prefs and utility examples

7/28/2019 Prefs and Utility Examples

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Preferences and Utility - Basic Review and Examples

Preference Relations

Preference relations are a handy way of talking about how people rank bundles of goods.

When we talk about preferences, we mainly use three binary relations: (strictly pre-ferred to), (indifferent between), and (weakly preferred to). We impose some generalassumption on preference relations to make them easier to deal with.

1. Complete - Completeness implies that given any two bundles in the set, our preferencerelation can compare and rank them.

2. Reflexive - Reflexivity requires that any bundle is at least as good as itself, i.e. x x.

3. Transitive - For x, y, and z, if x y and y z, then we must have x z. Prof.Schipper showed in class how this should hold for rational individuals.

Ideally, we would also have monotonicity, which implies that more of a good is better.

Example

Say we have a set that contains three different bundles x, y, and z and the relation at leastas good as. Is this transitive? Yes, because ifx y and y z, then it must be that x z(similarly for any other comparison order). Is it reflexive? Yes, because it makes sense tosay x is at least as good as x (or x x). Is it complete? Yes, because we can accuratelydescribe any possible relationship between the three bundles using the at least as good asrelation.

Preferences and Convexity

Well behaved preferences are convex. Dont confuse this with the convex of the functionworld - we can have a concave utility function that represents convex preferences. Rather,convexity in terms of preferences means if we have two bundles x and y,

ifx y then x + (1 )y x where 1 > > 0

The bunch of math there means that if we were to take bundles x and y, and create somebundle z that is made up of (for example) half of x and half ofy, z will be preferred to eitherx or y.

Example

Say we have bundles x = (30, 3) and y = (6, 15), and x y. Lets set = 0.33. Then aconvex combination z ofx and y would be 1

3(30, 3) + 2

3(6, 15) = (10 + 4, 1 + 10) = (14, 11). If

Prepared by Nick Sanders, UC Davis Graduate Department of Economics 2007


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the individual has convex preferences, it would mean that (14, 11) (30, 3) and (14, 11) (6, 15). Convex preferences imply the individual prefers mixes of goods to extremes.

Graphically, strict convexity of preferences means we can draw a straight line connectingany two points on an indifference curve and any point on that line will be preferred to anypoint on the indifference curve. Put another way, that implies any point on said line is in

the preferred set of the indifference curve.

x

y

x

y

x

y

Figure 1: The left two graphs could be indifference curves for convex preferences, but therightmost curve cannot.

Utility and Monotonic Transformations

We use utility functions to easily describe preferences. A utility function assigns numericalvalues to all bundles so that if x y, we have u(x) u(y). All that matters is the ordinal

ranking, not the cardinal ranking. The numerical values we get for utility functions onlymatter in that we can say one utility level is higher than another, but the actual values dontmean much. For example, ifu(x) = 100 and u(y) = 200, we cant say that y is twice as goodas x (thats a cardinal statement), only that y is preferred to x.

There can be multiple utility functions that describe the same set of preferences. Oneproperty of utility functions is that if u(x) is a valid utility function and f() is a monotoni-cally increasing transformation, then f(u(x)) is also a valid utility function. This will comein handy later when we need to deal with more complicated utility functions. Note that ithas to be a monotonically increasing function1.

1What would happen if we used a monotonic but decreasing function? The order of the bundle rankingwould change (what used to be larger positive numbers would now be larger negative numbers, so bundles

that used to give higher/more positive utilities would now give lower/more negative utilities), changing ourranking order and resulting in the new utility function no longer correctly representing the preferences.

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Example

Say we have only one good, and our utility function is u(x) = 4x2. Now define v = (u(x))2,g = 10ln(u(x)), and h =

u(x) (all of which are monotonically increasing transformations).

We can see a few example values in the table below. Note how the actual numeric change

in utility when moving from x = 1 to x = 2 varies a lot. How can these all be acceptableutility functions for the same preferences? Because regardless of their cardinal differences,their ordinal rankings always remain the same: x = 2 always gives higher utility than x = 1,and x = 10 always gives higher utility than x = 2 and so forth.

x u(x) v(x) g(x) h(x)1 4 16 13.86 22 16 256 27.73 43 36 1,296 35.84 6. . . . .

10 400 160,000 59.92 20

Utility Functions, Marginal Utility, and the Marginal Rate of Sub-stitution (MRS)

The MRS is the slope of the indifference curves. It mathematically explains at what rate theindividual is willing to trade between goods while still remaining on the same indifferencecurve.

If someone has convex preferences (and ideally monotonic) then we know theyll also havea decreasing marginal rate of substitution. Why? Convex preferences imply mixes are betterthan extremes. If you start with lots of x2 and little x1, your preference for mixes meansyoull be willing to give up a bunch of x2 for just a little

2 more x1. But as you move towardmore and more x1 and less and less x2, you start moving back to an extreme situation, soyoure willingness to trade x2 for x1 goes down, and your MRS decreases in absolute value.

We define marginal utility (MU) as the change in utility that results from small changesin consumption. Imagine a world of two goods. The marginal utility of good 1 would bethe change in utility resulting from a small change in the consumption of good one, withsimilar reasoning for good 2. With differentiable functions, we can say that the marginalutility of good 1 is the partial derivative of the utility function with respect to good 1 (andagain similar logic for good 2). Mathematically, this is to say

MU1 =u

x1and MU2 =

u

x2

2I know a bunch and a little arent exactly precise mathematical terms, but you get the idea.

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Marginal utilities are useful for more than just finding changes in utility. We also know that

MRS = MUx1MUx2

In words, the marginal rate of substitution is equal to the negative of the ratio of themarginal utilities. The MRS is maintained under monotonic transformations3.If we have autility function u() and a monotonically increasing transformation f() then the MRS ofu() is equal to the MRS of f(u()).

Example

Say we have u(x, y) = 4xy2. Then MUx = 4y2 and MUy = 8xy. The MRS is then

4y2

8xy=

y

2x. Now we apply the monotonically increasing transformation of v = ln(u(x, y)). Then

our new utility function is v(x, y) = ln(4xy2) = ln(4x) + 2ln(y), which means MUx =1

4x,

MUy =2

y, and the MRS is 1

4x/2y

= y2x

. Note how even though the marginal utilities

arent really comparable between utility functions, the MRS remains the same under themonotonically increasing transformation.

3Marginal utilities themselves are not unchanged under monotonic transformations. However, their ratio(which is the MRS) is unchanged. This is one reason that the MRS is particularly useful - while MU mayvary with the specification of the utility function, the MRS will remain constant.

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