Marshallian and Hicksian Demand derivation

1. CONSUMER DEMAND
By
Linda Chinenyenwa Familusi
1

2. Outline
1. Introduction
2. Income-consumption curve
3. Engel curve
4. Price-consumption curve
5. Marshallian demand function
6. Indirect utility function
7. Roy’s identity
8. Market demand
9. Hicksian demand function
10. Expenditure function
11. Shephard Lemma
2

3. Introduction: Consumer demand
• The consumer’s demand function is the function that gives the
optimal amounts of each of the goods as a function of the prices and
income faced by the consumer
• They tell us the best quantity of 𝑥𝑖 to consume when faced with
prices p and with available income M
• For each different set of prices and income, there will be a
different combination of goods that is the optimal choice of the
consumer.
3

4. Income - consumption curve
• As income level change, holding prices constant , the utility maximizing
consumption choice shift to the higher indifference curve allowed by new
income level.
• The point of consumer equilibrium shifts as well
• The line connecting the successive equilibria is called the income-
consumption curve of the combination of X and Y purchased at a given
price
•
Δ𝑥1(𝑝,𝑚)
Δ𝑚
> 0 normal good
•
Δ𝑥1(𝑝.𝑚)
Δ𝑚
< 0 inferior good
4

5. Income-consumption curve for a normal good: Positively sloped
Source: Salvatore, 2012
5

6. Income consumption curve for x and y being
inferior respectively
6
The curve is negatively sloped for inferior goods
Source: J. Singh

7. Engel curve
• An Engel curve is a function relating the equilibrium quantity
purchased of a commodity to the level of money income
• Engel curve describes how quantity of Y changes as income changes
holding all prices constant
• It is derived from the income-consumption curve
7

8. 8

9. 9

10. Price-consumption curve
• Holding income and price of other commodity constant, the utility-
maximizing choices changes as the price changes
• Connecting all points of utility –maximizing bundle at each new
budget line and hence new indifference curve, the line generated is
the price-consumption line.
• It is an important starting point to deriving ordinary demand curve
•
Δ𝑥1(𝑝1,𝑚)
Δ𝑝1
< 0 for normal good, demand is negatively sloped
•
Δ𝑥1(𝑝1,𝑚)
Δ𝑝1
> 0 for Giffen good, demand is positively slope
10

11. 11
Source: Mikroekonomie v AJ

12. Ordinary or Marshallian demand curve
• It is derived from the Price-consumption curve
• The Marshallian demand curve for a good relates equilibrium
quantities bought to the price of the good, assuming that all other
determinants are held constant
• A consumer’s Marshallian demand function specifies what the
consumer would buy in each price and wealth (or income), assuming
it perfectly solves the utility maximization problem
• Given the price-quantity relationship, the derived demand curve has
a negative slope for a normal good
12

13. 13
Source: http://www.tutorhelpdesk.com

14. Positively sloped demand- Giffen good
• For a giffen good, the change in price and resulting change in the
quantity demanded moves in the same direction
• If the price of x falls, the position of the consumer equilibrium shifts
in such a way that the quantity of x decreases
• If the price of x rises, the position of the consumer equilibrium shifts
in such a way that the quantity of x increases
14

15. 15
Source: J.Singh

16. Mathematical derivation of the Marshallian
demand curve
• It is derived from the utility maximizing problem
• Max U = xy; s.t 𝑚 = 𝑃𝑥 𝑥 + 𝑃𝑦y
• ℒ = 𝑓 𝑥, 𝑦, 𝜆 = 𝑥𝑦 + 𝜆 𝑚 − 𝑃𝑥 𝑥 − 𝑃𝑦 𝑦
• 𝑥∗(𝑚, 𝑃𝑥) =
𝑚
2𝑃 𝑥
Marshallian demand function for x
• 𝑦∗(𝑚, 𝑃𝑦) =
𝑚
2𝑃 𝑦
Marshallian demand function for y
16

