Utility Maximization
Abhishek Dureja
Teaching Fellow: Shreya Kapoor
Plaksha University
Utility Maximization and Choice
▶ The aim of the consumer is to maximize utility who are
constrained by limited incomes
▶ Let’s assume that there are n goods
▶ The utility function of the consumer is given by
Utility = U(x1 , x2 , ....xn )
subject to the budget constraint
I = p1 x1 + p2 x2 + ...... + pn xn
=⇒ I − p1 x1 − p2 x2 − ...... − pn xn = 0
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Utility Maximization and Choice
▶ Let’s set up the Lagrangian expression
L = U(x1 , x2 , ....xn ) + λ(I − p1 x1 − p2 x2 − ...... − pn xn )
▶ The FOCs are given by:
∂L ∂U
= − λp1 = 0
∂x1 ∂x1
∂L ∂U
= − λp2 = 0
∂x2 ∂x2
. .
∂L ∂U
= − λpn = 0
∂xn ∂xn
∂L
= I − p1 x1 − p2 x2 − ...... − pn xn = 0
∂λ
▶ These n + 1 equations can, in principle, be solved for the optimal
x1 , x2 , ..., xn and for λ
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Utility Maximization and Choice
Implications of first-order conditions
▶ The first-order conditions can be rewritten as
∂U
∂xi pi
∂U
=
∂xj
pj
=⇒ The conditions for an optimal allocation of income become
pi
MRS (xi for xj ) =
pj
=⇒ Slope of indifference curve = Slope of budget constraint
▶ Say if there are only two goods, x and y , then:
px dy MUx
=⇒ =− = MRS (of x for y ) =
py dx MUy
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Utility Maximization and Choice
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Utility Maximization and Choice
Interpreting the Lagrange multiplier
∂U ∂U ∂U
∂x1 ∂x2 ∂xn
λ= = = .... =
p1 p2 pn
▶ These equations state that
At the utility-maximizing point, each good purchased should
yield the same marginal utility per rupee spent on that good
▶ If this were not true, then
=⇒ One good would promise more marginal enjoyment per rupee
than some other good
=⇒ Funds (Income) are not optimally allocated
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Utility Maximization and Choice
Interpreting the Lagrange multiplier
∂U ∂U ∂U
∂x1 ∂x2 ∂xn
λ= = = .... =
p1 p2 pn
▶ An extra rupee should yield the same additional utility no matter
which good it is spent on
▶ The common value for this extra utility is given by the Lagrange
multiplier (λ) for the consumer’s budget constraint
=⇒ λ can be regarded as the marginal utility of an extra rupee of
consumption expenditure
=⇒ λ can be interpreted as the marginal utility of income
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Utility Maximization and Choice
Interpreting the Lagrange multiplier: Alternative way
∂U
∂xi MUxi
pi = =
λ λ
for every good i that is bought
▶ Since λ represents the marginal utility of an additional rupee of
income
The above ratio is the ratio of marginal utility and the Lagrange
multiplier
=⇒ The ratio signifies the extra utility value of one more unit of
good i as a proportion of the (common value of) marginal
utility of income
MU of good i
=⇒ Price of good i =
MU of income
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Utility Maximization and Choice
Interpreting the Lagrange multiplier: Alternative way
MU of good i
Price of good i =
MU of income (λ)
▶ Since the MU of income is common for all goods =⇒ A high price
for good i can only be justified if it also provides a great deal of
extra utility
▶ At the margin, therefore, the price of a good reflects an individual’s
willingness to pay for one more unit
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Utility Maximization and Choice
Cobb–Douglas Demand Functions
U(x , y ) = x α y β
▶ For what value of x and y can the consumer maximize utility
L = x α y β + λ(I − px x − py y )
∂L
= αx α−1 y β − λpx = 0
∂x
∂L
= βx α y β − λpy = 0
∂y
∂L
= I − px x − py y = 0
∂λ
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Utility Maximization and Choice
αy px
=⇒ =
βx py
α
=⇒ px x = py y
β
Substituting this in the budget constraint, we get
α
I= py y + py y
β
=⇒ βI = py y (α + β)
β I
=⇒ y∗ =
(α + β) py
α I
=⇒ x∗ =
(α + β) px
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Utility Maximization and Choice
Importance of α and β
▶ α and β indicate the importance of good x and y in the consumer’s
preferences
▶ α and β represent the relative share of income spent on good x and
y respectively
▶ Income spent on good x
α I
px x = px × ( )
α + β px
px x α
=⇒ =
I α+β
▶ Similarly,
py y β
=⇒ =
I α+β
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Utility Maximization and Choice
▶ Assume px = 1, py = 4, I = 8, α = β = 0.5
▶ Find the optimal values of x ∗ , y ∗ , and λ
▶ Solution?
