Budget Constraints and the Price Ratio

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 5

Plus office hours at 3-5pm in Econ 151!

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ATTENDANCE QUESTION

FRIDAY

THURSDAY

So much free boba at Landau!!!

Econ Major welcome back event

4-6pm

Free boba for the first 40 students!

Agenda for Today

Part I: Simple Budget Constraints

Part II: More Complexity, More Realism

Mathematics of the budget constraint
Geometry of a budget line
Interpreting the price ratio as the opportunity cost of good 1

Trading from an endowment
Kinked constraints

Part I: Simple
Budget Lines

Choice space:
all possible options

Feasible set:
all options available to you

Optimal choice:
Your best choice(s) of the ones available to you

Constrained Optimization

Choice Space
(all colleges plus alternatives)

Feasible Set
(colleges you got into)

Your optimal choice!

Preferences

Preferences describe how the agent ranks all options in the choice space.

For example, we'll assume that you could rank all possible colleges
(and other options for what to do after high school) based upon your preferences.

Preference Ranking

Found a startup

Harvard

Stanford

Play Xbox in parents' basement

Cal

Choice space

Feasible set

Optimal
choice!

Found a startup

Stanford

Cal

Harvard

Play XBox in parents' basement

Optimal choice is the highest-ranking option in the feasible set.

One type of feasible set: the budget set

Prices and Expenditure

Suppose each good has a constant price
(so every unit of the good costs the same)

p_1 = \text{price of good 1}

p_2 = \text{price of good 2}

x_1 = \text{quantity of good 1}

x_2 = \text{quantity of good 2}

p_1x_1 = \text{amount spent on good 1}

p_2x_2 = \text{amount spent on good 2}

p_1x_1 + p_2x_2 = \text{cost of buying bundle }X

Affordability

Suppose you have a given income $m$
to spend on goods 1 and 2.

Then bundle $X = (x_1,x_2)$ is affordable if

p_1x_1 + p_2x_2 \le m

The feasible set, or budget set, is the set of all affordable bundles.

Example: suppose you have $m = \$24$ to spend on two goods.

Good 1 costs $p_1 = \$4$ per unit.

Good 2 costs $p_2 = \$2$ per unit.

Is the bundle (2,4) affordable (in your budget set)? What about the bundle (4,6)?

Draw your budget set.

How would it change if the price of good 2 rose to $p_2' = \$6$ per unit?

How would it change if your income rose to $m' = \$32$?

m = \text{money income}

p_1 = \text{price of good 1}

p_2 = \text{price of good 2}

\text{horizontal intercept} = \frac{m}{p_1}

\text{vertical intercept} = \frac{m}{p_2}

\text{slope of budget line} = -\frac{p_1}{p_2}

Budget Line

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Holding income and the price of good 2 constant, an increase in the price of good 1 will cause the budget line to become:

steeper‎

flatter

‎it depends on the‎level of income

‎it depends on the‎ price of good 2‎

Interpreting the Slope of the Budget Line

Example:

Apples cost 50 cents each

Bananas cost 25 cents each

Slope of the budget line represents the opportunity cost of consuming good 1, as dictated by market prices.

-\frac{p_1}{p_2} = -2 \text{ bananas per apple}

In other words: it is the amount of good 2 the market requires you to give up in order to get another unit of good 1.

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If apples (good 1) cost $0.80 each,
and bananas (good 2) cost $0.20 each, what is the magnitude (absolute value) of the slope of the budget line?‎

Composite Goods

You have $100 in your pocket.

You see a cart selling apples (good 1) for $2 per pound.

Plot your budget line.
What is "good 2"?
What does the bundle (10,80) signify?
What is the slope of the budget line, and what are its units?

Part II: More Complicated (and Realistic) Budget Constraints

Trading from an Endowment

Good 1

Good 2

e_2

e_1

Note: lots of different notation for the endowment bundle!

