Download Comparative Statics: Analysis of Individual Demand and Labor Supply

Comparative
Statics: Analysis of
Individual Demand
and Labor Supply
Chapter 4
Slides by Pamela L. Hall
Western Washington University
©2005, Southwestern
Introduction


Rational households are never quite able to find local bliss
Ever-changing prices and income require households to
continuously adjust their commodity bundle
 Can study these changes by comparing one equilibrium position to
another

Comparative statics analysis
 Investigates a change in some parameters holding everything else
constant
• Called ceteris paribus
 With preferences held constant, individual indifference curves remain
fixed


Comparative statics is concerned with sensitivity of a
solution to changes in parameters
Will derive a household’s demand curve for each commodity
2
Introduction

Will investigate a change in income, holding all prices constant
 Develop Engel curves and Engel’s Law associated with income changes

Based on shapes (slopes) of demand and Engel curves
 Commodities are generally classified as normal, luxury, or inferior goods in
terms of income change

• Ordinary or Giffen goods for price change
Discuss how Slutsky equation considers total effect of a price change as
shown to be sum of substitution effect and income effect
 Illustrate theoretical possibility of a positively sloping demand curve

• Giffen’s Paradox
Discuss use of Slutsky equation to measure compensated price
changes and Laspeyres Index to measure Consumer Price Index (CPI)

Extend Slutsky equation to changes in price of another commodity
 Develop concepts of gross and net substitutes and gross and net
complements
3
Chapter Objective
To derive a household’s demand functions for
commodities it purchases and its labor
supply function
 Quantity demanded should generally decline
as price of a commodity increases
 Demand should generally increase with a rise in

income

We investigate underlying determinants for
this response of quantity demanded
4
Introduction




Determinants of a household’s supply of labor given a rise in wages may
result in supply increasing, declining, or remaining unchanged
Can develop aggregate (market) demand and supply
Market supply and demand functions will provide a foundation for
investigating efficient allocation of society’s resources
Applied economists estimate consumer demand and labor supply
functions
 To determine how responsive consumers and labor are to changes in prices,
wages, incomes, sales promotion, and various government programs


Consumer demand is a very large area in economics
Labor economics is also a large area
 Impacts of government welfare programs, working conditions, exploitation,
and unions
5
Derived Household Demand

With indifference curves representing preferences and
budget lines as income constraints,
 Can derive a theoretical relationship between price and a person’s
quantity demanded


Consider case of a price change in one of two commodities
x1 and x2
Budget line is
 I = p1 x 1 + p 2 x 2
• Where I, p1, and p2 represent income, per-unit price of commodity 1, and
per-unit price of commodity 2, respectively

Figure 4.1 is a graph of budget line
 With an x1-intercept of 5 units and an x2-intercept of 10 units
6
Figure 4.1 Derived demand for a
decrease in p1
7
Derived Household Demand

A household maximizes its utility for a given level of income
 At a point on budget line tangent with an indifference curve (commodity
bundle A)


Equilibrium bundle (at a given income level and prices) corresponds to
one point on household’s demand curve (point a)
Additional points on household’s demand curve for x1 are obtained
 By changing price of commodity x1 while holding income and price of x2
constant


For example, decreasing price of x1 from 2 to 1 (p1 = 1) results in budget
line tilting outward
With this new price for p1, commodity bundle C represents new
equilibrium level of utility maximization
8
Derived Household Demand

Further changes in p1 will result in additional tangencies of a budget line
 With an indifference curve and a corresponding point on household’s
demand curve


Connecting points results in household’s demand curve for commodity
x1
Each point on demand curve corresponds with a tangency point
between indifference curve and budget line
 Household is maximizing utility for a given income level and market price of


commodity x1
Illustrates how much of the commodity a household is willing and able to
purchase at a given price
As price of p1 declines household’s MRS(x2 for x1) declines
 How much it is willing to pay for an additional unit of x1 also declines

A decline in price results in an increase in a household’s level of utility
 Household’s purchasing power has increased as a result of this price decline
9
Shift in Demand versus a Change in
Quantity Demanded

Convenient to graph x1 as a function of its own price
 With understanding that income and all other prices are being held
constant



As illustrated in Figure 4.1, assuming two commodities and
considering a change in p1, then x1 = x1(p1|p2, I)
 Where p2 and I are being held constant
By varying p1, price consumption curve traces out locus of
tangencies
 Between budget line and indifference curve
Negatively sloped demand curve can be derived from this
price consumption curve
 Decrease in p1 will result in an increase in quantity demanded (a
movement along demand curve)
 A change in p2 or I will shift demand curve
10
Shift in Demand versus A Change in
Quantity Demanded


