RDP 2013-14: Reserves of Natural Resources in a Small Open Economy Appendix D: The Small Open Economy in General Equilibrium

D.1 Domestic Households

We assume a continuum of identical domestic households of unit measure who are able to self-insure with each other, and so the problem we describe is isomorphic to a model with a representative agent. Each household has identical preferences given by the utility function

where C_t is an aggregate consumption bundle containing domestic and imported goods, V_t is an external habit, and are the households' supply of labour to the non-resource and resource sectors respectively, ξ_c is the coefficient of relative risk aversion, ξ_h is a parameter governing the convexity of preferences with regard to the aggregate supply of labour, γ_h governs the elasticity of substitution between labour supplied in the resource and non-resource sectors, and ς is a scaling parameter used to obtain a well-defined steady state.

We include an external habit or ‘catching up with the Jones’ , see Abel (1990)) because it permits a more flexible representation of consumption preferences. In particular, it allows for a non-unitary intertemporal elasticity of substitution while still being consistent with a detrended stationary representation of the general equilibrium economy. We use a constant elasticity of substitution function for the disutlity of labour to capture the idea that working in the resource and non-resource sectors are not perfect substitutes, from the perspective of households, and so there can be relative wage dispersion between the resource and non-resource sectors.

Regarding financial markets, domestic households can trade in either of two nominal risk-less bonds denominated in the domestic, B_t, and foreign currencies, , respectively. Following Benigno and Thoenissen (2008), we assume that when domestic residents issue claims in foreign currency they must pay a premium on this borrowing, Φ_t. The household budget constraint for an individual household is given by

where are the aggregate profits (dividends) of the resource, non-resource, import and non-resource export goods sectors paid to household respectively, T is a lump-sum tax used to fund subsidies that undo the steady state distortions associated with monopolistic competition (discussed further below), and i_t and are the domestic and foreign borrowing interest rates (the latter being measured net of the risk premium). Assuming that domestic bonds are in zero net supply, it follows that in equilibrium

Equation (D2) is the marginal utility of consumption. Equations (D3) and (D4) are the standard Euler equations associated with ability to trade in domestic and foreign currency-denominated bonds and imply that the return from saving (or cost of borrowing) should be equal to the forgone (additional) consumption enjoyed in the current period. Equations (D5) and (D6) are the standard intra-temporal conditions ensuring that the marginal return from working in each sector is equivalent to the households' marginal disutility from working, and Equation (D7) implies that the household budget constraint will bind.

We assume that the risk-premium on foreign borrowing is described by the following relationship

That is, we assume that the risk premium is a function of both the domestic economy's capacity to repay its debt, and the percentage deviation between the real exchange rate and resource prices. We include the latter term to capture the idea that changes in resource prices and the real exchange rate can have direct effects on risk premia. The parameters φ_b and φ_s govern the relative importance of each of these effects and are estimated.

For their intra-temporal consumption decisions (choosing domestic and imported consumption good expenditure), each household solves

subject to:

where and are the prices of the non-traded and imported goods purchased respectively and are taken as given. Defining the shadow price of the aggregate consumption bundle as , the optimality conditions are given by

where the shadow price of consuming an additional bundle of non-traded and imported goods is given by

To find the consumption allocations within the non-traded goods bundle, a household solves

subject to:

From which the shadow price and consumption allocations are given by

Solving the analogous problem for the imported goods consumption allocations we have

D.2 Domestic Non-tradeable Producers

We assume a continuum of non-tradeable consumption producers on the unit interval. Each non-tradeable producer, indexed by i, has access to a linear production technology

where is a common non-traded technology, is the quantity of non-traded labour used by firm i, and χ_t can be interpreted as a cost-push shock. That is, firms have to pay more for the energy they use when resource prices rise.^[33] We assume that χ_t follows has the same AR(1) process as that modelled for resource prices, but that the direct response of non-traded firms' marginal costs to a change in resource prices is a free parameter to be estimated (ϒ)

For competitive structure, we assume non-traded firms operate under monopolistic competition and are subject to a Calvo pricing friction. For the fraction (1−ϕ_n) of firms able to set their price optimally, they solve

where

is the marginal cost of production for a domestic non-traded producer, is a measure of common non-traded demand and τ_n is a subsidy used to undo the steady state distortion associated with the assumption of monopolistic competition (this is funded by the lump-sum tax on households and simplifies the calculation of the steady state). A recursive formulation of the implied optimality conditions is

where is the optimal reset price for the firm. It should be noted that in equilibrium all firms will choose the same optimal reset price given that there will be a degenerate wage distribution in an equilibrium where all households are identical and that there are no idiosyncratic shocks. For the remaining fraction (ϕ_n) of firms not able to choose their price, they simply retain the price they offered in the previous period. Accordingly, a measure of non-traded goods prices, the shadow price of an extra bundle of non-traded consumption goods, is

It is straightforward to verify that the total profit of domestically owned non-traded producers is given by

D.3 Domestic Importing Firms

We assume a continuum of importing firms of unit measure who are owned by domestic households. Importing firms purchase final output from the foreign sector at the foreign currency price, , and use this output to produce a differentiated imported good. The real marginal cost, common to all importers, in domestic currency terms is

Assuming that importers operate under monopolistic competition, and that a Calvo pricing friction exists for importers resetting their domestic currency price, we have

where is the optimal reset price chosen by importers able to choose their price, and ϕ_o is the probability that any given firm will not be able to re-optimise its price in a given period and retains its previous period price. The shadow price relevant for imported goods is

