Commodity price shocks and real business cycles in a small commodity-exporting economy

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1 Commodity price shocks and real business cycles in a small commodity-exporting economy Valery Charnavoki February 8, Abstract This paper analyzes an effect of the world commodity price shocks on business cycles in a small commodity-exporting economy. It is shown that model with complete markets, separable preferences and no financial frictions cannot generate the main stylized facts regarding this economy, in particular, volatile and negatively correlated with commodity prices real exchange rate, Dutch disease and a positive effect of commodity prices on investment. To replicate these facts frictions in asset trade were introduced into the model. They create a wedge between stochastic discount factors and marginal rates of substitution in consumption damping international risk sharing. Keywords: commodity currency, real business cycles, real exchange rate, Dutch disease JEL classification: E3,F4,Q43 Department of Economics, Universidad Carlos III de Madrid, C/Madrid 6, 893, Getafe Madrid), Spain, vcharnav@eco.uc3m.es, Phone: +34)693658, Fax: +34)964939

2 Introduction The share of primary commodities in world output and trade has declined over the last century. But fluctuations in commodity prices still significantly affect economic activity, especially in countries, where primary commodities remain a main source of export earnings. In these countries commodity price movements have enormous impacts on real output, exchange rate, balance of payments and public finance, and, as a result, pose serious problems for the conduct of macroeconomic policy. Cashin, Cespedes, and Sahay 4) analyzed the data on the world prices of 44 commodities and national commodity export shares over 98-, and chose 58 commodityexporting countries, among which five developed Australia, Canada, Iceland, Norway and New Zealand). Most of these countries have small economy, in the sense that they cannot affect significantly the world prices and particularly prices of exported commodities). Dependence on the export of small number of primary goods with highly volatile and persistent world prices makes the business cycles dynamics in these countries different from those in countries with differentiated pattern of international trade. The economic literature emphasizes several important stylized facts about commodity-exporting economies. First, trade balances and current accounts in these economies are positively correlated with terms of trade and world prices of exported commodities. When commodity prices are high these countries export more than import accumulating foreign assets or decreasing foreign debt), whereas at the time of low commodity prices trade balances plummet. In particular, Kilian, Rebucci, and Spatafora 7) found that for the oilexporting economies trade balances are positively correlated with dynamics of oil prices. This comovement of commodity prices and external balances signalizes about the presence of international risk sharing and consumption smoothing, though risk sharing may be imperfect. Second, real exchange rates in commodity-exporting countries are highly volatile and negatively correlated with real commodity prices. An increase in commodity price results in appreciation of the real exchange rate, whereas its decrease is associated with real exchange rate depreciation. Cashin et al. 4) analyzed long-run cointegrating relationship between the real exchange rates and real prices of exported commodities for their sample of 58 commodity-exporting countries and found that for 9 of these countries there is statistically significant long-run relationship. Chen and Rogoff 3) revealed a long-run comovement of the real exchange rates and real commodity prices for three developed commodity-exporting countries: Australia, Canada and New Zealand. Cashin et al. 4) sample does not include OPEC countries and developing countries for which data are not available. Because of lack of data Cashin et al. 4) did not take into account other possible long-run factors affecting real exchange rate. In particular they did not consider the impact of productivity differential in tradable and nontradable sectors Balassa-Samuelson effect). Inclusion of these additional factors may increase a number of commodity currencies.

3 The third fact is related to the previous one. The empirical evidence suggests that the consumption differential across countries does not move in any systematic way with its relative price, i.e. the real exchange rate. It is in contrast to predictions of many international business cycle models assuming complete financial markets, which suggest that consumption across countries should be higher in the country where its price, converted into a common currency is lower. This collision is called in economic literature as consumption-real exchange rate anomaly or Backus-Smith puzzle Backus and Smith, 993). Although this puzzle is observed not only for commodity-exporting economies, for the last group it is especially pronounced. Negative, not predicted positive, correlation of the relative consumption and real exchange rate is often observed for these countries. Along with volatile and negatively correlated with commodity price real exchange rate it may be considered as a signal of imperfections in international risk sharing. Forth, fluctuations in the real exchange rate, induced by movements in commodity prices, result in cross-sectoral changes in commodity-exporting economy. That is one of the manifestations of Dutch disease. Rising commodity prices lead to appreciation of the real exchange rate and therefore to fall in price competitiveness and output of home manufacturing sector comparing to foreign, whereas output of home nontradable and commodity sectors increases. Stijns 3) provides some empirical evidence of sectoral changes in commodity-exporting economy after commodity price shocks using gravitational trade model. Bayoumi and Muhleisen 6) discuss this effect for Canada. And, finally, there is evidence of a positive relationship between commodity prices and investment in commodity-exporting economy Spatafora and Warner, 999). Appreciation of the real exchange rate, associated with the increase in commodity prices, leads to decrease in relative prices of investment goods, which are predominantly tradable and imported. As a result, investment demand increases. Much of this investment goes into the nontradable and commodity sector of the economy. A range of dynamic general equilibrium models were proposed in the economic literature to explain stylized facts about impact of the world commodity price shocks on business cycles in small commodity-exporting economy. The early attempt to model Dutch disease in two-country deterministic dynamic framework was made by Bruno and Sachs 98). Mendoza 995) analyzes the quantitative importance of terms of trade shocks in driving business cycles using a dynamic stochastic small open economy model. Kose ) extends Mendoza work by developing a richer production structure that captures several empirically relevant features of developing economies. Sosunov and Zamulin 7) study monetary policy in economy sick with Dutch disease. The last three models are constructed in line with new open economy macroeconomics paradigm and feature small open economy structure, multiply goods, exogenous world commodity price shocks and one non-risky internationally traded asset. However, these models have several drawbacks. 3

