ï»¿ WPS5943
Policy Research Working Paper 5943
Optimal Food Price Stabilization
in a Small Open Developing Country
Christophe Gouel
SÃ©bastien Jean
The World Bank
Development Research Group
Agriculture and Rural Development Team
January 2012
Policy Research Working Paper 5943
Abstract
In poor countries, most governments implement policies trade and storage policies. They show that an optimal
aiming to stabilize the prices of staple foods, which trade policy largely consists of subsidizing imports
often include storage and trade measures insulating their and taxing exports, which benefits consumers at the
domestic market from the world market. It is of crucial expense of producers. Import subsidies alleviate the non-
importance to understand the precise motivations and negativity of food storage. In other words, when stocks
efficiency of those interventions, because they can have are exhausted, subsidizing imports prevents domestic
consequences worldwide. This paper addresses those price spikes. One striking result: an optimal storage
issues by analyzing the case of a small, open developing policy on its own is detrimental to consumers, since its
country confronted by shocks to both the crop yield and stabilizing benefits leak into the world market and it
foreign price. In this model, government interventions raises the average domestic price. By contrast, an optimal
may be justified by the lack of an insurance market for combination of storage and trade policies results in a
food prices. Considering this market imperfection, the powerful stabilizing effect for domestic food prices.
authors design optimal public interventions through
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The author may be contacted at cgouel@worldbank.org.
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Produced by the Research Support Team
Optimal food price stabilization in a small open developing
countryâˆ—
Christophe Gouelâ€ SÃ©bastien Jeanâ€¡
Abstract
In poor countries, most governments implement policies aiming to stabilize the prices of staple
foods, which often include storage and trade measures insulating their domestic market from the
world market. It is of crucial importance to understand the precise motivations and eï¬ƒciency
of those interventions, because they can have consequences worldwide. This paper addresses
those issues by analyzing the case of a small, open developing country confronted by shocks to
both the crop yield and foreign price. In this model, government interventions may be justiï¬?ed
by the lack of an insurance market for food prices. Considering this market imperfection, the
authors design optimal public interventions through trade and storage policies. They show that
an optimal trade policy largely consists of subsidizing imports and taxing exports, which beneï¬?ts
consumers at the expense of producers. Import subsidies alleviate the non-negativity of food
storage. In other words, when stocks are exhausted, subsidizing imports prevents domestic price
spikes. One striking result: an optimal storage policy on its own is detrimental to consumers,
since its stabilizing beneï¬?ts leak into the world market and it raises the average domestic price.
By contrast, an optimal combination of storage and trade policies results in a powerful stabilizing
eï¬€ect for domestic food prices.
Keywords: food security, incomplete markets, storage, trade policy, export restrictions.
JEL classiï¬?cation: D52, F13, Q11, Q17, Q18.
Sector Board: Agriculture and Rural Development (ARD).
âˆ—
The authors thank Jean-Marc Bourgeon, Ã‰douard Challe, Christopher Gilbert, Michel Juillard, Will Martin
and Brian Wright for helpful comments. This research was generously supported by the Knowledge for Change
(KCP) Trust Fund and by the AgFoodTrade project, funded under the 7th Framework Programme for Research and
Development, DG-Research, European Commission. Much of this work was done while Christophe Gouel was a PhD
student at INRA and Ã‰cole Polytechnique. The views expressed here are the sole responsibility of the authors and do
not reï¬‚ect those of the European Commission or of the World Bank.
â€
The World Bank and CEPII (cgouel@worldbank.org)
â€¡
INRA and CEPII (sebastien.jean@inra.grignon.fr)
1
1 Introduction
In developing countries, staple foods frequently account for a signiï¬?cant share of poor householdsâ€™
budgets. Many poor people have limited possibilities to insure against adverse price shocks. Price
spikes are very problematic for poor households that are not self-suï¬ƒcient, and often jeopardize
their capacity to feed themselves. An important response of developing country governments to
expressions of this concern is to implement food price stabilization policies. Yet, the study of these
policies has been conï¬?ned mainly to closed economy contexts, with an emphasis on the role of
storage (see Wright, 2001, for a survey). From a theoretical standpoint, little is known about the
role of trade policy in price stabilization programs, despite their widespread use. Recourse to trade
policy to counter price volatility has been common in most Asian countries where stabilizing the
domestic price of rice is a central objective (Timmer, 1989, Islam and Thomas, 1996, Dorosh, 2008),
and also in Middle East and African countries in the case of wheat and to a lesser extent maize as
well as rice (Dorosh, 2009, Wright and Caï¬?ero, 2011). The question is probably less acute in Latin
America, where most countries are net exporters of grains, but it is not irrelevant, as witnessed by
the use by Chile of a price-band system for wheat and a few other food products (Bagwell and Sykes,
2004). More generally, based on a large-scale database on agricultural price distortions, Anderson
and Nelgen (2012) show that countries tend to vary their nominal rate of assistance to agriculture so
as to limit the eï¬€ects of variations in world prices on domestic prices.
Trade and trade policies are key aspects that need to be taken into account in considerations of
food security and price stabilization in developing countries. They raise numerous questions, the
most important perhaps being: How should storage and trade policies be combined to achieve price
stabilization? Increased reliance on national buï¬€er stocks is frequently suggested as a remedy for
developing countries faced with signiï¬?cant volatility in world prices. But is this a consistent policy
per se, independent of trade policy interventions? Dorosh (2008) suggests that greater reliance on
the world market allowed much more cost-eï¬€ective price stabilization in Bangladesh than in India;
the latter relied almost exclusively on huge public stocks and severe restrictions on imports. Can it
be taken for granted that greater trade openness, or more reactive trade policy, would reduce the
amounts of stocks needed to achieve a given stabilization target, and to what extent?
Export restrictions raise a number of additional questions. Most analysts of the 2007â€“08 food crisis
agree that trade policies played a signiï¬?cant role in fueling international price spikes (von Braun, 2008,
Mitra and Josling, 2009, Headey, 2011). In particular, export bans enforced by several rice exporters
seem to have contributed greatly to the astonishing price levels reached (Slayton, 2009). Noting the
similar situation in the 1973â€“74 crisis, Martin and Anderson (2012) emphasize the collective action
problem created by export restrictions: their use by some countries to provide shelter from price
spikes aggravates the problem for others (see also BouÃ«t and Laborde Debucquet, forthcoming). The
restrictions imposed by Russia on its cereal exports following a drought in 2010 can only add to
this concern. A ï¬?rst step towards coping with this problem is to achieve a better understanding
of the motivations and consequences of export restrictions. Based on Marshallian surplus analysis,
2
many authors conclude that such policies are harmful to the countries enacting them. Is this really
the case, or do export restrictions make economic sense for a small open economy? And, in this
case, is refraining from the imposition of export restrictions an important sacriï¬?ce for the country
concerned? Would speciï¬?c ï¬‚anking policies be preferable?
The way that uncertainty aï¬€ects trade theory results has been widely studied, as discussed in the
next section. However, a speciï¬?city of staple food products is that they are storable, and this is
not taken into account in most of these works. Storage and its consequences are the subject of a
separate strand in the literature, which includes some analyzes of its relationships with trade and
trade policy. Although several cases are studied, these studies do not identify optimal policies. In
contrast, the present paper proposes a design of optimal stabilization policies suitable for a small
open economy, within a rational expectations storage model using tools developed for the analysis of
optimal dynamic policies. The focus is on food security concerns in developing countries, assuming
consumers to be risk averse with no insurance possibilities and a country that is self-suï¬ƒcient on
average.
This is challenging because the combination of rational expectations and non-negative storage and
trade constraints renders the model (which does not admit closed-form solutions) problematic to
solve, and even more diï¬ƒcult to optimize along a dynamic path. To achieve model tractability and
identify stylized results requires some simpliï¬?cations. For this reason, we focus on consumersâ€™ risk
aversionâ€”most directly linked to food security concernsâ€”but overlook producersâ€™ risk aversion,
despite the signiï¬?cant proportion of poor farmers in developing countries. We disregard also supply
reaction which, while being a potentially important mechanism, is usually of limited quantitative
importance in the time frame of a price surge. We work with a single-country model also in the
interests of simplicity and, to obtain initial insights about the issue of export restrictions, we assess
the consequences for developing countries of refraining from imposing such restrictions.
2 Trade, uncertainty and storage: Related literature
Uncertainty is widely seen as potentially aï¬€ecting the main conclusions of trade theory. David
Ricardo (1821, Ch. 19) concluded that temporary tariï¬€s on cereals might be justiï¬?ed to avoid large
losses to farmers who after increasing their production, and so the required capital, to face a sudden
change in trade, such as wars, would suï¬€er a lot from an immediate return to the situation prevailing
before the crisis. The ï¬?rst formalization of this issue was achieved by Brainard and Cooper (1968).
Based on a portfolio approach, they showed that diversiï¬?cation in a primary producing country
decreases ï¬‚uctuations in national income, which increases national welfare if the country is risk averse.
Based on a comparable framework, including risk aversion in a context where productive choices are
made before uncertainty is resolved, several other papers challenge the idea of the optimality of free
trade under uncertainty (Batra and Russell, 1974, Turnovsky, 1974, Anderson and Riley, 1976).
Helpman and Razin (1978) point out that this result hinges crucially on the assumption of incomplete
risk-sharing markets. They show that the main results of Ricardian and Heckscher-Ohlin theories of
3
international trade, including the optimality of free trade, carry over to uncertain environments if
risk can be shared appropriately. In their model, this is the case because the stock market allows
households to diversify their capital, and cross-border trade in ï¬?nancial assets opens the possibility
for international risk-sharing arrangements.
Helpman and Razinâ€™s seminal contributions clarify decisively the conditions underlying potential
deviations from standard results and pave the way to numerous insightful elaborations. Yet there is
a variety of reasons why the conditions required for their results might not hold. For instance, in the
case that households need to invest their capital in a particular activity, without any possibility to
diversify, to insure, or to trade the corresponding risk. In this context, which is plausible especially
for rural households in developing countries, Eaton and Grossman (1985) show that optimal trade
policy for a small open economy is not free trade. On average, the optimal policy entails an anti-trade
bias. Similar conclusions emerge if market incompleteness is the result of lack of international trade
in ï¬?nancial assets (Feenstra, 1987). In a speciï¬?c-factor model with risk-averse factor owners, Cassing
et al. (1986) also show that a state-contingent tariï¬€ policy can increase the expected utility of all
agents. Newbery and Stiglitz (1984) provide another illustration of this potential insurance role of
trade restrictions, extending the analysis to a two-country model. Without insurance markets, they
show that free trade may be Pareto-inferior to no trade. Indeed, autarky directly links domestic
prices to domestic output, thus providing, for a unitary price elasticity of demand, a perfect income
insurance for farmers.
