## Abstract

The paper develops a model of a small economy that trades commodities whose world prices fluctuate exogenously, and studies its implications for monetary policy. It derives analytical characterizations of optimal Ramsey and flexible price allocations under both perfect risk sharing and financial autarky. This allows the paper to identify the crucial roles of production structure, price elasticities, and capital mobility in monetary policy evaluation. In a calibrated example, impulse-responses under PPI targeting track flexible price allocations closely, but can diverge greatly from Ramsey allocations when risk sharing is perfect and intratemporal elasticities are high. In those cases, policy rules that stabilize real exchange rates more than PPI targeting can deliver higher welfare. But PPI targeting is a clear winner under portfolio autarky.

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## Notes

An exception is Blanchard and Gali (2009).

Most prominently by Corsetti and Pesenti (2001), Benigno and Benigno (2003), Galí and Monacelli (2005), Faia and Monacelli (2008), and recently Monacelli (2013).

We have assumed η≠1. If η=1,

*C*_{ t }and*P*_{ t }(below) are Cobb Douglas.That is, assuming that the marginal utility of consumption in the rest of the world is proportional to

*C**_{ t }^{−σ}.In the case φ=0, the home aggregate is a nontraded good. We do not explore that case here, however.

For instance, if there are cost-push shocks or real rigidities.

All of the studies cited in the previous paragraph assume perfect risk sharing, with the exception of Corsetti and Pesenti (2001). But in the latter international asset markets play no role.

Some of the material in this section overlaps with Faia and Monacelli (2008), Catão and Chang (2013) and Monacelli (2013). However, we clarify and simplify the logic in several ways, which helps unifying the discussion and extend it to the many variants of our model, including the case of portfolio autarky.

It is easiest to define elasticities sequentially. Letting the elasticity of

*g*be denoted bythen Note that the elasticities are time varying, in general.

To see this point more clearly, suppose that, in fact, μ(1−υ)=1. Then the market flex price outcome would be which can be regarded as the optimality condition for a planner that does not exploit the terms-of-trade margin, that is, a planner that takes world prices as given, and that is also subject to a balanced trade constraint. The LHS would be the utility cost of an additional unit of consumption: for such a “price taking” planner, that unit would require producing

*P*_{ t }/*P*_{ ht }=*g*(*X*_{ t }*Z*_{ t }) units of home output, at labor cost*g*(*X*_{ t }*Z*_{ t })/*A*_{ t }. The RHS would be the utility cost of the additional output required to pay for the unit of consumption.Here, ɛ

_{ C }^{D}=(1−α)*g*(*X*_{ t }*Z*_{ t })^{η}*C*_{ t }/*D*_{ t }andThe rationale for a low η is that when the share of imported consumption is made up of food and or oil, goods with a low price elasticity of demand, it seems sensible to consider values η below one.

In practice, this typically involves a complex macro forecast apparatus by central banks and a choice of the relevant forecast horizon during which inflation is targeted. As discussed recently in central bank inflation reports around the world, the targeting horizon choice involves nontrivial trade-offs that have been mostly influenced by developments in global commodity markets. An analysis of targeting horizons is beyond the scope of this paper, however.

It should also be noted that if the export price volatility is extreme, as in recent times, strict application of this rule could entail unrealistically large fluctuations in interest rates. An alternative would be to aid the task of monetary policy with a fiscal-based stabilization instrument. One example would be using the proceeds of an export price stabilization fund, such as the copper stabilization fund in Chile, to purchase or sell domestic currency, so as to induce nominal exchange rate movements to offset those of world export prices. In this connection, an important issue is whether exacerbated variability of world export prices reflect cyclical or trend-like movements. Export price shocks with double-digit quarterly standard deviations (as opposed to the average 7 percent found in real-world data to which our baseline calibration is benchmarked) may reflect changes in the permanent component of commodity prices; offsetting those should not be, in principle, the sole responsibility of monetary policy. As such, the policy analysis of such large and persistent fluctuations is beyond the scope of our model.

Consistent with the analytical results discussed in Section II, when all elasticities equal and unitary, the numerical results show PPI dominance across the board. The case of σ=1 is not reported to save space.

Given that pegging the nominal exchange rate is widely regarded an inferior rule for commodity exporters (Cashin, Cespedes, and Ratna, 2004; Frankel, 2010, 2011), and to keep the table presentation concise, we do not report results for the peg rule.

