Transcript Lecture 3

Intermediate Microeconomics
Part I
CONSUMER THEORY (II)
Laura Sochat
Constrained optimisation
There are n goods consumed in quantities 𝑥1 , 𝑥2 …, 𝑥𝑛 making up a bundle
(𝑥1 , 𝑥2 …, 𝑥𝑛 ) ∈ 𝑋
The agent income is M, and the given market prices of each good are 𝑝1 , 𝑝2
…, 𝑝𝑛 .
Agent’s preferences are represented by a utility function U (i.e the agent has
rational preferences).
Preferences are monotonic and (generally) convex.
We have seen the graphical representation of optimisation.
The utility function is our objective function, and the budget set gives us the
constraint, and is given by:
𝐼
𝑃2
𝑃𝑛
𝑥1 𝑃1 + ⋯ + 𝑥𝑛 𝑃𝑛 ≤ 𝐼, or 𝑥1 = − 𝑥2 − ⋯ − 𝑥𝑛
𝑃
𝑃
𝑃
1
1
1
Optimisation with two goods
We have seen before the graphical method of solving for the optimising bundle of goods,
using the tangency condition:
𝑀𝑈1
= 𝑀𝑅𝑆
𝑀𝑈2
Let’s now look at the Lagrangean method
𝑀𝑎𝑥𝑖𝑚𝑖𝑠𝑒 𝑈(𝑥1 , 𝑥2 ) 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝐼 = 𝑃1 𝑥1 + 𝑃2 𝑥2
max 𝐿 = 𝑈 𝑥1 , 𝑥2 − λ( 𝑃1 𝑥1 + 𝑃2 𝑥2 − I)
𝑋,𝑌,λ
1.
Obtain the F.O.C.s
𝑑𝐿
𝑑𝑥1
=
𝜕𝑈
𝜕𝑥1
− λ𝑃1 = 0
𝑑𝐿
𝑑𝑥2
=
𝜕𝑈
𝜕𝑥2
− λ𝑃2 = 0
𝑑𝐿
𝑑λ
= 𝐼 − 𝑃1 𝑥1 − 𝑃2 𝑥2 = 0
Solve the system to find the values of 𝑥1 and 𝑥2 , for which
the Lagrangian is maximised, and the constraint holds.
Examples, and alternative method
Suppose that 𝐼 represents the consumer’s income, 𝑃𝑥 the price of good X, and 𝑃𝑌 , the price
of good Y. Assume that the consumer’s preferences for goods X and Y, are defined by the
following utility function:
a) 𝑈(𝑋, 𝑌) = 𝑋 𝑐 𝑌 𝑑
Solve for the optimal bundle of X and Y, and comment on the findings
An alternative method could be to substitute for X, or Y, within the utility function and solve
for the first order condition:
max 𝑈 𝑋, 𝑌 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝐼 = 𝑃𝑥 𝑋 + 𝑃𝑌 𝑌
𝑋,𝑌
a) 𝑈(𝑋, 𝑌) = 𝑋 1/3 𝑌 2/3 , also assume that 𝐼 = £24, 𝑃𝑥 = £4, and 𝑃𝑌 = £2
Given the values of 𝑃𝑥, 𝑃𝑦, and 𝐼, It is possible to solve for the optimal bundle.
Choosing between two types of taxes, using consumer
theory
Assuming the original budget constraint is given by
𝑃1 𝑥1 + 𝑃2 𝑥2 = I
Suppose the government wishes to raise tax revenue. Should they raise quantity tax (on
good 1)? Or income tax?