These diï¬€erent cases show that a departure from free trade may be motivated by risk-sharing
objectives, when other arrangements are not available. When dealing with food security in developing
countries, assuming incomplete insurance markets seems reasonable. Poor households have little
opportunity to insure against the real income risk associated with variable food prices, and poor
farmers (as well as many other poor workers) cannot diversify their income source, at least in the
short run. Since we focus on food security in a developing country, we adopt this assumption of
market incompleteness and assume consumers to be risk averse, with no insurance schemes available.
Dealing with food security requires accounting for the fact that staple food products are storable.
This is especially important because storage is a central feature of these markets and can be thought
of as an intertemporal risk-sharing arrangement. Studies of food security have long regarded storage
as a key feature. Early analyzes of storage-trade interactions relied upon idealized or arbitrary storage
technologies (Feder et al., 1977, Pelcovits, 1979, Bigman and Reutlinger, 1979, Reutlinger and Knapp,
1980). They tend to emphasize that trade is more cost-eï¬€ective than storage at promoting price
stability. While useful to ensure tractability, such simpliï¬?ed representations, not rooted in a consistent
description of agent behaviors, do not accurately reï¬‚ect the risk-sharing properties of storage. A
consistent modeling of these properties requires a proper accounting for agentsâ€™ expectations: the
best way to do this is to work in a rational expectations, inï¬?nite-horizon framework.
Apart from speciï¬?c analyzes of oil-related problems, where world prices are the main source of
uncertainty (Teisberg, 1981, Wright and Williams, 1982b), storage-trade interactions were ï¬?rst
studied, in a rational expectations, inï¬?nite-horizon framework, by Williams and Wright (1991, Ch. 9),
4
where a small open market is considered as the extreme case in a two-country model.1 In a series
of papers, Jha and Srinivasan (1999, 2001, Srinivasan and Jha, 2001) model Indian agricultural
markets in relation to the world. They consider the rice market, in which India is a large country,
and the wheat market, where India is a small country. In both markets, there are competitive private
storers. World prices are randomly generated without accounting for serial correlation. The authors
ï¬?nd international trade to be stabilizing even when the international price is more volatile than
the domestic price. Brennan (2003) considers the Bangladesh rice market. She shows that opening
the market to trade is as stabilizing as some public policies (such as subsidies to private storage or
price ceiling), and is without any ï¬?scal cost. In all these studies, welfare is measured by changes in
surpluses.
In contrast to the theoretical analyzes of trade under uncertainty mentioned above, storage-trade
models all focus on the assessment of given exogenous policies. They provide no hints as to what
the optimal policy would be. The present paper extends normative analyzes of trade theory in an
uncertain environment to an intertemporal framework with storage under rational expectations.
Since it is very diï¬ƒcult to design optimal policies in a dynamic setting, we consider a single-country
model. We follow Williams and Wrightâ€™s insight and represent the world price as generated by a
storage model and considered as exogenous to the country.
3 The model
We consider the market for a storable commodity in a small open economy. The world price is taken
as given and the per-unit transport cost is constant. Consumers are risk averse and domestic food
price volatility is driven by random output and a stochastic world price. Production is deï¬?ned by
exogenous stochastic shocks, so producers are not represented explicitly, but introduced later when
we account for the eï¬€ect of the policies on their welfare.
3.1 Consumers
The economy is populated with risk-averse consumers whose ï¬?nal demand for food has an isoelastic
speciï¬?cation: D (Pt ) = dPtÎ± Y Î· , where d > 0 is a parameter of normalization; Pt is period t price;
and Î±, with Î± < 0 and Î± = âˆ’1, and Î· = 1 are the price and income elasticities. Income, Y , is
assumed to be constant over time, which limits the number of state variables (see Section 3.4) and
allows a diagrammatic exposition of the results. Assuming there are only two goods and the second
good is the numeraire, integration of this demand function gives the following instantaneous indirect
1
Miranda and Glauber (1995) conï¬?rm Williams and Wrightâ€™s results and propose an improved numerical method.
Makki et al. (1996, 2001) present a policy application of this model; based on a three-country model including the EU,
the USA and the Rest of the World, they analyze the eï¬€ects of removing current policy distortions such as export
subsidies. Coleman (2009) extends Williams and Wrightâ€™s work by considering that trade takes time. The time to
ship then brings a new motive for stockpiling.
5
utility function (Hausman, 1981)
Y 1âˆ’Î· P 1+Î±
v (Pt , Y ) =
Ë† âˆ’d t . (1)
1âˆ’Î· 1+Î±
This utility function has relative risk aversion equal to the income elasticity of demand. To distinguish
Ë†
income elasticity from risk aversion, we follow Helms (1985a): we assume v (Pt , Y ) to be positive
and apply a monotone transformation to the indirect utility function,
[Ë† (Pt , Y )]1+Î¸
v
v (Pt , Y ) = , (2)
1+Î¸
with v (Pt , Y ) â†’ ln v (Pt , Y ) as Î¸ â†’ âˆ’1. This speciï¬?cation is still consistent with the isoelastic
Ë†
demand function, but its coeï¬ƒcient of relative risk aversion is
Y 1âˆ’Î·
Ï? (Pt , Y ) = Î· âˆ’ Î¸ , (3)
Ë†
v (Pt , Y )
with Î¸ indexing the degree of risk aversion.
For simplicity, the representative consumer is assumed to adopt hand-to-mouth behavior: he consumes
current income and does not save to smooth out ï¬‚uctuations. The dynamics are thus simpliï¬?ed,
since consumerâ€™s â€œcash on handâ€? does not have to be included as a state variable. This assumption
overlooks the role of self-insurance through saving. However, such self-insurance remains limited
in practice and falls short of providing protection comparable to what a complete market delivers,
due inter alia to borrowing constraints and to the rather large share of the budget accounted for by
staple food in many developing countries, especially for poor households.
Given the absence of saving, the consumer does not solve an intertemporal problem. At each period,
he is concerned only with current-period demand, which is not aï¬€ected by the degree of risk aversion.
Spatial and intertemporal arbitrages are thus independent from consumer risk aversion, which creates
the need for public intervention.
3.2 Storers
The single representative speculative storer is assumed to be risk neutral and acts competitively.
Storage transfers a commodity from one period to the next. Storing the quantity St from period t to
period t + 1 entails a purchasing cost, Pt St , and a storage cost, kSt , with k the unit physical cost
of storage. A (positive or negative) per-unit subsidy Î¶t for private storage is also considered. The
beneï¬?ts in period t are the proceeds from the sale of previous stocks: Pt Stâˆ’1 . The storer maximizes
his expected proï¬?t as stated by the following sum of cash ï¬‚ows
âˆž
V S (Stâˆ’1 , Pt , Î¶t ) = max âˆž Et Î² i [Pt+i St+iâˆ’1 âˆ’ (Pt+i + k âˆ’ Î¶t+i ) St+i ] , (4)
{St+i â‰¥0}i=0
i=0
6
where Et denotes the mathematical expectations operator conditional on information available at
time t, and Î² is the discount factor. The storerâ€™s problem can be expressed in a recursive form using
the following Bellman equation:
V S (Stâˆ’1 , Pt , Î¶t ) = max Pt Stâˆ’1 âˆ’ (Pt + k âˆ’ Î¶t ) St + Î² Et V S (St , Pt+1 , Î¶t+1 ) . (5)
St â‰¥0
This equation has three state variables: the price and the subsidy, whose dynamics are considered
exogenous by the storer, and the stock carried over from the previous year. Using the ï¬?rst-order
condition on St and the envelope theorem, and taking into account the possibility of a corner solution
(i.e., the non-negativity constraint of storage), this problem yields the following complementary
condition:2
St â‰¥ 0 âŠ¥ Î² Et (Pt+1 ) + Î¶t âˆ’ Pt âˆ’ k â‰¤ 0, (6)
which means that inventories are null when the marginal cost of storage is not covered by expected
marginal beneï¬?ts; for positive inventories, the arbitrage equation holds with equality. The storer
thus buys when present prices are low enough compared to their expected future level.
3.3 International trade
Since the model describes a homogeneous product for a small open economy, international trade
modeling collapses to two arbitrage conditions, between the domestic price on the one hand, and
export or import parity price on the other hand. Expressed in complementarity form,
Mt â‰¥ 0 âŠ¥ M
Pt âˆ’ Î½t âˆ’ (Ptw + Ï„ ) â‰¤ 0, (7)
Xt â‰¥ 0 âŠ¥ X
(Ptw âˆ’ Ï„ ) âˆ’ Pt âˆ’ Î½t â‰¤ 0, (8)
where Mt and Xt are imports and exports; Ptw is world price; and Ï„ represents per-unit import
M X
and export costs, assumed to be constant and identical. Î½t and Î½t denote (positive or negative)
per-unit taxes on imports and exports. The complementarity equations for trade (7)â€“(8) imply that
the domestic price is restricted to evolving in a moving band deï¬?ned by the world price, trade costs,
and trade taxes if any:
X M
Ptw âˆ’ Ï„ âˆ’ Î½t â‰¤ Pt â‰¤ Ptw + Ï„ + Î½t . (9)
3.4 Recursive equilibrium
Period t harvest is denoted H and is an i.i.d. random variable. The model has three state variables:
t
Stâˆ’1 , H and Ptw . In any time period, the ï¬?rst two can be combined into one variable, availability
t
2
Complementarity conditions in what follows are written using the â€œperpâ€? notation (âŠ¥). This means that both
inequalities must hold, and at least one must hold with equality. So a â‰¤ X âŠ¥ F (X) â‰¥ 0 is a compact formulation for
X = a â‡’ F (X) â‰¥ 0 and X > a â‡’ F (X) = 0.
7
(At ), the sum of production and private carry-over:
H
At = Stâˆ’1 + t . (10)
The other state variable, the world price, follows a continuous Markov chain, deï¬?ned in the next
section and characterized by the following transition function:
w
Pt+1 = f Ptw , w
t+1 , (11)
where w is the random production in the world market, which is assumed to be uncorrelated with
domestic production shocks.3
Market equilibrium can be written as
At + Mt = D (Pt ) + St + Xt , (12)
Based on the above, we can deï¬?ne the recursive equilibrium of the problem without public intervention:
M X
Deï¬?nition. In the absence of a stabilization policy (i.e., Î¶t = Î½t = Î½t = 0), a recursive equilibrium
is a set of functions, S (A, P w ), P (A, P w ), M (A, P w ) and X (A, P w ) deï¬?ning private storage, price,
import and export over the state {A, P w } and transition equations (10)â€“(11) such that (i) the storer
solves (4), (ii) trade obeys the arbitrage equations (7)â€“(8), and (iii) the market clears.
