1 / 124

Chapter 5

Chapter 5. INCOME AND SUBSTITUTION EFFECTS. Demand Functions. The optimal levels of x 1 , x 2 ,…, x n can be expressed as functions of all prices and income These can be expressed as n demand functions of the form:. x 1 * = d 1 ( p 1 , p 2 ,…, p n , I )

hiero
Télécharger la présentation

Chapter 5

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 5 INCOME AND SUBSTITUTION EFFECTS

  2. Demand Functions • The optimal levels of x1,x2,…,xn can be expressed as functions of all prices and income • These can be expressed as n demand functions of the form: x1* = d1(p1,p2,…,pn,I) x2* = d2(p1,p2,…,pn,I) • • • xn* = dn(p1,p2,…,pn,I)

  3. Aim • use the utility-maximization model to study how the quantity of a good that an individual chooses is affected by a change in that good’s price. This

  4. Key points of chap.1-4 • The envelope theorem(p33-36,44) y=f(x,a)

  5. Key points of chap.1-4 • Utility function Utility=U(X1, X2, . . . , Xn; other things) Cobb-Douglas Utility CES Utility

  6. Homogeneity • Euler theorem

  7. Key points of chap.1-4 • Utility Maximization Problem (UMP) • xi=x(p1,…,pn,I) ; indirect UF: v=U(x1,…,xn)=v(p1,…,pn,I) • dual: Expenditure Minimization Problem • X=x(p,u) • EF:E=E(p1,…,pn,U)

  8. Demand Functions • If there are only two goods (x and y), we can simplify the notation x* = x(px,py,I) y* = y(px,py,I) • Prices and income are exogenous • the individual has no control over these parameters

  9. Homogeneity • If we were to double all prices and income, the optimal quantities demanded will not change • the budget constraint is unchanged xi* = di(p1,p2,…,pn,I) = di(tp1,tp2,…,tpn,tI) • Individual demand functions are homogeneous of degree zero in all prices and income

  10. Homogeneity • With a Cobb-Douglas utility function utility = U(x,y) = x0.3y0.7 the demand functions are • Note that a doubling of both prices and income would leave x* and y* unaffected

  11. Homogeneity • With a CES utility function utility = U(x,y) = x0.5 + y0.5 the demand functions are • Note that a doubling of both prices and income would leave x* and y* unaffected

  12. Changes in Income • An increase in income will cause the budget constraint out in a parallel fashion • Since px/py does not change, the MRS will stay constant as the worker moves to higher levels of satisfaction

  13. C B A U3 U2 U1 Increase in Income • If both x and y increase as income rises, x and y are normal goods Quantity of y As income rises, the individual chooses to consume more x and y Quantity of x

  14. C B U3 U2 A U1 Increase in Income • If x decreases as income rises, x is an inferior good As income rises, the individual chooses to consume less x and more y Quantity of y Note that the indifference curves do not have to be “oddly” shaped. The assumption of a diminishing MRS is obeyed. Quantity of x

  15. Normal and Inferior Goods • A good xi for which xi/I 0 over some range of income is a normal good in that range • A good xi for which xi/I< 0 over some range of income is an inferior good in that range

  16. Changes in a Good’s Price • A change in the price of a good alters the slope of the budget constraint • it also changes the MRS at the consumer’s utility-maximizing choices • When the price changes, two effects come into play • substitution effect • income effect

  17. Changes in a Good’s Price • 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 individual’s “real” income and therefore he must move to a new indifference curve • the income effect

  18. Suppose the consumer is maximizing utility at point A. If the price of good x falls, the consumer will maximize utility at point B. B A U2 U1 Total increase in x Changes in a Good’s Price Quantity of y Quantity of x

  19. To isolate the substitution effect, we hold “real” income constant but allow the relative price of good x to change The substitution effect is the movement from point A to point C C Substitution effect Changes in a Good’s Price Quantity of y The individual substitutes good x for good y because it is now relatively cheaper A U1 Quantity of x

  20. The income effect is the movement from point C to point B B Income effect Changes in a Good’s Price Quantity of y The income effect occurs because the individual’s “real” income changes when the price of good x changes If x is a normal good, the individual will buy more because “real” income increased A C U2 U1 Quantity of x

