RDP 2008-07: A Medium-scale Open Economy Model of Australia Appendix B: The Linearised Model

This Appendix presents the full log-linearised model. Hat symbols on variables denote the log-deviations from steady-state values . Lower-case letters indicate that variables have been normalised with the trend level of technology, that is, Variables with no time subscript refer to steady-state values.

Nominal domestic, import and export prices are governed by Calvo (1983) contracts, augmented by indexation to the last period's inflation and the current (domestic) inflation target. The implied inflation dynamics are given by the following Phillips curve(s):

where s distinguishes between domestic (d), imported consumption (mc), imported investment (mi) and exported final domestic (x) goods sectors. , and denote the current perceived inflation target, firms' real marginal costs, and the time-varying shocks to the desired mark-ups in sector s, respectively. Parameters ρ_π, β, ξ_s and κ_s are the persistence of the inflation target shock; the discount factor; the Calvo parameter (that is, the probability that the firm is not allowed to re-optimise in period t); and the indexation parameter, respectively. If the indexation parameter κ_s is 0, the Phillips curve is purely forward-looking; and if κ_s = 1, prices are fully indexed to last period's inflation.

Marginal costs () for domestic firms are given by

where is the real rental rate of capital. This is derived from firms' optimal conditions (total payments for capital services should equal costs of hiring labour each period) and the assumption that firms finance part of their wage bill with funds borrowed one period prior (at ). Marginal cost is also a function of the labour input ; capital services ; the real wage ; and the gross effective nominal rate of interest rate paid by firms . Finally, and denote the permanent and stationary technology shocks, respectively. Marginal costs for consumption and investment good importers are given by

where is the relative price observed by the domestic exporters is the relative price between the domestically produced goods and the foreign goods; and and are the relative prices of imported consumption and investment goods.

Nominal wages are also subject to the Calvo adjustment mechanism, with indexation to the last period's CPI inflation , the current (domestic) inflation target , and the steady-state growth rate of technology (Adolfson et al 2007 assume that wages are indexed to the current realisation of technology; see also Altig et al 2005). This yields an equation for the real wage :

where and denote the Lagrangian multiplier and labour supply shock, respectively. and are labour income and payroll taxes. Parameters in (B5) are defined as follows:

where: ξ_w is the Calvo wage parameter (that is, the probability that the household is not allowed to re-optimise its wage); λ_w is the wage mark-up; and σ_L is the elasticity of labour supply. Note that η₁₂ and η₁₃ do not appear in Adolfson et al (2007).

Households have habit formation in their preferences (captured by the parameter b). Because of this, the marginal utility of consumption depends on current, lagged and expected future levels of consumption. The equilibrium condition for household consumption, , is

where is the consumption preference shock and is a consumption tax.

The equilibrium condition for investment (i_t) is given by

where: ,t is the hypothetical price of installed capital; denotes the investment-specific technology shock; and the parameter is the ‘slope’ of the investment adjustment cost function. The log-linearised version of households' money demand is given by

where: μ is the steady-state growth rate of money demand; and τ^k is a capital income tax. The log-linearised first-order condition for the physical stock of capital, , is

where δ is the rate of depreciation. The risk premium-adjusted uncovered interest rate parity condition is given by

It is assumed that the international financial markets are imperfectly integrated (holding foreign bonds carries a premium), under the specific modelling assumption that the net foreign asset position of the domestic economy and the risk premium shock enter into the parity condition (in which S_t is the nominal exchange rate; and R_t and denote the domestic and foreign nominal interest rates, respectively). The risk premium term is exogenous but the net asset position is an endogenous variable.

Current period resources can be consumed (domestically or exported), invested, or used to boost capital utilisation. The aggregate resource constraint can be written as

where: and are the relative price terms between the CPI and investment price indices to the domestic price level; is foreign output; is government expenditure; denotes commodity demand^[15]; is an asymmetric technology shock; ω_c is the share of imports in consumption; ω_i is the share of imports in investment; and η_c (η_i) is the elasticity of substitution between foreign and domestic consumption (investment) goods. Finally, λ_d is the domestic steady-state mark-up over factors of production and α is the share of capital in the production function.

The stock of physical capital follows

The degree of capacity utilisation (the difference between the physical capital stock and capital services) is given by

where σ_a is the capital utilisation rate.

The money demand function (that is, cash holdings, q) is given by

where: is a (household) money demand shock (assumed to be zero) and σ_q is the cash-money ratio.

The following identity relates money growth to domestic inflation and changes in real growth

The loan market clearing condition is

where: ν is the fraction of intermediate good firms' wage bill that is to be financed in advance; and is a (firms') money demand shock (assumed to be zero).

The law of motion for net foreign assets, , is

where: is the relative price of commodities ; and is the relative price between the home and foreign economy . The log-linearised relative prices are

where: is the relative price of imported consumption goods (with respect to domestic output price level); is the relative price of imported investment goods (to domestic output price level); is the price of (home) exports relative to foreign prices; and is the relative price of exports (in terms of foreign currency).

Monetary policy is modelled according to the following reaction function

The short-term interest rate is therefore a function of lagged CPI inflation , output , the real exchange rate and a monetary policy shock (ε_R,t). The CPI inflation measure is model-consistent but ignores indirect taxes

Output is given by .

The real exchange rate is given by .

Finally, employment follows

Footnote

It is assumed that commodity demand is completely inelastic. [15]