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I N T E R N E T C HA P T E R O N E
Microeconomics
Background for the
Study of Public Finance
He who has choice also has pain.
—German proverb
C
ertain tools of microeconomics are used throughout the text. They are briefly reviewed in
this appendix. Readers who have had an introductory course in microeconomics will likely
find this review sufficient to refresh their memories. Those confronting the material for the
first time may want to consult one of the standard introductory texts.1 The subjects covered
are demand and supply, consumer choice, and marginal analysis.
DEMAND AND SUPPLY
The demand and supply model shows how the price and output of a commodity are determined
in a competitive market. We discuss in turn the determinants of demand, supply, and their
interaction.
DEMAND
Which factors influence people’s decisions to consume certain goods? To make the problem
concrete, let us consider the specific case of coffee. A bit of introspection suggests that the
following factors affect the amount of coffee that people want to consume during a given
time period:
1. Price. We expect that as the price goes up, the quantity demanded goes down.
2. Income. Changes in income modify people’s consumption opportunities. It is hard to say
a priori, however, what effect such changes have on consumption of a given good. One
possibility is that as incomes go up, people use some of their additional income to purchase
more coffee. On the other hand, it may be that as incomes increase, people consume less
coffee, perhaps spending their money on cognac instead. We expect that changes in income
affect demand one way or the other, but in some cases it is hard to predict the direction
1 See, for example, Samuelson and Nordhaus (1989).
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of the change. If an increase in income increases the demand (other things being the
same), the good is called a normal good. If an increase in income decreases demand (other
things being the same), the good is called an inferior good.
3. Prices of related goods. Suppose the price of tea goes up. If people can substitute coffee for
tea, this increase in the price of tea increases the amount of coffee people wish to consume.
Now suppose the price of cream goes up. If people consume coffee and cream together, this
tends to decrease the amount of coffee consumed. Goods like tea and coffee are called substitutes; goods like coffee and cream are called complements.
4. Tastes. The extent to which people “like” a good affects the amount they demand. Not much
coffee is demanded by Mormons because their religion prohibits its consumption. Often, it
is realistic to assume that consumers’ tastes stay the same over time, but this is not always
the case. For example, when some scientists claimed that coffee might cause birth defects, it
presumably changed the tastes of pregnant women for coffee.
Hypothetical
Demand Curve for
Coffee
Price per pound of coffee
FIGURE IC1.1
We see, then, that a wide variety of things can affect demand. However, it is often useful
to focus on the relationship between the quantity of a commodity demanded and its price.
Suppose that we fix income, the prices of related goods, and tastes. We can imagine varying
the price of coffee and seeing how the quantity demanded changes under the assumption that
the other relevant variables stay at their fixed values. A demand schedule (or demand curve)
is the relation between the market price of a good and the quantity demanded of that good
during a given time period, other things being the same. (Economists often use the Latin for
“other things being the same,” ceteris paribus.)
A hypothetical demand schedule for coffee is represented graphically by curve Dc in Figure IC1.1. The horizontal axis measures pounds of coffee per year in a particular market, and
the price per pound is measured on the vertical. Thus, for example, if the price is $2.29 per
pound, people are willing to consume 750 pounds; when the price is only $1.38, they are willing
to consume 1,225 pounds. The downward slope of the demand schedule reflects the reasonable
assumption that when the price goes up, the quantity demanded goes down.
The demand curve can also be interpreted as an approximate schedule of “willingness
to pay,” because it shows the maximum price that people would pay for a given quantity.
For example, when people purchase 750 pounds per year, they value it at $2.29 per pound.
At any price more than $2.29, they would not willingly consume 750 pounds per year. If for
some reason people were able to
obtain 750 pounds at a price less than
$2.29, this would in some sense be a
“bargain.”
As already stressed, the demand
curve is drawn on the assumption that
all other variables that might affect
quantity demanded do not change.
What happens if one of them does?
Suppose, for example, that the price of
tea increases, and as a consequence,
$2.29
people want to buy more coffee. In Figure IC1.2, schedule Dc from Figure
$1.38
IC1.1 (before the increase) is reproDc
duced. As a consequence of the increase
in the price of tea, at each price of cof750
1,225
fee people are willing to purchase more
Pounds of coffee per year
coffee than they did previously. In
effect, then, an increase in the price of
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tea shifts each point on Dc to the right. The
collection of new points is D9c. Because D9c
shows how much people are willing to consume at each price (ceteris paribus), it is by
definition the demand curve.
More generally, a change in any variable
that influences the demand for a good—except
its own price—shifts the demand curve.2 (A
change in a good’s own price induces a movement along the demand curve.)
