2. Lecture 5: Demand Curve and Elasticity
Review: preference, budget line, optimal choice
Demand: relation between choice and price
Individual demand curve
Slope of demand curve and elasticity
Individual demand V.S. market demand
Demand and substitution
Demand and changes in income
Normal V.S. inferior goods
3. Review: preferences, budget line and
optimal choice
Up to now, we have the model of consumer behavior based on two basic
components:
Preference: represented by ICs and utility function (the representation is not
unique)
Budgets: determined by income and prices (recall the shifts of budget line)
Based on the two components, we derive the optimal choice of the consumer
The optimal consumption position is where the indifference curve is tangent to the
budget line.
In this lecture, we will derive the demand of the consumer: how does the
optimal choice change when prices and income vary?
4. Bridge optimal choice and demand
Studying how a choice responds to changes in the economic environment is known as
comparative statics
“Comparative” means that we want to compare two situations: before and after the
change in the economic environment.
“Statics” means that we are not concerned with any adjustment process that may be
involved in moving from one choice to another;
rather we will only examine the equilibrium choice (under different parameters/
exogenous variables)
In the case of the consumer, there are only two things in our model that affect the
optimal choice: prices and income.
The comparative statics questions in consumer theory therefore involve investigating
how demand changes when prices and income change.
5. Demand: relation between choice and
price
What happens to choice when prices vary?
Price variation impacts the slope of budget line
Price changes do not affect preferences
Can plot on the usual space of bundles the price-consumption curve
Budget line rotates, ICs stay unchanged
Optimal choice will change
7. Individual demand curve
Rather than the price consumption curve, we are more familiar with
representation of demand for one good as price varies
Consider space of quantity of one good (e.g. F) and price (e.g. PF )
Can plot (F, PF ) to describe how price and quantity demanded vary together
9. Slope of Demand Curve and Elasticity
Important feature of demand:
slope, indicates how sensitive qty demanded is to changes in prices
Recall from calculus that slope of curve P (X) is derivative
𝜕𝑃
𝜕𝑋
If demand curve is steep, it takes large changes in price to move quantity
demanded
Vice versa if demand curve is flat, small changes in price result in large
changes in quantity demanded
Slope of demand curve is inversely related to key concept of (Price) Elasticity
of Demand
10. Demand Curve and Elasticity
Elasticity of Demand for good X is:
Indicates intuitively the percent change in demand for X corresponding to one per cent change in
PX
High elasticity corresponds to large changes in demand for small change in price; vice versa for
low elasticity
Since slope of demand appears at the denominator in 𝐸 𝑋
𝐷
; there is inverse relationship: high
slope, low elasticity and vice versa
If 𝐸 𝑋
𝐷
is in absolute value between 0 and 1, we say demand is inelastic
If 𝐸 𝑋
𝐷
is is in absolute value greater than 1, we say demand is elastic
11. Difference with Demand as seen in ECON 101
In 101, start from market demand curve directly, as primitive
In this course, demand is derived from true primitives of choice
preferences and budgets
And is result of optimal choice by consumer(s)
Moreover, there’s a difference between individual vs. market demand
12. Individual vs. Market Demand
When discussing demand in Econ 101, typically draw Market Demand curve
But until now we described individual choice
Easy/intuitive connection between market and individual demand:
Market demand is just the sum of individual demand curves
If two individuals in market, say 1 and 2, denote X1 and X2 individual demands
Then, market demand is XM = X1+X2
13. Individual vs. Market Demand
Suppose demand curves are respectively 𝑃𝑋 = 5 −
2
3
𝑋1, and 𝑃𝑋 = 9 −
2
5
𝑋2
Rearrange to obtain functions of X
Then, sum up X1 and X2 to obtain function of XM
Rearrange again to have PX(XM) market demand curve
Simple graphical intuition: plot three curves, verify that mkt demand is
(horizontal) sum of other two
14. Price Changes, Demand and Substitution
Across Goods
Demand curve graphs what happens to X as own price PX changes
But, as PX changes, Y varies as well, as different relative price moves slope of
budget line and results in different optimal choice
Mathematically, can express Y as function of PX , and check derivative
𝜕𝑌
𝜕𝑃 𝑋
X and Y are substitutes if
𝜕𝑌
𝜕𝑃 𝑋
> 0; and complements if
𝜕𝑌
𝜕𝑃 𝑋
< 0
Makes rigorous our previous loose discussion of complementarity and substitutability
between goods
16. Example
Consumer has utility 𝑼 𝑿, 𝒀 = 𝟐 + 𝑿 𝒀 𝟐, 𝑴 = 𝟏𝟎, 𝑷 𝒀 = 𝟓
Does the utility function have decreasing MRS?
𝑀𝑅𝑆 𝑋𝑌 =
𝑌
2 2+𝑋
, decreasing in X. Hence, tangency sufficient for optimal choice
Express the optimal choice of both goods as a function of PX . Is it always
interior? If not, for what values of PX is it on the boundary?
