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7/28/2019 Economics 102 Lecture 6 Demand Rev
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8/3/200
Lecture 6: Theory of the Consumer:
Demand
Changes in income
Income changes and offer curves and Engel
curves
Changes in prices
Price offer curve and demand curves
Substitutes and complements
Inverse demand function
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Consumers demand functions give the
optimal amounts of each of the goods as
a function of the prices and income
faced by the consumer.
),,(
),,(
2122
2111
yppxx
yppxx
Change the consumers income, holding prices as
fixed
Increase in income shifts the budget line outward
in a parallel fashion
Effects on quantity demanded:
the quantity demanded increases as income increases
the quantity demanded decreases as income increases
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Income offer curve- set of bundles of goods that are
demanded at different levels of income.
Derived by connecting together the bundles that are
obtained as the budget line is shifted outwards.
Income offer curve is also known as the income
expansion path
If both goods are normal goods, then the income
expansion path will have a positive slope.
Engel curve graph of one of goods as a
function of income, with all prices being held
constant.
Derived by holding prices of good 1 and 2 fixed,
and then changing income, much like thederivation of the income expansion path.
We then plot the quantity demanded of one good
as a function of income
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x2
x1
Income ChangesFixed p
1
and p2
.
y < y < y
x2
x1
Income ChangesFixed p1 and p2.
y < y < y
x1x1
x1
x2x2x2
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x2
x1
Income ChangesFixed p
1
and p2
.
y < y < y
x1x1
x1
x2x2x
2
Income
offer curve
x2
x1
Income ChangesFixed p1 and p2.
y < y < y
x1x1
x1
x2x2x2
Income
offer curve
x1*
x2*
y
y
x1x1
x1
x2x2
x2
yy
y
yyy
Engel
curve;
good 2
Engel
curve;
good 1
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An example of computing the equations of
Engel curves; the Cobb-Douglas case.
The ordinary demand equations are
U x x x xa b( , ) .1 2 1 2
x aya b p
x bya b p
11
22
* *
( );
( ).
xay
a b px
by
a b p1
12
2
* *
( );
( ).
Rearranged to isolate y, these are:
ya b p
a x
ya b p
bx
( )
( )
*
*
1 1
22
Engel curve for good 1
Engel curve for good 2
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y
yx1*
x2*
y a b pa
x( ) *1 1Engel curve
for good 1
ya b p
b
x( ) *2
2
Engel curvefor good 2
Perfect complements
income offer curve- straight line through thequantities demanded since the consumer will alwaysconsume the same amount of each good no matterwhat.
The Engel curve is therefore a straight line with slope:
This is just derived from the demand for good 1:
21 pp
)/( 211 ppyx
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Rearranged to isolate y, these are:
y p p x
y p p x
( )
( )
*
*
1 2 1
1 2 2
Engel curve for good 1
x x yp p
1 21 2
* * .
Engel curve for good 2
Income Changes
x1
x2y < y < y
x1x1
x2x2
x2
x1 x1*
x2*
y
y x2x2
x2
yy
y
yyy
Engel
curve;
good 2
Engel
curve;
good 1
x1x1
x1
Fixed p1 and p2.
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Income Changes
x1*
x2*
y
y x2x2
x2
yyy
y
y
y
x1x1
x1
*
221 )( xppy
Engel
curve;
good 2
Engel
curve;
good 1
Fixed p1
and p2
.
*
121 )( xppy
Perfect substitutes
If the consumer is specializing in
the consumption of good 1. If his incomeincreases, so will his consumption of good 1.
Thus the income offer is the horizontal axis.
The Engel curve will be a straight line with a
slope of p1.
21 pp
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1
The perfect-substitution case.
The ordinary demand equations are
U x x x x( , ) .1 2 1 2
x p p yif p p
y p if p p1 1 2
1 2
1 1 2
0*( , , )
,
/ ,
x p p yif p p
y p if p p2 1 2
1 2
2 1 2
0*( , , )
,
/ , .
