RDP 2007-06: The Butterfly Effect of Small Open Economies 2. The Model

The model is a version of Galí and Monacelli's (2005) fully micro-founded, stochastic, dynamic, general equilibrium, sticky-price small open economy model. Some broad features of the model are: all output is tradable; prices are sticky as in Calvo (1983); there is full exchange rate pass-through; and there are complete securities markets.

We add foreign and domestic aggregate demand and supply shocks and keep the large economy in its structural form.^[5] Instead of working through the details of the derivation, which are in Galí and Monacelli, we present the key log-linear aggregate relations.

2.1 The Large Economy

Variables with a star superscript correspond to the large economy, which can be described with a standard set of new Keynesian closed economy equations.^[6]

Firms operate under monopolistic competition in the goods market and Calvo-price stickiness. Factor markets are competitive and goods are produced with a constant returns-to-scale technology. One can show that the Phillips curve in the large economy takes the form

where: stands for the foreign inflation rate; is the foreign output gap; is a foreign cost-push shock; the parameter κ is strictly positive and captures the degree of price rigidities; the household's discount factor, β, lies between zero and one; and E_t denotes expectations conditional on information at t.

The aggregate demand schedule (IS-curve) implies that the current level of the foreign output gap, , depends on 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 nominal short-term interest rate; σ is strictly positive and governs intertemporal substitution; is the persistence of ; is the persistence of ; and ϕ₁ defined for notational convenience, is , with φ > 0 governing the elasticity of labour supply.

Foreign monetary policy follows a Taylor rule of the form

where is an independent and identically distributed (iid) foreign monetary policy shock, with zero mean and standard deviation . Given the way in which the policy rule is written, and capture the short-run reaction of to the deviation of foreign inflation from target (assumed to be zero) and the foreign output gap. So, values of below unity correspond to violations of the Taylor principle and give rise to indeterminacy of the equilibrium.

The potential level of foreign output, , is the level that would prevail in the absence of nominal rigidities. For the large economy, it can be shown that the actual level of output, , and the output gap, , obey

Foreign exogenous processes evolve according to

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

2.2 The Small Open Economy

In the small open economy, the IS-curve implies that the output gap, x_t, is a function of its expected future value, the nominal interest rate, the expected rate of domestically produced goods inflation, the expected growth rate of foreign output, foreign and domestic aggregate demand shocks, and domestic total factor productivity. Following Galí and Monacelli (2005), one can show that the small open economy's IS-curve takes the form

where ρ_x and ρ_a are the persistence parameters of domestic demand and domestic productivity shocks, respectively. The parameters ρ_α, ϕ₂, ϕ₃ and ϕ₄ are functions of deep parameters. In particular,

where: α Є [0,1] captures the degree of openness; τ is the intratemporal elasticity of substitution between foreign and domestically produced goods; and ι is the elasticity of substitution across varieties of foreign goods.^[7]

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

where: governs the degree of price stickiness; and ν_π,t is a cost-push shock.

Monetary policy in the small economy is assumed to follow a Taylor rule that sets the nominal interest rate, r_t, in response to its own lagged value, the deviation of consumer price inflation, π_t, from its target (assumed to be zero), and the output gap, x_t:

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 foreign goods, p_f,t, in terms of the price of home goods, p_h,t. That is, s_t = p_f,t − p_h,t. Around a symmetric steady state the consumer price index is a weighted average of the form p_t = (1−α)p_h,t + αp_f,t. It is straightforward to show that p_t = p_h,t + αs_t. From this equation it follows that consumer price inflation and domestically produced goods inflation are linked by the expression

The nominal exchange rate, e_t, is defined as the price of foreign currency in terms of the domestic currency. The real exchange rate, q_t, in turn, is defined as It then follows that changes in the nominal exchange rate, Δe_t, can be decomposed into changes in the real exchange rate and consumer price inflation differentials

Positive values of Δe_t indicate a nominal depreciation of the domestic currency as the price of the foreign currency increases. Because the law of one price is assumed to hold , which implies that the terms of trade can also be written as . Combining these expressions, it is easy to show that the real exchange rate is proportional to the terms of trade. Thus,

Complete international securities markets, together with the market clearing conditions, lead to the following relationship between the terms of trade, s_t, and output differentials and demand shock differentials: ^[8]

The presence of the aggregate demand shock differential in Equation (14), (), alters the small economy's flexible price level of output, relative to Galí and Monacelli (2005).^[9] The relationship between the actual level of output, y_t, and the output gap, x_t, satisfies the following equation:

Finally, the exogenous domestic processes evolve according to

where: the shocks, ε_a,t, ε_π,t and ε_x,t are iid with zero mean and standard deviations and , respectively; the auto-regressive parameters, ρ_a,ρ_π and ρ_x, are less than unity in absolute value.

