Changes in Income;
Cost Minimization
Christopher Makler
Stanford University Department of Economics
Econ 50: Lecture 15
Today's Agenda
- Offer Curves
- Quasilinear Utility Maximization
- Cost Minimization
- Indirect Utility and Expenditure Functions
We might not make it...anything we don't get to we'll do on Friday!
x_1^*(p_1,p_2,m)\ \
Three Relationships
...its own price changes?
Movement along the demand curve
...the price of another good changes?
Complements
Substitutes
Independent Goods
How does the quantity demanded of a good change when...
...income changes?
Normal goods
Inferior goods
(possible) shift of the demand curve
Offer curves
- A parametric plot, not a functional relationship
- Show the bundles consumed in
good 1 - good 2 space (same as indifference curves and budget lines) - Can vary price to see if goods are complements/substitutes,
or income to see normal/inferior
x_1
x_1
x_2
p_1
DEMAND CURVE FOR GOOD 1
BL_{p_1 = 2}
BL_{p_1 = 3}
BL_{p_1 = 4}
2
3
4
"Good 1 - Good 2 Space"
"Quantity-Price Space for Good 1"
BL
PRICE OFFER CURVE
x_1
x_2
x_1
x_2
Complements: \(p_2 \uparrow \Rightarrow x_1^* \downarrow\)
What happens to the quantity of good 1 demanded when the price of good 2 increases?
Substitutes: \(p_2 \uparrow \Rightarrow x_1^* \uparrow\)
COMPLEMENTS:
UPWARD-SLOPING
PRICE OFFER CURVE
SUBSTITUTES:
DOWNWARD-SLOPING
PRICE OFFER CURVE
The Income Offer Curve
The Income Offer Curve
connects all the points a consumer would choose for different levels of income, holding the prices of the two goods constant.
x_1
x_2
x_1
x_2
Good 1 normal: \(m \uparrow \Rightarrow x_1^* \uparrow\)
What happens to the quantity of good 1 demanded when the income increases?
Good 1 inferior: \(m \uparrow \Rightarrow x_1^* \downarrow\)
BOTH NORMAL GOODS:
UPWARD-SLOPING
INCOME OFFER CURVE
ONE GOOD INFERIOR:
DOWNWARD-SLOPING
INCOME OFFER CURVE
How to Plot an Income Offer Curve
- Think about the "rule" that you plug into the budget line: e.g. tangency condition, ridge condition, "buy only good 1," "buy only good 1 if income is below a certain threshold," etc.
- That rule describes the income offer curve.
Utility Maximization
\text{Objective function: } x_1x_2
\text{Constraint: }p_1x_1 + p_2x_2 = m
Utility Maximization
\text{Objective function: } x_1x_2
\text{Constraint: }p_1x_1 + p_2x_2 = m
Plug tangency condition
into constraint:
\displaystyle{\text{Tangency condition: } x_2 = {p_1 \over p_2}x_1}
p_1x_1 + p_2 \left[ {p_1 \over p_2} x_1 \right] = m
p_1x_1 + p_1x_1 = m
x_1^*(p_1,p_2,m) = {m \over 2p_1}
Plug \(x_1^*\) back into tangency condition:
x_2^*(p_1,p_2,m) = {p_1 \over p_2} \left [{m \over 2p_1}\right] = {m \over 2p_2}
The Tangency Condition
In this case, the IOC is the tangency condition.
Quasilinear Optimization
Quasilinear Optimization
\text{Objective function: } 80 \ln x_1 + x_2
\text{Constraint: }p_1x_1 + x_2 = m
Plug tangency condition
into constraint:
\displaystyle{\text{Tangency condition: } x_1 = {80 \over p_1}}
p_1\left[ {80 \over p_1} \right] + x_2 = m
80 + x_2 = m
x_2 = m - 80
What does Lagrange find?
What is the optimal bundle?
\displaystyle{\text{Tangency condition: } x_1 = {80 \over p_1}}
\text{Lagrange finds: }x_2 = m - 80
\begin{aligned}
x_1^\star(p_1,m) &= \begin{cases}
{80 \over p_1} & \text{ if }m \ge 80\\ \\
{m \over p_1} & \text{ if }m \le 80
\end{cases}\\ \\
x_2^\star(p_1,m) &= \begin{cases}
m - 80 & \text{ if }m \ge 80\\ \\
0 & \text{ if }m \le 80
\end{cases}
\end{aligned}
Actual demand functions:
The IOC for a quasilinear utility function follows the rule for optimization: the tangency condition when it works, corner if not.
