Consumer Welfare 1

2 One way to evaluate the welfare cost of a price increase (from p x 0 to p x 1 ) would be to compare the expenditures required to achieve U 0 under these two situations expenditure at p x 0 = E 0 = E(p x 0,p y,U 0 ) expenditure at p x 1 = E 1 = E(p x 1,p y,U 0 )

3 Consumer Welfare In order to compensate for the price rise, this person would require a compensating variation (CV) of CV = E(p x 1,p y,U 0 ) - E(p x 0,p y,U 0 )

4 Consumer Welfare Quantity of x Quantity of y U1U1 A Suppose the consumer is maximizing utility at point A. U2U2 B If the price of good x rises, the consumer will maximize utility at point B. The consumer’s utility falls from U 1 to U 2

5 Consumer Welfare Quantity of x Quantity of y U1U1 A U2U2 B CV is the amount that the individual would need to be compensated The consumer could be compensated so that he can afford to remain on U 1 C

6 Consumer Welfare The derivative of the expenditure function with respect to p x is the compensated demand function

7 Consumer Welfare The amount of CV required can be found by integrating across a sequence of small increments to price from p x 0 to p x 1 –this integral is the area to the left of the compensated demand curve between p x 0 and p x 1

8 welfare loss Consumer Welfare Quantity of x pxpx x c (p x …U 0 ) px1px1 x1x1 px0px0 x0x0 When the price rises from p x 0 to p x 1, the consumer suffers a loss in welfare

9 The Consumer Surplus Concept Another way to look at this issue is to ask how much the person would be willing to pay for the right to consume all of this good that he wanted at the market price of p x 0

10 The Consumer Surplus Concept The area below the compensated demand curve and above the market price is called consumer surplus –the extra benefit the person receives by being able to make market transactions at the prevailing market price

11 Consumer Welfare Quantity of x pxpx x c (...U 0 ) px1px1 x1x1 When the price rises from p x 0 to p x 1, the actual market reaction will be to move from A to C x c (...U 1 ) x(p x …) A C px0px0 x0x0 The consumer’s utility falls from U 0 to U 1

12 Consumer Welfare Quantity of x pxpx x c (...U 0 ) px1px1 x1x1 Is the consumer’s loss in welfare best described by area p x 1 BAp x 0 [using x c (...U 0 )] or by area p x 1 CDp x 0 [using x c (...U 1 )]? x c (...U 1 ) A B C D px0px0 x0x0 Is U 0 or U 1 the appropriate utility target?

13 Consumer Welfare Quantity of x pxpx x c (...U 0 ) px1px1 x1x1 We can use the Marshallian demand curve as a compromise x c (...U 1 ) x(p x …) A B C D px0px0 x0x0 The area p x 1 CAp x 0 falls between the sizes of the welfare losses defined by x c (...U 0 ) and x c (...U 1 )

14 Consumer Surplus We will define consumer surplus as the area below the Marshallian demand curve and above price –shows what an individual would pay for the right to make voluntary transactions at this price –changes in consumer surplus measure the welfare effects of price changes

15 Welfare Loss from a Price Increase Suppose that the compensated demand function for x is given by The welfare cost of a price increase from p x = 1 to p x = 4 is given by

16 Welfare Loss from a Price Increase If we assume that V = 2 and p y = 2, CV = 2  2  2  (4) 0.5 – 2  2  2  (1) 0.5 = 8 If we assume that the utility level (V) falls to 1 after the price increase (and used this level to calculate welfare loss), CV = 1  2  2  (4) 0.5 – 1  2  2  (1) 0.5 = 4

17 Welfare Loss from Price Increase Suppose that we use the Marshallian demand function instead The welfare loss from a price increase from p x = 1 to p x = 4 is given by

18 Welfare Loss from a Price Increase If income ( I ) is equal to 8, loss = 4 ln(4) - 4 ln(1) = 4 ln(4) = 4(1.39) = 5.55 –this computed loss from the Marshallian demand function is a compromise between the two amounts computed using the compensated demand functions

Revealed preference 19

20 Revealed Preference and the Substitution Effect The theory of revealed preference was proposed by Paul Samuelson in the late 1940s The theory defines a principle of rationality based on observed behavior and then uses it to approximate an individual’s utility function