17. Mathematical derivation of the Marshallian
demand curve contd.
• ℒ = 𝑓 𝑥, 𝑦, 𝜆 = 𝑥𝑦 + 𝜆 𝑚 − 𝑃𝑥 𝑥 − 𝑃𝑦 𝑦
• FOC: ℒ 𝑥 = 𝑦 − 𝜆𝑃𝑥 = 0
• ℒ 𝑦 = 𝑥 − 𝜆𝑃𝑦 = 0
• ℒ 𝜆 = 𝑚 − 𝑃𝑥 𝑥 − 𝑃𝑦 𝑦 = 0
•
𝑦
𝑥
=
𝑃 𝑥
𝑃 𝑦
→ y = x
𝑃 𝑥
𝑃 𝑦
→Engel curve
• Substitute the Engel curve into budget constraint
• 𝑚 = 𝑃𝑥 𝑥 + 𝑃𝑦 (
𝑃 𝑥
𝑃 𝑦
)𝑥
17

18. Mathematical derivation of the Marshallian
demand curve contd.
• ℒ = 𝑓 𝑥, 𝑦, 𝜆 = 𝑥𝑦 + 𝜆 𝑚 − 𝑃𝑥 𝑥 − 𝑃𝑦 𝑦
• FOC: ℒ 𝑥 = 𝑦 − 𝜆𝑃𝑥 = 0
• ℒ 𝑦 = 𝑥 − 𝜆𝑃𝑦 = 0
• ℒ 𝜆 = 𝑚 − 𝑃𝑥 𝑥 − 𝑃𝑦 𝑦 = 0
•
𝑦
𝑥
=
𝑃 𝑥
𝑃 𝑦
→ y = x
𝑃 𝑥
𝑃 𝑦
→Engel curve
• Substitute the Engel curve into budget constraint
• 𝑚 = 𝑃𝑥 𝑥 + 𝑃𝑦 (
𝑃 𝑥
𝑃 𝑦
)𝑥
18

19. Mathematical derivation of the Marshallian
demand curve contd.
• 𝑚 = 2 𝑃𝑥 𝑥
• The demand for good x will be:
• 𝒙∗(𝒎, 𝑷 𝒙) =
𝒎
𝟐𝑷 𝒙
Marshallian demand function for x
• Substituting the demand for x into the Engel curve, we get:
• y =
𝑚
2𝑃 𝑥
𝑃 𝑥
𝑃 𝑦
=
𝑚
2𝑃 𝑥
𝑃 𝑥
𝑃 𝑦
• 𝒚∗
(𝒎, 𝑷 𝒚) =
𝒎
𝟐𝑷 𝒚
Marshallian demand function for y
19

20. Are good x and y normal goods?
Good x
• 𝒙∗
(𝒎, 𝑷 𝒙) =
𝒎
𝟐𝑷 𝒙
•
δ 𝒙∗(𝒎,𝑷 𝒙)
δ𝑷 𝒙
= -
𝒎
𝟐𝑷 𝒙
𝟐 < 0 normal good
•
δ 𝒙∗(𝒎,𝑷 𝒙)
δ𝒎
=
𝟏
𝟐𝑷 𝒙
> 0 normal good
Good y
• 𝒚∗(𝒎, 𝑷 𝒚) =
𝒎
𝟐𝑷 𝒚
•
δ 𝒚∗(𝒎,𝑷 𝒚)
δ𝑷 𝒚
= -
𝒎
𝟐𝑷 𝒚
𝟐 < 0 normal
good
•
δ 𝒚∗(𝒎,𝑷 𝒚)
δ𝒎
=
𝟏
𝟐𝑷 𝒚
> 0 normal good
20

21. Homogeneity of Marshallian demand
function
• Marshallian demand function is homogenous of degree zero in price
and income
• Homogeneity of degrees zero implies that the price and income
derivatives of demand for a good, when weighted by prices and
income, sum up to zero
21

