July18

New Schedule

� Monday – hw3

Tuesday – quiz2 on this week’s material

Friday – hw4 to be posted by Monday

next Monday – Midtermnext Monday – Midterm

� Thereafter

� Homework on Fridays

�Quizzes on Mondays

� Final still on Friday

� Extra office hours still on Monday and Friday mornings

Other clarifications

� I can’t move the projector image up � sit so you can see



� The right approach to bads is…

If U = X – YIf U = X – Y




If U = X – Y,If U = X – Y,

define Z = YH –Y where YH is high enough that it will never be consumed,




If U = X – Y,If U = X – Y,


define V = X + Z,




If U = X – Y,If U = X – Y,


define V = X + Z,

and graph V in the XZ-plane

Outline for Friday, July 18

� Remember

� Homework due Monday

�Quiz on Tuesday

� Review graphing� Review graphing

� Finding the optimum – exercises

� Perfect complements

� Perfect substitutes

Review graphing

� We’ll review

� what graphs we have available

� what axes they are graphed in

� It’ll be on the quiz� It’ll be on the quiz

Review graphing

� We’ll review

� what graphs we have available

� what axes they are graphed in

� It’ll be on the quiz� It’ll be on the quiz

� A bad graph:

Visualizing utility

x2

Next several slidesFrom Huangkai on SlideShare

U ≡ 6

U ≡ 4

U ≡ 2x1

Visualizing utility

U ≡ 6

Utility

U ≡ 6

U ≡ 5

U ≡ 4

U ≡ 3

U ≡ 2

U ≡ 1

x1

x2

Visualizing utility

x2

x1

Visualizing utility

x2

x1

Visualizing utility

x2

x1

Visualizing utility

x2

x1

Visualizing utility

x2

x1

Visualizing utility

x2

x1

x2

Visualizing utility

x1

Visualizing utility

x1

Visualizing utility

x1

Visualizing utility

x1

Visualizing utility

x1

Visualizing utility

x1

Visualizing utility

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Visualizing utility

x1

Visualizing utility

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Visualizing utility

Past several slidesFrom Huangkai on SlideShare

x1

How can we graph U = f(X,Y)

� Indifference maps XY plane



� Hold U constant – choose any utility level

� Find Y as a function of X

�Graph it�Graph it

� Repeat for other utility levels







� Utility functions XU or YU plane







� Utility functions XU or YU plane

� Hold X or Y constant – I’ll usually give a level

�Graph U = f(X, Y0) or U = f(X0, Y)

How can we graph X*,Y* as functions of I, PX, and PY

� Price-consumption curve XY-plane

How X* and Y* change with PX or PY



� You wont need to graph, but to interpret

� Is it a Giffen good or not?

� Is the other good a substitute or complement?� Is the other good a substitute or complement?



How X* and Y* change with PX or PYWhat if we only looked at X* or Y* alone?



How X* and Y* change with PX or PY� The demand curves XPX-plane

YPY-planeYPY-plane

� The “cross-demand” curves YPX-plane

XPY-plane




YPY-planeYPY-plane


XPY-plane

� Income-consumption curve XY-plane

How X* and Y* change with I




YPY-planeYPY-plane


XPY-plane


� Is the good normal, a luxury or inferior?




YPY-planeYPY-plane


XPY-plane


How X* and Y* change with I

� Engel curves XI-plane

YI-plane

Outline for Friday, July 18

� Remember

� Homework due Monday

�Quiz on Tuesday

� Review graphing� Review graphing

� Finding the optimum – exercises

� Perfect complements

� Perfect substitutes

Finding the optimum

� Mathematically, we’re solving

Find the bundle (X*,Y*) that

� maximizes U(X,Y) and

� satisfies PXX + PYY ≤ I� satisfies PXX + PYY ≤ I

Finding the optimum

� Mathematically, we’re solving

Find the bundle (X*,Y*) that

� maximizes U(X,Y) and

� satisfies PXX + PYY ≤ I

0

10

20

30

0 10 20

Un

its o

f g

oo

d Y

Units of good X

0

10

20

30

0 10 20

Un

its o

f g

oo

d Y

Units of good X

I1I2I3I4

I5

� satisfies PXX + PYY ≤ I

� That is, we’re finding the best affordable bundle

Finding the optimum

� We know that any optimum

�Must be on the budget line

�Must be on an indifference curve wholly above the budget line

IYPXP YX =+

Finding the optimum

� We know that any optimum

�Must be on the budget line

�Must be on an indifference curve wholly above the budget line

IYPXP YX =+

� In addition, if

�MRS exists

�MRT exists

� The solution is interior

Then we know the slopes must match

MRTMRS =

Finding the optimum

� When do we have

�MRS does not exist?