3.5 World price
Modeling world price dynamics is a crucial issue here. The level of the world price directly inï¬‚uences
domestic price through arbitrage with export and import parity prices, and may also matter for
expectations about its future level, by inï¬‚uencing storage decisions and domestic price expectations,
central to the issues considered here.
In single-country models, the world price is generally represented as a stochastic process following a
standard distribution, including in some cases ï¬?rst-order autocorrelation (Srinivasan and Jha, 2001,
Brennan, 2003). Such simpliï¬?cations are not consistent with the stylized facts on agricultural prices
(Deaton and Laroque, 1992), which appear to be correctly represented by a storage model (Caï¬?ero
et al., 2011). Taking full account of this storage-based representation of world prices would thus
require a two-country model, in which the rest of the world is modeled alongside the economy being
studied. The complexity cost of such an option would be high, and diï¬ƒcult to reconcile with our
objective of identifying optimal policies. Following Williams and Wright (1991, Ch. 9), a way out of
this dilemma is to consider the small-open economy model as a limit case of a two-country model
as the size diï¬€erence between the two countries increases. In this limit case, the small economy is
negligible compared to the big one, meaning that the rest of the world can be modeled without
3
It would not be hard to allow for correlated production shocks if needed.
8
paying attention to the small economy under study. In other words, overlooking the inï¬‚uence of
the economy on the world market allows world prices to be modeled as a separate process, which
is exogenous from the economyâ€™s point of view. This assumption of a price-taker economy leads
to a disregard for motivations potentially linked to the inï¬‚uence of government decisions on world
markets, and greatly simpliï¬?es the analysis.
World prices are thus assumed to result from a storage model with random inelastic production; they
are set as a result of a system of three equations equivalent to (6), (10) and (12), without import
and export variables, and without storage subsidy. This system has one state variable: availability.
All variables and functions corresponding to the world market are indicated with the superscript w.
Given the modelâ€™s structure, the observation of price allows us to deï¬?ne the state of the system; price
dynamics can thus be deï¬?ned as a continuous state Markov chain. Its expression can be derived
using equation (10) and the decision rules, P w (Aw ) and S w (Aw ). It gives
w
Pt+1 = P w Aw
t+1 (13)
= P w S w (Aw ) + Âµ
t
w
t+1 (14)
= P w S w (P w )âˆ’1 (Ptw ) + Âµ w
t+1 , (15)
from which equation (11) follows. Âµ is a scale parameter that allows a shift in world yield distribution.
In the following, we want to focus on economic mechanisms independent of structural diï¬€erences
between the economy and the rest of the world, so we consider a perfectly symmetrical situation in
which the world market is calibrated with the same parameter values as the domestic country (so in
the benchmark calibration Âµ = 1). Given that we consider an asymptotic situation in which the rest
of the world is inï¬?nitely larger than the country, trade does not aï¬€ect the world equilibrium and
the calibration does not need to account for any size diï¬€erence. The symmetry between the rest of
the world and the economy may seem arbitrary. For instance, the diversiï¬?cation of risks across the
many countries that are part of the rest of the world could be assumed to lead to a lower variability
of output. If the rest of the world is richer on average than the country considered, it could be
argued also that price elasticity should be assumed to be smaller in the rest of the world. This would
make sense, but this symmetry assumption means that international trade is not motivated by any
structural diï¬€erence, but only by the existence of country-speciï¬?c production shocks uncorrelated to
worldwide shocks. The sensitivity of the results to this assumption is analyzed in Section 6.6.
3.6 Calibration
The rational expectations storage model does not allow a closed-form solution. It has to be
approximated numerically. The numerical algorithm that we use is based on a projection method
and is described in detail in Appendix D.
The parameters are set such that, at the non-stochastic steady-state equilibrium, price, production,
consumption and availability are equal to 1, and imports and exports are equal to 0 (see Table 1 for
9
parameter values). As a result, the country is self-suï¬ƒcient in the steady state and no trade takes
place.
Table 1. Parameterization
Parameter Economic interpretation Assigned value
Î² Annual discount factor 0.95
Î· Income elasticity 0.5
Î± Own-price demand elasticity âˆ’0.4
Î³ Commodity budget share 0.15
Y Income 6.67
d Normalization parameter of demand function 0.39
Î¸ Parameter deï¬?ning risk aversion âˆ’2.62
k Physical storage cost 0.06
Ï„ Trade cost 0.2
Âµ Normalization parameter of world yield distribution 1
H
, w
Probability distribution of yield B (2, 2) Â· 0.5 + 0.75
An annual interest rate of 5% is used for discounting. Based inter alia on Korinek and Sourdin (2010),
we set trade costs to 20%. This is more than the average cost cited in this study for agricultural
products, reï¬‚ecting our focus on grains for poor countries.4
Seale and Regmi (2006) estimate elasticities for food consumption for 144 countries. From their
research, we choose cereal elasticities typical of low-income countries: âˆ’0.4 for price elasticity and
0.5 for income elasticity. We assume that consumers spend, at the steady state, Î³ = 15% of their
income on the staple, a value intermediate between what is observed for rice consumption in poor
and aï¬„uent households in Asia (Asian Development Bank, 2008). Since steady-state consumption
and price are equal to 1, income, which is assumed to be constant, is equal to the inverse of the
commodity budget share, 1/Î³. We assume at the steady state a relative risk aversion parameter of 2,
implying Î¸ = âˆ’2.62.
We follow Brennan (2003) and assume a per-unit storage cost of 6% of the steady-state price (i.e.,
k = 0.06). Combined with the opportunity cost, this physical storage cost entails an overall storage
cost at steady state equal to 11.3% of the steady-state price. There is no evidence on which to base
the choice of a particular probability distribution for yield at a country level. Thus, we assume
that the random productions, H and w , follow a beta distribution. The beta distribution has
the advantage of being empirically supported and popular in stochastic yield modeling at a local
level (see, among others, Nelson and Preckel, 1989, Babcock and Hennessy, 1996), and of having a
bounded support, which is computationally convenient. We assume the distribution to have shape
parameters 2 and 2, which makes it unimodal at 0.5, and symmetric. The distribution is translated
and rescaled to vary between 0.75 and 1.25, implying a coeï¬ƒcient of variation of 11.2%.
4
Noteworthily, international trade also frequently entails beyond-average domestic transport costs.
10
4 Dynamics without public policy
To understand the consequences of public policies, the situation without public intervention is a
natural and useful benchmark. Since the model used here diï¬€ers signiï¬?cantly from those discussed
in the literature so far, we analyze this benchmark case in some detail.
4.1 Price, trade and storage behavior
For a small open economy, without any trade taxes, domestic prices necessarily lie in a moving band
deï¬?ned by world prices plus or minus trade costs. Compared to a closed economy, this context
radically modiï¬?es storage behavior and its consequences. Abundant availability usually favors storage,
but here exporting is another potential proï¬?table outlet; when scarcity prevails, the stabilizing
eï¬€ect of selling inventories may be redundant in the face of the price ceiling imposed by import
competition.
A ï¬?rst salient feature is that there is no storage when the country imports (see Figure 1(a)). Indeed,
in this case, the domestic price is exactly equal to the world price plus trade costs, which allows
rewriting equation (6) as follows
Î² Et (Pt+1 ) âˆ’ Pt âˆ’ k = Î² Et (Pt+1 ) âˆ’ Ptw âˆ’ Ï„ âˆ’ k. (16)
Since the storage arbitrage condition (6) holds in the rest of the world (as assumed here given the
way world prices are determined),
w
âˆ’ Ptw âˆ’ k â‰¤ âˆ’Î² Et Pt+1 , (17)
which combined with (16) gives
w
Î² Et (Pt+1 ) âˆ’ Pt âˆ’ k â‰¤ Î² Et Pt+1 âˆ’ Pt+1 âˆ’ Ï„. (18)
Given (9), the domestic price is always inferior to the import parity price, which also holds in
expectations terms. As a result,
Î² Et (Pt+1 ) âˆ’ Pt âˆ’ k â‰¤ (Î² âˆ’ 1) Ï„ < 0. (19)
This implies that domestic storage is not proï¬?table while importing, because the expected value of
next yearâ€™s diï¬€erence between domestic and world prices cannot exceed trade costs.
This feature, emphasized inter alia by Williams and Wright (1991) in a two-country framework,
reï¬‚ects the fact that importing for storage never makes economic sense when intertemporal arbitrage
is the same at home and abroad: deferring until the next year the decision about whether importing
will be necessary or not is always preferable.
11
Storage and exports may coexist in cases where availability is relatively abundant. When the country
exports, the domestic price equals the export parity price and the storage arbitrage equation becomes
St â‰¥ 0 âŠ¥ Î² Et (Pt+1 ) âˆ’ Ptw + Ï„ âˆ’ k â‰¤ 0. (20)
For a high enough world price, the expected domestic price cannot be so high as to make speculation
proï¬?table: exporting is more proï¬?table than storing, and no storage takes place. The coexistence of
storage and exports is only observed for intermediate world price levels: high enough compared to
domestic prices as to make exporting proï¬?table, but not so high as to make the ï¬?rst unit of storage
less proï¬?table than exporting.5 The interrelations between storage and exports are illustrated also
by the storage rule, where exports are reï¬‚ected in ï¬‚at storage curves for relatively large availabilities.
In the left panel of Figure 1(a), this situation of non-zero storage is observed only with a world price
equal to 1.1. As soon as the world price reaches 1.2, exporting is always more proï¬?table than storing,
so that the storage rule is ï¬‚at throughout.
0.4
World price:
0.9 1.4 0.4
1
1.1
1.2 1.2 0.2
Trade
Stock
Price
0.2 1.3
1 0
0.8
0 âˆ’0.2
0.8 1 1.2 1.4 0.8 1 1.2 1.4 0.8 1 1.2 1.4
Availability Availability Availability
(a) Without public intervention
0.4
1.4 0.4
1.2 0.2
Trade
Stock
Price
0.2
1 0
0.8
0 âˆ’0.2
0.8 1 1.2 1.4 0.8 1 1.2 1.4 0.8 1 1.2 1.4
Availability Availability Availability
(b) Optimal trade policy
Figure 1. Stock, price and trade behavior. Negative trade values refer to imports.
5
The returns to storage are declining due to its negative inï¬‚uence on expected future domestic prices
12
0.4
1.4 0.4
1.2 0.2
Trade
Stock
Price
0.2
1 0
0.8
0 âˆ’0.2
0.8 1 1.2 1.4 0.8 1 1.2 1.4 0.8 1 1.2 1.4
Availability Availability Availability
(c) Optimal storage subsidy
0.4
1.4 0.4
1.2 0.2
Trade
Stock
Price
0.2
1 0
0.8
0 âˆ’0.2
0.8 1 1.2 1.4 0.8 1 1.2 1.4 0.8 1 1.2 1.4
Availability Availability Availability
(d) Optimal policy with both instruments
Figure 1. (continued )
Notably, the current world price aï¬€ects domestic storage even in the absence of trade. Indeed, world
prices are positively autocorrelated since they are generated by a storage model. As a result, a
higher current world price entails higher world price expectations for the next period. Accordingly,
storage outside trade situations moves slightly upward for higher world prices.
