1 Indifference Curves and Utility Maximization CHAPTER 6 Appendix © 2003 South-Western/Thomson Learning

2 Indifference Curves and Utility Maximization Marginal utility analysis requires some numerical measure of utility in order to determine the optimal consumption combinations Economists have developed another, more general, approach to utility and consumer behavior This approach does not require that numbers be attached to specific levels of utility

3 Indifference Curves and Utility Maximization All this new approach requires is that consumers be able to rank their preferences for various combinations of goods Specifically, the consumer should be able to say whether Combination A is preferred to combination B Combination B is preferred to combination A. or Both combinations are equally preferred

4 Consumer Preferences Indifference curve shows all combinations of goods that provide the consumer with the same satisfaction, or the same utility Thus, the consumer finds all combinations on a curve equally preferred Since each of the alternative bundles of goods yields the same level of utility, the consumer is indifferent about which combination is actually consumed

5 Exhibit 9: An Indifference Curve Suppose there are only two goods available: pizzas and movie videos Point a shows the consumption bundle consisting of 1 pizza and 8 video rentals Suppose a person is given a choice at point a or some other combination Holding utility constant, how many video rentals would a person be willing to give up to get a second pizza? Moving from point a to point b, we are willing to give up 4 videos to get a second pizza (total utility is the same at points a and b); the marginal utility of another pizza per week is just sufficient to compensate for the utility lost from decreasing video purchases by 4 movies per week Pizzas per week I d c b a Video rentals per week

6 Exhibit 9: An Indifference Curve In moving from point b to c, again total utility is constant; the person is willing to give up only 1 video for another pizza. Once at point c, the person is willing to give up another video only if they get two more pizzas in return  combination d consists of 5 pizzas and 2 videos Points a, b, c, and d can be connected to form the indifference curve, I, which represents possible combinations of pizza and videos that would keep the person at the same level of total utility. Combinations of goods along an indifference curve reflect some constant, though unspecified, level of total utility Pizzas per week I d c b a Video rentals per week

7 Indifference Curves For a person to remain indifferent among consumption combinations, the increase in utility from eating more pizza must just offset the decrease in utility from watching fewer videos Thus, along an indifference curve, there is an inverse relationship between the quantity of one good consumed and the quantity of another consumed  indifference curves slope down

8 Indifference Curves Indifference curves are also convex to the origin  they are bowed inward toward the origin The curve gets flatter as you move down it The marginal rate of substitution, or MRS, between pizza and videos indicates the number of videos that the consumer is willing to give up to get one more pizza, while maintaining the same level of total utility

9 Marginal Rate of Substitution The MRS measures the consumers willingness to trade videos for pizza  depends on the amount of each good the consumer is consuming at the time Mathematically, the MRS is equal to the absolute value of the slope of the indifference curve For example, in moving from combination a to combination b, the consumer is willing to give up 4 videos to get 1 more pizza  slope between these two points equals –4  MRS = 4; from b to c, MRS = 1

10 Marginal Rate of Substitution The law of diminishing marginal rate of substitution says that as a persons consumption of pizza increases, the number of videos that they are willing to give up to get another pizza declines This implies that the indifference curve has a diminishing slope  as we move down the indifference curve, the consumption of pizza increases and the marginal utility of additional pizza declines

11 Indifference Map We can use the same approach to generate a series of indifference curves, called an indifference map  graphical representation of a consumer’s tastes Each curve in the map reflects a different level of utility Exhibit 10 illustrates an indifference map for a particular consumer

12 Exhibit 10: An Indifference Map Each indifference curve represents a different level of utility and each consumer has a unique indifference map based on their preferences Because both goods yield marginal utility, the consumer prefers more of each, rather than less  curves farther from the origin represent greater consumption levels  higher levels of utility  total utility along I 2 higher than along I 1, I 3 higher than I 2, etc This can be seen by drawing a ray from the origin and following it to higher indifference curves The combination on each successive indifference curve reflects greater amounts of both goods V i d e o r e n t a l s p e r w e e k Pizzas per week I 1 I 2 I 3 I 4

13 Exhibit 11: Indifference Curves Do Not Intersect V i d e o r e n t a l s p e r w e e k 0 Pizzas per week k j i I' I If indifference curves crossed, such as point i, then every point on indifference curve I and every point on curve I', would have to reflect the same level of utility as at point i But point k is a combination with more pizza and more videos than point j  must represent a higher level of utility

14 Properties of Indifference Curves A particular indifference curve reflects a constant level of utility  the consumer is indifferent among all consumption combinations along a given curve If total utility is to remain constant, an increase in the consumption of one good must be offset by a decrease in the consumption of the other good  indifference curves slope downward Higher indifference curves represent higher levels of utility

15 Properties of Indifference Curves Because of the law of diminishing marginal rate of substitution, indifference curves are bowed in toward the origin Indifference curves do not intersect Indifference curves are a graphical representation of a consumer’s tastes for the two goods

16 Properties of Indifference Curves Once we have the consumer’s indifference may, we turn to the issue of how much of each good will be consumed? To answer this question, we must consider the relative prices of the two goods and the consumer’s income

17 Budget Line Budget line depicts all possible combinations of movies and pizzas, given prices and your budget Suppose movies rent for $4, pizza sells for $8, and the budget is $40 per week  if you spend the entire $40 on videos, consumer can purchase 10 videos, and if on pizzas person can afford 5 per week