4 First, a world economy is not modeled explicitly, like in two-country models. The foreign variables such as world commodity price, world interest rate and foreign demand on home tradable goods are introduced as exogenous stochastic shocks, and interrelations between them are either modeled through variance-covariance matrix or not specified at all. But changes in the world prices are not shocks per se. They reflect responses of the world economy to supply or demand shocks. For example, oil price growth can be result of decreasing world oil supply on the one side or increasing oil demand from growing manufacturing sector on the other. The effect of these world-wide shocks on small open economy may be different see Kilian, 6). So, the source of commodity price changes is important and should be modeled explicitly. Second, these models usually make very specific assumptions about production structure of economy. For example, in all of these models non-commodity tradable sector in home economy is completely ousted. In Mendoza 995) capital in non-tradable sector is specific factor of production. Although an asymmetry between home and foreign economies is important for study of commodity-exporting economies, it is possible to introduce it assuming different levels of productivity in home and foreign tradable, nontradable and commodity sectors. pattern of international trade observable in reality. A law of comparative advantages will create than a And finally, in these models there is only one non-risky internationally traded asset. An assumption of incomplete markets in stochastic framework of small open economy model results in steady state that depends on initial conditions and equilibrium dynamics that possess a random walk component. To induce stationarity standard models usually assume non-separable preferences or some form of frictions in assets trade Schmitt-Grohe and Uribe, 3). But these additional elements, not assumption of incomplete markets itself, are responsible for reproduction of many stylized facts in aforementioned models. Indeed, as Chari, Kehoe, and McGrattan ) show, the models with incomplete markets and without other frictions fail to solve Backus-Smith and exchange rate volatility puzzles. Besides, their dynamics is close to those generated by the models with complete markets. 3 So, an assumption of incomplete markets is not too crucial and working with complete markets model may be more convenient. In this paper a new real business cycles model of small commodity-exporting economy is proposed. The foreign world) economy is modeled explicitly, so all world prices are determined endogenously as a result of productivity and commodity shocks in foreign country. The only difference between home and foreign economies is different productivity levels in tradable, nontradable and commodity sectors, such that home economy appears 3 Benigno and Thoenissen 7) claim that a model with incomplete markets and nontradable intermediate goods can solve Backus-Smith puzzle, though it fails to generate volatile real exchange rate. Therefore, in this paper we report also impulse responses for the model with incomplete markets. Nevertheless, simulations show that dynamics of this model are very close to the model with complete markets. That is in line with results of Chari et al. ). 4

5 to be relatively commodity abundant. In the initial version of the model complete financial markets and no frictions in intertemporal trade are assumed. This model generates positive correlation between commodity price and trade balance, but cannot reproduce volatile and negatively correlated with world commodity price real exchange rate. The consumption differential moves in lockstep with real exchange rate, illustrating Backus-Smith puzzle. In this model international risk sharing is working well and frictions in intratemporal trade are not high enough to induce significant volatility in the real exchange rate. Because of that there is no manifestation of Dutch disease and investment effect. Next, the opposite case will be considered, when home country has no access to financial markets but intratemporal trade is allowed financial autarky). Because international risk sharing via trade in financial assets is not possible, commodity price shocks induce volatile real exchange rate, which is negatively correlated with commodity price and relative consumption. This model also predicts Dutch disease and investment effect. But assumption of financial autarky is too strict and contradicts observable fluctuations in trade balances. To increase a volatility of the real exchange rate and to damp risk sharing in the model with complete markets we need to break link between relative consumption and real exchange rate. There are several ways to do it. The first approach is to introduce occasionally binding borrowing constraints. These constraints may be exogenous, given, for example, by rule which prevents country to borrow or save beyond some limits. It is possible also to model these constraints endogenously assuming limited enforcement of international contracts, as in Kehoe and Perri ) and Alvarez and Jermann ). 4 The second strategy is to assume frictions in assets trade. For example, agents may face convex costs of holding assets in quantities different from some long-run level Schmitt- Grohe and Uribe, 3). 5 The last strategy is applied in this paper. To adjust the value of portfolio of international assets home households have to bear transaction costs. These costs may reflect portfolio management costs or the costs of legislative control on international capital movements. This model succeeded to replicate aforementioned stylized facts about commodity-exporting economy. 4 In the framework of our small open economy model two-sided limited commitment is not possible, because the value of autarky for foreign economy coincides with value of intertemporal trade. The case of one-sided limited commitment is not very interesting, since small economy can effectively achieve risk sharing under full commitment by accumulating assets, which would be seized in the case of default. 5 An alternative way to break link between relative consumption and real exchange rate and to increase volatility of the real exchange rate without introduction of frictions in assets trade is to relax an assumption of additive separable preferences. Nonseparable preferences create wedge between relative consumption and cross-country marginal rate of substitution of consumption though the latter is still equal to the real exchange rate. For example, Verdelhan 5) discusses this effect using habit-based preferences. 5

