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1 Chapter 5 INCOME AND SUBSTITUTION EFFECTS. 2 Objectives How will changes in prices and income influence influence consumers optimal choices? –We will.

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Presentation on theme: "1 Chapter 5 INCOME AND SUBSTITUTION EFFECTS. 2 Objectives How will changes in prices and income influence influence consumers optimal choices? –We will."— Presentation transcript:

1 1 Chapter 5 INCOME AND SUBSTITUTION EFFECTS

2 2 Objectives How will changes in prices and income influence influence consumers optimal choices? –We will look at partial derivatives

3 3 Demand Functions (review) We have already seen how to obtain consumers optimal choice Consumers optimal choice was computed Max consumers utility subject to the budget constraint After solving this problem, we obtained that optimal choices depend on prices of all goods and income. We usually call the formula for the optimal choice: the demand function For example, in the case of the Complements utility function, we obtained that the demand function (optimal choice) is:

4 4 Demand Functions If we work with a generic utility function (we do not know its mathematical formula), then we express the demand function as: x* = x(p x,p y, I ) y* = y(p x,p y, I ) We will keep assuming that prices and income is exogenous, that is: –the individual has no control over these parameters

5 5 Simple property of demand functions If we were to double all prices and income, the optimal quantities demanded will not change –Notice that the budget constraint does not change (the slope does not change, the crossing with the axis do not change either) x i * = d i (p x,p y, I ) = d i (2p x,2p y,2 I )

6 6 Changes in Income Since p x /p y does not change, the MRS will stay constant An increase in income will cause the budget constraint out in a parallel fashion (MRS stays constant)

7 7 What is a Normal Good? A good x i for which x i /I 0 over some range of income is a normal good in that range

8 8 Normal goods If both x and y increase as income rises, x and y are normal goods Quantity of x Quantity of y C U3U3 B U2U2 A U1U1 As income rises, the individual chooses to consume more x and y

9 9 What is an inferior Good? A good x i for which x i /I < 0 over some range of income is an inferior good in that range

10 10 Inferior good If x decreases as income rises, x is an inferior good Quantity of x Quantity of y C U3U3 As income rises, the individual chooses to consume less x and more y B U2U2 A U1U1

11 11 Changes in a Goods Price A change in the price of a good alters the slope of the budget constraint (p x /p y ) –Consequently, it changes the MRS at the consumers utility-maximizing choices When a price changes, we can decompose consumers reaction in two effects: –substitution effect –income effect

12 12 Substitution and Income effects Even if the individual remained on the same indifference curve when the price changes, his optimal choice will change because the MRS must equal the new price ratio –the substitution effect The price change alters the individuals real income and therefore he must move to a new indifference curve –the income effect

13 13 Sign of substitution effect (SE) SE is always negative, that is, if price increases, the substitution effect makes quantity to decrease and conversely. See why: 1) Assume p x decreases, so: p x 1 < p x 0 2) MRS(x 0,y 0 )= p x 0 / p y 0 & MRS(x 1,y 1 )= p x 1 / p y 0 1 and 2 implies that: MRS(x 1,y 1 )x 0

14 14 Changes in the optimal choice when a price decreases 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 falls, the consumer will maximize utility at point B. Total increase in x

15 15 Substitution effect when a price decreases U1U1 Quantity of x Quantity of y A To isolate the substitution effect, we hold utility constant but allow the relative price of good x to change. Purple is parallel to the new one Substitution effect C The substitution effect is the movement from point A to point C The individual substitutes good x for good y because it is now relatively cheaper

16 16 Income effect when the price decreases U1U1 U2U2 Quantity of x Quantity of y A The income effect occurs because the individuals real income changes (hence utility changes) when the price of good x changes C Income effect B The income effect is the movement from point C to point B If x is a normal good, the individual will buy more because real income increased How would the graph change if the good was inferior?

17 17 Subs and income effects when a price increases U2U2 U1U1 Quantity of x Quantity of y B A An increase in the price of good x means that the budget constraint gets steeper C The substitution effect is the movement from point A to point C Substitution effect Income effect The income effect is the movement from point C to point B How would the graph change if the good was inferior?

