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.