From preferences to numbers Cardinal v ordinal Examples MRS

Utility

Intermediate Micro

Lecture 4

Chapter 4 of Varian

From preferences to numbers Cardinal v ordinal Examples MRS

Preferences and decision-making

1. Last lecture: Ranking consumption bundles bypreference/indifference

2. Today: Assigning values (numbers) to bundles

3. Next lecture: Using values to model decisions

From preferences to numbers Cardinal v ordinal Examples MRS

Utility function

I Assume we know preferences, indifference curvesI Surveys, market data...

I Utility function u(x1, x2) is a function that describespreferences

I Original idea: measure of happiness with consumption bundleI Choice of u(·) must correctly give

1. Indifference curves2. Ranking of bundles

From preferences to numbers Cardinal v ordinal Examples MRS

Utility function

I u(x1, x2) > u(y1, y2)⇔ (x1, x2) � (y1, y2)

I Level sets of u(·) are indifference curvesI Level set is all pairs (x1, x2)

I so that u(x1, x2) = kI for a given k

From preferences to numbers Cardinal v ordinal Examples MRS

Utility function for perfect substitutes

I xa = minutes withAlbuquerque radiostation

I xb = minutes with Boiseradio station (sameplaylist)

I Indifference curves: lineswith slope = −1

From preferences to numbers Cardinal v ordinal Examples MRS

Utility function for perfect substitutes

I Indifference curve for 2hours of Albuquerqueradio

I This indifference curveI 120 = xa + xb

I General indifferencecurve

I k = xa + xbI A utility function

I u(xa, xb) = xa + xb

From preferences to numbers Cardinal v ordinal Examples MRS

Choosing u(x) given indifference curves

I Previous slide: u(x) forindifference curve =distance along xa-axisfrom origin

I u(xa, xb) = xa + xbI Book’s suggestion: u(x)

= distance along45-degree line fromorigin

I u(xa, xb) =√xa + xb

From preferences to numbers Cardinal v ordinal Examples MRS

Ordinal utility

u(xa, xb) = xa + xbu(xa, xb) =

√xa + xb

I Both are correctI Ordinal utility: The

utility function ranksbundles correctly

I Based solely on whatwe can observe aboutpreferences

From preferences to numbers Cardinal v ordinal Examples MRS

Cardinal utility

u(xa, xb) = xa + xbu(xa, xb) =

√xa + xb

I Only one, or neither, iscorrect

I Cardinal utility: Theutility function returnsthe right value(number) for eachbundle

I Assumes we canaccurately measureutility

I Can use to comparetwo individuals’ utility

From preferences to numbers Cardinal v ordinal Examples MRS

Monotonic transformation

I Monotonic transformation: Afunction whose outputs keep thesame order as its inputs

I If u1 > u2, then f (u1) > f (u2)

I Any function f with f ′(u) > 0for all (applicable) u

I Monotonic transformation of autility function has same ordinalproperties.

I Same indifference curvesI Same ranking of bundles

From preferences to numbers Cardinal v ordinal Examples MRS

Lexicographic preferences

Not all preferences allow for a utility functionExample: Lexicographic preferences

I (x1, x2) � (y1, y2) iffI [x1 > y1], orI [x1 = y1 and x2 > y2]

I Cookie Monster preferencesI 2 cookies and 1 apple better than 1 cookie and 50 applesI 2 cookies and 1 apple better than 2 cookies and 0 apples

From preferences to numbers Cardinal v ordinal Examples MRS

When is there a utility function?

I Utility function exists only ifpreferences are continuous

I Preferences are ”smooth”

I Continuous preferences giveat-least-as-good sets that areclosed

I A closed set includes itsboundaries

I Monotone, convexity notnecessary for utility function Lexicographic preferences

From preferences to numbers Cardinal v ordinal Examples MRS

Choosing a utility function

How we assigned utility function for perfect substitutes:

1. Draw indifference curves

2. Find function with level sets that look like indifference curves

3. Check fit/set constants

Next few slides: utility functions we will frequently work with

From preferences to numbers Cardinal v ordinal Examples MRS

Perfect substitutes

u(x1, x2) = ax1 + bx2

I Solve for x2, getindifference curvesstraight lines

I Slope of − ab

From preferences to numbers Cardinal v ordinal Examples MRS

Perfect compliments

u(x1, x2) = min{ax1, bx2}I Indifference curves are 90

degree angle aroundpoints where ax1 = bx2

I What are a and b whenx1 = left shoes, and x2 =right shoes?

From preferences to numbers Cardinal v ordinal Examples MRS

Quasilinear utility function

u(x1, x2) = v(x1) + x2

I v(·) is some function ofx1

I Indifference curves areparallel, vertical shifts ofone another

From preferences to numbers Cardinal v ordinal Examples MRS

Cobb-Douglas utility function

u(x1, x2) = xc1 xd2

c , d > 0

I Gives monotone, convexpreferences

I Easy to work with

From preferences to numbers Cardinal v ordinal Examples MRS

Cobb-Douglas example

u(x1, x2) = x31x52

Use monotonic transformations1st step: f (u) = u

18

u(x1, x2) = x381 x

582

2nd step: g(u) = ln(u)

u(x1, x2) =3

8ln(x1) +

5

8ln(x2)

From preferences to numbers Cardinal v ordinal Examples MRS

Marginal utility

I Effect of small increasein x1 on utility

I Marginal utility wrt x1I MUx1 = ∂u(x1,x2)

∂x1

I Can do same analysiswith x2

From preferences to numbers Cardinal v ordinal Examples MRS

Marginal utility - Example

u(x1, x2) = x1 + 2x2

(x1, x2) = (9, 8)

I Draw indifference curvethrough (9, 8)

I What is ∂u∂x1

?

I What is ∂u∂x2

?

From preferences to numbers Cardinal v ordinal Examples MRS

Marginal utility - Example

u(x1, x2) =√x1 + 2x2

(x1, x2) = (9, 8)

I Draw indifference curvethrough (9, 8)

I What is ∂u∂x1

?

I What is ∂u∂x2

?

From preferences to numbers Cardinal v ordinal Examples MRS

Marginal utility and ordinal utility

I MU changes with choice of utility functionI Even for monotonic transformations

I ∂u(x1,x2)/∂x1∂u(x1,x2)/∂x2

does not change with monotonic transformation

I This ratio is MRS, or (−1∗) slope of indifference curveI Recall MRS is |slope|

From preferences to numbers Cardinal v ordinal Examples MRS

MU and MRS

I Make small change dx1 to x1I Choose small change dx2 to x2 so u is unchanged

du = 0 =∂u

∂x1dx1 +

∂u

∂x2dx2

...(full proof on board and in book)

slope = −∂u/∂x1∂u/∂x2

I MRS = ∂u/∂x1∂u/∂x2

From preferences to numbers Cardinal v ordinal Examples MRS

MU and MRS

I Can also show monotonic transformations do not affectslope/MRS

I For 2 utility functionsI Same indifference curves ⇒ Same MRS (at all bundles)I Same MRS (at all bundles) ⇒ Same indifference curves

I Can observe MRS in real world (next chapter)

I Can find utility function that fits ordinal properties

From preferences to numbers Cardinal v ordinal Examples MRS

Calculating MRS

Example: For the following, calculate MRS as a function of x1 andx2, and find the value of the MRS at (1, 2):

1. u(x) = x31x22

2. u(x) =√x1 + 1

2x2