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Chapter 6
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PART III: FLEXIBLE-PRICE MACROECONOMICS
Chapter 6: Building Blocks of the Flexible-Price
Model
Questions
1. What is a full employment analysis?
2. What keeps the economy at full employment when wages and prices are
flexible?
3. What determines the level of consumption spending?
4. What determines the level of investment spending?
5. Why is the stock market a useful indicator of the likely future level of
investment spending?
6. What determines the level of net exports?
7. What determines the level of the exchange rate?
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In the past two chapters we have looked at long-run growth—at how the economy
develops and evolves over periods as long as generations. In this chapter we shift our
point of view and take instead a “snapshot” view of the economy, looking at it over such
a short period that its productive resources will be fixed. The key questions we will seek
to answer are:
1. What determines the equilibrium level of real GDP (Y)?
2. What economic forces keep real GDP (Y) at its equilibrium level?
3. What determines the composition of real GDP—that is, the division of production
and spending between consumption goods (C), investment goods (I government
purchases (G), and net exports (NX)?
To answer the first two questions we will assume that wages and prices are sufficiently
flexible that markets clear—that every buyer finds a willing seller and every seller finds a
willing buyer. This flexible-price assumption means most importantly that supply equals
demand in the labor market: no firms wishing to hire workers are left unsatisfied, and no
workers who are willing to work are left unemployed. This type of analysis is a fullemployment analysis.
In answering the third question, what determines the composition of spending, we will be
assembling the building blocks for he analysis of the following chapter. In Chapter 7 we
will put these building blocks together and show in detail how the answers to the three
key questions are consistent, and what are the economic forces that ensure that a flexibleprice economy reaches and stays at its equilibrium.
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6.1 Potential Output and Real Wages
In the flexible-price model of the macroeconomy to be developed in this section, two sets
of factors determine the levels of potential (and actual) output and of real wages; the
production function and the balance of supply and demand in the labor market.
The Production Function
Chapter 4 introduced the production function, the rule that tells us how much the
economy can produce given its available productive resources. In the Cobb-Douglas form
of the production function, we learned, potential output (Y*) is determined by (1) the size
of the labor force (L), (2) the economy's capital stock (K), (3) the efficiency of labor (E),
and (4) a parameter  that tells us how fast returns to investment diminish. The
production function tells us that potential output is:

Y*  K  (L E )1 
Figure 6.1 shows a graphical view of one slice of this production function—the
relationship between the capital stock K and potential output Y* holding labor L and the
efficiency of labor E fixed.
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Figure 6.1: The Production Function
Y, GDP
Production
Function
K, Capital
Stock
Legend: A graphical view of the production function. Holding the labor
force and the efficiency of labor constant, real GDP increases as the capital stock
increases. Because each successive addition to capital stock produces a smaller
increase in output, the production function is curved. The higher the level of
, the greater the curvature and the more rapidly the returns to investment
diminish.
In Chapters 4 and 5 all the variables in the production function had the subscript "t,"
indicating the year they referred to. Because we were looking at changes over time,
keeping track of what year we were talking about was important. In this chapter we are
looking at the economy at one instant, not over time, and so we do not have to keep track
of the year. To reduce clutter in the equations we will drop the subscripts and speak of the
labor force L, the capital stock K, the efficiency of labor E, and the level of potential
output Y*.
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The assumption in this chapter that wages and prices are flexible was commonly made by
the so-called “classical” economists who wrote back before World War II. Thus this
assumption is also called the classical assumption. The classical assumption guarantees
that markets work—that prices adjust rapidly to eliminate gaps between the quantities
demanded and the quantities supplied. Thus no businesses find their inventories of unsold
goods piling up. Thus there is full employment: everyone who wants a job (at the marketclearing level of wages) can get a job, and every business that wants to hire a worker (at
the market-clearing level of wages) can hire a worker. And because there is full
employment, actual output is equal to potential output: there is no gap between the
economy’s productive potential and the level of output the economy does produce.
The classical assumption made in this section means that this section is devoted to fullemployment flexible-price macroeconomics.
The flexible-price assumption it is not always a good one. Experience has shown that a
market economy does not always work well and does not always produce full
employment. So while the flexible-price assumption is key in this section, starting in
Section III we will drop it and make instead the “Keynesian” assumption that wages and
prices are sticky (see Table 6.1).
If the classical flexible-price assumption is not always a good one to make, why make it?
It is a good assumption if wages and prices are relatively flexible and have enough time
to adjust in order to balance supply and demand. The classical assumption simplifies the
analysis of several issues, making how the macroeconomy works easier to grasp. In
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general it is better to start with the simpler cases before looking at more complicated
ones. Moreover, the way an economy if the flexible-price assumption held provides a
useful baseline against which to assess economic performance. Nevertheless, we must
remember that this section presents only one model of the economy: the classical model.
The Keynesian sticky-price model behaves very differently in a number of ways.
Table 6.1: Classical Flexible-Price Analysis versus Keynesian Sticky-Price Analysis
Classical
Wages and prices
Expectations
Labor market
Effect of shocks to aggregate
demand
Fully flexible
Consistent with full
employment
Always in equilibrium with full
employment
Changing the composition but
not the level of GDP
Keynesian
Can be "sticky" or "fixed"
Volatile--can take a number of
forms
Can be out of equilibrium,
causing involuntary
unemployment
Changing the composition and
the level of GDP
The Labor Market
When markets work well, what keeps the economy at full employment and actual
production equal to potential output? The answer lies in the adjustment of prices and
supply and demand in the labor market. When the supply of and demand for labor
balance, real GDP will equal potential output.
Labor Demand
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Economists try to suppress every detail and difference that does not matter to the overall
result in order to simplify the analysis and focus it on the key, important factors that
count. Because differences between businesses will not matter, let’s think about an
economy with K typical--identical--competitive firms, each of which owns one unit of
the economy's capital stock. Each of these typical competitive firms hires L workers and
each firm pays each worker the same wage W. Each firm sells Y units of its product at a
per-unit price P.
The typical firm does not control either the wages it must pay or the prices it receives;
those are determined by the market. The firm tries to make as much money as it can. The
typical firm’s profits are simply its revenues minus its costs, and its only costs are the
wages it pays to workers. Therefore:
Profits = Revenues - Costs
Profits = P x Y - W x L
To figure out how many workers to hire, the firm follows two simple rules:
1. Hire workers to boost output.
2. Stop hiring when the extra revenue from the output produced by the last worker
hired just equals his or her wage.
The value of the output produced by the last worker hired is the product price P times the
marginal product of labor [MPL]. The cost of hiring the last worker is his or her wage W.
The firm will keep hiring until:
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P x MPL - W = 0
What is the MPL, the marginal product of labor? The typical firm owns one unit of
capital, and its output is what can be made by that single unit of capital and the firm’s
workers according to the production function:
Yfirm = F(1, Lfirm)
The marginal product of labor is the difference between what the firm can produce with
its current labor force Lfirm, and what it would produce if it hired one more worker (see
Figure 6.2):
MPL = F(1, Lfirm + 1) - F(1, Lfirm)
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Figure 6.2: The Firm’s Output as a Function of the Firm’s Employment
Output
Output as a function of labor hired
Increases firm output by the marginal product of labor
Hiring
one
extra
worker
Labor Employed
Legend: Holding the capital stock of the typical firm constant, each extra worker
the firm employs produces smaller and smaller increases in total output. As the
level
of employment increases, the marginal product of labor [MPL] decreases.
Because we want a form of the MPL that is simple to work with, we will consider the
Cobb-Douglas production function. The MPL for the Cobb-Douglas production function
is:
MPL  (K fi rm)  E 1 (Lfi rm 1)1  (K fi rm)  E1 (Lfi rm )1
Again, we take how much the firm would produce if it hired one more worker and
subtract how it produces now, with its current labor force. Since the firm has only one
unit of capital, we can rewrite this equation as:
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MPL  (1)  E1 (L firm  1)1  (1)  E 1 (Lfirm )1

