5 INCOME AND SUBSTITUTION EFFECTSBBM

Demand Functions

• The optimal levels of x

  1

  ,x

  2

  ,…,x

  n

  can be expressed as functions of all prices and income

• These can be expressed as n demand functions of the form:

  x ₁ * = d ₁ (p

₁

,p ₂ ,…,p _n ,I) x ₂ * = d ₂ (p

₁

,p ₂ ,…,p _n ,I)

• x _n
• = d _n

  (p

₁

,p ₂ ,…,p _n ,I)

Demand Functions

• If there are only two goods (x and y), we can simplify the notation

  x* = x(p _x ,p _y ,I) y* = y(p _x ,p _y ,I)

• Prices and income are exogenous
• – the individual has no control over these parameters

Homogeneity

• If we were to double all prices and income, the optimal quantities demanded will not change
• – the budget constraint is unchanged

  

x * = d (p ,p ,…,p ,I) = d (tp ,tp ,…,tp ,tI)

_{i i 1 2 n i 1 2 n}

• Individual demand functions are homogeneous of degree zero in all prices and income

  

Homogeneity

• With a Cobb-Douglas utility function

  utility = U(x,y) = x ^0.3 y ^0.7

  the demand functions are

  I 7 .

• 
• 
_y p y

  p x I 3 .

• Note that a doubling of both prices and income would leave x* and y* unaffected ^x
• With a CES utility function
• y
• y x y

   /

  1

   /

  I  

  p p p y

  1

  1

  I  

  x y x p p p x

  the demand functions are

  0.5

  0.5

  utility = U(x,y) = x

  

Homogeneity

  1

• Note that a doubling of both prices and income would leave x* and y* unaffected

  

Changes in Income

• An increase in income will cause the budget constraint out in a parallel fashion
• Since p

  /p does not change, the MRS

  x y

  will stay constant as the worker moves to higher levels of satisfaction

Increase in Income

• If both x and y increase as income rises,

  x and y are normal goods Quantity of y

  As income rises, the individual chooses to consume more x and y _{A B} C _{U 1 U}

₂

U ₃ Quantity of x

  

Increase in Income

• If x decreases as income rises, x is an inferior good

  As income rises, the individual chooses to consume less x and more y Quantity of y _C

  Note that the indifference curves do not have to be _{B U} “oddly” shaped. The ₃ assumption of a diminishing _{A U 2} MRS is obeyed.

  U ₁ Quantity of x

  

Normal and Inferior Goods

• A good x

  i for which x i /I  0 over some

  range of income is a normal good in that range

• A good x

  i i

  for which x /I < 0 over some range of income is an inferior good in that range

  

Changes in a Good’s Price

• A change in the price of a good alters the slope of the budget constraint
• – it also changes the MRS at the consumer’s utility-maximizing choices
• When the price changes, two effects come into play
• – substitution effect
• – income effect

  

Changes in a Good’s Price

• Even if the individual remained on the same indifference curve when the price changes, his optimal choice will change because the

  MRS must equal the new price ratio

• – the substitution effect
• The price change alters the individual’s

  “real” income and therefore he must move to a new indifference curve

• – the income effect

Changes in a Good’s Price

  Quantity of x Quantity of y U ₁ A

  Suppose the consumer is maximizing utility at point A.

  U ₂ B If the price of good x falls, the consumer will maximize utility at point B.

  Total increase in x

Changes in a Good’s Price

  Quantity of y To isolate the substitution effect, we hold “real” income constant but allow the relative price of good x to change The substitution effect is the movement from point A to point C The individual substitutes

  A C good x for good y _{U 1} because it is now relatively cheaper

  Quantity of x Substitution effect

  

Changes in a Good’s Price

Quantity of y

  The income effect occurs because the individual’s “real” income changes when

the price of good x changes

The income effect is the movement from point C to point B

  B If x is a normal good,

  A C _{U 2} the individual will buy

_U

₁ more because “real” income increased Quantity of x

  Income effect

  

Changes in a Good’s Price

Quantity of y

  An increase in the price of good x means that the budget constraint gets steeper The substitution effect is the

  C movement from point A to point C

  A The income effect is the

  B _{U 1} movement from point C _U

₂

to point B Quantity of x

  Substitution effect Income effect

  Price Changes for

Normal Goods

• If a good is normal, substitution and income effects reinforce one another
• – when price falls, both effects lead to a rise in quantity demanded
• – when price rises, both effects lead to a drop in quantity demanded

  

