The document summarizes key concepts related to consumer demand, including:
1. The income-consumption curve shows how consumption of goods changes as income increases, holding prices constant. It is positively sloped for normal goods and negatively sloped for inferior goods.
2. The Engel curve relates the quantity purchased of a good to income levels, holding other factors constant. It is derived from the income-consumption curve.
3. The price-consumption curve shows how consumption changes as the price of a good changes, holding income and other prices constant. It is negatively sloped for normal goods and positively sloped for Giffen goods.
4. The Marshallian demand curve relates the quantity demanded of
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It should be clear to readers that the classical economists did not formulate any specific theory of employment as such. They only laid down certain postulates which subsequently developed as a theory.
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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
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
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
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
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
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
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
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
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
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
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
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
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
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