RDP 2008-03: Monetary Transmission and the Yield Curve in a Small Open Economy 2. The Model

We extend the Galí and Monacelli (2005) small open economy model in two ways. First, we increase the set of equilibrium conditions in both the large and small economies to incorporate interest rates of longer maturities. Second, we add foreign and domestic demand shocks. Instead of working through the details of the derivation, which are in Galí and Monacelli, we discuss the log-linear aggregate equations and the role of the yield curve in the transmission mechanism.

2.1 The Large Economy

Variables with a star superscript correspond to the large economy, which obeys a standard set of New Keynesian closed economy equations.^[2] All variables are expressed in percentage deviations from their steady states.

The aggregate demand schedule links the current level of the foreign output gap, , to its expected future level, the ex-ante short-term real interest rate, foreign total factor productivity, , and a foreign aggregate demand disturbance, , as follows:

where: is the foreign short-term nominal interest rate; is the foreign inflation rate; σ is strictly positive and governs intertemporal substitution; is the persistence of is the persistence of ; and ø₁, defined for convenience, is , where the parameter φ > 0 captures the elasticity of labour supply.

It can be shown that in this model the theory of the term structure implied by optimising behaviour is the expectations hypothesis. Thus, the nominal interest rate at period t associated with a bond that promises to pay one unit of foreign currency at the end of period t + m − 1 is determined by

Firms operate in a monopolistically competitive goods market and are subject to Calvo-price stickiness. Factor markets are competitive and goods are produced with a constant returns-to-scale technology. These assumptions yield the New Phillips curve:

where: κ ≡ λ(σ+φ); λ ≡ (1-θ)(1-βθ)/θ; θ governs the degree of price stickiness; and β is the households' discount factor.

The foreign monetary authority is assumed to follow a Taylor-type rule of the form

where is an independent and identically distributed (iid) foreign monetary disturbance with zero mean and standard deviation .

The potential level of foreign output, , is the level that would prevail in the absence of nominal rigidities. So, in the large economy, the actual level of output, , and the output gap, , are such that

The technology shock, , and the demand shock, , follow autoregressive processes of the form

where: the persistence parameters, and , are less than unity in absolute value; and the shocks and are zero-mean iid disturbances with standard deviations and , respectively.

2.2 The Small Open Economy

The small economy's IS-curve links the output gap, x_t, to its expected future value, the one-period nominal interest rate, R_1,t, the expected rate of domestically produced goods inflation, E_tπ_h,t+1, the expected growth rate of foreign output, foreign and domestic aggregate demand disturbances, and total factor productivity, a_t. Following Galí and Monacelli (2005) the small open economy's IS-curve can be shown to take the form

where ρ_a and ρ_g are the persistence parameters of a_t and g_t. The parameters σ_α, ø₂, ø₃, and ø₄ are, in turn, functions of deeper parameters. In particular,

where α ∈ [0,1] is the share of foreign goods in the consumption basket, and therefore serves as a measure of openness. It is worth noting that for α = 0, the small economy's equations reduce to the standard set of closed economy equations discussed above. Thus, the small economy has all of the structural features of the large economy, overlayed, of course, by openness. Indeed, as discussed in Galí and Monacelli (2005), the linearised equations hold around a symmetric steady state. Finally, τ is the intratemporal elasticity of substitution between foreign and domestically produced goods, while i is the elasticity of substitution across varieties of foreign goods.

In equilibrium, the nominal interest rate at t, associated with a bond that promises to pay one unit of domestic currency at the end of period t + m − 1, is determined by

The dynamics of domestically produced goods price inflation, π_h,t, are governed by an analogous New Phillips curve

where κ_α ≡ λ(σ_α+φ).

Monetary policy in the small economy is also assumed to follow a Taylor-type rule of the form

where ε_r,t is an iid monetary policy shock with zero mean and standard deviation .

