INTERNATIONAL FINANCE (ECO397), FALL 2010 HOMEWORK 1 JACEK ROTHERT
1. Question 1 Go to OECD website and download the following quarterly series for Canada, Germany, Italy, France, UK, US and Japan: real and nominal GDP, real consumption, nominal and real exports and imports (a) For each country calculate: ρ(nx/y, y), σ(y), σ(c)/σ(y), σ(x), ρ(c, y). Also, calculate the average of these statistics across countries. (b) For each two countries i, j calculate: ρ(yi , yj ), ρ(ci , cj ) (with n countries you should have n(n − 1) country pairs). Also, calculate the average of theses statistics across all country pairs. where: y is the filtered series of log of real GDP, c is the filtered series of log of real consumption, nx/y is the filtered series of the ratio of net exports to GDP, all series filtered using the Hodrick-Prescott filter with penalty parameter 1600. 2. Question 2 There are two countries: Home and Foreign. State of the world is described by the history of events up to time t - st = (s0 , s1 , ..., st ), with s0 given. Time 0 probability of st occurring is π(st ). There are two goods: x and y, both tradable. Home country has endowment of good x only and foreign country has endowment of good y only. The total endowment of good x is X(st ) and of good y - Y (st ). A representative consumer in home country has preferences over aggregate consumption good c which is a CES composite of the two goods x and y: θ i θ−1 h θ−1 θ−1 c = G(x, y) = ωx θ + (1 − ω)y θ The aggregator in the foreign country is: θ h i θ−1 θ−1 θ−1 c∗ = G∗ (x∗ , y ∗ ) = ω ∗ x∗ θ + (1 − ω ∗ )y ∗ θ The period utility utility function is: u(c) =
c1−σ 1−σ
(a) Define an Arrow-Debreu equilibrium (ADE) and a sequential markets equilibrium (SME). (b) Show equivalence between ADE and SME. (c) Define the real exchange rate and derive a link between the real exchange rate and aggregate consumption. 1
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JACEK ROTHERT
(d) Add simple costs of shipping goods between countries, so that for each unit shipped only a fraction α arrives. Reevaluate the link derived between real exchange rates and aggregate consumption derived in part (a). Discuss the Obstfeld and Rogoff claim. (e) Suppose now only state uncontingent bonds are traded between countries. Derive the link between real exchange rates and aggregate consumption. Briefly discuss the implications of parts (b) and (c) for research in this area. 3. Question 3 - 2 country Real Business Cycle model There are two countries: Home and Foreign. State of the world is described by the history of events up to time t - st = (s0 , s1 , ..., st ), with s0 given. Time 0 probability of st occurring is π(st ). A stand-in household (HH) in each country has preferences of the form: XX c(st )1−σ β t π(st ) 1−σ t t s
Output in country i = H, F is: i
yti = ezt · kti
α
The productivity shocks zti follow a vector AR(1) process: H H H zt zt−1 t =A· + F ztF zt−1 F t ρHH ρF H where A = with all numbers being between -1 and 1. The ρ ρF F H HF shocks are i.i.d., normal with mean 0 and variance-covariance matrix F σHH σF H Σ= . The resource constraint in each country is: σHF σF F cit + xit + nxit = yti where x is investment. Capital in country i evolves according to: i 2 φ kt+1 i i i − 1 · kti kt+1 = (1 − δ)kt + xt − 2 kti where parameter φ measures adjustment costs to investment. One country’s export is another country’s import: F nxH t + nxt = 0 (a) Set up a planner’s problem for this economy with country’s weight being 1. (b) Derive FOCs and calculate steady state. (c) Log-linearize the FOCs around the steady state. (d) Set α = 0.33, δ = 0.025, β = 0.99, φ = 2.0. Use numbers from Table 3 in BKK, JPE 1992 paper (p. 761) for matrices A and Σ. Use the method of undeteremined coefficients to solve for decision rules (you do it using e.g. Uhlig’s toolkit: http://www2.wiwi.hu-berlin.de/institute/wpol/html/toolkit.htm, or Dynare: http://www.dynare.org/). (e) Calibrate parameter φ to match relative volatility of investment in either country σ(x)/σ(y).
INTERNATIONAL FINANCE (ECO397), FALL 2010 - HOMEWORK 1
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(f) Calculate the following statistics of HP-filtered simulated series: ρ(y H , y F ), ρ(cH , cF ), ρ(xH , xF ), ρ(nx/y, y), ρ(c, y), σ(y), σ(c)/σ(y), σ(x)/σ(y). Compare them with the moments from the data that you computed in Question 1.