What's Going on?
- Each point along the IOC is the optimal bundle for some budget line, defined for its level of income.
- We then plug the IOC condition into the budget line to find the optimal bundle for a particular level of income.
Cost Minimization
Utility Maximization
Cost Minimization
\max \ u(x_1,x_2)
\min \ p_1x_1 + p_2x_2
\text{s.t. }p_1x_1 + p_2x_2 = m
\text{s.t. }u(x_1,x_2) = U
Solution functions:
"Ordinary" Demand functions
x_1^*(p_1,p_2,m)
x_2^*(p_1,p_2,m)
Solution functions:
"Compensated" Demand functions
x_1^c(p_1,p_2,U)
x_2^c(p_1,p_2,U)
Utility Maximization
\text{Objective function: } x_1x_2
\text{Constraint: }p_1x_1 + p_2x_2 = m
Utility Maximization
\text{Objective function: } x_1x_2
\text{Constraint: }p_1x_1 + p_2x_2 = m
Plug tangency condition
into constraint:
\displaystyle{\text{Tangency condition: } x_2 = {p_1 \over p_2}x_1}
p_1x_1 + p_2 \left[ {p_1 \over p_2} x_1 \right] = m
p_1x_1 + p_1x_1 = m
x_1^*(p_1,p_2,m) = {m \over 2p_1}
Plug \(x_1^*\) back into tangency condition:
x_2^*(p_1,p_2,m) = {p_1 \over p_2} \left [{m \over 2p_1}\right] = {m \over 2p_2}
Marshallian (ordinary) demand functions
Cost Minimization
\text{Objective function: } p_1x_1 + p_2x_2
\text{Constraint: }x_1x_2 = U
Cost Minimization
\text{Objective function: } p_1x_1 + p_2x_2
\text{Constraint: }x_1x_2 = U
Plug tangency condition
into constraint:
\displaystyle{\text{Tangency condition: } x_2 = {p_1 \over p_2}x_1}
x_1 \left[ {p_1 \over p_2} x_1 \right] = U
{p_1 \over p_2} \times x_1^2 = U
x_1^c(p_1,p_2,U) = \sqrt{p_2U \over p_1}
Plug \(x_1^*\) back into tangency condition:
x_2^*(p_1,p_2,U) = {p_1 \over p_2} \left [\sqrt{p_2U \over p_1}\right] = \sqrt{p_1U \over p_2}
Hicksian (compensated) demand functions
Utility Maximization
Cost Minimization
\text{Objective function: } x_1x_2
\text{Objective function: } p_1x_1 + p_2x_2
\text{Constraint: }p_1x_1 + p_2x_2 = m
\text{Constraint: }x_1x_2 = U
\displaystyle{\text{Tangency condition: } x_2 = {p_1 \over p_2}x_1}
Solution functions:
"Ordinary" Demand functions
x_1^*(p_1,p_2,m) = {m \over 2p_1}
x_2^*(p_1,p_2,m) = {m \over 2p_2}
Solution functions:
"Compensated" Demand functions
x_1^c(p_1,p_2,U) = \sqrt{p_2U \over p_1}
x_2^c(p_1,p_2,U) = \sqrt{p_1U \over p_2}
Same tangency condition, different constraints
Utility Maximization, Cost Minimization, and the IOC
The IOC represents all
the utility-maximizing bundles
for various levels of income.
It also represents all
the cost-minimizing bundles
for various levels of utility
For a given price ratio \(p_1/p_2\):
(x_1^*,x_2^*): \text{intersection of IOC, BL}
(x_1^c,x_2^c): \text{intersection of IOC, IC}
Indirect Utility and Expenditure Functions: The Relationship between Utility and Money
Link to PowerPoint (start on slide 7)
Summary
To draw the IOC, we hold prices constant and vary income.
A change in income is represented by a movement along the IOC.
A change in prices is represented by a (possible) shift of the IOC
toward the good which is now relatively cheaper
(away from the good which is relatively more expensive)
Utility Maximization: intersection of the IOC and a budget line.
Cost Minimization: intersection of the IOC and an indifference curve.
Econ 50 | Lecture 15
By Chris Makler
Econ 50 | Lecture 15
Income Offer Curves; Cost Minimization