Under the assumption of a small economy, exporting implies a complete disconnect between domestic
prices and availability, as reï¬‚ected in the ï¬‚at segments of the price curves observed for large enough
availabilities for world prices equal to 1.1 and above, in the central panel of Figure 1(a). Similarly,
for limited availabilities, the domestic price is disconnected from availability when it reaches the
import trigger price, equal to the world price plus trade costs. In between these two cases, the price
curve takes a standard form in the presence of storage, with a strongly downward sloping curve
for availabilities below a given threshold under which no storage takes place, and a smoother curve
thereafter. As far as trade is concerned, assuming exogenous world prices implies that the net trade
curve has a unitary slope whenever trade is not zero.
13
A sample simulation of world and domestic prices illustrates the link between them (Figure 2). The
domestic price tends to be set to the import parity price when the world price is low, and to the
export parity price when the world price is high. Most world price spikes are imported through
trade to the domestic market.
2.0 World price
Export and import parity prices
Domestic price
1.5
Price
1.0
0.5
Year
Figure 2. Simulated history of prices without public intervention
4.2 Eï¬€ects of trade and storage costs
While commodity trade is usually seen as a way of smoothing production shocks in diï¬€erent markets,
in the present case, trade does not necessarily decrease price volatility. Actually, price instability is
the same when trade costs are zero and when they are prohibitive: in the ï¬?rst case, the domestic
price is exactly equal to the world price, while in the second case the economy behaves as if it were
a closed economy. Both situations are equivalent since we assume that world prices result from a
closed economy behavior, with the same parameters as those used for the economy studied.
When trade is costless, however, the strict link with world prices means that only storage in the
world market matters, and that storage in the small economy does not matter. Under prohibitive
trade costs, in contrast, only domestic storage matters. Intermediate cases exhibit less volatility,
with domestic and foreign storage combining to smooth price variability: domestic shortages can be
alleviated by imports while surpluses can be exported in the world market, but the country remains
partly shielded from world prices by trade costs. The result is a U-shaped relationship between trade
costs and volatility (Figure 3, left panel). Volatility is lowest for trade costs close to 0.23.6
This result contrasts with the inï¬‚uence of the domestic physical storage cost, which always increases
domestic price instability (Figure 3, right panel). For prohibitive storage costs, instability is slightly
lower than in the world market since trade costs partly insulate the country from worldwide instability.
6
Numerical results here and in the following sections are calculated over 1,000,000 sample observations from the
asymptotic distribution.
14
Coefficient of variation of domestic price
Coefficient of variation of domestic price
0.22 0.22
0.20 0.20
0.18 0.18
0.16 0.16
0.14 Benchmark calibration 0.14
0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.05 0.1 0.15 0.2 0.25
Trade cost Domestic physical storage cost
Figure 3. Dependence of price instability on trade and storage costs. The value of storage cost is
maintained equal to the benchmark when analyzing the inï¬‚uence of the trade cost, and reciprocally.
For zero physical storage costs, instability is still signiï¬?cant for various reasons: storage is proï¬?t
seeking and entails opportunity costs, so it will not provide for complete stabilization; and there is
some imported instability.
5 Optimal stabilization policy
In modeling the relationships between storage and trade policies, we do not want only to analyze
speciï¬?c cases, we want also to assess what would be the optimal use of these policies. To do this
we assume that government cannot commit to future policies and has to follow time-consistent
policies. Dynamic programming is used to derive the Markov-perfect equilibrium associated with
discretionary policy. To design such optimal stabilization policy, we need to formulate a meaningful
objective. The optimization problem of the discretionary optimal policy can then be stated.
5.1 Social welfare function
Policy design is usually based on maximizing the sum of all agentsâ€™ surpluses. This standard practice
is not valid here, since expected surplus is not a suitable measure of risk-averse consumersâ€™ welfare.
Helms (1985b) shows that, in contrast, ex ante equivalent variations are meaningful welfare indicators.
This corresponds to the amount of income that, in the price regime without intervention, would
bring the same change in expected utility as the intervention considered here. In the intertemporal
framework considered here, where savings are not taken into account, the temporal allocation of
this equivalent income is not neutral.7 Assuming equivalent variation to take the form of a constant
7
Gollier (2010) shows, in a diï¬€erent context, how savings behavior ensures equivalence between alternative patterns
of allocation of replacement income over time.
15
income ï¬‚ow, the corresponding amount EVt is implicitly deï¬?ned, at period t, by
âˆž
Et Ëœt
Î² i v Pt+i , Y + EVt âˆ’ v (Pt+i , Y ) = 0, (21)
i=0
Ëœt
where Pt+i refers to price in period t + i when policy intervention is stopped from t onward. A ï¬?rst-
order Taylor series expansion of the ï¬?rst term around the path followed without further intervention
then gives
âˆž
1 Ëœt
EVt â‰ˆ Et Î² i v (Pt+i , Y ) âˆ’ v Pt+i , Y , (22)
wt
i=0
where wt = Et âˆž Î² i vY Pt+i , Y / (1 âˆ’ Î²) is the discounted average of the future marginal utility
i=0
Ëœt
of income, in expected terms. Social welfare can be measured combining this ex ante equivalent
variation for consumers with the surplus of other agents:8
âˆž
1 Ëœt
Wt = Et Î² i v (Pt+i , Y ) âˆ’ v Pt+i , Y
wt
i=0
âˆž (23)
+ Et Î² i Pt+i H
t+i + Pt+i St+iâˆ’1 âˆ’ (Pt+i + k âˆ’ Î¶t+i ) St+i âˆ’ Cost t+i ,
i=0
where Cost t+i denotes period t + i ï¬?scal cost of public policies.
We neglect the distortionary cost caused by revenue collection, so that the ï¬?scal cost of policy
intervention is the cost of subsidizing private storage plus the net tax cost of trade policy:
M X
Cost t = Î¶t St âˆ’ Î½t Mt âˆ’ Î½t Xt . (24)
Using (10) and (24), the social welfare can be simpliï¬?ed to
ï£± ï£¼
âˆž Ëœt
ï£² v (Pt+i , Y ) âˆ’ v Pt+i , Y ï£½
Wt = Et Î²i M X
+ Pt+i At+i âˆ’ (Pt+i + k) St+i + Î½t+i Mt+i + Î½t+i Xt+i .
ï£³ wt ï£¾
i=0
(25)
The per-unit storage subsidy enters positively in private agentsâ€™ proï¬?t and negatively in public cost,
so it does not feature directly in the social welfare function.
While theoretically exact, this formulation of social welfare is not easily tractable in a dynamic
framework. In particular, it is not amenable to a standard recursive dynamic speciï¬?cation, due
to the time variability of wt . However, this time variability of the expected marginal utility of
income is limited in practice, and remains of second order.9 Numerical simulations can thus
8
Note that, were income elasticity and relative risk aversion equal to zero, this deï¬?nition would make wt equal to
1, so that the social welfare function below would actually be the classic sum of the surpluses.
9
In the simulations presented below, wt does not vary by more than one percent over time.
16
be greatly simpliï¬?ed by assuming wt to be constant over time and equal to its value at t = 0:
wt â‰ˆ E0 âˆž Î² i vY Pi0 , Y / (1 âˆ’ Î²) = w. This assumption is made throughout the simulations
i=0
Ëœ
presented below.
5.2 Optimization problem
The social welfare function is a natural objective for policy optimization. In stating this problem,
policy is assumed to start at period 0 and to be unanticipated. Commitment is unlikely in most
countries and especially in developing countries; the policy is thus assumed to be discretionary.
Three state-contingent instruments can be used to stabilize prices: a tax on, or a subsidy for private
storage, and a trade policy (import and/or export tax or subsidy). Initial state variables are taken
w
as being at their non-stochastic steady-state level (i.e., A0 = 1 and P0 = 1) and initial stocks are
assumed to be null.
Although subsidies to private storage have been used (often in the form of interest rate subsidies,
see Gardner and LÃ³pez, 1996), storage policies usually take the form of public storage. In many
countries, public storage were undertaken by parastatals that were given a monopoly on trade and
storage of grains (Rashid et al., 2008). An optimal subsidy to private storage can be interpreted as
the losses incurred by an eï¬ƒcient monopolistic public agency following an optimal storage rule. In
addition, our focus on storage subsidy removes the need to study the interaction between private
and public storage, which removes a numerical burden.
At each period, optimization entails maximizing the expected sum of the discounted social welfare
function subject to the constraints imposed by private agentsâ€™ behavior and the market equilibrium.
The optimization is carried out over current endogenous and control variables, taking future variables
as given:10
âˆž
max Et M X
Î² i v (Pt+i , Y ) + w Pt+i At+i âˆ’ (Pt+i + k) St+i + Î½t+i Mt+i + Î½t+i Xt+i
St â‰¥0,Pt ,Mt â‰¥0,
M X
Xt â‰¥0,At+1 ,Î¶t ,Î½t ,Î½t i=0
(26)
subject to equations (6)â€“(8) and (10)â€“(12), At and Ptw given, and anticipating St+i , Pt+i , Mt+i ,
M X w
Xt+i , At+i+1 , Î¶t+i , Î½t+i , Î½t+i , Pt+i for i â‰¥ 1.
The above problem deï¬?nes an optimal stabilization policy using both storage and trade taxes or
subsidies. A policy using only one of the two instruments can be deï¬?ned easily by removing from
the objective and constraints all occurrences of Î¶ to deï¬?ne a trade policy, or of Î½ M and Î½ X to deï¬?ne
a storage policy. In each case, solving the problem requires reformulating the constraints deï¬?ned as
complementary equations and writing the dynamic programming problem. See appendix for details.
10
This objective function is obtained, from (25), by multiplying the social welfare function by w, and subtracting the
Ëœt
term independent from policy choice reï¬‚ecting consumerâ€™s utility in the situation without intervention, v Pt+i , Y .
17
6 Characterization and consequences of optimal public
interventions
The analytical framework laid out above allows optimal public interventions to be characterized.
Three cases are considered here, corresponding to: optimal use of trade policy alone, of storage policy
alone, and of both policies jointly. We ï¬?rst describe the nature of these policy interventions, and
analyze their consequences for decision rules. We then assess the resulting consequences for welfare.