–
Show the results in a graph
Think about the limitations of this example in terms of:
–
–
Uniform income taxes
Uniform quantity taxes
Marshallian demand functions
Solutions to the earlier maximisation problem, the optimal values for the
quantity of goods consumed by the consumer can be expressed as a function
of prices and income. These are demand functions such as:
–
–
–
–
𝑥1∗ = 𝐷1 𝑝1 , 𝑝2 , … , 𝑝𝑛 , 𝐼
𝑥2∗ = 𝐷2 𝑝1 , 𝑝2 , … , 𝑝𝑛 , 𝐼
…
𝑥𝑛∗ = 𝐷𝑛 (𝑝1 , 𝑝2 , … , 𝑝𝑛 , 𝐼)
Prices and income, has before, are exogenous and the consumer has no
power over their values.
Marshallian demand functions are homogeneous of degree zero
Interpretation of the Lagrange Multiplier
It is interpreted as the marginal utility of an extra dollar of consumption
expenditure, that is the marginal utility of income:
𝜕𝑢
λ=
𝜕𝑥1
=
𝑝1
𝜕𝑢
𝜕𝑥2
=⋯=
𝑝2
𝜕𝑢
𝜕𝑥𝑛
𝑝𝑛
That is we can say that a dollar of extra income should increase the
consumer’s utility by λ.
𝜕𝑉
= λ∗
𝜕𝐼
Roy’s identity
Substituting for the Marshallian demands into the original utility function, we
obtain an expression for the actual level of utility obtained:
𝑣 𝑝1 , 𝑝2 , … , 𝑝𝑛 , 𝐼 = 𝑈(𝑥1∗ ( 𝑝1 , 𝑝2 , … , 𝑝𝑛 , 𝐼 , 𝑥2∗ ( 𝑝1 , 𝑝2 , … , 𝑝𝑛 , 𝐼 , … , 𝑥𝑛∗ ( 𝑝1 , 𝑝2 , … , 𝑝𝑛 , 𝐼 )
This is called the indirect utility function, and has the following properties:
–
–
–
It is non-increasing in every price, decreasing in at least one price
Increasing in Income
Homogeneous of degree zero in price and income
Roy’s identity- The envelop theorem
Consider the case of two goods. Taking the total derivative of the Indirect
utility function we get that (1):
𝜕𝑉
𝜕𝑈 𝜕𝑥1∗ 𝜕𝑈 𝜕𝑥2∗
=
+
𝜕𝑃1 𝜕𝑥1 𝜕𝑃1 𝜕𝑥2 𝜕𝑃2
Next we need to make use of two results found before:
–
The first order conditions tell us the value of the marginal utilities, which we can use in (1)
𝜕𝑈
𝜕𝑥1
–
𝜕𝑈
= λ𝑃1 , 𝜕𝑥 = λ𝑃2
2
Taking the total derivative of the budget constraint and substituting to obtain the following
This gives us an important result, Roy’s Identity
𝜕𝑉
𝜕𝑉
𝜕𝑃1
∗
= −λ𝑥1 → −
= 𝑥1∗
𝜕𝑉
𝜕𝑃1
𝜕𝐼
The Envelop Theorem
More generally, the result above can be assumed, by using the envelop
theorem. Consider the following maximisation problem:
max 𝑓(𝑥, 𝑦, 𝛽) 𝑠. 𝑡. 𝑔(𝑥, 𝑦, 𝛽) ≤ 𝑐
𝑥,𝑦