MPL  E 1 (Lfirm  1)1  (L firm )1

Note that the term inside the brackets looks like the rate-of-growth of a variable growing
by one unit (the firm’s labor force Lfirm) and then raised to a power (the term 1-). We
have a standard rule-of-thumb (look back in Chapter 2 at Box 2.3: Useful Mathematical
Tools) for finding the rate of growth of a variable raised to a power, which tells us that
the term inside the brackets is:
(L
fi rm

1)1  (Lfi rm)1   (1  ) 
1
L 

fi rm
So the MPL is:
MPL 
(1   )E1
(Lfirm )
There is nothing deep in this math. Indeed, the Cobb-Douglas function was carefully
tweaked so that it would yield such simple forms for quantities like the MPL. That is why
economists use it so often. If the Cobb-Douglas production function produced more
complicated expressions, we would not use it.
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Figure 6.3: The Typical Firm's Hiring Policy
Revenue: firm output times product price
Cost: wage times number of workers employed
Revenue as a function of labor hired
Profit-maximizing output
Labor costs
Profit
Profit-maximizing
employment
Labor Employed
Figure legend: The typical firm chooses to hire the number of workers that make
marginal revenue--the MPL times the product price P--equal to the wage W. At
that
point the revenue and cost curves are parallel, and profit is maximized.
As Figure 6.3 shows, the firm hires workers up to the point where the product price times
the marginal product of labor equals to the wage:
P x MPL - W = 0
Substituting for the MPL in this equation, we get
P
(1  )E 1
W
(Lfirm )
Next, we rearrange this equation to see that the typical firm's demand for workers is:
1
Lfirm
(1  )E1  
 
 (W / P) 
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Because there are K firms in the whole economy, total economy-wide employment is
equal to K times the typical firm’s demand for labor:
1

(1 )E1 

L K
 (W / P) 

d
Labor Market Equilibrium
In the last section we calculated economy-wide labor demand. But what is the labor
supply? The answer is simple: it is the number of workers who want to work. The labor
market will be in equilibrium when firms' total demand for workers is equal to the labor
force.
Can the labor market not be in equilibrium if wages and prices are flexible? Think about
what would happen if supply were not equal to demand. Suppose there are more workers
than firms wish to hire at current wages and prices. Then some of the unemployed will
underbid their employed fellow workers: offer to take their jobs and work for less. Those
workers who are employed will respond by offering to accept lower wages to keep their
jobs. The wage W will decline relative to the price level P, and the real wage W/P will
fall. As the real wage falls, firms will hire more workers.
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Figure 6.4: Equilibrium in the Labor Market
Legend: The equilibrium level of employment is equal to the labor force. At
the equilibrium level of the real wage, there is neither excess
demand for nor excess supply of labor.
Suppose firms want to hire more workers than there are people in the labor force. Some
firms will try to bid workers away from other firms by offering higher wages. The real
wage W/P will rise. As the real wage rises, employers will reduce the quantity of labor
they demand.
Thus in equilibrium, labor demand Ld will equal the labor force L (see Figure 6.4):
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1

(1  )E 1  

LL K
 (W / P) 
d
Labor demand is equal to the labor force when the real wage W/P is:

W
K
Y
 (1 )E1    (1   ) 
 L 
 L
P
and each of the K firms in the economy employs L/K workers. As long as wages and
prices are flexible enough for this adjustment process to work, the economy will remain
at full employment.
Note that a full employment economy is not necessarily the best or even a good economy.
The real incomes of those who don't own chunks of the capital stock are their real wages:
W/P = (1-) x (Y/L). If  is large, their real incomes will be small, and social welfare
may be low.
Employment and Output
When the labor market is in equilibrium the typical firm produces a level of output equal
to:
Yfirm  (1) (E)1 (L/ K)1 
Because there are K firms, total output Y is simply K times the typical firm’s output: Y =
K x Yfirm. This is the same potential output, Y* in the Cobb-Douglas form of the
production function analysis
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
1
Y  K  Yfi rm  K  (1) (E)
1
(L / K)
Final