Price Changes for

Inferior Goods

• If a good is inferior, substitution and income effects move in opposite directions
• The combined effect is indeterminate
• – when price rises, the substitution effect leads to a drop in quantity demanded, but the income effect is opposite
• – when price falls, the substitution effect leads to a rise in quantity demanded, but the income effect is opposite

  

Giffen’s Paradox

• If the income effect of a price change is strong enough, there could be a positive relationship between price and quantity demanded
• – an increase in price leads to a drop in real income
• – since the good is inferior, a drop in income causes quantity demanded to rise

  

A Summary

• Utility maximization implies that (for normal goods) a fall in price leads to an increase in quantity demanded
• – the substitution effect causes more to be purchased as the individual moves along an indifference curve
• – the income effect causes more to be purchased because the resulting rise in purchasing power allows the individual to move to a higher indifference curve

  

A Summary

• Utility maximization implies that (for normal goods) a rise in price leads to a decline in quantity demanded
• – the substitution effect causes less to be purchased as the individual moves along an indifference curve
• – the income effect causes less to be purchased because the resulting drop in purchasing power moves the individual to a lower indifference curve

  

A Summary

• Utility maximization implies that (for inferior goods) no definite prediction can be made for changes in price
• – the substitution effect and income effect move in opposite directions
• – if the income effect outweighs the substitution

  effect, we have a case of Giffen’s paradox

  

The Individual’s Demand Curve

• An individual’s demand for x depends on preferences, all prices, and income:

  

x* = x(p ,p ,I)

_{x y}
• It may be convenient to graph the individual’s demand for x assuming that income and the price of y (p ) are held

  y

  constant

  

The Individual’s Demand Curve

Quantity of y p _x

  As the price of x falls...

  …quantity of x _{p ’ x} demanded rises. _{p ’’’} p ’’ _{x x U 1 U 2} U _{3 x} x x x _{1 2 3 x’ x’’ x’’’ I = p ’ + p x y I = p ’’ + p x y} Quantity of x

  Quantity of x

  

The Individual’s Demand Curve

• An individual demand curve shows the relationship between the price of a good and the quantity of that good purchased by an individual assuming that all other determinants of demand are held constant

  

Shifts in the Demand Curve

• Three factors are held constant when a demand curve is derived
• – income
• – prices of other goods (p ) _y
• – the individual’s preferences
• If any of these factors change, the demand curve will shift to a new position

  

Shifts in the Demand Curve

• A movement along a given demand curve is caused by a change in the price of the good
• – a change in quantity demanded
• A shift in the demand curve is caused by changes in income, prices of other goods, or preferences
• – a change in demand

  

Demand Functions and Curves

• We discovered earlier that
• 
_y p
• 

  x p x

  I 3 .

  I 7 .

• If the individual’s income is $100, these functions become
• 
• 

  x p x

  30

  y p y

  70

  

Demand Functions and Curves

• Any change in income will shift these demand curves

  

Compensated Demand Curves

• The actual level of utility varies along the demand curve
• As the price of x falls, the individual moves to higher indifference curves

• – it is assumed that nominal income is held

  constant as the demand curve is derived

• – this means that “real” income rises as the

  price of x falls

  

Compensated Demand Curves

• An alternative approach holds real income

  (or utility) constant while examining reactions to changes in p

  x

• – the effects of the price change are “compensated” so as to constrain the individual to remain on the same indifference curve
• – reactions to price changes include only substitution effects

  

Compensated Demand Curves

• A compensated (Hicksian) demand curve shows the relationship between the price of a good and the quantity purchased assuming that other prices and utility are held constant
• The compensated demand curve is a two- dimensional representation of the compensated demand function _c

  

x* = x (p ,p ,U)

_{x y}

  x ^c …quantity demanded rises.

  

Compensated Demand Curves

Quantity of y

  Quantity of x Quantity of x p _x

  U ^'' _{2 x’’} ^{p x ’’} _x’’ ^{y x p p slope   ’} _x’ ^{p x y x p p slope '  } _x’ ^’’’ _x’’’

^p

^{x y x p p slope '' '  } _x’’’

Holding utility constant, as price falls...

