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!
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
DEMAND CURVE FOR GOOD 1
"Good 1 - Good 2 Space"
"Quantity-Price Space for Good 1"
PRICE OFFER CURVE
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.
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
Utility Maximization
Plug tangency condition
into constraint:
Plug \(x_1^*\) back into tangency condition:
The Tangency Condition
In this case, the IOC is the tangency condition.
Quasilinear Optimization
Quasilinear Optimization
Plug tangency condition
into constraint:
What does Lagrange find?
What is the optimal bundle?
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
Solution functions:
"Ordinary" Demand functions
Solution functions:
"Compensated" Demand functions
Utility Maximization
Utility Maximization
Plug tangency condition
into constraint:
Plug \(x_1^*\) back into tangency condition:
Marshallian (ordinary) demand functions
Cost Minimization
Cost Minimization
Plug tangency condition
into constraint:
Plug \(x_1^*\) back into tangency condition:
Hicksian (compensated) demand functions
Utility Maximization
Cost Minimization
Solution functions:
"Ordinary" Demand functions
Solution functions:
"Compensated" Demand functions
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 \(p_1/p_2\):
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
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