Price per pound of coffee
FIGURE IC1.2
Effect of an
Increase in the
Price of Tea on the
Demand for Coffee
3
D′c
Dc
Pounds of coffee per year
SUPPLY
Now consider the factors that determine the
quantity of a commodity that firms supply to
the market. We will continue using coffee as
our example.
1. Price. In many cases, it is reasonable to assume that the higher the price per pound of
coffee, the greater the quantity profit-maximizing firms are willing to supply.
2. Price of inputs. Coffee producers employ inputs to produce coffee—labour, land, and
fertilizer. If their input costs go up, the amount of coffee that they can profitably supply
at any given price goes down.
3. Conditions of production. The most important factor here is the state of technology. If
there is a technological improvement in coffee production, the supply increases. Other
variables also affect production conditions. For agricultural goods, weather is important.
Several years ago, for example, flooding in Latin America seriously reduced the coffee
crop.
Hypothetical
Supply Curve for
Coffee
Price per pound of coffee
FIGURE IC1.3
As with the demand curve, it is useful to focus attention on the relationship between the
quantity of a commodity supplied and its price, holding the other variables at fixed levels. The
supply schedule is the relation between market prices and the amount of a good that
Sc
producers are willing to supply during a
given time period, ceteris paribus.
A supply schedule for coffee is depicted
as Sc in Figure IC1.3. Its upward slope
reflects the assumption that the higher the
price, the greater the quantity supplied,
ceteris paribus.
When any variable that influences supply (other than the commodity’s own price)
changes, the supply schedule shifts. Suppose, for example, that the wage rate for
coffee-bean pickers increases. This increase
reduces the amount of coffee that firms are
willing to supply at any given price. The
Pounds of coffee per year
supply curve therefore shifts to the left. As
depicted in Figure IC1.4, the new supply
2 There is no need, incidentally, for D9c to be parallel to Dc. In general, this will not be the case.
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FIGURE IC1.5
Equilibrium in the
Coffee Market
S′c
Sc
curve is S9c. More generally, when any variable other than the commodity’s own price
changes, the supply curve shifts. (A change
in the commodity’s price induces a movement along the supply curve.)
EQUILIBRIUM
The demand and supply curves provide
answers to a set of hypothetical questions: If
the price of coffee is $2 per pound, how much
are consumers willing to purchase? If the
price is $1.75 per pound, how much are firms
willing to supply? Neither schedule by itself
tells us the actual price and quantity. But
Pounds of coffee per year
taken together, the schedules determine price
and quantity.
In Figure IC1.5 we superimpose demand
schedule Dc from Figure IC1.1 on supply schedule Sc from Figure IC1.3. We want to find
the price and output at which there is an equilibrium—a situation that tends to be maintained unless there is an underlying change in the system. Suppose the price is P1 dollars
per pound. At this price, the quantity demanded is Q1D and the quantity supplied is Q1S.
Price P1 cannot be maintained, because firms want to supply more coffee than consumers
are willing to purchase. This excess supply tends to push the price down, as suggested by
the arrows.
Now consider price P2. At this price, the quantity of coffee demanded, Q2D, exceeds the
quantity supplied, Q2S. Because there is excess demand for coffee, we expect the price to rise.
Similar reasoning suggests that any price at which the quantity supplied and quantity
demanded are unequal cannot be an equilibrium. In Figure IC1.5, quantity demanded equals
quantity supplied at price Pe. The associated output level is Qe pounds per year. Unless something else in the system changes, this price and output combination continues year after year.
It is an equilibrium.
Suppose something else does change:
For example, the weather turns bad, ruinSc
ing a considerable portion of the coffee
crop. In Figure IC1.6, Dc and Sc are reproduced from Figure IC1.5, and as before,
P1
the equilibrium price and output are Pe
and Qe, respectively. As a consequence of
the weather change, the supply curve shifts
to the left, say to S9c. Given the new supply
curve, Pe is no longer the equilibrium
price. Rather, equilibrium is found at the
intersection of Dc and S9c , at price P9e and
Pe
output Q9e . Note that, as one might expect,
P2
the crop disaster leads to a higher price
Dc
and smaller output—P9e . Pe and Q9e ,
Qe. More generally, a change in any variQ 1D Q 2S Q e Q 2D Q 1S
able that affects supply or demand creates
Pounds of coffee per year
a new equilibrium combination of price
and quantity.