Compensated & Uncompensated Demand

  p _x At p ’’, the curves intersect because

_x

the individual’s income is just sufficient to attain utility level U ₂ p ’’ _{x x x} c x’’

  Quantity of x

Compensated & Uncompensated Demand

  At prices above p , income _x2 p _x compensation is positive because the individual needs some help to remain _{p ’ x} on U ₂ p ’’ _{x x x} c x’ x*

  Quantity of x

Compensated & Uncompensated Demand

  Quantity of x p _x x

  ^’’ x ^{c p x ’’’} x*** x’’’ ^{p x}

  At prices below p _x2 , income compensation is negative to prevent an increase in utility from a lower price

  Compensated &

Uncompensated Demand

• For a normal good, the compensated demand curve is less responsive to price changes than is the uncompensated demand curve

• – the uncompensated demand curve reflects

  both income and substitution effects
• – the compensated demand curve reflects only substitution effects

  Compensated Demand

Functions

• Suppose that utility is given by
_{0.5 0.5} utility = U(x,y) = x
• The Marshallian demand functions are

  

x = I/2p y = I/2p

_{x y}
• The indirect utility function is

  I utility V ( , p , p )

  I   _{x y . 5 . 5} 2 p p _{x y}

  Compensated Demand

Functions

• To obtain the compensated demand functions, we can solve the indirect utility function for I and then substitute into the Marshallian demand functions _.
_{5 . 5} Vp _y

  Vp _x y  x  _{. 5 . 5} p p _{x y}

  

Compensated Demand

Functions
• Demand now depends on utility (V) rather than income
• Increases in p

  x

  reduce the amount of x demanded

  p Vp x  _{5 .} ^{5 .} _y ^x p

  Vp y 

• – only a substitution effect ^{5 . 5 . x y}

  

A Mathematical Examination

of a Change in Price

• • Our goal is to examine how purchases of good

  x change when p changes x

  

x/p x

• • Differentiation of the first-order conditions from

  utility maximization can be performed to solve for this derivative
• However, this approach is cumbersome and provides little economic insight

  A Mathematical Examination

of a Change in Price

• Instead, we will use an indirect approach
• Remember the expenditure function

  

minimum expenditure = E(p ,p ,U)

_{x y}
• Then, by definition

  

x (p ,p ,U) = x [p ,p ,E(p ,p ,U)]

c _{x y x y x y}
• – quantity demanded is equal for both demand functions when income is exactly what is needed to attain the required utility level

  A Mathematical Examination

of a Change in Price

  x ^c (p _x ,p _y ,U) = x[p _x

,p

_y ,E(p _x ,p _y ,U)]

x x x

^c p E
• We can differentiate the compensated demand function and get

  E x p x p x

     

  



 

     x x ^{c x} p E

  E x p x p x

     

     

    

  

A Mathematical Examination

of a Change in Price

  x x c x p E

  E x p x p x

     

     

    

• The first term is the slope of the compensated demand curve
• – the mathematical representation of the substitution effect

  

A Mathematical Examination

of a Change in Price

c

x x x E

     

  

p p E p

      x x x

• The second term measures the way in which changes in p affect the demand

  

x

  for x through changes in purchasing power

• – the mathematical representation of the income effect

  

The Slutsky Equation

• The substitution effect can be written as

  constant effect on substituti _ 

     

  



_{U x x} ^c p x p x
• The income effect can be written as

  x x p E x p E

  

E

x

 

      

     

    

  I effect income

  

The Slutsky Equation

  = x

• Note that E/p x
• – a $1 increase in p _x

  raises necessary expenditures by x dollars

• – $1 extra must be paid for each unit of x purchased

The Slutsky Equation

• The utility-maximization hypothesis shows that the substitution and income effects arising from a price change can be represented by

  x  substituti on effect income effect

    p

   _x x x x

     x

    p p

  I    _{x x U constant }

  

The Slutsky Equation

     

     

_

x x p x p x

I

  U x x constant

• The first term is the substitution effect
• – always negative as long as MRS is diminishing
• – the slope of the compensated demand curve must be negative

The Slutsky Equation

     

     

_

x x p x p x

I

  U x x constant

• The second term is the income effect
• – if x is a normal good, then x/I > 0
• the entire income effect is negative
• – if x is an inferior good, then x/I < 0
• the entire income effect is positive

  

A Slutsky Decomposition

• We can demonstrate the decomposition of a price effect using the Cobb-Douglas example studied earlier
• The Marshallian demand function for good x was .

  5 I x ( p , p , )

  

I

_{x y}  p _x

  

A Slutsky Decomposition

• The Hicksian (compensated) demand function for good x was _.
_{5 c} Vp _y x ( p , p , V ) _{x y}  _{. 5} p x<
• The overall effect of a price change on the demand for x is

  I

x .

  5    ₂

p p

   _{x x}

  

A Slutsky Decomposition

• This total effect is the sum of the two effects that Slutsky identified
• The substitution effect is found by differentiating the compensated demand function .

