Changes in Income;
Cost Minimization

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 13

Remember what you learned about demand and demand curves in Econ 1 / high school:

  • The demand curve shows the quantity demanded of a good at different prices
  • A change in the price of a good results in a movement along its demand curve
  • A change in income or the price of other goods results in a shift of the demand curve
    • If two goods are substitutes, an increase in the price of one will increase the demand for the other (shift the demand curve to the right).
    • If two goods are complements, an increase in the price of one will decrease the demand for the other (shift the demand curve to the left).
    • If a good is a normal good, an increase in income will increase demand for the good
    • If a good is an inferior good, an increase in income will decrease demand the good
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

Giffen goods

(possible) shift of the demand curve

x_1^*(p_1,p_2,m)\ \

Three Relationships

How does the quantity demanded of a good change when...

...income changes?

Normal Goods

Inferior Goods

When your income goes up,
demand for the good increases.

When your income goes up,
demand for the good decreases.

The income offer curve shows how the optimal bundle changes in good 1-good 2 space as income changes.

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.

The Tangency Condition

What happens when the price of a good increases or decreases?

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

PRICE OFFER CURVE

pollev.com/chrismakler

The "rule" for Cobb-Douglas is that you spend a certain fraction of your income on each good, regardless of prices or income.

 

What does this make the two goods?

 

Complements

Substitutes

Normal

Inferior

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.

The Tangency Condition

What happens when the price of a good increases or decreases?

The Tangency Condition when Lagrange sometimes works

What happens when income decreases?

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.

Worked Examples

Link to PowerPoint deck

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

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

Plug tangency condition back into constraint:

\displaystyle{\text{Tangency condition: } x_2 = {p_1 \over p_2}x_1}

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}

Utility Maximization, Cost Minimization,
and the IOC

When Lagrange Doesn't Work: Perfect Complements

\text{Objective function: } p_1x_1 + p_2x_2
\text{Constraint: }\min\left\{\frac{x_1}{2},\frac{x_2}{3}\right\} = U

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.