Suppose p1 < p2. Then
xy
p1
1
* x2
0* and
x2 0*
.y p x1 1*
and
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x2 0*
.y p x1 1
*
y y
x1* x2*0Engel curve
for good 1
Engel curve
for good 2
Homothetic preferences result in straight lineEngel curves or where the demand for the good goesup by the same proportion as income.
Homothetic preferences consumers preferencesdepend on the ratio of good 1 to good 2.
Given:
Then for any positive value of t:
),(),(s.t.),,(),,( 21212121 yyxxyyxx
),(),( 2121 tytytxtx
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Homothetic preferences
Three examples are perfect complements, perfectsubstitutes and Cobb-Douglas
If the consumer has homothetic preferences, thenthe income offer curves are all straight linesthrough the origin.
If preferences are homothetic, scaling income upor down by twill scale the quantity demanded bythe same amount.
The consumers MRS is the same anywhere on astraight line drawn from the origin.
Engel curves do not have to be straight lines. In
general, when income goes up, the demand for a
good goes up by a greater proportion than
income.
Luxury good when demand for the good
increases by a greater proportion than theincrease in income
Necessary good when the demand for that
good increases by a lesser proportion than the
increase in income.
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Quasilinear preferences Example of non-homothetic preferences
The utility function for these preferences take the form:
If an indifference curve is tangent to the budget line at
then another indifference curve must also be tangent at
for any constant k.
21)( xxvu
),( *2*
1 xx
),( *2*1 kxx
Quasilinear preferences Example of non-homotheticpreferences
Increasing income doesnt change the demand forgood 1 at all, and all the extra income goes entirely tothe consumption of good 2.
With quasilinear preferences, there is a zero incomeeffect for good 1.
Engel curve for good 1 is a vertical line, as youchange income, the demand for good 1 remainsconstant.
This kind of income offer curve is relevant for someitems that do not form a large amount of theconsumers budget.
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Quasilinear preferences are not homothetic.
For example,
U x x f x x( , ) ( ) .1 2 1 2
U x x x x( , ) .1 2 1 2
x2
x1
Each curve is a vertically shifted
copy of the others.
Each curve intersects
both axes.
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Income Changes; Quasilinear
Utilityx2
x1x1~
x1*
x2*y
y
x1~
Engel
curvefor
good 2
Engel
curve
for
good 1
Normal good- a good is a normal good when
the demand for the good increases as income
increases, and decreases when income
decreases.
the quantity demanded always changes the
same way as income changes:
A normal goods Engel curve is positively
sloping
0/1
mx
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Inferior good a good whose quantity
demanded decreases as income
increases.
examples are low quality goods
normality or inferiority of a good depends on
the income range that we are considering
An income inferior goods Engel curve is
negatively sloped.
x2
x1
Income Changes; Goods1 & 2 Normal
x1x1
x1
x2x2x2
Income
offer curve
x1*
x2*
y
y
x1x1
x1
x2x2
x2
yy
y
yyy
Engel
curve;
good 2
Engel
curve;
good 1
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Income Changes; Good 2 Is Normal,
Good 1 Becomes Income Inferior
x2
x1
Income Changes; Good 2 Is Normal,Good 1 Becomes Income Inferior
x2
x1
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Income Changes; Good 2 Is Normal,
Good 1 Becomes Income Inferior
x2
x1
Income Changes; Good 2 Is Normal,Good 1 Becomes Income Inferior
x2
x1
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Income Changes; Good 2 Is Normal,
Good 1 Becomes Income Inferior
x2
x1
Income Changes; Good 2 Is Normal,Good 1 Becomes Income Inferior
x2
x1
Income
offer curve
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2
Income Changes; Good 2 Is Normal,
Good 1 Becomes Income Inferior
x2
x1 x1*
yEngel curve
for good 1
Income Changes; Good 2 Is Normal,Good 1 Becomes Income Inferior
x2
x1 x1*
x2*
y
y
Engel curve
for good 2
Engel curvefor good 1
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Decrease the price of good 1 holding the price
of good 2 and income constant.
Depict changes via price offer curve and
ordinary demand curve
Price offer curve depicts the optimal choicesas the price of good 1 changes.
Demand curve shows for each level of p1 theoptimal level of consumption of good 1. The
demand curve is a plot of the demand function:
holding p2 and m constant.