  5 c .

  5 Vp  x y

   substituti on effect   1 .

  5 p p

   x x

A Slutsky Decomposition

• We can substitute in for the indirect utility function (V)

  2 ^{5 . 1 5 . 5 . 5 .}

  25 . )

( 5 .

5 . effect on substituti _{x x} ^{y y x} p p p p p

I I

      ^{ }

A Slutsky Decomposition

• Calculation of the income effect is easier

  2 25 .

  5 . 5 . effect income _{x x x} p p p x x

  I I

       

      

I

     

• Interestingly, the substitution and income effects are exactly the same size

Marshallian Demand Elasticities

• Most of the commonly used demand elasticities are derived from the Marshallian demand function x(p

  x

  ,p

  y

  ,I)

• Price elasticity of demand (e

  x,px

  )

  x p p x p p x x e x x x x p x _x

     

     /

  / ,

Marshallian Demand Elasticities

• Income elasticity of demand (e

  )

  x,I x / x x

  I   e

  

  

  I x , / x

  I I

  I

 

• Cross-price elasticity of demand (e

  )

  x,py p x / x x y

    e

     x , p _y p / p p x

  

 

y y y

  

Price Elasticity of Demand

• The own price elasticity of demand is always negative
• – the only exception is Giffen’s paradox
• The size of the elasticity is important
• – if e &lt; -1, demand is elastic _x,px &gt; -1, demand is inelastic
• – if e _x,px
• – if e = -1, demand is unit elastic _x,px

  Price Elasticity and Total

Spending

• Total spending on x is equal to

  total spending =p _x x

• Using elasticity, we can determine how total spending changes when the price of

  x changes ] 1 [

  ) ( _,     

      _{x p x x x x x} e x x p x p p x p

Price Elasticity and Total Spending

  ( p x ) x   x p x x [ e 1 ]

       x x , p _x p p

    x x

• The sign of this derivative depends on whether e is greater or less than -1

  x,px

• – if e &gt; -1, demand is inelastic and price and _x,px total spending move in the same direction
• – if e &lt; -1, demand is elastic and price and _x,px total spending move in opposite directions

  

Compensated Price Elasticities

• It is also useful to define elasticities based on the compensated demand function

  

Compensated Price Elasticities

• If the compensated demand function is

  x ^c = x ^c

(p

_x ,p _y ,U)

  we can calculate

• – compensated own price elasticity of demand (e _xc,px

  )

• – compensated cross-price elasticity of demand (e _xc,py

  )

  

Compensated Price Elasticities

• The compensated own price elasticity of demand (e

  xc,px

  ) is

  c ^x _x ^c _{x x} ^{c c c p x} x p p x p p x x e _x

     

     /

  / _,

• The compensated cross-price elasticity of demand (e _xc,py

  ) is c ^{y c c c c p x} x p p x p p x x e _y

     

     /

  / _,

  

Compensated Price Elasticities

• The relationship between Marshallian

  and compensated price elasticities can

  be shown using the Slutsky equation

       

       

   x x x p p x x p e p x x p x x c c x p x x x _x

  ,

I

• If s

  I , , , x x

c

p x p x e s e e _{x x}

  

 

  x

  = p

  x x/I, then

  

Compensated Price Elasticities

• The Slutsky equation shows that the compensated and uncompensated price elasticities will be similar if
• – the share of income devoted to x is small
• – the income elasticity of x is small

  

Homogeneity

• Demand functions are homogeneous of degree zero in all prices and income
• Euler’s theorem for homogenous functions shows that

      

  

 

 

    x p x

p

p x p _{y y x x}

I I

  

Homogeneity

• Dividing by x, we get

  I ^{, , , x p x p x}

e e e

_{y x}

    

• • Any proportional change in all prices and

  income will leave the quantity of x demanded unchanged

  

Engel Aggregation

• Engel’s law suggests that the income elasticity of demand for food items is less than one
• – this implies that the income elasticity of

  demand for all nonfood items must be

  greater than one

  

Engel Aggregation

• We can see this by differentiating the budget constraint with respect to income (treating prices as constant)

  x y   1 p p

      _{x y}

  I I   x x y y

  I I   1 p p s e s e

          _{x y x x , y y ,}

  I I I x

  I I y

  I  

  

Cournot Aggregation

• The size of the cross-price effect of a change in the price of x on the quantity of y consumed is restricted because of the budget constraint
• We can demonstrate this by differentiating the budget constraint with respect to p

  x

Cournot Aggregation

I

  x ^{y x x x} p y p x p x p p

       

