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 + 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”).   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(x 1,x 2 ) = x 1 x 2 represents a preference relation.  Consider the bundles (4,1), (2,3) and (2,2). 7

  U(x 1,x 2 ) = x 1 x 2, so U(2,3) = 6 > U(4,1) = U(2,2) = 4; that is, (2,3) (4,1)  (2,2).  8

 Define V = U 2.   Then V(x 1,x 2 ) = x 1 2 x 2 2 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.  9

 Define W = 2U   Then W(x 1,x 2 ) = 2x 1 x 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.  10

 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 function representing.  Clearly, V(x)>V(y) if and only if f(V(x)) > f(V(y)) by definition of increasing function. ~  ~  11

Fig

 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

U  6 U  4 U  2 x1x1 x2x2 15

U  6 U  5 U  4 U  3 U  2 U  1 x1x1 x2x2 Utility 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 Water x’ Units of water are goods Units of water are bads Around x’ units, a little extra water is a neutral. Utility function 18

 Consider V(x 1,x 2 ) = x 1 + x 2.  What does the indifference curve look like?  What relation does this function represent for these goods? 19

x1x1 x2x2 x 1 + x 2 = 5 x 1 + x 2 = 9 x 1 + x 2 = 13 V(x 1,x 2 ) = x 1 + x 2. These goods are perfect substitutes. 20

 Consider W(x 1,x 2 ) = min{x 1,x 2 }.  What does the indifference curve look like?  What relation does this function represent for these goods? 21

x2x2 x1x1 45 o min{x 1,x 2 } = min{x 1,x 2 } = 5 min{x 1,x 2 } = 3 W(x 1,x 2 ) = min{x 1,x 2 } 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(x 1,x 2 ) = f(x 1 ) + x 2 is linear in x 2 and is called quasi-linear.  E.g. U(x 1,x 2 ) = 2x 1 1/2 + x 2. 24

x2x2 x1x1 Each curve is a vertically shifted copy of the others. 25

 Any utility function of the form U(x 1,x 2 ) = x 1 a x 2 b with a > 0 and b > 0 is called a Cobb- Douglas utility function.  E.g. U(x 1,x 2 ) = x 1 1/2 x 2 1/2 (a = b = 1/2) V(x 1,x 2 ) = x 1 x 2 3 (a = 1, b = 3) 26

x2x2 x1x1 All curves are hyperbolic, asymptoting to, but never touching any axis. 27

 By a monotonic transformation V=ln(U): U( x, y) = x a y b implies V( x, y) = a ln (x) + b ln(y).  Consider another transformation W=U 1/(a+b) W( x, y)= x a/(a+b) y b/(a+b) = x c 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. 29

 E.g. if U(x 1,x 2 ) = x 1 1/2 x 2 2 then 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)  So its value is not particularly meaningful. 31

 The general equation for an indifference curve is U(x 1,x 2 )  k, a constant.  Totally differentiating this identity gives 32

We can rearrange this to Rearrange further: 33

 Recall that the definition of MRS: The negative of the slope of an indifference curve is its marginal rate of substitution. 34

 Therefore,  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 MU 1.  One unit of good 2 is worth MU 2.  Number of good 2 you are willing to sacrifice for a unit of good 1 is thus MU 1 / MU 2. 36

 Suppose U(x 1,x 2 ) = x 1 x 2. Then so 37

MRS(1,8) = - 8/1 = -8 MRS(6,6) = - 6/6 = -1. x1x1 x2x U = 8 U = 36 U(x 1,x 2 ) = x 1 x 2 ; 38

 A quasi-linear utility function is of the form U(x 1,x 2 ) = f(x 1 ) + x 2. Thus MRS for a quasi-linear function only depends on x 1. 39

 MRS = f ’(x 1 ) does not depend upon x 2 so the slope of indifference curves for a quasi- linear utility function is constant along any line for which x 1 is constant.  What does that make the indifference map for a quasi-linear utility function look like? 40

x2x2 x1x1 Each curve is a vertically shifted copy of the others. MRS is a constant along any line for which x 1 is constant. MRS = f(x 1 ’) MRS = f(x 1 ”) x1’x1’ x1”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(x 1,x 2 ) = x 1 x 2 the MRS = x 2 /x 1.  Create V = U 2 ; i.e. V(x 1,x 2 ) = x 1 2 x 2 2. What is the MRS for V? which is the same as the MRS for U. 43

 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. 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= TW – TT – 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