Price per pound of coffee
Effect of an
Increase in the
Wages of Coffee
Pickers on the
Supply of Coffee
Price per pound of coffee
FIGURE IC1.4
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SUPPLY AND DEMAND
FOR I NPUTS
FIGURE IC1.6
Price per pound of coffee
Effect of Bad
Weather on the
Coffee Market
S ′c
Sc
P ′e
Pe
Dc
Q′e Qe
Pounds of coffee per year
MEASURING
THE
SHAPES
5
OF
Supply and demand can be used not only
to investigate the markets for consumption goods but also the markets for inputs
into the production process. (Inputs are
sometimes referred to as factors of production.) For example, we could label the
horizontal axis in Figure IC1.5 “number of
hours worked per year” and the vertical
axis “wage rate per hour.” Then the schedules would represent the supply and
demand for labour, and the market would
determine wages and employment. Similarly, supply and demand analysis can be
applied to the markets for capital and for
land.
SUPPLY
AND
DEMAND CURVES
Clearly, the market price and output for a given item depend substantially on the shapes of its
demand and supply curves. Conventionally, the shape of the demand curve is measured by the
price elasticity of demand: the absolute value of the percentage change in quantity demanded
divided by the percentage change in price.3 If a 10 percent increase in price leads to a 2 percent
decrease in quantity demanded, the price elasticity of demand is 0.2. An important special case
is when the quantity demanded does not change at all with a price increase. Then the demand
curve is vertical and elasticity is zero. At the other extreme, when the demand curve is horizontal, then even a small change in price leads to a huge change in quantity demanded. By
convention, this is referred to as an infinitely elastic demand curve. Similarly, the price elasticity of supply is defined as the percentage change in quantity supplied divided by the percentage
change in price.
THEORY OF CHOICE
The fundamental problem of economics is that resources available to people are limited relative
to their wants. The theory of choice shows how people make sensible decisions in the presence
of such scarcity. In this section we develop a graphical representation of consumer tastes and
show how these tastes can best be gratified in the presence of a limited budget.
TASTES
We assume that an individual derives satisfaction from the consumption of commodities.4 Economists use the slightly archaic word utility as a synonym for satisfaction. Consider Oscar who
consumes only two commodities, marshmallows and donuts. (Using mathematical methods, all
the results for the two-good case can be shown to apply to situations in which there are many
commodities.) Assume further that for all feasible quantities of marshmallows and donuts, Oscar
3 The elasticity need not be constant all along the demand curve.
4 In this context, the notion of commodities should be interpreted very broadly. It includes not only items such
as food, cars, and compact disk players, but also less tangible things like leisure time, clean air, and so forth.
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Ranking
Alternative
Bundles
FIGURE IC1.8
Derivation of an
Indifference Curve
is never satiated—more consumption of either commodity always
produces some increase in his utilThese bundles are
ity. Economists believe that under
preffered to bundle a
most circumstances, this assumption is pretty realistic.
In Figure IC1.7, the horizontal
b
axis
measures
the number of donuts
h
a
7
consumed each day, and the vertical
f
axis shows daily marshmallow consumption. Thus, each point in the
These bundles are less
quadrant represents some bundle of
desirable than bundle a
marshmallows and donuts. For
example, point a. represents a bundle with seven marshmallows and
Donuts per day
5
five donuts.
Because Oscar’s utility depends
only on his consumption of marshmallows and donuts, we can also associate with each point in the quadrant a certain level of
utility. For example, if seven marshmallows and five donuts create 100 “utils” of happiness, then
point a is associated with 100 “utils.”
Some commodity bundles create more utility than point a, and others less. Consider point b
in Figure IC1.7, which has both more marshmallows and donuts than point a. Since satiation is
ruled out, b must yield higher utility than a. Bundle f has more donuts than a and no fewer marshmallows, and is also preferred to a. Indeed, any point to the northeast of a is preferred to a.
The same reasoning suggests that bundle a is preferred to bundle g, because the latter has
fewer marshmallows and donuts than the former. Point h is also less desirable than a, because
although it has the same number of marshmallows as a, it has fewer donuts. Point a is preferred
to any point southwest of it.
We have identified some bundles that yield more utility than a, and some that yield less.
Can we find some bundles that produce just the same amount of utility as point a? Presumably
there are such bundles, but we need more information about the individual to find out which
they are. Consider Figure IC1.8, where point a from Figure IC1.7 is reproduced. Imagine that
we pose the following question to Oscar:
“You are now consuming seven marshmallows and five donuts. If I take away one of
your donuts, how many marshmallows do
U0
I need to give you to make you just as satisfied as you were initially?” Suppose that
after thinking a while, Oscar (honestly)
answers that he would require two more
i
9
marshmallows. Then by definition, the
a
bundle consisting of four donuts and nine
7
j
marshmallows yields the same amount of
6
utility as a. This bundle is denoted i in Figure IC1.8.
U0
We could find another bundle of equal
utility by asking: “Starting again at point a,
suppose I take away one marshmallow.