       

  y y p p y p p x x x p p x p ^x _{x y} ^{x x} _{x x}

           

     

  I I

  I x x

^{p y y x p x x}

e s s e s _{, ,}

     x p y y p x x s e s e s _{x x}

  

  

_{, ,}

  

Demand Elasticities

• The Cobb-Douglas utility function is _{ }

  U(x,y) = x y (+=1)

• The demand functions for x and y are

  I I 

   x y

    p p _{x y}

  

Demand Elasticities

• Calculating the elasticities, we get

  x p I p   _{x x} e

  1 _{x , p x}

      

₂ p x p

  I  _{x x}  

      p _x

    p p x _{y y}

   e _{x , p y}     

p x x

   _y x

  I I   e

  1 _{x , I}

    

  I x p

  I

 

_x 

   

  

Demand Elasticities

• We can also show
• – homogeneity

  e e e

  1

  1        x , p x , p x , _{x y}

  I

• – Engel aggregation

  s e s e

  1

  1

  1            

  I I x x , y y ,

• – Cournot aggregation

  s e s e ( 1 ) s              x x , p y y , p _{x x} x

  

Demand Elasticities

• We can also use the Slutsky equation to derive the compensated price elasticity

              1 ) 1 (

  1 _{, , ,} I x x p x ^{c p x} e s e e _{x x}

• The compensated price elasticity depends on how important other goods (y) are in the utility function

  

Demand Elasticities

• The CES utility function (with  = 2,  = 5) is

  U(x,y) = x ^0.5 + y ^0.5

• The demand functions for x and y are

  ) 1 ( 1 _

    y x x p p p x

  I ) 1 (

  1 y x y p p p y _

   

  I

  

Demand Elasticities

• We will use the “share elasticity” to derive the own price elasticity

  x x x p x

x

x

x x p s e s p p s e

  , , 1   

   

• In this case,

  1

  

1

  1 _    y x x x p p x p s

  I

Demand Elasticities

• Thus, the share elasticity is given by _
_{1  1}

  p p p s p  p  _{y x y}

   _{x x x} e _{s , p x x}      _{   }

₁

_{2 1 1 1} p s ( 1 p p ) ( 1 p p ) 1 p p

      _{x x x y x y x y}

• Therefore, if we let p = p

  x y

  1  e e

  1

  1 1 .

  5       x , p s , p _{x x x}

  1

  1 

  

Demand Elasticities

• The CES utility function (with  = 0.5,  = -1) is _{-1 -1}

  U(x,y) = -x - y

• The share of good x is

  p x _x

  1 s _x   _{. 5 .  5} I 1 p p  _{y x}

  

Demand Elasticities

• Thus, the share elasticity is given by _.
_{5  1 . 5} .

  

5 p p

s p _{y x} p

   _{x x x} e _{s , p x x}    

_.

_{5 .    5 2 . 5 . 5 1} p s ( 1 p p ) ( 1 p p )

     _{x x y x y x . 5 .  5} . 5 p p _{y x}  _{. 5 .  5} 1 p p  _{y x}

• Again, if we let p

  = p

  

x y

.

  5 e e

  1 1 .

  75 _{x , p s , p x x x}      

  

Consumer Surplus

• An important problem in welfare economics is to devise a monetary measure of the gains and losses that individuals experience when prices change

Consumer Welfare

• One way to evaluate the welfare cost of a price increase (from p to p ) would be

  x0 x1

  to compare the expenditures required to achieve U under these two situations

  

expenditure at p = E = E(p ,p ,U )

_{x0 x0 y}

expenditure at p = E = E(p ,p ,U )

_{x1 1 x1 y}

  

Consumer Welfare

• In order to compensate for the price rise, this person would require a compensating variation (CV) of

  

CV = E(p ,p ,U ) - E(p ,p ,U )

_{x1 y x0 y}

  

Consumer Welfare

Quantity of x Quantity of y

  U ₁ A Suppose the consumer is maximizing utility at point A.

  U ₂ B If the price of good x rises, the consumer will maximize utility at point B.

  The consumer’s utility falls from U ₁ to U ₂

Consumer Welfare

  Quantity of y The consumer could be compensated so that he can afford to remain on U ₁ CV is the amount that the

  C individual would need to be

  A compensated

  B _{U 1} U ₂ Quantity of x

Consumer Welfare

• The derivative of the expenditure function with respect to p is the compensated

  x

  demand function

  E ( p , p , U )  x y c x ( p , p , U )

  



x y p