For determining import firm profits we define the alternative import price index

where total profit in the imported sector is given by

D.4 Non-resource Export Sector

Given the substantial interest in how resource sector developments can influence the non-resource export sector, we also assume a unit measure of domestically owned firms that engage in non-resource exporting (hereafter, exporters). An exporter, indexed by j, purchases a bundle of non-traded inputs from domestic producers and transforms this into a specialised export good. The demand for inputs from non-traded producer i by exporter j is given by

where (j) is the demand for exporter j's output. The real marginal cost for an exporter is

We assume the following (reduced-form) demand function for exports of type j

where is a measure of the common component of demand for non-resource exports, is foreign output, is the price of export type j in foreign currency terms, is an index of non-resource export prices in foreign currency terms, and is the foreign price index. Note that θ_x is the within sector elasticity of non-resource export demand, and that θ_* is the cross-sector (or common) elasticity of non-resource export demand. Consistent with Adolfson, Laséen, Lindé and Villani (2007), this formulation allows for both competition effects amongst firms within the exporting sector, and competition between the export sector as a whole and the rest of the world.

Assuming that exporters are monopolistically competitive, set their prices in foreign currency terms, and are subject to a Calvo pricing friction, a recursive formulation for determining their optimal reset price is

where the price index for non-resource exported goods is defined by

It will be useful for determining non-resource export firm profits to also define the alternative non-resource export price index

Total profit of exporters is thus

D.5 Monetary Policy

For domestic monetary policy we assume that the central bank follows a simple Taylor rule of the form

This rule is consistent with a forward-looking central bank that targets inflation, but also allows for gradual interest rate adjustment.

D.6 Market Clearing and the Rest of the World

For market clearing, supply must meet the demand for each non-traded good i

Aggregating demand and supply across the continuum of goods, it follows that

and where

Note that the common or non-idiosyncratic component of demand for non-traded goods is given by the sum of demand for non-traded consumption goods, demand for non-traded inputs which are then exported, and net demand for non-traded inputs used up in the exploration process.^[34]

For the rest of the world, we assume that prices and quantities admit the following VAR(p) representation

where is a vector collecting all foreign prices and quantities and ε_t is a 4×1 vector of reduced form shocks.

Definition 3. A small open economy general equilibrium with endogenous reserves, and under rational expectations, is given by sequences of quantities and prices that solve Equations (9) to (18) and (D2) to (D33) taking foreign quantities and prices as given by Equation (D34).

Definition 4. A small open economy general equilibrium with exogenous reserves, and under rational expectations, is given by sequences of quantities and prices that solve Equations (9), (11), (13) to (18), and (D2) to (D33) taking foreign quantities and prices as given by Equation (D34) and setting D_t = 0 and R_t = R for all t.

In view of the fact that the stock of natural reserves is potentially non-stationary, as highlighted in partial equilibrium, we still need to find a stationary representation of the above economy. Appendix D.7 identifies one stationary representation that has a locally stable solution.

D.7 Stationary Representation

Claim: A detrended representation of the general equilibrium economy in Definition (3) with a (locally) unique stable solution exists if

1. Log foreign demand and the log of the stock of domestic natural reserves are cointegrated

   

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2. The logs of resource sector and non-resource sector technology are also identically cointegrated with the log of the stock of domestic natural reserves

   

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and where |ϑ| < 1.

Verification. We verify this claim numerically. First, we claim that the following system is a detrended representation of the economy in Definition (3) with a unique stable solution (at the paramererisation of our model referred to in the main text):

and where

Inspecting the Blanchard-Kahn (BK) conditions of the above economy (when taking a first-order appoximation), we find that the BK conditions are not satisfied when ϑ = 1 (and so and become I(1) processes). Specifically, we find the presence of unit eigenvalues that are consistent with the linear approximation of this system admitting no stable solution. Imposing stationarity on either or was also not sufficient to satisfy the requirements for stability.

Some analytical intuition for this is as follows. Suppose that the above economy is a stationary representation and that in this equilibrium log reserves are an I(1) variable (as we demonstrated in partial equilibrium). In the absence of cointegration as specified above, the variables and become I(1) processes. Taking a first-order approximation of Equation (D72) it is clear that:

implying that will also be I(1) since will be I(0) under the claim that the above representation is stationary (when solved using a first-order approximation around the steady state). Taking a first-order approximation of Equation (D75):

But we see immediately that this cannot be an equilibrium solution since the right-hand side has a single variable that is I(1), while all other variables are I(0) under our claim. Consistent with this, the absence of cointegration between log foreign demand and log reserves is not consistent with the above economy admitting a unique stationary solution.

The intuition is similar when understanding why the log of domestic resource or the log of non-resource technology must also be cointegrated with the log of reserves. If either of these assumptions do not hold then inspecting the linear approximation of the above detrended representation makes it clear that some equations will represent a mixture of I(0) variables and a single I(1) variable, thus contradicting the claim that the above representation admits a unique stationary solution.

Footnotes

Note that this assumption is consistent with the use of commodity prices as a control for expected inflation in VARs that attempt to identify the effects of monetary policy shocks and address the so called ‘price puzzle’ (see, for example, Sims (1992)). It is also consistent with the observed correlation between commodity prices and inflation rates across countries that cannot be explained by correlation in real activity (see, for example, Gerard (2012)). [33]

To simplify calculation of the steady state, we assume that the government makes a constant lump-sum allocation of exploration inputs, , to the resource sector. One can think of this as analogous to tax incentives or government subsidies that encourage resource exploration. [34]