6 The rest of the paper is organized as follows. Section reports empirical evidence about business cycles dynamics for some commodity-exporting countries. Section 3 presents a framework for modeling a world economy. Section 4 discusses the model of small commodity-exporting economy with complete markets. Section 5 presents a model of financial autarky. Section 6 introduces frictions in assets trade into the model with complete markets. Section 7 discusses calibration of the model and computational algorithm. Section 8 presents steady-state equilibrium and undertakes simulation and impulse response analysis. Section 9 concludes. Empirical evidence for commodity-exporting countries In this section we briefly discuss an empirical evidence about business cycles in some commodity-exporting countries. We concentrate here on four developed economies, data for which are abundant: Australia, Canada, Norway and New Zealand. All these countries are included in Cashin et al. 4) sample. 6 Figures - illustrate some stylized facts about business cycles dynamics in these countries. The upper part of these figures shows relationship between relative consumption and effective real exchange rate. 7 The middle part illustrates dynamics of the real commodity price 8 and effective real exchange rate. And finally, the lowest part shows trade balance in % of GDP and real commodity price. Table summarizes the second moments of data for these countries, computed both for the levels and for HP-filtered series. 9 Data are presented both for the period of 97-8 and for the less turbulent subperiod of In what follows we will discuss the results for HP-filtered series. As the table shows, the real commodity price is very volatile. Its standard deviation varies from 7.95% in New Zealand to 4.6% in Norway. The effective real exchange rate is volatile and negatively correlated with the real commodity price. This negative correlation is more apparent for the recent period of and is more pronounced for Australia and New Zealand. These countries also have the highest volatility of the real 6 According to UN COMTRADE database the top three exported primary commodity two-digits) in 7 for these countries are the following the share in total export is reported). Canada: petroleum, petroleum products and related materials -.5%, gas, natural and manufactured - 7%, non-ferrous metals - 5.4%. Australia: metalliferous ores and metal scrap -.5%, coal, coke and briquettes -.4%, non-ferrous metals - 8.%. Norway: petroleum, petroleum products and related materials %, gas, natural and manufactured - 9.3%, non-ferrous metals - 7.7%. New Zealand: dairy products and birds eggs -.6%, meat and meat preparations -.%, vegetables and fruit - 5.3%. 7 Relative consumption is computed against country or block of countries with the highest share of mutual trade: Canada vs. US, Australia vs. OECD, Norway vs. EU and New Zealand vs. Australia. 8 The real commodity price is computed as price index of exported primary commodities in US dollars deflated by CPI in USA. 9 The levels of data contain nonstationary component reflecting low-frequency changes between countries, so HP-filtered data are more useful for our business cycles analysis. 6

7 exchange rate. Real commodity price is positively correlated with real GDP, investment except of Norway) and trade balance. For all countries Backus-Smith puzzle is observed. Besides, for Canada, Australia and New Zealand the correlation between the real exchange rate and relative consumption is negative for recent data. The most interesting pattern is shown by Norwegian data. Though the real commodity price index in this country is the most volatile in our sample, its effective real exchange rate has the lowest standard deviation. Besides, negative correlation between real exchange rate and commodity price is not so strong, -.4, comparing to -.4 in Australia. Investment in Norway is negatively correlated with real commodity price, that is in contrast to other developed commodity-exporting countries. Trade balance is closely linked with real commodity price. Consumption differential with EU is not correlated with the real exchange rate, in addition to that for the most recent data this correlation is positive. As will be shown in our paper all this facts may signal about better functioning of international risk sharing in Norway comparing to other countries. 3 Model of the world economy This section presents a general framework for modeling the world economy. 3. General description of the model The world economy is modeled as a continuum of small open economies represented by unit interval, as in Gali and Monacelli 5). All economies have the same preferences, technology and market structure. Assets markets are complete. A typical small economy produces three goods: differentiated tradable good, nontradable good and homogeneous tradable commodity. Tradable good can be used for consumption or investment either in domestic economy or abroad, whereas nontradable good can be invested or consumed only in domestic country. Commodity can be consumed, exported or used along with labor and capital as an input in production of tradable and nontradable goods. The firms producing tradable and nontradable goods and commodity endowments are owned by domestic households. World economy is affected by productivity shocks in tradable, nontradable and commodity sectors. We have computed real GDP using consumption deflator. 7

8 3. Households A typical small economy i is inhabited by representative household, which maximizes expected life-time utility: max {c t,l t,a t+ } E β t U c t i), l t i)) ) t= subject to a sequence of budget constraints expressed in units of domestic consumption good: c t i) + E t q t,t+ i)a t+ i) a t i) + w t i)l t i) + Π t i) ) where c t is a composite consumption good, l t denotes hours worked, w t is real wage, a t+ are holdings of Arrow-Debreu securities with price kernel q t,t+, Π t denotes profits from domestic firms and commodity endowment. All prices are given in terms of domestic consumption good and can be converted into prices in terms of consumption good of country j by division by bilateral real exchange rate e t i, j) the price of consumption good in j in terms of consumption good in country i). The budget constraint is supplemented with natural borrowing constraints to prevent Ponzi game: a t i) a 3) The composite consumption good c t is a CES basket of the tradable c T,t, nontradable c N,t goods and commodity c X,t : c t i) = α ɛct c ɛ ɛ T,t i) + α ɛ CN c ɛ ɛ N,t i) + α CT α CN ) ɛ ) ɛ ɛ c ɛ X,t i) ɛ 4) where ɛ denotes an elasticity of substitution between tradable and nontradable goods gross complementarity is assumed: ɛ < ) and α CT and α CN reflect the weights of tradable and nontradable goods in composite index. The index of tradable goods c T,t is in turn a CES basket of the home c H,t and foreign c F,t tradable goods: c T,t i) = α θch c θ θ H,t i) + α CH) θ ) θ θ c θ F,t i) θ 5) where θ is an elasticity of substitution between home and foreign tradable goods with ɛ < θ) and α CH reflects the home bias in consumption of tradable goods. So, an increase in bilateral real exchange rate e t i, j) reflects depreciation of the real exchange rate in country i and respectively its appreciation in country j. No arbitrage implies that e t i, j) = e t j, i) and e t i, j)e t j, k) = e t i, k). 8