),,( 2111 yppxx
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Two alternative effects:
the consumption of good 1 increases
the consumption of good 1 decreases
Ordinary good when the quantity demandedof the good increases as the price of the good isdecreased
Giffen good a good such that when its pricedeclines, the quantity demanded of the gooddeclines as well.
the reduction in the price of good 1 has freed upsome extra money that can be spent on otherthings, so much so that consumer decides toconsume more of the other good and reduce yourconsumption of this good
The price change is to some extent like an incomechange. Even though money income remains thesame, the change in the price of one good changespurchasing power and thus changes demand.
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Fixed p2
and y.
x1
x2
Fixed p2 and y.
x1
x2p1 price
offer
curve
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Fixed p2
and y.
x1
x2p1 price
offer
curve
x1*
Downward-sloping
demand curve
Good 1 is
ordinary
p1
Fixed p2 and y.
x1
x2
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Fixed p2
and y.
x1
x2 p1 price offercurve
Fixed p2 and y.
x1
x2 p1 price offercurve
x1*
Demand curve has
a positively
sloped part
Good 1 is
Giffen
p1
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Ordinarily, when the price of a good decreases,the demand for the good will increase and viceversa. Thus the demand curve has a negativeslope since price and consumption move inopposite directions.
Therefore:
01
1
p
x
x1
x2
p1 = p1
Fixed p2 and y.
p1x1 + p2x2 = y
Own-Price Changes
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Own-Price Changes
x1
x2
p1= p
1
p1 = p1
Fixed p2
and y.
p1x1 + p2x2 = y
Own-Price Changes
x1
x2
p1= p1p1=
p1
Fixed p2 and y.
p1 = p1
p1x1 + p2x2 = y
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x2
x1
p1 = p1
Own-Price ChangesFixed p
2
and y.
x2
x1x1*(p1)
Own-Price Changes
p1 = p1
Fixed p2 and y.
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x2
x1x1*(p1)
p1
x1*(p1)
p1
x1*
Own-Price ChangesFixed p
2
and y.
p1 = p1
x2
x1x1*(p1)
p1
x1*(p1)
p1
p1 = p1
x1*
Own-Price ChangesFixed p2 and y.
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x2
x1x1*(p1)
x1*(p1)
p1
x1*(p1)
p1
p1 = p1
x1*
Own-Price ChangesFixed p
2
and y.
x2
x1x1*(p1)
x1*(p1)
p1
x1*(p1)
x1*(p1)
p1
p1
x1*
Own-Price ChangesFixed p2 and y.
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x2
x1x1*(p1)
x1*(p1)
p1
x1*(p1)
x1*(p1)
p1
p1
p1 = p1
x1*
Own-Price ChangesFixed p
2
and y.
x2
x1x1*(p1) x1*(p1)
x1*(p1)
p1
x1*(p1)
x1*(p1)
p1
p1
p1 = p1
x1*
Own-Price ChangesFixed p2 and y.
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x2
x1x1*(p1) x1*(p1)
x1*(p1)
p1
x1*(p1)x1*(p1)
x1*(p1)
p1
p1
p1
x1*
Own-Price ChangesFixed p
2
and y.
x2
x1x1*(p1) x1*(p1)
x1*(p1)
p1
x1*(p1)x1*(p1)
x1*(p1)
p1
p1
p1
x1*
Own-Price Changes Ordinarydemand curve
for commodity 1Fixed p2 and y.
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x2
x1x1*(p1) x1*(p1)
x1*(p1)
p1
x1*(p1)x1*(p1)
x1*(p1)
p1
p1
p1
x1*
Own-Price Changes Ordinarydemand curve
for commodity 1Fixed p
2
and y.
x2
x1x1*(p1) x1*(p1)
x1*(p1)
p1
x1*(p1)x1*(p1)
x1*(p1)
p1
p1
p1
x1*
Own-Price Changes Ordinarydemand curve
for commodity 1
p1 price
offer
curve
Fixed p2 and y.
