**Problem 1**

Let's rewrite the story of Colleen and Bill in a way that highlights the role of relative price. Suppose that Bill and Colleen produce logs and bushels at the following rates:

Colleen | Bill | |
---|---|---|

bushels | 10 | 8 |

logs | 10 | 5 |

Suppose further that Bill and Colleen value bushels of food and logs equally, so that the price of one bushel equals the price of one log.

- Notice now that valuing bushels and logs equally, creates a situation where:
- Bill gains from trade with Colleen, but
- Colleen doesn't gain from trade with Bill.
- However, she doesn't lose by trading with Bill.
- Why doesn't Colleen gain from trade?

**ANSWER:** Colleen doesn't gain from trade because
her opportunity cost of gathering bushels of food is equal to the relative price of food:

$$1\frac{log}{bushel}=1\frac{log}{bushel}$$

Bill gains from trade because his opportunity cost of gathering bushels of food is less than the relative price of food:

$$\frac{5}{8}\frac{log}{bushel}<1\frac{log}{bushel}$$

- Leaving opportunity costs unchanged, how can the story be rewritten, so that both Bill and Colleen gain from trade?
**Hint:**How does the assumption that Bill and Colleen value bushels and logs equally prevent Colleen from gaining from trade (given the production rates above)?

**ANSWER:** If the relative price of food were between
$\frac{5}{8}\frac{log}{bushel}$
and
$1\frac{log}{bushel}$
,
then both Colleen and Bill would gain from trade.

**Problem 2**

Assume that there are only two countries in the world: Home and Foreign. Assume that there are 300 workers in Home and 100 workers in Foreign.

In Home, 2 units of labor are needed to produce one unit of wine and 3 units of labor are needed to produce one unit of cloth. In Foreign, 5 units of labor are needed to produce one unit of wine and 4 units of labor are needed to produce one unit of cloth.

(labor per unit of output)

Home | Foreign | |
---|---|---|

wine | 2 | 5 |

cloth | 3 | 4 |

Assume that each country's utility is given by the Cobb-Douglas function:

$$U(C,W)={C}^{0.5}\cdot {W}^{0.5}$$

Note that the demand functions associated with such a utility function are given by:

$$C({p}_{C},M)=\frac{1}{2}\cdot \frac{M}{{p}_{C}}$$

$$W({p}_{W},M)=\frac{1}{2}\cdot \frac{M}{{p}_{W}}$$

where $M$ is money income, ${p}_{C}$ is the price of cloth and ${p}_{W}$ is the price of wine.

- Find the autarky relative price of wine in each country. How much wine and cloth does each country consume in autarky?

**ANSWER:** In autarky, the relative price of wine equals the
opportunity cost of producing wine.

Home's opp. cost wine = $\frac{2}{3}\frac{cloth}{wine}$

Foreign's opp. cost wine = $\frac{5}{4}\frac{cloth}{wine}$

And because both countries share the same utility function, they share the same relative demand function. Combining the two individual demand functions above yields the following relative demand for wine in both countries:

$$\frac{W}{C}=\frac{1}{{p}_{W}/{p}_{C}}$$

Notice that the relative demand for wine is inversely related to the relative price of wine. As the relative price of wine increases, the relative quantity of wine demanded falls.

**Home's utility maximization**

When the countries are in autarky, the relative price of wine equals the opportunity cost of producing wine.

So given Home's opportunity cost of producing wine and Home's relative demand for wine, we know that in autarky, Home will maximize utility by consuming: $\frac{3}{2}\frac{wine}{cloth}$ , i.e. three units of wine for every two units of cloth.

And when the budget constraint is satisfied, that utility-maximizing relative quantity of wine implies consumption of 75 wine and 50 cloth.

**Foreign's utility maximization**

Similarly, in autarky, Foreign will maximize utility by consuming: $\frac{4}{5}\frac{wine}{cloth}$ , i.e. four units of wine for every five units of cloth.

When the budget constraint is satisfied, that utility-maximizing relative quantity of wine implies consumption of 10 wine and 12.5 cloth.

- Now assume that Home and Foreign engage in trade. Set world relative supply of wine equal to world relative demand for wine and solve for the world relative price of wine. (Hint: assume a particular pattern of specialization).

**ANSWER:** Because both countries share the same utility function,
they share the same relative demand for wine, which we obtained above by
combining the individual demand functions:

$$\frac{W}{C}=\frac{1}{{p}_{W}/{p}_{C}}$$

Obtaining the world relative supply of wine is both more complicated and very simple.

It's complicated because the relative supply function takes the shape of a step function. It's very simple because it's simple to calculate the steps.

Suppose that the relative price of wine was zero. In that case, the opportunity cost of producing wine would exceed the relative price of wine in both countries, so both Home and Foreign would only produce cloth (zero wine), so the relative supply of wine would be zero.

If the relative price of wine were $\frac{2}{3}\frac{cloth}{wine}$ , then it would still be lower than Foreign's opportunity cost of producing wine, but equal to Home's. So at a relative price of $\frac{2}{3}\frac{cloth}{wine}$ Foreign would not produce any wine (it would only produce cloth), while Home could produce either good.

And if the relative price of wine were $\frac{5}{4}\frac{cloth}{wine}$ , then it would still be higher than Home's opportunity cost of producing wine, but equal to Foreign's. So at a relative price of $\frac{5}{4}\frac{cloth}{wine}$ Home would only produce wine (zero cloth), while Foreign could produce either good.

And as shown in the graph above, the equilibrium relative quantity is: $\frac{3}{2}\frac{wine}{cloth}$ and the equilibrium relative price is: $\frac{2}{3}\frac{cloth}{wine}$ .

- Are Home and Foreign better off with free trade? Discuss.

**ANSWER:** Because the world equilibrium relative price of wine
would equal Home's opportunity cost of producing wine,
Home would neither gain nor lose from trade with Foreign.

But Foreign would gain from specializing in the production of cloth and trading for wine.

The new equilibrium relative price of $\frac{2}{3}\frac{cloth}{wine}$ is lower than Foreign's opportunity cost of $\frac{5}{4}\frac{cloth}{wine}$ , so specializing in cloth would enable Foreign to potentially consume more cloth and more wine than it did in autarky.

In the example here, Cobb-Douglas utility implies zero cross-price effects, so Foreign's consumption of cloth would remain unchanged at 12.5 units, but its consumption of wine would increase from 10 to 18.75 units.

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