individual demand comes from the interaction of an individual’s desires with the quantities of goods and services that he or she is able to afford. By desires, we mean the likes and dislikes of an individual. We assume that the individual is able to compare two goods (or collections of goods) and say which is preferred. We assume two things: (1) an individual prefers more to less and (2) likes and dislikes are consistent.

An example is shown in part (a) of Figure 31.1 "Individual Demand". (This example is taken from Chapter 4 "Everyday Decisions".) In this example, there are two goods: music downloads ($1 each) and chocolate bars ($5 each). The individual has income of $100. The budget line is the combination of goods and services that this person can afford if he spends all of his income. In this example, it is the solid line connecting 100 downloads and 20 chocolate bars. The horizontal intercept is the number of chocolate bars the individual could buy if all income was spent on chocolate bars and is income divided by the price of a chocolate bar. The vertical intercept is the number of downloads the individual could buy if all income was spent on downloads and is income divided by the price of a download. The budget set is all the combinations of goods and services that the individual can afford, given the prices he faces and his available income. In the diagram, the budget set is the triangle defined by the budget line and the horizontal and vertical axes.

An individual’s preferred point is the combination of downloads and chocolate bars that is the best among all of those that are affordable. Because an individual prefers more to less, all income will be spent. This means the preferred point lies on the budget line. The most that an individual would be willing to pay for a certain quantity of a good (say, five chocolate bars) is his valuation for that quantity. The marginal valuation is the most he would be willing to pay to obtain one extra unit of the good. The decision rule of the individual is to buy an amount of each good such that

The individual demand curve for chocolate bars is shown in part (b) of Figure 31.1 "Individual Demand". On the horizontal axis is the quantity of chocolate bars. On the vertical axis is the price. The demand curve is downward sloping: this is the law of demand. As the price of a chocolate bar increases, the individual substitutes away from chocolate bars to other goods. Thus the quantity demanded decreases as the price increases.

In some circumstances, the individual’s choice is a zero-one decision: either purchase a single unit of the good or purchase nothing. The unit demand curve shows us the price at which a buyer is willing to buy the good. This price is the same as the buyer’s valuation of the good. At any price above the buyer’s valuation, the individual will not want to buy the good. At any price below the buyer’s valuation, the individual wants to buy the good. If this price is exactly equal to the buyer’s valuation, then the buyer is indifferent between purchasing the good or not.

In other words, the individual’s decision rule is to purchase the good if the valuation of the good exceeds its price. This is consistent with the earlier condition because the marginal valuation of the first unit is the same as the valuation of that unit. The difference between the valuation and the price is the buyer surplus. (See Section 31.10 "Buyer Surplus and Seller Surplus" for more discussion.)

More Formally

Let p_d be the price of a download, p_c the price of a chocolate bar, and I the income of an individual. Think of prices and income in terms of dollars. Then the budget set is the combinations of downloads (d) and chocolate bars (c) such that

I ≥ p_d × d + p_c × c.

The budget line is the combinations of d and c such that

I = p_d × d + p_c × c.

In the graph, with downloads on the vertical axis, the equation for the budget line is

$$d=\frac{I}{p_d}-\frac{p_c}{p_d}\times c$$

You can use this equation to understand how changes in income and prices change the position of the budget line. You can also use this equation to find the vertical and horizontal intercepts of the budget line, along with the slope of −(p_c/p_d).

The individual purchases downloads up to the point where

MV_d = p_d

(where MV represents marginal valuation) and purchases chocolate bars up to the point where

MV_c = p_c.

Combining these expressions we get

$$-\frac{M{V}_c}{M{V}_d}=\frac{p_c}{p_d}\text{,}$$

which tells us that (minus) the ratio of marginal valuations equals the slope of the budget line. The ratio of marginal valuations is the rate at which an individual would like to trade one good for the other. The ratio of prices (the slope of the budget line) is the rate at which the market allows an individual to make these trades.

Figure 31.1 Individual Demand

31.2 Elasticity

Elasticity measures the proportionate change in one variable relative to the change in another variable. Consider, for example, the response of the quantity demanded to a change in the price. The price elasticity of demand is the percentage change in the quantity demanded divided by the percentage change in the price:

When the price increases (the percentage change in the price is positive), the quantity decreases, meaning that the percentage change in the quantity is negative. In other words, the law of demand tells us that the elasticity of demand is a negative number. For this reason we often use −(elasticity of demand) because we know this will always be a positive number.

• If −(elasticity of demand) > 1, demand is relatively elastic.
• If −(elasticity of demand) < 1, demand is relatively inelastic.

We can use the idea of the elasticity of demand whether we are thinking about the demand curve faced by a firm or the market demand curve. The definition is the same in either case.

If we are analyzing the demand curve faced by a firm, then we sometimes refer to the elasticity of demand as the own-price elasticity of demand. It tells us how much the quantity demanded changes when the firm changes its price. If we are analyzing a market demand curve, then the price elasticity of demand tells us how the quantity demanded in the market changes when the price changes. Similarly, the price elasticity of supply tells us how the quantity supplied in a market changes when the price changes. The price elasticity of supply is generally positive because the supply curve slopes upward.

The income elasticity of demand is the percentage change in the quantity demanded divided by the percentage change in income. The income elasticity of demand for a good can be positive or negative.

• If the income elasticity of demand is negative, it is an inferior good.
• If the income elasticity of demand is positive, it is a normal good.
• If the income elasticity of demand is greater than one, it is a luxury good.

