Demand Functions and Demand Curves

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 13

Welcome back alumni!

Ruru Hoong

Class of 2019

Worked for 1 year at Boston Consulting Group in London

Currently a 5th-year Ph.D. student in Economics at Harvard

Research interests are in...

TA for Econ 50

Today's Agenda

Part 1:
General Theory

Part 2:
Functional Forms and Behavior

  • Cobb-Douglas
  • Perfect Substitutes
  • Quasilinear
  • Demand Functions
  • Demand Curves

Last Class: What is the optimal bundle for a given budget line?

Today: What happens to the optimal bundle when prices/income change?

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BL1

We will be solving for the optimal bundle
as a function of income and prices:

The solutions to this problem will be called the demand functions. We have to think about how the optimal bundle will change when \(p_1,p_2,m\) change.

x_1^*(p_1,p_2,m)
x_2^*(p_1,p_2,m)

BL2

Specific Prices & Income

General Prices & Income

\text{Constraint: }2 x_1 + x_2 = 12

Plug tangency condition back into constraint:

Tangency Condition: \(MRS = p_1/p_2\)

\text{Constraint: }p_1x_1 + p_2x_2 = m
\text{Objective function: } x_1^{1 \over 2}x_2^{1 \over 2}
MRS(x_1,x_2) = {x_2 \over x_1}
{x_2 \over x_1}
=
{2 \over 1}
\Rightarrow x_2 = 2x_1
{x_2 \over x_1}
=
{p_1 \over p_2}
\Rightarrow x_2 = {p_1 \over p_2}x_1
2x_1 + 2x_1 = 12
x_1^* = 3
p_1x_1 + p_2\left[{p_1 \over p_2}x_1\right] = m
x_1^*(p_1,p_2,m) = {m \over 2p_1}
4x_1 = 12
2p_1x_1 = m
x_2^* = 2x_1^* = 6
x_2^*(p_1,p_2,m) = {m \over 2p_2}

Specific Prices & Income

General Prices & Income

\text{Constraint: }2 x_1 + x_2 = 12
\text{Constraint: }p_1x_1 + p_2x_2 = m
\text{Objective function: } x_1^{1 \over 2}x_2^{1 \over 2}
MRS(x_1,x_2) = {x_2 \over x_1}
x_1^* = 3
x_1^*(p_1,p_2,m) = {m \over 2p_1}
x_2^* = 2x_1^* = 6
x_2^*(p_1,p_2,m) = {m \over 2p_2}

OPTIMAL BUNDLE

DEMAND FUNCTIONS

(optimization)

(comparative statics)

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

The Demand Function Illustrates 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

(Wednesday)

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

Three Relationships

...its own price changes?

Movement along the demand curve

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

The demand curve for a good

shows the quantity demanded of that good

as a function of its own price

holding all other factors constant

(ceteris paribus)

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

Note: Maximum Possible Quantity Demanded

\overline x_1 = {m \over p_1}

Quantity of Good 1 \((x_1)\)

Price of Good 1 \((p_1)\)

All demand curves must be in this region

Quantity bought at each price if you spent all your money on good 1

x_1 = {m \over p_1}

Worked Examples

(link to PowerPoint)