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# Consumer Demand

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Marshallian and Hicksian Demand derivation

Marshallian and Hicksian Demand derivation

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### Consumer Demand

1. 1. CONSUMER DEMAND By Linda Chinenyenwa Familusi 1
2. 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. 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. 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. 5. Income-consumption curve for a normal good: Positively sloped Source: Salvatore, 2012 5
6. 6. Income consumption curve for x and y being inferior respectively 6 The curve is negatively sloped for inferior goods Source: J. Singh
7. 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. 8
9. 9. 9
10. 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. 11 Source: Mikroekonomie v AJ
12. 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. 13 Source: http://www.tutorhelpdesk.com
14. 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. 15 Source: J.Singh
16. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 32. Market demand contd. 32Source: J. Singh
33. 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. 34. Shifts in the market demand curve 34
35. 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. 36. Derivation of Hicksian demand or Compensated demand function 36Source: www.slideshare.net
37. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 51. Thank you !!! 51