6.1 Optimal trade policy
In the absence of any storage policy, the optimal use of trade policy can be characterized mainly as
comprising two types of interventions. The ï¬?rst consists of subsidizing imports when availability is
low. As illustrated in the central panel of Figure 1(b), the corresponding subsidies can be substantial,
with powerful trimming impacts on the upper tail of the price distribution. Thus, for a small open
economy, import subsidies are an eï¬ƒcient way to avoid very high prices: despite their cost, ad
valorem subsidies higher than 15% are shown to be optimal when domestic scarcity coincides with
high world prices. The use of import subsidies may raise questions, since this instrument is not
commonly used in practice. They appear in our framework because our benchmark situation is one
of free trade. In practice, even structural food importers tend to maintain positive barriers to trade,
motivated by political economy reasons (Grossman and Helpman, 1994) or the need to collect ï¬?scal
revenue. Given this alternative benchmark of positive tariï¬€s, import subsidies could be interpreted
as a tariï¬€ reduction for which we have much evidence (Anderson and Nelgen, 2012).
The second type of intervention consists of taxing exports when availability is abundant and world
prices are high. Such interventions decrease the export parity price and avoid importing price spikes
in the world market, through exports. The corresponding tax levels remain moderate, typically less
than 5% even for world prices as high as 1.4.
For high availability and a medium, although suï¬ƒciently high to make export proï¬?table, world price,
such as 1.2, optimal trade policy consists of subsidizing exports. The risk aversion of consumers
does not compel to trim high prices, but to reduce the variability around the mean. For a world
price equal to 1.2, the export parity price is 1, which is just below the mean price (Table 2). Slightly
subsidizing exports in this situation raises the domestic price closer to its mean.
For the asymptotic distribution of prices, such an optimal trade policy results in a decline in the
mean (by 2.4%) and a 20% decrease in the standard deviation. This reduced variability is strongly
asymmetrical, as illustrated by changes in quantiles: it stems mainly from a heavy reduction in the
high prices obtained thanks to import subsidies and export taxes (the top percentile of the price
distribution is driven down from 1.63 without policy to 1.41, a cut of one-third in the deviation from
the average price). The asymmetry typical of price distributions for storable commodities is greatly
reduced, with a skewness half that of the benchmark.
18
Table 2. Descriptive statistics on the asymptotic distribution
Benchmark Trade policy Storage subsidy Both instruments
Price
Mean 1.045 1.020 1.054 1.034
Coeï¬ƒcient of variation 0.173 0.141 0.159 0.121
Skewness 1.248 0.641 1.524 0.995
Correlation coeï¬ƒcient
with domestic shocks âˆ’0.474 âˆ’0.569 âˆ’0.414 âˆ’0.482
with world price 0.788 0.729 0.807 0.780
Quantiles
1% 0.790 0.787 0.837 0.837
25% 0.915 0.900 0.940 0.942
50% 1.000 1.015 1.001 1.008
75% 1.131 1.107 1.123 1.100
99% 1.628 1.410 1.625 1.406
Mean stocks 0.033 0.027 0.048 0.047
Mean imports 0.018 0.021 0.016 0.018
Mean export 0.028 0.024 0.031 0.028
Not surprisingly, these trade policy interventions result in increased imports and decreased exports.
Trimming the upper tail of the price distribution also reduces the proï¬?tability of storage. As a result,
the storage rule is moved to the right, and the mean stock level is reduced by approximately 20% in
the asymptotic distribution (Table 2).
6.2 Optimal storage policy
In a closed economy, an optimal storage policy motivated by consumer risk aversion under incomplete
insurance markets increases the level of storage compared to the situation without intervention
(Gouel, 2011). In the present context of a small open economy without any trade policy intervention,
this is true when storage does not compete directly with exporting, i.e., when the domestic price
remains above the export parity price, because availability is limited and/or the world price is not
too high. In this case, the storage subsidy increases storage and domestic price, contributing to
smoothing of intertemporal price variability in the domestic market.
When availability is abundant and the world price is high enough (e.g., P w â‰¥ 1.3), exporting is
signiï¬?cantly more proï¬?table than storage. This does not leave any room for a worthwhile storage
policy.
The storage subsidy is positive when the current social marginal value of availability (i.e., the
Lagrange multiplier in the market clearing equation (12), Ï‡t in appendix) is lower than its discounted,
expected future value. For a world price equal to 1.1 and for intermediate availabilities (between 1
and 1.2), storage takes place when exports are not proï¬?table (see Figure 1(c)). In this situation,
the small excess of availability with respect to steady-state consumption leads to a present social
19
marginal value of availability lower than its expected discounted value and thus to a positive subsidy
for storage. However, as soon as the domestic price reaches the export parity price, the social
marginal value of availability jumps to zero (when there is some trade, the market clearing equation
is not binding, since foreign demand and supply are inï¬?nitely elastic). Since stock levels are already
high (around 10% of steady-state consumption), the expected future marginal value of availability is
negative in this situation of glut, it then becomes optimal to tax storage. Hence the discontinuity
observed in the storage rule and in the export curve for an availability around 1.1. Beyond this level,
storage coexists with trade, and the optimal policy remains a storage tax (worth approximately
3%). This outcome is paradoxical: while public intervention is motivated by consumer risk aversion,
it tends to discourage storage. Intuitively, this is a context of overly abundant availability, where
dispensing with it through exports is socially preferable to retaining it through storage, even if the
storer would break even to do so.
When storage and export coexist, there is not always a storage tax. For high world prices and,
therefore, relatively high domestic prices, storage proï¬?tability is quickly driven down to zero.
However, the expected social marginal value of availability is positive in this situation, which justiï¬?es
subsidizing storage.
The impact of such an optimal storage policy on the asymptotic price distribution is strongly
constrained by foreign trade: as soon as the country exports or imports, its domestic price is
determined by the world price, since no trade policy intervention is assumed to take place. The
two main channels through which this optimal storage policy inï¬‚uences domestic prices are as
follows: storage subsidies increase domestic prices in situations of stock accumulation, when above-
average availability is combined with low to intermediate world price, with the exception of the
above-mentioned case of storage tax, when storage coexists with exports; ensuing higher stock levels
decrease prices when stocks are sold, but this situation often coincides with situations of international
trade, thus limiting the price fall.
As a result, an optimal public storage policy increases the mean price of the asymptotic distribution
(see Table 2). This seems paradoxical since the standard result obtained for a closed economy is
that introducing private storage or increasing levels of storage through public intervention depresses
average prices (Wright and Williams, 1982a, Miranda and Helmberger, 1988, Gouel, 2011). To
interpret this puzzling result, it is useful to consider such a stabilization policy through storage as a
sequence of transfers of demand from one period t1 to another t2 , when the price is higher (p1 < p2 ),
as suggested by Newbery and Stiglitz (1981, p. 251, theorem 2). Consumersâ€™ demand is reduced by
any additional storage, and it is increased by the same amount (in the absence of spoilage) when the
quantity stored is ï¬?nally sold. To see how demand transfer inï¬‚uences prices, let us diï¬€erentiate the
market clearing equation (12) with regard to stocks:
âˆ‚Pt âˆ‚(Xt âˆ’ Mt )
D (Pt ) +1+ = 0. (27)
âˆ‚St âˆ‚St
20
In a closed economy, the last term is pointless, and this equation collapses to âˆ‚Pt /âˆ‚St = âˆ’1/D (Pt ).
As soon as the demand function is convex, as in case of a constant elasticity function, this partial
derivative is an increasing function of prices, meaning that it is larger in t2 than in t1 . As a result,
modifying storage to operate a small transfer of consumption from t1 to t2 would cut the price in t2
by more than it would increase it in t1 . This explains why stabilization through additional storage
usually depresses mean prices in a closed economy.
In an open economy, in contrast, the last term on the left hand side of (27) is not zero. In particular,
it may be the case that the country exports when the consumption transfer takes place. In this
case, a small enough additional storage will not drive exports down to zero, meaning that they
will not change the domestic price, which will remain equal to the world price minus transport
cost. Conversely, when there is trade, selling additional stocks does not depress the domestic price.
This latter eï¬€ect dominates in practice, mainly because it means that the country cannot insulate
its market from episodes of very high world prices. In this case, domestic stocks are sold on the
world market (either directly or indirectly, when domestic consumption of domestic stocks displaces
imports). This is proï¬?table given the high price level, but it does not curb domestic prices. Hence,
the limited eï¬ƒciency of a storage policy for avoiding high-price episodes.
While this optimal storage policy reduces the standard deviation of prices by approximately 7%,
stabilization comes only from the increase in low prices, a paradoxical result for a public intervention
linked to consumer welfare. The upper quantiles of the asymptotic price distribution hardly change
compared to the benchmark, while the ï¬?rst percentile increases by 6% (Table 2).
The impact on trade is not trivial. The signiï¬?cantly higher average stock reduces the frequency of
scarce domestic availability and of the associated large imports. On average, imports are reduced
by 11%. For the same reason, abundant availabilities are more frequent, and they increase exports.
When storage coincides with exports, this export-enhancing eï¬€ect is magniï¬?ed by the storage tax
mentioned above, which reduces the demand for storage, thus increasing the volumes of domestic
output absorbed by the world market. On average, exports increase by 11%. This increased
importance of exports is also a strong driver of domestic price variability, as illustrated by the
increased correlation between domestic and world prices. On the whole, and despite the absence of
trade policy, this policy could be called opportunistic in trade terms, to the extent that the country
tends to favor storage when world prices are low, but to discourage it when they are high enough.
6.3 Optimal trade and storage policy
Combining optimally trade and storage policies allows a powerful stabilization policy to be devised,
with the standard deviation of domestic prices cut by more than a quarter compared to the benchmark.
This is not surprising given that trade policy is very eï¬ƒcient at preventing domestic prices from
reaching very high levels, while storage is a powerful tool to avoid excessively low prices. The
basics of an optimal policy mix thus consist of using import subsidies to trim the upper tail of the
distribution of domestic prices, and storage subsidies to trim the lower tail. The reduced variability
21
of domestic prices comes from both ends of the distribution, which get substantially closer to the
mean than in the benchmark (see quantiles in Table 2). This outcome is comparable to the outcome
obtained with storage policy for the lower quantiles, and with trade policy for upper quantiles.
The instruments are not independent though, and their interrelationships are especially strong when
the country both exports and stores, as is the case of abundant availability under intermediate world
prices (1.1 in Figure 1(d)). When both instruments are combined, the optimal policy consists of
subsidizing exports while subsidizing storage (for a world price of 1.1, the per-unit subsidy Î¶ is 0.04,
or 4% of the steady-state domestic price). Despite lower stocks, the result is a domestic price beyond
the price that would prevail without intervention, i.e., closer to the distribution mean. This shift
thus contributes to limiting price variability. For a higher world price, however, private storage is
still subsidized, but exports are taxed, which prevents domestic prices from reaching excessive levels.