22. Homogeneity of Marshallian demand
function contd.
• 𝑥∗(𝑚, 𝑃𝑥) =
𝑚
2𝑃 𝑥
• Increasing the prices and income by q
will yield:
• 𝑥∗(𝑞𝑚, 𝑞𝑃𝑥) =
𝑞𝑚
2𝑞𝑃 𝑥
• 𝑥∗
(𝑞𝑚, 𝑞𝑃𝑥) =
𝑞1−1 𝑚
2𝑃𝑥
• 𝑥∗(𝑞𝑚, 𝑞𝑃𝑥) =
𝑞0 𝑚
2𝑃𝑥
• 𝑥∗(𝑚, 𝑃𝑥) =
𝑚
2𝑃𝑥
• Example:
• m= 100, 𝑃𝑥 = 1, 𝑃𝑦 = 2
• 𝑥∗
(𝑚, 𝑃𝑥) =
𝑚
2𝑃 𝑥
=
100
2∗1
= 50
• Double the income and price
• m= 200, 𝑃𝑥 = 2, 𝑃𝑦 = 4
• 𝑥∗(𝑚, 𝑃𝑥) =
𝑚
2𝑃 𝑥
=
200
2∗2
= 50
22

23. Indirect utility function
• The optimal level of utility obtainable will depend indirectly on the
prices of a good being bought and the individual’s income
• Consumers usually think about their preferences in terms of what
they consume rather than the prices
• To find the optimal solution, we substitute the Marshallian demand
functions in the utility function, the resulting utility function is called
the indirect utility function Ψ (𝑃1, 𝑃2, … , 𝑃𝑛,m)
23

24. Indirect utility function contd.
• u = xy
• Rem: 𝑥∗(𝑚, 𝑃𝑥) =
𝑚
2𝑃 𝑥
𝑦∗(𝑚, 𝑃𝑦) =
𝑚
2𝑃 𝑦
• U(𝑥𝑖 𝑝, 𝑚 = Ψ (𝑃1, 𝑃2, … , 𝑃𝑛,m)
• Ψ (𝑃1, 𝑃2, … , 𝑃𝑛,m) =
𝑚
2𝑃 𝑥
.
𝑚
2𝑃 𝑦
=
𝒎 𝟐
𝟒𝑷 𝒙 𝑷 𝒚
→ The indirect utility function
24

25. Properties of indirect utility function
• Non-increasing in prices
• Non-decreasing in income
• Homogenous to degree zero in price and income
• Quasi-convex in prices and income
25

26. Indirect utility function: Non-increasing in prices
and non-decreasing in income
•
𝒎 𝟐
𝟒𝑷 𝒙 𝑷 𝒚
=
𝟏
𝟒
𝒎 𝟐 𝒑 𝒙
−𝟏 𝒑 𝒚
−𝟏 → The indirect utility function
•
𝒅Ψ
𝒅𝒑 𝒙
= −
𝟏
𝟒
𝒎 𝟐 𝒑 𝒙
−𝟐 𝒑 𝒚
−𝟏 < 0 (1)
•
𝒅Ψ
𝒅𝒑 𝒚
= −
𝟏
𝟒
𝒎 𝟐
𝒑 𝒙
−𝟏
𝒑 𝒚
−𝟐
< 0 (2)
•
𝒅Ψ
𝒅𝒎
=
𝟏
𝟐
𝒎 𝒑 𝒙
−𝟏 𝒑 𝒚
−𝟏 > 0 (3)
• This is a valid indirect utility function
26

27. Indirect utility function: Quasi-convex in price and
income
•
𝒅 𝟐Ψ
𝒅𝒑 𝒙
𝟐 = 𝒇 𝒙𝒙 =
𝟏
𝟐
𝒎 𝟐 𝒑 𝒙
−𝟑 𝒑 𝒚
−𝟏 ≥ 0
• 𝒇 𝒙𝒚 =
𝟏
𝟒
𝒎 𝟐 𝒑 𝒙
−𝟐 𝒑 𝒚
−𝟐
•
𝒅 𝟐Ψ
𝒅𝒑 𝒚
𝟐 = 𝒇 𝒚𝒚 =
𝟏
𝟐
𝒎 𝟐 𝒑 𝒙
−𝟏 𝒑 𝒚
−𝟑 ≥ 0
• 𝒇 𝒚𝒙 =
𝟏
𝟒
𝒎 𝟐 𝒑 𝒙
−𝟐 𝒑 𝒚
−𝟐
•
𝒅 𝟐Ψ
𝒅𝒎 𝟐 =
𝟏
𝟐
𝒑 𝒙
−𝟏
𝒑 𝒚
−𝟏
≥ 0
• 𝒇 𝒙𝒙 𝒇 𝒚𝒚 - 𝒇 𝒙𝒚
𝟐
≤ 0
𝒎 𝟒
𝟒𝒑 𝒙
𝟒 𝒑 𝒚
𝟒 -
𝒎 𝟒
𝟏𝟔𝒑 𝒙
𝟒 𝒑 𝒚
𝟒 ≤ 0 (answer is close to zero i.e. 0.18)
• This is a valid indirect utility function
27