The constant 𝛽 is given exogenously. We can solve the problem as usual:
𝐿 𝑥, 𝑦, 𝜆, 𝛽 = 𝑓 𝑥, 𝑦, 𝛽 + 𝜆(𝑐 − 𝑔(𝑥, 𝑦, 𝛽))
F.O.Cs are given by
𝜕𝐿 𝜕𝑓(𝑥 ∗ , 𝑦 ∗ , 𝛽)
𝜕𝑔
=
−𝜆
𝜕𝑥
𝜕𝑥
𝜕𝐿 𝜕𝑓(𝑥 ∗ , 𝑦 ∗ , 𝛽)
𝜕𝑔
=
−𝜆
𝜕𝑦
𝜕𝑦
𝑥∗, 𝑦∗, 𝛽
=0
𝜕𝑥
𝑥∗, 𝑦∗, 𝛽
=0
𝜕𝑦
Substituting for 𝑥 ∗ and 𝑦 ∗ into the objective function, we obtain the value
function: 𝐹 𝛽 = 𝑓(𝑥 ∗ (𝛽), 𝑦 ∗ (𝛽), 𝛽)
The Envelop Theorem
The value function is the maximised value of our objective function. Taking
the total derivative of the value function with respect to 𝛽:
𝑑𝐹(𝛽) 𝜕𝑓 𝜕𝑥 ∗ 𝜕𝑓 𝜕𝑦 ∗ 𝜕𝑓
=
+
+
𝑑𝛽
𝜕𝑥 𝜕𝛽 𝜕𝑦 𝜕𝛽 𝜕𝛽
∗
∗
𝑑𝐹(𝛽)
𝜕𝑔
𝜕𝑥
𝜕𝑔
𝜕𝑦
𝜕𝑓
= 𝜆∗
+ 𝜆∗
+
𝑑𝛽
𝜕𝑥 𝜕𝛽
𝜕𝑦 𝜕𝛽 𝜕𝛽
𝑑𝐹(𝛽)
𝜕𝑔 𝜕𝑥 ∗ 𝜕𝑔 𝜕𝑦 ∗
𝜕𝑓
∗
=𝜆
+
+
𝑑𝛽
𝜕𝑥 𝜕𝛽 𝜕𝑦 𝜕𝛽
𝜕𝛽
Differentiating the constraint, with respect to 𝛽
𝜕𝑔 𝜕𝑥 ∗ 𝜕𝑔 𝜕𝑦 ∗ 𝜕𝑔
𝜕𝑔 𝜕𝑥 ∗ 𝜕𝑔 𝜕𝑦 ∗
𝜕𝑔
+
+
=0→
+
=−
𝜕𝑥 𝜕𝛽 𝜕𝑦 𝜕𝛽 𝜕𝛽
𝜕𝑥 𝜕𝛽 𝜕𝑦 𝜕𝛽
𝜕𝛽
𝑑𝐹(𝛽) 𝜕𝑓
𝜕𝑔 𝜕𝐿
=
− 𝜆∗
=
𝑑𝛽
𝜕𝛽
𝜕𝛽 𝜕𝛽
Expenditure minimisation
We can find optimal decisions of our consumer using a different approach.
We can minimise the consumer’s expenditure subject to a given level of
utility that the consumer must obtain
– The goal and the constraint have been reversed.
This will be important to separate income and substitution effects.
The basic set up consists again of n goods making up a bundle, and each
good has a specific price.
The consumer has a utility target, say 𝑣, and his preferences are rational
The consumer is choosing 𝑥1 , 𝑥2 , … , 𝑥𝑛 to solve the following problem
𝑚𝑖𝑛 𝑝1 𝑥1 , 𝑝2 𝑥2 … 𝑝𝑛 𝑥𝑛
𝑠. 𝑡. 𝑈(𝑥1 , 𝑥1 , … , 𝑥1 ) ≥ 𝑣
Expenditure minimisation solution
Solution to the expenditure minimisation problem are called Hicksian
demand functions and take the form
𝐻1∗ (𝑝1, 𝑝2 , 𝑣)
• They are also called compensated demand, and represent the cost minimising value of
each good
Example- Solve for the Hicksian demand in the case of a Cobb Douglas Utility
function (where α + 𝛽 = 1)
𝑈 𝑥, 𝑦 = 𝑥 𝛼 𝑦 𝛽
Shepard’s Lemma
Remember Roy’s identity, obtained after solving for the utility maximisation
problem, and using the Envelop Theorem
Shepard’s Lemma is obtained the same way, and will recover the following
result. If the price of a good changes by a small amount, then demand
(compensated), will also change by a small amount, therefore the increased
cost of consumption will be equal to the compensated demand.
