McPeak

Lecture 6<= /span>

PPA 723

Constrained consumer = choice.

What is the optimal b= undle?

The bundle of goods t= hat makes the consumer as well off as possible given a particular budget constraint. The bundle consum= ed at the point where the indifference curve is tangent to the budget line.<= /o:p>

Graph with points a, = b, c, d.

B is the optimal bund= le.

C is not, as it is no= t in the opportunity set.

A is in the opportuni= ty set, but has less of good 1 and 2 than b, so must be worse.

D is in the opportuni= ty set, and has more of good one, less of good 2.&= nbsp; By looking at the indifference curves and knowing their properties, = we can rule it out.

Technical note:

A solution is an inte= rior solution if the optimal bundle involves the consumer consuming positive amo= unts of both goods.

A solution is a corner solution if the optimal bundle involves the consumer consuming zero of one = of the goods.

We will focus on inte= rior solutions.

For an interior solut= ion, the marginal rate of substitution equals the marginal rate of transformation. We know these expressions have oth= er forms, so we can write the expanded equality for the condition of an interi= or solution as follows:

₌

A nice way to think a= bout it is that the optimal bundle equates the marginal utility per dollar spent on good 1 with the marginal utility per dollar spent on good 2.

₌

Example:

Price of apples is $0= .50, price of oranges is $0.80.

Consumption bundle is= 4 apples and 10 oranges.

Income is $10.

MU_apples(apples=3D4, oranges=3D10)=3D3

MU_oranges(apples=3D4, oranges=3D10)=3D4

First, is the consume= r on the budget line at the consumption bundle in question?

What does a budget li= ne look like? ₌ or ₌, OK, so on the budget line (not above, not below).<= /o:p>

Draw with oranges on = y axis, apples on x axis.

What is MRS at observ= ed consumption bundle?

= -&nb= sp; MU_appl=
es/ MU_oranges , or -3/4 =3D-0.75

What is MRT at observ= ed consumption bundle? –price of apples/ price of oranges, or -0.5/0.8 or -0.625.

Market says I have to= give up .625 oranges to get 1 more apple. My preferences say I am willing to give up .75 oranges to get 1 more apple to be equally happy. I = can give up less than that and have some left over to make me move up to a high= er indifference curve.

Or, MRS at that point= says give up 3 oranges to get 4 applies, MRT says give up three oranges to get 4= .8 apples.

Why give up oranges t= o get more apples and not the other way around?&= nbsp; ₌

₌at optimal bundle, but

₌at current bundle.

I need to bring down = the marginal utility of apples, bring up the marginal utility of oranges.

By diminishing margin= al utility, I get higher marginal utility by consuming less of a good, lower marginal utility by consuming more of a good.

So to bring MU_ap=
ples down, I need to consume more apples.

To bring MU_orang=
es up, I need to consumer fewer oranges.

Thus more apples, few= er oranges (say 12 apples, 5 oranges which satisfies the budget constraint, and under the assumption that MU_apples (apples=3D12, oranges =3D 5) = =3D 2.75, MU_oranges (apples=3D12, oranges =3D 5) =3D 4.4).

Remember, MRS is the = rate at which a consumer is willing to give up one good to get more of the other. It comes from the underlying prefe= rences of the consumer.

MRT is the rate at wh= ich prices let a consumer give up one good to get more of the other. It comes from the underlying market conditions and has nothing to do with the consumer’s preferences.

Show corner solution.= In this case, the indifference cur= ve is not tangent to the budget line, since reality intrudes.

The underlying prefer= ences cause this, and since we take preferences as given, we can run across this. It happens when the con= sumer has a strong preference for one good as compared to the other.

Food stamp example.

Consider good 1 is fo= od, good two is all other goods. =

Food stamps are given= to the consumer, that can only be spent on food.&= nbsp; $100 worth let us say.

Consider extreme exam= ples, where all is spent on one good or the other again. However, note there is an issue ab= out the constrained nature of the food stamp program.

Graph.

Shape of indifference= curves matters here. Is the consumer better off getting $100 cash or $100 in food stamps?

Graph for equally wel= l off whether the transfer is in cash or in food stamps

Graph for worse off w= hen the transfer is in food stamps rather than cash.

Note in both cases, consumption of the other good has increased in spite of the fact that the p= rogram is targeted at food.

General result –= ; a subsidy program that constrains consumer choice will do worse or no better = than an unconstrained transfer.

Why do we constrain t= he transfer in the case of food stamps then?

What is the impact on= the budget constraint of a black market in food stamps that lets you get 30 cen= ts on the dollar face value?

Show graph.

(Chapter 5).

How do we derive a de= mand curve?

Remember that we bega= n the course by taking a supply curve and a demand curve as given. Now we are going to find out where= a demand curve comes from.

