# Demand Curves

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 9

### Last Week: Math

Demand curves:
plot demand for one good
in price-quantity space

Offer curves ("consumption loci"):
plot bundles chosen
in good 1-good 2 space

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

# 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

# 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

### 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.

# 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

### Inferior Goods

demand for the good increases.

demand for the good decreases.

# Demand Curve for Good 1

Plot relationship
between $$p_1$$ and $$x_1^*$$,
holding $$p_2$$ and $$m$$ constant
(ceteris paribus)

Change in $$p_1$$: movement along
the demand curve for good 1

x_1^*(p_1,p_2,m)

Change in $$p_2$$ or $$m$$: shift of
the demand curve for good 1

# Procedure

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?

Choose prices strategically and plot points.

# 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}

Leisure (R)

Consumption (C)

24
24 - L
M
M + \Delta C

You trade $$L$$ hours of labor for some amount of consumption, $$\Delta C$$.

You start with 24 hours of leisure and $$M$$ dollars.

L
\Delta C

You end up consuming $$R = 24 - L$$ hours of leisure,
and $$C = M + \Delta C$$ dollars worth of consumption.

R

# Last Time: MRS, MRT, and the “Gravitational Pull" towards Optimality

Better to produce
more good 1
and less good 2.

MRS
>
MRT
MRS
<
MRT

## “Gravitational Pull" Towards Optimality

Better to produce
more good 2
and less good 1.

These forces are always true.

In certain circumstances, optimality occurs where MRS = MRT.

MRS
>
MRT
MRS
<
MRT

# Constrained Optimization with One Variable

Think about maximizing each of these functions subject to the constraint $$0 \le x \le 10$$.

### Pause the video! Try to do these on your own.

Plot the graph on that interval; then find and plot the derivative $$f'(x)$$ on that same interval.

Which functions have a maximum at the point where $$f'(x) = 0$$? Why?

f(x) = 5 + 4x - x^2
f(x) = 10 - |2-x|
f(x) = 9 - (x-11)^2
f(x) = 1 + \tfrac{1}{5}(x-5)^2
f(x) = 10 - x
f(x) = 3

Sufficient conditions for an interior optimum characterized by $$f'(x)=0$$ with constraint $$x \in [0,10]$$

• $$f'(0) > 0$$
• $$f'(10) < 0$$
• $$f'(x)$$ continuous and strictly decreasing on $$[0,10]$$
f(x)
f'(x)
x
x
10
0
10
MRS
>
MRT
MRS
<
MRT

# Corner Solutions

## Corner Solution:

Optimal bundle contains
strictly positive quantities of both goods

Optimal bundle contains zero of one good
(spend all resources on the other)

If only consume good 1: $$MRS \ge MRT$$ at optimum

If only consume good 2: $$MRS \le MRT$$ at optimum

# Kinks (Discontinuities)

## Kinked Constraint:

Discontinuities in the MRS
(e.g. Perfect Complements utility function)

Discontinuities in the MRT
(e.g. homework question with two factories)

# Monotonicity and Convexity

If preferences are nonmonotonic,
you might be satisfied consuming something within the interior of your feasible set.

FISH

COCONUTS

PPF

If preferences are nonconvex,
the tangency condition might find a minimum rather than a maximum.

FISH

COCONUTS

PPF

## Conditions for Calculus to Work

avoids a satiation point within the constraint

At the left corner of the constraint, $$MRS > MRT$$

avoids a corner solution when $$x_1 = 0$$

Monotonicity (more is better)

At the right corner of the constraint, $$MRS < MRT$$

avoids a corner solution when $$x_2 = 0$$

MRS and MRT are continuous as you move along the constraint

avoids a solution at a kink

ensures FOCs find a maximum, not a minimum

Convexity (variety is better)

## Unit 1: Scarcity and Choice

Lectures 2 & 3: Derive feasible set (PPF)
from production functions and resource constraints.

Lectures 5 & 6: Solve constrained optimization problem:
find most preferred choice in the feasible set.

Lecture 4: Describe preferences using utility functions.

## Unit 2: Consumer Theory

Lecture 7: Derive feasible set (budget set)
as a function of prices and income.

Lectures 9-12: Analyze the comparative statics of how a consumer responds to changes in prices and income.

Lecture 8: Derive the demand function: the optimal bundle as a function of prices and income.

By Chris Makler

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