RDP 2008-04: A Small BVAR-DSGE Model for Forecasting the Australian Economy Appendix A: Deriving the IS Equation

In this appendix, to be explicit, we change the notation slightly, and distinguish variables which have been log-linearised. We denote log deviations from steady state as lower-case variables with a superscript ~.[21]

A.1 The Consumer's Problem

Let the consumer's utility be of the form in Lubik and Schorfheide (2005), namely

Compared to Lubik and Schorfheide (2005), a few minor modifications have been made: τ is now the intertemporal elasticity of substitution (rather than the coefficient of relative risk aversion); habits in aggregate consumption (Ct) have been removed; and technology is assumed to be common across all countries (At). Nt denotes the labour input.

We specify the consumer's budget constraint as:

where: Pt is the price of aggregate consumption; and Wt is the wage rate. For simplicity we have expressed the budget constraint in terms of bond holdings Bt and their return Rt.

A.2 First-order Conditions

We express the consumer's problem above as a Lagrangian, L :

The relevant first-order conditions are:

Eliminating λt yields

If Inline Equation, then this can be expressed as

which is the Euler equation.

A.3 Log-linearisation

We log-linearise Equation (A4) using the steady-state condition that β-1 = R (assuming no steady-state inflation):

A.4 The IS Equation

In order to obtain the IS equation we use two results from Galí and Monacelli (2002), namely their Equations (25) and (16):

Note that these have been modified to take into account differences in notation; they define the terms of trade as the price of imports relative to exports (and denote it by st), whereas qt is defined inversely to this. Similarly, they define the coefficient of relative risk aversion as σ, so τ = σ−1. Also, output (yt) has been normalised by technology, which is not necessary in their paper as technology is assumed to be stationary. From Equations (A6) and (A7), and using the market-clearing condition Inline Equation, yields their Equation (27):

We can substitute Equation (A8) into Equation (A5) for Inline Equation and Inline Equation, which yields:

Solving for Inline Equation:

Hence,

We can first difference Equation (A6) to obtain an expression for Inline Equation; substituting this into the equation above yields the IS equation:

In contrast to the IS equation in Lubik and Schorfheide (2007), the coefficients on the expected growth in the terms of trade and technology are positive (and the latter is not unity).

Footnote

This appendix draws on work by Jamie Hall. We also thank Adam Cagliarini for his assistance. [21]