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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 )

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Chapter 5

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  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

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