Figure 4.2 illustrates difference in a change in quantity demanded
versus a shift in demand
At bundle A, a decrease in p2 results in a movement from bundle A to B
 Shifts demand curve
• Shift depicted as a shift from point a on demand curve x1( p1| p2°, I) to point b on
x1( p1| p'2,I )

A decrease in p1 at bundle A
 Results in a movement from bundle A to C
• Causes a movement along demand curve x1( p1|p2°, I ) from point a to c


Change in quantity demanded
Change in either of the variables on axes causes a movement along a
curve
 Whereas change in any factor not on one of the axes causes a shift in curve
• For example, a change in income or preferences will shift a demand curve
11
Inverse Demand Curves

Demand functions depicted in Figures 4.1 and 4.2
are sometimes called inverse demand functions
 Dependent variable is on vertical axis and independent
variable is on horizontal axis


Price as dependent variable states what level of
quantity demanded for a commodity would have to
be for household to be willing to pay this price per
unit
Inverse demand functions represent price as a
function of quantity demanded
 As opposed to quantity demanded as a function of price
12
Figure 4.2 Shift in demand versus
change in quantity demanded
13
Generalizing For K Commodities




In general we can solve for optimal levels of x1*, x2*, … , xk*
and * as functions of all parameters (prices and income)
Quantities of x1, x2, … , xk demanded by the household will
depend on
 Shape of utility function (consumer preferences)
 p1, p2, … , pk and I
Demand functions state how much a household is willing
and able to consume of a commodity at given prices and
income
Mathematically, demand functions are represented as
14
Homogeneous of Degree Zero
Demand Functions

In many developing countries price and income indexing
occurs
 Due to high rates of inflation
• For example, if inflation is running at 10% annually, incomes are
automatically adjusted (indexed) upward by 10%

Keeps households’ purchasing power the same
 Does not change their demands for commodities
• Assuming no money illusion

When all prices and income change proportionately, optimal
quantities demanded would remain unchanged
 Slope and intercepts of budget constraint do not change
• Illustrated in Figure 4.3
15
Figure 4.3 Homogeneity of demand
functions
16
Homogeneous of Degree Zero
Demand Functions

Generally, if prices and income are multiplied by some
positive constant a, same budget constraint remains

Given no change in budget constraint from multiplying
prices and income by  > 0
 Quantity demanded by a household will also not change
• Called homogeneous of degree zero

In this case, consumer demand functions are homogeneous
of degree zero in all prices and income.
 Household demands are not affected by pure inflation
17
Numeraire Price



Result of homogeneous of degree zero demand functions
 Can divide all prices and income by one of the prices
Demand for a commodity depends on
 Price ratios (called relative prices)
 Ratio of money income to a price (called real income)
Picking any price, say p1, and multiplying demand function
by 1/p1 gives
 xj = xj(p1, p2,…, pk, I) = xj(1, p2/p1,…, pk/p1, I/p1)
 Where a is 1/p1
• Setting p1 = 1

Which is relative price to which all other prices and income are compared
 Called numeraire price
18
Changes in Income




A college graduate’s income will generally substantially
increase upon landing that first professional job
 Results in a change in purchasing power
An increase in income results in an expected increase in
purchases
 Represented by parallel shifts in budget lines
Only I has changed, so price ratio remains constant
Income consumption path, or income expansion path
 Curve intersecting all points where indifference curves are tangent
with budget lines (locus of utility-maximizing bundles)
 Every point on path represents demanded bundle at that level of
income
19
Engel Curves



From income expansion path, can derive a function that
relates income to demand for each commodity at constant
prices
Represented by Engel curves as illustrated in Figure 4.4
As income rises from I1 to I2 and then to I3
 Demand for x1 increases from x11 to x12 and then to x13
 Plotting this increased demand with rise in income yields an Engel
curve


Illustrate a relationship between demand for a commodity
and income
In Figure 4.4, Engel curve has a positive slope
 However Engel curves can have either positive or negative slopes
• Positively sloped Engel curves are called normal goods

An increase in income results in more of commodity being purchased
20
Figure 4.4 Income expansion path
and Engel curve …
21
Homothetic Preferences

If income expansion paths, and thus each Engel curve, are
straight lines (linear) through origin
 Household will consume same proportion of each commodity at
every level of income