Using the expenditure function (minimised objective function), we get the
following:
E 𝑝1 , 𝑝2 , … , 𝑝𝑛 , 𝑣 = 𝑝1 𝑥1∗ ( 𝑝1 , 𝑝2 , … , 𝑝𝑛 , 𝑣 , 𝑝2 𝑥2∗ 𝑝1 , 𝑝2 , … , 𝑝𝑛 , 𝑣 , … , 𝑝𝑛 𝑥𝑛∗ 𝑝1 , 𝑝2 , … , 𝑝𝑛 , 𝑣
𝜕𝐸
= 𝐻1∗ (𝑝1, 𝑝2 , 𝑣)
𝜕𝑝1
Connecting the two results- Two sides of the same coin
From utility maximisation, we obtained the Marshallian demands, from
which we can obtain the indirect utility function:
𝑣 𝒑𝟏 , 𝒑𝟐 , … , 𝒑𝒏 , 𝐼 = 𝑢(𝑥1∗ 𝑝1 , 𝑝2 , … , 𝑝𝑛 , 𝐼 , 𝑥2∗ 𝑝1 , 𝑝2 , … , 𝑝𝑛 , 𝐼 , … , 𝑥𝑛∗ 𝑝1 , 𝑝2 , … , 𝑝𝑛 , 𝐼 )
The indirect utility function tells us that utility indirectly depends on prices and
Income. It maps prices and income into maximum utility
From expenditure minimisation, we obtain the expenditure function, using
Hicksian demands:
In both cases, prices and income are given, and you choose the xs…
𝐸 𝒑𝟏 , 𝒑𝟐 , … , 𝒑𝒏 , 𝑣
= 𝑝1 𝐻1∗ 𝑝1 , 𝑝2 , … , 𝑝𝑛 , 𝐼 , 𝑝2 𝐻2∗ 𝑝1 , 𝑝2 , … , 𝑝𝑛 , 𝐼 , … , 𝑝𝑛 𝐻𝑛∗ 𝑝1 , 𝑝2 , … , 𝑝𝑛 , 𝐼
The constraint in the primal becomes the objective in the dual
Using the rational choice model to derive individual
demand: A change in the price of one good.
Remember the demand curve we have seen before, giving us relationship between the price
of a good and the quantity demanded of that good.
Price (P)
A
B
Demand
Quantity demanded
Using the rational choice model to derive individual
demand: A change in the price of one good.
Price (P)
Changing the price of good X, we obtain different budget lines-Using
rational consumer theory, we can find the optimal bundles corresponding
to the different budget line and obtain the price-consumption curve by
linking them
The price consumption curve
Price of fish
Quantity demanded
4
22
6
15
12
7
Quantity demanded
Using the rational choice model to derive individual
demand: A change in the price of one good.
Price (P)
12
Price of fish
Quantity demanded
4
22
6
15
12
7
6
4
Demand curve
7
15
22
Quantity demanded
A change in Income: The Income-Consumption curve and
the Engel curve
Recall the effect of a change in income on the budget constraint: It leads to a shift in the
budget constraint, and therefore to an increase in the feasible set.
All other goods
(£)
120
Income
Quantity demanded
120
12
90
8
60
5
The income consumption curve
90
𝐶
60
𝐵
𝐴
5
8
10 12
15
20
Fish (Kg/week)
A change in Income: The Income-Consumption curve and
the Engel curve
Income
The Engel curve
𝐶
120
𝐵
90
𝐴
Income
Quantity demanded
120
12
90
8
60
5
60
5
8
12
Fish (Kg/week)
Different types of goods
The income elasticity tells us how quantity demanded responds to a change in income. It is
given by:
𝜉=
–
Δ𝑥/𝑥
Δ𝐼/𝐼
=
Δ𝑥 I
Δ𝐼 𝑥
=
𝜕𝑦 𝑥
𝜕𝑥 𝐼
As income increases by 1%, quantity demanded increases by ξ%.