The mean price of the asymptotic distribution is slightly lower than in the benchmark, an outcome
that is between those for trade and storage policies. It is intermediate also in terms of the correlation
of domestic prices with world prices and with domestic shocks. The impact on foreign trade appears
intermediate between the impacts for each instrument individually, with a slight increase in mean
imports and a slight decrease in mean exports. While interventionist, and entailing potentially
signiï¬?cant export taxes, such an optimal policy is thus not trade-restrictive on average.
This policy leads to an average price transmission elasticity of 0.63 (obtained by regressing the
logarithm of domestic price on the logarithm of border price P t for non zero trade, see Appendix A
for P t deï¬?nition). This elasticity is consistent with econometric evidence. For instance, Anderson
and Nelgen (2012) ï¬?nd that for 75 countries the average short-run price transmission elasticities are
0.52, 0.47, 0.57 and 0.72, respectively, for rice, wheat, maize and soybean. However, contrary to what
happened during the recent food crisis, the optimal policy never involves a complete export ban.
6.4 Decomposition of welfare changes
By changing the distributions of prices, stabilization policies transfer both risk and resources across
agents. To understand the distributive eï¬€ects of these policies, welfare change can be decomposed
for each agent into eï¬ƒciency and transfer changes, where only the former matter for the economy as
a whole.11 This is done here for expected values at period 0.
Consumer gains are given by the ex ante per-period equivalent variation, EV 0 , which includes
two components: a transfer term corresponding to the change in mean expenditure, which exactly
matches the opposite change in the average revenues of the other agents, and an eï¬ƒciency term
corresponding to risk beneï¬?ts and changes in mean consumption. There is no closed-form solution
for the eï¬ƒciency term, which, following Newbery and Stiglitz (1981), is deï¬?ned by the diï¬€erence
11
See Newbery and Stiglitz (1981, Ch. 6 and 9) for a discussion of eï¬ƒciency and transfer gains from price
stabilization.
22
between the equivalent variation and the change in mean expenditures:
âˆž
EV 0 + (1 âˆ’ Î²) E0 Î² t âˆ† [Pt D (Pt )] , (28)
t=0
where the operator âˆ† before an expression refers to the diï¬€erence between its value after and before
public stabilization.
Changes in storersâ€™ proï¬?t, deï¬?ned by equation (4), contain an eï¬ƒciency term representing changes
in storage costs, E0 âˆž Î² t k âˆ† St , and two transfer terms: E0 âˆž Î² t âˆ†[Pt (Stâˆ’1 âˆ’ St )], a transfer
t=0 t=0
from other private agents, and E0 âˆž Î² t Î¶t St , a transfer from the government.
t=0
Producers are not represented explicitly since production is deï¬?ned by exogenous stochastic shocks
but they are aï¬€ected by policies, through price changes. Their welfare change is a transfer term
deï¬?ned by their average change in beneï¬?ts: E0 âˆž Î² t âˆ† Pt H .
t=0 t
International trade can be decomposed into four eï¬€ects on an agent labeled â€œshipperâ€? (in Table 3).
The mean change in imports and exports valued at domestic price, E0 âˆž Î² t âˆ†[Pt (Mt âˆ’ Xt )], is a
t=0
transfer from private domestic agents; the mean change in trade costs, Ï„ E0 âˆž Î² t âˆ†(Mt + Xt ), is
t=0
an eï¬ƒciency term; the mean change of imports and exports valued at the world price, E0 âˆž Î² t t=0
âˆ†[Ptw (Xt âˆ’ Mt )], which we call trade balance, is an eï¬ƒciency term accounting for the possibility to
buy foreign goods allowed by an export surplus in the food commodity; and there is a transfer from
government through the trade policy instrument.
Government is only involved in transfers to private storers through subsidies and to the shipper
through trade policy.
This decomposition is illustrated for the three optimal policies in Table 3.12 It is striking that total
gains are small in comparison to transfers: to protect consumers from price ï¬‚uctuations, public
intervention induces comparatively large changes for other agents. These transfers stem mainly
from two eï¬€ects. The ï¬?rst one is the change in the mean price. A lower mean price, as a result of
optimal use of trade policy or both instruments, beneï¬?ts consumers at the expense of producers. The
reverse is true under storage policy alone. Changes in the covariance between prices and production
shocks also originate transfers between producers and consumers but their importance is more
limited. Changes in the foreign trade balance valued at domestic prices are the second main source
of transfers, from shipper to consumers when the balance increases. Once trade costs and possible
taxes are taken into account, changes in the trade balance aï¬€ect the consumer, not the shipper
(whose proï¬?t is assumed to remain zero throughout). The ï¬?scal cost of policies is an additional
source of transfer: trade policy intervention generates ï¬?scal revenue, while storage subsidies entail
costs. While the magnitude of these ï¬?scal eï¬€ects is limited compared to other eï¬€ects, the cost of a
storage policy is not negligible.
12
Welfare terms are all discounted inï¬?nite sums. They are calculated by transformation to a recursive formulation
and value function iteration.
23
Table 3. Decomposition of welfare impacts of optimal policies on transitional dynamics (as
percentage of the steady-state commodity budget share.)
Trade policy Storage subsidy Both instruments
Consumers 2.47 âˆ’0.94 1.05
Eï¬ƒciencya 1.05 âˆ’0.35 0.52
Consumption expendituresb 1.42 âˆ’0.59 0.53
Producersb âˆ’2.53 1.08 âˆ’0.92
Storers 0 0 0
Transfersb âˆ’0.03 âˆ’0.04 âˆ’0.10
Storage costsa 0.03 âˆ’0.08 âˆ’0.08
Storage subsidyb âˆ’ 0.12 0.17
Shipper 0 0 0
Transfersb 1.15 âˆ’0.45 0.49
Trade costsa 0.03 âˆ’0.02 0.01
Trade balancea âˆ’1.06 0.47 âˆ’0.35
Trade policyb âˆ’0.12 âˆ’ âˆ’0.14
Government 0.12 âˆ’0.12 âˆ’0.03
Storage subsidyb âˆ’ âˆ’0.12 âˆ’0.17
Trade policyb 0.12 âˆ’ 0.14
Total 0.06 0.03 0.10
a
Eï¬ƒciency terms. Total gains are equal to their sum.
b
Transfer terms. They sum to zero, except for the numerical approximations, and do not contribute to total gains.
Eï¬ƒciency gains stem from four terms. Two of them, storage and trade costs, are comparatively small
in all cases. Changes in storage costs are positive for trade policy, reï¬‚ecting the decrease in storage
resulting from trade policy intervention. For the other two policies, they are negative as storage
increases. Most eï¬ƒciency eï¬€ects correspond to the remaining two eï¬€ects, namely consumersâ€™ eï¬ƒciency
gains and trade balance changes. Consumersâ€™ eï¬ƒciency gains are comprised of two components,
not decomposed in Table 3: the gains originating in the change in mean consumption (i.e., the
traditional welfare triangle in a surplus analysis); and the reduction in the risk premium. This latter
is necessarily positive since instability decreases with all policies, while the former may be positive
or negative depending on the mean price change, which reveals the mean consumption change. With
respect to the distribution of these various gains, the policies considered stand in stark contrast.
An optimal storage subsidy without trade policy intervention has counter-intuitive impacts. While
public intervention is motivated by consumersâ€™ risk aversion, it actually results in eï¬ƒciency losses for
consumers: decreased price volatility is more than compensated by reduced mean consumption due
to higher prices. The policy is still beneï¬?cial socially, because the risk-premium decreases, but it does
not increase consumersâ€™ welfare. In the absence of trade policy, a storage subsidy makes consumers
worse oï¬€, since stocks are accumulated mostly when prices are aï¬€ected by stock accumulation, but
sold when the economy is connected to the world market (see the eï¬€ect on quantiles in Table 2).
With an optimal trade policy, consumers enjoy signiï¬?cant eï¬ƒciency gains, resulting from decreases
in both the mean price and price variability, in particular from less frequent price spikes; but this
24
result is at the cost of a signiï¬?cantly deteriorated trade balance. The order of magnitude of both
eï¬€ects is 1%, with a net gain to the economy worth 0.06%.
While intermediate between single-instrument policies, the impacts of an optimal combination of
trade and storage policies are closer to those found for trade policy. Consumers enjoy eï¬ƒciency gains,
mainly reï¬‚ecting the policyâ€™s eï¬€ectiveness in preventing price spikes, and the cost is a deterioration
in the trade balance. However, public intervention is more eï¬€ective in this case: while consumersâ€™
eï¬ƒciency gains do not reach 1%, total social gains reach 0.1%. With smaller transfers, this policy
achieves more gains. This ï¬?nding illustrates the strong complementarity between trade and storage
policies.
6.5 Consequences of a discipline on export restrictions
The use of trade policies to cope with price volatility is potentially problematic because these are
non-cooperative policies, with potential negative consequences for partner countries. In particular,
when a country uses trade policy to insulate its domestic market from extreme world prices values,
its action may further increase the level of world prices. If all trading countries apply this policy,
the collective-action problem may reach the point where individual countriesâ€™ eï¬€orts to insulate
themselves from high world prices cancel each other out (Martin and Anderson, 2012). The use of
export restrictions during the 2007â€“08 food crisis epitomizes this concern; it played a signiï¬?cant role
in price upsurges and spurred calls for multilateral disciplines in the use of such instruments (Mitra
and Josling, 2009, is an example).
The small-open economy setup adopted here does not allow explicit analysis of the problem in
terms of international cooperation. Nonetheless, this problem is rooted in distorted excess supply
curves in countries undertaking price-insulation policies, and these curves can be characterized here
based on the expected value of net exports, for a given world price level. Based on the asymptotic
distribution, these curves are plotted in Figure 4 for the benchmark case of no policy intervention and
under diï¬€erent optimal policy settings. Since policy intervention is justiï¬?ed here by consumersâ€™ risk
aversion, while assuming away producersâ€™ risk aversion, the most meaningful deviations correspond
to high world price levels. An optimal trade policy alone decreases excess supply for all world prices,
and the deviation is signiï¬?cant for world prices above 1.2, thus conï¬?rming the prior that the use of
trade policy may increase high world price episodes. When trade policy is optimally combined with
storage policy, though, deviations from the benchmark are less pronounced. Expected excess supply
is signiï¬?cantly reduced only for world prices above 1.35, and it is slightly increased for intermediate
world price levels, due to the positive eï¬€ect of storage policy on asymptotic availability. The latter
eï¬€ect explains why excess supply is always increased when only storage policy is applied.
These results conï¬?rm that trade policy intervention may signiï¬?cantly distort excess supply curves.