28. Indirect utility function: Homogenous to degree
zero in price and income
• Ψ (𝑃𝑥, 𝑃𝑦,m) =
𝒎 𝟐
𝟒𝑷 𝒙 𝑷 𝒚
→ The indirect utility function
• Ψ (q𝑃𝑥, q𝑃𝑦, 𝑞m) =
𝒒 𝟐 𝒎 𝟐
𝟒𝒒𝑷 𝒙 𝒒𝑷 𝒚
=
𝒒 𝟐 𝒎 𝟐
𝒒 𝟐 𝟒𝑷 𝒙 𝑷 𝒚
(q>0)
• Ψ (q𝑃𝑥, q𝑃𝑦, 𝑞m) =
𝒒 𝟐−𝟐 𝒎 𝟐
𝟒𝑷 𝒙 𝑷 𝒚
• Ψ (q𝑃𝑥, q𝑃𝑦, 𝑞m) =
𝒒 𝟎 𝒎 𝟐
𝟒𝑷 𝒙 𝑷 𝒚
• Ψ (𝑷 𝒙, 𝑷 𝒚,m) =
𝒎 𝟐
𝟒𝑷 𝒙 𝑷 𝒚
→ The indirect utility function
• This is a valid indirect utility function
28

29. Roy’s Identity for deriving Marshallian demand for
good x
• x(𝑃𝑥, 𝑃𝑦, 𝑚) = −
𝛿Ψ 𝛿𝑝 𝑥
𝛿Ψ 𝛿𝑚
• x(𝑃𝑥, 𝑃𝑦, 𝑚) = −
−𝟏
𝟒
𝒎 𝟐 𝒑 𝒙
−𝟐 𝒑 𝒚
−𝟏
𝟏
𝟐
𝒎 𝒑 𝒙
−𝟏 𝒑 𝒚
−𝟏
• x(𝑃𝑥, 𝑃𝑦, 𝑚) =
𝟏
𝟐
𝒎 𝟐−𝟏 𝒑 𝒙
−𝟐+𝟏 𝒑 𝒚
−𝟏+𝟏
• 𝒙∗(𝒎, 𝑷 𝒙) =
𝒎
𝟐𝑷 𝒙
Marshallian demand function for x
29

30. Roy’s Identity for deriving Marshallian demand for
good y
• y(𝑃𝑥, 𝑃𝑦, 𝑚) = −
𝛿Ψ 𝛿𝑝 𝑦
𝛿Ψ 𝛿𝑚
• y(𝑃𝑥, 𝑃𝑦, 𝑚) = −
−𝟏
𝟒
𝒎 𝟐 𝒑 𝒙
−𝟏 𝒑 𝒚
−𝟐
𝟏
𝟐
𝒎 𝒑 𝒙
−𝟏 𝒑 𝒚
−𝟏
• y(𝑃𝑥, 𝑃𝑦, 𝑚) =
𝟏
𝟐
𝒎 𝟐−𝟏 𝒑 𝒙
−𝟏+𝟏 𝒑 𝒚
−𝟐+𝟏
• 𝒚∗
(𝒎, 𝑷 𝒚) =
𝒎
𝟐𝑷 𝒚
Marshallian demand function for y
30

31. Market demand
• This is the aggregates of consumer demand
• It gives the total quantity demanded by all consumers at each prices,
holding total income and prices of other goods constant
• We assume that both individuals face the same prices and each person is a
price taker
• Each persons demand depends on her own income
• The demand is downward sloping
31

32. Market demand contd.
32Source: J. Singh

33. Shifts in the market demand curve
• The change in price will result in a movement along the market
demand curve
• Whereas change in other determinants of demand will result in a shift
in the marker demand curve to a new position
• Eg rise in income , rise in price of substitute
33

