RDP 2015-07: A Multi-sector Model of the Australian Economy Appendix A: The Log-linearised Model

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