1 
 (K) (E)
1
(L)
 (K) (LE)1  Y *
The conclusion is clear: if markets work well—if wages and prices are flexible and adjust
to balance supply and demand, and if markets are competitive—then the actual level of
production in the economy Y will equal to the economy’s potential output Y*, as Figure
6.5 shows.
Figure 6.5: In a Full-Employment Economy, Real GDP Equals Potential Output
Real GDP
National Income
Production Function
Potential Output
Labor
Force
Number of Workers
Employed
Legend: When the economy is at full employment, the level of employment is
equal
to the labor force, and real GDP is equal to potential output.
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6.2 Domestic Spending
In Chapters 2 and 3 we saw that national income is divided into four components:
1. Consumption spending (C)
2. Investment spending (I)
3. Government purchases (G)
4. Net exports, the balancing item (NX).
These four components add up to national income, which according to the circular-flow
principles is the same as real GDP, Y:
C + I + G + NX = Y
In this section we will look at the determinants of the three domestic components of
spending, C, I, and G. we will discuss international trade in a later section.
Figure 6.6: The Four Components of Spending Add Up to Real GDP
Consumption Spending
Investment Spending
Real GDP
Government Purchases
Net Exports
Legend: The circular flow principle guarantees that the four components of
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spending—consumption, investment, government purchases, and net exports--add
up to real GDP.
6.2.1 Consumption Spending
Households earn all the income in the economy. The wages earned by the labor of their
workers plus the profits--rent, interest, dividends, and retained earnings--earned by their
property together add up to national income, which at this level of analysis is the same
thing as real GDP. So we use the letter “Y” for both, relying on the circular flow
principle to ensure that whatever businesses produce and sell shows up as income for
households.
The Household’s Decisions
Households earn all the income in the economy. The wages earned by the labor of their
workers plus the profits--rent, interest, dividends, and retained earnings--earned by their
property together add up to national income, which at this level of analysis is the same
thing as real GDP. So we use the letter “Y” for both, relying on the circular flow
principle to ensure that whatever businesses produce and sell shows up as income for
households.
Households pay some of their income to the government in net taxes--taxes less transfer
payments to households from the government--which we will write T. To keep things
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simple, throughout this book assume that net taxes paid are equal to the constant average
tax rate t multiplied by national income:
T=txY
What is left after households pay their taxes is their disposable income, written YD
YD = Y - T = (1-t)Y
Households save some of their income to boost their wealth and spend in the future.
Write "S" for this amount of private savings. Households spend the rest of their income-everything that is not paid to the government in taxes or saved--buying consumption
goods:
C = YD - S = Y - T - S
Consumption spending C--purchases by households for their use, from pine nuts and
flour to washing machines and automobiles--adds up to two-thirds of GDP.
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Figure 6.7: From National Income to Consumption Spending
National Income
Minus Taxes
Disposable Income
Minus Household
(and Business)
Savings
Consumption Spending
Legend: Subtract taxes from national income to get disposable income.
Subtract savings (households plus business) from disposable income to get
consumption spending.
Consumption spending is a function of disposable income YD:
C = C0 + Cy x YD = C0 + Cy x (1-t)Y
Final
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Consumption spending is equal to a baseline level of consumption (the value of the
parameter C0) plus a fraction (the parameter Cy) times disposable income—YD, equal to
the fraction (1-t) times total income Y.
Figure 6.8: Other Determinants of Consumption Spending
Income
Distribution
Demography
Household Wealth
Tolerance
for
Risk
Consumption
Spending
Permanent vs. Transitory
Income
Expectations of
Income Growth
Real Interest Rate
Relative Optimism
or Pessimism
Notice that in writing down this consumption function we have once again followed
economists' principle (or vice) of ruthless simplification. In this complicated world
consumption spending does not mechanically and directly depend on disposable income
alone. Other factors affect consumption spending. Consumption changes in response to
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changes in the real interest rate, in households' total stock market and real estate wealth,
in the demographic structure of the population, consumers' relative optimism, and on
whether consumers see changes in disposable income as transitory or permanent. (If
consumers expect an income increase to be transitory, they will save most of it and only
use a little bit for higher consumption spending.; if they expect an income increase to be
permanent, they will use most of the increase to support higher consumption spending.)
But here and throughout the book--to keep our analysis simple--we sweep all these
complications under the rug. We think only about baseline consumption C0, the marginal
propensity to consume Cy, and disposable income YD as the determinants of consumption
spending.
6.2.1.2 The Marginal Propensity to Consume
The baseline level of consumption—the parameter C0--is the amount that households
would spend on consumption goods if they had no income at all. It is the amount by
which they would draw down their savings and decrease their wealth in the absence of
income in order to try to keep body and soul together.
The marginal propensity to consume—also called the MPC, and the same thing as the
parameter Cy in the consumption function—is the amount by which consumption
spending rises in response to a $1 increase in disposable income. We are sure that Cy will
be greater than zero: if incomes rise, households will use some of their extra income to
boost consumption spending. We are sure that Cy will be less than one: as incomes rise,
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households will increase their savings as well—they will not spend all of their extra
income on consumption goods.
The value of the marginal propensity to consume also depends on how long people
expect the change in income to last. As is discussed in the first appendix to this chapter, if
people expect the change in income to be permanent, then the MPC is likely to be
relatively large. If people expect the change in income to be transitory—and income next
year to revert back to its normal pattern—they are likely to save most and spend little of
this transitory windfall, and the MPC is likely to be relatively small.
Figure 6.9: The Consumption Function
Consumption
Spending
Consumption Function
Resulting increase in consumption
= marginal propensity to consume [MPC]
One unit
= Cy
increas e in
dispos able
income
C0
Disposable Income Y
d
Legend: Consumption spending depends on the level of disposable income
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and on two parameters: Cy, the marginal propensity to consume [MPC], and C0,
the
baseline level of consumption. If we know these parameters and disposable
income YD, then we can graph what the level of consumption spending will be for
each possible level of disposable income
Box 6.1--Example: Calculating Consumption from Income
If we know the parameters C0 (the baseline level of consumption), Cy (the
marginal propensity to consume, or MPC) and t (the tax rate), we can calculate
what the level of consumption spending C is for any level of total national income
Y using our equation:
C = C0 + Cy x YD = C0 + Cy x (1-t)Y
Suppose that the tax rate t is 25%, that total national income Y is $10 trillion, that
the baseline level of consumption C0 is $2 trillion, and that the marginal
propensity to consume Cy is 0.6. Then we first calculate disposable income: how
much households have after paying their taxes. Disposable income is equal to (1t)Y, which for these parameter values and this level of national income is:
(1 – 0.25) x $10 trillion = $7.5 trillion
We can then calculate consumption directly:
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C = C0 + Cy x $7.5 trillion
C = $2 trillion + 0.6 x $7.5 trillion
C = $6.5 trillion
What would happen if disposable income were boosted from $7.5 trillion to $8
trillion? Consumption spending would rise from $6.5 trillion to $6.8 trillion—an
amount equal to the marginal propensity to consume, 0.6, times the change in
disposable income, $500 billion.
6.2.2 Investment Spending
Investment spending is, on average, about 15% of GDP. But investment spending is the
most volatile component of GDP: it fluctuates the most.
Fluctuations in economy-wide investment spending have two sources. First, the higher is
the real interest rate, the lower is investment spending. A higher real interest rate makes it
more expensive for firms to undertake investment projects, and so they undertake fewer
of them. Second, the higher is business manager and investor confidence--what John
Maynard Keynes called their "animal spirits"--the higher is investment spending. The
more optimistic are managers and investors, the more willing they are to bet their jobs
and fortunes on the proposition that an expansion of productive capacity or some other
investment will pay off.
Box 6.2--Details: What Is Investment?
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When economists use the term "investment," they mean something different from
what most people mean by the term. Most people use it to mean activities such as
buying a stock or bond, a certificate of deposit, or commodity futures. But such
activities do not directly increase the economy's capital stock or have any place in
the national income and product accounts.
When economists use the terms "investment" or "investment spending," they are
talking about transactions that add to the capital stock and increase potential
output: (a) the purchase and installation of new business machinery and
equipment, (b) the construction and purchase of a new building (or the repair of
an old one), and (c) a change in business inventories.
Box 6.3--Details: Kinds of Investment.
The first distinction economists draw is between "gross investment" and "net
investment." Gross investment is the total sum of spending on machines, on
building structures (houses, factories, office buildings, as well as roads, dams, and
bridges), and on additions to inventories.
Some of gross investment adds to the capital stock of machines, goods-in-process,
buildings, and other structures that amplifies productivity. The rest of gross
investment replaces worn-out and obsolete pieces of capital. That amount of gross
investment spending that increases the capital stock is called "net investment."
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That amount of investment spending that replaces obsolete and worn-out capital is
called "depreciation" or "capital consumption."
A second--independent--division of investment is into:
4. Residential construction
5. Non-residential construction
6. Equipment investment
7. Inventory investment
To some degree, these four subcategories of investment have different
determinants and different consequences. But here we seek to simplify in order to
construct a useful model, so we ignore these differences.
6.2.2.1 Why Firms Invest
A business invests because its managers believe that the investment will be profitable: the
appropriately-discounted return on the investment must be greater than the investment's
cost. If the interest rate is higher, then the appropriately-discounted return on the
investment will be lower. If the interest rate is higher, fewer potential investment projects
will be profitable. Thus a lower interest rate leads to higher investment spending.
How many investment projects will a higher interest rate discourage? How much lower
will investment be if the interest rate is higher? To analyze that question is beyond the
scope of the main thread of this book. (This chapter's appendix B, however, sketches out
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Final
the key tool of present value that is used in finance to help assess whether an investment
project is worth undertaking at a particular interest rate.)
The interest rate most relevant for determining investment is the long-term real interest
rate. Investments affect the business's costs for a long time to come. An investment gives