  21. The substitution effect is the movement from point A to point C C The income effect is the movement from point C to point B Substitution effect Income effect Changes in a Good’s Price Quantity of y An increase in the price of good x means that the budget constraint gets steeper A B U1 U2 Quantity of x

  22. Price Changes forNormal 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

  23. Price Changes forInferior 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

  24. Giffen’s 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

  25. 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

  26. A Summary • Utility maximization implies that (for normal goods) a rise in price leads to a decline in quantity demanded • the substitution effect causes less to be purchased as the individual moves along an indifference curve • the income effect causes less to be purchased because the resulting drop in purchasing power moves the individual to a lower indifference curve

  27. 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 Giffen’s paradox

  28. The Individual’s Demand Curve • An individual’s demand for x depends on preferences, all prices, and income: x* = x(px,py,I) • It may be convenient to graph the individual’s demand for x assuming that income and the price of y (py) are held constant

  29. …quantity of x demanded rises. px’ px’’ px’’’ U3 x U2 U1 x1 x2 x3 x’ x’’ x’’’ I = px’ + py I = px’’ + py I = px’’’ + py The Individual’s Demand Curve Quantity of y As the price of x falls... px Quantity of x Quantity of x

  30. The Individual’s Demand Curve • An individual 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

  31. Shifts in the Demand Curve • Three factors are held constant when a demand curve is derived • income • prices of other goods (py) • the individual’s preferences • If any of these factors change, the demand curve will shift to a new position

  32. 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

  33. Demand Functions and Curves • We discovered earlier that • If the individual’s income is $100, these functions become

  34. Demand Functions and Curves • Any change in income will shift these demand curves

  35. Compensated 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

  36. Compensated Demand Curves • An alternative approach holds real income (or utility) constant while examining reactions to changes in px • the effects of the price change are “compensated” so as to constrain the individual to remain on the same indifference curve • reactions to price changes include only substitution effects

  37. 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* = xc(px,py,U)

  38. …quantity demanded rises. px’ px’’ px’’’ xc U2 x’ x’’ x’’’ x’ x’’ x’’’ Compensated Demand Curves Holding utility constant, as price falls... Quantity of y px Quantity of x Quantity of x

  39. At px’’, the curves intersect because the individual’s income is just sufficient to attain utility level U2 px’’ x’’ Compensated & Uncompensated Demand px x xc Quantity of x

  40. At prices above px2, income compensation is positive because the individual needs some help to remain on U2 px’ x’ x* Compensated & Uncompensated Demand px px’’ x xc Quantity of x

  41. At prices below px2, income compensation is negative to prevent an increase in utility from a lower price px’’’ x*** x’’’ Compensated & Uncompensated Demand px px’’ x xc Quantity of x

  42. 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

  43. Compensated Demand Functions • Suppose that utility is given by utility = U(x,y) = x0.5y0.5 • The Marshallian demand functions are x = I/2pxy = I/2py • The indirect utility function is

  44. Compensated Demand Functions • To obtain the compensated demand functions, we can solve the indirect utility function for I and then substitute into the Marshallian demand functions

  45. Compensated Demand Functions • Demand now depends on utility (V) rather than income • Increases in px reduce the amount of x demanded • only a substitution effect

  46. A Mathematical Examination of a Change in Price • Our goal is to examine how purchases of good x change when px changes x/px • Differentiation of the first-order conditions from utility maximization can be performed to solve for this derivative • However, this approach is cumbersome and provides little economic insight

  47. A Mathematical Examination of a Change in Price • Instead, we will use an indirect approach • Remember the expenditure function minimum expenditure = E(px,py,U) • Then, by definition xc(px,py,U) = x[px,py,E(px,py,U)] • quantity demanded is equal for both demand functions when income is exactly what is needed to attain the required utility level

  48. A Mathematical Examination of a Change in Price • We can differentiate the compensated demand function and get xc(px,py,U) = x[px,py,E(px,py,U)]

  49. 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

  50. A Mathematical Examination of a Change in Price • The second term measures the way in which changes in px affect the demand for x through changes in purchasing power • the mathematical representation of the income effect

More Related