The cross-price elasticity of demand tells us how the quantity demanded of one good changes when the price of another good changes.

• If the cross-price elasticity of demand is positive, the goods are substitutes.
• If the cross-price elasticity of demand is negative, the goods are complements.

In general, we can use elasticity whenever we want to show how one variable responds to changes in another variable.

31.3 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 firm demands fewer hours of labor input—that is, a firm’s labor 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 firm shifts rightward: the quantity of labor demanded is higher at a given real wage.

The labor market equilibrium is shown in Figure 31.2 "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 31.2 Labor Market Equilibrium

31.4 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 31.3 "Choices over Time" shows examples of these choices over a two-year horizon. The individual earns income this year and next. The combinations of consumption that are affordable and that exhaust all of an individual’s income are shown on the budget line, which in this case is called an intertemporal budget constraint. The slope of the budget line is equal to (1 + 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 31.3 "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 (b) of Figure 31.3 "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:

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’s lifetime.

Key Insights

• Over a lifetime, an individual’s 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 31.3 Choices over Time

31.5 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 $1.00 that you will receive next year is as follows:

If the nominal interest rate is 10 percent, then the nominal interest factor is 1.1, so $1 next year is worth $1/1.1 = $0.91 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:

The flow benefit depends on the asset. For a bond, the flow benefit is a coupon payment. For a stock, the flow benefit is a dividend payment. For a fruit tree, the flow benefit is the yield of a crop.

If an asset (such as a bond) yields a payment next year of $10 and has a price next year of $90, then the “flow benefit from asset + price of the asset next year” is $100. The value of the asset this year is then $\frac{\$100}{\text{nominal interest factor}}\text{.}$ If the nominal interest rate is 20 percent, then the value of the asset is $100/1.2 = 83.33.

We discount nominal flows using a nominal interest factor. We discount real flows (that is, flows already corrected for inflation) using a real interest factor, which is equal to (1 + real interest rate).

31.6 The Credit Market

An individual’s choices over time determine how much he or she will borrow or lend. In particular, individual loan supply is upward sloping: when the real interest rate increases, a typical household will supply a greater quantity of funds to the credit market. Market loan supply is obtained by adding together the individual loan supplies of everyone in an economy. We use the terms “credit” and “loans” interchangably.

The demand for credit comes from households and firms that are borrowing. Market loan demand is obtained by adding together all the individual demands for loans. When real interest rates increase, borrowing is more expensive, so the quantity of loans demanded decreases. That is, loan demand obeys the law of demand.

Borrowers and lenders interact in the credit market (or loan market), which is illustrated in Figure 31.4 "The Credit Market (or Loan Market)". Credit 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. All gains from trade through loans are exhausted in equilibrium.

Key Insight

• As the real interest rate increases, more loans are supplied, and fewer loans are demanded.

Figure 31.4 The Credit Market (or Loan Market)

31.7 Expected Value

Probability is the percentage chance that something will occur. For example, there is a 50 percent chance that a tossed coin will come up heads. We say that the probability of getting the outcome “heads” is 1/2. There are five things you need to know about probability:

1. The list of possible outcomes must be complete.
2. The list of possible outcomes must not overlap.
3. If an outcome is certain to occur, it has probability 1.
4. If an outcome is certain not to occur, it has probability 0.
5. If we add together the probabilities for all the possible outcomes, the total must equal 1.

The expected value of a situation with financial risk is a measure of how much you would expect to win (or lose) on average if the situation were to be replayed a large number of times. You can calculate expected value as follows:

• For each outcome, multiply the probability of that outcome by the amount you will receive.
• Add together these amounts over all the possible outcomes.

For example, suppose you are offered the following proposal. Roll a six-sided die. If it comes up with 1 or 2, you get $90. If it comes up 3, 4, 5, or 6, you get $30. The expected value is

Most people dislike risk. They prefer a fixed sum of money to a gamble that has the same expected value. Risk aversion is a measure of how much people want to avoid risk. In the example we just gave, most people would prefer a sure $50 to the uncertain proposal with the expected value of $50.

Suppose we present an individual with the following gamble:

• With 99 percent probability, you lose nothing.
• With 1 percent probability, you lose $1,000.

The expected value of this gamble is −$10. Now ask the individual how much she would pay to avoid this gamble. Someone who is risk-neutral would be willing to pay only $10. Someone who is risk-averse would be willing to pay more than $10. The more risk-averse an individual, the more the person would be willing to pay.

The fact that risk-averse people will pay to shed risk is the basis of insurance. If people have different attitudes toward risky gambles, then the less risk-averse individual can provide insurance to the more risk-averse individual. There are gains from trade. Insurance is also based on diversification, which is the idea that people can share their risks so it is much less likely that any individual will face a large loss.

31.8 Correcting for Inflation

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.

1. Select your deflator. In most cases, the Consumer Price Index (CPI) is the best deflator to use. You can find data on the CPI (for the United States) at the Bureau of Labor Statistics website (http://www.bls.gov).
2. Select your base year. Find the value of the index in that base year.
3. 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 1.
4. For each year, divide the value in the nominal data series by the number you calculated in step 3. This gives you the value in “base year dollars.”

Table 31.2 "Correcting Nominal Sales for Inflation" shows an example. We have data on the CPI for three years, as listed in the second column. The price index is created using the year 2000 as a base year, following steps 1–3. Sales measured in millions of dollars are given in the fourth column. To correct for inflation, 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 inflation each year (the price index is increasing over time), real sales do not increase as rapidly as nominal sales.

