July15

Outline for Tuesday, July 15

� New documents

�Quiz study sheet

� Hw2, quiz1 answers

� Hw3� Hw3

� Recap Monday

� Indifference map examples

� The budget constraint

� The best affordable bundle

Indifference maps

� Indifference maps, like contour maps, show level sets – bundles that the consumer is indifferent between

� We know that

� Indifference sets are curves, not thick bands� Indifference sets are curves, not thick bands

� Indifference curves do not cross

� Indifference curves slope down

Marginal Rate of Substitution

� The MRS is the slope of the indifference curve. It tells us what tradeoff the consumer will accept

0∆X

∆Y<=MRS

� It typically decreases with higher X

∆X


� Imperfect substitutes are have diminishing MRS

� Perfect substitutes have the same, finite, MRS at every point

� If Y is a neutral good, the MRS is infinite at all points� If Y is a neutral good, the MRS is infinite at all points

� Perfect complements have infinite MRS at low X and zero MRS at high X


� Imperfect substitutes are have diminishing MRS

� Perfect substitutes have the same, finite, MRS at every point

� If Y is a neutral good, the MRS is infinite at all points� If Y is a neutral good, the MRS is infinite at all points

� Perfect complements have infinite MRS at low X and zero MRS at high X

� If Y is a bad, we can reinterpret it as a good

� If they won’t pay for it, they might pay not to have it

� Paying to get trash � paying to not get trash

Marginal utility

� “More is better”, but how much better?

� Look at the slope of the utility-Y graph for fixed X

∆Y

∆UMU

Y=∆Y

MUY

=

Marginal utility

� “More is better”, but how much better?

� Look at the slope of the utility-Y graph for fixed X

∆Y

∆UMU

Y=

� It tells us how much Y is worth on the margin

� It is usually decreasing in Y, except perhaps for

� Risk – probability of safety

� Threshold goods – the last episode of Lost

� The same holds for X

∆YMU

Y=

Marginal utility � MRS

� The MRS tells us what tradeoff’s between X and Y are acceptable

0∆X

∆YMRS <=

� The MUX and MUY tell us how much X and Y are worth at the margin

∆X

∆X

∆UMU

∆Y

∆UMU

XY==


� The MRS tells us what tradeoff’s between X and Y are acceptable

0∆X

∆YMRS <=

� The MUX and MUY tell us how much X and Y are worth at the margin

� To combine them, use ∆U = 1 in the MU’s

∆X

∆X

∆UMU

∆Y

∆UMU

XY==


� Take a bundle (X,Y) and suppose

� ∆Y gives one utile

� ∆X gives one utile





� Then� Then

�MUY = 1/∆Y

�MUX = 1/∆X





� Then� Then

�MUY = 1/∆Y

�MUX = 1/∆X

� And the consumer will accept the tradeoff

∆Y for ∆X





� Then� Then

�MUY = 1/∆Y

�MUX = 1/∆X

� And the consumer will accept the tradeoff

∆Y for ∆X

�Which is equivalent to

1/MUY for 1/MUX


� The consumer will accept the trade-off

∆Y for ∆X

� Which is equivalent to

1/MUY for 1/MUX1/MUY for 1/MUX

� The marginal rate of substitution tells us which trade-offs will be accepted, so…


� The consumer will accept the trade-off

∆Y for ∆X

� Which is equivalent to

1/MUY for 1/MUX1/MUY for 1/MUX

� The marginal rate of substitution tells us which trade-offs will be accepted, so…

Y

X

MU

MU

∆X

∆Y−==MRS


� New documents

�Quiz study sheet


� Hw3� Hw3

� Recap Monday




Graphing utility

� What types of goods are X and Y if…

� U = XY

� U = X+Y

� U = X-Y� U = X-Y

Graphing utility


� U = XY imperfect substitutes

� U = X+Y perfect substitutes

� U = X-Y Y is a bad, X is a good� U = X-Y Y is a bad, X is a good

�U = X+(Y0-Y) Choose Y0 high enough

Graphing utility






� We can graph them by…

Graphing utility






� We can graph them by…

� Fixing U0

� Finding Y as a function of X

�Graphing for several U0

Finding MU and MRS

� What is the approximate MRS of the fourth unit?

X 1 2 3 4 5 6

Y 20 16 13 11 10 9

Finding MU and MRS


� Take the average…

X 1 2 3 4 5 6

Y 20 16 13 11 10 9


∆+

∆≈ +−

112

1 YYMRS

Finding MU and MRS



X 1 2 3 4 5 6

Y 20 16 13 11 10 9


( )2

312

2

1

112

1

−=−−=

∆+

∆≈ +− YY

MRS

Finding MU and MRS



X 1 2 3 4 5 6

Y 20 16 13 11 10 9


�We don’t need utility – only tradeoffs – to find MRS

( )2

312

2

1

112

1

−=−−=

∆+

∆≈ +− YY

MRS

Finding MU and MRS


� Take the average… 3/2

X 1 2 3 4 5 6

Y 20 16 13 11 10 9

� Take the average… 3/2

� What is the approximate MUY of the fourth unit?

Y 1 2 3 4 5 6 7 8 9

Utility 15 35 50 62 70 74 76 77 77

Finding MU and MRS


� Take the average… -3/2

X 1 2 3 4 5 6

Y 20 16 13 11 10 9



� Take the average…10

Y 1 2 3 4 5 6 7 8 9

Utility 15 35 50 62 70 74 76 77 77

Finding MU and MRS



X 1 2 3 4 5 6

Y 20 16 13 11 10 9



� Take the average…10

� What is MUX at (1,4) if the cross-section above is for X=1?