▶ x ∗ = 4, y ∗ = 1, λ = 0.25, and Utility = 2
▶ Since λ = 0.25
=⇒ Each small change in income will increase utility by
approximately one-fourth of that amount
▶ If income increases by 1%
=⇒ New I = 8.08, x ∗ = 4.04, y ∗ = 1.01, utility =
4.040.5 1.010.5 = 2.02
▶ Hence a |0.08 increase in income increased utility by 0.02 (as
predicted by the fact that λ = .25)
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Utility Maximization and Choice
Second order conditions
▶ The FOCs have the condition for the turning points
▶ But how do we make sure that the turning points provided by the
FOCs are indeed utility maximizing?
▶ We need the second-order conditions for this
▶ Also we know from unconstrained optimization that concave
functions provide a maxima
▶ Thus, quasi-concavity of the utility function makes sure that
the optimal solution provided by FOCs are utility maximizing
=⇒ Utility is maximized if utility function function is
quasi-concave
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Utility Maximization and Choice
Second order conditions
▶ A function f , f : S → R, is said to be quasi-concave if ∀x , y ∈ S
f (tx + (1 − t)y ) ≥ Min(f (x ), f (y )), where t ∈ [0, 1]
=⇒ A function is quasi-concave if its upper contour set is
convex
▶ A function f , f : S → R, is said to be concave if ∀x , y ∈ S
f (tx + (1 − t)y ) ≥ tf (x ) + (1 − t)f (y )), where t ∈ [0, 1]
▶ Every concave function is quasi-concave
▶ But every quasi-concave function may not be concave
▶ Any function which rises monotonically until it reaches a global
maximum and then monotonically decreases is quasi-concave
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Utility Maximization and Choice
▶ The FOCs provide solution to utility maximization problem of the
consumer
▶ The FOCs however provide solution which is an interior
solution
=⇒ The quantities of both the goods are positive i.e. x ∗ > 0
and y ∗ > 0
=⇒ Positive quantities of both the goods are consumed
▶ But, in certain situations individual preferences may be such that
utility is maximized by choosing to consume no amount of one
of the goods
▶ For instance: Case of neutral goods – If a consumer likes Maggi
(x ) but not Tea (y ) then it may be optimal to choose
Maggi (x ∗ ) = M
pmaggi and Tea (y ∗ ) = 0
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Utility Maximization and Choice
Corner Solution
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Utility Maximization and Choice
Corner solution
▶ Any point on the budget constraint where positive amounts of y are
consumed yields a lower utility than does point E
▶ At E the budget constraint is not precisely tangent to the
indifference curve U2
▶ Instead, at the optimal point – The budget constraint is flatter than
at U2
=⇒ The rate at which x can be traded for y in the market is
lower than the individual’s psychic trade off rate (the MRS)
▶ At prevailing market prices the individual is more than willing
to trade away y to get extra x
▶ The lower bound for the consumption of good y is zero, thus at
optimal y ∗ = 0
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Utility Maximization and Choice
Corner solution
▶ It is necessary to amend the FOCs for a utility maximum to
allow for corner solutions
▶ Earlier the conditions for utility maximum were:
∂L ∂U
= − λpi = 0 ∀ i = 1, 2, ...n
∂xi ∂xi
▶ The modified condition to allow for a corner solution is:
∂L ∂U
= − λpi ≤ 0 ∀ i = 1, 2, ...n
∂xi ∂xi
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Utility Maximization and Choice
Corner solution
▶ If
∂L ∂U
= − λpi < 0 ∀ i = 1, 2, ...n
∂xi ∂xi
=⇒ x∗i = 0
▶ The reason is that
∂U
∂xi
pi >
λ
