Transcript Intermediate Microeconomic Theory

Intermediate Microeconomic Theory
Buying and Selling
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An Endowment Economy
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We have now developed a theory of
choice.
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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.
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So consider a “desert island” economy (a
“Robinson Crusoe” economy).
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An Endowment Economy
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Key feature of this simple economy, is that
there is no money, only goods.
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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
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For simplicity, assume there are only
two goods on island:
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coconut milk
mangos
Budget Set:
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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
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How would Al’s budget set change if 1gallon
coconut milk could be traded for 1/2 lb. of
mangos and vice versa?
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How about if 1gallon coconut milk could be
traded for 2 lbs. of mangos?
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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
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Preferences:
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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.
qm
slope = -1
4
8
qc
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Buying and Selling in an Endowment Economy
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Market Participation:
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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
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So in an endowment economy with 2 goods,
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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.
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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
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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
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Clearly what matters is relative price.
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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.
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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
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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).
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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
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Are there historic examples of numeraire
goods in primitive economies?
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In what way did numeraire type goods
come up in NYT article on barter goods in
Russia?
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Note: Numeraire goods are completely
distinct from composite goods.
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Buying and Selling in an Endowment Economy Analytically
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Let’s consider Al again.
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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) = qcaqmb
How do we analytically describe Al’s behavior?
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What is general form of his budget constraint?
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What is general expression for his optimal bundle?
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Buying and Selling in an Endowment Economy Analytically
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So for generic Cobb-Douglas preferences u(qc,qm) = qcaqmb, 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:
a p1w1  (1) w2
q1 ( p1 , w1 , w2 ) 
ab
p1
b p1w1  (1) w2
q2 ( p1 , w1 , w2 ) 
ab
1
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Buying and Selling in an Endowment Economy Analytically
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Define: qcA(pc, wcA,wmA) as Al’s gross demand for coconut milk
qmA(pc, wcA,wmA) as Al’s gross demand for mangos.
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If qcA(pc, wcA,wmA) – wcA > 0, Al buys coconut milk, or is net
demander of coconut milk,
If qcA(pc, wcA,wmA) – wcA < 0, Al sells coconut milk, or is net supplier
of coconut milk.
Analogue holds for mangos.
Also note that:
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If qcA(pc, wcA,wmA) – wcA > 0, then qmA(pc, wcA,wmA) – wmA < 0, and
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If qmA(pc, wcA,wmA) – wmA > 0, then qcA(pc, wcA,wmA) – wcA < 0
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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
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Example:
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Let his preferences be captured by U= qc0.5qm0.5 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?
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What will this mean about whether Al is a net demander or net supplier of
mangos?
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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
Al’s Gross Demands when 1 coconuts
can be traded for ½ lb mango
(pc = 0.50)
Al’s Gross Demands when 1 coconut
can be traded for 2 mangos
(pc = 2)
qm
qm
8
10
4
4
qmA
qmA
8
qcA
5
8
12
qc
16
qcA
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Buying and Selling in an Endowment Economy Analytically
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Example (alternate numeraire):
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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
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Suppose:
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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.
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If price of gal. of coconut milk fell pc = 1,
would Bob still be net demander of
coconut milk?
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What if price of gal. of coconut milk rose
to pc = 3, would Bob still be a net
demander of coconut milk?
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