# RDP 2015-11: Unprecedented Changes in the Terms of Trade: Online Appendix 2 Derivation of Equilibrium Conditions

## 2.1 Households

The household's problem is to choose a sequence for Ct, LH,t, LN,t, LX,t, Ih,t, IN,t, IX,t, KH,t+1, KN,t+1, KX,t+1, Bt+1, to maximise expected lifetime utility subject to the budget constraint and capital accumulation equation. The Lagrangian for this problem is:

The first order conditions for the household's problem are:

and

where is the discount factor on nominal claims and Qj,t = Φtt is Tobin's Q for sector j.

Demand for each type of consumption good:

and for investment goods:

## 2.2 Firms

### 2.2.1 Commodity Firms

Given prices, commodity firms choose LX,t and KX,t to maximise profits, given by:

The resulting quantities are:

Note that this implies that Γx,t = 0.

### 2.2.2 Tradeable Firms

Profits for tradeable firms are given by:

The optimality conditions of the cost minimisation problem are:

where MCt denotes real marginal cost.

When setting prices, the problem of firm i is to choose PH,t(i) to maximise:

subject to the demand constraint that: YH,t(i) = YH,t. Optimal price setting implies that:

### 2.2.3 Non-tradeable Firms

Profits for non-tradeable firms are given by:

The optimality conditions of the cost minimisation problem are:

where MCN,t denotes real marginal cost.

The firm's pricing problem is to choose PN,t(i) to maximise:

Optimal price setting implies that the inflation rate of non-tradeable goods is given by:

## 2.3 Importing Firms

Profits for importing firms are given by

Optimal price setting implies that the inflation rate of imported goods is given by: