Chapter 4 Utility

1. Chapter 4 Utility Course: Microeconomics Text: Varian’s Intermediate Microeconomics

2. Introduction • Last chapter we talk about preference, describing the ordering of what a consumer likes. • For a more convenient mathematical treatment, we turn this ordering into a mathematical function.

3. Utility Function • A utility function U: R+nR maps each consumption bundle of n goods into a real number that satisfies the following conditions: x’ x” U(x’) > U(x”) x’ x” U(x’) < U(x”) x’ ~ x” U(x’) = U(x”). p p

4. Utility Function • Not all theoretically possible preference have a utility function representation. • Technically, a preference relation that is complete, transitive and continuous has a corresponding continuous utility function.

5. Utility Function • Utility is an ordinal (i.e. ordering) concept. • The number assigned only matters about ranking, but the sizes of numerical differences are not meaningful.

6. Example • Consider only three bundles A, B, C. • The following three are all valid utility function of the preference.

7. Utility Functions • There is no unique utility function representation of a preference relation. • Suppose U(x1,x2) = x1x2 represents a preference relation. • Consider the bundles (4,1), (2,3) and (2,2).

8. Utility Functions • U(x1,x2) = x1x2, soU(2,3) = 6 > U(4,1) = U(2,2) = 4;that is, (2,3) (4,1) ~ (2,2). p

9. Utility Functions • Define V = U2. • Then V(x1,x2) = x12x22 and V(2,3) = 36 > V(4,1) = V(2,2) = 16so again(2,3) (4,1) ~(2,2). • V preserves the same order as U and so represents the same preferences. p

10. Utility Functions • Define W = 2U + 10. • Then W(x1,x2) = 2x1x2+10 so W(2,3) = 22 > W(4,1) = W(2,2) = 18. Again,(2,3) (4,1) ~ (2,2). • W preserves the same order as U and V and so represents the same preferences. p

11. f f ~ ~ Utility Functions • If • U is a utility function that represents a preference relation and • f is a strictly increasing function, • then V = f(U) is also a utility functionrepresenting . • Clearly, V(x)>V(y) if and only if f(V(x)) > f(V(y)) by definition of increasing function.

12. Fig. 4.1

13. Utility Function • As you will see, for our analysis of consumer choices, an ordinal utility is enough. • If the numerical differences are also meaningful, we call it cardinal. • e.g. money, weight, height are cardinal • Cardinal utility can be useful in some areas, such as preference under uncertainty.

14. Utility Functions & Indiff. Curves • An indifference curve contains equally preferred bundles. • Equal preference  same utility level. • Therefore, all bundles in an indifference curve have the same utility level.

15. Utility Functions & Indiff. Curves x2 U º 6 U º 4 U º 2 x1

16. Utility Functions & Indiff. Curves Utility U º 6 U º 5 U º 4 U º 3 x2 U º 2 U º 1 x1

17. Goods, Bads and Neutrals • A good is a commodity which increases utility (gives a more preferred bundle) when you have more of it. • A bad is a commodity which decreases utility (gives a less preferred bundle) when you have more of it. • A neutral is a commodity which does not change utility (gives an equally preferred bundle) when you have more of it.

18. Goods, Bads and Neutrals Utility Utilityfunction Units ofwater aregoods Units ofwater arebads Water x’ Around x’ units, a little extra water is a neutral.

19. Some Other Utility Functions and Their Indifference Curves • Consider V(x1,x2) = x1 + x2. • What does the indifference curve look like? • What relation does this function represent for these goods?

20. Special Cases x2 x1 + x2 = 5 13 x1 + x2 = 9 9 x1 + x2 = 13 5 V(x1,x2) = x1 + x2. 5 9 13 x1 These goods are perfect substitutes.

21. Some Other Utility Functions and Their Indifference Curves • Consider W(x1,x2) = min{x1,x2}. • What does the indifference curve look like? • What relation does this function represent for these goods?

22. Perfect Complementarity Indifference Curves x2 45o W(x1,x2) = min{x1,x2} min{x1,x2} = 8 8 min{x1,x2} = 5 5 3 min{x1,x2} = 3 5 3 8 x1

23. Perfect Substitutes and Perfect Complements • In general, utility function for perfect substitutes can be expressed as u(x , y) = ax + by • Utility function for perfect complement can be expressed as: u(x , y) = min{ ax , by }for constants a and b.

24. Some Other Utility Functions and Their Indifference Curves • A utility function of the form U(x1,x2) = f(x1) + x2is linear in x2 and is called quasi-linear. • E.g. U(x1,x2) = 2x11/2 + x2.

