Individual Demand
Lecture 3A
UNSW ECON5101
June 6, 2023
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Motivation
Elements of a decision problem:
What is feasible?
What is desirable?
We can now put them together to talk about choice. We will:
1 solve the consumer’s optimisation problem to derive their individual demand function. (now)
2 look at some comparative statics: how consumer demand changes as exogenous variables change.
(next)
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Assumptions about Individuals and Preferences
Behavioural postulate: An individual always chooses their most preferred alternative among available
choices.
In a market environment where the consumption space consists of two perfectly divisible goods
(X = R2+), and the price vector p = (p1, p2) is given,
their feasible set is B(p,m) = {x ∈ R2+ : p1x1 + p2x2 ≤ m},
their preferences are described by the rational preference relation ⪰,
they will choose a bundle that is both affordable and weakly preferred to all other affordable
bundles.
Assume that preferences are continuous.
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Rational Choice: Graphically
For now, also assume strictly convex and
strongly monotone preferences.
We will choose the bundle that lies on the
highest indifference curve we can ”reach.”
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Rational Choice: Graphically
At the optimal bundle (x∗1 , x
∗
2 ),
the indifference curve it belongs to is tangent
to the budget constraint,
the budget is fully exhausted.
This rests on the assumptions we made on the
previous slide, and won’t always be true otherwise.
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Rational Choice
At the optimal bundle (x∗1 , x
∗
2 ),
the indifference curve it belongs to is tangent to the budget constraint,
the budget is fully exhausted.
We will now see how to derive optimal choice more formally, as the solution to a constrained
optimisation problem. We will:
discuss the assumptions we need for the two above conditions to characterise the optimum.
discuss how to derive optimal choice in those circumstances.
do some examples, including those in which our assumptions don’t hold.
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1 Constrained Utility Maximisation
Assumptions
Individual Demand
Cobb-Douglas Utility Functions
2 Optimisation with Other Common Utility Functions
Violation of Assumptions
Quasilinear Utility Functions
Perfect Substitutes
Perfect Complements
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Constrained Utility Maximisation
We need a systematic approach to the consumer’s optimisation problem. Instead of the rational
preference relation ⪰, we will work with a utility function that represents it.
Utility Maximisation Problem (UMP)
Given price p = (p1, p2) with p1, p2 > 0, a consumer with income m > 0 solves the following problem:
max
x1,x2≥0
u(x1, x2) subject to p1x1 + p2x2 ≤ m.
Assumptions: The utility function u : R2+ → R is continuous, differentiable, strictly increasing, and
strictly concave.
Notation: For convenience, I will occasionally write (x1, x2) using the vector boldface, x.
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Assumptions
u is continuous
Ensures the existence of a solution to the UMP.
Since we assumed our preferences are rational and continuous, the Utility Representation Theorem
tells us that we will be able to represent them with a continuous utility function.
u is strictly increasing
Ensures the consumer spends all their income (Walras’ Law), i.e., p1x
∗
1 + p2x
∗
2 = m.
This is guaranteed when preferences are strongly monotone.
u is strictly concave
Ensures a unique solution the UMP.
This is guaranteed when preferences are strictly convex.
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Assumptions
Claim: If u is strictly concave, then there is a unique solution to the UMP.
Proof (not assessed):
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Assumptions
Claim: If preferences are strictly convex, then a utility function u which represents these preferences is
strictly concave..
Proof (not assessed):
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Individual Demand
Definition 1: Demand
For each price-income combination, the consumer solves the UMP to obtain an optimal consumption
bundle. The function that relates the optimal quantity of good i to prices and income is called the
demand for good i , xi (p,m).
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Deriving Individual Demand
The Lagrangian for the UMP, given prices p and income m > 0 is
L = u(x)− λ(p1x1 + p2x2 −m).
Our assumptions that u is strictly concave, strictly increasing, and continuously differentiable ensure
that first order conditions are necessary and sufficient for constrained optima.