=⇒ For any good whose price (pi ) exceeds its marginal value
to the consumer will not be purchased i.e. xi∗ = 0
▶ Individuals will not purchase goods that they believe are not worth
the money
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Utility Maximization and Choice
Perfect Substitutes
▶ Hence the optimal condition is
∂L ∂U
= − λpi ≤ 0 ∀ i = 1, 2, ...n
∂xi ∂xi
and if
∂L ∂U
= − λpi < 0 ∀ i = 1, 2, ...n
∂xi ∂xi
=⇒ xi∗ = 0
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Utility Maximization and Choice
Perfect Substitutes
▶ What is the optimal bundle for preferences that are perfect
substitutes
U(x , y ) = x + y
▶ Since one good can be exchanged for other (as goods are identical
in use and can be used interchangeably)
=⇒ Intuitively, we should purchase the good which is the
cheapest
Hence, if p1 < p2 =⇒ Makes sense to spend all income on good
1
Hence, if p1 > p2 =⇒ Makes sense to spend all income on good
2
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Utility Maximization and Choice
Perfect Substitutes
▶ What if p1 = p2 ?
▶ In such a case it does not matter which good the chooses
▶ Consumer can either spend entire income on consuming good 1 or
consuming good 2 or any combination thereof
▶ Demand function for good 1
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Utility Maximization and Choice
Perfect Substitutes
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Utility Maximization and Choice
Perfect Substitutes
▶ What is the optimal bundle for preferences that are perfect
substitute type with MRS not equal to one
U(x , y ) = αx + βy
▶ For optima, Slope of MRS = Slope of budget line = p1
p2
▶ But the MRS is α
β → which is constant
▶ Hence, for optimum α p1
β = p2
=⇒ The ICs and budget line will overlap
=⇒ Consumer can consume any quantity of good 1 and good
2 if MRS = pp12
=⇒ Optimal demand = t ( pM1 , 0) + (1-t) (0, pM2 ) where t ∈ [0, 1]
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Utility Maximization and Choice
Perfect Substitutes
▶ However if MRS > p1
p2 ,
MU1 p1
=⇒ MU2 > p2
MU1 MU2
=⇒ p1 > p2
=⇒ Consumer prefers good 1 more than good 2
Spend entire income on good 1
=⇒ Optimal demand = ( pM2 , 0)
▶ If MRS < p1
p2
=⇒ Consumer prefers good 2 more than good 1
=⇒ Spend entire income on good 2
=⇒ Optimal demand = (0, pM2 )
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Utility Maximization and Choice
Perfect Complements
▶ How will the optimal demand function look like if
U(x , y ) = Min(x , y )
▶ We know the optimal is where x = y given the consumer’s budget
constraint
px x + py y = M
For optimum, x = y
=⇒ px x + py x = M
M
=⇒ x ∗ = y ∗ =
(px + py )
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Utility Maximization and Choice
Perfect Complements
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Utility Maximization and Choice
Perfect Substitutes
M
x∗ = y∗ =
(px + py )
▶ Since the goods are perfect complements and have to be used
in fixed proportion (1:1, in this case)
=⇒ the two goods become one joint good with price (px + py )
▶ Thus the optimal demand function is ( (p M , M )
x +py ) (px +py )
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Utility Maximization and Choice
Perfect Complements
▶ How will the demand function look like for more general fixed
proportions
U(x , y ) = Min(αx , βy )
▶ For optimum αx = βy =⇒ y = α
βx
▶ Given the budget constraint
px x + py x = M
α
=⇒ px x + py x = M
β
βM
=⇒ x ∗ =
(βpx + αpy )
αM
=⇒ y ∗ =
(βpx + αpy )
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Utility Maximization and Choice
Perfect Substitutes
▶ Since the goods are combined in fixed proportions, represented