This calculation uses the CPI, which is an example of a price index. To see how a price index like the CPI is constructed, consider Table 31.3 "Constructing a Price Index", which shows a very simple economy with three goods: T-shirts, music downloads, and meals. The prices and quantities purchased in the economy in 2012 and 2013 are summarized in the table.

To construct a price index, you must choose a fixed basket of goods. For example, we could use the goods purchased in 2013 (12 T-shirts, 60 downloads, and 5 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 CPI and gross domestic product (GDP) deflator 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 inflation rate.

We also correct interest rates for inflation. 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 inflation, we use the Fisher equation:

For more details, see Section 31.25 "The Fisher Equation: Nominal and Real Interest Rates" on the Fisher equation.

31.9 Supply and Demand

The supply-and-demand 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 firm, taking prices as given, will produce at a level such that

Marginal cost usually increases as a firm produces more output. Thus an increase in the price of a product creates an incentive for firms to produce more—that is, the supply curve of a firm 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 firm’s 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 firms 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

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 cheaper—that 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 significant number of households.

A market equilibrium is a price and a quantity such that the quantity supplied equals the quantity demanded at the equilibrium price (Figure 31.5 "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 firms, 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 profitable.

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 31.5 Market Equilibrium

Figure 31.5 "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.

31.10 Buyer Surplus and Seller Surplus

If you buy a good, then you obtain buyer surplus. If you did not expect to obtain any surplus, then you would not choose to buy the good.

• Suppose you buy a single unit of the good. Your surplus is the difference between your valuation of the good and the price you pay. This is a measure of how much you gain from the exchange.
• If you purchase many units of a good, then your surplus is the sum of the surplus you get from each unit. To calculate the surplus from each unit, you subtract the price paid from your marginal valuation of that unit.

If you sell a good, then you obtain seller surplus. If you did not expect to obtain any surplus, you would not sell the good.

• Suppose you sell a single unit of a good. Your surplus is equal to the difference between the price you receive from selling the good and your valuation of the good. This valuation may be a measure of how much you enjoy the good or what you think you could sell it for in some other market.
• If you sell many units of a good, then the surplus you receive is the sum of the surplus for each unit you sell. To calculate the surplus from selling each unit, you take the difference between the price you get for each unit sold and your marginal valuation of that extra unit.

Buyer surplus and seller surplus are created by trade in a competitive market (Figure 31.6 "A Competitive Market"). The equilibrium price and the equilibrium quantity are determined by the intersection of the supply and demand curves. The area below the demand curve and above the price is the buyer surplus; the area above the supply curve and below the price is the seller surplus. The sum of the buyer surplus and the seller surplus is called total surplus or the gains from trade.

Buyer surplus and seller surplus can also arise from individual bargaining (Figure 31.7 "Individual Bargaining"). When a single unit is traded (the case of unit demand and unit supply), the total surplus is the difference between the buyer’s valuation and the seller’s valuation. Bargaining determines how they share the gains from trade. The quantity of trades, indicated on the horizontal axis, is either zero or one. The valuations of the buyer and the seller are shown on the vertical axis. In this case, the valuation of the buyer ($3,000) exceeds the valuation of the seller ($2,000), indicating that there are gains from trade equal to $1,000. How these gains are shared between the buyer and seller depends on the price they agree on. In part (a) of Figure 31.7 "Individual Bargaining", the buyer gets most of the surplus; in part (b) of Figure 31.7 "Individual Bargaining", the seller gets most of the surplus.

Key Insights

• Buyer surplus and seller surplus are created by trade.
• Buyer surplus is the difference between the marginal value of a good and the price paid.
• Seller surplus is the difference between the price received and the marginal value of a good.

Figure 31.6 A Competitive Market

Figure 31.7 Individual Bargaining

31.11 Efficiency and Deadweight Loss

The outcome of a competitive market has a very important property. In equilibrium, all gains from trade are realized. This means that there is no additional surplus to obtain from further trades between buyers and sellers. In this situation, we say that the allocation of goods and services in the economy is efficient. However, markets sometimes fail to operate properly and not all gains from trade are exhausted. In this case, some buyer surplus, seller surplus, or both are lost. Economists call this a deadweight loss.

The deadweight loss from a monopoly is illustrated in Figure 31.8 "Deadweight Loss". The monopolist produces a quantity such that marginal revenue equals marginal cost. The price is determined by the demand curve at this quantity. A monopoly makes a profit equal to total revenue minus total cost. When the total output is less than socially optimal, there is a deadweight loss, which is indicated by the red area in Figure 31.8 "Deadweight Loss".

Deadweight loss arises in other situations, such as when there are quantity or price restrictions. It also arises when taxes or subsidies are imposed in a market. Tax incidence is the way in which the burden of a tax falls on buyers and sellers—that is, who suffers most of the deadweight loss. In general, the incidence of a tax depends on the elasticities of supply and demand.

A tax creates a difference between the price paid by the buyer and the price received by the seller (Figure 31.9 "Tax Burdens"). The burden of the tax and the deadweight loss are defined relative to the tax-free competitive equilibrium. The tax burden borne by the buyer is the difference between the price paid under the tax and the price paid in the competitive equilibrium. Similarly, the burden of the seller is the difference between the price in the competitive equilibrium and the price received under the equilibrium with taxes. The burden borne by the buyer is higher—all else being the same—if demand is less elastic. The burden borne by the seller is higher—all else being the same—if supply is less elastic.

The deadweight loss from the tax measures the sum of the buyer’s lost surplus and the seller’s lost surplus in the equilibrium with the tax. The total amount of the deadweight loss therefore also depends on the elasticities of demand and supply. The smaller these elasticities, the closer the equilibrium quantity traded with a tax will be to the equilibrium quantity traded without a tax, and the smaller is the deadweight loss.

Key Insights

• In a competitive market, all the gains from trade are realized.
• If sellers have market power, some gains from trade are lost because the quantity traded is below the competitive level.
• Other market distortions, such as taxes, subsidies, price floors, or price ceilings, similarly cause the amount to be traded to differ from the competitive level and cause deadweight loss.

Figure 31.8 Deadweight Loss

Figure 31.9 Tax Burdens

31.12 Production Possibilities Frontier

The production possibilities frontier shows the combinations of goods and services that an economy can produce if it is efficiently using every available input. A key component in understanding the production possibilities frontier is the term efficiently. If an economy is using its inputs in an efficient way, then it is not possible to produce more of one good without producing less of another.

Figure 31.10 "The Production Possibilities Frontier" shows the production possibilities frontier for an economy producing web pages and meals. It is downward sloping: to produce more web pages, the production of meals must decrease. Combinations of web pages and meals given by points inside the production possibilities frontier are possible for the economy to produce but are not efficient: at points inside the production possibilities frontier, it is possible for the economy to produce more of both goods. Points outside the production possibilities frontier are not feasible given the current levels of inputs in the economy and current technology.

The negative slope of the production possibilities frontier reflects opportunity cost. The opportunity cost of producing more meals is that fewer web pages can be created. Likewise, the opportunity cost of creating more web pages means that fewer meals can be produced.

The production possibilities frontier shifts over time. If an economy accumulates more physical capital or has a larger workforce, then it will be able to produce more of all the goods in an economy. Further, it will be able to produce new goods. Another factor shifting the production possibilities frontier outward over time is technology. As an economy creates new ideas (or receives them from other countries) on how to produce goods more cheaply, then it can produce more goods.

Key Insights

• The production possibilities frontier shows the combinations of goods and services that can be produced efficiently in an economy at a point in time.
• The production possibilities frontier is downward sloping: producing more of one good requires producing less of others. The production of a good has an opportunity cost.
• As time passes, the production possibilities frontier shifts outward due to the accumulation of inputs and technological progress.

Figure 31.10 The Production Possibilities Frontier

31.13 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 31.4 "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 goods—tomatoes and beer—in two countries: Guatemala 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 Guatemala.

In Guatemala, the opportunity cost of 1 kilogram of tomatoes is 2 liters of beer. To produce an extra kilogram of tomatoes in Guatemala, 6 hours of labor time must be taken away from beer production; 6 hours of labor time is the equivalent of 2 liters of beer. In Mexico, the opportunity cost of 1 kilogram of tomatoes is 1 liter of beer. Thus the opportunity cost of producing tomatoes is lower in Mexico than in Guatemala. This means that Mexico has a comparative advantage in the production of tomatoes. By a similar logic, Guatemala has a comparative advantage in the production of beer.

Guatemala 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.

31.14 Costs of Production

The costs of production for a firm are split into two categories. One type of cost, fixed costs, is independent of a firm’s output level. A second type of cost, variable costs, depends on a firm’s level of output. Total costs are the sum of the fixed costs and the variable costs.

The change in costs as output changes by a small amount is called marginal cost. It is calculated as follows:

Because fixed costs do not depend on the quantity, if we produce one more unit, then the change in total cost and the change in the variable cost are the same. Marginal cost is positive because variable costs increase with output. Marginal cost is usually increasing in the level of output, reflecting the diminishing marginal product of factors of production.

For example, suppose that total costs are given by

Here the fixed cost is 50, and the variable cost is 10 times the level of output. In this example, marginal cost equals 10. These costs are shown in Table 31.5.

We sometimes divide fixed costs into two components: entry costs, which are the one-time fixed costs required to open a new business or set up a new plant, and fixed operating costs, which are the fixed costs incurred regularly during the normal operation of a business.

Some costs are sunk costs; once incurred, these costs cannot be recovered. Such costs should be ignored in forward-looking business decisions. Other costs are partially or fully recoverable costs. For example, if a firm purchases an asset that can be resold, then the cost of that asset is recoverable.

Figure 31.11 Cost Measures

Figure 31.11 "Cost Measures" shows these various measures of costs. It is drawn assuming a fixed cost of 50 and variable costs given by

For this example, marginal cost is positive and increasing.

31.15 Pricing with Market Power

The goal of the managers of a firm is to maximize the firm’s profit.

We can think of a firm as choosing either the price to set or the quantity that it sells. Either way, the firm faces a demand curve and chooses a point on that curve that maximizes its profits. In reality, most firms choose the price of the good that they sell. However, it is often simpler to analyze a firm’s behavior by looking at the quantity that it chooses.

Profits are maximized (Figure 31.12 "Markup Pricing") when the extra revenue from selling one more unit of output (marginal revenue) is equal to the extra cost of producing one more unit (marginal cost). The firm’s decision rule is to select a point on the demand curve such that

We can rearrange this condition to obtain a firm’s pricing rule:

Figure 31.12 "Markup Pricing" illustrates this pricing decision. The markup depends on the price elasticity of demand. When demand is relatively inelastic, firms have a lot of market power and set a high markup. This is not a “plug-and-play” formula because both the markup and marginal cost depend, in general, on the price that a firm chooses. However, it does provide a useful description of a firm’s decision.