4 5
7
Donuts per day
How many more donuts must I give you to
keep you as well off as you originally
Marshmallows per day
FIGURE IC1.7
Marshmollows per day
6
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FIGURE IC1.10
An Indifference
Map
Marshmallows per day
An Indifference
Curve with a
Diminishing
Marginal Rate of
Substitution
were?” Assume the answer is two donuts.
Then the bundle with six marshmallows and
seven donuts, denoted j in Figure IC1.8,
must also yield the same amount of utility as
bundle a.
m
i
We could go on like this indefinitely—
start
at point a, take away various amounts
n
of one commodity, find out the amount of
the other commodity required for compensation, and record the results on Figure
IC1.8. The outcome is curve U0U0, which
ii
shows all points that yield the same amount
p
of utility. U0U0 is referred to as an indifferq
U0
ence curve, because it shows all consumption bundles among which the individual is
Donuts per day
indifferent.
By definition, the slope of a curve is the
change in the value of the variable measured
on the vertical axis divided by the change in the variable measured on the horizontal—the “rise
over the run.” The slope of an indifference curve has an important economic interpretation. It
shows the rate at which the individual is willing to trade one good for another. For example,
in Figure IC1.9, around point i, the slope of the indifference curve is m/n. But by definition of
an indifference curve, n is just the amount of donuts that Oscar is willing to substitute for
sacrificing m marshmallows. For this reason, the absolute value of the slope of the indifference
curve is often referred to as the marginal rate of substitution of donuts for marshmallows.5
This is abbreviated MRSdm.
As drawn in Figure IC1.9, the marginal rate of substitution declines as we move down along
the indifference curve. For example, around point ii, MRSdm is p/q, which is clearly smaller than
m/n. This makes intuitive sense. Around point i, Oscar has a lot of marshmallows relative to
donuts and is therefore willing to give up
quite a few marshmallows in return for an
additional donut—hence a higher MRSdm.
On the other hand, around point ii, Oscar
U2
U1
has a lot of donuts relative to marshmallows,
U0
so he is unwilling to sacrifice a lot of marshk
mallows in return for yet another donut. The
decline of MRSdm as we move down along the
indifference curve is called a diminishing
b
marginal rate of substitution.
a
Recall that our construction of indifferU2
ence curve U0U0 was based on bundle a as a
starting point. But point a was chosen arbiU1
trarily, and we could just as well have started
Increasing utility
U0
at any other point in the quadrant. In Figure
IC1.10, if we start with point b and proceed
Donuts per day
in the same way, we generate indifference
curve U1U1. Or starting at point k we generate
U0
Marshmallows per day
FIGURE IC1.9
7
5 As noted later, marginal means additional or incremental. The indifference curve’s slope shows the marginal
rate of substitution because it indicates the rate at which the individual would be willing to substitute marshmallows for an additional donut.
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Budget Constraint
Marshmallows per day
FIGURE IC1.11
L
Feasible bundles
10
Infeasible bundles
r
8
s
O
5 6
N
Donuts per day
indifference curve U2U2. Note that any
point on U2U2 represents a higher level of
utility than any point on U1U1, which in
turn is preferred to any point on U0U0. If
Oscar is interested in maximizing his utility, he tries to reach the highest indifference
curve that he can.
The entire collection of indifference
curves is referred to as the indifference
map. The indifference map tells us everything there is to know about the individual’s preferences.
BUDGET CONSTRAINT
Basic setup. Suppose that marshmallows
(M) cost 3 cents apiece, donuts (D) cost
6 cents, and Oscar’s weekly income is 60 cents. What options does Oscar have? Whatever
amounts he purchases must satisfy the equation
3 3 M 1 6 3 D 5 60.
(IC1.1)
In words, expenditures on marshmallows (3 3 M) plus expenditures on donuts (6 3 D) must
equal income (60).6 Thus, for example, if M 5 10, then to satisfy Equation (IC1.1), D must
equal 5 (3 3 10 1 6 3 5 5 60). Alternatively, if M 5 8, then D must equal 6 (3 3 8 1 6 3
6 5 60).
Let us represent Equation (IC1.1) graphically. The usual way is to graph a number of points
that satisfy the equation. This is straightforward once we recall from basic algebra that (IC1.1)
is just the equation of a straight line. Given two points on the line, the rest of the line is determined by connecting them. In Figure IC1.11, point r represents 10 marshmallows and 5 donuts,
and point s represents 8 marshmallows and 6 donuts. Therefore, the line associated with Equation (IC1.1) is LN, which passes through these points. By construction, any combination of
marshmallows and donuts that lies along LN satisfies Equation (IC1.1). Line LN is known as
the budget constraint or the budget line. Any point on or below LN (the shaded area) is feasible because it involves an expenditure less than or equal to income. Any point above LN is
impossible because it involves an expenditure greater than income.