9 Foreign tradable good c F,t is an aggregate of all imported tradable goods: c F,t i) = ) θ c θ θ θ F,t i, j)dj 6) Households allocate their consumption by solving expenditure minimization problem taking prices of goods as given). 3.3 Firms The markets for all goods in the model are perfectly competitive. All the firms and commodity endowments are assumed to be owned by domestic households. Technology in tradable and nontradable sectors is symmetric across all the world. A typical firm producing nontradable good has the following production function: y N,t i) = φ N ) ξ A N,t i)k β NK N,t ) ξ i)l β ξ NK N,t i) + φ ξ ξ N x ξ ξ N,t i) ) ξ 7) where A N,t is a productivity in the nontradable sector, k N,t, l N,t and x N,t are capital, labor and commodity inputs, ξ is elasticity of substitution between value added and commodity input and φ N denotes share of commodity input in total costs of nontradable firm. This firm maximizes the expected discounted value of profits: max {k N,t+,l N,t,x N,t,i d N,t} E t= with profits Π N,t determined as: β t U c c t i), l t i)) U c c i), l i)) Π N,ti) 8) Π N,t i) = p N,t i)y N,t i) p I,t i)i d N,ti) w t i)l N,t i) p X,t i)x N,t i) 9) where p N,t, p I,t and p X,t denote prices of nontradable good, investment and commodity respectively. Maximization is subject to constraints by production function 7) and by the following capital transition equation: ) i d N,t i) k N,t+ i) = δ)k N,t i) + k N,t i)ψ k k N,t i) where δ is depreciation rate and i d N,t denotes new investment into nontradable sector with installation costs determined by function Ψ k ). The capital adjustment costs are Capital adjustment costs function Ψ k ) has to satisfy the following properties in steady state: Ψ k δ) = δ, Ψ k δ) = and Ψ k δ) = d <. ) 9

10 introduced into this model to avoid excessive equilibrium volatility of investment. An initial capital stock in nontradable sector k N, i) is given. A typical firm in tradable sector solves a similar problem. It maximizes the expected discounted value of profits: max {k T,t+,l T,t,x T,t,i d T,t} E t= with profits Π T,t determined as: β t U c c t i), l t i)) U c c i), l i)) Π T,ti) ) Π T,t i) = p H,t i)y T,t i) p I,t i)i d T,ti) w t i)l T,t i) p X,t i)x T,t i) ) where k T,t, l T,t and x T,t are capital, labor and commodity inputs and p H,t is price of domestic tradable good. Maximization is constrained by production function and capital transition equation: y T,t i) = φ T ) ξ A T,t i)k β T K T,t ) ξ i)l β ξ T K T,t i) k T,t+ i) = δ)k T,t i) + k T,t i)ψ k i d T,t i) k T,t i) ξ ) ξ + φ ξ ξ T x ξ T,t i) ) where A T,t is a productivity in the tradable sector, φ T is share of commodity input in total costs of tradable firm, δ is depreciation rate and i d T,t denotes new investment into tradable sector with installation costs Ψ k ). An initial capital stock in tradable sector k T, i) is given. Investment good i t is a CES basket of the tradable i T,t and non-tradable i N,t goods: 3) 4) i t i) = α ɛit i ɛ ɛ T,t i) + α IT ) ɛ ) ɛ ɛ i ɛ N,t i) ɛ 5) where α IT reflects the weight of tradable goods in composite investment index. It is assumed that investments are relatively intensive in using of tradable goods comparing to consumption, so α IT α CT > α IN α CN with α IN = α IT. Investment of tradable goods i T,t is a CES basket of the home i H,t and foreign i F,t tradable goods: i T,t i) = α θih i θ θ H,t i) + α IH) θ ) θ θ i θ F,t i) θ 6) where i F,t i) = ) θ i θ θ θ F,t i, j)dj 7) Firms allocate their investment demand by solving expenditure minimization problem

11 taking prices of goods as given). Production in commodity sector is exogenously determined and does not incur costs. Therefore, the profits of this sector are product of commodity endowment X t and its price p X,t : Π X,t i) = p X,t i)x t i) 8) Total profits from nontradable, tradable and commodity sectors are distributed across domestic households, so: Π t i) = Π N,t i) + Π T,t i) + Π X,t i) 9) 3.4 Market clearing conditions All goods, factors and assets markets clear at any time and any contingency. Markets of tradable and nontradable goods: y N,t i) = c N,t i) + i N,t i), i ) y T,t i) = c H,t i) + c F,t j, i)dj + i H,t i) + Markets of commodity, capital and labor: X t i)di = c X,t i)di + x T,t i)di + i F,t j, i)dj, i ) x N,t i)di ) i t i) = i d T,ti) + i d N,ti), i 3) l t i) = l T,t i) + l N,t i), i 4) Assets markets: e t j, i)a t i)di =, j 5) 3.5 Equilibrium In this section we characterize an equilibrium of the world economy. The optimal first-order conditions from the problem )-3), characterizing the savings and consumption-leisure decisions of the representative household are given by the following equations: q t,t+ i)u c c t i), l t i)) = βu c c t+ i), l t+ i)) 6) w t i)u c c t i), l t i)) = U l c t i), l t i)) 7)

