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 ζt 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 JN,t refers to investment that produces capital for use in the non-tradeable sector. It is distinct from IN,t 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:
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 γH,t = γH and so on.
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 ZX,t is a stationary sector-specific TFP shock:
At is a stationary technology shock that is common across sectors:
and Zt 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
κt 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:
ZN,t is a stationary sector-specific TFP shock:
and Zt 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:
ZH,t is a stationary sector-specific TFP shock:
and Zt 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
PF,t(i). The parameter ϛ represents
a subsidy to imported firms, funded by lump-sum taxation. We set the subsidy
equal to
ϛ = (θf − 1)/θf,
thereby ensuring that markups in this sector are zero in equilibrium.
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:
