RDP 2015-07: A Multi-sector Model of the Australian Economy Appendix A: The Log-linearised Model
May 2015 – ISSN 1448-5109 (Online)
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This appendix provides the log-linearised model. Hat symbols denote log deviations from steady-state values (i.e. = lnΧ_{t} − lnΧ). Lower-case letters indicate that variables have been normalised by the trend level of technology (i.e. x_{t} = Χ_{t} /Z_{t} ). Variables without a time subscript refer to steady-state values.
The equilibrium condition for household consumption is:
where is the Lagrangian multiplier associated with the household's consumption choice.
The household labour supply decision in each sector is:
for j ∈ {n,m,z}.
The household's choice of investment in each sector is:
for j ∈ {n,m,z}, where is the shadow price of installed capital.
The production function for each sector is:
for j ∈ {n,m,z}.
The equilibrium condition for capital in each sector is:
for j ∈ {n,m,z}.
Capital in each sector accumulates according to the law of motion:
for j ∈ {n,m,z}.
The household's choice of domestic bond holdings is:
The definition of the household's labour bundle is:
The Phillips curves for the non-tradeable and non-resource tradeable sectors are:
for j ∈ {n,m} where .
Marginal costs in the non-tradeable and non-resource tradeable sectors are given by:
for j ∈ {n,m}, where p_{j,t} refers to the relative price of good j, that is p_{j,t} = ln[P_{j,t}/P_{t}].
The optimal choices of labour in the tradeable and non-resource tradeable sectors are:
for j ∈ {n,m}.
The optimal choices of resource inputs in the tradeable and non-resource tradeable sectors are:
for j ∈ {n,m}.
The Phillips curve for the imported good retailer is:
where .
Marginal costs for the imported good retailer are given by:
Hours worked in the resources sector is given by:
Capital used in the resources sector is given by:
The law of motion for the domestic price of resources ensures that:
Foreign demand for domestic non-resource exports is:
Domestic demand for non-tradeable goods, domestic non-resource tradeable goods and imported goods is:
for j ∈ {n,m,f}
Market clearing in the domestic non-resource tradeable sector requires that:
Market clearing for the resources sector requires that:
The definition of domestic final demand is:
where is aggregate investment, defined as:
Nominal value added is given by:
Real value added is given by:
The current account equation is:
Note that we linearise the net foreign asset-to-GDP ratio, b*, rather than log-linearise this value to reflect the fact that it can take both positive and negative values.
Uncovered interest rate parity requires that:
The risk premium on foreign borrowing evolves according to:
The definition of the real exchange rate is:
The definition of consumer price inflation implies that:
for j ∈ {n,m,f} where is the relative price of good j, which evolves according to:
The domestic monetary policy reaction function is:
The foreign economy is characterised by three equations: an IS curve, a Phillips curve and a monetary policy reaction function.
The autoregressive shock processes are given by:
A.1 Observation Equations
The observation equations link the model variables to the observed variables in the data.
GDP growth:
Consumption growth:
Investment growth:
Non-tradeable value added growth:
Non-resource tradeable value added growth:
Resource value added growth:
Non-resource exports growth:
Resource exports growth:
Inflation:
Non-tradeable inflation:
Cash rate:
Nominal exchange rate growth:
Resource prices growth:
Foreign output growth
Foreign inflation
Foreign interest rates