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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. 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

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

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

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

  7. 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

  8. 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

  9. 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

  10. 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

  11. 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

  12. Changes in a Good’s Price • Even if the individual remains on the same indifference curve, his optimal choice will change because the MRS must equal the new price ratio • the substitution effect • The individual’s “real” income has changed and he must move to a new indifference curve • the income effect

  13. 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

  14. Substitution effect Changes in a Good’s Price Quantity of y 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 The individual substitutes x for y because it is now relatively cheaper A C U1 Quantity of x

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

  16. 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

  17. Price Changes – Normal Goods • If a good is normal, substitution and income effects reinforce one another • when p : • substitution effect  quantity demanded  • income effect  quantity demanded  • when p: • substitution effect  quantity demanded  • income effect  quantity demanded 

  18. Price Changes – Inferior Goods • If a good is inferior, substitution and income effects move in opposite directions • when p : • substitution effect  quantity demanded  • income effect  quantity demanded  • when p: • substitution effect  quantity demanded  • income effect  quantity demanded 

  19. 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

  20. A Summary • 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

  21. A Summary • 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

  22. A Summary • 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

  23. 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

  24. …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

  25. 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

  26. 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

  27. 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

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

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

  30. 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

  31. 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 force the individual to remain on the same indifference curve • reactions to price changes include only substitution effects

  32. 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)

  33. …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

  34. 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

  35. At prices above px’, 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

  36. At prices below px’”, 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

  37. 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

  38. 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

  39. 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

  40. 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

  41. The Response to a Change in Price • What happens to purchases of good x change when px changes? x/px • Differentiation of the FOCs from utility maximization could be used • this approach is cumbersome and provides little economic insight

  42. The Response to a Change in Price • We will use an indirect approach using 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

  43. The Response to a Change in Price • We can differentiate the compensated demand function and get xc(px,py,U) = x[px,py,E(px,py,U)]

  44. The Response to a Change in Price • The first term is the slope of the compensated demand curve • the mathematical representation of the substitution effect

  45. The Response to 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

  46. The Slutsky Equation • The substitution effect can be written as • The income effect can be written as

  47. The Slutsky Equation • Note that E/px = x(px,py,I) • a $1 increase in px raises necessary expenditures by x dollars • $1 extra must be paid for each unit of x purchased

  48. The Slutsky Equation • The utility-maximization hypothesis shows that the substitution and income effects arising from a price change can be represented by

  49. 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

  50. 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

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