Two aspects of line LN are worth noting. First, the horizontal and vertical intercepts of the
line have economic interpretations. By definition, the vertical intercept is the point associated
with D 5 0. At this point, Oscar spends all his 60 cents on marshmallows, buying 20 (5 60 4 3)
of them. Hence, distance OL is 20. Similarly, at point N, Oscar consumes zero marshmallows,
but can afford a binge consisting of 10 (5 60 4 6) donuts. Distance ON is therefore 10. In
short, the vertical and horizontal intercepts represent bundles in which Oscar consumes only
one of the commodities.
The slope also has an economic interpretation. To calculate the slope, recall that the “rise”
(OL) is 20 and the “run” (ON) is 10, so the slope (in absolute value) is 2. Note that 2 is the
ratio of the price of donuts (6 cents) to the price of marshmallows (3 cents). This is no accident.
The absolute value of the slope of the budget line indicates the rate at which the market permits
an individual to substitute marshmallows for donuts. Because the price of donuts is twice the
price of marshmallows, Oscar can trade two marshmallows for each donut.
6 If Oscar is a utility maximizer, he will not throw away any of his income.
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9
To generalize this discussion, suppose that the price per marshmallow is Pm, the price per
donut is Pd, and income is I. Then in analogy to Equation (IC1.1), the budget constraint is
PmM 1 PdD 5 I.
(IC1.2)
If M is measured on the vertical axis and D on the horizontal, the vertical intercept is I/Pm and
the horizontal intercept is I/Pd. The slope of the budget constraint, in absolute value, is Pd/Pm.
A common mistake is to assume that because M is measured on the vertical axis, the absolute
value of the slope of the budget constraint is Pm/Pd. To see that this is wrong just divide the
rise (I/Pm) by the run (I/Pd): (I/Pm) 4 (I/Pd) 5 Pd/Pm. Intuitively, Pd must be in the numerator
because its ratio to Pm shows the rate at which the market permits one to trade M for D.
FIGURE IC1.13
Effect on the
Budget Constraint
of a Change in
Relative Prices
Marshmallows per day
Effect on the
Budget Constraint
of a Decrease in
Income
Marshmallows per day
FIGURE IC1.12
Changes in prices and income. The budget line shows Oscar’s consumption opportunities given
his current income and the prevailing prices. What if any of these change? Return to the case
where Pm 5 3, Pd 5 6, and I 5 60. The associated budget line, 3M 1 6D 5 60, is drawn as LN
in Figure IC1.12. Now suppose that Oscar’s income falls to 30. Substituting into Equation (IC1.2),
the new budget line is described by 3M 1
6D 5 30. To graph this equation, note that
the vertical intercept is 10 and the horizontal intercept is 5. Denoting these two points
20 L
in Figure IC1.12 as R and S, respectively,
and recalling that two points determine a
Original budget
line, we find that the new budget conconstraint
straint is RS. The slope of RS in absolute
value is 2, just like that of LN. This is
because the relative prices of donuts and
10 R
marshmallows have not changed. When
income changes but relative prices do not,
Budget constraint
a parallel shift in the budget line is induced.
after income falls
If income decreases, the constraint shifts
in; if income increases, it shifts out.
S
N
Return again to the original conO
5
10
Donuts per day
straint, 3M 1 6D 5 60, which is reproduced in Figure IC1.13 as LN. Suppose
that the price of D increases to 12, but
everything else stays the same. Then by
Equation (IC1.2), the relevant budget constraint is 3M 1 12D 5 60. To graph this
20 L
new constraint, we begin by noting that it
has a vertical intercept of 20, which is the
Original budget
same as that of LN. Because the price of
constraint
M has stayed the same, if Oscar spends all
his money only on M, then he can buy just
as much as he did before. The horizontal
intercept, how-ever, is changed. It is now
Budget constraint
at five donuts (5 60 4 12), a point denoted
after Pd increases
T in Figure IC1.13. The new budget constraint is then LT. The slope of LT in absolute value is 4 (5 20 4 5), reflecting the
S
N
fact that the market now allows each indiO
5
10
Donuts per day
vidual to trade four marshmallows per
donut.
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More generally, when the price of one
commodity changes and other things stay
the same, the budget line pivots along the
axis of the good whose price changes. If
the price goes up, the line pivots in; if the
price goes down, the line pivots out.
Utility
Maximization
Subject to a
Budget Constraint
Marshmallows per day
FIGURE IC1.14
U0 U1 U2
L
E1
M1
ii
i
EQUILIBRIUM
U2
The indifference map shows what Oscar
wants to do; the budget constraint shows
iii
what he can do. To find out what Oscar
actually does, they must be put together.
