Responses to Price Changes: Complements and Substitutes

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 12

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

...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)

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

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^*(p_1,p_2,m)\ \

Three Relationships

...the price of another good changes?

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

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.

Independent

Demand not related

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

CES Utility

= \begin{cases}\infty & \text{ if } x_1 < x_2 \\ 0 & \text{ if } x_1 > x_2 \end{cases}
-\infty
1
0
MRS = \left(x_2 \over x_1\right)^\infty
MRS = {x_2 \over x_1}
MRS = 1
r
u(x_1,x_2) = \min\{x_1,x_2\}
u(x_1,x_2) = x_1x_2
u(x_1,x_2) = x_1 + x_2

PERFECT
SUBSTITUTES

PERFECT
COMPLEMENTS

INDEPENDENT

PERFECT
SUBSTITUTES

u(x_1,x_2) = (x_1^r+x_2^r)^{1 \over r}

Constant Elasticity of Substitution (CES) Utility

MRS = \left(x_2 \over x_1\right)^{1-r}
u(x_1,x_2) = (x_1^r+x_2^r)^{1 \over r}

Constant Elasticity of Substitution (CES) Utility

MRS = \left(x_2 \over x_1\right)^{1-r}
-\infty
1
r

COMPLEMENTS: \(r < 0\)

SUBSTITUTES: \(r > 0\)

-1
MRS = \left(x_2 \over x_1\right)^2
{1 \over 2}
MRS = \left(x_2 \over x_1\right)^{1 \over 2}
u(x_1,x_2) = (x_1^{-1}+x_2^{-1})^{-1}
u(x_1,x_2) = \left(x_1^{1 \over 2} + x_2^{1 \over 2}\right)^2
  • 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

Plotting Offer Curves

Parametric Functions

x(t) = t\sin t
y(t) = t\cos t
0 \le t \le 2\pi

How to Plot a Price Offer Curve

  • Pick a few values of the price which is being varied along the offer curve.
  • Be sure to think of extreme cases (when the price approaches zero or infinity)
  • Find the optimal bundle for each of those prices
  • Connect the dots
  • Do not try to find an equation  

Worked Examples

Link to PowerPoint deck