the business that makes it a real, physical asset: the ability to make commodities and earn
purchasing power that provides command over real goods and services. To calculate
whether investment projects are worthwhile, be sure to compare apples to apples.
Discount the (long-term, real, risky) profits from undertaking an investment project by
the appropriate (long-term, real, risky) measure of the interest rate.
6.2.2.2 The Investment Function
We model this inverse relationship between the level of investment spending and the
long-term real risky interest rate by an investment function:
I = I0 - Ir x r
We set investment spending I equal to the baseline level of investment (the value of the
parameter I0) minus the real interest rate r times the slope-of-the-investment-function
parameter Ir.
Notice the pattern used for parameters so far: C0, Cy, I0, Ir. This should make the symbols
used in algebraic equations easier to remember and much clearer. The capital letter in the
name of the parameter tells you which variable is on the left-hand-side of the equation in
which this parameter appears: a “C” means that this parameter is part of an equation
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determining the level of consumption spending C; an “I” means that this parameter is part
of an equation determining the level of investment spending. The subscript tells you
which variable the parameter multiplies in that equation: Ir is the amount by which
investment spending I changes in response to a change in the real interest rate r.
Once again this particular function is an enormous simplification of what goes on in the
real world. In the real world firms’ investment decisions depend not only on the real
interest rate but also on how much money firms have available. Total profits are an
important determinant of investment. In the real world some components of investment—
the building of residential structures, for example—are very sensitive to changes in real
interest rates. Other components of investment—inventory investments by small firms
that have little access to outside sources of funding, for example—are not.
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Figure 6.10: The Investment Function
Investment I
I
0
A one-unit increase in
the real interes t rate...
...generates a decreas e in real investment spending of I r
Real Interest Rate r
Box 6.4—Economic Policy: How to Boost Investment.
As we saw in chapters 4 and 5, a high level of savings and investment is one of
the keys to having a prosperous economy. The higher the share of investment
spending in GDP, the higher is the steady-state capital-output ratio and the richer
the economy will be.
Governments seeking to boost investment have two major tools at their disposal.
First, they can lower real interest rates (or induce the central bank to lower real
interest rates). If real interest rates are lower, more investment projects will be
undertaken and investment spending will be higher. Second, governments can try
to make the baseline level of investment higher by attempting to make private
Chapter 6
30
Final
decision makers more optimistic. They can exhort and reassure—although this
sometimes backfires, as in President Herbert Hoover’s repeated declarations
during the Great Depression that “prosperity is just around the corner.” More
important, they can try to reduce or eliminate sources of risk. Confidence that the
economy will be stable, and that risks will be managed, is perhaps the best way to
boost investment by encouraging optimism.
Box 6.5—Example: How Investment Responds to a Change in Interest Rates.
From the parameters I0 (the baseline level of investment) and Ir (the
responsiveness of investment to a change in real interest rates) we can calculate
what the level of investment spending will be for each possible value of the real
interest rate r.
For example, suppose that I0 is $2 trillion, and that Ir is $100 billion. Then we can
use the equation:
I = I0 - Ir x r
to calculate that if the real interest rate is 5%, then the level of investment
spending will be $1.5 trillion:
I = $2 trillion - $100 billion x 5 = $1.5 trillion
If the real interest rate is 10%, the level of investment spending will be $1 trillion:
I = $2 trillion - $100 billion x 10 = $1 trillion
And if the real interest rate is 0%, the level of investment spending will be $2
trillion:
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31
Final
I = $2 trillion - $100 billion x 0 = $2 trillion
Box 6.6—Details: The Stock Market
An alternative way of writing down the investment function—not used in this book
because it would complicate our equations and models—is to make the level of
investment a function of the level of the stock market. To see why, think about what
determines stock market values. Most investors in the stock market face a choice
between holding stocks--shares of ownership of a corporation that also give you
ownership of that corporation's profits or earnings--or holding bonds: a piece of paper
that represents a that pays interest. If you invest your money in bonds, you earn the
real interest rate r. If you invest in shares of stock, your return is equal to your share
of the profits of the companies in which you have invested.
When expected future profits are high, investors will find stocks more attractive and
will bid up their prices. The stock market will rise. When the real interest rate falls,
investors will find stocks more attractive and will bid up their prices. The stock
market will rise.
But it is also the case that when expected future profits are high, businesses will
invest more. And when the real interest rate falls, businesses will find investment
projects cheaper and will invest more. The same things that determine the stock
market determine the level of investment. The stock market and investment move
together: what raises or lowers one raises or lowers the other.
Chapter 6
32
Final
The only significant difference is that causes of fluctuations in investment affect the
stock market first and investment spending second. The stock market is thus a very
useful leading indicator of investment spending. Thus keep a close watch on the stock
market if you want to forecast the level of investment spending.
6.2.3 Government Purchases
The federal government buys the labor of government employees--judges, air traffic
controllers, custom inspectors, FBI agents, National Oceanic and Atmospheric
Administration researchers, and others--military hardware, sections of the interstate
highway system, and other goods and services. All of these make up the government
purchases component of GDP. Such government purchases of goods and services add up
to about 25 percent of GDP.
Notice that government spending is larger than government purchases. The government
also spends money not buy buying goods and services but by transferring money to
citizens: Social Security payments, disability payments, food stamps, and other transfer
payments. These transfer payments are not themselves demand for final goods and
services. Therefore they do not show up directly in GDP, or in the government purchases
total G. Where do transfer payments show up in the NIPA? As negative taxes: the
variable T--taxes--is net taxes, taxes less transfer payments, the net amount by which the
government reduces disposable income.
Chapter 6
33
Final
Figure 6.11: Government Purchases, Transfer Payments, and Taxes
Transfer Pay ments
Households
Businesses
Government
Tax es
Government Purchas es
Goods and Services
Purchased
In this book we set net taxes, T, taxes less transfers equal to the average tax rate t times
GDP Y:
T  tY
We do not inquire into what determines either the amount of government purchases G or
the net tax rate t: that is left for the political scientists. We do look at what happens when
government spending G, or the tax rate t changes
6.3 International Trade
The final component of GDP is net exports--the difference between gross exports and
imports. Gross exports are made up of goods and services that are produced here in the
Chapter 6
34
Final
home country and then sold. GDP is a measure of production. Exports are part of
production. So gross exports need to be counted in GDP.
But we also need to subtract imports from GDP. Not all the goods and services that make
up consumption, investment, and government purchases are produced domestically.
Consumption and the other spending categories include spending on consumption goods
made here and on consumption goods made abroad. So adding up C and I and G
overestimate for U.S. made products. By adding net rather than gross exports to C+I+G,
we (a) take account of goods made here that are sold to foreigners and don't show up in
C+I+G, and (b) correct the sum of C, I, and G for the amount by which it overstates
domestic demand for the U.S.-made goods.
Figure 6.12: Gross Exports, Imports, and Net Exports
Gross Exports
-Minus-
Gross Imports
-Equals-
Net Imports
Chapter 6
35
Final
6.3.1 Gross Exports
The volume of gross exports from here to abroad depends on two things. First, it depends
on the value of real GDP in our trading partners—call it Yf, f standing for "foreign" and
Y standing for "GDP". Second, it depends on the value of the real exchange rate—call it
. The higher is the value of the real exchange rate—the more expensive is foreign
currency—the cheaper do home-produced goods appear to foreigners, and the more of
them they buy.
Thus we write down our function for gross exports GX:
GX  Xyf  Y  X   
f
Here, just as in the investment and consumption equations above, X and Xyf are
parameters. The X shows that they are parameters that help determine gross exports. Xyf
is the increase in exports generated by an increase in GDP abroad. It is the proportion of
foreign income spent on our exports. X is the increase in exports produced by a rise in
the real exchange rate A higher value of the exchange rate means that foreign currency
is more valuable. Domestically-produced goods are cheaper to foreigners. So foreigners
buy more of them.
Chapter 6
36
Final
Figure 6.13: Gross Exports and the Real Exchange Rate
Figure: During the decade of the 1980s the value of the U.S. dollar
increased and then collapsed--thus the number we use to denote the real exchange
rate, the value of foreign currency, fell from a value 20 percent above its 1973
level to a value 25 percent below its 1973 level, and then rose back. As the real
exchange rate fell, U.S. exports became relatively more expensive to foreigners,
and so exports fell. As the real exchange rate rose in the late 1980s, U.S. exports
became relatively cheaper, and so exports rose.
Source: Economic Report of the President (Washington, DC: Government
Chapter 6
37
Final
Printing Office), 1999 edition.
Box 6.7--Details: The J-Curve
A complication: trade links across countries take time to create, time to modify,
and time to destroy. A depreciation of the U.S. real exchange rate will cause an
increase in foreign purchases of U.S. goods--but it will take a year or so before a
change in the real exchange rate has an effect on the volume of trade. In the shortrun a depreciation of the dollar may see a fall, not a rise, in exports--a
phenomenon economists call the "J-curve," because someone once thought that
the plot of exports over time looked a little like a J. In the 1980s, the real
exchange rate--the value of foreign currency--began to rise very steeply in 1986,
but real exports in 1986 were flat. It was not until 1987 and 1988 that the
increased competitiveness of U.S. exporters led to an export boom. But in this
chapter we ignore these lags.
Chapter 6
38
Final
Figure: The J-Curve
Legend: The J-curve in action: export volumes lag real exchange
rate movements by a year and a half.
Source: Economic Report of the President (Washington, DC: Government
Printing Office), 1999 edition.
6.3.2 Imports and Net Exports
Our demand for imports--for products produced abroad--depends on our real GDP: the
higher is real GDP and thus total national income, the more imports we want to buy.
Chapter 6
39
Final
Our demand for quantities of imports also depends on the real exchange rate : the more
appreciated the real exchange rate and the higher the value of the home currency, the
cheaper foreign-made goods seem and the more of them we want. But we are interested
in the value of imports. And when the exchange rate is appreciated, the home-currency
price of each imported good is cheap. These two effects roughly cancel each other out. So
simplify by modeling gross imports IM as equal to a constant share—a share determined
by the parameter IMy --of real GDP Y.
IM  IMy  Y
Net exports NX are the difference between gross exports and imports. Thus they depend
on the real exchange rate, ; on real GDP abroad, Yf; and on real GDP here at home, Y:
NX  GX  IM  Xyf  Y
f
 X