Varian uses $\omega$, some other people use $x_1^E$

x_2

x_1

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

You start out with some endowment E

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

\Delta x_1

\Delta x_2

\Delta x_1 = e_1 - x_1

\Delta x_2 = x_2 - e_2

Buying and Selling

Good 1

Good 2

e_2

e_1

x_2

x_1

If you can't find someone to trade good 1 for good 2 directly, you could sell 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?

p_1 \Delta x_1

p_2 \Delta x_2

\Delta x_1

\Delta x_2

\Delta x_1 = e_1 - x_1

\Delta x_2 = x_2 - e_2

p_1 (e_1 - x_1)

p_2 (x_2 - e_2)

Buying and Selling

Good 1

Good 2

e_2

e_1

x_2

x_1

\Delta x_1

\Delta x_2

\Delta x_1 = e_1 - x_1

\Delta x_2 = x_2 - e_2

p_1 (e_1 - x_1)

p_2 (x_2 - e_2)

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

p_2x_2 - p_2e_2 = p_1e_1 - p_1x_1

p_1x_1 + p_2x_2 = p_1e_1 +p_2e_2

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

e_2

e_1

p_1x_1 + p_2x_2 = p_1e_1 +p_2e_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 $x_2 = 0$:

x_2 = e_2 + {p_1e_1 \over p_2}

x_1 = e_1 + {p_2e_2 \over p_1}

Endowment Budget Line

Good 1

Good 2

e_2

e_1

p_1x_1 + p_2x_2 = p_1e_1 +p_2e_2

e_2 + {p_1e_1 \over p_2}

e_1 + {p_2e_2 \over p_1}

Liquidation value of your endowment

\hat m

Divide both sides by $p_2$:

{p_1 \over p_2}x_1 + x_2 = {p_1 \over p_2} e_1 + e_2

{\hat m \over p_2} =

Divide both sides by $p_1$:

x_1 + {p_2 \over p_1}x_2 = e_1 + {p_2 \over p_1}e_2

{\hat m \over 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

p_1x_1+p_2x_2=p_1e_1+p_2e_2

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.

{p_1 \over p_2}x_1 + x_2 = {p_1 \over p_2} e_1 + e_2

x_1 + {p_2 \over p_1}x_2 = e_1 + {p_2 \over p_1}e_2

Effect of a Change in Prices

What happens if the price of good 1 doubles?

What happens if both prices double?

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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.

Kinked Constraints

Remember: the slope of the budget line is the price ratio (opportunity cost of good 1)
Straight budget line: every unit of every good costs the same amount
Kinked budget constraint: prices of at least one good are different over different portions of the constraint

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.

40 \text{ tickets} \times \$25/\text{ticket} = \$1000

\$1000 + \$1200 = \$2200

1200

2200

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

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

\$1200 \div \$60/\text{ticket} = 20\text{ tickets}

40\text{ tickets} + 20\text{ tickets} = 60\text{ tickets}

You can sell tickets for $25 each,
or buy additional tickets for $60 each.

Tiered Pricing for Electricity Rates

kWh

Money

Suppose you start out with $m$ dollars.
What does your budget constraint look like?

300

$m$

Slope = $0.10/kWh

Slope = $0.20/kWh

Palo Alto has a tiered pricing policy for electricity rates. While it's changed in recent years, it used to be the something like the following:

First 300 kWh per month: $0.10/kWh
After that: $0.20/kWh

$m-30$

Let $x_1$ be kWh of electricity per month, and $x_2$ be money spent on other things.

Should you give cash or a gift card?

What do you think about this "proof"?

Are you only ever going to give cash from now on?

Summary and Next Steps

Budget constraints divide the choice space into affordable and unaffordable sets
The slope of the budget line at any point is the pricr ratio $p_1/p_2$, which represents the opportunity cost of good 1, in terms of good 2.
A simple "budget line" with constant prices $p_1$ and $p_2$, and exogenous income $m$, has the equation $p_1x_1 + p_2x_2 = m$
And endowment budget line with constant prices $p_1$ and $p_2$, and an endowment of $(e_1,e_2)$, has the equation $p_1x_1 + p_2x_2 = p_1e_1 + p_2e_2$
With more complex constraints, have to think about how the price ratio differs over the constraint, and what the coordinates of relevant points are
Next: find the most preferred point on the constraint
In section: learn about the Lagrange method for constrained optimization

Budget Constraints and the Price Ratio

Agenda for Today

Part I: Simple Budget Lines

Constrained Optimization

Preferences

One type of feasible set: the budget set

Prices and Expenditure

Affordability

The feasible set, or budget set, is the set of all affordable bundles.

Budget Line

Interpreting the Slope of the Budget Line

Composite Goods

Part II: More Complicated (and Realistic) Budget Constraints

Trading from an Endowment

Buying and Selling

Buying and Selling

Endowment Budget Line

Endowment Budget Line

Endowment Budget Line

Effect of a Change in Prices

Kinked Constraints

Different Prices for Buying and Selling

Tiered Pricing for Electricity Rates

Should you give cash or a gift card?

Summary and Next Steps

Part I: Simple
Budget Lines