Demand Functions and Demand Curves

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 8

Today's Agenda

Part 1: Brief Overview

Part 2: Worked Examples

Review of what demand is

Demand functions

Demand curves

Complements and Substitutes

Normal and Inferior Goods

Cobb-Douglas

Perfect Complements

Perfect Substitutes

Quadratic Quasilinear

Section: Three goods!

 

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
  • The demand curve represents the marginal benefit of an additional unit,
    or alternatively the marginal willingness to pay for another unit
  • 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

Demand Curve for Good 1

\text{Fix }p_2\text{ and }m
\text{Plot }(x_1^*(p_1),p_1)

Demand Functions

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

"The demand curve shows the quantity demanded of a good at different prices"

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

"The demand curve represents the marginal benefit of an additional unit,
or alternatively the marginal willingness to pay for another unit"

\mathcal{L}(x_1,x_2,\lambda)=
u(x_1,x_2)+
(m - p_1x_1 - p_2x_2)
\lambda
\frac{\partial \mathcal{L}}{\partial x_1} = \frac{\partial u}{\partial x_1} - \lambda p_1

Let's look at the FOC with respect to good 1:

= 0 \Rightarrow \lambda = \frac{MU_1}{p_1}

Solve for \(p_1\):

\lambda = \text{bang for your buck, in }\frac{\text{utils}}{\$}
  • 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

Own-Price Elasticity

What is the effect of a 1% change
in the price of good 1 \((p_1)\) on the quantity demanded of good 1 \((x_1^*)\)?

x_1^*(p_1,p_2,m)

no change

perfectly inelastic

less than 1%

inelastic

exactly 1%

unit elastic

more than 1%

elastic

  1. How does \(x_1^*\) change with \(p_1\)?
    • ​​Own-price elasticity
    • Elastic vs. inelastic
  2. How does \(x_1^*\) change with \(p_2\)?
    • ​​Cross-price elasticity
    • Complements vs. substitutes
  3. How does \(x_1^*\) change with \(m\)?
    • ​​Income elasticity
    • Normal vs. inferior goods
x_1^*(p_1,p_2,m)

Three Relationships

Cross-Price Elasticity

What is the effect of an increase
in the price of good 2 \((p_2)\) on the quantity demanded of good 1 \((x_1^*)\)?

x_1^*(p_1,p_2,m)

no change

independent

decrease

complements

increase

substitutes

  1. How does \(x_1^*\) change with \(p_1\)?
    • ​​Own-price elasticity
    • Elastic vs. inelastic
  2. How does \(x_1^*\) change with \(p_2\)?
    • ​​Cross-price elasticity
    • Complements vs. substitutes
  3. How does \(x_1^*\) change with \(m\)?
    • ​​Income elasticity
    • Normal vs. inferior goods
x_1^*(p_1,p_2,m)

Three Relationships

Substitutes

Complements

When the price of one good goes up, demand for the other increases.

When the price of one good goes up, demand for the other decreases.

  1. How does \(x_1^*\) change with \(p_1\)?
    • ​​Own-price elasticity
    • Elastic vs. inelastic
  2. How does \(x_1^*\) change with \(p_2\)?
    • ​​Cross-price elasticity
    • Complements vs. substitutes
  3. How does \(x_1^*\) change with \(m\)?
    • ​​Income elasticity
    • Normal vs. inferior goods
x_1^*(p_1,p_2,m)

Three Relationships

Income Elasticity

What is the effect of an increase
in income \((m)\) on the quantity demanded of good 1 \((x_1^*)\)?

x_1^*(p_1,p_2,m)

decrease

good 1 is inferior

increase

good 1 is normal

  1. How does \(x_1^*\) change with \(p_1\)?
    • ​​Own-price elasticity
    • Elastic vs. inelastic
  2. How does \(x_1^*\) change with \(p_2\)?
    • ​​Cross-price elasticity
    • Complements vs. substitutes
  3. How does \(x_1^*\) change with \(m\)?
    • ​​Income elasticity
    • Normal vs. inferior goods
x_1^*(p_1,p_2,m)

Three Relationships

Normal Goods

Inferior Goods

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

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

  1. How does \(x_1^*\) change with \(p_1\)?
    • ​​Own-price elasticity
    • Elastic vs. inelastic
  2. How does \(x_1^*\) change with \(p_2\)?
    • ​​Cross-price elasticity
    • Complements vs. substitutes
  3. How does \(x_1^*\) change with \(m\)?
    • ​​Income elasticity
    • Normal vs. inferior goods
x_1^*(p_1,p_2,m)

Three Relationships

Things to Think About

Think about how the behavior described by the demand function translates into the overall shape of the demand curve:

  • Are there discontinuities/cutoff prices where behavior changes?
  • What happens as the price gets really high, or approaches zero?
  • What fraction of income is being spent on this good? 

The reason we use different utility functions is because people's relationship with prices depends on the nature of their preferences.

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}

Econ 50 | Fall 2021 | 8 | Demand

By Chris Makler

Econ 50 | Fall 2021 | 8 | Demand

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