Back Matter

## A Appendix

### A. Proofs of Propositions and Corollaries

#### A.1. Proof of Proposition 1

Proof. Combine (22) and (23) to derive

$\begin{array}{ll}{\stackrel{^}{p}}_{N,t}=-\left[\frac{\left(\psi +\sigma \right)\left({\alpha }_{F}-{\kappa }_{F}\right)}{\left(1+\eta \psi \right)\left(1-{\alpha }_{N}\right)+\eta \left(\psi +\sigma \right){\alpha }_{N}}\right]{\stackrel{^}{p}}_{F,t}^{*}.\hfill & \hfill \left(\text{A}.1\right)\end{array}$

Note that uniqueness follows from the facts that (22) and (23) are, respectively, strictly increasing and strictly decreasing in ${\stackrel{^}{p}}_{N,t}$ Using (28) and (A.1), we can solve for ${\stackrel{^}{\pi }}_{t}^{FA}$ in terms of $\Delta {\stackrel{^}{p}}_{F,t}^{*}$ and obtain (29). ▪

#### A.2. Proof of Proposition 2

Proof. Combine (22) and (24) to derive

$\begin{array}{ll}{\stackrel{^}{p}}_{N,t}=-\left[\frac{\left(\psi +\sigma \right){\alpha }_{F}}{\left(1+\eta \psi \right)\left(1-{\alpha }_{N}\right)\sigma +\left(\psi +\sigma \right){\alpha }_{N}}\right]{\stackrel{^}{p}}_{F,t}^{*}.\hfill & \hfill \left(\text{A}.2\right)\end{array}$

Note that uniqueness follows from the facts that (22) and (24) are, respectively, strictly increasing and strictly decreasing in ${\stackrel{^}{p}}_{N,t}$. Using (28) and (A.2), we can solve for ${\stackrel{^}{\pi }}_{t}^{CM}$ in terms of $\Delta {\stackrel{^}{p}}_{F,t}^{*}$ and obtain (30). ▪

#### A.3. Proof of Proposition 3

Proof. The proof has two parts. First we prove the existence of a unique equilibrium (stability). Second we apply the method of undetermined coefficients to derive the analytical solution.

To prove equilibrium uniqueness, we rewrite equations (22), (25) and (26) as the system

$\begin{array}{ll}{E}_{t}{\stackrel{^}{x}}_{t+1}=\Psi \stackrel{^}{{x}_{t}}+\Upsilon {\stackrel{^}{p}}_{F,t}^{*},\hfill & \hfill \left(\text{A}.3\right)\end{array}$

where ${\stackrel{^}{x}}_{t}=\left[{\stackrel{^}{p}}_{N,t},{\stackrel{^}{b}}_{t-1}^{*}{\right]}^{\prime },$,

$\mathrm{\psi }=\left[\underset{\begin{array}{c}-{\vartheta }^{-1}\hfill \end{array}}{1+\gamma \nu \left(1-{\alpha }_{N}\right)}-\underset{\begin{array}{c}\hfill {R}^{*}\hfill \end{array}}{\tau \nu \left(1-{\alpha }_{N}\right){R}^{*}}\right],\text{\hspace{0.17em}}\gamma \equiv \frac{\tau }{\vartheta },\text{\hspace{0.17em}}\vartheta =\frac{\left(\psi +\sigma \right)}{\left(1+\eta \psi \right)\left(1-{\alpha }_{N}\right)+\left(\psi +\sigma \right){\alpha }_{N}\eta },\text{\hspace{0.17em}}\tau =\frac{\left(\psi +\sigma \right)}{\left(1+\eta \psi \right)\left(1-{\alpha }_{N}\right)\sigma +\left(\psi +\sigma \right){\alpha }_{N}},$

and the form of ϒ is omitted since it is not required for the stability analysis. The characteristic polynomial associated with Ψ is given by

$\begin{array}{ll}P\left(\mathrm{a}\right)={\mathrm{a}}^{2}-\left[1+{R}^{*}+\gamma \nu \left(1-{\alpha }_{N}\right)\right]\mathrm{a}+{R}^{*}\hfill & \hfill \left(\text{A}.4\right)\end{array}$

satisfying

$P\left(1\right)=-\gamma \nu \left(1-{\alpha }_{N}\right)<0\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\text{\hspace{0.17em}}\text{and}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\text{\hspace{0.17em}}P\left(-1\right)=2\left(1+{R}^{*}\right)+\gamma \nu \left(1-{\alpha }_{N}\right)>0.$