by α and β
=⇒ The price of the joint good is (βpx + αpy )
▶ Thus the optimal demand function βM αM
(βpx +αpy ) , (βpx +αpy ) )
▶ Notice the demand functions depend upon exogenous
parameters like the prices of the good and the consumer’s
income
▶ The demand function for any good can then be written as
x ∗ = x (p1 , p2 , ..., pn , M)
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Demand
Introduction
▶ Given the individual’s preferences and budget constraint, we can
determine the consumer’s optimal bundle
▶ Given the utility function, U = U(x1 , x2 , ....xn ), we can
determine the optimal demand functions:
x1∗ = x1 (p1 , p2 , ...., pn , M)
x2∗ = x2 (p1 , p2 , ...., pn , M)
.
.
xn∗ = xn (p1 , p2 , ...., pn , M)
given prices (p1 , p2 , ...., pn ) and consumer’s income M
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Demand
Introduction
▶ For instance: if U(x , y ) = x 0.3 y 0.7
=⇒ x ∗ = 0.3M
px and y ∗ = 0.7M
py
Both x ∗ and y ∗ depend upon prices px , py , and income M
=⇒ x ∗ = x (px , py , M) ; y ∗ = y (px , py , M)
▶ In general, if U(x , y ) = x α y β
α M
=⇒ x ∗ = α+β Px
β M
=⇒ y ∗ = α+β Py
▶ What happens if the prices and income are both doubled?
▶ Will the optimal demand bundle change?
▶ No, the optimal bundle (x ∗ , y ∗ ) will not change
33 / 57 ▶ Only the relative income and prices matter
Demand
Introduction
▶ Demand functions are homogeneous of degree zero
▶ A function, f (x1 , x2 , ...xn ), is said to be homogeneous of degree k, if
f (tx1 , tx2 , ....., txn ) = t k f (x1 , x2 , ...xn )
=⇒ If all the arguments of the function are made t-times, then the
value of function becomes t k times
▶ If k=0,
=⇒ f (tx1 , tx2 , ....., txn ) = t 0 f (x1 , x2 , ...xn ) = f (x1 , x2 , ...xn )
=⇒ Increasing (changing) all inputs by t-times does not change the
value of function
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Demand
Introduction
▶ If k=1
=⇒ f (tx1 , tx2 , ....., txn ) = t 1 f (x1 , x2 , ...xn ) = tf (x1 , x2 , ...xn )
=⇒ Increasing (changing) all inputs by t-times makes the value of
function t-times
▶ For instance, if k=1, =⇒ Doubling all the inputs will double the
output (Constant returns to scale)
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▶ Demand functions are homogeneous of degree zero in all prices and
income
=⇒ xi∗ = xi (p1 , p2 , ...., pn , M) = xi (tp1 , tp2 , ...., tpn , tM)
for any t > 0
=⇒ Changing prices of all goods and income in the same
proportion will not change the optimal consumption bundle
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Demand
Choosing Taxes – The Lump Sum Principle
▶ We know that the government can distort the budget constraint by
imposing taxes (or by providing subsidies)
▶ If the government wants to raise the tax revenue
=⇒ It will impose a tax
▶ The government has two possible ways to tax the consumer:
i Quantity tax: Tax on the amount consumed of a good
ii Lump sum tax: A lump sum tax is just a fixed tax on income
▶ If the government wants to raise a certain amount of revenue, is it
better to raise it via a quantity tax or an (income) lump sum
tax
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Demand
Choosing Taxes – The Lump Sum Principle
▶ Suppose that the original budget constraint is
px x + py y = I
▶ Given income and prices, the consumer chooses the bundle (x ∗ , y ∗ )
and gets U3 utility
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Demand
Choosing Taxes – The Lump Sum Principle
▶ Let’s analyze the imposition of a quantity tax at a rate of t on good
1
▶ From a consumer’s viewpoint – it is an increase in the price of good
x by an amount t
▶ What is the new budget constraint?