A good is said to be normal, if ξ>0, the quantity demanded of a normal good increases (decreases)
as income increases decreases)
A good is said to be inferior, if ξ<0, the quantity demanded of an inferior good decreases
(increases) as income increases (decreases)
A good is said to be a luxury good if ξ>1
A good is said to be a necessary good if ξ<1
Income elasticities and Income consumption curves
Assume income increases; The budget
constraint shifts to the right.
𝑋2
𝐼𝐶𝐶3
𝐼𝐶𝐶1 : Both goods are normal, quantity
demanded of both goods has increased
following the increase in income
𝐼𝐶𝐶2 : 𝑋1 is a normal good, while 𝑋2 is
inferior. Quantity demanded of good 2
has fallen following the increase in
income
𝐼𝐶𝐶3 : Good 2 in normal, while good 1 is
inferior.
𝐼𝐶𝐶1
𝐼𝐶𝐶2
𝑋1
Difference preferences: What would the Engel curves look
like?
Perfect substitutes
Perfect complements
Homothetic preferences
Quasilinear preferences
The Engel curve when one of the good is both normal and
inferior
All other goods (£)
𝐼3
Income
𝐼3
The income consumption curve
𝐼2
The Engel curve
𝐶
𝐼2
𝐶
𝐵
𝐼𝐶3
𝐼1
𝐼1
From 𝐼1 to 𝐼2 , the increase
in income lead the
consumer to demand
more of X.
From 𝐼2 to 𝐼3 , however,
the increase in income
lead the consumer to
demand less of X.
𝐴
𝐵
𝐴
𝐼𝐶2
𝐼𝐶1
𝑋1 𝑋3 𝑋2
The income consumption curve
X
𝑋1 𝑋3 𝑋2
The Engel curve
𝑋
The effect of a change in the prices of goods: The income
and substitution effects
From the law of demand, we know that an increase (decrease) in the price a good leads to
an decrease (increase) in the quantity demanded of that good. We can divide the total
effect of a price change into two effects:
The substitution effect refers to the change in the relative price of the good. As the price of a
good rises (falls), other goods become relatively cheaper (more expensive), making them
more (less) attractive to the consumer.
–
Even if the consumer was to stay on the same indifference curve, optimisation will lead to the
consumer having to equate the marginal rate of substitution to the new price ratio
The income effect refers to the change in real income from a rise (fall) in the price of one
good. The consumer is now poorer (richer), leading to a change in quantity demanded.
–
The individual cannot stay on the same indifference curve and will have to move to a new one
The income and substitution effects (Hicks) : A normal good
All other goods (£)
All other goods (£)
Assume that we compensate the
consumer, by providing him with
enough money to achieve the
same level of utility than before
the price of fish increased. We
draw an imaginary budget
constraint tangent to the old IC.
𝐵
𝐶
𝐶
𝐴
𝐴
𝐼𝐶1
𝐼𝐶1
𝐼𝐶2
𝐼𝐶2
𝑋3
𝑋1
Fish (Kg/week)
𝑋3
𝑋2
𝑋1
Income effect Substitution effect
Fish (Kg/week)
The income and substitution effects (Hicks) : An inferior good
The income elasticity of an inferior good being negative, the income effect from a price
increase will be positive, while the substitution effect is still negative.
All other goods (£)
All other goods (£)
𝐵
𝐴
𝐴
𝐶
𝐼𝐶1
𝐶
𝐼𝐶2
𝑋3
𝑋1
𝑋2 𝑋3
X
Income effect
X
𝑋1
Total effect
Substitution effect
The income and substitution effect (Hicks) : A giffen good
All other goods (£)
Suppose the price of 𝑋1 falls,
leading to a new (rotated) BL. 𝑋1
being an inferior good, the
substitution effect will lead to the
consumer consuming more of good
1, while the income effect will lead
the consumer to consume less of
the good.
𝐶
𝐼𝐶2
𝐴
In this situation, the substitution
effect is completely offset by the
income effect.
