# Trading from an Endowment

Christopher Makler

Stanford University Department of Economics

Econ 51: Lecture 2

## Today's Agenda

Part 1: Exchange Optimization

Part 2: Comparative Statics

The endowment budget line

Optimal choice with a constant price

Different prices for buying and selling

Optimization at general prices

Gross demand and net demand

Net demand and net supply

Applications: Labor Supply, Intertemporal Choice, Risk and Uncertainty

# Marginal Rate of Substitution

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Suppose you were indifferent between the following two bundles:

Starting at bundle X,

you would be willing

to give up 4 bananas

to get 2 apples

Let apples be good 1, and bananas be good 2.

Starting at bundle Y,

you would be willing

to give up 2 apples

to get 4 bananas

Visually: the MRS is the magnitude of the slope

of an indifference curve

### Econ 50

Consumer starts with money,

can buy a bundle of goods.

### Econ 51

Consumer starts with a bundle,

can trade one good for another

Straight trade/barter

Sell some of one good for money,

use the money to buy the other good

# Trading from an Endowment

Good 1

Good 2

*Note: lots of different notation for the endowment bundle!*

Β

*Varian uses \(\omega\), some other people use \(x_1^E\)*

Suppose you'd like to move from that endowment to some other bundle X

You start out with some endowment E

This involves *trading*Β some of your good 1 to get some more good 2

# Buying and Selling

Good 1

Good 2

If you can't find someone to trade good 1 for good 2 directly, you could *sell*Β some of your good 1 and use the money to buy good 2.

Suppose you sell \(\Delta x_1\) of good 1 at price \(p_1\). How much money would you get?

Suppose you wanted to buy \(\Delta x_2\) of good 2 at price \(p_2\). How much would that cost?

# Buying and Selling

Good 1

Good 2

If the amount you get from selling good 1 exactly equals the amount you spend on good 2, then

monetary value of \(E\)

at market prices

monetary value of \(X\)

at market prices

(Basically: you can afford any bundle with the same monetary value as your endowment.)

# Endowment Budget Line

Good 1

Good 2

If you sell all your good 1 for \(p_1\),

how much good 2 can you consume?

If you sell all your good 2 for \(p_2\),

how much good 1 can you consume?

If \(x_1 = 0\):

If \(x_2 = 0\):

# Endowment Budget Line

Good 1

Good 2

Liquidation value of your endowment

Divide both sides by \(p_2\):

Divide both sides by \(p_1\):

In other words: the endowment budget line is just like a normal budget line,

but the amount of money you have is the liquidation value of your endowment.

# Endowment Budget Line

Divide both sides by \(p_2\):

Divide both sides by \(p_1\):

The budget line **only depends on the price ratio \({p_1 \over p_2}\),
not the individual prices.**

# Effect of a Change in Prices

What happens if the price of good 1 doubles?

What happens if **both** prices double?

pollev.com/chrismakler

Bob has an endowment of (8,8) and can buy and sell goods 1 and 2. What happens to his endowment budget line if the price of good 1 decreases? You may select more than one answer.

# Optimization

Optimization problem with money

Optimization problem with an endowment

Procedure is exactly the same - we just have a different equation for the budget constraint.

pollev.com/chrismakler

Bob has the endowment (8,8) and the utility function $$u(x_1,x_2)=x_1x_2$$If he faces prices \(p_1 = 10\) and \(p_2 = 5\), what is his optimal choice?

# Optimization: Income vs. Endowment

# Recall: The βGravitational Pull" Argument

Indifference curve is steeper

than the budget constraint

Can increase utility by

moving to the **right**

along the budget constraint

Indifference curve is flatter

than the budget constraint

Can increase utility by

moving to the **left**

along the budget constraint

Good 1

Good 2

pollev.com/chrismakler

Suppose Alison has the endowment (12,2) and the utility function $$u(x_1,x_2)=x_1x_2$$ If the price of good 2 is 6, for what price of good 1 will she be willing to sell some of her good 1?Β

# Different Prices for Buying and Selling

Tickets

Money

If you sell all your tickets,

how much money will you have?

If you spend all your money on additional tickets, how many tickets will you have?

Suppose you have 40 tickets and $1200.

1200

40

2200

Slope = \(p^{\text{sell}}\) = $25/ticket

Slope = \(p^{\text{buy}}\) = $60/ticket

You can sell tickets for $25 each,

or buy additional tickets for $60 each.

60

# Part II: Comparative Statics

# Gross Demands and Net Demands

The **total** quantity of a good

you want to consume (i.e. end up with)

at different prices.

## Gross Demand

The **transaction**Β you want to engage in

(the amount you want to buy or sell)

at different prices.

## Net Demand

Is this positive or negative?

Positive: you are a **net demander**Β of good 1.

Negative: you are a **net supplier**Β of good 1.

# Most Important Takeaways

The **endowment budget line**Β depends only on the **price ratio**, not on individual prices.

Whether you're a net demander or supplier depends on the relationship between the **price ratio**Β and the **MRS at the endowment.**

## Applications

### Labor supply

"Good 1" = time

"Good 2" = money

**Working:**Β selling time for money

### Intertemporal Choice

"Good 1" = money in the present

"Good 2" = money in the future

**Saving:**Β selling current money to

get more money in the future

**Borrowing:**Β selling future money to get more money in the present

"Goods" =

money in different

"states of the world"

**Trading:**Β betting, investing, insurance

### Uncertainty and Risk

# Application 1:

Labor Supply

# Leisure-Consumption Tradeoff

Leisure (R)

Consumption (C)

You trade \(L\) hours of labor for some amount of consumption, \(\Delta C\).Β

You start with 24 hours of leisure and \(M\) dollars.

You end up consuming \(R = 24 - L\) hours of leisure,

and \(C = M + \Delta C\) dollars worth of consumption.

# Selling Labor at a Constant Wage

Leisure (R)

Consumption (C)

You sell \(L\) hours of labor at wage rate \(w\).

You start with 24 hours of leisure and \(M\) dollars.

You earn \(\Delta C = wL\) dollars in addition to the \(M\) you had.

...and you consume \(R = 24 - L\) hours of leisure.

# Budget Line Equation

Leisure (R)

Consumption (C)

This is just an endowment budget line

# Optimal Supply of Labor

Preferences are over the two "good" things: **leisure**Β and **consumption**

We've just derived the budget constraint in terms of leisure and consumption as well:

Maximize utility as usual, with one caveat:

you can only *sell*Β your leisure time, not buy it.

# When will labor supply be zero?

Remember: you only want to sell good 1 (in this case, your time) if

#### Econ 51 | 02 | Trading from an Endowment

By Chris Makler

# Econ 51 | 02 | Trading from an Endowment

Building the Basis of Exchange Equilibrium

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