In other words, x∗ = (x∗1 , x
∗
2 ) is a solution to the above if and only if there exist λ ≥ 0 such that,
∂L
∂x1
=
∂u(x)
∂x1
− λp1 = 0, (1)
∂L
∂x2
=
∂u(x)
∂x2
− λp2 = 0, (2)
∂L
∂λ
= −(p1x1 + p2x2 −m) = 0. (3)
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Individual Demand
Rearranging (1) and (2), we have
∂u(x)
∂x1
= λp1 =⇒ ∂u(x)
∂x1
1
p1
= λ,
∂u(x)
∂x2
= λp2 =⇒ ∂u(x)
∂x2
1
p2
= λ.
Therefore, we have
∂u(x)
∂x1
1
p1
=
∂u(x)
∂x2
1
p2
,
∂u(x)
∂x1
∂u(x)
∂x2
=
p1
p2
,
MU1(x)
MU2(x)
=
p1
p2
,
MRS1,2(x) =
p1
p2
.
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Individual Demand
Because of the assumptions we made about our
preferences, at the optimal bundle (x∗1 , x
∗
2 ),
the indifference curve it belongs to is tangent
to the budget constraint,
i.e., their slopes are equal:
MRS1,2(x∗) =
p1
p2
.
the budget is fully exhausted,
i.e., since u is strictly increasing, we have
m = p1x
∗
1 + p2x
∗
2
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Individual Demand
The solution implies that for any bundle x where
MU1(x
∗)
MU2(x∗) choosing a different bundle.
E.g., at (xˆ1, xˆ2),
MU1(xˆ)
MU2(xˆ)
>
p1
p2
,
MU1(xˆ)
p1
>
MU2(xˆ)
p2
.
The marginal value per dollar spent on good 1
exceeds that of good 2. The consumer would be
better off increasing their consumption of good 2
and reducing their consumption of good 1.
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Cobb-Douglas Utility Functions
Recall that Cobb-Douglas preferences are
continuous, strictly convex, and strongly monotone.
Therefore, our optimisation technique will produce
a unique solution.
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Example 1a: Cobb-Douglas
Suppose that Sophia has $100. She wants to consume apples (xA), which cost $4 each and bananas
(xB), which cost $5 each. Suppose that her utility function is u(xA, xB) = 3x0.25A x
0.75
B .
a Show that her optimal consumption of apples and bananas is (x∗A, x
∗
B) = (6.25, 15).
b Verify that the marginal value per dollar of each good is the same at the optimal bundle
(x∗A, x
∗
B) = (6.25, 15).
c Verify that the budget is fully exhausted at the optimal bundle.
d Verify that the budget is also fully exhausted at the bundle (xˆA, xˆB) = (5, 16).
e Verify that marginal value per dollar for apples is higher than the marginal value per dollar for
bananas at the bundle (xˆA, xˆB) = (5, 16).
f Verify that u(xˆA, xˆB) < u(x
∗
A, x
∗
B).
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Example 1b: Cobb-Douglas
For prices p1, p2 and income m > 0, find demand functions x1(p1, p2,m) and x2(p1, p2,m) given the
utility function u(x1, x2) = 3x
0.25
1 x
0.75
2 .
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From 2 to L goods
Two-good case: if both commodities are consumed, we have an interior solution at which the
following holds:
MRS1,2 =
MU1
MU2
=
p1
p2
.
Generic L-good case: when any two commodities, j and k, are consumed in positive quantities at the
optimum, we have
MRSj,k =
MUj
MUk
=
pj
pk
.
NB: this may not be true if the quantity consumed of either good is zero at the optimum.
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1 Constrained Utility Maximisation
Assumptions
Individual Demand
Cobb-Douglas Utility Functions
2 Optimisation with Other Common Utility Functions
Violation of Assumptions
Quasilinear Utility Functions
Perfect Substitutes
Perfect Complements
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Violation of Assumptions
These preferences are not monotonic.
Looking for a bundle on the budget line at
which an indifference curve is tangent to the
budget line yields bundle x .
Bundle y is feasible and more preferred.
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Violation of Assumptions
These preferences are not monotonic and not
convex.
Looking for a bundle on the budget line at
which an indifference curve is tangent to the
budget line yields bundles x and y .
Only bundle y is the solution to the UMP.
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Quasilinear Utility Functions
For some sets of price and income, the solution will
be interior, and we find the demand function in the
usual way.
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Quasilinear Utility Functions
For other sets of price and income, the solution will
be a corner solution. Seeking the solution in the
usual way would lead to a negative quantity of one
good. Instead, we consume exclusively the good
with the higher marginal value per dollar.