18 18 Price Changes for Normal Goods If a good is normal, substitution and income effects reinforce one another –when price falls, both effects lead to a rise in quantity demanded –when price rises, both effects lead to a drop in quantity demanded

19 19 Price Changes for Inferior Goods If a good is inferior, substitution and income effects move in opposite directions The combined effect is indeterminate –when price rises, the substitution effect leads to a drop in quantity demanded, but the income effect is opposite –when price falls, the substitution effect leads to a rise in quantity demanded, but the income effect is opposite

20 20 Giffens Paradox If the income effect of a price change is strong enough, there could be a positive relationship between price and quantity demanded –an increase in price leads to a drop in real income –since the good is inferior, a drop in income causes quantity demanded to rise

21 21 A Summary Utility maximization implies that (for normal goods) a fall in price leads to an increase in quantity demanded –the substitution effect causes more to be purchased as the individual moves along an indifference curve –the income effect causes more to be purchased because the resulting rise in purchasing power allows the individual to move to a higher indifference curve Obvious relation hold for a rise in price…

22 22 A Summary Utility maximization implies that (for inferior goods) no definite prediction can be made for changes in price –the substitution effect and income effect move in opposite directions –if the income effect outweighs the substitution effect, we have a case of Giffens paradox

23 23 Compensated Demand Functions This is a new concept It is the solution to the following problem: –MIN P X X+ P Y Y –SUBJECT TO U(X,Y)=U 0 Basically, the compensated demand functions are the solution to the Expenditure Minimization problem that we saw in the previous chapter After solving this problem, we obtained that optimal choices depend on prices of all goods and utility. We usually call the formula: the compensated demand function x* = x c (p x,p y,U), y* = y c (p x,p y,U)

24 24 Compensated Demand Functions x c (p x,p y,U 0 ), and y c (p x,p y,U 0 ) tell us what quantities of x and y minimize the expenditure required to achieve utility level U 0 at current prices p x,p y Notice that the following relation must hold: p x x c (p x,p y,U 0 )+ p y y c (p x,p y,U 0 )=E(p x,p y,U 0 ) –So this is another way of computing the expenditure function !!!!

25 25 Compensated Demand Functions There are two mathematical tricks to obtain the compensated demand function without the need to solve the problem: –MIN P X X+ P Y Y –SUBJECT TO U(X,Y)=U 0 One trick(A) (called Shephards Lemma) is using the derivative of the expenditure function Another trick(B) is to use the marshallian demand and the expenditure function

26 26 Compensated Demand Functions Sheppards Lema to obtain the compensated demand function Intuition: a £1 increase in p x raises necessary expenditures by x pounds, because £1 must be paid for each unit of x purchased. Proof: footnote 5 in page 137

27 27 Trick (B) to obtain compensated demand functions

28 28 Trick (B) to obtain compensated demand functions Suppose that utility is given by utility = U(x,y) = x 0.5 y 0.5 The Marshallian demand functions are x = I /2p x y = I /2p y The expenditure function is

29 29 Substitute the expenditure function into the Marshallian demand functions, and find the compensated ones: Another trick to obtain compensated demand functions

30 30 Compensated Demand Functions Demand now depends on utility (V) rather than income Increases in p x changes the amount of x demanded, keeping utility V constant. Hence the compensated demand function only includes the substitution effect but not the income effect

31 31 Roys identity It is the relation between marshallian demand function and indirect utility function Proof of the Roys identity…

32 32 Proof of Roys identity

33 33 Demand curves… We will start to talk about demand curves. Notice that they are not the same that demand functions !!!!

34 34 The Marshallian Demand Curve An individuals demand for x depends on preferences, all prices, and income: x* = x(p x,p y, I ) It may be convenient to graph the individuals demand for x assuming that income and the price of y (p y ) are held constant

35 35 x …quantity of x demanded rises. The Marshallian Demand Curve Quantity of y Quantity of x pxpx x p x U2U2 x2x2 I = p x + p y x p x U1U1 x1x1 I = p x + p y x p x x3x3 U3U3 I = p x + p y As the price of x falls...

36 36 The Marshallian Demand Curve The Marshallian demand curve shows the relationship between the price of a good and the quantity of that good purchased by an individual assuming that all other determinants of demand are held constant Notice that demand curve and demand function is not the same thing!!!

