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Consumer Demand
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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Consumer Demand
- 1. CONSUMER DEMAND By Linda ChinenyenwaFamilusi 1
- 2. Outline 1. Introduction 2. Income-consumptioncurve 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 - consumptioncurve • 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 fora normal good: Positively sloped Source: Salvatore, 2012 5
- 6. Income consumption curvefor x and y being inferior respectively 6 The curve is negatively sloped for inferior goods Source: J. Singh
- 7. Engel curve • AnEngel 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
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- 10. Price-consumption curve • Holdingincome 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
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- 12. Ordinary or Marshalliandemand 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
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- 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
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- 16. Mathematical derivation ofthe 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 ofthe 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 ofthe 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 ofthe 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 xand 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 Marshalliandemand 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 Marshalliandemand 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 functioncontd. • 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 indirectutility 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 forderiving 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 forderiving 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 • Thisis 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 themarket 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 themarket demand curve 34
- 35. Hicksian demand orCompensated 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 Hicksiandemand or Compensated demand function 36Source: www.slideshare.net
- 37. Mathematical derivation ofHicksian 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 orCompensated 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 orCompensated 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 Hicksiandemand 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 • Atoptimal 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 theexpenditure 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 expenditurefunction 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: Homogenousto 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: Increasingin 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: Concavein 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 theindirect 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,,𝑝𝑦,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 theMarshallian 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 • Practicalapproach 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.