Lecture 1: The Theory of Consumer Choice

Alfonso Irarrazabal
August, 2023

1
Introduction
Overview

In this lecture, we will cover three topics

◦ Budget Constraint
◦ Preferences and Utility
◦ Choice

2
Budget Constraint
Budget Constraint

◦ Suppose the consumer can choose between two goods x1 and

x2 with prices p1 and p2 . Suppose the consumer has money to
spend m.
◦ The budget constraint for the consumer is

p 1 x1 + p 2 x2 ≤ m

◦ To find the budget line we solve for x2 as a function of x1

(with equality in the budget equation).

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Budget Constraint

◦ The budget line is the set of bundles that cost m.

◦ The slope of the budget line measures the opportunity cost of
consuming good 1, ie: to consume one more unit of good 1
she has to give up p2 /p1 units of good 2.

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Budget Constraint: A Change in Income

◦ An increase in income m implies that the consumer now can

afford more of both goods.

5
Budget Constraint: A Change in Price

◦ An increase in p1 moves the budget line inward.

6
Preferences and Utility
Preferences

◦ Consumer theory can be described in terms of preferences.

We will describe preferences using indifference curves.
◦ The shaded area represents those bundles weakly preferred to
(x1 ,x2 ). The indifference curve is the set of bundles for which
the consumer is indifferent.

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Utility

◦ A utility function is a way to assign a number to every

possible consumption bundle such that more preferred bundles
get assigned a larger number to less-preferred bundles.
◦ If a consumer has utility function u(x1 , x2 ) then she will be
indifferent between two bundles if they are assigned the same
utility.
◦ An example of such utility is u(x1 , x2 ) = x1 x2 .
◦ Say you consume x1 = 2 and x2 = 3, then you get utility
u(x1 , x2 ) = 6.
◦ If you consume x1 = 2 and x2 = 5, then you get utility
u(x1 , x2 ) = 10, a higher utility than before.

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Utility: Monotonic Transformation

◦ A monotone transformation of a utility function is a utility

function. For example, if u(x) is a utility function, and f (y ) is
a monotone transformation, then f (u(x)) is a utility.
◦ Examples of monotone transformations are multiplication for a
constant (f (u) = 3u), raising to power (f (u) = u 3 ), taking
logs (f (u) = log(u)) etc.
◦ Consider utility u(x1 , x2 ) = x1 x2 and a monotonic
transformation log(u), then log (u(x1 , x2 )) = log(x1 ) + log(x2 )
is a utility that represents the same preferences as
u(x1 , x2 ) = x1 x2 .

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Utility Function

◦ Consider several indifference curves. How can we know that a

utility function can represent indifferences curves?
◦ If preferences are monotonic then the line through the origin
must intersect each indifference curve only once.

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Utility: Plot Preferences

◦ Suppose the utility is given by u(x1 , x2 ) = x1 x2 . We can

compute the indifference curves as x2 = xk1 .
◦ How would the indifference curve of u a (x1 , x2 ) = x12 x22 look
like? Answer: the same! as u a (x1 , x2 ) = u(x1 , x2 )2

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Utility: Perfect Substitutes

◦ Two goods are perfect substitutes if the consumer is willing to

substitute one good for the other at a constant rate.
◦ Preferences for perfect substitutes can be represented by a
utility function of the form u(x1 , x2 ) = ax1 + bx2 .
◦ For example u(x1 , x2 ) = x1 + x2 . Then, the indifference curve
is x2 = k − x1

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Utility: Perfect Complements

◦ Perfect complements are goods that are always consumed in

fixed proportions.
◦ Preferences for perfect complements can be represented by a
utility function of the form u(x1 , x2 ) = min(ax1 , bx2 ).

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Utility: Cobb-Douglas Preference

◦ Cobb-Douglas utility has the following form u(x1 , x2 ) = x1c x2d .

◦ This preference is very popular in economics/finance and it is
a utility that generates a well-behaved preference.

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Marginal Utility and MRS

◦ Consider a consumer who is consuming two goods (x1 , x2 ).

How does this consumer’s utility change as we give her a little
more of good 1?
◦ This rate of change is called the marginal utility with respect
to good 1.
◦ Mathematically,
∂u(x1 , x2 )
MU1 =
∂x1
◦ For example, for u(x) = x1 x2 we have that MU1 = x2 and
MU2 = x1 .

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Marginal Rate of Substitution

◦ A related concept, is the slope of the indifference curve (at a

point) or marginal rate of substitution (MRS).
◦ The MRS can be interpreted as the rate at which a consumer
is just willing to substitute a small amount of good 2 for good
1.
◦ Mathematically,

∂u/∂x1 MU1
MRS = =
∂u/∂x2 MU2

◦ For example, for u(x) = x1 x2 we have that MRS = x2 /x1 .

◦ Similarly, for u(x) = log x1 + x2 we have that MRS = 1/x1 .

