McPeak

Lecture 5 =

PPA 723

Consumer choice.=

Model premises;<= /o:p>

1)&n= bsp; Tastes for a consumer are given. They may = vary across consumers, but for our purposes, are treated as fixed for a given individual. We don’t re= ally care where they came from.

2)&n= bsp; There are c= onstraints on the choices and individual can make, either through regulation or budget limits.

3)&n= bsp; The consumer makes choices that maximize their well-being.

The goal is to unders= tand and predict behavior.

Very flexible, very b= road.

How does a consumer c= hoose one bundle of goods over another?

We assume they are gu= ided by their preferences.

Preferences obey three properties by assumption.

1)&n= bsp; Completenes= s. For any two bundles of goods, call= them bundle A and bundle B, one and only one of the following statements is true= :

a.&n= bsp; A is prefer= red to B

b.&n= bsp; B is prefer= red to A

c.&n= bsp; The consume= r is indifferent between A and B.

[meaning, no “I dunno’s”]

2)&n= bsp; Transitivit= y. Preferences are logically consistent. If I prefer bundl= e A to bundle B, and prefer bundle B to bundle C, then I also prefer bundle A to bundle C. <= /p>

[meaning, if I prefer cranberry juice to orange juice, and orange juice to apple juic= e, then it is safe to assume I prefer cranberry juice over apple juice]

3)&n= bsp; More is bet= ter than less. Assume the commodi= ty in question is a “good” rather than a bad, and that there is free disposal if all else fails.

[meaning, if you see me drinking cranberry juice, it is fair to assume that another g= lass will make me no worse off if I can consume it, give it to somebody, or pour= it down the sink]

With these three assu= mptions, we can proceed to map a consumer’s preferences.

First, if we are give= n a point that corresponds to a consumption level of two goods (our bundle if composed of two goods), we can partition space to tell us where the consume= r is better off, where the consumer is worse off, and where we need to know more info.

Graph here.

We can put more preci= sion on preference maps through the use of indifference curves. These tell us the different combin= ations of the two goods in question that make the consumer equally well off. The consumer is indifferent betwee= n any of the bundles that define the line – hence the name.

Indifference curves a= re the contours on our “map of happiness”. Each traces out a line that tells = us consumption bundles that take us to the same “happiness height”.

Indifference curve properties:

1)&n= bsp; Bundles on a curve further from the origin (consumption bundle equals zero for both good= s) are preferred to those on curves closer to the origin.

2)&n= bsp; There exist= s a curve passing through every bundle.

3)&n= bsp; Indifference curves can not cross.

4)&n= bsp; Indifference curves slope downward (and for our purposes are convex to the origin though this last bit need not hold).

Draw indifference cur= ve.

The slope of the indi= fference curve is defined as the marginal rate of substitution.

This tells us the max= imum amount of one good the consumer will sacrifice to obtain one more unit of t= he other good.

The trade off for a m= arginal increase in one good in terms of a marginal decrease in the other.

MRS=3D

(assuming rise is mea= sured in good two, run in good one).

What sign should this= have?

We assume indifferenc= e curves are convex to the origin, reflecting a diminishing marginal rate of substitution. The less you currently are consuming of a good, the more valuable an additional unit is = to you. More on this= in a moment.

MRS=3D

What does the indiffe= rence curve for perfect substitutes look like?&n= bsp; [price chopper milk, byrne dairy milk]

Graph.

What does the indiffe= rence curve for perfect complements look like? [ left glove, right glove]

Graph.

Most things fall in b= etween.

Utility. A numerical value that reflects the relative ranking of various bundles of goods.

How many “utils= ” of happiness does a given market bundle give you?

Ordinal, rather than cardinal, which means the absolute number of utils and the distance in utils between two bundles is not meaningful in and of itself. It orders the bundles.<= /span>

Utility is often repr= esented by a utility function, and we assume utility is concave to the origin.=

Utility in two goods = has three axes to be concerned about: good one, good two, and the utility derived from the combination of = the goods. For example:

For x₁ fro= m 1 to 10, and x₂ from 1 to 10

3-d view looks like a= wedge from an upside down bowl.

Note the contours, th= at is like the indifference curve.