12 supplemented with transversality condition: lim E tq t,t+t i)a t+t i) = 8) T Consumption demand functions for tradable, nontradable goods and commodity, derived from the expenditure minimization problem of the representative household are: c t i) = α ɛct c ɛ ɛ T,t i) + α ɛ CN c ɛ ɛ N,t i) + α CT α CN ) ɛ ) ɛ ɛ c ɛ X,t i) c T,t i) = α CT p ɛ T,t i)c ti) 3) c N,t i) = α CN p ɛ N,t i)c ti) 3) c X,t i) = α CT α CN )p ɛ X,t i)c ti) 3) Consumption demand functions for home and foreign tradable goods are given by: c T,t i) = α θch c θ θ H,t i) + α CH) θ ) θ θ c θ F,t i) θ 33) ) θ ph,t i) c H,t i) = α CH c T,ti) 34) p T,t i) ) θ pf,t i) c F,t i) = α CH ) c T,ti) 35) p T,t i) ) θ c F,t i) = c θ θ θ F,t i, j)dj 36) ) θ ei, j)ph,t j) c F,t i, j) = c F,ti) 37) p F,t i) The first-order intertemporal conditions of the profit maximization problems of nontradable 7)-) and tradable )-4) firms have the following form: U c c t+ i), l t+ i)) p KN,t+ i) βe t U c c t i), l t i)) p KN,t i) + r N,t+i) δ + Λ N,t+ i)) = 38) βe t U c c t+ i), l t+ i)) U c c t i), l t i)) ɛ 9) p KT,t+ i) p KT,t i) + r T,t+i) δ + Λ T,t+ i)) = 39) where r N,t+ and r T,t+ are real returns to capital, p KN,t = p I,t /Ψ k p I,t /Ψ i d ) T,t i d ) k k T,t are prices of capital and Λ N,t = Ψ N,t k k N,t id N,t k N,t Ψ k k T,t ) i d ) T,t k T,t i d ) N,t k N,t i d ) N,t k N,t and p KT,t = and Λ T,t = i d Ψ T,t k id T,t k T,t Ψ k are marginal effects of increasing capital in period t on installation costs in period t for nontradable and tradable firms respectively. Nontradable and tradable firms demands on capital, labor and commodity inputs are

13 determined by the following equations: ) rn,t i)p KN,t i) k N,t i) = β NK v N,ti) 4) p V N,t i) ) rt,t i)p KT,t i) k T,t i) = β T K v T,ti) 4) p V H,t i) ) wt i) l N,t i) = β NK ) v N,ti) 4) p V N,t i) ) wt i) l T,t i) = β T K ) v T,ti) 43) p V H,t i) ) ξ px,t i) x N,t i) = φ N y N,ti) 44) p N,t i) ) ξ px,t i) x T,t i) = φ T y T,ti) 45) p H,t i) v N,t i) = A N,t i)k β NK N,t i)l β NK N,t i) 46) v T,t i) = A T,t i)k β T K T,t i)l β T K T,t i) 47) ) ξ pv N,t i) v N,t i) = φ N ) y N,ti) 48) p N,t i) ) ξ pv H,t i) v T,t i) = φ T ) y T,ti) 49) p H,t i) where v N,t and v T,t denote value added of nontradable and tradable firms respectively with prices p V N,t and p V T,t. 3

14 Investment demand functions: i t i) = α ɛit i ɛ ɛ T,t i) + α IT ) ɛ ) ɛ ɛ i ɛ N,t i) ɛ 5) ) ɛ pit,t i) i T,t i) = α IT i t i) 5) p I,t i) ) ɛ pn,t i) i N,t i) = α IT ) i t i) 5) p I,t i) i T,t i) = α θih i θ θ H,t i) + α IH) θ ) θ θ i θ F,t i) θ 53) ) θ ph,t i) i H,t i) = α IH i T,ti) 54) p IT,t i) ) θ pf,t i) i F,t i) = α IH ) i T,ti) 55) p IT,t i) ) θ i F,t i) = i θ θ θ F,t i, j)dj 56) ) θ ej, i)ph,t j) i F,t i, j) = i F,ti) 57) p F,t i) The law of one price for commodity prices and for prices of Arrow-Debreu securities no arbitrage conditions): p X,t i) = e t i, j)p X,t j) 58) q t,t+ i) = e ti, j) e t+ i, j) q t,t+j) 59) The trade balance of country i is given by: tb t i) = p V H,t i)v T,t i) + p V N,t i)v N,t i) + p X,t i)x t i) c t i) p I,t i)i t i) 6) tb t i) = E t q t,t+ i)a t+ i) a t i) 6) Equations 6)-6) together with capital transition equations ), 4), production functions 7), 3) and market clearing conditions )-5) characterize equilibrium of the model of the world economy. 4 Model of small commodity exporting economy with complete markets Now we modify the modeling framework introduced in the previous section to present the model of small commodity-exporting economy. We keep here an assumption of complete assets markets. 4

15 Among continuum of small open economies there is one economy of measure zero, called home economy. All other foreign) economies are completely symmetric: they have the same productivities in nontradable and tradable sectors, A N,t and A T,t, and commodity endowments, Xt, for any period and contingency. 3 Home country is assumed to be commodity abundant in steady state: A T /A T > A N/A N > X /X 6) where A T A T ), A N A N ) and X X ) denote steady-state productivity of tradable and nontradable firms and commodity endowments in home foreign) country respectively. Representative households and firms problems for these economies and market clearing conditions are given in )-5). Since home economy has measure zero it does not affect an equilibrium in foreign economies. A symmetry across foreign countries implies in turn a symmetry of their equilibrium allocations and prices. 4 So, we can treat typical foreign economy as a world economy and instead of continuum of economies consider only two: small home economy and representative foreign economy. To find an equilibrium of this model we will proceed in two steps. Initially, we compute an equilibrium of the representative foreign economy. It provides us with the world prices of commodity p X,t, foreign tradable good p T,t and Arrow-Debreu securities q t,t+ and foreign demands on tradable consumption c T,t and tradable investment i T,t. These prices and allocations are contingent on foreign productivity and commodity shocks A T,t, A N,t and X t. An equilibrium of home economy is computed in second stage, taking the world demand and prices as given. markets are given in Appendix A. All equilibrium conditions of the model with complete In assumptions of complete markets and frictionless assets trade the first-order optimal conditions 6) for home and foreign representative households and the law of one price 59) for Arrow-Debreu securities imply that: e t+ U c c t+, l t+ ) e t U c c t, l t ) Therefore, = U c c t+, lt+) U c c t, lt ) 63) e t U c c t, l t ) = ϑu c c t, l t ) 64) where ϑ is constant in any period and contingency and reflects an initial wealth position 3 Hereinafter the variables related to foreign economies are marked with asterix. 4 In particular, symmetric equilibrium for the world economy implies that c F,t i) = c F,t i, j) = c F,t, i F,t i) = i F,t i, j) = i F,t, p T,ti) = p H,t i) = p F,t i) = p T,t and, consequently, c T,ti) = c H,t i) + c F,t i) = c T,t and i T,ti) = i H,t i) + i F,t i) = i T,t for all foreign economies i and j. So, in equilibrium differentiated foreign tradable goods can be treated as one homogeneous foreign tradable good. Besides, equilibrium financial asset holdings of foreign economy are null for any period and contingency: a t i) =. 5