34. Shifts in the market demand curve
34

35. Hicksian demand or Compensated demand
function
• It finds the cheapest consumption bundle that achieves a given utility
level and measures the impact of price changes for fixed utility.
• Hicksian demand curve shows the relationship between the price of a
good and the quantity purchased on the assumption that other prices
and utility are held constant
35

36. Derivation of Hicksian demand or Compensated demand
function
36Source: www.slideshare.net

37. Mathematical derivation of Hicksian demand or
Compensated demand function
• min 𝐸 = 𝑃𝑥 𝑥 + 𝑃𝑦y
• s.t. U(x,y) = xy
• ℒ = 𝑓 𝑥, 𝑦, 𝜆 = 𝑃𝑥 𝑥 + 𝑃𝑦y + 𝜆 𝑢 − 𝑥𝑦
• FOC: ℒ 𝑥 = 𝑃𝑥 − 𝜆𝑦 = 0
• ℒ 𝑦 = 𝑃𝑦 − 𝜆𝑥 = 0
• ℒ 𝜆 = 𝑢 − 𝑥𝑦 = 0
•
𝑃 𝑥
𝑃 𝑦
=
𝑦
𝑥
→ y = x
𝑃 𝑥
𝑃 𝑦
→Engel curve
• Substitute the Engel curve into utility function
37

38. Hicksian demand or Compensated demand
function
• 𝑢 = 𝑥𝑦 → u = x(x
𝑃 𝑥
𝑃 𝑦
)
• u = 𝑥2(
𝑃 𝑥
𝑃 𝑦
) → 𝑥2 = u
𝑃 𝑦
𝑃𝑥
• Square root both sides: 𝑥2 = u
𝑃 𝑦
𝑃 𝑥
• 𝑥2 = u
𝑃 𝑦
𝑃 𝑥
• 𝒙 𝒄
∗(𝑷 𝒙, 𝑷 𝒚, 𝒖) =
𝑷 𝒚
𝑷 𝒙
𝒖 or
𝑷 𝒚
𝑷 𝒙
𝒖 𝟎.𝟓 Hicksian demand function for x
38

39. Hicksian demand or Compensated demand
function
• Substitute the Hicksian demand for x in the Engel curve: y = x
𝑃 𝑥
𝑃 𝑦
• y =
𝑃 𝑦
𝑃 𝑥
𝑢 0.5 𝑃𝑥
𝑃 𝑦
→ 𝑝 𝑦
0.5
𝑝 𝑥
−0.5
𝑝 𝑥
1
𝑝 𝑦
−1
𝑢0.5
• y =𝑝 𝑥
0.5
𝑝 𝑦
−0.5
𝑢0.5
• 𝑦𝑐
∗(𝑃𝑥, 𝑃𝑦, 𝑢) =
𝑃 𝑥
𝑃 𝑦
𝑢 or
𝑃 𝑥
𝑃 𝑦
𝑢 𝟎.𝟓 Hicksian demand function for y
39

40. Homogeneity of Hicksian demand function
• Hicksian demand function is homogenous of degree zero in price
• Increasing all prices by q:
• 𝑦𝑐
∗ 𝑞𝑃𝑥, 𝑞𝑃𝑦, 𝑢 =
𝑞𝑃𝑥
𝑞𝑃 𝑦
𝑢 0.5
• 𝑦𝑐
∗(𝑞𝑃𝑥, 𝑞𝑃𝑦, 𝑢) =
𝑞0.5
𝑞0.5
𝑃 𝑥
𝑃 𝑦
𝑢 0.5
• 𝑦𝑐
∗
(𝑞𝑃𝑥, 𝑞𝑃𝑦, 𝑢) = 𝑞0.5−0.5 𝑃 𝑥
𝑃 𝑦
𝑢 0.5
= 𝑞0 𝑃 𝑥
𝑃 𝑦
𝑢 0.5
• 𝑦𝑐
∗
(𝑃𝑥, 𝑃𝑦, 𝑢) =
𝑃𝑥
𝑃 𝑦
𝑢 or
𝑃𝑥
𝑃 𝑦
𝑢 0.5
Hicksian demand function for y
40