25. Quasi-linear Indifference Curves x2 Each curve is a vertically shifted copy of the others. x1

26. Some Other Utility Functions and Their Indifference Curves • Any utility function of the form U(x1,x2) = x1ax2bwith a > 0 and b > 0 is called a Cobb-Douglas utility function. • E.g. U(x1,x2) = x11/2 x21/2 (a = b = 1/2) V(x1,x2) = x1 x23 (a = 1, b = 3)

27. Cobb-Douglas Indifference Curves x2 All curves are hyperbolic,asymptoting to, but nevertouching any axis. x1

28. Cobb-Douglas Utility Function • By a monotonic transformation V=ln(U):U( x, y) = xayb implies V( x, y) = a ln (x) + b ln(y). • Consider another transformation W=U1/(a+b)W( x, y)= xa/(a+b) y b/(a+b)= xc y 1-cso that the sum of the indices becomes 1.

29. Marginal Utilities • Marginal means “incremental”. • The marginal utility of commodity i is the rate-of-change of total utility as the quantity of commodity i consumed changes; i.e.

30. Marginal Utilities • E.g. if U(x1,x2) = x11/2 x22 then

31. Marginal Utility • Marginal utility is positive if it is a good, negative if it is a bad, zero if it is neutral. • Its value changes under a monotonic transformation: (Consider the differentiable case) • So its value is not particularly meaningful.

32. Marginal Utilities and Marginal Rates of Substitution • The general equation for an indifference curve is U(x1,x2) º k, a constant. • Totally differentiating this identity gives

33. Marginal Utilities and Marginal Rates of Substitution We can rearrange this to Rearrange further:

34. Marginal Utilities and Marginal Rates of Substitution • Recall that the definition of MRS: The negative of the slope of an indifference curve is its marginal rate of substitution.

35. Marginal Utilities and Marginal Rates of Substitution • Therefore, • The Marginal Rate of Substitution is the ratio of marginal utilities.

36. Marginal Utilities and Marginal Rates of Substitution • MRS also means how many quantities of good 2 you are willing to sacrifice for one more unit of good 1. • One unit of good 1 is worth MU1. • One unit of good 2 is worth MU2. • Number of good 2 you are willing to sacrifice for a unit of good 1 is thus MU1 / MU2.

37. Marg. Utilities & Marg. Rates-of-Substitution; An example • Suppose U(x1,x2) = x1x2. Then so

38. Marg. Utilities & Marg. Rates-of-Substitution; An example U(x1,x2) = x1x2; x2 8 MRS(1,8) = - 8/1 = -8 MRS(6,6) = - 6/6 = -1. 6 U = 36 U = 8 x1 1 6

39. Marg. Rates-of-Substitution for Quasi-linear Utility Functions • A quasi-linear utility function is of the form U(x1,x2) = f(x1) + x2. Thus MRS for a quasi-linear function only depends on x1.

40. Marg. Rates-of-Substitution for Quasi-linear Utility Functions • MRS = f ’(x1) does not depend upon x2 so the slope of indifference curves for a quasi-linear utility function is constant along any line for which x1 is constant. • What does that make the indifference map for a quasi-linear utility function look like?

41. Marg. Rates-of-Substitution for Quasi-linear Utility Functions x2 MRS =f(x1’) Each curve is a vertically shifted copy of the others. MRS = f(x1”) MRS is a constantalong any line for which x1 isconstant. x1 x1’ x1”

42. Monotonic Transformations & Marginal Rates-of-Substitution • Applying a monotonic (increasing) transformation to a utility function representing a preference relation simply creates another utility function representing the same preference relation. • What happens to marginal rates of substitution when a monotonic transformation is applied?

43. Monotonic Transformations & Marginal Rates-of-Substitution • For U(x1,x2) = x1x2 the MRS = x2/x1. • Create V = U2; i.e. V(x1,x2) = x12x22. What is the MRS for V?which is the same as the MRS for U.

44. Monotonic Transformations & Marginal Rates-of-Substitution • More generally, if V = f(U) where f is a strictly increasing function, then Thus the MRS does not change with monotonic transformation. So, the same preference with different utility functions still show the same MRS.

45. Example: Means of Transportation • Consider the goods are the attributes of each mode of transportation • TW=total walking time • TT=total time of trip on the bus/car • C= total monetary cost • Different means of transportation have different values of the above “goods”, forming different “bundles”.

46. Example • E.g. walking all the way involves a high TW and low C, Traveling on a taxi has a high C, but low TW and TT. Taking public transport may be something in between. • We can estimate (with suitable econometrics methods) a utility function that represents people’s preferences if we know their choices and TW, TT and C.

47. Example • Domenich and McFadden (1975) use the linear form (recall what it represents?) and have the following function:U= -0.147TW – 0.0411TT – 2.24C • We can obtain the MRS. E.g. Commuters are willing to substitute 3 minutes of walking for 1 minute of walking. • How much is one willing to pay to shorten the trip (on vehicle) for one minute?

48. Summary • This chapter we introduce utility function as a way to represent preference numerically. • One prefers a bundle of higher utility than a bundle of lower utility. • Utility function is ordinal, and is invariant to increasing transformation. • The ratio of marginal utility is also the marginal rate of substitution.

49. What’s next? • Chapter 2: Budget constraint -what is affordable/feasible • Chapter 3 / 4: Preference / Utility -what one likes more • Chapter 5: Choice -choose the one with highest utility under budget constraint