3

4

5

Finding the optimum

(S)

Sandwiches

� When does MRS not exist?

When the indifference map has “kinks”

0

1

2

5/28/2008M. L. Williams, Department of Economics, PSU

Ketchup (K)

2 4 6 8 10

Sandwiches

map has “kinks”

3

4

5

Finding the optimum

(S)

Sandwiches

� For perfect complements…

The solution will always be on the orange line, where

0

1

2


Ketchup (K)

2 4 6 8 10

Sandwiches

orange line, where MRS is not defined

Finding the optimum

� Perfect complements have utility functions of the form

},min{ bYaXU =

Finding the optimum


� The important thing to remember is that the

},min{ bYaXU =

� The important thing to remember is that the consumer will always consume X and Y together

bYaX =

Finding the optimum


� The important thing to remember is that the

},min{ bYaXU =

� The important thing to remember is that the consumer will always consume X and Y together

� Together with the budget equation, this allows us to find the optimum

bYaX =

IYPXP YX =+

Finding the optimum

bYaX = IYPXP YX =+

Finding the optimum

� Suppose

� a = 3, b = 4

bYaX = IYPXP YX =+

� a = 3, b = 4

� I = 70, PX = 1, PY = 1

� What is the consumer’s optimum?

Finding the optimum

� The consumer only spends money on burgers and fries, and only enjoys the combination if there are 10 fries per burger.

Finding the optimum

� The consumer only spends money on burgers and fries, and only enjoys the combination if there are 10 fries per burger.

� Draw the consumer’s indifference map and find the optimum if the consumer has $300 to spend, as found earlier (as a function of prices).

3

4

5

Finding the optimum

(S)

Sandwiches

� For perfect complements…

Forget MRS = MRT

0

1

2


Ketchup (K)

2 4 6 8 10

Sandwiches

Solve by keeping X and Y at the complementary ratio

Finding the optimum

� When do we have


�MRT does not exist?

�QuotasQuotas

� Coupons

�Otherwise distorted budget sets

Finding the optimum

� When do we have


�MRT does not exist?

� A corner solution?� A corner solution?

� Neutral goods

� Perfect subsitutes

Finding the optimum

� X is neutral if U is only a function of Y

� U = Y

� U = 20Y2

Finding the optimum

� X is neutral if U is only a function of Y

� U = Y

� U = 20Y2

� Note that all these functions represent the same � Note that all these functions represent the same consumer preferences

Finding the optimum

� Perfect substitutes have utility functions of the form U = aX + bY

Finding the optimum

� Perfect substitutes have utility functions of the form U = aX + bY

� We know that all money will be spent on one or the other good

PIYPIX /or / ==

other good

� Whichever gives higher utility

YXPIYPIX /or / ==

Finding the optimum

� U = 5X + 6Y

� I = 30, P = 1, P = 1

YXPIYPIX /or / == IYPXP YX =+

� I = 30, PX = 1, PY = 1

� What is the optimal bundle?

Finding the optimum

� U = 5X + 6Y

� I = 30, P = 1, P = 1


� I = 30, PX = 1, PY = 1


�We can buy 30 of one or the other

� Everything will be spent on Y

Finding the optimum

� U = 5X + 6Y

� I = 30, P = 1, P = 1


� I = 30, PX = 1, PY = 1




� I = 30, PX = 1, PY = 2. Now what’s optimal?

Finding the optimum

� U = 5X + 6Y

� I = 30, P = 1, P = 1


� I = 30, PX = 1, PY = 1




� I = 30, PX = 1, PY = 2. Now what’s optimal?

� Compare the utility of 30X with that of 15Y

Finding the optimum

� Leisure is priced by the consumer’s wage PLConsumption is priced by the market P

LC4U +=

Consumption is priced by the market PC“Income” is how much could be earned 24PL

Finding the optimum

� Leisure is priced by the consumer’s wage PLConsumption is priced by the market P

LC4U +=

Consumption is priced by the market PC“Income” is how much could be earned 24PL

What is the optimal bundle if I=10, PL = 1, PC = 5

Date post:	24-May-2015
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July18

Business