Download Choice with Certainty Part II File

Consumer Choice Under Certainty
Part
II
Agenda:
VI. Review of the key consumption relationship
VII. Our FIRST numerical example!
VII. Utility functions – play dough economics
VIII. So where DO demand curves come from?
IX. Demand vs. quantity demanded
X.
Whew…That’s it!
Putting it Together: The Optimal Bundle!
Why not here?
Tangent:
where a line touches a curve at one point.
The rate of change of the curve =
the rate of change of the line
Why not here?
Marginal rate of substitution
Y
px

X
py
Slope of the budget line
Check your understanding: Why is it Px/Py and not Py/Px?
Putting it Together: The Optimal Bundle Numerical Example
You love coffee and bagels for breakfast.
Currently 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.
1. What is the marginal rate of substitution?
A: 2 coffee/1 bagel
2. What is the budget constraint?
A: 1C + 2B = 6
Test 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?
px 2C $2
Y



X
p y 1B $1
1C + 2B = 6
Step #3: Use what you know to get where you need to go!
This is just math: two equations & two unknowns!
1. Cross multiply & simplify to express C
in terms of B.
2C $2

1B $1
2C  2 B
CB
2. Substitute into the Budget Constraint!
$1( B)  $2 B  $6
$3B  $6
B2
3. Substitute the answer for B into the Budget Constraint.
2 suggestions:
1. Use $ for prices so you keep your units
clear!
2. Check your answer by substituting your
answer into the budget constraint!
$1C  $2(2)  $6
$1C  $2
C2
TEST YOURSELF!
What if you were ONLY willing to have
coffee and bagels for breakfast at a ration
of one (1) bagel to two (2) cups of
coffee? In other words, coffee and
bagels are perfect complements to you!
Hint: The MRS must stay constant at 2.
What do these indifference curves look like?
Answer: 1.5 Bagels and 3 cups of coffee
NOTE: optimal bundle MUST have a ratio of 2:1!
Utility Functions
A function is a mapping from X to one Y
U  f (S , F )
Functional forms
U  SF
U SF
U  S F 
Utility is ordinal, not cardinal!
We can draw indifference
curves from a utility function
Indifference Curves & Utility Functions
Indifference curves
Utility curves
Y
px

X
py
Different notation
Same meaning!
U x px

U y p y
A little math…a lot of different interpretations!
U x px

U y p y
The ratio of marginal utilities equals the
price ratio, or
Marginal benefit = Marginal cost
U x U y

px
py
The marginal utility per dollar for
each good must be equal
Y
px

X
py
Y
X

px
py
The marginal rate of substitution
equals the slope of the budget line
The opportunity cost per dollar for
each good must be equal
Putting it Together Again: Optimal Bundle Example revisited
You like coffee and bagels with the
marginal utility of coffee = 1/2 bagels and
the marginal utility of bagels = 2 coffee.
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. What
is your optimal bundle?
What is the key relationship that defines the optimal bundle?
MU x U x px
2C
2
1



 C  B
MU y U y p y
1/ 2B 1
2
1(1/ 2) B  2B  6
B  2.4
1C  2(2.4)  6
U x U
C  1.2

How can you check that this is correct?
px
y
py
So Where DO Demand Curves Come From?
composite
good
Indifference Curves
Budget Constraints
Price-consumption curve
Price
KEY:
Pay attention
to what is on
the X and Y
axes!
What is the price of shelter at this budget constraint?
quantity
What is Elasticity?
Price
Q P0
P 
Point elasticity
P Q0
Arc elasticity
quantit
y
Q2  Q1
Q2  Q1  2

A 
P2  P1
 P2  P1  2
Change
Mid-point
Key things to remember about elasticity:
1. Elasticity is NOT CONSTANT along the demand curve.
2. If quantity does not change, demand is perfectly ELASTIC.
3. If price does not change, demand is perfectly INELASTIC.
TEST YOURSELF: 1)Compute the arc elasticities between each of the
points in the graph above. 2) Draw demand curves for points 2 & 3.
What would cause a shift in Demand?
A change in income
A change in preferences (shape of the indifference curves)
A change in population (aggregate demand)
Check your understanding: Write some examples in your notes of shifts in demand.
Try to draw the graphs too!
Yikes! That was a lot of stuff!
The key relationship:
Y
px

X
py
U x px

U y p y
The intuition:
We consume to the point where the marginal cost (price) =
the marginal benefit
PRACTICE!!