978-0393123982 Chapter 32 Solution Manual Part 1

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Chapter 32 NAME

Exchange

Introduction. The Edgeworth box is a thing of beauty. An amazing

amount of information is displayed with a few lines, points and curves.

In fact one can use an Edgeworth box to show just about everything

there is to say about the case of two traders dealing in two commodities.

Economists know that the real world has more than two people and more

than two commodities. But it turns out that the insights gained from this

model extend nicely to the case of many traders and many commodities.

So for the purpose of introducing the subject of exchange equilibrium, the

Edgeworth box is exactly the right tool. We will start you out with an

example of two gardeners engaged in trade. You will get most out of this

example if you fill in the box as you read along.

Example: Alice and Byron consume two goods, camelias and dahlias.

Alice has 16 camelias and 4 dahlias. Byron has 8 camelias and 8 dahlias.

They consume no other goods, and they trade only with each other. To

Dahlias Byron

0 6 12 18 24

Alice Camelias

Any feasible allocation of ﬂowers between Alice and Byron is fully

described by a single point in the box. Consider, for example, the alloca-

tion where Alice gets the bundle (15,9) and Byron gets the bundle (9,3).

This allocation is represented by the point A=(15,9) in the Edgeworth

box, which you should draw in. The distance 15 from Ato the left side of

the box is the number of camelias for Alice and the distance 9 from Ato

392 EXCHANGE (Ch. 32)

It is also useful to mark the initial allocation in the Edgeworth box,

which, in this case, is the point E=(16,4). Now suppose that Alice’s

utility function is U(c, d)=c+2dand Byron’s utility funtion is U(c, d)=

cd. Alice’s indifference curves will be straight lines with slope −1/2. The

The Pareto set or contract curve is the set of points where Alice’s

indifference curves are tangent to Byron’s. There will be tangency if the

slopes are the same. The slope of Alice’s indifference curve at any point is

−1/2. The slope of Byron’s indifference curve depends on his consumption

Some problems ask you to find a competitive equilibrium. For an

economy with two goods, the following procedure is often a good way to

calculate equilibrium prices and quantities.

•Since demand for either good depends only on the ratio of prices of

•With the price of good 1 held at 1, calculate each consumer’s demand

•Write an equation that sets the total amount of good 2 demanded by

•Solve this equation for the value of p2that makes the demand for

•Plug this price into the demand functions to determine quantities.

Example: Frank’s utility function is U(x1,x

2)=x1x2and Maggie’s is

U(x1,x

2)=min{x1,x

2}. Frank’s initial endowment is 0 units of good 1

Set p1= 1 and find Frank’s and Maggie’s demand functions for good

2 as a function of p2. Using the techniques learned in Chapter 6, we

find that Frank’s demand function for good 2 is m/2p2,wheremis his

income. Since Frank’s initial endowment is 0 units of good 1 and 10 units

NAME 393

budget constraint implies that Maggie’s demand function for good 2 is

m/(1 + p2). Since her endowment is 20 units of good 1 and 5 units of

good 2, her income is 20 + 5p2. Therefore at price p2, Maggie’s demand

Solving this equation, one finds that the equilibrium price is p2=2. At

the equilibrium price, Frank will demand 5 units of good 2 and Maggie

will demand 10 units of good 2.

32.1 (0) Morris Zapp and Philip Swallow consume wine and books.

Morris has an initial endowment of 60 books and 10 bottles of wine. Philip

has an initial endowment of 20 books and 30 bottles of wine. They have

no other assets and make no trades with anyone other than each other.

For Morris, a book and a bottle of wine are perfect substitutes. His utility

function is U(b, w)=b+w,wherebis the number of books he consumes

and wis the number of bottles of wine he consumes. Philip’s preferences

are more subtle and convex. He has a Cobb-Douglas utility function,

U(b, w)=bw. In the Edgeworth box below, Morris’s consumption is

measured from the lower left, and Philip’s is measured from the upper

right corner of the box.

0204060

Books

PhilipWine

Morris

Blue curve

Red curve

Black

line

(a) On this diagram, mark the initial endowment and label it E.Usered

ink to draw Morris Zapp’s indifference curve that passes through his initial

394 EXCHANGE (Ch. 32)

(b) At any Pareto optimum, where both people consume some of each

good, it must be that their marginal rates of substitution are equal. No

matter what he consumes, Morris’s marginal rate of substitution is equal

bP.Use black ink on the diagram above to draw the locus of Pareto

optimal allocations.

