1. INCOME AND SUBSTITUTION EFFECTS

2. Demand Functions
The optimal levels of x1,x2,…,xn can be expressed as functions of all prices and income
These can be expressed as n demand functions of the form:
x1* = d1(p1,p2,…,pn,I)
x2* = d2(p1,p2,…,pn,I) •
xn* = dn(p1,p2,…,pn,I)

3. Demand Functions
If there are only two goods (x and y), we can simplify the notation
x* = x(px,py,I)
y* = y(px,py,I)
Prices and income are exogenous
the individual has no control over these parameters

4. Changes in Income
An increase in income will cause the budget constraint out in a parallel fashion
Since px/py does not change, the MRS will stay constant as the worker moves to higher levels of satisfaction

5. 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 C U3 B U2 A U1
Quantity of x

6. 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 U3
Note that the indifference
curves do not have to be
“oddly” shaped. The
assumption of a diminishing
MRS is obeyed. B U2 A U1
Quantity of x

7. Normal and Inferior Goods
A good xi for which xi/I  0 over some range of income is a normal good in that range
A good xi for which xi/I < 0 over some range of income is an inferior good in that range

8. 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

9. 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

10. Changes in a Good’s PriceU1 A
Suppose the consumer is maximizing
utility at point A.
Quantity of y U2 B
If the price of good x falls, the consumer
will maximize utility at point B.
Total increase in x
Quantity of x

11. 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 C
The substitution effect is the movement
from point A to point C
The individual substitutes
good x for good y
because it is now
relatively cheaper A
Substitution effect U1
Quantity of x

12. 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 B
The income effect is the movement
from point C to point B
If x is a normal good,
the individual will buy
more because “real”
income increased
Income effect A C U2 U1
Quantity of x

13. Changes in a Good’s Price
Quantity of y
An increase in the price of good x means that
the budget constraint gets steeper C
The substitution effect is the
movement from point A to point C
Substitution effect A
Income effect
The income effect is the
movement from point C
to point B B U1 U2
Quantity of x

14. 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

15. 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

16. 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

17. The Individual’s Demand Curve
An individual’s demand for x depends on preferences, all prices, and income:
x* = x(px,py,I)
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

18. The Individual’s Demand Curve
Quantity of y
As the price
of x falls... px x’
px’ U1 x1
I = px’ + py
x’’
px’’ U2 x2
I = px’’ + py
x’’’
px’’’ x3 U3
I = px’’’ + py x
…quantity of x
demanded rises.
Quantity of x
Quantity of x

19. 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

20. Demand The Cobb-Douglas utility function is
U(x,y) = xy (+=1)
The demand functions for x and y are

21. Demand Functions and Curves
If the individual’s income is $100, these functions become
Any change in income will shift these demand curves

22. Uncompensated 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
know as an uncompensated or Marshallian Demand

23. Compensated Demand Curves
An alternative approach holds real income (or utility) constant while examining reactions to changes in px
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
Also know as Hicksian Demand

24. 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
x* = xc(px,py,U)

25. Compensated Demand Curves
Holding utility constant, as price falls...
Quantity of y px x’
px’ xc
…quantity demanded
rises. U2
x’’
px’’
x’’’
px’’’
Quantity of x
Quantity of x

26. Compensated & Uncompensated Demandpx
x’’
px’’
At px’’, the curves intersect because
the individual’s income is just sufficient to attain utility level U2 x xc
Quantity of x

27. A Mathematical Examination of a Change in Price
xc (px,py,U) = x[px,py,E(px,py,U)]
We can differentiate the compensated demand function and get

28. The Slutsky Equation The substitution effect can be written as
The income effect can be written as
Note that E/px = x
a $1 increase in px raises necessary expenditures by x dollars

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

30. Marshallian Demand Elasticities
Most of the commonly used demand elasticities are derived from the Marshallian demand function x(px,py,I)
Price elasticity of demand (ex,px)

31. Elasticities Income elasticity of demand (ex,I)
Cross-price elasticity of demand (ex,py)

32. 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 ex,px < -1, demand is elastic
if ex,px > -1, demand is inelastic
if ex,px = -1, demand is unit elastic

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

34. Homogeneity Dividing by x, we get
Any proportional change in all prices and income will leave the quantity of x demanded unchanged

35. Engel Aggregation
We can see this by differentiating the budget constraint with respect to income (treating prices as constant)

36. 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