18 Exhibit 12: A Budget Line Given the prices and budget, the budget line meets the vertical axis at 10 videos and meets the horizontal axis at 5 pizzas These two intercepts are then connected to form the budget line. The budget line defines all possible combinations of the two goods, pizza and videos, that can be purchased, given prices and income  can be thought of as a consumption possibilities frontier At the point where the budget line meets the vertical axis, the maximum number of videos you can rent equals income divided by the video rental price = I / p v and for the horizontal axis is I / p p Slope of the budget line indicates what it costs the consumer in terms of foregone video rentals to get another pizza

19 Summary The indifference curve indicates what the consumer is willing to buy The budget line shows what the consumer is able to buy When the indifference curve and the budget line are combined, we find the quantities of each good the consumer is both willing and able to buy Exhibit 13 illustrates this

20 Exhibit 13: Utility Maximization The utility-maximizing consumer will select a combination along the budget line that lies on the highest attainable indifference curve Given prices and income, this occurs at point e, where I 2 just touches, or is tangent to, the budget line  buy 3 pizzas at $8 each and rent 4 videos at $4 each Other attainable combinations along the budget line reflect lower levels of utility. For example, point a is on the budget line  a combination that can be purchased. However, point a lies on a lower indifference curve. Exhibit 13: Utility Maximization V i d e o r e n t a l s p e r w e e k Pizzas per week 4 I 2 I 1 I 3 e a

21 Consumer Equilibrium Consumer equilibrium occurs where the slope of the indifference curve is equal to the slope of the budget line Recall that the absolute value of the slope of the indifference curve is the marginal rate of substitution, and the absolute value of the slope of the budget line equals the price ratio

22 Consumer Equilibrium Thus, MRS = P p / P v Further, the marginal rate of substitution of pizzas for video rentals can be found from the marginal utilities of pizza and video  MRS = MU p / MU v

23 Consumer Equilibrium In fact, the absolute value of the slope of the indifference curve equals MU p /MU v and the slope of the budget line equals p p / p v  the equilibrium condition for the indifference curve approach can be written as

24 Effects of a Change in Price What happens to the consumer’s equilibrium consumption when there is a change in price? The answer can be found by using indifference curve approach to derive the demand curve Exhibit 14 illustrates this process

25 Exhibit 14: Effect of a Drop in Price of Pizza The initial equilibrium here is at point e with 3 pizzas and 4 videos. If the price of pizza falls from $8 to $6 per unit, other things constant, the price drop means that if the entire budget were devoted to pizza, the consumer could purchase 6.67 pizzas (40 / 6) Thus, even though money income has remained the same, real income has increased because of the lower price of pizza. Since the rental price of videos has not changed, the maximum number of videos that can be rented remains the same at 10. As a result of the price change, the lower end of the budget line rotates out from 5 to 6.67 After the price change, the new equilibrium occurs at e", where pizza purchases increase from 3 to 4, and, as it happens, video rentals remains at Pizzas per week 4 e I e" Videos per week

26 Exhibit 14: Effect of a Drop in Price of Pizza V i d e o s p e r w e e k Pizzas per week 4 e I (b) (a) P r i c e p e r p i z z a $ Pizzas per week e e" 6 D The demand curve is shown in the lower panel and depicts how price and quantity demanded are related Specifically, when the price of pizza falls from $8 per unit to $6 per unit, other things assumed constant, the quantity demanded increases from 3 to 4 Since the consumer is on a higher indifference curve at e", the consumer is clearly better off after the price reduction  the consumer surplus has increased

27 Income and Substitution Effects The law of demand was initially explained in terms of an income effect and a substitution effect With indifference curve analysis we have the analytical tools to examine these two effects more precisely Exhibit 15 illustrates this process

28 Exhibit 15: Substitution and Income Effects Pizzas per week V i d e o r e n t a l s p e r w e e k 0 4 e I I* e* 5 Suppose the price of pizza falls from $8 to $4, other things constant  consumer can now purchase a maximum of 10 pizzas with a budget of $40. The budget intercept line rotates out from 5 to 10 pizzas. The increase in the quantity of pizzas demanded can be broken down into the substitution and the income effect of a price change. When the price of pizza falls, the change in the ratio of the price of pizza to the price of video rentals shows up through the change in the slope of the budget line. To derive the substitution effect, let’s assume that you must maintain the same level of utility after the price change as before  consumer’s utility level has not changed but the relative prices you face have changed.

29 Exhibit 15: Substitution and Income Effects A new budget line reflecting the change in relative prices, not a change in utility is shown by the dashed line CF. Given the new set of relative prices, the consumer would increase the quantity of pizza demanded to the point on the indifference curve I is just tangent to the dashed budget line  utility is at the initial level with the new relative prices, but adjust income so that utility remains the same The consumer moves down along the indifference curve I to point e', renting fewer videos but buying more pizzas. These changes reflect the substitution effect of lower prices of pizza Since consumption bundle e' represents the same level of utility as consumption bundle e, the consumer is neither better or worse off at point e'. Pizzas per week V i d e o r e n t a l s p e r w e e k 0 4 e I I* e* 5 e' 4 F C Substitution effect

30 Exhibit 15: Substitution and Income Effects But at point e', the consumer has not spent the full budget. The consumer’s real income has increased because of the lower price of pizza  he is able to attain point e* on indifference curve I*. At this point, the consumer buys 5 pizzas and rent 5 videos. Because prices are held constant during the move from e’ to e*, the change in consumption is due solely to a change in real income  the change in the quantity demanded from 4 to 5 reflects the income effect of the lower pizza price The substitution effect is shown by the move from point e to point e' in response to a change in the relative price of pizza, with utility held constant along I. The income effect is shown by the move from e' to e* in response to a change in real income with relative prices constant Pizzas per week V i d e o r e n t a l s p e r w e e k 0 4 e I I* e* 5 e' 4 F C Substitution effect Income effect