37 37 Shifts in the Demand Curve Three factors are held constant when a demand curve is derived –income –prices of other goods (p y ) –the individuals preferences If any of these factors change, the demand curve will shift to a new position

38 38 Shifts in the Demand Curve A movement along a given demand curve is caused by a change in the price of the good –a change in quantity demanded A shift in the demand curve is caused by changes in income, prices of other goods, or preferences –a change in demand

39 39 Compensated Demand Curves An alternative approach holds utility constant while examining reactions to changes in p x –the effects of the price change are compensated with income so as to constrain the individual to remain on the same indifference curve –reactions to price changes include only substitution effects (utility is kept constant)

40 40 Marshallian Demand Curves The actual level of utility varies along the demand curve As the price of x falls, the individual moves to higher indifference curves –it is assumed that nominal income is held constant as the demand curve is derived –this means that real income rises as the price of x falls

41 41 Compensated Demand Curves A compensated (Hicksian) demand curve shows the relationship between the price of a good and the quantity purchased assuming that other prices and utility are held constant The compensated demand curve is a two- dimensional representation of the compensated demand function x* = x c (p x,p y,U)

42 42 xcxc …quantity demanded rises. Compensated Demand Curves Quantity of y Quantity of x pxpx U2U2 x p x x x x x x Holding utility constant, as price falls...

43 43 Compensated & Uncompensated Demand for normal goods Quantity of x pxpx x xcxc x p x At p x, the curves intersect because the individuals income is just sufficient to attain utility level U 2

44 44 Compensated & Uncompensated Demand for normal goods Quantity of x pxpx x xcxc p x x*x*x At prices above p x, income compensation is positive because the individual needs some help to remain on U 2 As we are looking at normal goods, income and substitution effects go in the same direction, so they are reinforced. X includes both while X c only the substitution effect. That is what drives the relative position of both curves

45 45 Compensated & Uncompensated Demand for normal goods Quantity of x pxpx x xcxc p x x***x p x At prices below p x2, income compensation is negative to prevent an increase in utility from a lower price As we are looking at normal goods, income and substitution effects go in the same direction, so they are reinforced. X includes both while X c only the substitution effect. That is what drives the relative position of both curves

46 46 Compensated & Uncompensated Demand For a normal good, the compensated demand curve is less responsive to price changes than is the uncompensated demand curve –the uncompensated demand curve reflects both income and substitution effects –the compensated demand curve reflects only substitution effects

47 47 Relations to keep in mind Sheppards Lema & Roys identity V ( p x,p y,E( p x,p y,U o )) = U 0 E( p x,p y,V( p x,p y,I 0 )) = I 0 x c ( p x,p y,U 0 )=x( p x,p y,I 0 )

48 48 A Mathematical Examination of a Change in Price Our goal is to examine how purchases of good x change when p x changes x/ p x Differentiation of the first-order conditions from utility maximization can be performed to solve for this derivative

49 49 A Mathematical Examination of a Change in Price However, for our purpose, we will use an indirect approach Remember the expenditure function minimum expenditure = E(p x,p y,U) Then, by definition x c (p x,p y,U) = x [p x,p y,E(p x,p y,U)] –quantity demanded is equal for both demand functions when income is exactly what is needed to attain the required utility level

50 50 A Mathematical Examination of a Change in Price We can differentiate the compensated demand function and get x c (p x,p y,U) = x[p x,p y,E(p x,p y,U)]

51 51 A Mathematical Examination of a Change in Price The first term is the slope of the compensated demand curve –the mathematical representation of the substitution effect

52 52 A Mathematical Examination of a Change in Price The second term measures the way in which changes in p x affect the demand for x through changes in purchasing power –the mathematical representation of the income effect

53 53 The Slutsky Equation The substitution effect can be written as The income effect can be written as

54 54 The Slutsky Equation A price change can be represented by

55 55 The Slutsky Equation The first term is the substitution effect –always negative as long as MRS is diminishing –the slope of the compensated demand curve must be negative

56 56 The Slutsky Equation The second term is the income effect –if x is a normal good, then x/I > 0 the entire income effect is negative –if x is an inferior good, then x/I < 0 the entire income effect is positive

57 57 A Slutsky Decomposition We can demonstrate the decomposition of a price effect using the Cobb-Douglas example studied earlier The Marshallian demand function for good x was

58 58 A Slutsky Decomposition The Hicksian (compensated) demand function for good x was The overall effect of a price change on the demand for x is

59 59 A Slutsky Decomposition This total effect is the sum of the two effects that Slutsky identified The substitution effect is found by differentiating the compensated demand function

60 60 A Slutsky Decomposition We can substitute in for the indirect utility function (V)

61 61 A Slutsky Decomposition Calculation of the income effect is easier By adding up substitution and income effect, we will obtain the overall effect


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