Advanced Macroeconomics II
Summer semester 2016/2017
Warsaw School of Economics
Instructor: Marcin Kolasa
marcin.kolasa(at)sgh.waw.pl
Assignment 2
Due date: 10th May 2017, 9.50 a.m.
1. Consider two economies of equal size where firms produce consumption goods only
using production technology that is linear in local labor, i.e. yt = At lt and yt∗ = A∗t lt∗ .
Households in both countries maximize expected value of discounted utility streams,
lt1+ϕ
lt∗1+ϕ
with the latter given by u(ct , lt ) = log ct − 1+ϕ
and u(c∗t , lt∗ ) = log c∗t − 1+ϕ
. The
cα c1−α
∗
c∗α c∗1−α
∗
H,t F,t
H,t F,t
∗
consumption baskets are defined as ct = αα (1−α)
1−α and ct = α∗α∗ (1−α∗ )1−α∗ , where
α > α∗ (home bias). International financial markets are complete, all prices are fully
∗
∗
.
and PH,t = et PH,t
flexible and the law of one price holds, i.e. PF,t = et PF,t
(a) Using the model’s equilibrium conditions, write the real exchange rate as a function of relative productivity (and model parameters) only. HINT: Write the real
exchange rate as a function of relative productivity and relative labor input, and
the domestic to foreign labor ratio as a function of relative productivity and the
real exchange rate, then merge the derivations. (2p)
(b) Now solve analytically for output, consumption and labor in each of the two
countries as functions of productivity (and model parameters) only. Use these
derivations to calculate the effects on these variables and on the real exchange
rate of a 1% permanent change in At , assuming the following parametrization:
ϕ = 2, α = 0.8, α∗ = 0.2. Express your results as percent deviations from the
initial state in which A0 = A∗0 = 1. (1p)
2. Consider a small open economy New Keynesian model in which households maximize
c1−θ
lt1+ϕ
t
the expected discounted sum of utility flows given by u(ct , lt ) = 1−θ
− 1+ϕ
, where the
consumption basket is the following CES aggregate of home-made and imported goods
η−1
η−1 η
1
1
η
+ (1 − α) η cF,tη η−1 . Households have access to one-period nominal bonds
ct = α η cH,t
that pay risk-free (gross) rate of return Rt . Local firms produce only consumption
goods and sell them at domestic and foreign markets so that the goods market clearing
can be written as yt = cH,t + c∗H,t , where the demand for domestic exports is given
−γ ∗
P
yt and the rest of the world output yt∗ and prices Pt∗ are
by c∗H,t = (1 − α) etH,t
Pt∗
exogenous to the home economy (small open economy assumption). The law of one
∗
price holds so that PF,t = et Pt∗ and PH,t = et PH,t
.
1
(a) Find the relationship between the terms of trade and the real exchange rate. Write
it in the log-linearized form. (0.5p)
(b) Derive the dynamic IS curve for this economy by writing it in the following loglinearized form
∗
ŷt = Et {ŷt+1 } − a1 (R̂t − Et π̂t+1 ) + a2 Et ∆q̂t+1 + a3 Et ∆ŷt+1
where a1 , a2 and a3 depend on deep model parameters. Use the following steadystate relationships: q̄ = 1 and ȳ = ȳ ∗ . (1.5p)
(c) Show that if international financial markets are complete, the dynamic IS curve
can be written as
ŷt = Et {ŷt+1 } − b1 (R̂t − Et π̂t+1 ) + b2 Et ∆q̂t+1
where b1 and b2 depend on deep model parameters. HINT: Note that the small
open economy assumption implies c∗t = yt∗ . (1p)
2