Key Insights

• When marginal cost is higher, a firm sets a higher price.
• When demand is more inelastic (so a firm has more market power), the markup is higher, so a firm sets a higher price.
• When demand is perfectly elastic, the markup is 1, and the firm sets its price equal to marginal cost. This is the case of a competitive market.
• Any price you see has two components: the marginal cost and the markup. When a price changes, one or both of these must have changed.

Figure 31.12 Markup Pricing

31.16 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:

1. Begin at an equilibrium point where the quantity supplied equals the quantity demanded.
2. 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.
3. Determine the direction of this shift.
4. After shifting the curve, find the new equilibrium point.
5. Compare the new and old equilibrium points to predict how the exogenous event affects the market.

Figure 31.13 "A Shift in the Demand Curve" and Figure 31.14 "A Shift in the Supply Curve" show comparative statics in action in the market for Curtis Granderson replica shirts and the market for beer. In Figure 31.13 "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 31.14 "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 fixed 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 fixed market supply curve.
• If the market supply curve shifts, then the new and old equilibrium points lie on a fixed market demand curve.

Figure 31.13 A Shift in the Demand Curve

Figure 31.14 A Shift in the Supply Curve

31.17 Production Function

The production function characterizes the output of a firm given the inputs it uses. The link between inputs and output is shown Figure 31.15 "The Production Function". The production function combines a firm’s physical capital stock, labor, raw materials (or intermediate inputs), and technology to produce output. Technology is the knowledge (the “blueprints”) that the firm possesses, together with managerial skills.

Production functions generally have two important properties:

1. Positive marginal product of an input
2. Diminishing marginal product of an input

By input, we mean any of the factors of production, such as physical capital, labor, or raw materials. The marginal product of an input is the extra output obtained if extra input is used. In this conceptual exercise, all other inputs are held fixed so that we change only one input at a time.

The first property asserts that additional output will be obtained from additional units of an input. Adding another machine, another worker, some more fuel, and so on, increases the output of a firm. A positive marginal product does not necessarily mean that the extra output is profitable: it might be that the cost of the extra input is high relative to the value of the additional output obtained.

The second property explains how the marginal product of an input changes as we increase the amount of that input, keeping the quantities of other inputs fixed. An additional unit of an input will (usually) increase output more when there is a small (rather than a large) amount of that input being used. For example, the extra output obtained from adding the first machine is greater than the additional output obtained from adding the 50th machine.

A simple production function relating output to labor input is shown in Figure 31.16 "Labor Input in the Production Function". This figure illustrates the two properties of positive and diminishing marginal product of labor. As more labor is added, output increases: there is a positive marginal product of labor (that is, the slope of the relationship is positive). But the extra output obtained from adding labor is greater when the labor input is low: there is diminishing marginal product of labor. From the graph, the slope of the production function (which is the marginal product of labor) is greater at low levels of the labor input.

Key Insights

• The production function shows the output produced by a firm given its inputs.
• The production function displays two important properties: positive marginal product and diminishing marginal product.

Figure 31.15 The Production Function

Figure 31.16 Labor Input in the Production Function

31.18 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 payoff that depends on the actions chosen by everyone in the game.

A Nash equilibrium is an action for each player that satisfies two conditions:

1. The action yields the highest payoff for that player given her predictions about the other players’ actions.
2. The player’s predictions of others’ actions are correct.

Thus a Nash equilibrium has two dimensions. Players make decisions that are in their own self-interests, and players make accurate predictions about the actions of others.

Consider the games in Table 31.6 "Prisoners’ Dilemma", Table 31.7 "Dictator Game", Table 31.8 "Ultimatum Game", and Table 31.9 "Coordination Game". The numbers in the tables give the payoff to each player from the actions that can be taken, with the payoff of the row player listed first.

• Prisoners’ dilemma. The row player chooses between the action labeled Up and the one labeled Down. The column player chooses between the action labeled Left and the one labeled Right. For example, if row chooses Up and column chooses Right, then the row player has a payoff of 0, 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’s 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 (x) 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 (x = 0).
• 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.
• 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 Down/Right is also a Nash equilibrium. Both players prefer Up/Left, but it is possible to get stuck in a bad equilibrium.

31.19 Externalities and Public Goods

Some economic transactions have effects on individuals not directly involved in that transaction. When this happens, we say there is an externality present. An externality is generated by a decision maker who disregards the effects of his actions on others. In the case of a positive externality, the individual’s actions increase the welfare of others (for example, research and development by firms). In the case of a negative externality, an individual’s actions decrease the welfare of others (for example, pollution).

Economic outcomes are not efficient when externalities are present. So the government may be able to improve on the private outcome. The possible remedies are as follows:

• Subsidies (in the case of positive externalities) and taxes (in the case of negative externalities)
• The creation of markets by the government

If people are altruistic, then they may instead take into account others’ welfare and may internalize some of the effects of their actions.

We typically see externalities associated with nonexcludable goods (or resources)—goods for which it is impossible to selectively deny access. In other words, it is not possible to let some people consume the good while preventing others from consuming it. An excludable good (or resource) is one to which we can selectively allow or deny access. If a good is nonexcludable or partially excludable, there are positive externalities associated with its production and negative externalities associated with its consumption.

We say that a good is a rival if one person’s consumption of the good prevents others from consuming the good. Most of the goods we deal with in economics are rival goods. A good is nonrival if one person can consume the good without preventing others from consuming the same good. Knowledge is a nonrival good.