N
In Figure IC1.14, we superimpose the
O
indifference
map from Figure IC1.10 onto
Donuts per day
D1
10
budget line LN from Figure IC1.11. The
problem is to find the combination of
M and D that maximizes Oscar’s utility subject to the constraint that he cannot spend more
than his income.
Consider first bundle i on U2U2. This bundle is ruled out, because it is above LN. Oscar
might like to be on indifference curve U2U2, but he simply cannot afford it. Next consider
point ii, which is certainly feasible, because it lies below the budget constraint. But it cannot
be optimal, because Oscar is not spending his whole income. In effect, at bundle ii, he just
throws away money that could have been spent on more marshmallows and/or donuts.
What about point iii? It is feasible, and Oscar is not throwing away any income. Yet he can
still do better in the sense of putting himself on a higher indifference curve. Consider point E1,
where Oscar consumes D1 donuts and M1 marshmallows. Because it lies on LN, it is feasible.
Moreover, it is more desirable than bundle iii, because E1 lies on U1U1, which is above U0U0.
Indeed, no point on LN touches an indifference curve that is higher than U1U1. Therefore, the
bundle consisting of M1 and D1 maximizes Oscar’s utility subject to budget constraint LN. E1
is an equilibrium because unless something else in the system changes, Oscar will continue to
consume M1 marshmallows and D1 donuts day after day.
Note that at the equilibrium, indifference curve U1U1 just barely touches the budget line.
Intuitively, this is because Oscar is trying to achieve the very highest indifference curve he can
while still keeping on LN. In more technical language, line LN is tangent to curve U1U1 at
point E1. This means that at point E1 the slope of U1U1 is equal to the slope of LN.
This observation suggests an equation to characterize the point of utility maximization.
Recall that by definition, the slope of the indifference curve (in absolute value) is the marginal
rate of substitution of donuts for marshmallows, MRSdm. The slope of the budget line (in absolute value) is Pd/Pm. But we just showed that at equilibrium, the two slopes are equal, or
U1
U0
MRSdm 5 Pd /Pm.
(IC1.3)
7
Equation (IC1.3) is a necessary condition for utility maximization. That is, if the consumption bundle is not consistent with Equation (IC1.3), then Oscar could do better by reallocating
his income between the two commodities. Intuitively, MRSdm is the rate at which Oscar is
willing to trade M for D, while Pd/Pm is the rate at which the market allows Oscar to trade
M for D. At equilibrium, these two rates must be equal.
7 The equation holds only if some of each commodity is consumed. If the consumption of some commodity is
zero, then Equation (IC1.3) need not be satisfied.
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FIGURE IC1.16
Change in Relative
Prices with No
Effect on Donut
Consumption
Marshmallows per day
Effect on
Equilibrium of a
Change in Relative
Prices
Now let us suppose that the price of marshmallows falls by some amount. Figure IC1.15
V
reproduces the equilibrium point E1 from Figure IC1.14. As we showed earlier, when a price
changes (ceteris paribus) the budget line pivots
along the axis of the good whose price has
U1 U4
L
changed. Because Pm falls, the budget line LN
pivots around N to a point that is higher on the
E2
M2
vertical axis. The new budget line is VN. Given
M1
that Oscar now faces budget line VN, E1 is no
U4
E1
longer an equilibrium. The fall in Pm creates
new opportunities for Oscar, and utility maxiU1
mization requires that he take advantage of
them. Specifically, subject to budget line VN,
N
Oscar maximizes utility at point E2, where he
O
D1 D2
Donuts per day
consumes M2 marshmallows and D2 donuts.
At the new equilibrium, the amounts of
both D and M increase relative to the amounts
consumed at the old equilibrium (D2 . D1
and M2 . M1). In effect, the price decrease in
marshmallows allows Oscar to purchase more
V
marshmallows and still have money left to
purchase more donuts. While this is common,
it need not always be the case. The change
L
depends on the tastes of the particular indiE2
M2
vidual. Suppose that Bert faces exactly the
same prices as Oscar and also has the same
income. Bert’s indifference map and budget
M1
E1
constraints are depicted in Figure IC1.16. For
Bert, donut consumption is totally unchanged
by the decrease in the price of marshmallows.
On the other hand, Ernie’s preferences, depicted
in Figure IC1.17, are such that a fall in Pm
O
D2 = D1
N Donuts per day
leaves the amount of marshmallows the same,
and only the amount of donuts increases.
Thus, without information about the individual’s indifference map, we cannot predict just how he or she will respond to a change in
relative prices.