Tangency condition + budget line yields 𝑋 =
100
3𝑃 𝑋
−
4
3
; (derive Y using budget line
again); for 𝑃𝑋 ≥ 25, X = 0; so boundary solution
Find the elasticity of demand for X; is demand elastic?
𝐸 𝑋
𝐷
=
100
100−4𝑃 𝑋
, in abs>1 as long as PX > 0
17. Demand and Changes in Income
Changes in income M shift the budget line, will move optimal choice and
hence demand
As M varies, we can plot in space of bundles the income-consumption curve
Quantity consumed of each good could increase or decrease as M varies
19. Normal vs. Inferior Goods
Similar to demand curve, can plot curve in the space of demand X; income M:
it’s called an Engel curve
As M increases X could either increase or decrease
If
𝜕𝑀
𝜕𝑋
> 0, good X is normal good; if
𝜕𝑀
𝜕𝑋
< 0, good X is inferior good
Intuition for inferior goods: as you get richer, substitute for something better
Think potatoes, beef
When normal, Engel curve slopes upward, when inferior Engel curve slopes
downward
21. Normal vs. Inferior Goods
Whether a good is inferior or not depends on the income level that we are
examining
It might very well be that very poor people consume more potatoes as their
income increases.
But after a point, the consumption of potatoes would probably decline as
income continued to increase.
Since in real life the consumption of goods can increase or decrease when
income increases, it is comforting to know that economic theory allows for
both possibilities.
22. Inferior and Normal Goods
Given a starting level of income M; can all goods be inferior?
No
optimal choice must be on budget line
When one good is inferior, i.e. demand decrease as income increase, another good
must increase with income, otherwise the indifference curve is moving SW, which
cross the budget line
Can a good be inferior at all levels of income?
No
If we start at zero income – as we can get a little more than zero income, but we
cannot get negative number of good
We can at least afford some good while we can buy nothing with zero income
23. Example (contd.)
Consumer has utility 𝑼 𝑿, 𝒀 = 𝟐 + 𝑿 𝒀 𝟐, 𝑷 𝒀 = 𝟓, 𝑷 𝑿 = 𝟒
Derive expression of Engel curve for both goods
The Engel curve is the relation between X/Y and M
For each good, we have an Engel curve. So for questions asking Engel curve on
a x-y plane, there should always be two Engel curves
To derive the Engel Curve, we use tangent condition to develop the relation
between X and Y
And then we use the budget line condition to derive the relation between M
and X (or Y)
24. Example (contd.)
Consumer has utility 𝑼 𝑿, 𝒀 = 𝟐 + 𝑿 𝒀 𝟐, 𝑷 𝒀 = 𝟓, 𝑷 𝑿 = 𝟒
Derive expression of Engel curve for both goods
First, calculate MRS:
𝜕𝑈
𝜕𝑋
= 𝑌2
𝜕𝑈
𝜕𝑌
= 2 2 + 𝑋 𝑌
𝑀𝑅𝑆 𝑋𝑌 =
𝑌
2 2 + 𝑋
Then we have the tangent condition:
𝑀𝑅𝑆 𝑋𝑌 =
𝑌
2 2 + 𝑋
=
𝑃𝑋
𝑃𝑌
=
4
5
According to the equation, we have 8𝑋 = 5𝑌 + 16
25. Example (contd.)
Consumer has utility 𝑼 𝑿, 𝒀 = 𝟐 + 𝑿 𝒀 𝟐, 𝑷 𝒀 = 𝟓, 𝑷 𝑿 = 𝟒
Derive expression of Engel curve for both goods
we have 8𝑋 = 5𝑌 + 16 (1) by tangent condition
We know at optimal choice, the consumer use up budget, which gives
4𝑋 + 5𝑌 = 𝑀 (2)
By (1) and (2), we have
𝑀 = 12𝑋 − 16
𝑀 =
15
2
𝑌 + 8
These are the two Engel curve we derive
26. Example (contd.)
Consumer has utility 𝑼 𝑿, 𝒀 = 𝟐 + 𝑿 𝒀 𝟐, 𝑷 𝒀 = 𝟓, 𝑷 𝑿 = 𝟒
Are X, Y inferior at any level of income M?
No
𝑀 = 12𝑋 − 16
𝑀 =
15
2
𝑌 + 8
We take derivative of M on X and Y respectively, it gives us 12 and 15/2, both
are positive and are constant
So there is no point X, Y are inferior to M
27. Summary
The consumer’s demand function for a good will in general depend on the
prices of all goods and income.
The relation between price and choice derives individual demand curve,
which will potentially give us the market demand curve
A normal good is one for which the demand increases when income increases.
An inferior good is one for which the demand decreases when income
increases.
If the demand for good 1 increases when the price of good 2 increases, then
good 1 is a substitute for good 2. If the demand for good 1 decreases in this
situation, then it is a complement for good 2.
The relation between income and choice derives Engel curves
