Macroeconomics
Tomohiro Hirano
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Notations are the same as the lecture notes.
Problem 1
We consider a CES production function.
ππ‘ = π΄ (πΌπΎπ‘
πβ1
π + (1 β πΌ)πΏπ‘
πβ1
π )
π
πβ1
.
Q1: As π β 1, prove the Cobb-Douglas production function π΄πΎπ‘
πΌπΏπ‘
1βπΌ. (10 marks)
Q2: As π β 0, prove the Leontief production function ππ‘ = π΄ min(πΎπ‘, πΏπ‘). (10 marks)
Q3: The profit maximization problem is given by
max
πΎπ‘,πΏπ‘
ππ‘ = ππ‘ β π
π‘πΎπ‘ β π€π‘πΏπ‘
By solving the profit maximization problem, derive the definition of the value of π
mathematically. (10 marks)
Problem 2
The utility maximization problem is given by
max
π1π‘,π2π‘,π π‘
π’π‘ = (π1
1
π(π1π‘)πβ1
π + π2
1
π(π2π‘)πβ1
π )
π
πβ1
subject to
π1π‘ + π π‘ = π€π‘ + π
π2π‘ = (1 + ππ‘+1)π π‘
Q4: By solving the maximization problem, characterize the saving function depending on the
value of π, i.e., there are three cases. (30 marks)
Q5: By solving the maximization problem, derive the definition of the value of π
mathematically. (10 marks)
Problem 3
Consider a CES utility function.
π’π‘ = (π1
1
π(π1π‘)πβ1
π + π2
1
π(π2π‘)πβ1
π )
π
πβ1
Q6: Derive π’π‘ as π β 1. (10 marks)
Problem 4
Consider the following CES production function.
ππ‘ = π΄ (πΌ (πΎπ‘
β1
)
πβ1
π
+ (1 β πΌ) (πΏπ‘
β2
)
πβ1
π
)
π
πβ1
Q7: Derive factor prices π
π‘ and π€π‘. (10 marks)
Q8: Compute the values of π
π‘ and π€π‘, respectively, as π β 0. (10 mark