This is especially true for high prices, reï¬‚ecting the use of export taxes. Considering multilateral
disciplines on export restrictions thus makes sense. To evaluate what would be at stake in the case of
such disciplines, we assess the consequences, for a developing country, of deviating from its optimal
25
1.6
World price 1.4
1.2
Benchmark
Trade policy
1.0
Storage subsidy
Both instruments
0.8
0 0.05 0.1 0.15
Trade (E(X t âˆ’ M t |P tw))
Figure 4. Asymptotic excess supply curves. Smoothing splines of the trade-world price relationship
over 1,000,000 observations of the asymptotic distribution.
policy by committing not to use any export restrictions (given existing disciplines, we also assume
export subsidies to be prohibited).
Banning export taxes prevents the country from insulating against high world prices when domestic
availability is large enough. This substantially increases the occurrence of high domestic prices,
with an upper tail of the price distribution closer to the no-intervention benchmark than to the
undisciplined optimal policy (the 1, 25, 50 ,75 and 99 quantiles are respectively at 0.84, 0.94, 1.00,
1.11 and 1.61). Since the optimal storage subsidy decreases the occurrence of low prices by increasing
stock levels, the overall eï¬€ect is a slight increase in the mean price (up to 1.048), compared to the
benchmark, rather than a decline resulting from an undisciplined optimal policy.
With a limited decrease in volatility (the coeï¬ƒcient of variation of domestic price is 0.15), welfare
eï¬€ects on consumers are dominated by the increased mean price, so that consumers lose as a result of
this policy, as in the case of only a storage policy. Export disciplines thus entail signiï¬?cant transfers
from consumers to producers. Overall, the gains for the whole economy are more than halved
compared to the undisciplined optimal policy. In sum, export taxes appear to be key components of
price-stabilizing policies that beneï¬?t consumers: the stakes of possible multilateral disciplines on
export restrictions would be high for a number of poor countries.
6.6 Optimal policy when the country is not self-suï¬ƒcient on average
So far, we have considered a situation where the country trades with the rest of the world because
yield shocks are distinct in the two regions. Now we consider in addition that the mean price in
autarky diï¬€ers from the mean world price, so that trade will take place even in the absence of
stochastic shocks. For convenience, the change is implemented through Âµ, which parameterizes the
world yield distribution (see equation (14)). Its default value is 1 and we consider two alternatives, 0.9
26
and 1.1â€”a 10% increase or decrease in the world mean productionâ€”where the country is respectively
a structural exporter or a structural importer.
For Âµ = 0.9 the deterministic steady-state price in the world market is equal to 1.3, whereas in
autarky the steady-state price is still 1 in the domestic market. Thus, when open to trade, the
country exports and has a mean price higher than its mean autarkic price (see Table 4). Storage
behavior is consistent with what was previously described for the trade/storage relationship: the
mean stock level is very low when the country is an importer, and is much higher if it is an exporter.
At the aggregate level, the parameter Âµ does not seem to aï¬€ect the optimal policy pattern, which
consistently increases stock levels and produces very limited changes in average trade. As mentioned
above, the optimal policyâ€™s trade eï¬€ect consists of distorting the distribution of trade, not in isolating
the economy from the world market, which would be too costly. In the benchmark case, the
distribution distortion is achieved mainly through the use of import subsidies and export taxes.
With Âµ = 0.9, export taxes and subsidies become the most important trade policy instruments. For
prices higher than the mean price, export restrictions are used to avoid importing high prices from
the world market. For prices below the mean price, exports are subsidized so as to move domestic
price closer to its mean. For Âµ = 1.1, the behavior is similar but applied to imports. Imports are
subsidized when the domestic price is above its mean and taxed when it is below it. So the optimal
behavior is to exploit the world market so as to stabilize the domestic price around its mean.
Table 4. Sensitivity to the assumption of self-suï¬ƒciency
Structural exporter (Âµ =0.9) Self-suï¬ƒcient (Âµ =1)a Structural importer (Âµ =1.1)
No Optimal No Optimal No Optimal
Policy Policy Policy Policy Policy Policy
Descriptive statistics
Mean price 1.230 1.208 1.045 1.034 0.915 0.913
CV of price 0.216 0.134 0.173 0.121 0.156 0.109
Mean stocks 0.090 0.120 0.033 0.047 0.013 0.020
Mean imports 0.003 0.004 0.018 0.018 0.050 0.048
Mean exports 0.074 0.074 0.028 0.028 0.008 0.008
Welfare
Consumers gains 1.34 1.05 0.25
Producers gains âˆ’1.38 âˆ’0.92 âˆ’0.10
Government 0.27 âˆ’0.03 âˆ’0.09
Total gains 0.23 0.10 0.07
Notes: Columns No Policy and Optimal Policy display respectively results for the model without public intervention
and for the optimal policy using the two instruments.
a
Benchmark
The welfare analysis shows that policy intervention is more beneï¬?cial from a social point of view for
structural exporters, while the gains to structural importers are sharply reduced in comparison to
the benchmark. If anything, this only adds to the policy concerns surrounding the use of export
27
restrictions, since structural exporters seem to have potentially serious motivations to persist in
using them.
7 Conclusion
To our knowledge, this paper is the ï¬?rst attempt to design optimal dynamic food price stabilization
policies in an open economy setting. The model can only be solved numerically, and tractability
requires simple speciï¬?cation. Our analysis focuses on the optimal use of trade and storage policies
on the food market of a small developing economy, where public intervention is justiï¬?ed by the lack
of insurance against price volatility for risk-averse consumers. For the sake of tractability, the model
overlooks supply reaction and producersâ€™ risk aversion. The framework developed here, however,
could be applied to other cases, in terms of both its parameterization and speciï¬?cation.
Our results show that, for the case of a normally self-suï¬ƒcient country, an optimal trade policy
in the presence of risk aversion includes subsidizing imports and taxing exports in periods of high
domestic prices. This policy truncates the upper half of the distribution and is not ï¬?scally costly
since the proceeds from export taxation cover the ï¬?scal cost of import subsidies. Import subsidies
alleviate the traditional limit of food storage: its non-negativity. When stocks are zero, subsidizing
imports prevents price spikes.
When stabilization is pursued through storage subsidies only, it does not improve consumersâ€™ welfare.
Additional storage increases low prices through additional demand for stockpiling, but it is not
eï¬€ective at preventing price spikes. In a small open economy, price spikes occur often when the world
price is high, in which case any additional stock is sold on the world market. While domestic prices
are stabilized to some extent, the potential beneï¬?ts for consumers are wiped out by the increase
in the mean price. Such a policy improves the countryâ€™s trade balance by giving it more resources
to export when the world price is relatively high, but it does not beneï¬?t consumers. Since storage
policies are generally seen as a way to help consumers, these results sound a warning that storage
policies designed without any ï¬‚anking trade policy might be inconsistent: the limited insulation
provided by trade costsâ€”especially when they are relatively smallâ€”does not allow any independent
food price policy to be pursued. In contrast, a well-designed combination of trade and storage taxes
and subsidies can be a cost-eï¬ƒcient price-stabilizing policy.
These policies have an important common limitation. They produce distributive welfare eï¬€ects
that are much larger than total gains. Reducing consumersâ€™ risk bearing by manipulating prices
in an open economy may thus face strong opposition. This result emphasizes the drawbacks of
stabilization policies compared to policies targeting speciï¬?cally poor households.
Our results show that an optimal combination of trade and storage policies trims both the lower
and upper parts of the domestic prices distribution. The optimal policy identiï¬?ed here should thus
remain welfare-increasing even when producers are risk averse, since they should value the trimming
28
of low prices. The same goes for the issue of supply elasticity: since expected prices are not strongly
modiï¬?ed, supply reaction should remain limited.
We ï¬?nd that export taxes are important for designing price-stabilizing policies that beneï¬?t consumers.
This result should not be understood as a plea for export restrictions, the destabilizing eï¬€ects of
which are well documented and are a legitimate and serious source of concern at the multilateral
level (Mitra and Josling, 2009). Nonetheless, this analysis may provide a better understanding of
why export restrictions are such a frequent occurrence. It might also help to gauge the stakes of
potential multilateral disciplines. For the poorest countries, in particular, banning export restrictions
altogether may be politically diï¬ƒcult, at least if no substantial compensating measures are oï¬€ered.
The collective action problem created by export restrictions certainly deserves closer scrutiny. In
any case these constraints, disturbing as they are, are better acknowledged than ignored.
A First-order conditions of the optimal policy problem
To solve the optimal policy problem presented in Section 5, we need to reformulate the complemen-
tarity equations (6)â€“(8) since they cannot enter directly as constraints in a maximization problem.
We restate these equations as a combination of inequalities and equations. For equation (6), we
introduce a positive slack variable, Ï†, with its associated complementarity slackness conditions
Ï†t = Pt + k âˆ’ Î² Et (Pt+1 ) âˆ’ Î¶t , (29)
St Ï†t = 0. (30)
To limit the number of equations and variables, the two trade policy instruments are merged into
X M
one with Î½t = Î½t âˆ’ Î½t , which is equivalent since each instrument is redundant when the other is
active. The equations governing trade, (7)â€“(8), can be restated as
Xt [Ptt âˆ’ (Ptw âˆ’ Ï„ )] = 0, (31)
Mt [(Ptw + Ï„ ) âˆ’ Ptt ] = 0, (32)
Ptt = Pt + Î½t , (33)
where Xt â‰¥ 0, Mt â‰¥ 0, and Ptw âˆ’ Ï„ â‰¤ Ptt â‰¤ Ptw + Ï„ .
Discretionary equilibria are known to be diï¬ƒcult to characterize since they involve functional
equations, often called generalized Euler equations (Klein et al., 2008, Ambler and Pelgrin, 2010).
A model setting with occasionally binding constraints is even more complex because ï¬?rst-order
conditions cannot be derived (see appendix C). For optimal policies with both instruments and
with storage subsidy alone, this diï¬ƒculty can be sidestepped by noting that the policy maker acts
identically under commitment and under discretion. This peculiarity makes it possible to use the
solution to the problem under commitment, more easily computed, as a solution to the problem
under discretion.