We have undertaken further sensitivity analysis which suggests that these results—and in particular the dominance of expected CPI targeting for intermediate values of η and σ under complete markets—are broadly robust to reducing the persistence of commodity price shocks. As one might expect, however, if the persistence of commodity price is very small, expected CPI targeting loses its welfare dominance. But that would be highly unrealistic, as commodity price shocks are well known to be highly persistent.

See Catão and Chang (2011) for cross-country evidence on deviations from target inflation in the wake of commodity price shocks and their relation to structural breaks in Taylor rules.

## References

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Catão, L., and R. Chang, 2010, “World Food Prices and Monetary Policy,” NBER Working Paper 16563.

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Catão, L., and R. Chang, 2013, “World Food Prices, the Terms of Trade-Real Exchange Rate Nexus, and Monetary Policy.” Unpublished Manuscript.

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## Additional information

^{*}Luis Catão is Senior Economist and Acting Deputy Chief at the Open Economy Macroeconomics Division in the Research Department of the International Monetary Fund. Roberto Chang is a Professor at the Department of Economics, Rutgers University, New Brunswick, NJ, US. The authors thank Marola Castillo for assistance beyond the call of duty. They are also indebted to Oliver Blanchard, Pierre Olivier Gourinchas, two anonymous referees, participants of the IMF Review—Central Bank of Turkey Conference on Policy Responses to Commodity Price Movements, Istanbul, April 2012, and especially their discussant Stefan Laséen for extensive and useful comments and suggestions. Any error is only the authors’. The views expressed in this paper are the authors’ alone and do not necessarily reflect those of the IMF or IMF policy.

## Appendix

### Appendix

This Appendix shows how the analysis of Section II can be extended to the full model, allowing for imported inputs and the homogeneous exports sector. The key is to recognize that the Ramsey planner must choose imported inputs and competitive exports efficiently. Hence, in particular, the planner will choose labor effort in the competitive export sector so as to equate marginal product in that sector to the implicit real wage, given by the marginal rate of substitution between labor effort and consumption:

that is,

where for convenience we have suppressed time subscripts and the dependence of *H* on exogenous shocks.

Likewise, the planner will choose the imported input *M* efficiently so

which implies that *Y*_{
h
}=ALψ(*C,N,X*), where

### A.I. Perfect Risk Sharing

If risk sharing is perfect, the planner's problem is to choose *C*,*N*,*L*,*L*_{
x
}, and *X* to maximize *u*(*C*)−*v*(*N*) subject to the risk-sharing constraint (4), the labor supply constraint (15), efficient choice of labor input in the homogenous exports sector:

and equilibrium in the market for the home aggregate

Some tedious but straightforward derivations yield the optimality condition

where ɛ_{
C
}^{H} denotes the partial elasticity of *H* with respect to *C*, and so on.

This generalizes Equation (20) and, as the reader can check, it coincides with it if =0 and *L*_{
x
}=0. This and the four constraints completely characterize the Ramsey allocation. The optimality condition is interpreted in a similar way as in the simpler case.

Expressing the elasticities in full, the expression simplifies drastically and I get:

The LHS is, as before, the utility increase due to a 1 percent real depreciation. This increases the demand for the home aggregate by ɛ_{
X
}^{Θ}, but the overall increase in the demand for labor is less because the associated cost in the implicit real wage induces substitution toward imported inputs (the *L*/*N*

term) and a reduction in employment in the homogeneous exports sector (the term).

### A.II. Financial Autarky

To express the financial autarky constraint *V*_{
t
}/*P*_{
t
}=*C*_{
t
}, note that optimality requires (*P*_{
ht
}/*P*_{
t
})*Y*_{
ht
}−(*P*_{
mt
}/*P*_{
t
})*M*_{
t
}=1− )*P*_{
ht
}*Y*_{
ht
}/*P*_{
t
}, and that *P*_{
xt
}/*P*_{
t
}=*X*_{
t
}*Z*_{
xt
}. Then one can define real value added as (suppressing time subscripts):

The planner's problem is to choose *C*,*N*,*L*,*L*_{
x
}, and *X* to maximize *u*(*C*)−*v*(*N*) subject to the labor supply constraint (15), efficient choice of labor input in the homogenous exports sector (30), the financial autarky constraint:

and equilibrium in the market for the home aggregate

where *D*(*C,X*) is the RHS of Equation (17), as defined in Section II.

After some tedious work, the optimality condition can be expressed as follows:

where ϒ is an auxiliary variable defined by:

Note that the elasticities are functions of stochastic variables, and hence will be generally time varying. Indeed one can show that ɛ_{
C
}^{ψ}= σ, ɛ_{
N
}^{ψ}= ϕ, ɛ_{
X
}^{ψ}=− while

where