The terms of trade, s_t , are defined (from the perspective of the large economy) as the price of goods produced in the large economy, , relative to the price of small economy goods, p_h,t. The nominal exchange rate, e_t, is defined as the price of foreign currency in terms of the domestic currency. That is, . Around a symmetric steady state, the consumer price index of the small economy is a weighted average of the form .^[3] It is straightforward to show that p_t = p_h,t + αs_t, which implies that consumer price inflation and domestically produced goods inflation are linked by the expression below:

The real exchange rate, q_t, in turn, is defined as . It follows that changes in the nominal exchange rate, Δe_t, can be decomposed into changes in the real exchange rate and the differential in consumer price inflation.

Combining these expressions, it is easy to show that the real exchange rate is proportional to the terms of trade as follows:

Complete international securities markets together with market clearing, imply the following relationship between the terms of trade, s_t, and output differentials and demand shock differentials: ^[4]

The presence of the aggregate demand-shocks differential, , in Equation (15), alters the small economy's flexible price level of output, relative to Galí and Monacelli (2005). The relationship between the actual level of output, y_t, and the output gap, x_t, satisfies^[5]

Finally, exogenous domestic processes evolve according to

where the shocks ε_a,t, and ε_g,t are iid with zero-mean and standard deviations and , respectively. The persistence parameters ρ_a and ρ_g are, as before, less than unity in absolute value.

2.3 The Transmission Mechanism

The linearised dynamics of the model, as we mentioned above, are valid around a symmetric steady state in which the condition of uncovered interest rate parity holds:

Equation (19), however, is not an independent equilibrium condition since it can be recovered from the Euler equations for consumption and the international risk-sharing condition, Equation (15). The expectations hypothesis, Equations (2) and (9), combined with uncovered interest parity, relate foreign and domestic interest rates of equivalent maturities as follows:

This highlights the fact that the expected path of the nominal exchange rate plays a central role to the extent that it governs the degree to which movements in foreign rates are reflected in domestic rates. In the extreme case in which the small economy is closed, movements in foreign rates would not translate into movements in domestic rates.^[6]

While Equations (2) and (9) are independent equilibrium conditions, they are nevertheless redundant for the determination of the equilibrium; that is, the equilibrium has a representation without reference to the equations that determine the yield curve.

However, this is not to say that long-term nominal rates are not central for the transmission of monetary policy, nor that they do not contain important information. As emphasised by Rotemberg and Woodford (1997), in sticky-price models it is the ex-ante long-term real interest rate that matters for aggregate demand. In the small open economy version of the model, it also happens to be the ex-ante long-term real interest rate that matters for aggregate demand, although the economy's openness alters the relevant measure of the long-term real rate as well as the interest rate sensitivity of aggregate demand. To see this, take the IS-curve for the small economy, Equation (8), set all disturbances to zero for simplicity, and assume that the large economy is in steady state. This implies that

Advance the equation one period, take expectations and substitute the resulting expression to obtain

Repeating this operation m times and using Equation (9) we can write

since in a stationary equilibrium, E_tx_t+m is approximately zero for large m. Equation (21) implies that the current level of the output gap depends on an ex-ante long-term real interest rate, measured in domestically produced goods price inflation, magnified by maturity and scaled by the economy's intertemporal substitution. If the economy were closed, Equation (21) becomes , because for α = 0, we have that σ_α = σ and π_h,t = π_t. Thus, the sticky-price small open economy model puts long-term nominal interest rates at the very heart of the transmission mechanism in much the same way as the closed economy sticky-price model: the expectations hypothesis implies that monetary policy influences long-term nominal interest rates and nominal rigidities mean that policy will therefore influence real activity.

Footnotes

Ireland (2004) and Woodford (2003) contain detailed discussions of the New Keynesian closed economy model. [2]

This relationship implies complete and contemporaneous pass-through from the nominal exchange rate to domestic prices. While not realistic, this assumption is used for simplicity. It is not surprising that, in the empirical section later in this paper, the model fails to match the low contemporaneous correlation between the exchange rate and inflation observed in the Australian data. [3]

Because the demand shock, g_t, enters the household's lifetime expected utility – – it follows that demand shocks enter the international risk-sharing condition as in Equation (15). [4]

One can show that the level of potential output in the small economy is given by . If aggregate demand shocks were absent from our model, the expression for the output gap collapses back to that of Galí and Monacelli (2005). [5]

This would hold if capital markets were closed, which in this model would necessarily be the case if α = 0. [6]