More generally, a change in prices and/or income changes the position of the budget constraint. The individual then reoptimizes—finds the point that maximizes utility subject to the new
budget constraint. This usually involves the selection of a new commodity bundle, but without
information on the individual’s tastes, one cannot know for sure exactly what the new bundle
looks like. We do know, however, that as long as the individual is a utility maximizer, the new
bundle satisfies the condition that the price ratio equal the marginal rate of substitution.
Marshmallows per day
FIGURE IC1.15
11
DERIVATION
OF
DEMAND CURVES
There is a simple connection between the theory of consumer choice and individual demand
curves. Recall from Figure IC1.15 that at the original price of marshmallows—call it Pm1—
Oscar consumed M1 marshmallows. When the price fell to Pm2, Oscar increased his marshmallow consumption to M2. This pair of points may be plotted as in Figure IC1.18.
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Repeating this experiment for various
prices of marshmallows, we find the quantity
of marshmallows demanded at each price,
holding fixed money income, the price of
donuts, and tastes. By definition, this is the
demand curve for marshmallows, shown as
Dm in Figure IC1.18. Thus, we see how the
demand curve is derived from the underlying
indifference map.
Change in Relative
Prices with No
Effect on
Marshmallow
Consumption
Marshmallows per day
FIGURE IC1.17
V
L
M1 = M2
E1 E2
SUBSTITUTION AND
INCOME EFFECTS
Figure IC1.19 depicts the situation of Grover,
who initially faces budget constraint WN, and
O
D1 D2
N Donuts per day
maximizes utility at point E1 on indifference
curve i, where he consumes D1 donuts. Suppose now that the price of donuts increases.
Grover’s budget constraint pivots from WN to WZ, and at the new equilibrium, point E2 on
indifference curve ii, he consumes D2 donuts.
Just for hypothetical purposes, suppose that at the new equilibrium E2, the price of donuts
falls back to its initial level, but that simultaneously, Grover’s income is adjusted so that he is
kept on indifference curve ii. If this hypothetical adjustment were made, what budget constraint
would Grover face? Suppose we call this budget constraint XY. We know that XY must satisfy
two conditions:
• Because Grover is kept on indifference curve ii, XY must be tangent to indifference curve ii.
• The slope (in absolute value) must be equal to the ratio of the original price of donuts to
the price of marshmallows. This is because of the stipulation that the price of donuts is at
its original value. Recall, however, that the slope of WN is the ratio of the original price of
donuts to the price of marshmallows. Hence, XY must have the same slope as WN; that is, it
must be parallel to WN.
Demand Curve for
Marshmallows
Derived from an
Indifference Map
Price per marshmallow (Pm)
FIGURE IC1.18
P m1
P m2
O
Dm
M1
M2
Marshmallows per day
In Figure IC1.19, XY is drawn to satisfy these two conditions—the line is
parallel to WN and is tangent to indifference curve ii. If Grover were confronted with constraint XY, he would
maximize utility at point Ec, where his
consumption of donuts is Dc.
Why should this hypothetical
budget line be of any interest? Because
drawing line XY helps us break down
the effect of the change in the price of
donuts into two components, the first
from E1 to Ec, and the second from
Ec to E2.
1. The movement from E1 to Ec is
generated by the parallel shift of
WN down to XY. But recall from
Figure IC1.12 that such parallel
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Substitution and
Income Effects of
a Price Change
movements are associated with
changes in income, holding relative prices constant. Hence, the
W
movement from E1 to Ec is in effect
due to a change in income, and is
called the income effect of the
price change.
X
2. The movement from Ec to E2 is a
E2
M2
consequence purely of the change in
the relative price of donuts to marshE1
mallows. This movement shows that
M1
Ec
Grover substitutes marshmallows for
Mc
i
ii
donuts when donuts become more
N
Z
Y
expensive. Hence, the movement
O
D2
Dc
D1 Donuts per day
from Ec to E2 is called the substituSubstitution Income
tion effect. Since the movement
effect
effect
from Ec to E2 involves compensating
income (in the sense of changing
income to stay on the same indifference curve), the movement from
Ec to E2 is sometimes called the compensated response to a change in price. If we wish to
keep utility at the level represented by indifference curve ii, we measure the substitution effect
by moving along ii. If, alternatively, we had wanted to keep utility at the level enjoyed along
indifference curve i, we could have measured the substitution effect along indifference curve i
instead. In any case, the compensated response to a price change shows how the price change
affects quantity demanded when income is simultaneously altered so that the level of utility
is unchanged.
Marshmallows per day
FIGURE IC1.19
13
Intuitively, when the price of donuts increases two things happen:
• The increase in price reduces the individual’s real income—his or her ability to afford commodities. When income goes down, the quantity purchased generally changes, even without
any change in relative prices. This is the income effect.