16 of home economy comparing to foreign economy. So, given correction for wealth differences, marginal rate of substitution between home and foreign consumption has to be equal to their relative price, i.e. real exchange rate. In other words, other things being equal, consumption is higher in the country where its price, converted into a common currency, is lower. 5 It is in contrast to empirical evidence, which suggests that there are no any systematic comovement between relative consumption and real exchange rate. This collusion is well known in economy literature and is called consumption-real exchange rate anomaly or Backus-Smith puzzle Backus and Smith, 993). It should be emphasized that condition 64) does not imply in general perfect international risk-sharing, understood as cross-country comovement of consumption in response to productivity and commodity shocks. As Brandt, Cochrane, and Santa-Clara 6) stress, risk sharing requires frictionless goods markets. They argue: Suppose that Earth trades assets with Mars by radio, in complete and frictionless capital markets. If Mars enjoys a positive shock, Earth-based owners of Martian assets rejoice in anticipation of their payoffs. But trade with Mars is still impossible, so the real exchange rate between Mars and Earth must adjust exactly to offset any net payoff. In the end, Earth marginal utility growth must reflect Earth resources, and the same for Mars. Risk sharing is impossible. If the underlying shocks are uncorrelated, the exchange rate variance is the sum of the variances of Earth and Mars marginal utility growth, and we measure a zero risk sharing index despite perfect capital markets. At the other extreme, if there is costless trade between the two planets teletransportation), and the real exchange rate is therefore constant, marginal utilities can move in lockstep. With constant exchange rates, we measure a perfect risk sharing index of one. In the model introduced here there are nontradable goods and consumption of tradable goods is home-biased. So, the productivity shocks in tradable and nontradable sectors result in volatility of the real exchange rate, reflecting imperfection of risk sharing. Nonseparability of consumption and leisure in utility together with nontradability of labor also imply some degree of real exchange volatility. 5 Nonseparability of consumption and leisure in utility implies that correlation between relative consumption and real exchange rate is not perfect in contrast to correlation between cross-country marginal rate of substitution and real exchange rate. Nevertheless, simulation results show that this effect is negligible. 6

17 5 Model of financial autarky In the model with complete markets international risk sharing is not too bad despite the presence of frictions in international trade. In other words, these frictions are not sufficient to induce an observed high volatility of the real exchange rate in commodity-exporting economies. International capital markets are prefect and provide enough insurance against productivity and commodity shocks. Now we assume that international financial markets disappeared and home economy has no access to international borrowing and saving, but is allowed to trade intratemporally. This is the case of financial autarky which to some extent is opposite extreme to the model with complete markets. Home representative household in this model maximizes expected life-time utility: max {c t,l t} E β t U c t, l t ) 65) t= subject to a sequence of budget constraints expressed in units of domestic consumption good: c t w t l t + Π t 66) where c t is a composite consumption good, l t denotes hours worked, w t is real wage and Π t denotes profits from home firms. The foreign representative household s problem, problems of firms and market clearing conditions except of the absent assets market) are analogous to the previous case. All equilibrium conditions for the model of financial autarky are given in Appendix B. As in the model with complete markets world prices are completely determined in the equilibrium of world economy. Taking into account that in complete markets equilibrium representative foreign economy has zero asset holdings, both equilibria, i.e. complete markets and financial autarky, are equivalent for this economy. The law of one price for financial assets and consequently condition 63) are not satisfied in the model of financial autarky. Instead, an equilibrium for home economy is determined by balanced trade condition: tb t = p H,t c H,t + i H,t) + p X,t X c X,t x T,t x N,t ) p F,t c F,t + i F,t ) = 67) where p H,t c H,t + i H,t ) and p X,tX c X,t x T,t x N,t ) are respectively export of home tradable goods and commodity, p F,t c F,t + i F,t ) is import of foreign tradables. To compare this model with complete markets case it is useful to compute time and state varying variable ϑ t reflecting the wedge between cross-country marginal rate of 7

