Demand Functions and Demand Curves
Christopher Makler
Stanford University Department of Economics
Econ 50: Lecture 11
Let's address the elephant in the room...
Production functions for with intermediate goods
Today's Agenda
Part 1:
General Theory
Part 2:
Functional Forms and Behavior
- Cobb-Douglas
- Perfect Complements
- Perfect Substitutes
- Quasilinear
- Demand Functions
- Demand Curves
Last Two Classes: 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
(next week)
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
(on PowerPoint)
pollev.com/chrismakler
Econ 50 | Spring 23 | Lecture 11
By Chris Makler
Econ 50 | Spring 23 | Lecture 11
Demand Functions and Demand Curves