2.3 Calibration

The benchmark calibration of the model yields a unique REE and resembles that of Galí and Monacelli (2005).^[10] Our calibration is loosely based on data from the US and Australia and falls within the range of chosen values in the literature. The values assigned to the structural parameters are summarised in Table 1.

Table 1: Benchmark Calibration

Price stickiness	θ = 0.75
Discount factor	β = 0.99
Intertemporal elasticity of substitution	σ = 1.50
Share of foreign goods in CPI basket	α = 0.40
Elasticity of substitution between foreign varieties	τ = 1.1
Elasticity of substitution between domestic and foreign goods	ι = 1.2
Elasticity of labour supply	φ = 2.0
Interest rate smoothing	ρr = 0.90
Output gap response	αx = 0.001
Inflation response	απ = 0.125
Interest rate smoothing (large)	= 0.90
Output gap response (large)	= 0.001
Inflation response (large)	= 0.125

The shape – but not the size – of the impulse responses are invariant to the standard deviations of the fundamental disturbances. The set of optimal policy results, however, are sensitive to these values.

The exogenous processes described by Equations (5), (6), and (7) and their domestic counterparts are known in the literature to be highly persistent.^[11] We chose to 0.95, to 0.96, to 0.98, and ρ_α, , and to 0.95.

Given the parameter values in Table 1, we set the standard deviations of the shocks in two steps. First, we calibrate the standard deviations of the large economy's shocks as follows: is set to 0.007 as suggested by Cooley and Prescott (1995).

Then, , and are chosen to minimise the sum of squares deviations of the theoretical standard deviations of the interest rate, inflation, and the output gap from empirical counterparts.^[12] The interest rate in the data is taken to be the quarterly average of the Federal funds rate, foreign inflation is measured as the quarterly growth rate of the US consumer price index, and the foreign output gap is measured as log deviations of US real quarterly GDP per capita from a linear trend over the sample period 1980:Q1–2006:Q4. This strategy yields the values summarised in Table 2.

Table 2: Benchmark Calibration – Foreign Shocks

= 0.01982
= 0.00700
= 0.00002
= 0.00071

Second, we take the large economy's parameter values as given and calibrate the standard deviation of the small economy's shocks in a similar way. The value of σ_a is also set to 0.007, and , and are set to minimise the sum of squares deviations of the theoretical standard deviations of the small economy's interest rate, consumer price inflation, and the output gap from their empirical counterparts. For the small economy we use Australian data and take these to be the quarterly average of the nominal cash rate, the quarterly growth rate of the consumer price index, and log deviations of real quarterly GDP per capita from a linear trend; once again, all series are taken over the same sample period as before. This procedure yields the values summarised in Table 3.

Table 3: Benchmark Calibration – Domestic Shocks

= 0.03713
σa = 0.00700
σr = 0.00066
= 0.00002

Footnotes

The terms foreign and large are used interchangeably. [5]

See Goodfriend and King (1997), Clarida, Galí and Gertler (1999), Woodford (2003) and Ireland (2004) for discussions of the new Keynesian closed economy model. [6]

We refer the reader to Galí and Monacelli (2005) for the non-linear expressions that contain these structural parameters. [7]

Demand shocks, ν_x,t, enter the household's lifetime expected utility as follows: . Thus, one can show that aggregate demand disturbances enter the international risk-sharing condition as in Equation (14). [8]

One could 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's. [9]

Galí and Monacelli set σ = τ = ι = 1. For this special case, the small economy's real marginal cost is completely insulated from movements in foreign output. We chose to select a more general calibration, although our main findings hold in this special case as well. [10]

See, for example, Ireland (2004). [11]

The criterion that we seek to minimise is of the form: , where σ_i stands for the model-generated standard deviation of variable i, and for its empirical counterpart. [12]