Assumes prices are held fixed
Homothetic preferences
 Preferences resulting in consuming same proportion of commodities
as income increases
 Commodities are scaled up and down in same proportion as income
changes
22
Luxury and Necessary Goods

Can further divide normal goods into luxury and necessary goods
 If income expansion path bends toward one commodity or the other

Figure 4.5 illustrates income expansion path bending toward commodity
x1
 Making x1 a luxury good

x1 is a luxury good if (x1/I)I/x1 > 1
 As income increases, household spends proportionally more of its income
on x1

• Examples: fine wines and silk suits
Necessary good
 If, as income increases, household spends proportionally less of its income
on a commodity
• Examples are gasoline and textbooks
23
Figure 4.5 Income expansion path and
Engel curve for a luxury good, x1
24
Luxury and Necessary Goods

As household receives more income, it
wishes to consume more of both types of
commodities
 But proportionally more of luxury good than of
necessary good

For two-commodity case, if one commodity is
a luxury good
 Other must be a necessary good
25
Inferior Goods
Negatively sloped Engel curve is associated
with a backward-bending income expansion
path
 With an increase in income, a household
actually wants to consume less of one of the
commodities
 In Figure 4.6, as income increases
 Consumption of x1 declines

• Called an inferior good and is defined as x1/I < 0

Examples include cheap wine and used books
26
Figure 4.6 Income expansion path and
Engel curve for an inferior good x1
27
Engel’s Law


Relationship between income and consumption of specific
items has been studied since 18th century
Engel was first to conduct such studies
 Developed generalization about consumer behavior
• Proportion of total expenditure devoted to food declines as income rises



Has been verified in numerous subsequent studies
Engel’s Law appears to be such a consistent empirical
finding that some economists have suggested proportion of
income spent on food might be used as an indicator of
poverty
Families that spend more than say 40% of their income on
food might be regarded as poor
28
Changes in Price
If p1 is allowed to vary holding p2 and I fixed,
budget line will tilt
 Illustrated in Figure 4.7
 Locus of tangencies will sweep out a price
consumption curve
 Curve connecting all tangencies between

indifference curves and budget lines for
alternative price levels
29
Figure 4.7 Price consumption curve
and demand curve …
30
Ordinary Goods
Price consumption curve for an ordinary
good is illustrated in Figure 4.7
 Where xj ÷ pj < 0 defines an ordinary good
 Demand curve derived from price
consumption curve has a negative slope
 Indicates inverse relationship between a

commodity’s own price and quantity consumed
31
Giffen Goods

Slope of a demand curve could be positive
 xj ÷ pj > 0
• Defines a Giffen good

Decrease in p1 results in a decrease in
demand for x1
 Shown in Figure 4.8
32
Figure 4.8 Price consumption curve
for a Giffen good
33
Substitution and Income Effects

Determinants of whether a commodity is ordinary good or Giffen good
 Depend on direction and magnitude of substitution and income effects


Figure 4.9 shows effects for case of an own price change where p1
decreases
Initial budget constraint
 I = p1x1 + p2x2

New budget constraint
 I = p‘1x1 + p2x2

• Where p'1 < p°1
Price decrease in p1 results in increased quantity demanded of x1
 Shown in Figure 4.9
• Increase in quantity demanded is total effect of price decline

Total effect = x1/p1 < 0
 Can be decomposed into substitution and income effects
34
Figure 4.9 Substitution and income
effects …
35
Substitution Effect
In 2002, automobiles in Canada generally
cost from 20% to 35% less than in US
 With fall of trade barriers and harmonizing of
environmental and safety regulations
 Only major differences between new cars made

for sale in Canada and those made for sale in
U.S.
• Speedometers and odometers
36
Substitution Effect

U.S. automobile buyers attempt to substitute Canadian cars
for U.S. ones
 Illustrates substitution effect (also called Hicksian substitution)
• As an incentive for consumers to purchase more of a lower-priced
commodity (Canadian cars) and less of a higher-priced commodity (U.S.
cars)


Given a change in price of one commodity relative to another
To determine substitution effect
 Hold level of utility constant at initial utility level, U°
• Consider price change for x1
 If a household were to stay on same indifference curve
• Consumption patterns would be allocated to equate MRS to new price
ratio
37
Substitution Effect

Represented in Figure 4.9 by a budget line parallel
to new budget line
 But tangent to initial indifference curve
• Point B, illustrates a household’s equilibrium for a level of utility
U° with p1' as the price of x1



