By solving the maximization problem, derive the definition of the value of 𝜃 mathematically.

Quiz 2 Topics in Macroeconomics

Tomohiro Hirano

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Notations are the same as the lecture notes.
Problem 1
We consider a CES production function.
𝑌 𝑡 = 𝐴(𝛼𝐾𝑡 𝜎−1 𝜎 + (1 − 𝛼)𝐿𝑡 𝜎−1 𝜎 )
𝜎 𝜎−1
.
Q1: As 𝜎 → 1, prove the Cobb-Douglas production function 𝐴𝐾𝑡𝛼𝐿𝑡 1−𝛼. (10 marks)
Q2: As 𝜎 → 0, prove the Leontief production function 𝑌 𝑡 = 𝐴min(𝐾𝑡,𝐿𝑡). (10 marks)
Q3: The profit maximization problem is given by
max 𝐾𝑡,𝐿𝑡
𝜋𝑡 = 𝑌 𝑡 − 𝑅𝑡𝐾𝑡 − 𝑤𝑡𝐿𝑡
By solving the profit maximization problem, derive the definition of the value of 𝜎 mathematically. (10 marks)
Problem 2
The utility maximization problem is given by
max 𝑐1𝑡,𝑐2𝑡,𝑠𝑡
𝑢𝑡 = (𝑎1 1 𝜃(𝑐1𝑡)
𝜃−1 𝜃 + 𝑎2 1 𝜃(𝑐2𝑡)
𝜃−1 𝜃 )
𝜃 𝜃−1
subject to
𝑐1𝑡 + 𝑠𝑡 = 𝑤𝑡 + 𝑒 𝑐2𝑡 = (1 + 𝑟𝑡+1)𝑠𝑡 Q4: By solving the maximization problem, characterize the saving function depending on the value of 𝜃, i.e., there are three cases. (30 marks)

Q5: By solving the maximization problem, derive the definition of the value of 𝜃 mathematically. (10 marks)

Problem 3
Consider a CES utility function.
𝑢𝑡 = (𝑎1 1 𝜃(𝑐1𝑡)
𝜃−1 𝜃 + 𝑎2 1 𝜃(𝑐2𝑡)
𝜃−1 𝜃 )
𝜃 𝜃−1
Q6: Derive 𝑢𝑡 as 𝜃 → 1. (10 marks)
Problem 4
Consider the following CES production function.
𝑌 𝑡 = 𝐴(𝛼( 𝐾𝑡 ℎ1
)
𝜎−1 𝜎
+ (1 − 𝛼)(
𝐿𝑡 ℎ2
)
𝜎−1 𝜎
)
𝜎 𝜎−1
Q7: Derive factor prices 𝑅𝑡 and 𝑤𝑡. (10 marks)
Q8: Compute the values of 𝑅𝑡 and 𝑤𝑡, respectively, as 𝜎 → 0. (10 marks)