ECON 7030 Microeconomic Analysis
Lecture 4: Consumer Choice
University of Queensland
Semester 1, 2022
Outline
Previous Lectures
• Indifference Curves — contours of the utility function
• Budget Set — collection of affordable consumption
bundles
• Budget Line — frontier of the budget set
This Lecture Utility Maximisation
1 Graphical method
2 Algebraic method
3 Featured Example
Part I
Utility Maximisation: Graphical Method
Utility Maximisation
• We say that a consumer’s choice of consumption bundle is
rational if it maximises the consumer’s utility subject to
his/her budget constraint
• A bundle is utility maximising (subject to budget constraint) if
it yields the highest utility among all bundles in the budget set
• Put differently, a bundle A is utility maximising if any bundle
that is strictly preferred to A is not affordable
• For expositional neatness, I often refer to the utility
maximising bundle simply as the “optimal bundle”
Utility Maximisation on a Graph: Part 1
x
y
BL U0
A
Key:
Affordable
Strictly
Prefers to A
• Bundle A is not optimal — there are affordable bundles
strictly better than A
Utility Maximisation on a Graph: Part 2
x
y
BL
U1
A
Key:
Affordable
Strictly
Prefers to A
• Bundle A is optimal — any bundle that is strictly preferred to
A is not affordable
Utility Maximisation on a Graph: Part 3
• In other words, if bundle A is utility maximising subject to the
budget constraint, then:
1 A must be affordable
2 The set of bundles that are strictly preferred to A is separated
from the budget set
The Tangency Condition
x
y
BL U1
A
Key:
Affordable
Strictly
Prefers to A
• When
1 The utility function (hence the indifference curves) is
differentiable; and
2 The optimal bundle does not occur on either axes,
the previous condition implies that the optimal bundle is
where the indifference curve passing through the bundle is
tangent to the budget line
Tangency Condition
• Tangency of the budget line and the indifference curve at A
means that the two lines have the same slope at A, that is
Slope of Budget Line = Slope of Indifference Curve at A
− Px
Py
= MRS at A
− Px
Py
= −MUx(A)
MUy (A)
Px
Py
=
MUx(xA, yA)
MUy (xA, yA)
Tangency Condition: Interpretation
x
y
BL
U0
U1
A
Willing to give up Y for X exactly as
required, no further adjustment
More willing to give
up Y for X than
required, give up
more Y for X
Less willing to give
up Y for X than
required, give up
more X for Y
• MRS measures how many units of good Y a consumer is
willing to give up for an additional unit of good X
• Relative price measures how many units of good Y a
consumer needs to give up for an additional unit of good X
Tangency Condition: Interpretation 2
• We can also rearrange the tangency condition into
MUx
Px
=
MUy
Py
• The LHS says how much utility I gain from the last dollar on
good X
• The RHS says how much utility I gain from the last dollar on
good Y
• Hence the tangency condition says that the marginal utility
from the last dollar on each good must be the same
• If MU from last dollar on X is strictly bigger than that on Y ,
can re-allocate the last dollar from Y to X
• If MU from last dollar on X is strictly smaller than that on Y ,
can re-allocate the last dollar from X to Y
Finding the Optimal Bundle
• The tangency condition gives us
MUx(x , y)
MUy (x , y)
=
Px
Py
• Meanwhile, since an optimal bundle is on the budget line,
Pxx + Pyy = M
• This gives us two equations in two unknowns (x and y)
• We can solve for the optimal bundle by solving these two
equations simultaneously
Finding the Optimal Bundle: Numerical Example
• Recall the utility function from last lecture:
U(x , y) = x
1
2 y
1
2
• Suppose
Px = 2
Py = 3
M = 12
• Goal: Find the optimal consumption bundle
Finding the Optimal Bundle: Numerical Example
• Recall from last lecture that the MRS of this utility function is
−y/x
(Or you can do it again. . . )
• Hence the tangency condition gives us
y
x
=
Px
Py
=
2
3
• Meanwhile, the budget line is
Pxx + Pyy = M
2x + 3y = 12
Finding the Optimal Bundle: Numerical Example
• We are solving for x and y in
y
x
=
2
3
(1)
2x + 3y = 12 (2)
• Rearranging Equation (1)
3y = 2x
• Substitute this into Equation (2)
2x + 2x = 12
x = 3
• Finally back out y :
y =
2x
3
=
2× 3
3
= 2.
