Putting it Together: The Optimal Bundle!

x

y

Y p

X p

Tangent: where a line touches a curve at one point.

The rate of change of the curve = the rate of change of the line

Check your understanding: Why is it Px/Py and not Py/Px?

Why not here?

Why not here?

Marginal rate of substitution Slope of the budget line

Putting it Together: The Optimal Bundle Numerical ExampleYou love coffee and bagels for breakfast. You’re willing to give up two cups of coffee for one bagel (with cream cheese). The price of coffee is $1 and the price of a bagel is $2. You have $6 to spend on breakfast and you can buy fractions. Please put coffee on the Y-axis.

1. What is the marginal rate of substitution?A: 2 coffee/1 bagel

2. What is the budget constraint?

A: $1C + $2B = $6Test yourself: draw the budget line – what are the X and Y intercepts and what do they mean?

3. What your optimal breakfast?

REMEMBER the Process of Problem Solving!

Step #1: Where do you need to go?What does the answer look like?A QUANTITY of Coffee AND a QUANTITY of Bagels (B,C)

Step #2: What do you know?

2 $2

1 $1x

y

pY C

X p B

$1C + $2B = 6Step #3: Use what you know to get where you need to go!

This is just math: two equations & two unknowns!

$1( ) $2 $6

$3 $6

2

B B

B

B

2 $2

1 $1$2 $2

C

BC B

C B

1. Cross multiply & simplify to express C in terms of B.

2. Substitute into the Budget Constraint!

3. Substitute the answer for B into the Budget Constraint.

$1 $2(2) $6

$1 $2

2

C

C

C

2 suggestions: 1. Use $ for prices so you keep your units

clear!2. Check your answer by substituting your

answer into the budget constraint!

What if you were ONLY willing to have coffee and bagels for breakfast at a ratio of one (1) bagel to two (2) cups of coffee? In other words, coffee and bagels are perfect complements to you! What is your optimal breakfast?

Hint: The MRS must stay constant at 2 no matter what the prices are!

What do these indifference curves look like?

$1(2 ) $2 $6

$4 $6

1.5

B B

B

B

MRS is the rate at which we substitute Y (coffee) for X (bagels)

$1( ) $2(1.5) $6

$1 $3 $6

$1 $3

3

C

C

C

C

Your answer MUST have a ratio of 2 coffee per bagel!