The profit maximization problem is given by max 𝐾𝑡,𝐿𝑡 𝜋𝑡 = 𝑌𝑡 − 𝑅𝑡𝐾𝑡 − 𝑤𝑡𝐿𝑡 By solving the profit maximization problem, derive the definition of the value of 𝜎 mathematically. By solving the maximization problem, characterize the saving function depending on the value of 𝜃, i.e., there are three cases. By solving the maximization problem, derive the definition of the value of 𝜃 mathematically.

Problem 1
We consider a CES production function.
𝑌𝑡 = 𝐴 (𝛼𝐾𝑡
𝜎−1
𝜎 + (1 − 𝛼)𝐿𝑡
𝜎−1
𝜎 )
𝜎
𝜎−1
.
Q1: As 𝜎 → 1, prove the Cobb–Douglas production function 𝐴𝐾𝑡 𝛼𝐿𝑡 1−𝛼. (10 marks)

Q2: As 𝜎 → 0, prove the Leontief production function 𝑌𝑡 = 𝐴 min(𝐾𝑡, 𝐿𝑡). (10 marks)

Q3: The profit maximization problem is given by max
𝐾𝑡,𝐿𝑡
𝜋𝑡 = 𝑌𝑡 − 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.
𝑌𝑡 = 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)