𝐵
𝐼𝐶1
Total effect
Substitution effect
Income effect
𝑋1
How to calculate the effects?
STEP 1
Utility maximisation
–
Allows us to find the initial optimising bundle of goods chosen by the consumer at initial prices
STEP 2
Expenditure minimisation
–
Allows us to maintain the level of utility fixed at initial level, while minimising expenditure at
new prices
STEP 3
Utility maximisation
–
Allows us to calculate the income effect from the consumer’s maximisation problem at the new
set of prices
Compensated Hicksian demand
The compensated demand is the solution obtained from the expenditure
minimisation problem (subject to a fixed level of utility).
It gives us the smallest possible expenditure at the old level utility- It is often
called the compensated demand, as it accounts only for the substitution
effect
𝑥 ∗ = 𝐻(𝑝𝑥 , 𝑝𝑦 , 𝑢)
The own price demand curve derived before, the Marshallian demand, is the
uncompensated demand curve. It accounts for both the income and the
substitution effect
Compensated Hicksian demand
The compensated
Hicksian demand can be
derived as shown on the
graph to the left.
The effect of the price
change are compensated
so as to force the
individual to remain on
the same indifference
curve.
The income and substitution effects (Slutsky) : A normal good
All other goods (£)
All other goods (£)
Assume that we compensate
the consumer, by providing
him with enough money to
achieve the same purchasing
power than before the price of
fish increased. We draw an
imaginary budget constraint
tangent to go through the
original optimal bundle.
𝐵
𝐶
𝐶
𝐴
𝐴
𝐼𝐶1
𝐼𝐶1
𝐼𝐶2
𝐼𝐶2
𝑋2
𝑋1
X
𝑋2
𝑋3
𝑋1
Income effect Substitution effect
X
An algebraic interpretation: The substitution effect
As seen in the graph above, the ‘pivoted’ budget line represents a situation where the
consumer has been compensated to ensure its purchasing power remained unchanged (at
the new set of prices, the consumer can still consume the initial optimal bundle)
Consider the general situation the price of good 𝑥1 changes from 𝑝1 to 𝑝1′
How can we calculate the amount of money income needed to keep that initial bundle
affordable?
𝐼′ = 𝑝1′ 𝑥1 + 𝑝2 𝑥2
𝐼 = 𝑝1 𝑥1 + 𝑝2 𝑥2
𝐼′ − 𝐼 = 𝑥1 (𝑝1′ − 𝑝1 )
The substitution effect is the change in demand for a good when its price changes, and at
the same time, money income is compensated.
∆𝑥1𝑠 = 𝑥1 𝑝1′ , 𝑚′ − 𝑥1 (𝑝1 , 𝑚)
An algebraic interpretation: The Income effect
Consider still a change is the price of good 𝑥1 , from 𝑝1 to 𝑝1′
The income effect will be the change in the demand for the good, when we change income
from 𝑚′ to 𝑚, while holding the price of the good at the new level. That is:
∆𝑥1𝑛 = 𝑥1 𝑝1′ , 𝑚 − 𝑥1 𝑝1′ , 𝑚′
What can you say about the direction of the income effect, based on the type of good, good
1 is?
What about the sign of the substitution effect?
An algebraic interpretation: The Slutsky equation
Putting the two together, we obtain the Slutsky identity:
∆𝑥1 = 𝑥1 𝑝1′ , 𝑚′ − 𝑥1 (𝑝1 , 𝑚) + (𝑥1 𝑝1′ , 𝑚 − 𝑥1 𝑝1′ , 𝑚′
∆𝑥1 = 𝑥1 𝑝1′ , 𝑚 − −𝑥1 𝑝1 , 𝑚 = ∆𝑥1𝑠 + ∆𝑥1𝑛
While we often see the Slutsky identity in terms of absolute changes, it is often useful to
look at it in terms of rate of change:
∆𝑥1 ∆𝑥1𝑠 ∆𝑥1𝑚
=
−
𝑥1
∆𝑝1 ∆𝑝1 ∆𝑚
Marshallian demand elasticities
The price elasticity of demand 𝑒𝑥,𝑝𝑥 measures the proportionate change in
quantity demanded in response to a proportionate change in a good’s own
price. Apart from the exception of a Giffen good, the own price elasticity of
demand is always negative.
