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 (C_t) have been removed; and technology is assumed to be common across all countries (A_t). N_t denotes the labour input.

We specify the consumer's budget constraint as:

where: P_t is the price of aggregate consumption; and W_t is the wage rate. For simplicity we have expressed the budget constraint in terms of bond holdings B_t and their return R_t.

A.2 First-order Conditions

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

The relevant first-order conditions are:

Eliminating λ_t yields

If , 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 s_t), whereas q_t is defined inversely to this. Similarly, they define the coefficient of relative risk aversion as σ, so τ = σ⁻¹. Also, output (y_t) 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 , yields their Equation (27):

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

Solving for :

Hence,

We can first difference Equation (A6) to obtain an expression for ; 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]