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Utility: MRS

◦ The MRS is the slope of the indifference curve (a negative

number)

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Convexity of Indifference Curves

◦ The indifference curve is convex if any line joining two points

above u1 is also above u1 . A convex utility has diminishing
MRS everywhere.
◦ In (b) the curve does not have diminishing MRS everywhere
(concave indifference curve)

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Choice: Utility Maximization
Utility: Optimal Choice and MRS

◦ The optimal choice of the consumer means to find the best

bundle the consumer can afford.
◦ The figure shows that at (x1∗ , x2∗ ) the consumer is achieving
the highest utility given the budget constraint.

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Solving for the Optimal Choice

◦ One way to solve it is by using

p1
MRS(x1∗ , x2∗ ) =
p2

that is where the slope of the indifference curve (MRS) equals

the slope of the budget constraint (p1 /p2 ).
◦ You can solve for (x1∗ , x2∗ ) using the ”tangency” condition
MRS(x1∗ , x2∗ ) = pp12 and the equation for the budget constraint
p1 x1 + p2 x2 = m.

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An example

1/2 1/2
◦ Consider the utility u(x1 , x2 ) = x1 x2 and the budget
constraint 2x1 + x2 = 100.
1
◦ It is helpful to take logs u t (x1 , x2 ) = 2 log(x1 ) + 12 log(x2 ).
◦ Then, optimality implies
x2 p1
=
x1 p2
and 2x1 + x2 = 100.
◦ Replacing the optimality condition x2 = 2x1 into the budget
constraint gives x1 = 25 and x2 = 50.

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Optimal Choice: Perfect Substitutes

◦ If p2 > p1 , then x1∗ = m/p1 and x2∗ = 0.

◦ If p2 < p1 , then x1∗ = 0 and x2∗ = m/p2 .
◦ If p2 = p1 , then any bundle that satisfies the budget
constraint is optimal.

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Optimal Choice: Perfect Complements

◦ Optimal choice is independent of prices.

◦ Once you know the proportion under which goods are
consumed, you can use the budget constraint to solve for
quantities.

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Utility Maximization

◦ We can also solve the U-max problem by using Lagrangian

method. Suppose the consumer chooses to consumer x1 and
x2 in order to maximize utility u(x1 , x2 ) subject the budget
constraint.
◦ This problem can be written as

max u(x1 , x2 )
x1 ,x2

st:
p 1 x1 + p 2 x2 = m

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U-max: Lagrangian

◦ We setup an auxiliary Lagrangian function

L = u(x1 , x2 ) − λ(p1 x1 + p2 x2 − m)

where λ denotes the Lagrangian multiplier.

◦ The optimal choice (x1∗ , x2∗ ) satisfies the three first-order
conditions:
∂L
= ux1 (x1∗ , x2∗ ) − λp1 = 0
∂x1
∂L
= ux2 (x1∗ , x2∗ ) − λp2 = 0
∂x2
∂L
= p1 x1∗ + p2 x2∗ − m = 0
∂λ
∂u(x1∗ ,x2∗ ) ∂u(x1∗ ,x2∗ )
where ux1 (x1∗ , x2∗ ) = ∂x1 and ux2 (x1∗ , x2∗ ) = ∂x2

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Utility Maximization: An Example

◦ Let’s consider the Cobb Douglas utility u(x1 , x2 ) = x1c x2d .

Since utility function are only defined up to a monotonic
transformation it is convenient to take logs.
◦ This way the Lagrangian for this problem can be written as

max c log x1 + d log x2

x1 ,x2

st:
p 1 x1 + p 2 x2 = m

◦ The trick is usually to solve for λ first.

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Utility Maximization: An Example

◦ To solve for optimal x1∗ and x2∗ we setup an auxiliary

Lagrangian function

L = u(x1 , x2 ) − λ(p1 x1 + p2 x2 − m)

where λ denotes the Lagrangian multiplier.

◦ The optimal choice (x1∗ , x2∗ ) satisfies the three first-order
conditions:
c
− λp1 = 0
x1
d
− λp2 = 0
x2
p 1 x1 + p 2 x2 − m = 0

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Utility Maximization: An Example

◦ Use the first two FOCs

c = λp1 x1
d = λp2 x2

◦ Plug the demands x1 = c/(λp1 ) and x2 = c/(λp2 ) into the

budget constraint to get an expression of λ

c + d = λ(p1 x1 + p2 x2 − m) = λm

which gives
c +d
λ=
m

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Utility Maximization: An Example

◦ Substitute into the FOCs for x1 and x2

c m
x1 =
c +d p1
c m
x2 =
c +d p2
◦ The Cobb Douglas utility has the property that the fraction of
the expenditure spent on good x1 is constant
p 1 x1 p1 c m c
= =
m m c + d p1 c +d
◦ Similarly, the fraction of her income that the consumer spends
on good 2 is d/(c + d).

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