1. MICROECONOMICS Principles and Analysis Frank Cowell
Exercise 5.7
MICROECONOMICS
Principles and Analysis
Frank Cowell
March 2007

2. Ex 5.7: Question
purpose: Illustration of labour supply model with very simple preferences
method: Consumer optimisation with endogenous budget constraint

3. Ex 5.7: Worker’s problem Basic constraints on worker are
x is consumption
`y is non-labour income
w is wage rate
ℓ is labour supply
Worker’s problem can therefore be written as
found by substituting from above into utility function

4. Ex 5.7: Worker’s optimum Take log of maximand to get
a log(wℓ +`y) + [1  a] log(1  ℓ)
Differentiate with respect to ℓ
This is zero if
 wℓ + aw + [a  1]`y = 0
which implies ℓ = a + [a  1]`y / w
But this only makes sense if ℓ is non-negative
requires w ≥ [1  a]`y / a
so optimal labour supply is

5. Ex 5.7: Interior maximum w ≥ [1  a]`y / a 0 < ℓ* < 1 `y x 1  ℓ
(leisure, consumption)-space x
Max value of leisure
Indifference curves
Non-labour income
Budget constraint
Optimum
w ≥ [1  a]`y / a
0 < ℓ* < 1
slope = w `y
1  ℓ
1  ℓ* 1

6. Ex 5.7: Corner solution • w < [1  a]`y / a ℓ* = 0 `y x 1  ℓ 1
Indifference curves as before x
Non-labour income
Budget constraint
Optimum
w < [1  a]`y / a
ℓ* = 0 `y •
1  ℓ 1

7. Ex 5.7: Points to note Treat leisure as a conventional good
Has a natural upper bound
where labour is zero
Have to allow for corner solution
where person chooses not to work
Type of equilibrium depends on
wage rate
non-labour income