View from above: (exc= el puts the label on the rhs, but same idea here).

This makes it easy to= pick out the indifference curve shape.

If we pick one value = of good one and consider how utility changes as we increase good two, we get someth= ing like this (using the info in the surface map).

(side view of utility function, hold other good constant)

Utility increases as = we consume more of the good, but it is assumed by a decreasing rate. Decreasing marginal utility.<= /o:p>

Distinguish between decreasing utility and decreasing marginal utility.

From 0 to 1, utility increased by 0.7. =

From 1 to 2, utility increased by 0.5.

From 2 to 3, utility increased by 0.4.

Marginal utility: _{Note =
that
this is assumed positive.}

The marginal rate of substitution (MRS) is equal to the negative of the ratio of the marginal utilities.

At some level, you ju= st have to accept this.

_{, but change in Utility =3D0 on an indifference curve, so
rearrange terms.}

For a more detailed e= xplanation, see the following.

From a calculus point of vie= w, it makes sense:

U=3DU(x₁, x₂)

The total derivative:

dU=3D(dU/dx₁)dx= ₁+(dU/dx₂)dx₂

If we move along an indiff= erence curve, we know that dU=3D0, so that: (dU/dx₁)dx₁-(dU/dx₂)dx₂= and we rearrange terms from there.

However, we don’t ha= ve that calculus option, so my best intuitive try here:

Take yourself back to the = 3-d graph.

If we change x₁= positively, it will have a positive impact on utility (change in x₁ moves us sideways, but also up). If we= want to get back down to the utility level we started at, we need to change x2 negatively, until it moves us back down on utility (change in x₂ moves us sideways, but also down). So loosely stated, we= have to balance the total impact of the change x₁had on us in 3-d spa= ce by changing x₂ enough to bring us back down to the same utility level as we started at, but at a different consumption bundle.
Budget constraints.

Assume away savings a= nd dynamic aspects, and to keep things simple, assume that the budget constrai= nt is the income received in a given period.

Now we have two goods= , so let’s define Y as income, p₁as the price of good one, x1 as the quantity of good one consumed, p₂as the price of good two, = and x₂as the quantity of good two consumed.

p_1* x_=
1 +p_2* x₂=3DY

The budget line/ budg= et constraint.

The plot of all possi= ble combinations of goods x₁ and x₂ that can be purchased= at given levels of Y, p₁and p₂

The opportunity set i= s all bundles on or below the bundle line.

Y=3D100, p_{1 =3D10,and p₂ =3D 5}

p_1* x_=
1+ p_2* x₂=3DY

10_* x_=
1+ 5_* x₂=3D100

Max x₁=3D(= Y/ p₁)=3D(100/10)=3D10

Max x₂=3D(= Y/ p₂)=3D(100/5)=3D20

Draw this, note oppor= tunity set.

Slope? Change in x2/change in x1 =3D-20/1= 0=3D-2

Or<= /p>

-p1/p2, -10/5 =3D-2.<= o:p>

Why?

As graphed, x2 is a f= unction of x1. So we can write:<= /o:p>

x₂=3D(Y/ p= ₂)-( p₁ / p₂)* x₁

x₂=3D(100/= 5)-( 10 / 5)* x₁
The marginal rate of transformation is the slope of the budget line. It reflects the trade offs the mar= ket imposes on the consumer in terms of the amount of one good the consumer must give up to obtain more of the other good.

What is the impact of= a change in price on the budget line? If price of good one increases from 10 to 20. Slope is now -4.=

20_* x_=
1+ 5_* x₂=3D100

Draw.

What is the impact if= the price of good one decreases to 5 from 10?&= nbsp; Slope is now -1. =

5_* x_1=
+ 5_* x₂=3D100

Draw.

What is the impact of= a change in income on the budget line?

10_* x_=
1+ 5_* x₂=3D150

Income goes to $150 f= rom $100. Does this change the slope? No – outward shi= ft.

What happens to the b= udget line if income doubles? What happens if the price of all goods is halved? Takes you to the same place.

10_* x₁₊ 5_* x2=3D100

10_* x_=
1+ 5_* x₂=3D200

10_* x_=
1+ 5_* x₂=3D100

5_* x_1=
+ 2.5_* x₂=3D100