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EC 352: Intermediate Microeconomics, Lecture 6
Economics 352: Intermediate Microeconomics
Notes and Sample Questions
Chapter 6: Demand Relationships Among Goods
This chapter will use the consumer theory we have established to study demand
relationships across goods.
Substitutes and Complements
We will start with the concepts of pairs of goods being substitutes or complements. To
review, substitutes are two goods that can replace one another, such as butter and
margarine. To be more precise, when the price of one rises, demand for the other
increases. Examples include:
butter and margarine
Budweiser beer and Miller beer
beef and chicken
caffeine and sleep
electric light and candle light
old cars and new cars
bicycles and cars
Complements are goods that are typically used together. To be more precise, when the
price of one falls, demand for the other rises. Examples include:
electricity and computers
computer hardware and computer software
roads and automobiles
left shoes and right shoes
beer and salty snacks
It is worth noting that when the price of salty snacks falls, demand for beer rises. This is
why taverns will very often give patrons salty snacks for free.
Substitutes and complements can be illustrated in the standard consumer diagram.
Consider the change in the quantity of x demanded when the price of y falls. If the
quantity of x demanded rises, the two goods are complements, of the quantity of x
demanded falls, the two goods are substitutes:
EC 352: Intermediate Microeconomics, Lecture 6
Two graphs showing the effect of a decrease in the price of good y on the quantity of
good x that a consumer demands under the assumption that the two goods are
complements and under the assumption that they are substitutes.
Mathematically, the effect of a change in the price of y on the quantity demanded of x
can be shown as:
(
)
∂x p x , p y , I
∂x
∂x
=
− y⋅
∂p y
∂p y
∂I
U=U
This comes from the Slutsky stuff from the never-ending Chapter 5 as follows:
(
) [
(
xc px , py , U = x px , py , E px , py , U
∂x c
∂x
∂x ∂E
=
+
⋅
∂p y ∂p y ∂E ∂p y
But
∂E
∂x ∂x
= y and
=
so we have:
∂p y
∂E ∂I
∂x c
∂x
∂x
=
+ y⋅
∂p y ∂p y
∂I
or
)]
EC 352: Intermediate Microeconomics, Lecture 6
∂x
∂x c
∂x
− y⋅
=
∂I
∂p y ∂p y
or the total effect of a change in the price of y on the quantity demanded of x is equal to a
 ∂x c 
 and an income effect  − y ⋅ ∂x  .
substitution effect 
 ∂p y 
∂I 



As the book points out, the term y represents how much of good y this person consumes.
As consumption of y increases, the income effect is likely to become much larger. If x is
a normal good and y is large, then the income effect will be a large negative number and
these two goods will likely be substitutes regardless of the substitution effect.
For example, if a person spent a large portion of their income on food and the price of
food rises, they will likely reduce their consumption of all other normal goods to avoid
starving to death (a huge income effect) even if some other goods would otherwise be
complements for food.
Try Example 6.2
You should work through Example 6.2 in the textbook. It would help to know that ln x is
d ln x 1
the natural log of x and
= .
dx
x
Net versus Gross – Some Terminology
The book distinguishes between net substitutes or complements and gross substitutes or
complements.
The basic idea is that net refers only to the substitution effect
∂x
∂p y
while gross
U=U
refers to the sum of substitution and income effects. The important difference is that the
net or substitution effect is symmetric. That is, it is the same for
∂y
. Put another way:
∂p x U = U
∂x
∂p y
as it is for
U=U
EC 352: Intermediate Microeconomics, Lecture 6
∂x
∂p y
=
U=U
∂y
∂p x U = U
With the income effect included, however, we are looking at whether things are gross
substitutes or gross complements, and if one of the goods makes up a larger share of
consumption, then there will be a larger income impact from a change in the price of that
good and the symmetry that existed in the net (substitution effect only) relationship won’t
exist in the gross (substitution plus income effect) relationship.
The book presents this analysis for the case of many (more than two) goods by calling the
goods involved xi and xj. I’ve called them x and y here for the sake of familiarity, but
this analysis of net and gross effects is only technically correct when there are more than
two goods.
Composite Commodities
The analysis gets tough when the number of goods increases. Economists typically avoid
this difficulty by looking at one good specifically and lumping all other goods together
into a composite commodity, which it may help to call “all other stuff.”
For example, if you wanted to analyze demand for gasoline, you might use the graph:
A graph showing a budget line and an indifference curve in the case
where the two goods are gasoline and a composite good named “All other
Stuff”.
where the price of all other stuff is given by some sort of price index, and is usually set to
paos=1, just to make the math easier. Income and the price of gasoline are adjusted so that
this price of all other stuff (AOS) makes since. Also, since paos=1, the quantity of AOS
EC 352: Intermediate Microeconomics, Lecture 6
consumed is also equal to expenditures on AOS, so it is sometimes said that expenditures
on AOS rather than the quantity of AOS (which isn’t well defined) is the other good.
Try Example 6.3
Example 6.3 has two things going for it.
First, it uses a CES rather than a Cobb-Douglas utility function, so the math is a bit
different. You should try to work through from the utility function to the Marshallian
demand functions, just to be sure you can do it.
Second, and more importantly, this example shows that if you have a model with three
goods (x, y and z) and you analyze the result of a change in income on x, you will get
results identical to what you get from a model with only two goods (x and h, where h is a
composite of y and z). That is, making a composite good out of y and z didn’t change the
results for x. This lends credibility to the idea of modeling consumer behavior using
composite goods to simplify the task.
Practice Problems
Illustrate the following in consumer diagrams.
1. Increase in the price of x, x and y are substitutes.
2. Increase in the price of x, x and y are complements.
3. Decrease in the price of x, x and y are substitutes.
4. Decrease in the price of x, x and y are complements.
5. Increase in the price of y, x and y are substitutes.
6. Increase in the price of y, x and y are complements.
7. Decrease in the price of y, x and y are substitutes.
8. Decrease in the price of y, x and y are complements.