RDP 2009-01: Currency Misalignments and Optimal Monetary Policy: A Re-examination 2. The Model

The model is nearly identical to CGG's. It is based on two countries of equal size, while CGG allow the population of the countries to be different. Since the population size plays no real role in their analysis, the model here is simplified along this dimension. But two significant generalisations are made. The first is to allow for different preferences in the two countries. Home agents may derive greater utility from goods produced in the home country. Home households put a weight of on home goods and on foreign goods (and vice versa for foreign households.) This is a popular assumption in the open economy macroeconomics literature, and can be considered as a shortcut for modelling ‘openness’. That is, a less open economy puts less weight on consumption of imported goods, and in the limit the economy becomes closed if it imports no goods. The second major change, as already noted, is to allow for goods to be sold at different prices in the home and foreign countries.

The model assumes two countries, each inhabited with a continuum of households, normalised to a total of one in each country. Households have utility over consumption of goods and disutility from provision of labour services. In each country, there is a continuum of goods produced, each by a monopolist. Households supply labour to firms located within their own country, and get utility from all goods produced in both countries. Each household is a monopolistic supplier of a unique type of labour to firms within its country. Trade in a complete set of nominally denominated contingent claims is assumed.

Monopolistic firms produce output using only labour, subject to technology shocks.

In this section, no assumptions are made about how wages are set by monopolistic households or prices are set by monopolistic firms. In particular, prices and wages may be sticky, and there may be LCP or PCP for firms. The period loss function is derived for the policy-maker, which expresses the loss (relative to the efficient outcome) in terms of within-country and international price misalignments and output gaps. This loss function applies under various assumptions about how prices are actually set, and so is more general than the policy rules subsequently derived which depend on the specific nature of price and wage setting.

All households within a country are identical. It is assumed that in each period their labour supplies are identical. This assumption rules out staggered wage setting as in Erceg et al (2000), because in that model there will be dispersion in labour input across households that arises from the dispersion in wages. The set-up is consistent with sticky wages, but not wage dispersion. However, it is entirely straightforward to generalise the loss functions derived to allow for wage dispersion following the steps in Erceg et al. This is not done so that the model is more directly comparable to CGG.

2.1 Households

The representative household, h, in the home country maximises

C_t (h) is the consumption aggregate. We assume Cobb-Douglas preferences:

If v = 1, home and foreign preferences are identical as in CGG. There is home bias in preferences when v > 1.

In turn, C_Ht(h) and C_Ft(h) are CES aggregates over a continuum of goods produced in each country:

N_t(h) is an aggregate of the labour services that the household sells to each of a continuum of firms located in the home country:

Households receive wage income, W_t(h)N_t(h), and aggregate profits from home firms, Γ_t. They pay lump-sum taxes each period, T_t. Each household can trade in a complete market in contingent claims (arbitrarily) denominated in the home currency. The budget constraint is given by:

where D(h,∇^t) represents household h's pay-offs on state-contingent claims for state ∇^t. Z(∇^t+1 | ∇^t) is the price of a claim that pays one dollar in state ∇^t+1, conditional on state ∇^t occurring at time t.

In this equation, P_t is the exact price index for consumption, given by:

P_Ht is the home-currency price of the home aggregate good and P_Ft is the home-currency price of the foreign aggregate good. Equation (6) follows from cost minimisation. Also, from cost minimisation, P_Ht and P_Ft are the usual CES aggregates over prices of individual varieties, f:

Foreign households have analogous preferences and face an analogous budget constraint.

Because all home households are identical, we can drop the index for the household and use the fact that aggregate per capita consumption of each good is equal to the consumption of each good by each household. The first-order conditions for consumption are given by:

In Equation (11), an index for the state at time t is made explicit for the purpose of clarity. is the normalised price of the state-contingent claim. That is, it is defined as Z(∇^t+1|∇^t) divided by the probability of state ∇^t+1 conditional on state ∇^t.

Note that the sum of Z(∇^t+1|∇^t) across all possible states at time t + 1 must equal 1/R_t, where R_t denotes the gross nominal yield on a one-period non-state-contingent bond. Therefore, taking a probability-weighted sum across all states of Equation (11), we have the familiar Euler equation:

Analogous equations hold for foreign households. Since contingent claims are (arbitrarily) denominated in home currency, the first-order condition for foreign households that is analogous to Equation (11) is:

Following the unfortunate notation of CGG, E_t is the nominal exchange rate, defined as the home-currency price of foreign currency, and should not be mistaken for the conditional expectations operator. As noted above, at this stage the labour input of all households is assumed to be the same, so N_t = N_t(h).

2.2 Firms

Each home good, Y_t (f) is made by firm f according to a production function that is linear in the labour input. These are given by:

Note that the productivity shock, A_t, is common to all firms in the home country. N_t(f) is a CES composite of individual home-country household labour, given by:

where the technology parameter, η_t, is stochastic and common to all home firms.

Profits are given by:

In this equation, P_Ht(f) is the home-currency price of the good when it is sold in the home country and is the foreign-currency price of the good when it is sold in the foreign country. C_Ht(f) is aggregate sales of the good in the home country:

Sales of the same good in the foreign country, , is defined analogously. It follows that . The subsidy for using labour is τ_t.

There are analogous equations for , with the foreign productivity shock given by , the foreign technology parameter shock given by , and foreign subsidy given by .

2.3 Equilibrium

Goods market-clearing conditions in the home and foreign countries are given by:

S_t and are used to represent the price of imported goods relative to locally produced goods in the home and foreign countries, respectively (the inverse of the standard definition for the terms of trade):

Equations (11) and (13) provide the familiar condition that arises in open economy models with a complete set of state-contingent claims when PPP does not hold:

This condition equates the marginal rate value of a dollar for home and foreign households, in terms of its purchasing power over aggregate consumption in each country.

Total employment is determined by output in each industry:

where