Consumer Preferences

1. Consumer Preferences Chapter 2

2. Introduction • Consumers are interested in consuming commodities that satisfy wants • Food, shelter, and clothing • Without these commodities there would probably be no happiness • Not all commodities bring happiness • Bad water, leaky roofs, smelly clothes

3. Introduction • Because consumers are faced with limited resources, not all their wants can be satisfied • Must choose which wants to satisfy from limited resources • Consumer preferences are central to how consumers behave to satisfy as many wants as possible

4. Aim of Chapter • To investigate how consumer preferences are employed by consumers in making their individual commodity choices • Allows us to relate individual consumer preferences to society’s overall commodity choices • No central authority is required for determining overall consumer choices • Decentralized consumer choice underlies an efficient allocation of resources • Will investigate properties associated with a number of utility functions • Main objective—to derive demand functions based on consumer preferences, prices, and income

5. Aim of Chapter • Demand • How much of a commodity consumers are willing and able to purchase at a given price • May be willing and able to purchase 5 pounds of bananas at 59¢ per pound • Can afford to purchase more bananas, but you are not interested in purchasing more • As price of a commodity declines • Quantity demanded for commodity increases

6. Law of Demand • While at supermarket, you notice a special on bananas for 39¢ per pound • You are now willing to purchase 7 pounds instead of 5 pounds • Figure 2.1.2 illustrates law of demand • Inverse relationship between price and quantity demanded • Increase price, quantity demanded declines • Decrease price, quantity demanded increases

7. Figure 2.1 Law of Demand

8. Introduction • To determine consumer demand, large firms employ applied economists to model demand for their products • For example, American Express employs hundreds of applied economists to model demand for their credit cards • Characteristics of consumers likely to default, commit fraud, or always pay their minimum balances • Demand functions are based on assumption • Consumers attempt to maximize their satisfaction by consuming commodities • Out of an infinite combination of commodities, they choose a particular set of commodities that maximize satisfaction • Maximization is constrained by limited resources

9. Commodity Bundles and Household Preferences • Foundation of consumer theory • Model of consumer or household preferences • Each household is a unit of society attempting to maximize its happiness based on its preferences for commodities • Commodity • Particular good or service delivered at a specific time and at a specific location • Commodities consumed at different times and locations should be viewed as distinct commodities • However, in practice, economic models often involve some aggregation over time and location (space) • Assumption • Commodities being aggregated are sufficiently similar

10. Commodity Bundles and Household Preferences • Problem facing a household • Deciding how much of each available commodity it should consume • Household’s objective • Maximize satisfaction from commodities it consumes • Given prices of all commodities and household’s limited resources • Monetary constraint (income) • For developing models of consumer behavior, economists abstract by assuming a finite number of k commodities • k could be restricted to just two commodities or be unrestricted

11. Commodity Bundles and Household Preferences • Level of consumption may be zero for some commodities • For example, consumers generally consume zero levels of antique buses or trips into outer space • A bundle comprising all the k commodities a household may consume is called a commodity bundle, and is represented as

12. Two-Commodity Assumption • Often assume a household is faced with a choice of only two commodities (x1 and x2) • These commodities could be either • Only two commodities a household consumes or • Only two commodities it can vary • All other commodities are fixed in terms of some given quantity • Represent the commodity bundle as • Where all commodities to right of the bar | are considered fixed

13. Two-Commodity Assumption • Two-commodity assumption can be generalized by assuming one of the commodities is a composite commodity—numeraire commodity—composed of all other commodities • In a graph of two commodities, x1 and x2, a commodity bundle is a point in commodity space • Set of all possible commodity bundles • A household cannot consume a negative amount of a commodity • Commodity space is represented by nonnegative quadrant

14. Figure 2.2 Commodity space

15. Preference Relation • Objective of a household • To consume commodity bundle that yields highest satisfaction it can afford • A household’s choice of preferred commodity bundle depends on • Commodity prices and household’s limited income • Tastes and preferences of household • Can be summarized by the preference relation, “is preferred to or indifferent to,” written as & • Indifference between x and y means household would be just as satisfied consuming x as it would be consuming y

16. Figure 2.3 Gaps of indecision in the commodity space

17. Preference Ordering • Preference relation provides a method for modeling how a household orders or ranks a set of bundles • From most to least desirable • Often done unconsciously • But with large purchases it is also done consciously • Such as buying an automobile • Without preference ordering a household cannot determine its preferred consumption bundle • Two assumptions—preference axioms—are required to order a set of bundles • These assumptions are basic axioms in consumer theory • Axiom • An assumption that is generally accepted as true

18. Axiom 1: Completeness • If x and y are any two commodity bundles, a household can always specify exactly one of following • Commodity bundles within an area of household indecision cannot be ordered in terms of a household’s preferences • Completeness Axiom precludes areas of indecision • Assuming household members completely understand contents of each bundle and can always make up their minds • Households generally can make up their minds within their range of common experience

19. Axiom 2: Transitivity • States that a household’s preferences for alternative commodity bundles cannot be cyclical • Example • Partying Friday night is preferred to a Saturday football game • Saturday football game is preferred to going to church on Sunday • Going to church cannot be preferred to Friday partying • Necessary for any discussion of preference maximization • Without this axiom, households cannot order commodity bundles from most to least desirable • Rationality means households can ordinally rank a set of commodity bundles to maximize their satisfaction of wants given their limited resources

20. Utility Functions • Political theorist Jeremy Bentham introduced a ranking of commodity bundles • Represented by a utility function (U) • Utility • Ability or power of a commodity or commodity bundle to satisfy wants when a household consumes the commodity or bundle. • Utility functions • Indicate how a household ranks commodity bundles • Assigns numerical value to each level of satisfaction associated with each commodity bundle • Higher the preference ranking, larger the number assigned • Household then determines which bundle maximizes this utility function • Given household’s limited income and fixed commodity prices

21. Utility Functions • For graphical representation we assume that only two commodities can vary • Hold all other commodities constant • Utility function is represented as • U =U(x1, x2|x3, . . . , xk) • Suppressing fixed commodities x3 … xk • U = (x1, x2)

22. Characteristics of Utility Functions • Utility functions and indifference curves derived from them can take on a number of shapes • Depending on particular assumptions concerning a household’s preferences • Classical shape of a utility function assumes a commodity is desirable • Greater amounts of the commodity increase utility • Stated in Axiom 3 • Nonsatiation

23. Figure 2.4 Utility curve for the function U(x) = ln x

24. Axiom 3: Nonsatiation • More of a commodity is preferred to less • Household can always do a little bit better by consuming more of a commodity • Such a commodity is termed a good (or desirable) commodity • As opposed to a bad (or undesirable) commodity • Results in a decline in utility as more of the commodity is consumed • More of the bad commodity is not preferred to less • Garbage by definition is a bad commodity • One that yields negative utility

25. Axiom 3: Nonsatiation • Assuming utility function U(x) is differentiable • Nonsatiation Axiom requires that all first-order partial derivatives of utility function be positive • Increasing consumption of any commodity increases utility • Holding consumption of all other commodities constant • In other words, nobody is content with their current level of any commodity • For example, assuming a household ranks commodities by utility function • U =U(x1, x2, . . . , xk) then • Represents extra utility obtained from consuming slightly more of xj • Holding amounts consumed of all other commodities constant

26. Axiom 3: Nonsatiation • Value of marginal utility depends on point at which partial derivative is to be evaluated • How much x1, x2, …, xk household is currently consuming • Only sign of MU is important • Actual magnitude is meaningless since utility is an ordinal ranking • Figure 2.5 illustrates result of Nonsatiation Axiom for x1 and x2 • Every point or bundle within positive quadrant represents a commodity bundle • States that given an initial commodity bundle x, every commodity bundle with more of at least one commodity will be preferred to x • Shaded area in Figure 2.5 represents preferred set of bundles

27. Figure 2.5 Nonsatiation, more is preferred to less

28. Axioms • Axioms 1 and 2 provide necessary assumptions for household preference ordering • Determine indifference sets of commodity bundles • Within each set a household receives same level of satisfaction • A household is indifferent between any two bundles within an indifference set • For example, a household may be willing to give up a six pack of beer for a pound of candy with no change in satisfaction • Axiom 3 provides direction of increasing utility given a change in a commodity bundle • These three axioms are all that is necessary for determining utility-maximizing bundle

29. Table 2.1 Preference axioms

30. Indifference Curves • Indifference sets are a set of curves • Each indifference curve is a locus of points (commodity bundles) that yield same level of utility • Every point along an indifference curve represents a different combination of two commodities • Each combination is equally satisfactory to a household • Each combination yields same level of total utility • Commodity bundles can be represented by an indifference space • Analogous to a relief map • Contour lines or curves represent equal levels of satisfaction or utility instead of equal elevation

31. Indifference Curves • Figure 2.6 shows an indifference space for a household consuming two commodities, x1 and x2 • Each of the indifference curves yield different levels of utility (U1,U2, and U3) • Commodity bundles x and y are on same indifference curve • Yield same level of utility even though they represent different combinations of commodities • More of commodity x2 and less of commodity x1 are consumed at x than y • Commodity bundle z is on a higher indifference curve and yields a higher level of total utility • According to Nonsatiation Axiom, increasing either or both of the commodities shifts household to higher and higher indifference curves • Until household approaches global bliss • Some households will never reach global bliss • In this case, there will be an infinite number of indifference curves

32. Figure 2.6 Indifference space, MRS(x2 for x1)

33. Indifference Curves • Between any two indifference curves are a finite number of curves • Household will not be able to distinguish between two indifference sets that are very close to each other • As sets diverge distinction will become apparent • Indifference curves cannot intersect • Transitivity Axiom is violated if indifference curves intersect

34. Figure 2.7 Indifference curves cannot intersect

35. Marginal Rate of Substitution (MRS) • The (generally) negative slope of an indifference curve implies • If a household is forced to give up some x1, it must be compensated by an additional amount of x2 to remain indifferent • A measure for this substitution is marginal rate of substitution (MRS) • Defined as negative of indifference curve slope • For the two commodities x1 and x2 in Figure 2.6 • Where U = constant (dU = 0) indicates that utility is being held constant as slope changes • Represents a movement along an indifference curve • Marginal rate of substitution (x2 for x1) is defined as

36. Figure 2.8 Indifference space, MRS(x1 for x2)

37. Marginal Rate of Substitution (MRS) • How much a household is willing to pay in terms of x2 in order to consume more of x1 is measured by MRS(x1 for x2) • Results in flipping the axes • MRS is directly related to a household’s marginal utilities for each commodity • Extra utility obtainable from consuming slightly more x1, x2, … , xk is sum of additional utility provided by each of these increments

38. Marginal Rate of Substitution (MRS) • Taking total differential of U = U(x1, x2, …, xk) gives • Concept of MRS changes level of only two commodities (say, x1 and x2), keeping household indifferent (dU = 0) • Implies that all dx’s are equal to zero except dx1 and dx2

39. Marginal Rate of Substitution (MRS) • Rearranging terms yields an application of Implicit Function Theorem • Illustrates relationship of MRS to ratio of marginal utilities

40. Strictly Convex Indifference Curves • Generally, most people prefer a combination of beer and munchies • Vs. all beer and no munchies or all munchies and no beer • Strictly convex indifference curves indicate this type of trade-off • Based on Axiom 4 • Diminishing Marginal Rate of Substitution

41. Axiom 4: Diminishing Marginal Rate of Substitution (Strict Convexity) • Diminishing MRS exists when value of MRS(x2 for x1) approaches zero as x1 increases • In Figure 2.9, as x1 increases, slope of indifference curve tends to zero • Becomes less negative • Because MRS is the negative of the slope • MRS decreases toward zero as x1 increases • For very low values of x1, a household is willing to give up a larger amount of x2 to get another unit of x1

42. Axiom 4: Diminishing Marginal Rate of Substitution (Strict Convexity) • In Figure 2.9 household prefers average bundle z to two extreme bundles x and y (a > b) • Assumes indifference curves form a convex set of commodity bundles • Yield at least same level of utility represented by an indifference curve • As x1 increases, household is willing to give up less of x2 to obtain one more unit of x1 • Similarly, as x2 increases, household is willing to give up less of x1 to obtain one more unit of x2

43. Figure 2.9 Indifference curves with diminishing marginal rate of substitution

44. Axiom 4: Diminishing Marginal Rate of Substitution (Strict Convexity) • A set of points is a convex set if • Any two points within set can be joined by a straight line contained completely within the set • Can determine an implication of Diminishing MRS Axiom • Suppose a household is indifferent between following bundles • Diminishing MRS Axiom states combination bundle will be preferred

45. Axiom 4: Diminishing Marginal Rate of Substitution (Strict Convexity) • Strict convexity is equivalent to assumption of diminishing MRS • Provided linear combinations of the commodities are possible • Implication of diminishing MRS • Well-balanced, diversified bundles of commodities are preferred to bundles that are heavily weighted toward one commodity • Household prefers averages to extremes

46. Strictly Concave Indifference Curves • If a household’s preferences were represented by strictly concave preferences • Household would prefer extremes to averages • Generally, households attempt to diversify • Rules out these concave preferences • Some rational choices imply concave preferences • Examples • Preferring either alcohol or driving to consuming both together • Preferring grades of all As instead of some combination of As and Bs

47. Figure 2.10 Indifference curves with concave preferences

48. Imperfect Substitutes • Indifference curves represent an individual household’s preferences for commodities • Suppose a household may prefer consuming relatively more of commodity x2 over commodity x1 • For household to be willing to give up a small amount x2 • Would have to be given a relatively large quantity of x1 • Results in relatively flat indifference curves • Imperfect substitutes are characterized by the four preference axioms • Most household preferences for most commodities fall into this category

49. Figure 2.11 Indifference curves where commodity x2 is relatively more preferred to x1

50. Perfect Substitutes • A utility function of form U =ax1 + bx2 represents preferences associated with perfect substitutes • a and b are positive parameters • For perfect substitutes • Slope of indifference curve does not change as one commodity is substituted for another • In general, perfect substitutes are associated with MRS = constant • Violates Diminishing MRS Axiom • Indifference curves are parallel straight lines with a constant slope • An example of perfect substitutes may be two different brands of cola