RDP 2013-07: An Empirical BVAR-DSGE Model of the Australian Economy Appendix B: Steady State

Superscript bars denote steady state values. We log-linearise around a deterministic steady state where the real exchange rate,, is normalised to 1, and net foreign assets equal 0. Also, real commodity prices are exogenous to the small economy, so we set . Steady state inflation is assumed to be 0. In this section we provide the details necessary to calculate the steady state quantities which appear in the log-linearised model or were used in calibration.

B.1 Cash Rate

In the steady state, from the consumer's first-order condition for bond holdings .

B.2 Prices

Import prices

In the steady state, retail import prices are a constant mark-up over nominal marginal costs (alternatively, real marginal costs are constant):

Note also that

Domestic good prices

From the expression for the CPI we have:

Using the expression for real import prices above, we obtain:

Real marginal costs in the domestic goods sector

Similar to import prices, we have

B.3 Investment

Domestic goods gross rental rate

Tobin's Q in steady state is 1. From the equation for Tobin's Q we can obtain:

Export sector gross rental rate

Similarly:

Capital to output ratio, domestic good sector

From the factor demand equation

Investment to output ratio, domestic good sector

From the capital accumulation equation

Capital to output ratio, export sector

Investment to output ratio, export sector

Note for the measurement equation for investment , and that . An expression for the is developed below. We can then express and .

B.4 Trade

In steady state we have net foreign assets as a share of GDP equal to 0. This implies balanced trade. Further, we have

From which we can obtain:

Note that , so we have

B.5 Consumption

Lagrange multiplier on the budget constraint

Imports

We have the following demand function:

Domestic goods

Similarly, for domestic goods.

Consumption share of domestic goods

From the market clearing for domestic goods we have:

From this we can obtain

B.6 Labour

Labour, domestic goods

From the labour supply of consumers we can obtain an expression for the labour share:

From the labour demand of firms we have

Equating these, and using results from above, we obtain:

Similarly,

These can be substituted into

B.7 Real Money Balances

Note that from money demand we have

The can be obtained by rearranging the production function of domestic good firms to be:

As we have an expression for above we can solve for , and hence .

B.8 Value added

In steady state we have

The coefficients in the log-linearised expression for value-added are:

and

The ratios to value added, such as those in the net foreign assets equation, can easily be derived, e.g. . Similarly, . To determine the compensation of employees to nominal value added, i.e. , which were used in the calibration but not in the steady state of the model, the labour demand equations can be utilised, together with the previous expressions for the share of value added constituted by each sector's output and relative prices. In the domestic good sector and . For the export sector we have .