   IMy  Y 
6.3.3 The Exchange Rate
The exchange rate is probably the most important determinant of net exports. We have
just gone through how the exchange rate affects net exports. But what determines the
exchange rate?
Consider those foreign exchange speculators whose job it is to trade currencies and make
money. They spend their days glued to computer terminals. They watch the prices of
bonds flash across the screen, buying and selling bonds and stocks of different countries
Chapter 6
40
Final
and governments denominated in different currencies--dollars, euros, pounds, yen, pesos,
ringgit, and more than one hundred others. Their lives are ruled by greed and fear:
Greed: Suppose a trader sees a gap between real interest rates paid on bonds of U.S.
companies denominated in dollars $ and the bonds of German companies dominated in
euros ¤. If American companies are the ones paying higher interest rates, there is money
to be made by selling German companies' bonds, buying American companies' bonds
instead, and pocketing the interest. If German companies are the ones paying the higher
interest rates, there is money to be made by selling American companies' bonds and
buying German companies' bonds instead.
Fear: Suppose that the trader has bought--is long--American companies' bonds, and the
dollar then depreciates. The higher interest payments will immediately be swallowed up
by the exchange rate-driven loss of value. And if today's value of the currency is highly
appreciated--if the currency is very valuable relative to long-run historical trends--the
fear that exchange rates will return to normal relationships and impose large foreign
exchange losses will be immense.
Chapter 6
41
Final
Figure 6.14: Greed and Fear in Foreign Exchange Markets
Fear: a portfolio invested
in dollar-denominated securities
when the dollar is overvalued may
suddenly suffer a large capital loss if
the dollar loses value on the foreign
exchanges
Foreign
Exchange
Traders
Greed: a portfolio invested in
dollar-denominated
securities earns higher
interest rates
The greater the difference in interest rates in favor of dollar-denominated securities, the
higher the greed factor. And the higher the greed factor, the more appreciated must the
dollar be for fear to balance greed. The enormously-liquid, enormously-high-volume
foreign exchange markets settle at the point where the greed and fear balance.
Thus the real exchange rate  is equal to the average trader's opinion 0 of what the
exchange rate should be if there were no difference in interest rates, minus a parameter r
times the interest rate difference between home real interest rates r and foreign real
interest rates rf:
   0   r  r  r f 
The longer are interest rate differentials expected to continue, and the more slowly are
real exchange rates expected to move back to trend, the higher will be r: the larger will
be the effect of a given interest rate differential on the exchange rate.
Chapter 6
42
Final
Remember: the exchange rate is the value of foreign currency: if foreign currency
becomes more valuable, the exchange rate rises. Thus an appreciation or revaluation of
the dollar is a reduction in the value of the exchange rate. Thus a depreciation or
devaluation of the dollar is an increase in the value of the exchange rate.
We can take our equation for net exports:
NX  GX  IM  Xyf  Y
f
 X

   IMy  Y 
And substitute into it our equation for what the value of the exchange rate is. The result
is:
NX  GX  IM  Xyf  Y
f
 X
  0  X   r  r X   r  r  IMy  Y 
f

This equation tells us directly how interest rates at home and abroad affect net exports. It
is often more useful to look at this equation—with the exchange rate removed. With the
exchange rate removed, we have for a moment one fewer thing to keep track of. This is
an example of one of economists' standard techniques: tweak the model to rid it of as
many complicating variables as possible. Such reductions in the number of variables that
you have to keep track of are called--for natural reasons--"reduced forms."
6.4 Conclusion
This chapter has begun to analyze a flexible-price full-employment economy in the short
run. This short run is a time short enough in which neither labor nor capital stocks have
an opportunity to change, and thus potential output does not grow. Nevertheless, this
Chapter 6
43
Final
short run is long enough for wages and prices to be flexible enough for supply to match
demand. This is most important in the labor market: in the analysis of this chapter there is
no unemployment, for there are always jobs available for workers willing to work at the
market-clearing wage.
In a flexible-price full-employment economy, the level of real GDP is equal to
potential output. The balance of supply and demand in the labor market keeps the
economy at full employment. If firms’ demand for labor is less than the labor force,
wages will fall. As wages fall, firms’ demand for labor will rise. If labor demand is
greater than the labor force, firms competing for labor will raise wages. And higher
wages will soon curb labor demand.
We know the level of real GDP: it is equal to potential output. But how is real GDP
distributed among the different categories of demand? How many export goods are
produced? How many consumption goods? How many investment goods? The
division of real GDP among categories of products (or, what is the same thing, the
division of national income among categories of spending) is determined categoryby-category. Each category of spending has a different set of determinants. Thus each
of the building blocks of our analysis looks somewhat different.
The level of consumption spending C is determined by many things, but the most
important of them are the level of disposable income YD, the baseline level of
consumption C0, and households’ marginal propensity to consume (MPC) Cy. The
level of disposable income is determined by the level of national income Y and the
tax rate t:
Chapter 6
44
Final
YD = (1-t)Y
Putting these determinants together produces the consumption function:
C = C0 + Cy(1-t)Y
The level of investment spending I is primarily determined by two factors. The first is
businesses' degree of optimism: could they increase their profits by investing? The
second is the real interest rate. The higher the real interest rate, the lower is the level
of investment spending. Our simple investment function is thus:
I = I0 – Ir x r
The stock market is a useful indicator of the likely future level of investment
spending because its value, too, depends on the general degree of optimism about
future profits and on the real interest rate. The stock market and investment move
together: what raises or lowers one raises or lowers the other, usually affecting the
stock market first and investment spending second. The stock market is thus a very
useful leading indicator of investment spending.
The exchange rate  is determined by (a) foreign exchange traders' view 0 of the
long-run equilibrium level of the exchange rate, and (b) the interest rate differential r
- rf between investments at home and abroad. Net exports are determined by (a) the
level of the exchange rate  and the sensitivity of exports to the exchange rate X, (b)
the level of national income Y and the share IMy of national income spent on imports
(which together determine the level of imports), and (c) the level of foreign income
Yf and the share Xyf of foreign income spent on our exports (which affects the level of
exports). Putting all these together gives us the last of our building-block equations,
Chapter 6
45
Final
the equation describing the function for net exports:
NX = Xyf x Yf + X x 0 + X x r (r – rf) – IMy x Y
Needless to say, the short-run analysis begun in this chapter is not complete. This chapter
has presented only the building blocks of the analysis: understanding how the economy
reaches and restores its equilibrium has to be deferred until chapter 7. This chapter has
taken a partial, snapshot view of the economy. It has not discussed the impact of changes
in policy and the economic environment on economic growth--that was done in chapters
4 and 5. Refer back to them to analyze how changes in savings ultimately affect
productivity and material standards of living in the long run. Moreover, this chapter has
ignored the nominal financial side of the economy--money, prices, and inflation-completely. That will be covered in chapter 8.
Last and most important, it has maintained the flexible-price full-employment
assumption. We will see what happens when that assumption no longer holds—and when
the economy suffers from unemployment—in the chapters that follow chapter 8.
6.5 Chapter Summary
6.5.1 Main Points
When the economy is at full employment, the level of real GDP is equal to
potential output: the level of output generated by the aggregate production
Chapter 6
46
Final
function of chapter 4, given the current stocks of labor and capital and the current
level of the efficiency of labor.
When wages and prices are flexible, the working of the labor market keeps the
economy at full employment: if labor demand is less than the labor force, falling
wages raise employment; if labor demand is greater than the labor force, rising
wages soon curb labor demand.
The level of consumption spending is determined by many things, but the most
important of them is the level of disposable income.
The level of investment spending is primarily determined by businesses' degree of
optimism--could they increase their profits by investing?--and by the real interest
rate.
The stock market is a useful indicator of the likely future level of investment
spending because its value, too, depends on the general degree of optimism about
future profits and on the real interest rate.
The exchange rate is determined by (a) foreign exchange traders' view of the
long-run equilibrium level of the exchange rate, and (b) the interest rate
differential between investments at home and abroad.
Net exports are determined by (a) the level of the exchange rate, (b) the level of
real GDP (which determines the level of imports), and (c) the level of real GDP
Chapter 6
47
abroad (which affects the level of exports).
6.5.2 Important Concepts
Production function
Labor market
Labor supply
Labor market equilibrium
Marginal product of labor [MPL]
Firm profit maximization
Real wage
National income identity
Potential output
Disposable income
Consumption function
Marginal propensity to consume [MPC]
Investment function
Real interest rate
Government purchases
Transfer payments
Net taxes
Gross exports
Gross imports
Final
Chapter 6
48
Final
Net exports
Real exchange rate
Foreigners
6.5.3 Analytical Exercises
1. What, in the full-employment model, determines the level of real GDP?
2. What makes labor demand equal to the labor force in the economy as a whole?
3. What determines how much labor a typical firm decides to hire?
4. What happens if the parameter C0 in the consumption function rises?
5. What happens to the level of investment spending if the interest rate falls?
6. What happens to the level of investment spending if businesses become more
optimistic about future profits?
7. What happens to net exports if foreign exchange speculators become more optimistic
about the long-run real value of the domestic currency?
8. What happens to net exports if interest rates abroad rise?
9. What happens to net exports if national income rises?
Chapter 6
49
Final
10. Does an increase in the tax rate make consumption spending more sensitive or less
sensitive to changes in real GDP?
6.5.4 Policy-Relevant Exercises [to be updated every year…]
1. Suppose an economy with the standard Cobb-Douglas production function:

Y*  K  (L E )1 
has a value of diminishing returns to scale parameter =1/3, a value of the labor force L
equal to 100 million workers, a value of the capital stock K equal to $40 trillion, and an
efficiency of the labor force E equal to $50000. What is the value of potential output Y*?
What is the value of potential output per worker, Y*/L? What is the market-clearing real
wage (in dollars per year) at which the economy is at full employment, with neither
unemployed workers nor excess demand for labor?
2. If an economy at full employment with an unchanging diminishing returns to scale
parameter  has output per worker growing at 3% per year, at what rate must the real
wage be growing in order to maintain full employment?
3. What would you expect has happened to real investment in the United States over the
past five years given that the real value of the stock market has doubled?
4. Over the past decade foreign exchange speculators have become much more confident
in the long-run value of the dollar. What would you suspect has happened to net exports
over the past decade?
Chapter 6
50
Final
5. Consumers whose stock market wealth has multiplied over the past decade have
recently started pulling money out of the stock market to enhance their standard of living.
What kind of shift in which parameter of the consumption function could be used to
capture and model this phenomenon?
6.A. Appendix: A Closer Look at Consumption
6.A.1. Permanent and Transitory Income
One of the most important of the factors omitted from our consumption function was the
distinction between permanent and transitory income. Your permanent income is the
average level that you expect your level of income to be in the future. Your transitory
income is the difference between your income now and your permanent income. Milton
Friedman was the very first to point out that not this year's income but "permanent
income" was likely to be the main determinant of consumption.
Think of a consumer trying to decide how much to spend in two periods only--"now" and
the "future." Suppose that you can save (or borrow) in the present for the future at a real
interest rate r. And suppose that the “now” and the “future” are not necessarily the same
length of time: that the future period is some parameter  times the length of the present
period.
Chapter 6
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Final
In the present the typical consumer decides how much to spend and how much to save.
Income Ynow, consumption Cnow, and saving S are linked by:
S = Ynow – Cnow
In the future the consumer adds his or her future income to savings—which have in the
meantime grown because they have earned real interest at rate r--and spends the total:
you can't take it with you, after all:
 x Cfuture =  x Yfuture + (1+r)S
Combining these two equations produces what economists call the present value form of
the consumer's budget constraint: /(1+r)
Cnow + [Cfuture/(1+r)] = Ynow + [Yfuture/(1+r)]
Total consumption spending now and in the future must equal total income in the two
periods, where future income and consumption count a little bit less—are “discounted”
by the real interest rate—in order to turn them into “present values” before adding them
to current consumption. Because savings earn interest, you must put aside only 1/(1+r)
dollars today in order to be able to spend 1 dollar on consumption in the future. Hence 1
dollar of consumption—or income—in the future has a “present value” of only 1/(1+r)
dollars today.
We can show this budget constraint on a diagram that plots "now" on the horizontal and
"future" on the vertical axis. By borrowing or saving the consumer can redistribute
consumption across time:
Chapter 6
52
Final
The Budget Constraint
Legend: You can read off income in the present and the future from the
circle showing the consumer's income in both periods. By borrowing and
lending, the consumer is free to choose levels of consumption in the
present and the future corresponding to any point along the budget
constraint line. The slope of the budget line corresponds to the real interest
rate. The higher the real interest rate, the steeper the line: the more you can boost
consumption in the future by cutting back and saving today.
A representative consumer modeled by an economist will try to arrange his or her
consumption now and in the future to maximize his or her utility. And any representative
consumer in a model built by an economist will have a very simple utility function to
maximize, like:
Chapter 6
53
Final
U = [Cnow] x [Cfuture]
Where —which looks like a fish swimming downwards—is the Greek letter gamma,
and is the parameter of the utility function. It governs how much the consumer values
consumption now as opposed to consumption in the future.
If the marginal utility of consumption today is more than (1+r)/ times the marginal
utility of consumption in the future, the consumer can increase his or her total utility a bit
by cutting consumption in the future by an average of (1+r)/ dollars, reducing savings
now by 1 dollar, and increasing consumption now by 1 dollar. If the marginal utility of
consumption today is less than (1+r)/ times the marginal utility of consumption in the
future, the consumer can increase his or her total utility a bit by boosting consumption in
the future by (1+r)/ dollars, increasing savings now by 1 dollar, and cutting consumption
now by 1 dollar. Thus if the consumer if behaving like a proper agent in an economist’s
model, it must be that (writing “MUC” as an abbreviation for the marginal utility of
consumption):
MUCnow
1 r

MUCfuture

What is the marginal utility of consumption [MUC] now? Just as with the marginal
product of labor, it is the change (in utility) produced by adding one more unit of
consumption now:



MUCnow  Cnow 1  C future

1
 C
  C future

now
which can be simplified into:



MUCnow  Cnow 1  Cnow  Cfuture


1 
1 

Chapter 6
54
Final
Once again we can use our rule of thumb for the growth rate of a quantity raised to a
power—in this case Cnow growing at the proportional rate of 1/Cnow—and thus evaluate
the marginal utility of consumption. It is:
MUCnow   Cnow
 1


 C future
1
Similarly, the marginal utility of consumption in the future is:


MUCfuture  1  Cnow   C future


Thus if the consumer if behaving like a proper agent in an economist’s model, it must be
that:
1r

 Cnow
 1



 C future
1 

1  Cnow  C future


 C future
1   Cnow
or:
C future 
1  r  1   

 Cnow
This equation tells us that the consumer spends a fraction  of the present value of his or
her total income on current consumption:
Yfuture 