If a good is both nonexcludable and nonrival, it is a public good.

31.20 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, firms, and governments that wish to purchase goods, services, or financial assets denominated in the currency of another economy. For example, if a US auto importer wants to buy a German car, the importer must 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 denominated 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 prices—by 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 denominated in currencies other than the euro

The demand for euros comes from the following:

• Households and firms in noneuro countries that wish to buy goods and services from Europe
• Investors in noneuro countries that wish to buy assets (government debt, stocks, bonds, etc.) that are denominated in euros

Figure 31.17 "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.

Figure 31.17 The Foreign Exchange 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 firms 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 depreciated relative to the euro. From the perspective of the euro, the depreciation of the dollar represents an appreciation of the euro.

31.21 Growth Rates

If some variable x (for example, the number of gallons of gasoline sold in a week) changes from x₁ to x₂, then we can define the change in that variable as Δx = x₂ − x₁. But there are difficulties with this simple definition. The number that we calculate will change, depending on the units in which we measure x. If we measure in millions of gallons, x will be a much smaller number than if we measure in gallons. If we measured x 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 %Δx to mean the percentage change in x and define it as follows: %Δx = (x₂ − x₁)/x₁. A percentage change equal to 0.1 means that gasoline consumption increased by 10 percent. Why? Because 10 percent means 10 “per hundred,” so 10 percent = 10/100 = 0.1.

Very often in economics, we are interested in changes that take place over time. Thus we might want to compare gross domestic product (GDP) between 2012 and 2013. Suppose we know that GDP in the United States in 2012 was $14 trillion and that GDP in 2013 was $14.7 trillion. Using the letter Y to denote GDP measured in trillions, we write Y₂₀₁₂ = 14.0 and Y₂₀₁₃ = 14.7. If we want to talk about GDP at different points in time without specifying a particular year, we use the notation Y_t. 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 GDP in 2013 is calculated as follows:

The growth rate equals 5 percent. In general, we write %ΔY_t+1 = (Y_t+1 − Y_t)/Y_t. Occasionally, we use the gross growth rate, which simply equals 1 + the growth rate. So, for example, the gross growth rate of GDP equals Y₂₀₁₃/Y₂₀₁₂, or 1.05.

There are some useful rules that describe the behavior of percentage changes and growth rates.

The Product Rule. Suppose we have three variables, x, y, and z, and suppose

Then

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 z to obtain

If we take the product rule and subtract %Δz from both sides, we get the following:

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

Then

For example, if y = x², then the growth rate of y is twice the growth rate of x. If $y=\sqrt{x}\text{,}$ then the growth rate of y is half the growth rate of x (remembering that a square root is the same as a power of ½).

31.22 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 flipping of a two-sided coin. Although you might think it would be easier just to flip 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 31.10.

The third column creates the two events of heads and tails that we normally associate with a coin flip. To generate this last column, we adopted a rule: if the random number was less than 0.5, we termed this a “tail” and assigned a 0 to the draw; otherwise we termed it a “head” and assigned a 1 to the draw. The choice of 0.5 as the cutoff for heads reflects the fact that we are considering the flips of a fair coin in which each side has the same probability: 0.5.

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 flip 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 define 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 flipping exercise?” It is natural to say 0.5, 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 31.10 and calculating the mean of the outcome. We let k_i be the outcome of draw i. For example, from the table, k₁ = 1 and k₅ = 0. Then the formula for the mean if there are N draws is μ = Σ_ik_i/N. Here Σ_ik_i means the sum of the k_i outcomes. In words, the mean, denoted by μ, is calculated by adding together the draws and dividing by the number of draws (N). In the table, N = 10, and the sum of the draws of random numbers is about 51.0. Thus the mean of the 10 draws is about 0.51.

We can also calculate the mean of the heads/tails column, which is 0.6 since heads came up 6 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 σ², is calculated as σ² = Σ_i(k_i − μ)²/(N). 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 (k_i) and the mean (μ) when calculating the variance. This means that values 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 μ = 0.51. 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 0.118 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 0.34. The standard deviation is greater than the variance since the variance is less than 1.

31.23 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, etc.). 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 GPAs also had high scores on the SAT or ACT exam. In this instance, we would say that the GPA and SAT/ACT scores were positively correlated: looking across your classmates, when a person’s GPA is higher than average, that person’s SAT or ACT score is likely to be higher than average as well.

As another example, consider the relationship between a household’s income and its expenditures on housing. If you conducted a survey across households, it is likely that you would find 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 GDP), and the fraction of workers without jobs, called the unemployment rate. Over long periods of time, when GDP is above average (the economy is doing well), the unemployment rate is below average. In this case, GDP 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 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’s announcement is not followed by turbulence.

But—obviously—this does not mean that we could solve the turbulence problem by turning off the public address system. The pilot’s 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’s announcement.

We noted earlier that real GDP and unemployment are negatively correlated. When real GDP 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 recessions, 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 GDP. Neither do we necessarily think that high real GDP causes low unemployment. Instead, based on economic theory, there are other influences that affect both real GDP and unemployment.

31.24 The Credit (Loan) Market (Macro)

Consider a simple example of a loan. Imagine you go to your bank to inquire about a loan of $1,000, to be repaid in one year’s time. A loan is a contract that specifies three things:

1. The amount being borrowed (in this example, $1,000)
2. The date(s) at which repayment must be made (in this example, one year from now)
3. The amount that must be repaid

What determines the amount of the repayment? The lender—the bank—is a supplier of credit, and the borrower—you—is 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 macroeconomics, 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 demanders 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:

1. The private saving of households and firms
2. The saving of governments (in the case of a government surplus)
3. The saving of foreigners (when there is a flow 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:

1. The borrowing of households and firms to finance purchases, such as housing, durable goods, and investment goods
2. The borrowing of governments (in the case of a government deficit)
3. The borrowing of foreigners (when there is a flow 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 firm borrows to purchase capital. It is also true if it uses internal funds (retained earnings) to finance investment because the firm could always put those funds into an interest-bearing 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 firms will borrow less.

Equilibrium in the market for loans is shown in Figure 31.18 "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 flows into and out of the financial sector in the circular flow diagram.

The Main Uses of This Tool

• Chapter 19 "The Interconnected Economy"
• Chapter 24 "Money: A User’s Guide"
• Chapter 25 "Understanding the Fed"
• Chapter 26 "Inflations Big and Small"
• Chapter 29 "Balancing the Budget"

Figure 31.18 The Credit Market

31.25 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 denominated in dollars, and your promised payments are denominated in dollars. These dollar flows must be corrected for inflation 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 inflation.

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:

To find the real interest rate, we take the nominal interest rate and subtract the inflation rate. For example, if a loan has a 12 percent interest rate and the inflation rate is 8 percent, then the real return on that loan is 4 percent.

In calculating the real interest rate, we used the actual inflation 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 inflation rate that will occur in the future is not known with certainty. Instead, the borrower and lender use their expectations of future inflation to determine the interest rate on a loan. From that perspective, we use the following formula:

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.

31.26 The Aggregate Production Function

The aggregate production function describes how total real gross domestic product (real GDP) in an economy depends on available inputs. Aggregate output (real GDP) depends on the following:

• Physical capital—machines, production facilities, and so forth that are used in production
• Labor—the number of hours that are worked in the entire economy
• Human capital—skills and education embodied in the workforce of the economy
• Knowledge—basic scientific knowledge, and blueprints that describe the available production processes
• Social infrastructure—the general business, legal and cultural environment
• The amount of natural resources available in an economy
• Anything else that we have not yet included

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 31.19 shows the relationship between output and capital, holding fixed the level of other inputs. This figure 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 31.19

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 31.20. 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.

Figure 31.20

31.27 The Circular Flow of Income

The circular flow of income describes the flows of money among the five main sectors of an economy. As individuals and firms buy and sell goods and services, money flows among the different sectors of an economy. The circular flow of income describes these flows of dollars (pesos, euros, or whatever). From a simple version of the circular flow, we learn that—as a matter of accounting—

This relationship lies at the heart of macroeconomic analysis.

There are two sides to every transaction. Corresponding to the flows of money in the circular flow, there are flows of goods and services among these sectors. For example, the wage income received by consumers is in return for labor services that flow from households to firms. The consumption spending of households is in return for the goods and services that flow from firms to households.

A complete version of the circular flow is presented in Figure 31.21. (Chapter 18 "The State of the Economy" contains a discussion of a simpler version of the circular flow with only two sectors: households and firms.)

Figure 31.21

The complete circular flow has five sectors: a household sector, a firm sector, a government sector, a foreign sector, and a financial sector. Different chapters of the book emphasize different pieces of the circular flow, and Figure 31.21 shows us how everything fits together. In the following subsections, we look at the flows into and from each sector in turn. In each case, the balance of the flows into and from each sector underlies a useful economic relationship.

31.28 Growth Accounting

Growth accounting is a tool that tells us how changes in real gross domestic product (real GDP) 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:

In this equation, a is just a number. For example, if a = 1/3, the growth in output is as follows:

Growth rates can be positive or negative, so we can use this equation to analyze decreases in GDP 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 31.21 "Growth Rates") to the Cobb-Douglas aggregate production function (discussed in Section 31.26 "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 cannot, 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,

So if we know the number a, we are done—we 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 (1 − 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 GDP. For most economies, a is in the range of about 1/3 to 1/2.

31.29 The Solow Growth Model

The analysis in Chapter 21 "Global Prosperity and Global Poverty" is (implicitly) based on a theory of economic growth known as the Solow growth model. Here we present two formal versions of the mathematics of the model. The first 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 long-run 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.

31.30 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 flow is that real gross domestic product (real GDP) measures three things: the production of firms, the income earned by households, and total spending on firms’ output. The aggregate expenditure model focuses on the relationships between production (GDP) and planned spending:

Planned spending depends on the level of income/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 production—particularly if they are expected to persist—mean 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 31.22 "Planned Spending in the Aggregate Expenditure Model" where we suppose for simplicity that there is a linear relationship between spending and GDP. The equation of the line is as follows:

Figure 31.22 Planned Spending in the Aggregate Expenditure Model

The intercept in Figure 31.22 "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 (GDP) were zero. We expect that this will be positive for two reasons: (1) 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 (2) the government would spend money even if GDP were zero.

The slope of the line in Figure 31.22 "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 31.23. On the horizontal axis is the level of real GDP. On the vertical axis is the level of spending as well as the level of GDP. There are two lines shown. The first is the 45° line, which equates real GDP on the horizontal axis with real GDP on the vertical axis. The second line is the planned spending line. The intersection of the spending line with the 45° line gives the equilibrium level of output.

Figure 31.23

31.31 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:

The equation tells us that there are two reasons for rising prices. The first is because the output gap is negative. The output gap is the difference between potential output and actual output:

A positive gap means that the economy is in recession—below potential output. If the economy is in a boom, then the output gap is negative.

The second reason for rising prices is that autonomous inflation is positive. Autonomous inflation refers to the inflation 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 GDP is greater than potential output, the output gap is negative, so there is upward pressure on prices in the economy. The inflation rate will exceed autonomous inflation. By contrast, when real GDP is less than potential output, the output gap is negative, so there is downward pressure on prices. The inflation rate will be below the autonomous inflation rate. The “inflation sensitivity” tells us how responsive the inflation rate is to the output gap.

The output gap matters because, as GDP 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 aside—that is, even when an economy is at potential output—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 inflation is positive. Figure 31.24 "Price Adjustment" shows the price adjustment equation graphically.

Figure 31.24 Price Adjustment

31.32 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 upward-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:

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 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 consumption-smoothing 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 31.25 "The Consumption Function" shows this relationship.

Figure 31.25 The Consumption Function

31.33 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.

31.34 The Life-Cycle Model of Consumption

The life-cycle 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’s entire lifetime. The model has two key components: the lifetime budget constraint and individual choice given that constraint.

Consider the consumption/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 income—again the same in every year. According to the life-cycle model of consumption, the individual first calculates the discounted present value (DPV) of lifetime income:

(If the real interest rate is zero, then the DPV calculation simply involves adding income flows 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:

31.35 Aggregate Supply and Aggregate Demand

The aggregate supply and aggregate demand (ASAD) model is presented here. To understand the ASAD 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 GDP) decreases when the price level increases. The downward sloping aggregate demand curve does not follow from the microeconomic “law of demand.” 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 GDP.

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 firm’s 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 increases, firms 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 GDP equals potential GDP, and real GDP 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 31.26 "Aggregate Supply and Aggregate Demand in the Long Run".

Figure 31.26 Aggregate Supply and Aggregate Demand in the Long Run

In the short run, output is determined by aggregate demand at the existing price level. Prices need not be at their long-run equilibrium levels. If they are not, then output will not equal potential output. This is shown in Figure 31.27 "Aggregate Supply and Aggregate Demand in the Short Run".

Figure 31.27 Aggregate Supply and Aggregate Demand in the Short Run

The short-run 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 long-run equilibrium level of prices, output is below potential output. The price level adjusts over time to its long-run level, according to the price-adjustment equation.

31.36 The IS-LM Model

The IS-LM model provides another way of looking at the determination of the level of short-run real gross domestic product (real GDP) in the economy. Like the aggregate expenditure model, it takes the price level as fixed. But whereas that model takes the interest rate as exogenous—specifically, a change in the interest rate results in a change in autonomous spending—the IS-LM model treats the interest rate as an endogenous variable.

The basis of the IS-LM 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 (GDP) such that the money market is in equilibrium. This creates the LM curve. The model also finds combinations of interest rates and output such that the goods market is in equilibrium. This creates the IS curve. The equilibrium is the interest rate and output combination that is on both the IS and the LM curves.

LM Curve

The LM 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, firms, 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 31.28 "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 25 "Understanding the Fed".) Figure 31.28 "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 31.28 Money Market Equilibrium

To trace out the LM 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 31.29 "A Change in Income" shows the money market equilibrium at two different levels of real GDP. 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 LM curve is upward sloping: higher real GDP is associated with higher interest rates. At each point along the LM curve, money supply equals money demand.

We have not yet been specific 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 LM curve, however, we put the real interest rate on the axis, as shown in Figure 31.30 "The LM Curve". The simplest way to think about this is to suppose that we are considering an economy where the inflation 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 inflation by noting that changes in the inflation rate will shift the LM curve. Changes in the money supply also shift the LM curve.

Figure 31.29 A Change in Income

Figure 31.30 The LM Curve

IS Curve

The IS curve relates the level of real GDP 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 GDP equals spending. The IS curve is shown in Figure 31.29 "A Change in Income". We label the horizontal axis “real GDP” since, in the short run, real GDP is determined by aggregate spending. The IS curve is downward sloping: as the real interest rate increases, the level of spending decreases.

Figure 31.31 The IS Curve

In fact, we derived the IS curve in Chapter 25 "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 GDP comes from the aggregate expenditure model. Given a particular level of the interest rate, the aggregate expenditure model determines the level of real GDP. 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 GDP.

Equilibrium

Combining the discussion of the LM 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 31.32. This just combines the LM curve from Figure 31.30 "The LM Curve" and the IS curve from Figure 31.31 "The IS Curve". The crossing of these two curves is the combination of the interest rate and real GDP, denoted (r*,Y*), such that both the money market and the goods market are in equilibrium.

Figure 31.32

Equilibrium in the IS-LM Model.

Comparative Statics

Comparative statics results for this model illustrate how changes in exogenous factors influence 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 influence 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 31.33 "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 LM curve, ensuring money market equilibrium. One source of variations in autonomous spending is fiscal policy. Autonomous spending includes government spending (G). Thus an increase in G leads to an increase in output and interest rates as shown in Figure 31.33 "A Shift in the IS Curve".

Figure 31.33 A Shift in the IS Curve

Variations in the real money supply shift the LM curve, as shown in Figure 31.34 "A Shift in the LM Curve". If the money supply decreases, then the LM curve shifts in. This leads to a higher real interest rate and lower output as the LM curve shifts along the fixed IS curve.

Figure 31.34 A Shift in the LM Curve