Exchange Rate Economics: Theories and Evidence

The new open economy macroeconomics part 2 281
exchange rate pass-through,while countries with relatively high volatility of money
growth have relatively high pass-through rates.
11.4 Stochastic versions of the new open economy
macroeconomic model: exchange rate volatility
issues revisited
Obstfeld and Rogoff (2000c) use a stochastic version of NOEM model to generate
a variant of the forward-looking monetary model considered in Chapter 4. The
basic difference here is that the introduction of uncertainty means that the forwardlooking
reduced form features a risk premium term. Risk,however,has a more
pervasive effect in this model,affecting the price-setting behaviour of individual
producers and on expected output and international trade flows.
In a stochastic context Obstfeld and Rogoff demonstrate that the solution for
the expected value of the log of (world) consumption is:
Ec = 1
1+ρ
{
log
( θ − 1
θ
)
− E log κ − 1 [
2 σ κ 2 2
− 2n(1 − n)σs − 2 − 1 ]
2 (1 − ρ)2
× σ 2
c − 2n(1 − n)(σ ks − σ k ∗ s) − 2 [nσ kc + (1 − n)σ k ∗ c]
}
,
(11.15)
where of terms not familiar from the previous chapter,the σi
2 terms represent the
variance of consumption, c,productivity,k,and the exchange rate,s,respectively,
and the σ ij terms represent the corresponding covariance terms. Needless to say,the
distinguishing factor between this expression and its certainty equivalent expression
is that none of these uncertainty terms appear in the certainty equivalent version.
Assuming all shocks are log-normally distributed allows Obstfeld and Rogoff to loglinearise
the first-order conditions for consumption and money and using the firstorder
conditions for money they are able to derive an equation for the exchange
rate of the following form:
s t = īε ∞∑
( ) 1 s−t [
E t m s − ms ∗ 1 + īε 1 + īε
+ v s − vs
∗ ]
,(11.16)
īε
s=t
where the risk premium, v t − vt ∗ ,is given by:
( )
v t − vt ∗ = 1 2
σp 2 ∗ ,t − σ p,t
2 + ρ(σ cp ∗ ,t − σ cp,t ),(11.17)
and where the subscript p denotes the log of the price level and the overbar above
the interest rate term reflects a non-stochastic steady-state value (which arises
because the non-linearity of the money equilibrium condition makes it necessary to
approximate it in the neighbourhood of a non-stochastic steady state),and ε is the
consumption elasticity of the demand for money. The term involving [v s − v ∗ s ]∞ s=t
is referred to as the ‘level’ risk premium,and is not exactly equal to the standard
forward market risk premium because of the existence of 1/ _ i ε. There are two