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For Cobb-Douglas preferences
Take
Then the ordinary demand functions for
commodities 1 and 2 are
U x x x xa b( , ) .1 2 1 2
x p p ya
a b
y
p1 1 2
1
*( , , )
x p p yb
a b
y
p2 1 2
2
*( , , ) .
and
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x p p y aa b
yp
1 1 21
*( , , )
x p p yb
a b
y
p2 1 2
2
*( , , ) .
and
Notice that x2* does not vary with p1 so the p1price offer curve is flat and the ordinary demand
curve for commodity 1 is a rectangular hyperbola.
x1*(p1) x1*(p1)
x1*(p1)
x2
x1
p1
x1*
Own-Price Changes Ordinarydemand curve
for commodity 1
is
Fixed p2 and y.
x
bya b p
2
2
*
( )
xay
a b p1
1
*
( )
xay
a b p1
1
*
( )
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For a perfect-complements utility function
U x x x x( , ) min , .1 2 1 2Then the ordinary demand functions
for commodities 1 and 2 are
x p p y x p p yy
p p1 1 2 2 1 2
1 2
* *( , , ) ( , , ) .
With p2 and y fixed, higher p1causes smaller x1* and x2*.
p x x yp
1 1 22
0 , .* *Asp x x1 1 2 0 , .* *As
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Fixed p2
and y.
x1
x2
p1
x1*
Fixed p2 and y.
x
yp p
2
1 2
*
xy
p p1
1 2
* x1
x2
p1
xy
p p1
1 2
*
p1 = p1
y/p2
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p1
x1*
Fixed p2
and y.
x
y
p p
2
1 2
*
xy
p p1
1 2
* x1
x2
p1
p1
p1 = p1
x
y
p p1
1 2
*
y/p2
p1
x1*
Fixed p2 and y.
x
y
p p
2
1 2
*
xy
p p1
1 2
* x1
x2
p1
p1
p1
xy
p p1
1 2
*
p1 = p1
y/p2
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p1
x1*
Ordinary
demand curve
for commodity 1is
Fixed p2
and y.
x
y
p p
2
1 2
*
xy
p p1
1 2
*
xy
p p1
1 2
*.
x1
x2
p1
p1
p1
y
p2
y/p2
For a perfect-substitutes utility
function
U x x x x( , ) .1 2 1 2Then the ordinary demand functions
for commodities 1 and 2 are
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x p p yif p p
y p if p p1 1 2
1 2
1 1 2
0*( , , )
,
/ ,
x p p yif p p
y p if p p2 1 2
1 2
2 1 2
0*( , , )
,
/ , .
and
Fixed p2 and y.
x2
x1
p1
x1*
Fixed p2 and y.
x2 0*
xy
p1
1
*
p1
p1 = p1 < p2
xy
p1
1
*
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Fixed p2
and y.
x2
x1
p1
x1*
Fixed p2
and y.
x2 0*
xy
p1
2
*
p1
p1 = p1 = p2
x1 0*
xy
p2
2
*
0 1
2
x yp
*
p2 = p1
Fixed p2 and y.
x2
x1
p1
x1*
Fixed p2 and y.
xy
p
2
2
*
x1 0*
p1
p1
x1 0*
p2 = p1
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Fixed p2
and y.
x2
x1
p1
x1*
Fixed p2
and y.
p1
p2 = p1
p1
xy
p1
1
*
0 1
2
x yp
*
y
p2
p1 price
offer
curve
Ordinary
demand curve
for commodity 1
Discrete good - Suppose that good 1 is a
discrete good.
At some high price, consumption will bezero
At some low price, consumption will beone unit
At some price r1, the consumer will beindifferent between consuming good 1or not consuming it.
r1 is called the reservation price.
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Demand behavior can be described by asequence of reservation prices at whichthe consumer is just willing to purchaseanother unit of the good.
at a price r1, he is just willing topurchase one unit,
at a price r2, he is just willing topurchase another unit.