18 substitution of consumption and real exchange rate: e t U c c t, l t ) = ϑ t U c c t, lt ) 68) This wedge induces additional volatility of the real exchange rate and solves Backus- Smith puzzle, breaking link between relative consumption and real exchange rate. Nevertheless, financial autarky is too restrictive assumption. So, it would be interesting to consider intermediate case, when home country can borrow and save abroad but there are frictions in assets trade. 6 Model with portfolio adjustment costs Now we assume that to adjust the value of portfolio of international assets home representative household have to bear transaction costs. These costs may reflect portfolio management costs or the costs related to control on international capital movements. Home representative household in this model maximizes expected life-time utility: max {c t,l t,a t+ } E β t U c t, l t ) 69) t= subject to a sequence of budget constraints expressed in units of domestic consumption good: c t + E t q t,t+ a t+ + p N,t Ψ a E t q t,t+ a t+ a t ) a t + w t l t + Π t 7) where c t is a composite consumption good, l t is hours worked, w t is real wage, a t+ denotes portfolio of Arrow-Debreu securities with price kernel q t,t+, Ψ a ) are portfolio adjustment costs and Π t are profits from home firms. We assume that all portfolio adjustment costs are paid in nontradable goods and Ψ a ) is convex function satisfying the following properties: Ψ a z) and Ψ a z) = if and only if z =, zψ az) and Ψ az) >. The foreign representative household s problem, problem for firms and market clearing conditions 6 are the same as in the complete markets model. by: The first order intertemporal optimal condition for representative household is given q t,t+ = β U c c t+, l t+ ) + p N,t+ Ψ a,t+ U c c t, l t ) + p N,t Ψ a,t 7) where Ψ a,t = Ψ a E t q t,t+ a t+ a t ). 6 Given that portfolio adjustment costs are paid in home nontradable good, the market clearing condition for this good is now: y N,t = c N,t + i N,t + Ψ a E t q t,t+ a t+ a t ). 8

19 where This condition implies that: e t U c c t, l t ) = ϑ t U c c t, l t ) 7) ϑ t = ϑ + p N,t Ψ a,t 73) So, in this model there is the wedge between cross-country marginal rate of substitution and real exchange rate. This wedge implies negative correlation between real exchange rate and world commodity price. Indeed, negative commodity shock in foreign economy results in commodity price growth and trade balance surplus. Since ϑ t is decreasing function of trade balance, real exchange rate have to decrease appreciate). Besides, this model generates additional volatility of the real exchange rate and solves Backus-Smith puzzle, breaking link between relative consumption and real exchange rate. But, in contrast to financial autarky, it allows international borrowings and savings. 7 Calibration The model was calibrated for Canadian economy, whereas the parameters of the world economy were approximated using US data. Calibrated values of parameters are summarized in Table. The breakdown of industries into commodity, tradable and nontradable sectors according to International Standard Industrial Classification ISIC, Rev. 3) is presented in Table 3. The commodity sector includes agriculture, mining and manufacturing industries producing raw materials wood and pulp, metals, chemicals, petroleum and coal products, etc.). The tradable sector consists of the rest of manufacturing food and tobacco, textile, machinery and equipment manufacturing, etc.) and services with significant share of international trade transport, financial intermediation, computer and other business services). The nontradable sector includes the rest of services, public utilities and construction. 7. Preferences and technology The following functional form for the utility function is assumed in this paper: U c, l) = c γ l) γ) σ σ This functional form implies that consumption and leisure are non-separable. The discount factor β is set to.99 to match 4% annual real rate in steady state. The risk 9

20 aversion parameter σ is equal to. The labor share parameter µ is set to.38, so the fraction of working time in steady-state is 3%. The elasticity of substitution between commodity, tradable and nontradable goods ɛ is set to.74 following Mendoza 995). We assume that elasticity of substitution between home and foreign tradable goods θ is equal to.5. The shares of tradables a CT =.3) and nontradables a CN =.73) in consumption basket were computed using data of input-output tables for Canada. 7 The weight of home tradable good in tradable consumption a CH is set to.68. Corresponding shares for investment good are a IT =.38 and a IH =.3. So, the share of tradable goods in investment is higher than in consumption. Besides, consumption of tradables in Canada is home-biased, whereas investments of tradables are predominantly foreign. The shares of capital in value added, b T K and b NK, computed using data of inputoutput tables for Canada, are found to be approximately the same across sectors and equal to.34. The weights of commodity input costs in total costs for tradable φ T and nontradable φ N sectors are set to % and % respectively. So, tradable sector uses commodity more intensively than nontradable sector in production process, what makes it more vulnerable to the commodity price hikes. Elasticity of substitution between value added and commodity input ξ is set to.69 for both sectors. That is in line with estimate of Rotemberg and Woodford 996). We assume also that the capital stocks in both sectors depreciate at the same rate of.5% per quarter. The steady state productivity levels in foreign tradable A T and nontradable A N sectors were computed, assuming that in steady state all prices of foreign goods in terms of foreign consumption good) are equal to and normalizing steady state foreign commodity endowment X to. Both values are found to be equal to 3.3. To compute the productivity levels in tradable A T and nontradable A N sectors in home country we used estimates of multi-factor productivity gap between US and Canada for different industries, reported by Rao, Tang, and Wang 4). According to these estimates productivity levels in Canada constitute 79% and 9% of those in US for tradable and nontradable sectors respectively. Implied values of A T and A N are equal to.47 and.88. Home economy is also assumed to be commodity abundant, so in steady state X = Capital and portfolio adjustment costs Capital adjustment costs are symmetric across sectors. The following functional form is assumed in this paper: Ψ k z) = ψ + ψ z ψ. The parameters are set to guarantee, that in steady state equilibrium Ψ k δ) = δ, Ψ k δ) = and Ψ k δ) = d <. The value of d is chosen so as to ensure a volatility of investment relative to that of GDP in excess of 3, as in the US data. 7 OECD Input-Output Database, 6 edition.