• The optimal bundle is (3, 2)
A More General Example
• Consider a general Cobb-Douglas utility function:
U(x , y) = xαyβ
• And we have the budget line
Pxx + Pyy = M
• Goal: Find the optimal consumption bundle (x , y) (In terms
of α, β, Px , Py and M)
Step 1: Find MRS
• Since I only need the MRS, I am going to use the trick in last
lecture and take natural log of the utility function. Let
V (x , y) = lnU(x , y) = α ln x + β ln y
• This gives me
MUVx =
∂V
∂x
= α
1
x
MUVy =
∂V
∂y
= β
1
y
• Hence we have
MRS = −MU
V
x
MUVy
= − αy
βx
Step 2: Equate MRS with Relative Price
• Next I am going to use the tangency condition:
MRS = −Px
Py
αy
βx
=
Px
Py
αPyy = βPxx
Pyy =
β
α
Pxx
Step 3: Solve the Simultaneous Equations (part 1)
• Now we have two equations (tangency condition and budget
line) in two unknowns (x and y):
Pyy =
β
α
Pxx
Pxx + Pyy = M
• Substitute out the Pyy in the budget line with the tangency
condition:
Pxx +
β
α
Pxx = M
Pxx
α + β
α
= M
x =
α
α + β
M
Px
Step 3: Solve the Simultaneous Equations (part 2)
• Finally we find y :
Pyy =
β
α
Pxx
y =
β
α
Px
Py
x
=
β
α
Px
Py
α
α + β
M
Px
=
β
α + β
M
Py
Expenditure Shares for Cobb-Douglas Utility
• Cobb-Douglas Demands are:
x =
α
α + β
M
Px
y =
β
α + β
M
Py
• For x , it can be interpreted as: Take a αα+β fraction of the
income, spend it all on x
• Similarly for y
• Notice that the fractions, αα+β and
β
α+β , are fixed solely by
the utility function
• We can normalised α and β so that they add up to 1 (which
is again a monotonic transformation) — in which case α and
β are commonly known as Expenditure Shares
The Tangency Condition Does Not Always Hold
x
y
U1
A
BL
Key:
Affordable
Strictly
Prefers to A
• A is an optimal bundle despite MRS at A 6= Relative Price
• The consumer is still willing to get more Y by reducing X ,
but the amount of X is already zero
Yet Another Example
x
y
U1
A
BL
Key:
Affordable
Strictly
Prefers to A
• Again, A is optimal but the tangency condition does not hold
• The consumer is still willing to get more X by reducing Y at
A, but the amount of Y is already zero
Corner Solutions
• The last two examples are known as Corner Solutions —
where the optimal bundle lies on either one of the axes
Optimal
Bundle
Location of
Bundle
The Steeper
Line
The Flatter
Line
x = 0, y > 0 y -axis Budget Line Indifference
Curve
x > 0, y = 0 x-axis Indifference
Curve
Budget Line
Utility Function May Not Be Differentiable
x
y
U1
BL
A
Key:
Affordable
Strictly
Prefers to A
• A is the optimal bundle
• But the tangency condition does not hold as the utility
function is not differentiable at A
What to do
• When we have a corner solution, or a utility function that is
not differentiable at critical points, we cannot use the
tangency condition to find the optimal bundle
• However, in such cases, a graphical approach is often quite
easy
• There are some examples in Tutorial Exercise 3
Part II
Utility Maximisation: Algebraic Method
An Algebraic Approach
• What we have discussed is adequate for a two-good world, an
algebraic approach does not add anything to it
• However, I do want to introduce you to the algebraic method
of dealing with a utility maximisation because
1 Other people do it
2 It is the starting point of doing something more advanced
• In your assessments, you will always have the choice — you
can use the tangency condition or a graph as discussed earlier,
or you can use the Lagragian method, which I am going to
talk about
Utility Maximisation
max
x≥0,y≥0
U(x , y)
s.t. Pxx + Pyy ≤ M
• The above is read as:
“maximise U(x , y) by choosing x and y (both of which must
be weakly positive) subject to the budget constraint”
• Some terminology:
Objective function The function to be maximised
Constraints (In)equalities that the solution must satisfy
Choice variables Variables to be solved, also known as
endogenous variables
Parameters Variables given, also known as exogenous
variables
Constrained Maximisation
• Had the budget constraint not been there, we could have
dealt with the utility maximisation problem as an
unconstrained maximisation
• But the budget constraint is getting into our way
• One possibility is to rearrange the budget constraint and
substitute it into the utility function — but that is clumsy
• So we are going to use ”the Lagrangian” which is tailored for
constrained maximisation
The Lagrangian
max
x ,y ,λ
L = U(x , y) + λ (M − Pxx − Pyy)
• We transform the constrained maximisation
• Idea: λ measures how “tight” the constraint is
• If the constraint is not binding (“loose”), λ is zero
• If the constraint is binding (“tight”), M − Pxx − Pyy is zero
• The new unconstrained maximisation is called a Lagrangian
• λ is known as the Lagrange Multiplier, it is an added choice
variable
Lagrangian: First Order Conditions (FOC)
max
x ,y ,λ
L = U(x , y) + λ (M − Pxx − Pyy)
• We then (partially) differentiate the Lagrangian with respect
to each of the choice variables and set the derivatives to
zeroes to get the First Order Conditions (FOC):
[x ] :
∂L
∂x
= Ux(x , y)− λPx = 0
[y ] :
∂L
∂y
= Uy (x , y)− λPy = 0
[λ] :
∂L
∂λ
= M − Pxx − Pyy = 0
How about the Second Order Conditions (SOC)?
