International Finance, Fall 2017 Homework

International Finance, Fall 2017 Homework#1 Answer

1. Borrow USD at the interest rate 𝑖∗, exchange into CNY, invest in China to earn 𝑖𝑡, while simultaneously sell the amount  𝑆𝑡(1 + 𝑖𝑡)  CNY forward at  𝐹𝑡+1. By the inequality given in the question, you can earn arbitrage profit.[pic 1]

1. Given the monetary policy rule, we can subtract the foreign Taylor rule from the Home one, and rearrange, to get

r̂∗ − 𝑟𝑡 = 𝜎(𝜋∗ − 𝜋𝑡) − 𝐸𝑡(𝜋∗        − 𝜋𝑡+1) + (𝜈∗ − 𝑣𝑡)

𝑡        𝑡        𝑡+1        𝑡

Substituting this real interest differential into the expression for the real exchange rate on top of slides 23, lecture notes part III, and rearrange, we get

q̂𝑡 = 𝜎(𝜋∗ − 𝜋𝑡) + (𝜎 − 1) ∑∞

𝐸𝑡(𝜋∗        − 𝜋𝑡+𝑗) − ∑∞

𝐸𝑡 𝜌̂𝑡+𝑗 + ∑∞

𝐸𝑡 (𝑣∗        − 𝑣𝑡+𝑗)

𝑡        𝑗=1

𝑡+j

𝑗=0

𝑗=0

𝑡+𝑗

Thus given a contemporaneous increase in inflation at Home, the real exchange rate is going to appreciate.

1. The optimization problem can be set up as maximizing the negative of the cost, subject to the constraint that the consumer has to gain one unit of utility:

max −𝑃1𝐶1 − 𝑃2𝐶2 𝑠. 𝑡. 𝐶𝛼𝐶1−𝛼 = 1

1   2

The first order conditions for this problem are

P1 = 𝜆𝛼𝐶𝛼𝐶1−𝛼𝐶−1, P2 = 𝜆(1 − 𝛼)𝐶𝛼𝐶1−𝛼𝐶−1

1   2        1        1   2        2

Where 𝜆 is the Lagrangian multiplier associated with the constraint. Using the constraint

𝐶𝛼𝐶1−𝛼 = 1 to write the two first order conditions as

1 2

P1𝐶1 = 𝜆𝛼, P2𝐶2 = 𝜆(1 − 𝛼)

Note that these equations imply P1𝐶1/(𝑃2𝐶2) = 𝛼/(1 − 𝛼), which proves (b). Note also that adding the two equations gives us the cost of the consumption basket that yields one unit of utility when the consumer is choosing optimally: P1𝐶1 + 𝑃2𝐶2 = 𝜆 ≡ 𝑃.

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