41. Expenditure function
• At optimal levels of utility, the consumer spends all the income at
disposal.
• Income = expenditure
• We allocate income in such a way as to achieve a given level of utility
with minimum expenditure for a particular set of prices
• To find the optimal solution, we substitute the Hicksian demand
functions into the expenditure function
41

42. Derivative of the expenditure function
• Substitute the Hicksian demand functions into the objective function:
m = 𝑃𝑥 𝑥 + 𝑃𝑦y → Expenditure equation
• Rem: 𝑥 𝑐
∗
(𝑃𝑥, 𝑃𝑦, 𝑢) =
𝑃 𝑦
𝑃 𝑥
𝑢 0.5
and 𝑦𝑐
∗
(𝑃𝑥, 𝑃𝑦, 𝑢) =
𝑃 𝑥
𝑃 𝑦
𝑢 0.5
• 𝑚∗ = 𝑃𝑥
𝑃 𝑦
𝑃 𝑥
𝑢 0.5 + 𝑃𝑦
𝑃 𝑥
𝑃 𝑦
𝑢 0.5
• Simplifying:
• 𝒎∗
= (𝟐 𝒖 𝟎.𝟓
𝒑 𝒙
𝟎.𝟓
𝒑 𝒚
𝟎.𝟓
) → The Expenditure function
• Or 𝒎∗=2( 𝒖 𝑷 𝒙 𝑷 𝒚) 𝟎.𝟓
42

43. Properties of expenditure function
1. e(p,u) is homogenous to degree one in price
2. e(p,u) is strictly increasing in u, and non-decreasing in price
3. e(p,u) is concave in price
43

44. Expenditure function: Homogenous to degree
one in price
• 𝒎∗=2( 𝒖 𝑷 𝒙 𝑷 𝒚) 𝟎.𝟓 → The Expenditure function
• Let the prices be increasing by q:
• 𝒎∗
( 𝒒𝑷 𝒙, 𝒒𝑷 𝒚, u) = 2( 𝒖 𝒒𝑷 𝒙 𝒒𝑷 𝒚) 𝟎.𝟓
• 𝒎∗( 𝒒𝑷 𝒙, 𝒒𝑷 𝒚, u) = 2( 𝒖 𝑷 𝒙 𝑷 𝒚) 𝟎.𝟓 𝒒 𝟎.𝟓+𝟎.𝟓
• 𝒎∗( 𝒖, 𝒒𝑷 𝒙 , 𝒒𝑷 𝒚) → The Expenditure function
• This is a valid expenditure function
44

45. Expenditure function: Increasing in u, and non-
decreasing in p
• 𝒎∗ = (𝟐 𝒖 𝟎.𝟓 𝒑 𝒙
𝟎.𝟓 𝒑 𝒚
𝟎.𝟓) → The Expenditure function
•
𝑑𝑚∗
𝑑𝑝 𝑥
= 𝑢0.5 𝑝 𝑥
−0.5 𝑝 𝑦
0.5 > 0 → Shephard lemma (4)
•
𝑑𝑚∗
𝑑𝑝 𝑦
= 𝑢0.5
𝑝 𝑥
0.5
𝑝 𝑦
−0.5
> 0 → Shephard lemma (5)
•
𝑑𝑚∗
𝑑 𝑢
= 𝑢−0.5 𝑝 𝑥
0.5 𝑝 𝑦
0.5 > 0 (6)
• This is a valid expenditure function
45

46. Expenditure function: Concave in p
•
𝒅 𝟐 𝒎∗
𝒅𝒑 𝒙
𝟐 = 𝒇 𝒙𝒙 = −𝟎. 𝟓 𝒖 𝟎.𝟓
𝒑 𝒙
−𝟏.𝟓
𝒑 𝒚
𝟎.𝟓
≤ 0
• 𝒇 𝒙𝒚 = 𝟎. 𝟓 𝒖 𝟎.𝟓
𝒑 𝒙
−𝟎.𝟓
𝒑 𝒚
−𝟎.𝟓
•
𝒅 𝟐 𝒎∗
𝒅𝒑 𝒚
𝟐 = 𝒇 𝒚𝒚 = −𝟎. 𝟓 𝒖 𝟎.𝟓
𝒑 𝒙
𝟎.𝟓
𝒑 𝒚
−𝟏.𝟓
≤ 0
• 𝒇 𝒚𝒙 = 𝟎. 𝟓 𝒖 𝟎.𝟓
𝒑 𝒙
−𝟎.𝟓
𝒑 𝒚
−𝟎.𝟓
• 𝒇 𝒙𝒙 and 𝒇 𝒚𝒚 ≤ 0
• 𝒇 𝒙𝒙 𝒇 𝒚𝒚 − 𝒇 𝒙𝒚
𝟐
≥ 0
•
𝒖
𝟒𝒑 𝒙 𝒑 𝒚
-
𝒖
𝟒𝒑 𝒙 𝒑 𝒚
≥ 0 This is a valid expenditure function
46

