# RDP 2015-11: Unprecedented Changes in the Terms of Trade: Online Appendix 1 The Model

August 2015

This section outlines the non-linear model.

## 1.1 Households

The representative household maximises its expected lifetime utility:

where *ζ _{t}* is an intertemporal preference shock:

The household's utility function is given by:

where is a shock to labour supply:

Households maximise subject to the budget constraint:

and the capital accumulation equations:

where:

Note that *J _{N,t}* refers to investment that produces capital for use
in the non-tradeable sector. It is distinct from

*I*which refers to non-tradeable goods used to produce investment goods via the investment goods aggregator. The interest rate that the household receives on foreign bonds follows the process:

_{N,t}
where *b** is the steady-state net foreign asset-to-GDP ratio, is a risk-premium shock
that follows the process:

and follows the process

The household's consumption bundle is a CES aggregate of traded and non-traded goods, while the traded goods is itself a CES aggregate of home- and foreign-produced traded goods:

Note that the Cobb-Douglas specification allows us to assume that
*γ _{H,t}* =

*γ*and so on.

_{H}The non-traded, home-produced traded and imported consumption goods are themselves bundles of imperfectly substitutable goods:

The price indices corresponding to the consumption goods aggregates are:

The investment good is similarly a CES aggregate of non-traded and traded goods:

The price indices corresponding to the investment goods aggregates are:

The non-traded, home-produced traded and imported investment goods are themselves bundles of imperfectly substitutable goods:

## 1.2 Production

### 1.2.1 Commodity Firms

Commodity firms produce using the Cobb-Douglas production function:

where *Z _{X,t}* is a stationary sector-specific TFP shock:

*A _{t}* is a stationary technology shock that is common across
sectors:

and *Z _{t}* is a labour-augmenting technology shock that is common
across sectors whose growth rate follows:

We assume that the foreign price of commodities follows

where is the foreign price level and
*κ _{t}* follows the process:

The law of one price holds and so the domestic price of commodities is:

and the firm takes this as given when making its production decisions.

### 1.2.2 Non-tradeable Firms

Non-tradeable firms sell differentiated products, which they produce using the Cobb-Douglas production function:

*Z _{N,t}* is a stationary sector-specific TFP shock:

and *Z _{t}* is a labour-augmenting technology shock defined above.

Firms can only change prices at some cost, following a Rotemberg (1982) pricing mechanism:

The output of the non-traded sector is an aggregate of the output of each of the non-traded firms

### 1.2.3 Tradeable Firms

Tradeable firms produce using the Cobb-Douglas production function:

*Z _{H,t}* is a stationary sector-specific TFP shock:

and *Z _{t}* is a labour-augmenting technology shock defined above.

Firms can only change prices at some cost, following a Rotemberg (1982) pricing mechanism:

The output of the non-traded sector is an aggregate of the output of each of the non-traded firms

## 1.3 Importing Firms

Importing firms purchase foreign good varieties at the price and sell them in the domestic market at price
*P _{F,t}*(

*i*). The parameter

*ϛ*represents a subsidy to imported firms, funded by lump-sum taxation. We set the subsidy equal to

*ϛ*= (

*θ*

_{f}− 1)

*/θ*, thereby ensuring that markups in this sector are zero in equilibrium.

_{f}Importing firms can only change prices at some cost, following a Rotemberg [1982] pricing mechanism:

## 1.4 Foreign Sector

The rate of foreign goods price inflation is , which follows the process,

We also assume that foreign demand for the domestically produced tradable, , follows the process below;

where

and follows the process:

## 1.5 Relative Prices and Current Account

In what follows it will be convenient to define a number of relative prices:

Nominal net exports are given by:

And the current account equation is given by:

## 1.6 Monetary Policy

## 1.7 Market Clearing

Investment goods:

Labour market:

Non-tradeable goods:

Tradeables:

Imports:

Nominal GDP:

and Real GDP: