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International Finance, Fall 2017 Homework

Autor:   •  April 15, 2018  •  Coursework  •  638 Words (3 Pages)  •  113 Views

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International Finance, Fall 2017 Homework#1 Answer

  1. Borrow USD at the interest rate π‘–βˆ—, exchange into CNY, invest in China to earn 𝑖𝑑, while simultaneously sell the amount  𝑆𝑑(1 + 𝑖𝑑) Β CNY forward at  𝐹𝑑+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

rΜ‚βˆ— βˆ’ π‘Ÿπ‘‘Β = 𝜎(πœ‹βˆ— βˆ’ πœ‹π‘‘) βˆ’ 𝐸𝑑(πœ‹βˆ—Β Β Β Β Β Β Β Β βˆ’ πœ‹π‘‘+1) + (πœˆβˆ— βˆ’ 𝑣𝑑)

𝑑        𝑑        𝑑+1        𝑑

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

q̂𝑑 = 𝜎(πœ‹βˆ— βˆ’ πœ‹π‘‘) + (𝜎 βˆ’ 1) βˆ‘βˆž


𝐸𝑑(πœ‹βˆ—Β Β Β Β Β Β Β Β βˆ’ πœ‹π‘‘+𝑗) βˆ’ βˆ‘βˆž


𝐸𝑑 πœŒΜ‚π‘‘+𝑗 + βˆ‘βˆž


𝐸𝑑 (π‘£βˆ—Β Β Β Β Β Β Β Β βˆ’ 𝑣𝑑+𝑗)

𝑑        𝑗=1


𝑑+j


𝑗=0


𝑗=0


𝑑+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 βˆ’π‘ƒ1𝐢1Β βˆ’ 𝑃2𝐢2 𝑠. 𝑑. 𝐢𝛼𝐢1βˆ’π›Ό = 1

1 Β  2

The first order conditions for this problem are

P1Β = πœ†π›ΌπΆπ›ΌπΆ1βˆ’π›ΌπΆβˆ’1, P2Β = πœ†(1 βˆ’ 𝛼)𝐢𝛼𝐢1βˆ’π›ΌπΆβˆ’1

1 Β  2Β Β Β Β Β Β Β Β 1Β Β Β Β Β Β Β Β 1 Β  2Β Β Β Β Β Β Β Β 2

Where πœ† is the Lagrangian multiplier associated with the constraint. Using the constraint

𝐢𝛼𝐢1βˆ’π›Ό = 1 to write the two first order conditions as

1 2

P1𝐢1Β = πœ†π›Ό, P2𝐢2Β = πœ†(1 βˆ’ 𝛼)

Note that these equations imply P1𝐢1/(𝑃2𝐢2) = 𝛼/(1 βˆ’ 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: P1𝐢1Β + 𝑃2𝐢2Β = πœ† ≑ 𝑃.

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