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Course: Microeconomics Text: Varian’s Intermediate Microeconomics 1 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. 2 A utility function U: R+nR maps each consumption bundle of n goods into a real number that satisfies the following conditions: x’ x” U(x’) > U(x”) p x’p x” U(x’) < U(x”) x’ ~ x” U(x’) = U(x”). 3 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. 4 Utility is an ordinal (i.e. ordering) concept. The number assigned only matters about ranking, but the sizes of numerical differences are not meaningful. 5 Consider only three bundles A, B, C. The following three are all valid utility function of the preference. 6 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). 7 U(x1,x2) = x1x2, so U(2,3) = 6 > U(4,1) = U(2,2) = 4; that is, (2,3) (4,1) ~ (2,2). p 8 Define V = U2. Then V(x1,x2) = x12x22 and V(2,3) = 36 > V(4,1) = V(2,2) = 16 so again (2,3) (4,1) ~ (2,2). V preserves the same order as U and so represents the same preferences. p 9 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 10 If U is a utility function that represents a preference relationf and ~ f is a strictly increasing function, then V = f(U) is also a utility function representing f . ~ if and only if Clearly, V(x)>V(y) f(V(x)) > f(V(y)) by definition of increasing function. 11 12 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. 13 An indifference curve contains equally preferred bundles. Equal preference same utility level. Therefore, all bundles in an indifference curve have the same utility level. 14 x2 U6 U4 U2 x1 15 Utility U6 U5 U4 U3 U2 x2 U1 x1 16 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. 17 Utility Utility function Units of water are goods Units of water are bads x’ Water Around x’ units, a little extra water is a neutral. 18 Consider V(x1,x2) = x1 + x2. What does the indifference curve look like? What relation does this function represent for these goods? 19 x2 x 1 + 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. 20 Consider W(x1,x2) = min{x1,x2}. What does the indifference curve look like? What relation does this function represent for these goods? 21 x2 45o W(x1,x2) = min{x1,x2} 8 min{x1,x2} = 8 5 min{x1,x2} = 5 3 min{x1,x2} = 3 3 5 8 x1 22 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. 23 A utility function of the form U(x1,x2) = f(x1) + x2 is linear in x2 and is called quasi-linear. E.g. U(x1,x2) = 2x11/2 + x2. 24 x2 Each curve is a vertically shifted copy of the others. x1 25 Any utility function of the form U(x1,x2) = x1a x2b with a > 0 and b > 0 is called a CobbDouglas utility function. E.g. U(x1,x2) = x11/2 x21/2 (a = b = 1/2) V(x1,x2) = x1 x23 (a = 1, b = 3) 26 x2 All curves are hyperbolic, asymptoting to, but never touching any axis. x1 27 By a monotonic transformation V=ln(U): U( x, y) = xa y b 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-c so that the sum of the indices becomes 1. 28 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. MU i U xi 29 E.g. if U(x1,x2) = x11/2 x22 then MU 1 MU 2 U x1 U x2 1 2 1 / 2 x1 2x 1/ 2 1 2 x2 x2 30 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) MU i f (U ) xi f ' (U ) U xi So its value is not particularly meaningful. 31 The general equation for an indifference curve is U(x1,x2) k, a constant. Totally differentiating this identity gives U x1 dx1 U x2 dx2 0 32 U x1 dx1 U x2 dx2 0 We can rearrange this to U x2 dx2 U x1 dx1 Rearrange further: d x2 d x1 U / x1 U / x2 . 33 d x2 d x1 U / x1 U / x2 . Recall that the definition of MRS: The negative of the slope of an indifference curve is its marginal rate of substitution. MRS d x2 d x1 U 34 Therefore, MRS U / x1 U / x2 MU 1 MU 2 The Marginal Rate of Substitution is the ratio of marginal utilities. 35 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. 36 Suppose U(x1,x2) = x1x2. Then U x1 U x2 so MRS ( 1)( x2 ) x2 ( x1 )( 1) x1 d x2 d x1 U / x1 U / x2 x2 . x1 37 U(x1,x2) = x1x2; x2 MRS x2 x1 8 MRS(1,8) = - 8/1 = -8 MRS(6,6) = - 6/6 = -1. 6 U = 36 U=8 1 6 x1 38 A quasi-linear utility function is of the form U(x1,x2) = f(x1) + x2. U x1 U f ( x1 ) MRS d x2 d x1 x2 1 U / x1 U / x2 f ( x1 ). Thus MRS for a quasi-linear function only depends on x1. 39 MRS = f ’(x1) does not depend upon x2 so the slope of indifference curves for a quasilinear 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? 40 x2 MRS = f(x1’) Each curve is a vertically shifted copy of the others. MRS = f(x1”) x1’ x1” MRS is a constant along any line for which x1 is constant. x1 41 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? 42 For U(x1,x2) = x1x2 the MRS = x2/x1. Create V = U2; i.e. V(x1,x2) = x12x22. What is the MRS for V? 2 V / x1 2 x1 x2 x2 MRS V / x2 2 1 2 2x x x1 which is the same as the MRS for U. 43 More generally, if V = f(U) where f is a strictly increasing function, then V / x1 f (U ) U / x1 MRS V / x2 U / x1 U / x2 f ' (U ) U / x2 . Thus the MRS does not change with monotonic transformation. So, the same preference with different utility functions still show the same MRS. 44 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”. 45 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. 46 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? 47 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. 48 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 49