Quick Reminder of the Theory of Consumer Choice

Professor Roberto Chang

Rutgers University

January 2007

• Reminder of Theory of Consumer Choice, as given by Mankiw, Principles of Economics, chapter 21, and other elementary textbooks.

A Canonical Problem

• Consider the problem of a consumer that may choose to buy apples (x) or bananas (y)

• Suppose the price of apples is px and the price of bananas is py.

• Finally, suppose that he has I dollars to spend.

The Budget Set

• The budget set is the set of options (here, combinations of x and y) open to the consumer.

• Given our assumptions, the total expenditure on apples and bananas cannot exceed income, i.e.

px x + py y ≤ I

• Rewrite

px x + py y = I

as

y = I/py – (px/py) x

This is the budget line

Apples (x)

Bananas (y)

O I/px

I/py

Apples (x)

Bananas (y)

O I/px

I/py

Apples (x)

Bananas (y)

Budget Line:

px x + py y = I

(Slope = - px/py)

O I/px

I/py

• If I increases, the new budget line is higher and parallel to the old one.

Apples (x)

Bananas (y)

O I/px

I/py

Apples (x)

Bananas (y)

O I/px

I/py

I’/px

I’/py

I’ > I

• If px increases, the budget line retains the same vertical intercept, but the horizontal intercept shrinks

Apples (x)

Bananas (y)

O I/px

I/py

Apples (x)

Bananas (y)

O I/px

I/py

I/ px’

px’ > px

Preferences

• Now that we have identified the options open to the consumer, which one will he choose?

• The choice will depend on his preferences, i.e. his relative taste for apples or bananas.

• In Economics, preferences are usually assumed to be given by a utility function.

Utility Functions

• In this case, a utility function is a function U = U(x,y) , where U is the level of satisfaction derived from consumption of (x,y).

• For example, one may assume that

U = log x + log y

or that

U = xy

Indifference Curves

• It is useful to identify indifference curves. An indifference curve is a set of pairs (x,y) that yield the same level of utility.

• For example, for U = xy, an indifference curve is given by setting U = 1, i.e.

1 = xy

• A different indifference curve is given by the pairs (x,y) such that U = 2, i.e. 2 = xy

x

y

Utility = u0

Three Indifference Curves

x

y

Utility = u0

Utility = u1

Three Indifference Curves

Here u1 > u0

x

y

Utility = u0

Utility = u1

Utility = u2

Three Indifference Curves

Here u2 > u1 > u0

Properties of Indifference Curves

• Higher indifference curves represent higher levels of utility

• Indifference curves slope down

• They do not cross

• They “bow inward”

Optimal Consumption

• In Economics we assume that the consumer will pick the best feasible combination of apples and bananas.

• “Feasible” means that (x*,y*) must be in the budget set

• “Best” means that (x*,y*) must attain the highest possible indifference curve

Apples (x)

Bananas (y)

O I/px

I/py

Consumer Optimum

Apples (x)

Bananas (y)

O I/px

I/py

x*

y* C

Consumer Optimum

Apples (x)

Bananas (y)

O I/px

I/py

x*

y* C

Consumer Optimum

Key Optimality Condition

• Note that the optimal choice has the property that the indifference curve must be tangent to the budget line.

• In technical jargon, the slope of the indifference curve at the optimum must be equal to the slope of the budget line.

The Marginal Rate of Substitution

• The slope of an indifference curve is called the marginal rate of substitution, and is given by the ratio of the marginal utilities of x and y:

MRSxy = MUx/ MUy

• Recall that the marginal utility of x is given by ∂U/∂x

• Quick derivation: the set of all pairs (x,y) that give the same utility level z must satisfy U(x,y) = z, or U(x,y) – z = 0. This equation defines y implicitly as a function of x (the graph of such implicit function is the indifference curve). The Implicit Function Theorem then implies the rest.

• Intuition: suppose that consumption of x increases by Δx and consumption of y falls by Δy. How are Δx and Δy to be related for utility to stay the same?

• Increase in utility due to higher x consumption is approx. Δx times MUx

• Fall in utility due to lower y consumption = -Δy times MUy

• Utility is the same if MUx Δx = - MUy Δy, i.e. Δy/ Δx = - MUx/ MUy

• For example, with U = xy,

MUx = ∂U/∂x = y

MUy = ∂U/∂y = x

and

MRSxy = MUx/ MUy = y/x

• Exercise: Find marginal utilities and MRSxy

if U = log x + log y

• Back to our consumer problem, we knew that the slope of the budget line is equal to the ratio of the prices of x and y, px/py. Hence the optimal choice of the consumer must satisfy:

MUx/ MUy = px/py

Numerical Example

• Let U = xy again, and suppose px = 3, py = 3, and I = 12.

• The budget line is given by

3x + 3y = 12

• Optimal choice requires MRSxy = px/py, that is,

y/x = 3/3 = 1

• The solution is, naturally, x = y = 2.

Changes in Income

• Suppose that income doubles, i.e. I = 24. Then the budget line becomes

3x + 3y = 24

• The MRS = px/py condition is the same, so now

x = y = 4

x

y

O

C

I/px

I/py

x

y

O

C

I/px

I/py

An increase in income

I’ > I

I’/px

I’/py

x

y

O

C

I/px

I/py

An increase in income

I’ > I

I’/px

I’/py

C’

• In the precious slide, both goods are normal. But it is possible that one of the goods be inferior.

x

y

O

C

I/px

I/py

An increase in income, Good y inferior

I’ > I

I’/px

I’/py

C’

Changes in Prices

• In the previous example, suppose that px falls to 1.

• The budget line and optimality conditions change to

x + 3 y = 12

y/x = 1/3

• Solution: x = 6, y = 2.

x

y

O

C

I/px

I/py

x

y

O

C

Effects of a fall in px

px > px’

I/px I/px’

I/py

x

y

O

C’C

Effects of a fall in px

px > px’

I/px I/px’

I/py

• If x is a normal good, a fall in its price will result in an increase in the quantity purchased (this is the Law of Demand)

• This is because the so called substitution and income effects reinforce each other.

x

y

O

C’C

I/px I/px’

I/py

x

y

O

C’C

Substitution vs Income Effects

I/px I/px’

I/py

C’’