Since P(1) < 0 and P(−1) > 0 then following Azariadis (1993) we can infer that both eigenvalues of Ψ, i.e., a1 and a2, are on the same side of −1 and on different sides of 1. The only possibility is that one eigenvalue is in (−1, 1) and the other one in (1, ∞). Thus the steady state is a saddle. Without loss of generality, assume that a2 is the explosive eigenvalue—i.e., a2 ∈ (1, ∞)—while a1 is non-explosive—i.e., a1 ∈ (−1, 1). Then, since there is one non-predetermined variable, ${\stackrel{^}{p}}_{N,t}$, and one predetermined variable ${\stackrel{^}{b}}_{t-1}^{*}$, we can use the results by Blanchard and Kahn (1980) to conclude that there exists a unique rational expectations equilibrium for $\left\{{\stackrel{^}{p}}_{N,t},{\stackrel{^}{b}}_{t-1}^{*}\right\}$. Using this and (28) we can also conclude that there is a unique equilibrium for ${\stackrel{^}{\pi }}_{t}$.

To obtain the analytical solutions for ${\stackrel{^}{\pi }}_{t}$ we combine equations (22), (25), (26) and (27) and rewrite the model as the system

$\begin{array}{ll}\Theta {\stackrel{^}{x}}_{t}=\mathrm{\Omega }{E}_{t}{\stackrel{^}{x}}_{t+1}+\Gamma {\stackrel{^}{x}}_{t-1}+\mathrm{\Pi }{\stackrel{^}{p}}_{F,t}^{*},\hfill & \hfill \left(\text{A}.5\right)\end{array}$

where now ${\stackrel{^}{x}}_{t}=\left[{\stackrel{^}{p}}_{N,t},{\stackrel{^}{b}}_{t}^{*}{\right]}^{\prime }$, while

$\mathrm{\Theta }=\left[\begin{array}{cc}\hfill 1\hfill & \hfill -\tau \nu \left(1-{\alpha }_{N}\right)\hfill \\ \hfill 1\hfill & \hfill \vartheta \hfill \end{array}\right],\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\text{\hspace{0.17em}}\mathrm{\Omega }=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right],\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\text{\hspace{0.17em}}\mathrm{\Gamma }=\left[\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \vartheta {R}^{*}\hfill \end{array}\right],$

and

$\Pi =\left[\begin{array}{c}\hfill -\tau {\alpha }_{F}\left(1-{\rho }_{{p}_{F}^{*}}\right)\hfill \\ \hfill -\vartheta \left({\alpha }_{F}-{\kappa }_{F}\right)\hfill \end{array}\right].$

Following the undetermined coefficient methods (see Christiano, 2002), the Minimal State Variable (MSV) representation of the solution corresponds to

$\begin{array}{cc}\hfill {\stackrel{^}{p}}_{N,t}=𝔢{\stackrel{^}{b}}_{t-1}^{*}+e{\stackrel{^}{p}}_{F,t}^{*}\text{\hspace{0.17em}}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\text{and}\text{\hspace{0.17em}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}{\stackrel{^}{b}}_{t-1}^{*}=a{\stackrel{^}{b}}_{t-1}^{*}+c{\stackrel{^}{p}}_{F,t}^{*},\hfill & \hfill \left(\text{A}.6\right)\end{array}$

and it can be written in a compact form as:

$\begin{array}{cc}\hfill {\stackrel{^}{x}}_{t}=A{\stackrel{^}{x}}_{t-1}+B{\stackrel{^}{p}}_{F,t}^{*}\text{\hspace{0.17em}}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\text{and}\text{\hspace{0.17em}}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}{\stackrel{^}{p}}_{F,t}^{*}={\rho }_{{p}_{F}^{*}}{\stackrel{^}{p}}_{F,t-1}^{*}+{∊}_{{p}_{F}^{*},t},\hfill & \hfill \left(\text{A}.7\right)\hfill \end{array}$

with

$A=\left[\begin{array}{cc}\hfill 0\hfill & \hfill 𝔢\hfill \\ \hfill 0\hfill & \hfill a\hfill \end{array}\right],\text{\hspace{0.17em}}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\text{and}\text{\hspace{0.17em}}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}B=\left[\begin{array}{c}\hfill e\hfill \\ \hfill c\hfill \end{array}\right].$

Iterating forward the MSV (A.7) and using it to eliminate all the forecasts ${E}_{t}{\stackrel{^}{x}}_{t+1}$ as well as ${\stackrel{^}{x}}_{t}$ in the model (A.5), we obtain

$\left[\Omega {A}^{2}-\Theta A+\Gamma \right]{\stackrel{^}{x}}_{t-1}+\left[\Omega AB+{\rho }_{{p}_{F}^{*}}\Omega B-\Theta B+\Pi \right]{\stackrel{^}{p}}_{F,t}^{*}=0,$

which defines the following mappings:

$\begin{array}{cc}\hfill \Omega {A}^{2}-\Theta A+\Gamma =0\text{\hspace{0.17em}}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\text{and}\text{\hspace{0.17em}}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\Omega AB+{\rho }_{{p}_{F}^{*}}\Omega B-\Theta B+\Pi =0.\hfill & \hfill \left(\text{A}.8\right)\hfill \end{array}$

These mappings define a set equations for the elements of the matrices A and B. In particular, a has to solve a quadratic equation P(a) = 0, where P(a) is the same as the polynomial defined in (A.4). From our stability analysis we know that the steady state is a saddle, while from (A.6) we know that a is the coefficient of ${\stackrel{^}{b}}_{t-1}^{*}$. Therefore, we choose the sable root of P(a) = 0, i.e., a = a1 ∈ (−1, 1), where

$a=\frac{1}{2}\left\{1+{R}^{*}+\gamma \nu \left(1-{\alpha }_{N}\right)-\sqrt{{\left[1+{R}^{*}+\gamma \nu \left(1-{\alpha }_{N}\right)\right]}^{2}-4{R}^{*}}\right\}.$

Note that a is real since [1 + R* + γv(1 − αN)]2 − 4R* = (R* − 1)2 + 2(1 + R*)γv(1 − αN) + [γv(1—αN)]2 > 0. The mappings (A.8) also imply the following expressions for 𝔡, and 𝔢, in terms of a:

$𝔡=-\vartheta \left(a-{R}^{*}\right),$

and

$e=\left[\frac{1-{\rho }_{{p}_{F}^{*}}}{1-{\rho }_{{p}_{F}^{*}}+{R}^{*}+\gamma \nu \left(1-{\alpha }_{N}\right)-a}\right]\left(-\tau {\alpha }_{F}\right)+\left[1-\frac{1-{\rho }_{{p}_{F}^{*}}}{1-{\rho }_{{p}_{F}^{*}}+{R}^{*}+\gamma \nu \left(1-{\alpha }_{N}\right)-a}\right]\left[-\vartheta \left({\alpha }_{F}-{\kappa }_{F}\right)\right],$

which together with (28), (A.6), and the definitions ${\Phi }_{{p}_{F}^{*}}^{CM}=\tau {\alpha }_{F}$, and ${\Phi }_{{p}_{F}^{*}}^{FA}=\vartheta \left({\alpha }_{F}-{\kappa }_{F}\right)$, can be combined to obtain the analytical expression (31) for ${\stackrel{^}{\pi }}_{t}^{IM}$. Finally, ${\omega }_{{p}_{F}^{*}}\in \left[0,1\right)$, follows from the facts that R* + γv(1 − αN) − a > 0—since a ∈ (−1, 1), γ > 0, 1 − αN > 0, R* > 1 and $\underset{v\to 0}{\mathrm{lim}}{\omega }_{{p}_{F}^{*}}=0$. ▪

#### A.4. Proof of Corollary 1

Proof. The proof of this corollary follows from our results in Proposition 1, 2 and 3. Points a) and b) follow from setting ${\Phi }_{{p}_{F}^{*}}^{FA}={\alpha }_{F}$ and ${\Phi }_{{p}_{F}^{*}}^{CM}={\alpha }_{F}$, while c) follows from a) and b) combined with the fact that the first-round effects under IM are just a convex combination of these effects under FA and CM. ▪

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We have benefited from comments and conversations with Andy Berg, Ed Buffie, Doug Gollin, Peter Montiel, and Steve O’Connell and participants of the 2014 CSAE conference at Oxford University. All errors remain ours. This paper is part of a research project on macroeconomic policy in low-income countries supported by U.K.’s Department for International Development (DFID), and it should not be reported as representing the views of the International Monetary Fund or of DFID.

To our knowledge, this advice has its origins in policy decisions taken by the Bundesbank in the 1970s, which in the face of shocks to the international price of oil raised its one-year-ahead inflation objective to accommodate the inflation caused by the shock (see Bernanke et al., 1999). The standard advice likely became more clearly articulated with the adoption of inflation targeting regimes, as these helped focus the policy discussion on better understanding the sources of inflation and tailoring the policy response accordingly. From an academic perspective, although not explicitly stated, the advice can be traced back to the seminal work of Robert Gordon (1975).

See Woodford (2003), among others.

“Managing Inflation in an Era of Commodity Price Volatility.” Opening Remarks by the Deputy Managing Director, Mr Naoyuki Shinohara, at a Joint ADB-IMF-Reserve Bank of India Seminar, New Delhi, India May 3, 2013. Available at: https://www.imf.org/external/np/speeches/2013/050313.htm

Our focus on asset markets is inspired by De Paoli (2009b), who shows the key role that the latter plays for the design of monetary policy in small open economy models. We make the related point that the asset structure also matters for first-round effects.

Most New—Keynesian small open economy models follow the seminal work by Gali and Monacelli (2005), which assumes complete markets.