• Just as in a single-variable unconstrained maximisation, the
Second Order Condition (SOC) guarantees that the FOC is
characterising a local maximum, rather than a local minimum
or an inflection point
• Because we have 3 choice variables, the SOC involves a
Hessian Matrix
• For assessment purposes, you don’t have to worry too much
about SOC
Back to the FOC
• We have 3 equations:
Ux(x , y)− λPx = 0
Uy (x , y)− λPy = 0
M − Pxx − Pyy = 0
in 3 unknowns: x , y , λ
• Unless the Hessian Matrix is singular (don’t worry about what
this is), this system of equations can be solved
• Economists are often more interested in knowing that there is
a solution, rather than finding the solution, so we are happy
now
Another Look at the FOC
• The 3 equations can be written as
Ux(x , y) = λPx
Uy (x , y) = λPy
M − Pxx − Pyy = 0
• If I divide the first two equations I get:
Ux(x , y)
Uy (x , y)
=
Px
Py
which is the tangency condition!
• Needless to say, the third equation is the budget line
• So we have got exactly the same equations as in the tangency
condition approach
λ as Marginal Utility of Income
• Moreover, I can also write the first two equations as
Ux(x , y)
Px
= λ
Uy (x , y)
Py
= λ
• Recall that the two fractions on the LHS are the extra utility
the consumer gets from the last dollar spent on x and y ,
respectively
• Hence λ can be interpreted as the marginal utility of income
— how much extra utility can the consumer gets if he has one
more dollar of income
How About Corner Solutions?
• Algebraically, a corner solution has to do with the
non-negativity constraints (i.e., x ≥ 0, y ≥ 0) which I omitted
• Strictly speaking, we should also put Lagrange multipliers on
those two constraints, and a corner solution occurs when one
of the two Lagrange multipliers on the non-negativity
constraints is strictly positive (i.e., when a non-negativity
constraint is binding)
• As for the case where the utility function is not differentiable,
the Lagrangian method cannot be used either
Part III
Featured Example
Featured Example: Water Fluoridation
• A city council is deciding whether to fluoridate the council
water
• A consultancy report says that doing so will improve oral
hygiene of the residents: the benefit, when translated into
money term, is estimated to be $10 per resident
• Feeling that the benefit is significant, the council decided to
go ahead
• But after water fluoridation, the residents brush their teeth
less! The improvement in oral hygiene is not as big as
estimated
• What would you say as an economist?
Setting Up the Scene
• For simplicity, suppose there are only two goods: oral hygiene
(good x) and money for all other goods (good y)
• The price of oral hygiene is Px and the price of money for all
other goods is 1
• The representative resident has a money income of M
• For simplicity, also assume that, without fluoride in water, if
an individual spends no money on oral hygiene s/he gets 0
units of it
Budget Line and Indifference Curve
x
y
M
slope= −Px
M
Px
U0
A
M+m
Px
m
Px
m
Px
B
U1
C
Water Fluoridation: Budget Line
• Suppose water fluoridation improve oral hygiene of a resident
(without the resident doing anything) by a value (i.e.,
quantity times price) of m
• Then the improvement of oral hygiene in terms of quantity of
oral hygiene must be m/Px
• Thus we shift the budget line horizontally by m/Px (reference:
the food bank example from last week)
Consultancy Report Benefits Estimtion
• If residents do not change any of their behaviour, they will be
at point B
• What is the gain in their utility?
• If the change in x is small, then the gain in utility is
approximately
m
Px︸︷︷︸
gain in x
MUx = m
MUx
Px︸ ︷︷ ︸
MU of income
= mλ
Residents Re-optimise
• But residents are not going to stay at point B
• Drawing in the new indifference curves, we predict that they
move to point C — substituting some of the gain in oral
hygiene with money for all other goods
• Note: The diagram is drawn assuming that “money for all
other goods” is a normal good (i.e., not an inferior good),
which seems to be a reasonable assumption
Two Quick Remarks
1 At point A (and point C), the tangency condition implies
MUx
Px
= MUy = λ
2 The value of the total change in consumption must be equal
to m
PxxC + yC = m + PxxA + yA
Px(xC − xA) + (yC − yA) = m
Gain in Utility
• If the change in x is small, then the gain in utility is
approximately
(xC − xA)MUx + (yC − yA)MUy
= Px(xC − xA)MUx
Px
+ (yC − yA)MUy
= λ [Px(xC − xA) + (yC − yA)]
= λm
• This is the same as the estimates without taking into account
the change in behaviour!
Wait a Minute, you say
• But we do know that the residents are better off adjusting
their behaviour!
• This is coming from the approximation error (the MU’s are
different at points A, B and C , and I am using those at point
A for the whole time)
• If m is small, the approximation error will be small
• In any case, the consultancy report estimate is a lower bound
on the gain in utility — residents can gain more than it by
re-optimising
Lessons Learnt
1 A subsidy in kind leads to changes in behaviour, and can
(partially) off-set the subsidy
2 However, the benefits calculated assuming no behavioural
change is a good approximation of welfare gain
• At an optimal bundle, the gain in utility from increasing either
good is the same
• Therefore the form of which the consumer “cash in” the utility
gain (through good x or y) does not matter
Summary
• Utility Maximisation – Tangency Method
• Utility Maximisation — Lagrangain
• Corner solution and non-differentiable utility