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Intermediate Microeconomic Theory

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Intermediate Microeconomic Theory

Buying and Selling

- We have now developed a theory of choice.
- Given this theory, we can already consider the role of prices and markets in an economy.

- 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).

- 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.

- 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?

- How would Al’s budget set change if 1gallon coconut milk could be traded for 1/2 lb. of mangos and vice versa (i.e., 1 gal coconut milk “costs” ½ lb mangos)?
- How about if 1gallon coconut milk could be traded for 2 lbs. of mangos (i.e., 1 gal coconut milk “costs” 2 lbs mangos)?
- How would Al’s budget set be affected by the above price changes if his endowment was 10 gal. coconut milk, 0 mangos?

- Preferences:
- Suppose the utility Al gets from coconut milk and mangos is given by some utility function u(qc,qm) that exhibits DMRS.
- Further suppose that by consuming his endowment he gets utility of u(8, 4) and his MRS at (8,4) is -1.

qm

4

slope = -1

8 qc

- Market Participation:
- Suppose a market opened up where 1 gal. milk costs 1/2 lb. of mangos (or equivalently, 1 lb of mangos costs 2 gal of milk).
- What would Al do? Would this market make Al better off?
- Suppose instead a market opened up where 1 gal. milk costs 2 lbs. of mangos (or equivalently, 1 lb of mangos costs 1/2 gal of milk).
- What would Al do? Would this market make Al better off?

- 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.

- What relative price (i.e. terms of trade) would cause Al to be neither a buyer or a seller of coconuts?

- 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.

- We have been calculating the price of a gallon of coconut milk in terms of 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.

- 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.

- 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)

- 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.

- 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) = qcaqmb

- How do we analytically describe Al’s behavior?
- What is general form of his budget constraint?
- So what is general expression for his optimal bundle?

- 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:

- Define: qcA(pc,wcA,wmA) as Al’s gross demand for coconut milk qmA(pc,wcA,wmA) as Al’s gross demand for mangos.
- 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:
- If qcA(pc,wcA,wmA) – wcA > 0, then qmA(pc,wcA,wmA) – wmA < 0, and
- If qmA(pc,wcA,wmA) – wmA > 0, then qcA(pc,wcA,wmA) – wcA < 0
- Intuitively, if Al is buying coconut milk, he must be selling mangos, and vice versa.

- Example:
- Let his preferences be captured by U= qc0.5qm0.5 and
endowment be given by wc = 8 and wm = 4.

- Let mangos be 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?

- Let his preferences be captured by U= qc0.5qm0.5 and

Al’s Gross Demands when 1 gal coconut milk costs ½ lb mango (pc = 0.50)

Al’s Gross Demands when 1 gal. coconut milk costs 2 mangos (pc = 2)

qm

8

4

qm

10

4

qmA

qmA

8 16

qcA 5 8 12 qc

qcA

- 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?

- 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)

- Suppose we don’t know anything else about Bob’s preferences other than at these prices Bob is a net demander of coconut milk.
- If price of gal. of coconut milk fell pc = 1, can we know whether Bob will 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?