21 Revealed Preference and the Substitution Effect Consider two bundles of goods: A and B If the individual can afford to purchase either bundle but chooses A, we say that A had been revealed preferred to B Under any other price-income arrangement, B can never be revealed preferred to A

22 Revealed Preference and the Substitution Effect Quantity of x Quantity of y A I1I1 Suppose that, when the budget constraint is given by I 1, A is chosen B I3I3 A must still be preferred to B when income is I 3 (because both A and B are available) I2I2 If B is chosen, the budget constraint must be similar to that given by I 2 where A is not available

Negative Substitution Effect and Revealed Preference 23

24 Negativity of the Substitution Effect Suppose that an individual is indifferent between two bundles: C and D Let p x C,p y C be the prices at which bundle C is chosen Let p x D,p y D be the prices at which bundle D is chosen

25 Negativity of the Substitution Effect Since the individual is indifferent between C and D –When C is chosen, D must cost at least as much as C p x C x C + p y C y C ≤ p x C x D + p y C y D –When D is chosen, C must cost at least as much as D p x D x D + p y D y D ≤ p x D x C + p y D y C

26 Negativity of the Substitution Effect Rearranging, we get p x C (x C - x D ) + p y C (y C -y D ) ≤ 0 p x D (x D - x C ) + p y D (y D -y C ) ≤ 0 Adding these together, we get (p x C – p x D )(x C - x D ) + (p y C – p y D )(y C - y D ) ≤ 0

27 Negativity of the Substitution Effect Suppose that only the price of x changes (p y C = p y D ) (p x C – p x D )(x C - x D ) ≤ 0 This implies that price and quantity move in opposite direction when utility is held constant –the substitution effect is negative

28 Strong Axiom of Revealed Preference If commodity bundle 0 is revealed preferred to bundle 1, and if bundle 1 is revealed preferred to bundle 2, and if bundle 2 is revealed preferred to bundle 3,…,and if bundle K-1 is revealed preferred to bundle K, then bundle K cannot be revealed preferred to bundle 0

Elasticity 29

30 Marshallian Demand Elasticities Most of the commonly used demand elasticities are derived from the Marshallian demand function x(p x,p y, I ) Price elasticity of demand (e x,px )

31 Marshallian Demand Elasticities Income elasticity of demand (e x, I ) Cross-price elasticity of demand (e x,py )

32 Compensated Price Elasticities The compensated own price elasticity of demand (e x c,px ) is The compensated cross-price elasticity of demand (e x c,py ) is

Relationship between Marshallian and compensated price elasticities 33

34 The relationship between Marshallian and compensated price elasticities can be shown using the Slutsky equation If s x = p x x/ I, then

35 Compensated Price Elasticities The Slutsky equation shows that the compensated and uncompensated price elasticities will be similar if –the share of income devoted to x is small –the income elasticity of x is small

36 Chapter 6 DEMAND RELATIONSHIPS AMONG GOODS

37 The Two-Good Case The types of relationships that can occur when there are only two goods are limited But this case can be illustrated with two- dimensional graphs

38 Gross Complements Quantity of x Quantity of y x1x0 y1 y0 U1 U0 When the price of y falls, the substitution effect may be so small that the consumer purchases more x and more y In this case, we call x and y gross complements  x/  py < 0

39 Gross Substitutes Quantity of x Quantity of y In this case, we call x and y gross substitutes x1x0 y1 y0 U0 When the price of y falls, the substitution effect may be so large that the consumer purchases less x and more y U1  x/  py > 0

40 A Mathematical Treatment The change in x caused by changes in p y can be shown by a Slutsky-type equation substitution effect (+) income effect (-) if x is normal combined effect (ambiguous)

41 Substitutes and Complements For the case of many goods, we can generalize the Slutsky analysis for any i or j – this implies that the change in the price of any good induces income and substitution effects that may change the quantity of every good demanded

42 Substitutes and Complements Two goods are substitutes if one good may replace the other in use –examples: tea & coffee, butter & margarine Two goods are complements if they are used together –examples: coffee & cream, fish & chips

43 Gross Substitutes and Complements The concepts of gross substitutes and complements include both substitution and income effects –two goods are gross substitutes if  x i /  p j > 0 –two goods are gross complements if  x i /  p j < 0