(px + t)x + py y = I
▶ A quantity tax on a good increases the price perceived by the
consumer
▶ The new budget line is steeper and slope is px +t
p2
▶ Assume the consumer chooses (x1 , y1 ) as the new optimal bundle
=⇒ The utility achieved by the consumer is U1 (where U1 < U3 )
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Demand
Choosing Taxes – The Lump Sum Principle
▶ Since (x1 , y1 ) is the new optimal bundle
=⇒ It must satisfy the new budget constraint
(px + t)x1 + py y1 = I
=⇒ The tax revenue collected is tx1 = T
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Demand
Choosing Taxes – The Lump Sum Principle
41 / 57
Demand
Choosing Taxes – The Lump Sum Principle
▶ Alternatively, if the consumer is charged an equivalent income tax T
=⇒ The new income is
I ′ = I − tx1
▶ The new budget line is
px x + py y = I ′
=⇒ It is parallel to the original budget line
▶ (x1 , y1 ) is still affordable
▶ But, the new optimal bundle is (x2 , y2 )
▶ This bundle lies on a higher IC and provides utility U2
(U2 > U1 )
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Demand
Choosing Taxes – The Lump Sum Principle
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Demand
Indirect Utility Function
▶ Given the individual’s preferences and budget constrained, we can
determine the consumer’s optimal bundle
▶ Given the utility function, U = U(x1 , x2 , ....xn ), we can determine
the optimal demand functions:
x1∗ = x1 (p1 , p2 , ...., pn , M)
x2∗ = x2 (p1 , p2 , ...., pn , M)
.
.
xn∗ = xn (p1 , p2 , ...., pn , M)
given prices (p1 , p2 , ...., pn ) and consumer’s income M
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Demand
Indirect Utility Function
▶ When we plug in the optimal values of optimal xi∗ in the utility
function,
=⇒ We get the maximum utility a consumer can achieve given the
prices and income
Maximum utility = U[x1∗ (p1 , p2 , ...pn , I), x2∗ (p1 , p2 , ...pn , I), ....,
xn∗ (p1 , p2 , ...pn , I)]
= V (p1 , p2 , ...pn , I)
▶ Because of the individual’s desire to maximize utility given a budget
constraint
=⇒ The optimal level of utility obtainable will depend
indirectly on the prices of the goods and the individual’s
income
▶ This dependence is reflected by the indirect utility function V
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Demand
Indirect Utility Function
= V (p1 , p2 , ...pn , I)
▶ If either prices or income were to change, the level of utility
that could be attained would also be affected
▶ For instance, if U(x , y ) = x 0.5 y 0.5
1 M
=⇒ x ∗ = 2 Px
1 M
=⇒ y ∗ = 2 Py
M
=⇒ V (px , py , M) = U(x ∗ , y ∗ ) = (X ∗ )0 .5(y ∗ )0.5 =
2px0.5 py0.5
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Demand
Indirect Utility Function
M
=⇒ V (px , py , M) = U(x ∗ , y ∗ ) = (X ∗ )0.5 (y ∗ )0.5 =
2px0.5 py0.5
▶ When px = 1, py = 4, and M = 8 =⇒ V = 4
2 =2
▶ If prices change =⇒ the maximum utility will also change
▶ We can use the indirect utility function to know the optimal level of
utility
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Demand
Expenditure Minimization
▶ Every constrained maximum problem has an associated dual
constrained minimum problem
▶ For the case of utility maximization, there is an alternative dual
which is the expenditure minimization problem
▶ Expenditure minimization: Allocating income in such a way so
as to achieve a given utility level with the minimal expenditure
▶ Either we can maximize utility for a given level of income and get
optimal bundle
▶ Alternatively we can fix the level of utility and see what is the
minimum expenditure we need to incur, by choosing an optimal
bundle (x , y ), to achieve that utility level
▶ The solution to both the problems will yield in the sample