Ingredients: Budget line, Indifference curves,<= span style=3D'mso-spacerun:yes'> Variation

Go back to our basic = budget line:

p_1* x_=
1 +p_2* x₂=3DY

first with:

Y=3D100, p_{1 =3D2.5nd p₂
=3D 10}

Then consider

Y=3D100, p_{1 =3D5,and p₂ =3D 10}

Y=3D100, p_{1 =3D10and p₂
=3D 10}

On the same graph.

X2<= /p>

[graph] =

= ; &n= bsp; = ; &n= bsp; = ; &n= bsp; X1

Draw some indifferenc= e curves on this.

X2<= /p>

= ; &n= bsp; = ; &n= bsp; = ; &n= bsp; = ; X1

The line connecting t= hese points traces out the price consumption curve (5,5) ; (10,5) ; (20, 5). In this case, it is a horizontal l= ine due to the simple form I picked for utility.

= &= nbsp; &nbs= p; Q

We can summarize the information in this form.

All else constant, ho= w does a change in price impact the quantity demanded for a given good. Sound familiar? This is the derivation of the individual’s demand curve for a particular good.

The line traced out b= y this, the price consumption curve, will reflect the individual’s underlying preferences. For a different individual (one who has less of a preference for x2 than the individual just considered), we would get the following result.

X2<= /p>

= ; &n= bsp; = ; &n= bsp; = ; &n= bsp; X1

Q= of goods	P= rice
(6.4)	10
(14,3)	5
(28.3)	2.5

Now, if we take each individual’s demand curve and sum them horizontally, we end up with t= he market demand.

At a price of 10, con= sumer 1 wants 5 and consumer 2 wants 6. 11

At a price of 5, cons= umer 1 wants 10 and consumer 2 wants 14. 24

At a price of 2.5, co= nsumer 1 wants 20 and consumer 2 wants 28. 48

What about a change in income?

p_1* x_=
1 +p_2* x₂=3DY

first with:

Y=3D50, p_{1 =3D10,and p₂ =3D 5}

Y=3D100, p_{1 =3D10,and p₂ =3D 5}

Y=3D150, p_{1 =3D10,and p₂ =3D 5}

The line connecting t= he optimal bundles defined by increasing income is the Income – Consumpt= ion curve.

At a given price of 1= 0 for good one, we can draw these quantities on a price quantity graph to illustr= ate the idea of demand shifts.

The graph traced out = by placing income on the y axis and quantity on the x axis is called an Engel curve. I traces out the relationship between the quantity demanded of a single good and income.

Remember the idea of = income elastiticites from before. He= re is where they come from. A norma= l good has a positive income elasticity, an inferior good has a negative income elasticity.

What is the formula f= or an income elasticity?

(% change in q divide= d by % change in income)

A good can be normal = for one person and inferior for another.

A good can be normal = for one person at one point in their life and inferior at another.

A good can be normal = at one level of income and inferior at another.

We can block out spac= e from an optimal bundle to areas where both goods are normal, one good is normal = and the other inferior.

Can both goods be inf= erior?

Show on graph.

What is the impact of= a price change?

First, we identify th= e change in the quantity demanded as a result of a price change as the total effect.

We decompose this tot= al effect into two distinct components.

1)&n= bsp; The substit= ution effect. If utility and the pr= ice of the other good are held constant, as the price of one good rises, consumers substitute between the two goods to reflect the new price ratio.=

Since MRS=3DMRT, pric= e change leads to a change in MRT, the consumer will adjust the consumption bundle a= long the indifference curve to re-equate MRS and MRT.

2)&n= bsp; The income effect. A change in price cha= nges the consumers buying power. R= ecall the example of how a having of prices is equivalent to a doubling of income. This is the core idea= here. If the price of good one goes down= all else held constant, that is going to increase the consumer’s buying p= ower which is like increasing their income.

Total effect =3D subs= titution effect + income effect.

Since indifference cu= rves are downward sloping, can we predict the sign of the substitution effect? The change in price and the substi= tution effect should have opposite signs. An increase in price leads to less of the good being consumed. A decrease in price leads to more = of a good being consumed.

Show graph.

However, the income e= ffect depends on whether the good is a normal or an inferior good.

A Giffen good is one = for which a decrease in price causes quantity demanded to fall (the income effe= ct outweighs the substitution effect, making the total effect negative). Not observed in reality, but an in= teresting example.

Show graphs for subst= itution effect for normal good, inferior good, and Giffen good.

For edification / inoculation:

This graphically represents what in economics is known= as the Slutsky equation.

It describes the relationship between two different ap= proaches to deriving the demand function: One, a money income held constant derivation (the Marshallian), and a second utility held constant derivation (the Hicksian).

Show three graphs for a price decrease: One, S+, I+,T+= ; Two, S+, I-, T+; Three S+, I-, T-.