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Reservation prices can be described in
terms of the original utility function.
r1 satisfies the condition:
r2 satisfies the condition:
),1(),( 1rmumou
)2,2(),1( 22 rmurmu
For quasilinear utilities - the formulas
describing the reservation prices are simpler:
)1(
,rforsolving
,)1()0(:asequationfirstthewritecanwe,0)0(and
2)(),(
1
1
1
121
vr
rmvmmv
v
xxvxxu
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Similarly, we can write the second equation as:
Proceeding in this manner, the reservation price for
the third unit is just:
)1()2(
:grearranginandtermscanceling
,2)2()1(
2
22
vvr
rmvrmv
)2()3(3 vvr
In each case, the reservation price measures theincrement in utility necessary to induce theconsumer to choose an additional unit of the good.
Assumption of convex preferences mean that thesequence of reservation prices must decrease:
Given any price p, we just find where it falls in thelist of reservation prices.
321 rrr
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Suppose p is between r6 and r7.
r6 > p means that the consumer is willingto give up p pesos per unit bought to get 6units of good 1
p > r7 means that the consumer is not
willing to give up p pesos to get the 7thunit of good 1.
Substitutes If the demand for good 1 goesup when the price of good 2 goes up, then wesay that good 1 is a substitute for good 2:
Complements If the demand for good 1goes down as the price of good 2 increases,we say that good 1 is a complementto good2:
.0
2
1
p
x
.02
1
p
x
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A perfect-complements example:
xy
p p1
1 2
*
x
p
y
p p
1
2 1 22
0*
. so
Therefore commodity 2 is a gross complement
for commodity 1.
p1
x1*
p1
p1
p1
y
p2
Increase the price of
good 2 from p2 to p2
and
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p1
x1*
p1
p1
p1
y
p2
Increase the price of
good 2 from p2 to p2
and the demand curve
for good 1 shifts inwards
-- good 2 is a
complement for good 1.
A Cobb- Douglas example:
xby
a b p2
2
*
( )
so
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A Cobb- Douglas example:
xby
a b p2
2
*
( )
x
p2
1
0*
.
so
Therefore commodity 1 is neither a grosscomplement nor a gross substitute for
commodity 2.
As long as we have a downward sloping
demand curve, it is meaningful to speak of the
inverse demand function.
Inverse demand function is the demandfunction viewing price as a function of
quantity.
Therefore, for each quantity, what would be the
level of prices such that the consumer would
choose that level of consumption.
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p1
x1*
p1
Given p1, what quantity isdemanded of commodity 1?
p1
x1*
p1
Given p1, what quantity is
demanded of commodity 1?
Answer: x1 units.
x1
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p1
x1*x1
Given p1, what quantity isdemanded of commodity 1?
Answer: x1 units.
The inverse question is:
Given x1 units are
demanded, what is the
price of
commodity 1?
p1
x1*
p1
x1
Given p1, what quantity is
demanded of commodity 1?
Answer: x1 units.
The inverse question is:
Given x1 units are
demanded, what is the
price ofcommodity 1?
Answer: p1
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A Cobb-Douglas example:
xay
a b p1
1
*
( )
is the ordinary demand function and
pay
a b x
1
1
( ) *
is the inverse demand function.
A perfect-complements example:
xy
p p1
1 2
* is the ordinary demand function and
py
xp1
1
2*is the inverse demand function.
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An economic interpretation
For well-behaved preferences, the optimal choice
must satisfy the condition that the absolute value of
the MRS equals the price ratio:
This says that at the optimal level of demand for
good 1, we must have:
At the optimal level of good 1, the price of good 1 is
proportional to the absolute value of the MRS
between good 1 and good 2.
2
1
p
pMRS
MRSpp 21
Suppose that p2 =1, the equation states that at theoptimal level of demand, the price of good 1measures how much the consumer is willing to giveup of good 2 in order to gain a little of good 1.
If we think of good 2 as the amount of money tospend on other goods, then we can think of the MRSas being how many pesos the individual would bewilling to give up in order to have more of good 1.
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MRS is therefore the marginal willingness topay . the marginal willingness to pay is just the price of
the good.
At each quantity x1, the inverse demand functionmeasures how many pesos a consumer is willingto give up for a little more of good 1.
Marginal willingness to pay, i.e., in the senseof the marginal willingness to sacrifice good 2for good 1 is decreasing as we increase theconsumption of good 1.