Δ𝑥/𝑥
Δ𝑥 𝑝𝑥
𝜕𝑥 𝑝𝑥
𝑒𝑥,𝑝𝑥 =
=
=
Δ𝑝𝑥 /𝑝𝑥 Δ𝑝𝑥 𝑥
𝜕𝑝𝑥 𝑥
Cross price elasticity of demand, 𝑒𝑥,𝑦 , measures the proportionate change in
quantity demanded in response to a proportionate change in the price of
some other good
Δ𝑥/𝑥
Δ𝑥 𝑝𝑦
𝜕𝑥 𝑝𝑦
𝑒𝑥,𝑝𝑦 =
=
=
Δ𝑝𝑦 /𝑝𝑦 Δ𝑝𝑦 𝑥
𝜕𝑝𝑦 𝑥
Application: Labor-Leisure choice
Consider a consumer choosing how to spend his time. He has a choice between working,
and consuming leisure (N).
𝐻 = 24 − 𝑁
The consumer spends his total income on a variety of goods (a composite good) which costs
£1 per unit.
–
How many goods the consumer buys depends on how much he earns; so does the cost the leisure.
When the consumer isn’t working, he is losing earnings.
The consumer’s utility depends on how many goods he buys, and how many hours he
spends not working (consuming leisure)
𝑈 = 𝑈(𝑌, 𝑁)
The consumer’s total income is given by :
𝑌 = 𝑤𝐻 = 𝑤(24 − 𝑁)
Where 𝑤 represents hourly wage
Application: Labor-Leisure choice
The slope of the
budget constraint
is given by -𝑤1 ,
the price of one
extra unit of
leisure is an hour
of foregone
earnings working.
Composite good per day
(£)
Time constraint
At point A, the consumer’s
optimal choice is to consume 16
hours of leisure, and work for 8
hours.
𝐼𝐶1
𝐴
𝑌1
0
24
24
0
Leisure hours per day
Work hours per day
Application: Labor-Leisure choice
We can now derive a demand curve for leisure. Increasing the wage from 𝑤1 to 𝑤2 , we obtain a
new rotated budget constraint and a new optimal bundle of work and leisure (12, 12).
Composite good per day
(£)
Time constraint
𝑤2
𝐵
𝑌2
Wage per hour (£)
𝐵
𝐴
𝑤1
Demand for leisure
𝐴
𝑌1
0
24
𝑁𝐵 = 12
𝐻𝐵 = 12
𝑁𝐴 = 16
𝐻𝐴 = 8
24
0
Leisure hours per day
Work hours per day
𝑁𝐵 = 12
𝑁𝐴 = 16
Application: Labor-Leisure choice
Wage per hour (£)
Wage per hour (£)
Supply of labor
𝑤2
𝐵
𝐵
𝑤2
𝐴
𝐴
𝑤1
𝑤1
Demand for leisure
𝑁𝐵 = 12
𝑁𝐴 = 16
𝐻𝐴 = 8
𝐻𝐵 = 12
Application: Labor-Leisure choice – Income and substitution
effects
Composite good per day
(£)
Time constraint
C
B
A
𝑌1
Substitution effect
Total effect 24
Income effect 0
Leisure hours per day
Work hours per day
From A to B is the substitution
effect: At the higher wage, leisure is
now more expensive. The consumer
will substitute leisure for work.
From B to C is the income effect,
with the now higher wage, the
consumer consumes more leisure.
What does this tell you about
leisure?
What would happen if leisure
becomes an inferior good after the
wage increases above a certain
threshold?