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Example 2a: Quasilinear Utility Functions
Tabitha consumes only tuna (x1) and money (x2). The price of tuna is $5 and the price of money is $1.
Suppose that she has $20 and that her utility function is u(x1, x2) = 5 ln(x1) + x2. What is her optimal
consumption bundle?
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Example 2b: Quasilinear Utility Functions
Tabitha consumes only tuna (x1) and money (x2). The price of tuna is $5 and the price of money is $1.
Suppose that she has $3 and that her utility function is u(x1, x2) = 5 ln(x1) + x2. What is her optimal
consumption bundle?
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Example 2c: Quasilinear Utility Functions
For prices p1, p2 and income m > 0, find demand functions x1(p1, p2,m) and x2(p1, p2,m) given the
utility function u(x1, x2) = 5 ln(x1) + x2.
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Perfect Substitutes
These preferences have utility functions of the form
u(x1, x2) = ax1 + bx2.
Since the indifference curves are straight lines, the
MRS will either:
a never be equal to the price ratio, or
b always be equal to the price ratio.
Here, we have case (a), where for all bundles
(x1, x2),
MU1(xˆ)
MU2(xˆ)
<
p1
p2
=⇒ MU1(xˆ)
p1
<
MU2(xˆ)
p2
.
Hence, we have a corner solution where we should
consume exclusively good 2.
If the inequality were flipped, our corner solution
would tell us to consume exclusively good 1.
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Perfect Substitutes
These preferences have utility functions of the form
u(x1, x2) = ax1 + bx2.
Since the indifference curves are straight lines, the
MRS will either:
a never be equal to the price ratio, or
b always be equal to the price ratio.
Here, we have case (b), where for all bundles
(x1, x2),
MU1(xˆ)
MU2(xˆ)
=
p1
p2
=⇒ MU1(xˆ)
p1
=
MU2(xˆ)
p2
.
Hence, any bundle (x1, x2) that satisfies the budget
line p1x2 + p2x2 = m is an optimal solution.
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Example 3a: Perfect Substitutes
Billy likes to drink coffee (c) and tea (t). He consumes these beverages purely for their caffeine
content, thus finds them to be perfect substitutes. Indeed, Billy thinks that two cups of tea are always
as good as one cup of coffee; in other words, one valid utility representation of Billy’s preferences is
u(c , t) = 2c + t. If each cup of coffee costs $6 and each cup of tea costs $2, how much of each good
should Billy consume if he has $30 to spend?
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Example 3b: Perfect Substitutes
u(c , t) = 2c + t. What if each cup of coffee costs $6 and each cup of tea costs $5, how much of each
good should Billy consume if he has $30 to spend?
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Example 3c: Perfect Substitutes
u(c , t) = 2c + t. What if each cup of coffee costs $6 and each cup of tea costs $3, how much of each
good should Billy consume if he has $30 to spend?
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Example 3d: Perfect Substitutes
For prices p1, p2 and income m > 0, find demand functions x1(p1, p2,m) and x2(p1, p2,m) given the
utility function u(x1, x2) = 2x1 + x2.
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Perfect Complements
These preferences have utility functions of the form
u(x1, x2) = min{ax1, bx2}.
This utility function is non-differentiable at the kink
of each indifference curve.
Note that, when x2 >
a
b x1, the marginal values per
dollar are:
MU1(x1,x2)
p1
> 0,
MU2(x1,x2)
p2
= 0,
so we should be consuming more of good 1.
Similarly, when x2 <
a
b x1, the marginal value per
dollar of good 2 is higher, and we should consume
more of it.
Hence, the bundle on the budget line satisfying
x2 =
a
b x1 is the optimal bundle.
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Example 4: Perfect Complements
For prices p1, p2 and income m > 0, find demand functions x1(p1, p2,m) and x2(p1, p2,m) given the
utility function u(x1, x2) = min{x1, 2x2}.
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Summary
You should be able to:
recall the assumptions usually imposed on utility functions in a UMP.
solve standard UMPs involving utility functions satisfying the four assumptions.
identify utility functions which violate some of the four assumptions.
solve UMPs for some commonly used utility functions that violate some of the four assumptions.
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