Note that the presence of food in the basket drives a wedge between the real exchange rate and the inverse of the relative price of non-traded goods: the former can appreciate even though the relative price of non-traded goods decreases. This point has been emphasized by Catao and Chang (2010).

Once ϕ1 = 1, the value of ϕ2 does not matter: access to a complete set of contingent assets makes incomplete (non-contingent) assets redundant.

See International Monetary Fund (2011), among others. The policy objective in this case is to avoid persistent effects on inflation. The first-round or direct effects—which also include the effects associated with the use of oil as an intermediate production input—capture changes in relative prices in the economy and therefore their impact on headline inflation should be short-lived. In contrast, the second-round effects involve increases in prices that are more persistent, including those that result from pressures to preserve real wage levels.

See for instance Aoki (2001), among others, in the context of the New Keynesian literature.

The exchange rate is assumed to be flexible.

To see this, use ${\overline{P}}_{N,t}={P}_{N,0}$ and set δ = 1/ in equation (14) and divide both sides of this equation by PN,0 to obtain

$1=\frac{{E}_{t}{\sum }_{j=0}^{\infty }{\left(\beta \theta \right)}^{j}{\lambda }_{t+j}\left[{y}_{N,t+j}\frac{{W}_{t+j}}{{P}_{N,t+j}}\right]}{{E}_{t}{\sum }_{j=0}^{\theta }{\left(\beta \theta \right)}^{j}{\lambda }_{t+j}\left[{y}_{N,t+j}\right]},$

which holds if:

$\frac{{W}_{t+j}}{{P}_{N,t+j}}=1\text{\hspace{0.17em}}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\text{for}\text{\hspace{0.17em}}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}j=0,1,\dots$

See Backus and Smith (1992).

Here ${\stackrel{^}{b}}_{t}^{*}$ indicates deviations of ${b}_{t}^{*}$ from its steady-state value (0) in percent of steady state consumption.

In this case, the CPI-based real exchange rate appreciates by the same magnitude as the nominal exchange rate: $\Delta {\stackrel{^}{s}}_{t}=\Delta {\stackrel{^}{S}}_{t}$. Note that while inflation does not change, the increase in international food prices has real effects, namely on the composition of trade. The real appreciation increases consumption of the generic traded good and a “generic” trade deficit opens up, and the opposite occurs for food. Overall trade remains balanced, however.

Of course, the opposite holds if the intertemporal elasticity is bigger than the intratemporal one (1/σ > η)—i.e., the three consumption goods are Edgeworth complements.

As before if instead 1/σ > η then $\partial |{\Phi }_{{p}_{F}^{*}}^{FA}|/\partial \psi >0$, $\partial {\Phi }_{{p}_{F}^{*}}^{CM}/\partial \psi >0$ and $\partial {\omega }_{{p}_{F}^{*}}/\partial \psi <0$.

If instead 1/σ > η then $\partial |{\Phi }_{{p}_{F}^{*}}^{FA}|/\partial {\alpha }_{N}>0$ and $\partial {\Phi }_{{p}_{F}^{*}}^{CM}/\partial {\alpha }_{N}<0$, but $\partial {\omega }_{{p}_{F}^{*}}/\partial \psi \underset{>}{\overset{<}{=}}0$.

On Uganda, see “Consumer Price Index April 2001,” available at www.ubos.org. On Ghana see “Time Series P1,” available at www.statsghana.gov.gh. On Kenya see “CPI December 2008,” available at www.knbs.or.ke/consumerpriceindex.php.

Data is available on http://wits.worldbank.org. We define food trade as consisting of the following categories: live animals except fish, meat and preparations, dairy products and eggs, fish/shellfish, cereals/cereal preparation, vegetables and fruit, sugar/sugar/honey, coffee/tea/cocoa/spices, animal feed, miscellaneous food products, beverages, tobacco/manufactures, oil seeds/oil fruits, crude animal/vegetable matters, animal/vegetable oil/fat/wax.

Trade costs may render certain staples effectively non-tradable within a certain price band (see Bergin and Glick, 2009). Trade restrictions have similar effects. Non-tradability may be endogenous, with large changes in international prices increasing the tradability of certain food items. For simplicity, we treat tradability in food as exogenous.

Relaxing the restriction that the elasticity of substitution between food and non-food is the same as the elasticity of substitution between different types of food does not change this result. The proof is available upon request.

There are some works that, without disentangling these effects, have tried to relate the overall inflationary impact of commodity price shocks to a broad range of structural characteristics and policy frameworks, across countries. See for instance, Gelos and Ustyugova (2012).

On the First-Round Effects of International Food Price Shocks: the Role of the Asset Market Structure
Author: Mr. Rafael A Portillo and Luis-Felipe Zanna