Bundle A represents consumer equilibrium for the
same level of utility as bundle B
Decrease in p1 results in consumer purchasing
more of x1 and less of x2 with level of utility
unchanged
Movement from bundle A to bundle B is substitution
effect
38
Compensated Law of Demand

In Figure 4.9, Strict Convexity Axiom makes it
impossible for a tangency point representing the
new price ratio (bundle B) to occur left of bundle A
 If p1 decreases, implying p1/p2 decreasing
• MRS also decreases

Only way for MRS to decrease is for x1 to increase and x2 to
decrease
 Thus, decreasing x1’s own price holding utility constant
results in
• Consumption of x1 increasing


Own substitution effect is always negative
Implying x1/p1|dU=0 < 0
 Known as Compensated Law of Demand
 Where price and quantity always move in opposite directions
for a constant level of utility
39
Income Effect


In general, a change in the price of a commodity a
household purchases changes purchasing power of
household’s income
 Called income effect
For example, an increase in price of prescription drugs
decreases ability to purchase both drugs and food
 Decreased ability to purchase same level of commodities represents
a decline in purchasing power
• Has same effect as if household experienced a change in income

A price decline has effect of increasing a household’s
purchasing power or real income
40
Income Effect

Figure 4.9 illustrates a price decline of p1 with utility remaining constant
 Results in an increase in real income or purchasing power
 Represented by a parallel outward shift in budget line associated with the


new price
Equilibrium tangency shifts from point B to C, which is the measurement of
the income effect
Mathematically, given the budget constraint I = p1x1 + p2x2
 Change in income from a change in p1, holding consumption of commodities
x1 and x2 constant, is
• I/p1 = x1
 Substitution effect will equal total effect if, given a decline in p1, income also
falls by x1
• If income is not reduced, then this decline in p1 represents an increase in real
income
 Specifically, a decline in p1 results in an increase in IR of x1
• I/p1 = -x1

Minus sign results from condition that a change in price and a change in real income
move in opposite directions
41
Income Effect

Change in real income depends on how much x1 a
household is consuming
 If purchases of x1 are small, impact of a price change will
be minor

Partial derivative x1/I may be either positive
(normal good) or negative (inferior good)
 Sign of income effect is indeterminate
42
Slutsky Equation

Combining equations for substitution and income effects yields
 Called Slutsky equation
• Mathematically defines substitution and income effects



Sum is total effect of a price change
Total effect is a movement from bundle A to bundle C or a movement along demand
curve for a change in p1 (Figure 4.9)
Substitution effect defines a change in slope of budget line
 Would motivate a household to choose bundle B if choices had been
confined to those on original indifference curve

Income effect defines movement from B to C resulting from a change in
purchasing power
 p1 decreases
• Implies an increase in real income
43
Slutsky Equation



If x1 is a normal good, a household will demand more of it in
response to increase in purchasing power
Own substitution effect always holds
 x1/p1|U=constant < 0
Normal good results in a negative income and negative
substitution effect
 Total effect is sum of these two negative effects, so it also is negative
• x1/p1 < 0, an ordinary good

Known as Law of Demand
 Demand for a commodity will always decrease when its price increases
 If demand increases with an increase in income
44
Slutsky Equation






Figure 4.10 represents income and substitution effects for a price
increase in commodity x1
Results in an inward tilt of budget line with a new equilibrium tangency
point C
Substitution effect represents price increase holding level of utility
constant
Bundle B is a tangency of a budget line, given new increase in price,
with initial indifference curve
Movement from bundle A to bundle B is substitution effect
Increase in price of p1 results in a decrease in purchasing power or real
income
 Income effect measuring this decrease in real income is represented by a
parallel leftward shift in budget line

• Income effect is negative and reinforces negative substitution effect
Total effect is sum of substitution and income effect representing a
movement from A to C
45
Figure 4.10 Substitution and income
effects for an increase in p1
46
Slutsky Equation
Not all commodities are normal goods
 Some commodities are inferior
 Rise in income will yield a decrease in their

consumption
 Income effect is positive
• Will partially or completely offset substitution effect

If income effect does not completely offset negative
substitution effect
 Total effect will still be negative
47
Slutsky Equation




