4.1 Ordinal versus cardinal utility

Transcription

1 Microeconomics I. Antonio Zabalza. Universit of Valencia 1 Micro I. Lesson 4. Utilit In the previous lesson we have developed a method to rank consistentl all bundles in the (,) space and we have introduced a concept the indifference curve- to help us in this analsis. Now we introduce a related concept to rank bundles the utilit function- that will be useful to solve the equilibrium of the consumer in terms of calculus. 4.1 Ordinal versus cardinal utilit In the past, utilit was conceived as a quantitative measure of a person s welfare out of consuming goods. Now it is recognised that utilit cannot be quantified because of the impossibilit of interpersonal comparisons. So we do now as we did in Lesson 3, we onl rank bundles. Utilit function: A wa of assigning a number to ever possible consumption bundle such that morepreferred bundles get assigned larger numbers than less-preferred bundles. (, ) (, ) if and onl if U(, ) U(, ) The onl propert about the numbers the utilit function generates which is important is how it orders the bundles; how it ranks them. The size of the difference between the numbers assigned to each

2 Microeconomics I. Antonio Zabalza. Universit of Valencia 2 bundle does not matter. Thus we talk about ordinal utilit. Since onl the ranking matters, there can be no unique wa to assign utilit to bundles. If U(,) represents one wa of ranking goods, 2U(,) is equall acceptable: it ranks bundles in the same manner. Multipling b 2 is an eample of a monotonic transformation. Eample: U= Bundle U= A B C Ranking: B>A>C U=2 Ranking: B>A>C Bundle U=2 A B C 1 2 4

3 Microeconomics I. Antonio Zabalza. Universit of Valencia 3 A monotonic transformation is a wa of transforming one set of numbers into another set of numbers in a wa that the order of the numbers is preserved. If the original utilit function is U(,), we represent a monotonic transformation b f [ U(, )]. The propert the function f[.] has to have is that If U > U f( U ) > f( U) Eamples of monotonic transformations: f(u) = 2U f(u) = 3U f(u) = U+2 f(u) = U+10 f(u) = 5+3U f(u) = U 3 (What about f(u) = U 2?) See that to preserve the order, f(u) must be a strictl increasing function of U. Utilit functions have indifference curves too; the are the level curves in the space (,) of the three dimensional function U=f(,). The indifference curves of a monotonic transformation of a utilit function are the same as the indifference curves of the original utilit function, onl that the numbers attached to each indifference curve are different.

4 Microeconomics I. Antonio Zabalza. Universit of Valencia From utilit functions to indifference curves If ou are given a utilit function U(,), it is eas to derive a given indifference curve from it: simpl plot all points (,) such that U(,) equals a constant. Eamples: U(,)= k= =k/ Rectangular hperbola k=3 k=2 k=1 U(, ) = 2 2 Notice that since cannot be negative (we are in the positive quadrant), = ( ) preserves the same order. So this utilit function is a monotonic transformation of the previous function. The formula for the indifference curve is:

5 Microeconomics I. Antonio Zabalza. Universit of Valencia 5 φ = φ 12 = φ 2 2 = 12 / This is the same indifference curve map as before, onl that the levels of the indifference curves are the squared of the previous levels Rectangular hperbola φ=9 φ=4 φ=1 Perfect substitutes (blue pencils, red pencils) U(, ) = + k = + = k Slope -1

6 Microeconomics I. Antonio Zabalza. Universit of Valencia 6 Perfect substitutes but at differents proportions: for eample, suppose for the consumer is twice as valuable as. U(, ) = 2+ k = 2+ = k 2 Slope -2 In general, U(, ) = a+ b k = a+ b k a = b b This is a utilit function in which the consumer values as much as a/b units of.

7 Microeconomics I. Antonio Zabalza. Universit of Valencia 7 Perfect complements (left shoe, right shoe) { } U(, ) = min, If I have 2 (two right shoes) and 1 (one left shoe) it is like if I had onl one pair of shoes: I get the same utilit as with 1 and 1. Slope The proportion need not be 1 to 1. Sa, a consumer uses alwas 1 (cup of cofee) with 2 (two sugars), then 1 u (, ) = min, 2 Slope 2

8 Microeconomics I. Antonio Zabalza. Universit of Valencia 8 In general { a b} U(, ) = min, The slope of the ais is then a/b. Check ou understand the function min{}. Cobb-Douglas utilit function a b U(, ) = ; a > 0; and b > 0 This is a well known function which generates well behaved indifference curves (smooth, negative and conve). a>b a<b Good is relativel preferred to good Good is relativel preferred to good Cobb-Douglas functions are frequentl used in production theor, where instead of utilit we talk of output, and instead of goods we talk of inputs.

9 Microeconomics I. Antonio Zabalza. Universit of Valencia Marginal Utilit and the MRS Consider a consumer that consumes the bundle (,). How does this consumer s utilit change when we maintain the amount of and give him a little more of? The change in utilit per unit of change in is the marginal utilit of ( ). U U( +, ) U (, ) = = measures b how much utilit changes when we change b a small amount holding constant. In the limit, if the change in is infinitesimal, = δ From the definition of marginal utilit it follows that the change in utilit that results from a small increase in, holding constant is: U = We have the same sort of definitions for good ;. marginal utilit of ( ) U U(, + ) U (, ) = = = δ U =

10 Microeconomics I. Antonio Zabalza. Universit of Valencia 10 The utilit function can be used to measure the MRS defined in the previous lesson. Suppose that, for a given utilit function, both and change. In general, when we change the quantities consumed of and, the level of utilit will change. The total change in utilit will be the sum of the change in utilit generated b the change in plus the change in utilit generated b the change in. U = + Suppose additionall that this change in and is a movement along a given indifference curve. This means that after the movement, the level of utilit must be the same, and that U = 0. Therefore, Or 0 = + = But, measured along a given indifference curve, is (minus) the MRS. Therefore, MRS = = The MRS can be measured b the ratio of the respective marginal utilities of the two goods.

11 Microeconomics I. Antonio Zabalza. Universit of Valencia 11 Anne (Some calculus) U = U (, ) (1) = = δ = = δ (, ) δ (, ) δ Totall differentiating (1) we find du = d+ d δ δ du = d+ d Along an indifference curve du = 0. Therefore, 0 = d+ d d = d d MRS = = d