21 The portfolio adjustment costs function is Ψ a z) = ψz. The parameter ψ is set so to guarantee that a standard deviation of trade balance to GDP ratio is equal to.9, as in the data for Canada. 7.3 Commodity and productivity shocks The commodity and productivity shocks follow independent AR) processes: log A T,t = ρ T log A T,t + u T,t log A N,t = ρ N log A N,t + u N,t log X t = ρ X log X t + u X,t log A T,t = ρ T log A T,t + u T,t log A N,t = ρ N log A N,t + u N,t log Xt = ρ X log Xt + u X,t Serial correlation coefficients are assumed to be symmetric across sectors and countries and equal to ρ =.8. Standard deviations of innovations in tradable and nontradable sectors are set to satisfy proportion two to one in both countries. Innovations in commodity endowment are twice more volatile than productivity shocks in tradable sector for home economy and three times more volatile for foreign economy. 7.4 Algorithm The equilibrium conditions of the model are log-linearized around steady state. A corresponding system of linear expectational difference equations is solved by generalized Schur decomposition QZ) algorithm see Klein, ; Sims, ) realized in Dynare 8 package. 8 Results This section reports a steady state solution and simulation results for different variants of the model of small commodity exporting economy. 8. Steady-state equilibrium Given a cross-country asymmetry of the model it is interesting to consider its steadystate equilibrium characterizing long-run structural differences between home and foreign 8

22 economies. In steady state both economies have balanced trade, so the corresponding equilibria are the same for all variants of the model. A steady state solution of the model is presented in Table 4. total GDP in this table are given in terms of domestic consumption good. 9 Relative prices and A steady state equilibrium conforms to stylized facts about commodity exporting economy, such as Canada. First, international trade flows fit well with law of comparative advantages: home country exports commodity whereas import of foreign tradable goods exceeds export of home tradables. Price ratio between domestic tradables and commodity in home country is higher than in foreign country. Second, production in home economy is skewed to commodity and nontradable sector. Nontradable sector uses relatively more labor, capital and commodity goods comparing to tradable sector. Abundance of commodity in home country implies also that both sectors use commodity more intensively than in foreign country. And finally, steady-state real exchange rate is equal to.78. So, international price level in home economy is higher than abroad. In other words, prices of nontradable and domestic tradable goods expressed in the same currency will be higher in home country. It should be mentioned that this result is far from general for commodity-exporting countries. Figure 3 illustrates dynamics of international price levels in four developed countries: Australia, Canada, Norway and New Zealand. Though for Norway prices are usually significantly higher than in USA an average for 97-8 is close to 3% of US level), in Canada and Autralia there is parity an averages are % and 97% of US level respectively) and New Zealand has lower international price level 86% of US level). It is known also, that all developing countries have low international price level despite the fact that many of them are commodity exporting countries. It is not so difficult to explain. According to well-known Balassa-Samuelson hypothesis, countries with higher relative productivity in tradable sector compared to nontradable sector usually have higher price level. In our model tradable sector includes both commodity and non-commodity producing industries. Though commodity-exporting countries are more productive in commodity sector they may be significantly less productive in non-commodity producing tradable sector. If the second factor dominates the first, international price level in these countries will be lower than abroad. Besides, important role in our model plays degree of substitutability in consumption of home and foreign non-commodity tradable goods. In the case of low elasticity of 9 In other words, home prices and GDP are presented in terms of home consumption, whereas foreign prices and GDP in terms of foreign consumption. i.e. value of import Remember, that real exchange rate is defined in our paper as price of foreign consumption in terms of home consumption. So, international price level in home economy is inverse of the real exchange rate. According to World bank s International Comparison Program ICP) for 5 even rich OPEC countries have lower than in USA price levels: 73% in Kuwait, 75% in Qatar and 64% in Saudi Arabia.

23 substitution prices of these goods may significantly deviate from the law of one price 3 without strong effect on demand. Therefore, relatively higher prices of home tradable goods may compensate their low relative productivity dumping Balassa-Samuelson effect. If home and foreign tradable goods were perfect substitutes, the law of one price between these goods would be satisfied precisely and international price level in home country would be lower. So, for example, increasing elasticity of substitution θ from.5 to 5 results in international price level in home country equal to % of foreign which is close to Canada average) comparing to initial 8% inverse of.78). 8. Business cycles statistics Here we present the second moments of time series generated by different variants of the model of small commodity-exporting economy and compare them with actual data. To compute these statistics we simulated the model times. After that we took logs of all variables 4 and detrended them with HP-filter. Adjusted time series were used for computation of standard deviations and correlations between variables. We calculated means and standard errors of these statistics along simulations. Quarterly data for Canada in were used to compute second moments of actual data. 5 Table 5 summarizes basic results. Simulated standard deviations conform well with data. Volatility of GDP in home country varies between.55 and.7 across models that is close to actual value of.78 in Canada. The highest volatility of GDP is generated by the model of financial autarky. Standard deviation is equal to.7, comparing with.55 in complete markets case. It could be explained by asymmetric effect of the world commodity price changes on home GDP in commodity and other sectors under complete markets assumption. In this model negative effect of commodity shock on foreign economy is transmitted to home economy, where production of tradable and non-tradable goods decreases. At the same time, value of commodity endowment in home country increases as a result of higher commodity price. Since in financial autarky the risk sharing does not work, the first effect is negligible and an increase of GDP in commodity sector is not compensated by decrease of production in other sectors. Therefore, total GDP of home economy is more volatile in the model of financial autarky. Consumption and employment are less volatile whereas investment is more volatile than GDP in all variants of the model, that conforms to standard results in business cycles literature. For the model with portfolio adjustment costs standard deviations of consumption and investment are close to data,.99 and 4.3 versus.9 and 4. respectively. 3 One of the assumption in Balassa-Samuelson theory. 4 except of net export, which is presented as a share of GDP. 5 We used data of OECD Economic Outlook provided by EcoWin database. 3