Cnow    Ynow 

1  r 

And he or she spends the rest on future consumption. Once again, there is nothing
especially “deep” in this simple result. Economists use this particular utility function
often because it produces simple results.
Chapter 6
55
Final
Figure: Consumption Smoothing
Legend: Consumers try to smooth consumption over time. If their income is
unusually high in the present, they will spend little of the excess and save most of
it.
Then a $1 increase in transitory income--in Ynow but not in Yfuture--would lead to a 
dollar increase in consumption today. But a $1 increase in permanent income--in both
Ynow and Yfuture--would generate an increase in consumption in the present of:
 

Cnow    1 
 1 r 
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Thus the marginal propensity to consume is much, much larger if the change in income is
a change in permanent income than if it is a change in transitory income.
Figure: An Increase in Transitory Income
Legend: A change in transitory income--a change in income in the
present but not the future--leads to a change in consumption in the present
that is only a small fraction of the change in today’s income.
6.A.2 The Vietnam War Surtax
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In the late 1960s President Lyndon Johnson proposed and Congress passed the Vietnam
War income surtax. The surtax was a 10% increase in federal taxes imposed in an attempt
to reduce consumption spending and so reduce inflationary pressures during the Vietnam
War. But President Johnson sold the surtax to the Congress (and to the public) by
promising that it would be a short-term, temporary measure with no permanent effects.
Figure: Effects of the Vietnam War Surtax on Consumption
Legend: The Vietnam War income tax surcharge had next to no
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effect on consumption because it was broadly seen as a
transitory change in tax policy.
Source: Economic Report of the President (Washington, DC: Government
Printing Office), 1999 edition.
He was convincing. Because everyone believed that this tax increase was short-term and
temporary, it had no effect on consumers' beliefs about their permanent income.
Everyone saw it as a change in their transitory income only. And so it had next to no
effect on consumption spending.
6.A.3 Consumption and the Real Interest Rate
An increase in the real interest rate makes saving more profitable: it means a higher rate
of return earned on wealth saved and invested. Consumer saving is equal to after-tax
income minus consumption. Does this mean that consumption spending is powerfully
affected by the real interest rate, and that an increase in the real interest rate decreases
consumption spending?
Probably not. An increase in the real interest rate does increase the rate of return on
savings, and this does induce a consumer to substitute savings for consumption in the
present. But return to the expression for consumption spending now—Cnow—derived
earlier in this appendix:
Yfuture 

Cnow    Ynow 

1  r 

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If current income is large relative to future income—as it is if we are thinking of people
saving for their retirement—then a change in the real interest rate r has no effect on
current consumption spending at all. Thus it has no effect on current saving—the
difference between current income and current consumption—at all.
Why not? Because an increase in savings doesn’t just make it more attractive to substitute
savings for consumption today. An increase in the real interest rate also increases
consumers' total lifetime wealth: their permanent income is boosted because they earn
higher returns on the money that they do save. This higher permanent income increases
consumption in the present--and so reduces current savings. Which effect dominates? Is
the income effect stronger, or is the substitution effect stronger? For consumers with low
future incomes, the two effects almost cancel out. For consumers with high future
incomes, they do indeed save more. But consumers with high future incomes had little
reason to save to begin with, so even a large proportional increase in their saving has
little effect on total economy-wide saving.
As a result, most economists think that these two effects roughly balance each other.
They believe that changes in real interest rates have a small negative effect on
consumption spending—and a small positive effect on savings. But the effect of the real
interest rate on consumption spending is not large enough to be worth the extra
complication it would add to our models here.
.
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6.B. Appendix: Present Value and Investment
How should you decide whether an investment is worth making? Suppose that you sit on
the investment committee of a business, and that brought before you is a proposal to
make a $100 million investment this year that will pay off by creating $130 million worth
of real inflation-adjusted value five years hence.
If the real interest rate is five percent, and if you took that $100 million and invested it in
the bond market, after one year you would have $100 x (1 + 5%) = $105 million. After
two years you would have $105 x (1 + 5%) = $110.25 million; after three years you
would have $110.25 x (1 + 5%) = $115.7625 million… and after five years $127.63
million of real inflation-adjusted purchasing power. Thus you make more money in riskadjusted inflation-adjusted expected-value terms by undertaking the investment project
than by undertaking the next best alternative. So the investment project is worthwhile.
If the real interest rate is six percent, however, the answer would be different. At six
percent after five years, you would have $100 x (1.06)5 = $133.8 million in inflationadjusted purchasing power. The investment project loses money compared to the nextbest alternative. So it should not be undertaken.
Such comparisons are easier if you use an economists' concept called present value. The
present value is the amount of wealth that you would have to set aside today and invest at
the real interest rate to generate some particular amount of purchasing power in the
future. If the real interest rate is 5% per year, to have $130 million of inflation-adjusted
purchasing power in five years you would have to take $101.85 million today, put it
aside, and let it compound in the bond market: $101.85 x (1.05)5 = $130 million.
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Thus the present value of $130 million in five years at an interest rate of 5% is $101.85
million.
To calculate the present value $PV of a sum of inflation-adjusted purchasing power
$SUM to be received n years in the future at an interest rate of r percent per year, you can
discount the future sum back to the present at the rate r by using the formula:
$PV 
$SUM
(1 r)n
because $PV in the bond market at an interest rate of r for n years will then compound to
$SUM. (If whether or not the $SUM will actually be paid in the future is subject to more
than the usual amount of risk found in the bond market, the present value will be lower:
you can either risk-adjust the $SUM to a lower value or risk-adjust the discount rate r by
adding a risk premium , and discounting it at rate r + .)
With present value, the decisions of business investment committees become easier. One
investment project will be a better use of resources than another only if the first has a
higher present value than the second.
Most investment projects don't yield returns in the shape of one, single, lump-sum
payment n years into the future. Most yield a stream of profits each year for a prolonged
period. Thus more useful than the formula for the present value of a $SUM n years in the
future is the $STREAM formula:
$PV 
$STREAM
r
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for the present value of a stream of payments each year from now into the indefinite
future.
Think of how much a flow of real purchasing power of $1 million per year each year into
the indefinite future is worth. If you wanted to receive such an annual flow, how much
would you have to put into the bond market million today? $1/r million invested in the
bond market yields an annual flow of real purchasing power of $1 million per year. Thus
an investment project that you expect to yield a cash flow of $STREAM in real
purchasing power per year each year has a present value of $STREAM/r.
You can see from these financial formulas how important the real interest rate is for
determining whether investments are worthwhile or not. If an investment project
promises a long-running stream of returns—as in the example above—a small change in
the real interest rate can have an enormous impact on present value.
But these topics are pursued further in finance courses, not here.