(d) At the equilibrium prices you found in the last part of the question,

what is the value of Philip Swallow’s initial endowment? 50. At these

(e) At the competitive equilibrium prices that you found above, Morris’s

income is 70. Therefore at these prices, the cost to Morris of con-

(f) Suppose that an economy consisted of 1,000 people just like Morris

and 1,000 people just like Philip. Each of the Morris types had the same

endowment and the same tastes as Morris. Each of the Philip types had

the same endowment and tastes as Philip. Would the prices that you

found to be equilibrium prices for Morris and Philip still be competitive

NAME 395

32.2 (0) Consider a small exchange economy with two consumers, Astrid

and Birger, and two commodities, herring and cheese. Astrid’s initial

endowment is 4 units of herring and 1 unit of cheese. Birger’s initial en-

dowment has no herring and 7 units of cheese. Astrid’s utility function is

U(HA,C

A)=HACA. Birger is a more inﬂexible person. His utility func-

tion is U(HB,C

B)=min{HB,C

B}.(HereHAand CAare the amounts

of herring and cheese for Astrid, and HBand CBare amounts of herring

and cheese for Birger.)

(a) Draw an Edgeworth box, showing the initial allocation and sketching

in a few indifference curves. Measure Astrid’s consumption from the lower

0246

Cheese

BirgerHerring

Astrid

eBlue curves

Red curves

Black

line

(b) Use black ink to show the locus of Pareto optimal allocations. (Hint:

Since Birger is kinky, calculus won’t help much here. But notice that

because of the rigidity of the proportions in which he demands the two

goods, it would be inefficient to give Birger a positive amount of either

good if he had less than that amount of the other good. What does that

of herring. Write an expression for the amount of herring that Birger

will demand at these prices. 7/(p+1) (Hint: Since Birger initially

owns 7 units of cheese and no herring and since cheese is the numeraire,

the value of his initial endowment is 7. If the price of herring is p,how

many units of herring will he choose to maximize his utility subject to his

budget constraint?)

396 EXCHANGE (Ch. 32)

32.3 (0) Dean Foster Z. Interface and Professor J. Fetid Nightsoil ex-

change platitudes and bromides. When Dean Interface consumes TIplat-

itudes and BIbromides, his utility is given by

UI(BI,T

I)=BI+2

TI.

When Professor Nightsoil consumes TNplatitudes and BNbromide, his

utility is given by

TN.

Dean Interface’s initial endowment is 12 platitudes and 8 bromides. Pro-

fessor Nightsoil’s initial endowment is 4 platitudes and 8 bromides.

04812

Bromides

Platitudes

Nightsoil

Interface

eRed curve

Pencil curve

Blue line

3.2

(a) If Dean Interface consumes TIplatitudes and BIbromides, his mar-

(b) On the contract curve, Dean Interface’s marginal rate of substitution

equals Professor Nightsoil’s. Write an equation that states this condition.

NAME 397

(d) But we also know that along the contract curve it must be that TI+

(e) Solving these two equations in two unknowns, we find that everywhere

(f) In the Edgeworth box, label the initial endowment with the letter

E. Dean Interface has thick gray penciled indifference curves. Profes-

sor Nightsoil has red indifference curves. Draw a few of these in the

Edgeworth box you made. Use blue ink to show the locus of Pareto op-

(g) Find the competitive equilibrium prices and quantities. You know

what the prices have to be at competitive equilibrium because you know

what the marginal rates of substitution have to be at every Pareto

32.4 (0) A little exchange economy has just two consumers, named Ken

and Barbie, and two commodities, quiche and wine. Ken’s initial endow-

ment is 3 units of quiche and 2 units of wine. Barbie’s initial endowment

is 1 unit of quiche and 6 units of wine. Ken and Barbie have identical util-

ity functions. We write Ken’s utility function as, U(QK,W

K)=QKWK

and Barbie’s utility function as U(QB,W

B)=QBWB,whereQKand

WKare the amounts of quiche and wine for Ken and QBand WBare

amounts of quiche and wine for Barbie.

(a) Draw an Edgeworth box below, to illustrate this situation. Put quiche

on the horizontal axis and wine on the vertical axis. Measure goods for

Ken from the lower left corner of the box and goods for Barbie from the

upper right corner of the box. (Be sure that you make the length of the

398 EXCHANGE (Ch. 32)

Ken Quiche

Wine Barbie

Black line

Red

curve

Blue

curve

Pareto

efficient

points

(b) Use blue ink to draw an indifference curve for Ken that shows alloca-

good, Ken’s marginal rate of substitution between quiche and wine must

(d) On your graph, show the locus of points that are Pareto efficient.

(Hint: If two people must each consume two goods in the same proportions

(e) In this example, at any Pareto efficient allocation, where both persons

consume both goods, the slope of Ken’s indifference curve will be −2.

(f) In competitive equilibrium, Ken’s consumption bundle must be 2