Y 1 2 3 4 5 6 7 8 9

Utility 15 35 50 62 70 74 76 77 77

Setting up a basketball match

� The rules: neither team can have more

� Than 5 players on the court

� Players on the court than the other team

� Than 10 players total – for substitutes� Than 10 players total – for substitutes

A has B has On Court

10 5 5

3 5 3

1 4 1


� The promoter wants more people on the court, to sell more tickets. What is his indifference map?



� The A team coach wants more players than B. What is his indifference map?is his indifference map?



� The A team coach wants more players than B. What is his indifference map?is his indifference map?

�What is his indifference map if we change the bad, B, into a good?


� New documents

�Quiz study sheet


� Hw3� Hw3

� Recap Monday




Budget set

� Given $100, and facing prices PX = $10, PY = $5, what can the consumer afford?

Budget set


� How does this set change if…

� the consumer gets $200� the consumer gets $200

Budget set




� PX = $5

Budget set




� PX = $5 and PY = $2.50

Budget set




� PX = $5

� There is a quota of 5 units on Y

Budget set




� PX = $5


� The consumer gets a non-saleable coupon for 10 X

Budget set




� PX = $5


� The consumer gets a non-saleable coupon for 10 X

� We know from supply and demand that coupons and quotas can change the price, too

Budget set

� Suppose food stamps can only be spent on food and may not be resold

Budget set


� Given I = $1000, PFood = $10, PNot= $5…

� Suppose the consumer is given $100 of food stamps. � Suppose the consumer is given $100 of food stamps. What is their budget line?

Budget set



� Suppose the consumer is given $100 of food stamps� Suppose the consumer is given $100 of food stamps

� Suppose those $100 of food stamps can be resold at half their value on the black market. What is the new budget set?

Budget set


� Suppose the government sets individual food quotas of 50 units. What is the budget set?

Budget set


� Suppose the government sets individual food quotas of 50 units

� Suppose those 50 units can be sold at $5 each on the black market. What is the new budget set?

Budget set

� The budget set includes all affordable bundles

� The edge of the budget set is the budget line

Budget set



� The slope of the budget line is the Marginal Rate of TransformationTransformation

Y

X

P

P

X

YMRT −=

∆

∆=

Budget set




� Tells us what tradeoff the consumer can make between X and Y

Y

X

P

P

X

YMRT −=

∆

∆=

Budget set




� Tells us what tradeoff the consumer can make between X and Y

� Recall that MRS tells us what tradeoff the consumer is willing to make

Y

X

P

P

X

YMRT −=

∆

∆=


� New documents

�Quiz study sheet


� Hw3� Hw3

� Recap Monday




The best affordable bundle

� The consumer will choose a bundle in the budget set that gives them the highest utility


� The consumer will choose a bundle in the budget set that gives them the highest utility. We call this

� The consumer’s optimum or

� The solution to the consumer’s problem� The solution to the consumer’s problem



� Because more is better, the optimum will be on the budget linebudget line



� Because more is better, the optimum will be on the budget linebudget line

� The optimum must be on an indifference curve that is never below the budget line

� Everything above an indifference curve gives greater utility

� So something better would be affordable if the indifference curve crossed the budget line


� The optimum will be on

� the budget line

� an indifference curve that does not cross the line



� the budget line


� If the optimum has X>0 and Y>0, we call it interior� If the optimum has X>0 and Y>0, we call it interior



� the budget line



� In this case, the slopes of the budget line, the MRT, and that of the indifference curve, the MRS, must be the same



� the budget line



� In this case, the slopes of the budget line, the MRT, and that of the indifference curve, the MRS, must be the same:

� The tradeoff the consumer makes is no better or worse than the tradeoff the consumer takes



� the budget line


� If the optimum has X>0 and Y>0, we call it interior � If the optimum has X>0 and Y>0, we call it interior and have MRS = MTS at the optimum

MTSP

P

MU

MUMRS

Y

X

Y

X =−=−=



� the budget line


� If the optimum has X>0 and Y>0, we call it interior � If the optimum has X>0 and Y>0, we call it interior and have MRS = MTS at the optimum

� So, if this equality does not hold, either

� The consumer is not rational or

�One of the goods is not consumed

MTSP

P

MU

MUMRS

Y

X

Y

X =−=−=


� If the optimum has X>0 and Y>0, we call it interior and have MRS = MTS at the optimum

� If the optimum has X=0, then PX/PY must be too high for the consumer to make any tradeoff, even at high for the consumer to make any tradeoff, even at zero




Y

X

Y

X

MU

MU

P

P>




MRTMRSMU

MU

P

P

Y

X

Y

X >⇔>




� Neutral goods

� Perfect substitutes

MRTMRSMU

MU

P

P

Y

X

Y

X >⇔>


� If the optimum has Y=0, the consumer is not willing to trade X for even one unit of Y at current prices

MRSMRTMU

MU

P

PYY >⇔> MRSMRT

MUPXX

>⇔>


� If the optimum has Y=0, the consumer is not willing to trade X for even one unit of Y at current prices

MRSMRTMU

MU

P

PYY >⇔>

� When X = 0 or Y = 0, we have a corner solution, as opposed to an interior solution

MRSMRTMUP

XX

>⇔>


� If X>0 and Y>0 MRS = MRT

� If X=0 MRS > MRT

� If Y=0 MRS < MRT


� If X>0 and Y>0 MRS = MRT

� If X=0 MRS > MRT

� If Y=0 MRS < MRT

� This makes sense because

�MRS is the rate at which we want to trade Y for X

�MRT is the rate at which we can trade Y for X

� Both are inconveniently negative


� New documents

�Quiz study sheet


� Hw3� Hw3

� Recap Monday




Date post:	23-Jun-2015
Category:	Technology
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July15

Technology