• The increase in the price of donuts makes donuts less attractive relative to marshmallows,
inducing the substitution effect.
Any change in prices can be broken down into an income effect and a substitution effect.
We could repeat the exercise depicted in Figure IC1.19 for any change in the price of
marshmallows. Suppose that for each price, we find the compensated quantity of donuts
demanded and make a plot with price on the vertical axis and donuts on the horizontal. This
plot is called the compensated demand curve for donuts. Note that the ordinary demand curve
discussed at the beginning of this appendix shows how quantity demanded varies with price,
holding I fixed, where I is income measured in dollars. In contrast, the compensated demand
curve shows how quantity demanded varies with price, holding the level of utility fixed.
MARGINAL ANALYSIS IN ECONOMICS
In economics, the word marginal usually means additional or incremental. Suppose, for example,
the annual total benefit per citizen of a 50-kilometre road is $42, and the annual total benefit
of a 51-kilometre road is $43.50. Then the marginal benefit of the 51st kilometre is $1.50
($43.50 − $42.00). Similarly, if the annual total cost per person of maintaining a 50-kilometre
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road is $38, and the total cost of a 51-kilometre road is $40, then the marginal cost of the 51st
kilometre is $2.
Economists focus a lot of attention on marginal quantities because they usually convey the
information required for rational decision making. Suppose that the government is trying to
decide whether to construct the 51st kilometre. The key question is whether the marginal benefit is at least as great as the marginal cost. In our example, the marginal cost is $2 while the
marginal benefit is only $1.50. Does it make sense to spend $2 to create $1.50 worth of benefits?
The answer is no, and the extra kilometre should not be built. Note that basing the decision
on total benefits and costs would have led to the wrong answer. The total cost per person of
the 51-kilometre road ($40) is less than the total benefit ($43.50). Still, it is not sensible to build
the 51st kilometre. An activity should be pursued only if its marginal benefit is at least as large
as its marginal cost.8
Another example of marginal analysis: Farmer McGregor has two fields. The first is planted
in wheat and the second in corn. McGregor has seven tons of fertilizer to distribute between
the two fields and wants to allocate the fertilizer so that his total profits are as high as possible.
The relationship between the amount of fertilizer and total profitability for each crop is depicted
in Table IC1.1. Thus, for example, if six tons of fertilizer were devoted to wheat and one ton
to corn, total profits would be $503 (5 $178 1 $325).
To find the optimal allocation of fertilizer between the fields, it is useful to compute the
marginal contribution to profits made by each ton of fertilizer. The first ton in the wheat field
increases profits from $0 to $100, so the marginal contribution is $100. The second ton increases
profits from $100 to $150, so its marginal contribution is $50. The complete set of computations
for both crops is recorded in Table IC1.2.
TABLE IC1.1
Total Profit
Tons of Fertilizer
0
1
2
3
4
5
6
TABLE IC1.2
Wheat
$ 0
100
150
170
175
177
178
Corn
$ 0
325
385
415
435
441
444
Marginal Profit
Tons of Fertilizer
1
2
3
4
5
6
Wheat
$100
50
20
5
2
1
Corn
$325
60
30
20
6
3
8 If the marginal cost of an action just equals its marginal benefit, one is indifferent between taking the action
and not taking it.
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15
Suppose that McGregor puts two tons of fertilizer on the wheat field and five tons on the
corn field. Is this a profit-maximizing allocation? To answer this question, we must determine
whether any other allocation would lead to higher total profits. Suppose that one ton of fertilizer were removed from the corn field and devoted instead to wheat. As a consequence of
removing the fertilizer from the corn field, profits from corn would go down by $6. But at the
same time, profits from the wheat field would increase by $20 (the marginal profit associated
with the third ton of fertilizer in the wheat field). Farmer McGregor would therefore be
$14 richer on balance. Clearly, it is not sensible for McGregor to put two tons of fertilizer on
the wheat field and five tons on the corn, because he can do better (by $14) with three tons
devoted to wheat and four to corn.
Is this latter allocation optimal? To answer, note that at this allocation, the marginal profit
of fertilizer in each field is equal to $20. When the marginal profitability of fertilizer is the same
in each field, there is no way that fertilizer can be reallocated between fields to increase total
profit. In other words, total profits are maximized when the marginal profit in each field is the
same. Readers who are skeptical of this result should try to find an allocation of the seven tons
of fertilizer such that the total profit is higher than the $605 ($170 1 $435) associated with the
allocation at which the marginal profits are equal.
In general, if resources are being distributed across several activities, maximization of total
returns requires that marginal returns in each activity be equal.9
9 More precisely, this result requires that the marginal returns be diminishing, as they are in Table IC1.2.
In most applications, this is a reasonable assumption.
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