Figure 4.11 illustrates these effects for inferior and
ordinary goods
Movement from A to B is substitution effect
Decline in price results in an increase in
consumption of x1 holding utility constant
 Illustrates negative substitution effect
Income effect, movement from B to C, partially
offsets negative substitution effect
 If this positive income effect does not completely offset
negative substitution effect, total effect is still negative
• Results in an ordinary but inferior good
48
Figure 4.11 Substitution and income effects
for an inferior and ordinary good, x1
49
Giffen’s Paradox

If x1 is an inferior good
 Sign of total effect can be either positive or negative
• Substitution effect is negative and income effect is positive

Positive income effect can be large enough to
produce result illustrated in Figure 4.12
 Demand curve has a positive slope
• Increase in price results in an increase in quantity demanded

English applied economist Robert Giffen claims to
have observed effect in 19th-century Ireland
 An increase in price of potatoes resulted in an increase in
consumption of potatoes
50
Figure 4.12 Giffen’s paradox
51
Giffen’s Paradox

Potatoes were a large part of total expenditure in 19th-century Ireland
 An increase in I would lead to an increase in meat and a decrease in potato
consumption
 An increase in price of potatoes resulted in a decrease in real income with
an associated decrease in meat and an increase in potato consumption



• Shown in Figure 4.12
Own substitution effect is still negative
An increase in p1 reduces consumption of x1, holding utility constant,
from A to B
Income effect is positive
 Commodity x1 is an inferior commodity
 Offsets negative substitution effect
• If it completely offsets substitution effect, total effect is positive

Defining a Giffen good
52
Giffen’s Paradox

Size of income effect depends on proportion
of income spent on commodity
 More of a household’s income spent on a
commodity, the larger will be the income effect
 Generally, commodities associated with a large
proportion of a household’s income are normal
goods rather than inferior goods
• Very rare to encounter a Giffen good
53
Summary of Ordinary, Normal,
Inferior, and Giffen Goods



Relationships among an ordinary good, a normal
good, an inferior good, and a Giffen good are
illustrated in Figure 4.13 for an increase in p1
 Initially a household is consuming commodity bundle A
Price increase in p1 results in negative own
substitution effect
 Movement from A to B
If x1 is a normal good, income effect will reinforce
negative substitution effect
 Further decrease consumption of x1
54
Figure 4.13 Summary of ordinary,
normal, inferior, and Giffen goods
55
Summary of Ordinary, Normal,
Inferior, and Giffen Goods


If x1 is an inferior good, income effect will partially
or completely offset negative own substitution effect
If income effect completely offsets negative own
substitution effect, a Giffen good results
 A necessary condition for a Giffen good is an inferior
good


If income effect does not completely offset negative
own substitution effect, an ordinary good results
 Price increase in x1 yields a decline in consumption of x1
Giffen good must be an inferior good
 Ordinary good could be either an inferior or a normal
good
56
Compensated Price Changes

In very cold winters, price per therm for natural gas usually rises as a
result of increased demand
 Especially hard on poor, who spend a larger proportion of their income on
heating


Government programs provide direct income compensation to poor to
offset higher heating costs
To determine how much compensation is required, we rewrite the
Slutsky equation as
 Substitution Effect = Total Effect – Income Effect
 x1/p1|U=constant = x1/p1 + (x1/I)x1
 Diminishing MRS Axiom assures own substitution effect will always be
negative

Substitution effect, also called a compensated price effect, is sum of two
components that are observed (revealed) in markets
 Indicates effect from a pure price change
• Level of consumption is adjusted for any beneficial or adverse effects from a
change in real income, so utility remains unchanged
57
Compensated Price Changes




Income effect is removed from total effect, resulting
in pure price effect
As a result of higher price for natural gas,
households’ real incomes have declined
 Reduces their satisfaction
Compensating households for this loss in
satisfaction by increasing their income results in
compensated price effect
 Utility then returns to same level prior to price rise
Compensated price effect is calculated from
estimates of observed total effect and income effect
58
Hicks versus Slutsky Compensation

Compensated price effect is called Hicks
compensation
 Based on Pareto’s discussion of compensation
• Holds utility constant
 Slutsky compensation is an alternative type of
compensation
• Instead of adjusting income to same level of utility
prior to a price change

Slutsky compensation adjusts income so a household can
purchase original consumption bundle
 Keeps household’s purchasing power constant
59
Hicks versus Slutsky Compensation


Figure 4.14 compares Hicks vs Slutsky compensation
An increase in price of x1 results in Hicks compensation
bundle B
 Budget line with new price ratio is tangent to original indifference
curve