47. Relationship between the indirect utility function
and the expenditure function
• Ψ (𝑃1, 𝑃2, … , 𝑃𝑛,m) = u =
𝑚2
4𝑃 𝑥 𝑃 𝑦
→ The indirect utility function
• Rearrange to make m the subject:
• 𝑚2= 𝑢4𝑃𝑥 𝑃𝑦
• Square root both sides
• 𝑚∗(𝑃𝑥, 𝑃𝑦, 𝑢)= (2 𝑢0.5 𝑝 𝑥
0.5 𝑝 𝑦
0.5) → The Expenditure function
• Or 𝑚∗
=2( 𝑢 𝑃𝑥 𝑃𝑦)0.5
47

48. Shephard Lemma
•
𝛿𝑚(𝑝 𝑥,,𝑝 𝑦,𝑢)
𝛿𝑝 𝑥
= 𝑥 𝑐
(𝑝 𝑥, 𝑝 𝑦, 𝑢) →
2 𝑢0.5 𝑝 𝑥
0.5 𝑝 𝑦
0.5
𝛿𝑝 𝑥
→ Shephard lemma
• 𝑥 𝑐
(𝑝 𝑥, 𝑝 𝑦, 𝑢) = 𝑢0.5
𝑝 𝑥
−0.5
𝑝 𝑦
0.5
• 𝑥 𝑐
∗(𝑃𝑥, 𝑃𝑦, 𝑢) =
𝑃 𝑦
𝑃 𝑥
𝑢 or
𝑃 𝑦
𝑃 𝑥
𝑢 0.5 Hicksian demand function for x
•
𝛿𝐸(𝑝 𝑥,,𝑝 𝑦,𝑢)
𝛿𝑝 𝑦
= 𝑦 𝑐(𝑝 𝑥,, 𝑝 𝑦, 𝑢) →
2 𝑢0.5 𝑝 𝑥
0.5 𝑝 𝑦
0.5
𝛿𝑝 𝑦
→ Shephard lemma
• 𝑦 𝑐
(𝑝 𝑥, 𝑝 𝑦, 𝑢) = 𝑢0.5
𝑝 𝑥
0.5
𝑝 𝑦
−0.5
• 𝑦𝑐
∗(𝑃𝑥, 𝑃𝑦, 𝑢) =
𝑃 𝑥
𝑃 𝑦
𝑢 or
𝑃 𝑥
𝑃 𝑦
𝑢 0.5 Hicksian demand function for y
48

49. Comparison between the Marshallian and Hicksian
demand function
Marshallian demand function
• It’s a function of p and m
• Measures the changes in
demand when income is held
constant
• Measures the total effect
Hicksian demand function
• It’s a function of p and u
• Measures the changes in
demand when utility is held
constant.
• Measures the change in demand
along an indifference curve
• Measures the substitution effect
49
Marshallian effect – Hicksian effect = income effect .
This is the difference between the two demand function

50. Further reading
• Practical approach to microeconomic theory: For graduate students in
Applied economics
• Varian, H.R. (2010).Intermediate microeconomics: A modern approach (8th
ed.). New York: W.W Norton & Company, Inc.
• Varian, H.R. (1992).Microeconomic analysis (3rd ed.). New York: W.W
Norton & Company, Inc.
• Wainwright, K.J. (2013).Marshall and Hicks: Understanding the
• Salvatore, Dominick. Microeconomics (PDF). Archived from the
original (PDF) on October 20, 2012.ordinary and compensated demand
50

51. Thank you !!!
51