optimal bundle (x ∗ , y ∗ )
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Demand
Expenditure Minimization
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Demand
Expenditure Minimization
▶ The expenditure minimization problem is analogous to the primary
utility-maximization problem, but the goals and constraints of
the problems have been reversed
▶ Assume the consumer wants to attain the utility level U2 (This is the
constraint)
▶ The consumer needs to minimize the expenditure needed to
achieve this level of utility
=⇒ This minimum expenditure problem represents the objective
problem
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Demand
Expenditure Minimization
▶ Three possible expenditure amounts E1 , E2 , and E3 are shown as
three budget constraint lines in the figure.
▶ Expenditure level E1 is clearly too small to achieve U2 , hence it
cannot solve the dual problem
▶ With expenditures given by E3 , the individual can reach U2 (at
either of the two points B or C), but this is not the minimal
expenditure level required.
▶ Rather, E2 clearly provides just enough total expenditures to reach
U2 (at point A), and this is in fact the solution to the dual problem.
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Demand
Expenditure Minimization
▶ Both the primary utility-maximization approach and the dual
expenditure-minimization approach yield the same solution
(x ∗ , y ∗ )
▶ They are simply alternative ways of viewing the same process
▶ Often the expenditure-minimization approach is more useful,
however, because expenditures are directly observable, whereas
utility is not
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Demand
Expenditure Minimization
A mathematical statement
▶ The individual’s dual expenditure-minimization problem is to
choose x1 , x2 , ..., xn to minimize
Total expenditure = E = p1 x1 + p2 x2 + ...... + pn xn
subject to the constraint
Utility = Ū = U(x1 , x2 , ....xn )
▶ The optimal amounts of (x1 , x2 , ..., xn ) chosen in this problem will
depend on the prices of the various goods (p1 , p2 , ..., pn ) and on
the required utility level Ū
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Demand
Expenditure Minimization
▶ If any of the prices were to change or if the individual had a
different utility target, then another commodity bundle would
be optimal
▶ This dependence can be summarized by an expenditure
function
▶ Expenditure function: The individual’s expenditure function shows
the minimal expenditures necessary to achieve a given utility
level for a particular set of prices
Minimal Expenditures = E (p1 , p2 , ...., pn , U)
▶ Recall the Indirect Utility function was:
V (p1 , p2 , ...pn , I)
=⇒ Expenditure function and the indirect utility function are
inverse functions of one another
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Demand
Expenditure Minimization
▶ For the Cobb-Douglas utility function
U(x , y ) = x 0.5 y 0.5
The indirect utility function is given by:
M
=⇒ V (px , py , M) = U(x ∗ , y ∗ ) =
2px0.5 py0.5
▶ Thus the expenditure function is (replace M with E and V with U)
E (px , py , U) = 2px0.5 py0.5 U
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Demand
Expenditure Minimization
E (px , py , U) = 2px0.5 py0.5 U
▶ To achieve utility target, U = 2, given px = 1 and py = 4, the
expenditure required is E = 8
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References
Chapter 4
Snyder, Christopher and Nicholson, Walter. (2012). Microeconomic
Theory: Basic Principles and Extensions
Chapter 5
Hal R. Varian, Intermediate Microeconomics: A Modern Approach
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