Slutsky compensation adjusts income to where budget line
intersects original commodity bundle (A) prior to price
change
 Requires ability to just purchase this original bundle A
 Budget line cuts original indifference curve
• By additional substitution a household can obtain bundle C

Yields a higher level of utility than bundles A and B
60
Figure 4.14 Hicks and Slutsky
compensation
61
Hicks versus Slutsky Compensation

In the limit, as price change tends to zero, Hicks
and Slutsky compensations are identical
 Will usually not matter which type of compensation is
used
• Provided price change is small

Hicks compensation has desirable property for
policy analysis
 Compensating a household to point where level of
satisfaction is unaffected by a price change
 Such compensation is not directly revealed in market

Slutsky compensation is revealed
 Can provide an approximation for Hicks compensation
62
Figure 4.15 Excise tax on gasoline
with an income tax rebate
63
Laspeyres Index
Used by United States and other countries to
define Consumer Price Index (CPI)
 A Slutsky compensation index
 Computes change in quantity of commodities
consumed using initial set of prices as
weights
 CPI states, in percentage, amount of income
required in current year to purchase same
consumption bundle in some base year

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Laspeyres Index

Calculating CPI with Laspeyres index does not account for
households adjusting to price changes by changing their
consumption bundle
 Adjusting households’ income for any price increases with CPI may
result in households increasing their utility
 Since Laspeyres index does not account for possibility of households
substituting other commodities, it biases CPI upward
 For small relative price changes, bias should not be large
• Given the good approximation of Slutsky compensation to Hicks
compensation

Congressional Budget Office estimates CPI has grown
faster than cost of living by between 0.2 and 0.8 percentage
points annually
 Has cost federal government $300 billion over past few decades
• Social Security payments, civil service pensions, and earned income tax
credits are all based on CPI
65
Changes in the Price of another
Commodity

When apples are harvested in the fall their
price declines
 Likely to have an effect on amount of oranges a
shopper purchases

Can use Slutsky equation to express this
change in x1 (oranges) when p2 (price of
apples) changes
 x1/p2 = x1/p2|U=constant – (x1/I)x2
66
Changes in the Price of another
Commodity

Total effect = substitution effect + income effect
 Income effect represents x1/I times change in income


as a result of changing p2
Sign of x1/p2|U=constant is generally indeterminate for
three or more commodities
In a two-commodity case, this cross substitution effect
will be positive
• Assuming diminishing MRS
• When utility is held constant, a decrease in price of x2 will tend to
cause purchases of x1 to decrease due to diminishing MRS

Shown in Figure 4.16
67
Figure 4.16 Gross substitutes and
complements
68
Changes in the Price of another
Commodity





Given strict convexity of indifference curves
 Impossible for this tangency to occur at the right of bundle A
If p2 decreases, implying p1/p2 increasing, MRS also
increases
 Only way for MRS to increase is for x1 to decrease and x2 to increase
Decreasing p2, holding utility constant, results in decreasing
consumption of x1
 Cross substitution effect is always positive for a two-commodity case
 Implies x1/p2|dU=constant, given two commodities
If x1 is a normal good, income effect is negative
Total effect may be either positive or negative
 Depending on relative strengths of cross substitution and income
effects
69
Substitutes and Complements



Magnitude and signs of substitution and income
effects will determine whether commodities are
substitutes or complements
Two commodities are substitutes if one commodity
may, as a result of a price change, replace the
other
 Examples are two brands of cola or gasoline
Two commodities are complements if one
commodity is consumed with another good
 Examples are pancakes and syrup, gasoline and
automobiles
70
Substitutes and Complements




Figure 4.16 illustrates gross substitute and gross
complement relationship for a price decrease in x2
Because only two commodities are considered, for both
cases, cross substitution effect is positive
 A decrease in price of x2 results in a decrease in consumption of x1
Income effect overwhelms positive cross substitution effect
for gross complement case
Two commodities are independent goods
 If a price change in one commodity does not affect consumption of
another
• Example: bottled water and laundry detergent
71
Substitutes and Complements





Can use price consumption curve to determine whether
commodities are gross substitutes or complements
Negatively sloping price consumption curve indicates that
commodity x2 is a gross substitute for x1
If price consumption curve has a positive slope a fall in price
of x1 results in an increase in consumption of x2
 x2 is a gross complement for x1
For independent goods
 Price consumption curve is horizontal for a change in price of x1
It is possible for xj to be a substitute for xi and at the same
time for xi to be a complement of xj
 Presence of income effects can produce this paradoxical result
72