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Intermediate Microeconomic Theory

Buying and Selling

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An Endowment Economy

We have now developed a theory of choice. Given this theory, we can already consider

the role of markets and prices.

As is the norm in economic theory, we start with the simplest possible world and build up. So consider a “desert island” economy (a

“Robinson Crusoe” economy).

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An Endowment Economy

Key feature of this simple economy, is that there is no money, only goods.

Specifically, an individual is “endowed” with a given amount of various goods.

If there is a market, an individual can potentially choose to trade some of his endowed amount of one good for more of another.

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An Endowment Economy

For simplicity, assume there are only two goods on island: coconut milk mangos

Budget Set: Suppose Al (one of the inhabitants) has

endowment of wc = 8 and wm = 4 (8 gallons of coconut milk and 4 lbs. of mangos).

If there were no “markets” on the island, how would we graphically depict Al’s budget set?

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Budget Set in an Endowment Economy

How would Al’s budget set change if 1gallon coconut milk could be traded for 1/2 lb. of mangos and vice versa?

How about if 1gallon coconut milk could be traded for 2 lbs. of mangos?

How would Al’s budget set be affected by the above price changes if his endowment was 10 gal. coconut milk, 0 mangos?

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Buying and Selling in an Endowment Economy Preferences:

Suppose the utility Al gets from coconut milk and mangos is given by some utility function u(qc,qm).

Further suppose that by consuming his endowment he gets utility of u(8, 4) and his MRS at (8,4) is -1.

8 qc

qm

4

slope = -1

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Buying and Selling in an Endowment Economy Market Participation:

Suppose a market opened up where Al could “buy” 1 gal. milk for 2 lbs. of mangos and vice versa .

What would Al do? Would this market make Al better off?

Suppose instead a market opened up where Al could “buy” 1 gal. coconut milk for 1/2 lb. of mangos and vice versa.

What would Al do? Would this market make Al better off?

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Buying and Selling in an Endowment Economy So in an endowment economy with 2 goods,

If an individual chooses to consume a bundle with more of good 1 than he is endowed with (and therefore less of good 2 than he is endowed with), he must be a buyer of good 1 and a seller of good 2.

If an individual chooses to consume a bundle with less of good 1 than he is endowed with (and therefore more of good 2 than he is endowed with), he must be a seller of good 1 and a buyer of good 2.

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Buying and Selling in an Endowment Economy What relative price (i.e. terms of trade) would cause Al to

be neither a buyer or a seller of coconuts?

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Buying and Selling in an Endowment Economy Clearly what matters is relative price.

We have been calculating the price of a gallon of coconut milk in terms of lbs of mangos e.g. 1 more gal coconut milk costs X lbs of mangos.

Note: this system could be adopted for any number of goods. 1 lb of fish costs Y lbs of mangos 2 sharpened stones cost Z lbs. of mangos.

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Buying and Selling in an Endowment Economy Therefore, for a market with K goods, we only need K-1 prices,

and make one good a numeraire (a good we compute every other good’s price relative to).

So we have been using lbs of mangos as numeraire, meaning pc = 2 implies one more gal coconut milk costs 2 lbs mangos.

What would be “cost” of another lb of mangos if mangos are numeraire?

Alternatively, we could use coconut milk as numeraire good, then pm = 1/2 implies that one would need to trade 1/2 gal coconut milk for one more lb. of mangos.

What would be “cost” of another gal of coconut milk now?

Note that regardless of which good we select as numeraire, relative terms of trade are the same (i.e. 2 lbs mangos traded for 1 gal coconut milk is equivalent to 1 lb mangos traded for ½ lb coconut milk)

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Buying and Selling in an Endowment Economy Are there historic examples of numeraire

goods in primitive economies?

In what way did numeraire type goods come up in NYT article on barter goods in Russia?

Note: Numeraire goods are completely distinct from composite goods.

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Buying and Selling in an Endowment Economy Analytically Let’s consider Al again.

Let his endowment be given by {wc ,wm} Suppose mangos are the numeraire good and the relative price of

coconuts is pc.

Suppose Al’s preferences are captured by a generic Cobb-Douglas utility function u(qc,qm) = qc

aqmb

How do we analytically describe Al’s behavior? What is general form of his budget constraint?

What is general expression for his optimal bundle?

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Buying and Selling in an Endowment Economy Analytically So for generic Cobb-Douglas preferences u(qc,qm) = qc

aqmb, with

endowment {w1,w2} and relative prices such that one more unit of good 1 costs p1 units of good 2 (the numeraire), the optimal bundle will again be given by the corresponding demand functions, which will now be:

1

)1(),,(

)1(),,(

2112112

1

2112111

wwp

ba

bwwpq

p

wwp

ba

awwpq

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Buying and Selling in an Endowment Economy Analytically Define: qc

A(pc, wcA,wm

A) as Al’s gross demand for coconut milk qm

A(pc, wcA,wm

A) as Al’s gross demand for mangos.

If qcA(pc, wc

A,wmA) – wc

A > 0, Al buys coconut milk, or is net demander of coconut milk,

If qcA(pc, wc

A,wmA) – wc

A < 0, Al sells coconut milk, or is net supplier of coconut milk.

Analogue holds for mangos.

Also note that: If qc

A(pc, wcA,wm

A) – wcA > 0, then qm

A(pc, wcA,wm

A) – wmA < 0, and

If qmA(pc, wc

A,wmA) – wm

A > 0, then qcA(pc, wc

A,wmA) – wc

A < 0

Intuitively, if Al is buying coconut milk, he must be selling mangos, and vice versa.

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Buying and Selling in an Endowment Economy Analytically Example:

Let his preferences be captured by U= qc0.5qm

0.5 and

endowment be given by wc = 8 and wm = 4. Suppose mangos numeraire and the relative price of coconut milk in terms of

lbs of mangos is pc = 2

What will be Al’s gross and net demands for coconut milk?

What will this mean about whether Al is a net demander or net supplier of mangos?

What if the relative price of coconuts (in terms of mangos) dropped to pc = 0.50?

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Buying and Selling in an Endowment Economy

qcA 5 8 12 qc

8 16

qm

8

4

Al’s Gross Demands when 1 coconut can be traded for 2 mangos (pc = 2)

Al’s Gross Demands when 1 coconuts can be traded for ½ lb mango(pc = 0.50)

qmA

qcA

qmA

qm

10

4

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Buying and Selling in an Endowment Economy Analytically Example (alternate numeraire):

What would happen if we used coconut milk as numeraire, with pm = 0.5, but let Al’s endowment again be given by wc = 8 and wm = 4?

What will be Al’s gross and net demands for coconut milk?

What will this mean about whether Al is a net demander or net supplier of mangos?

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Revealed Preference

Suppose: Bob is endowed with 4 gal. coconut milk

and 4 lbs. mangos. Current price of 1 gal. coconut milk in

terms of lbs of mangos is 2 (i.e. pc = 2) At these prices, we know Bob is a net

demander of coconut milk.

If price of gal. of coconut milk fell pc = 1, would Bob still be net demander of coconut milk?

What if price of gal. of coconut milk rose to pc = 3, would Bob still be a net demander of coconut milk?