29
The equivalence between commitment and discretion for these two policies arises because the storage
subsidy allows full control of the only intertemporal trade-oï¬€, namely the storage decision. Formally,
following Marcet and Marimon (2011), the optimal policy problem under commitment can be
expressed as a saddle-point functional equation problem:
J (At , Ptw , Î»tâˆ’1 ) = min max v (Pt , Y ) + w [Pt At âˆ’ (Pt + k) St + Î½t (Xt âˆ’ Mt )]
Î¦t â„¦t
+ Ï‡t [At + Mt âˆ’ D (Pt ) âˆ’ St âˆ’ Xt ]
+ Î»t (Ï†t + Î¶t âˆ’ Pt âˆ’ k)
+ Î»tâˆ’1 Pt
S
+ Î´t St Ï†t (34)
M
+ Î´t Mt (Ptw + Ï„ âˆ’ Ptt )
X
+ Î´t Xt (Ptt âˆ’ Ptw + Ï„ )
+ Îºt (Ptt âˆ’ Pt âˆ’ Î½t )
H
+ Î² Et J St + t+1 , f Ptw , w
t+1 , Î»t ,
where â„¦t = {St â‰¥ 0, Pt , Mt â‰¥ 0, Xt â‰¥ 0, Ptw âˆ’ Ï„ â‰¤ Ptt â‰¤ Ptw + Ï„, Î¶t , Ï†t â‰¥ 0, Î½t } and Î¦t = {Ï‡t , Î»t ,
S M X
Î´t , Î´t , Î´t , Îºt }. From the ï¬?rst-order conditions and the envelope theorem, and following some
manipulation, the recursive equilibrium under an optimal stabilization policy can be characterized
by the following system of complementarity conditions:13
S
St : St â‰¥ 0 âŠ¥ âˆ’w [Pt + k âˆ’ Î² Et (Pt+1 )] âˆ’ Ï‡t + Î² Et (Ï‡t+1 ) + Î´t Ï†t â‰¤ 0, (35)
Pt : vP (t) + wD (Pt ) âˆ’ Ï‡t D (Pt ) âˆ’ Î»t + Î»tâˆ’1 = 0, (36)
Mt : Mt â‰¥ 0 âŠ¥ M
âˆ’wÎ½t + Ï‡t + Î´t (Ptw + Ï„ âˆ’ Ptt ) â‰¤ 0, (37)
X
Xt : Xt â‰¥ 0 âŠ¥ wÎ½t âˆ’ Ï‡t + Î´t âˆ’ Ptw + Ï„ ) â‰¤ 0,
(Ptt (38)
Ptt : Ptw âˆ’ Ï„ â‰¤ Ptt â‰¤ Ptw + Ï„ âŠ¥ M X
âˆ’Î´t Mt + Î´t Xt (39)
Î¶t : Î»t = 0, (40)
S
Ï†t : Ï†t â‰¥ 0 âŠ¥ Î»t + Î´ t St â‰¤ 0, (41)
Î½t : Ptt âˆ’ Pt âˆ’ Î½t = 0, (42)
Ï‡t : At + Mt = D (Pt ) + St + Xt , (43)
Î»t : Ï†t + Î¶t âˆ’ Pt âˆ’ k + Î² Et (Pt+1 ) = 0, (44)
S
Î´t : St Ï†t = 0, (45)
M
Î´t : Mt (Ptw + Ï„ âˆ’ Ptt ) = 0, (46)
X
Î´t : Xt (Ptt âˆ’ Ptw + Ï„ ) = 0. (47)
13
Here the â€œperpâ€? notation (âŠ¥) is extended to situations with two complementarity constraints. The expression
a â‰¤ X â‰¤ b âŠ¥ F (X) is a compact formulation for X = a â‡’ F (X) â‰¥ 0, X âˆˆ]a, b[â‡’ F (X) = 0, X = b â‡’ F (X) â‰¤ 0.
30
The transition from one period to the next is still governed by equations (10)â€“(11). Combining
equations (37)â€“(38) and (46)â€“(47) gives Î½t = Ï‡t /w for positive trade. It means that the optimal
trade policy is to base trade decisions on the total marginal value of the commodity (the sum of
private marginal value, domestic price, and social marginal value, Ï‡t /w) instead of the price.
Usually, the time-inconsistency of policies under commitment shows up in the lagged Lagrange
multipliers. Here, the multiplier Î» is always null (equation (40)), which demonstrates that there is
no commitment problem. The equilibrium, therefore, is recursive in its natural state variables and is
time-consistent. It is identical under commitment and under discretion.
B First-order conditions of the optimal storage subsidy
The ï¬?rst-order conditions of the optimal storage subsidy are obtained similarly, except that variables
Î½ and P t have to be dropped from (34). From the ï¬?rst-order conditions with both instruments,
equations (35), (40), (41), and (43)â€“(45) remain valid. The following ï¬?rst-order conditions must be
considered in addition:
Pt : Ptw âˆ’ Ï„ â‰¤ Pt â‰¤ Ptw + Ï„ âŠ¥ M X
vP (t) + w (At âˆ’ St ) âˆ’ Ï‡t D (Pt ) âˆ’ Î´t Mt + Î´t Xt , (48)
Mt : M t â‰¥ 0 âŠ¥ M
Ï‡t + Î´t (Ptw + Ï„ âˆ’ Pt ) â‰¤ 0, (49)
X
Xt : Xt â‰¥ 0 âŠ¥ âˆ’Ï‡t + Î´t (Pt âˆ’ Ptw + Ï„ ) â‰¤ 0, (50)
M
Î´t : Mt (Ptw + Ï„ âˆ’ Pt ) = 0, (51)
X
Î´t : Xt (Pt âˆ’ Ptw + Ï„ ) = 0. (52)
C Characterization of the optimal trade policy
For the optimal trade policy, the solutions under commitment and under discretion being diï¬€erent, we
have to characterize the discretionary solution directly. Since we focus on a Markovian equilibrium,
price can be characterized by a function of the state variables: Pt = Ïˆ (A, Ptw ). Using this function
Ïˆ to characterize price expectations, the value function is deï¬?ned by the following Bellman equation
J (At , Ptw ) = min max v (Pt , Y ) + w [Pt At âˆ’ (Pt + k) St + Î½t (Xt âˆ’ Mt )]
Î¦t â„¦t
+ Ï‡t [At + Mt âˆ’ D (Pt ) âˆ’ St âˆ’ Xt ]
H
+ Î»t Î² Et Ïˆ St + t+1 , f Ptw , w
t+1 + Ï†t âˆ’ Pt âˆ’ k
S
+ Î´ t S t Ï†t
(53)
M
+ Î´t Mt (Ptw + Ï„ âˆ’ Ptt )
X
+ Î´t Xt (Ptt âˆ’ Ptw + Ï„ )
+ Îºt (Ptt âˆ’ Pt âˆ’ Î½t )
H
+ Î² Et J St + t+1 , f Ptw , w
t+1 .
31
In this setting with occasionally binding constraints, we cannot assume Ïˆ to be diï¬€erentiable
everywhere. Thus, in theory, we cannot derive the ï¬?rst-order conditions of this problem since
they would imply derivatives of Ïˆ. But since, in practice, Ïˆ is approximated by a spline, which is
diï¬€erentiable everywhere, we can solve the dynamic programming problem numerically by solving
the ï¬?rst-order conditions of the approximated problem:
w S
St : St â‰¥ 0 âŠ¥ âˆ’w (Pt + k) âˆ’ Ï‡t + Î²Î»t Et ÏˆA At+1 , Pt+1 + Î´t Ï†t + Î² Et (wPt+1 + Ï‡t+1 ) â‰¤ 0, (54)
Pt : vP (t) + wD (Pt ) âˆ’ Ï‡t D (Pt ) âˆ’ Î»t = 0, (55)
Mt : Mt â‰¥ 0 M
âŠ¥ âˆ’wÎ½t + Ï‡t + Î´t (Ptw + Ï„ âˆ’ Ptt ) â‰¤ 0, (56)
Xt : Xt â‰¥ 0 X
âŠ¥ wÎ½t âˆ’ Ï‡t + Î´t (Ptt âˆ’ Ptw + Ï„ ) â‰¤ 0, (57)
Ptt : Ptw âˆ’Ï„ M X
â‰¤ Ptt â‰¤ Ptw + Ï„ âŠ¥ âˆ’Î´t Mt + Î´t Xt , (58)
S
Ï†t : Ï†t â‰¥ 0 âŠ¥ Î»t + Î´t St â‰¤ 0, (59)
Ï‡t : At + Mt = D (Pt ) + St + Xt , (60)
Î»t : Î² Et (Pt+1 ) + Ï†t âˆ’ Pt âˆ’ k = 0, (61)
S
Î´t : St Ï†t = 0, (62)
M
Î´t : Mt (Ptw + Ï„ âˆ’ Ptt ) = 0, (63)
X
Î´t : Xt (Ptt âˆ’ Ptw + Ï„ ) = 0, (64)
Î½t : Ptt âˆ’ Pt âˆ’ Î½t = 0. (65)
D Numerical algorithm
The numerical algorithm used here is inspired by Fackler (2005) and Miranda and Glauber (1995). It
is a projection method with a collocation approach. Since several models are solved, we present here
a general method that can be applied to all of them. Following Fackler (2005), rational expectations
problems can be expressed using three groups of equations. State variables s are updated through a
transition equation:
Ë™ Ë™
s = g(s, x, e), (66)
where x are response variables, e are stochastic shocks, and next-period variables are indicated with
a dot on top of the character. Response variables are deï¬?ned by solving a system of complementarity
equilibrium equations:
x(s) â‰¤ x â‰¤ x(s) âŠ¥ f (s, x, z). (67)
Response variables can have lower and upper bounds, x and x, which can themselves be function
of state variables.14 For generality, equilibrium equations have been expressed as complementarity
problems. In cases where response variables have no lower and upper bounds, equation (67) simpliï¬?es
14
It is the case in equation (39) where lower and upper bounds on Ptt are functions of Ptw .
32
to a traditional equation:
f (s, x, z) = 0. (68)
z is a variable representing the expectations about next period and is deï¬?ned by
Ë™ Ë™ Ë™
z = Ee [h(s, x, e, s, x)] .
Ë™ (69)
One way to solve this problem is to ï¬?nd a function that approximates well the behavior of response
variables. We consider a spline approximation of response variables,
x â‰ˆ Î¦ (s, Î¸) , (70)
where Î¸ are the parameters deï¬?ning the spline approximation. To calculate this spline, we discretize
the state space (using, for the storage-trade model, 41 points for availability and 23 for world price),
and the spline has to hold exactly for all points of the grid.
The expectations operator in equation (69) is approximated through a Gaussian quadrature (with
5 points on each dimension), which deï¬?nes a set of pairs {el , wl } in which el represents a possible
realization of shocks and wl the associated probability. Using this discretization, and equations (66)
and (69)â€“(70), we can express the equilibrium equation (67) as
x(s) â‰¤ x â‰¤ x(s) âŠ¥ f s, x, wl h (s, x, el , g (s, x, el ) , Î¦ (g (s, x, el ) , Î¸)) . (71)
l
For a given spline approximation, Î¸, and a given s, equation (71) is a function of x and can be solved
using a mixed complementarity solver.
Once all the above elements are deï¬?ned, we can proceed to the algorithm, which runs as follows:
1. Initialize the spline approximation, Î¸0 , based on a ï¬?rst-guess, x0 .
2. For each point of the grid of state variables, si , solve for xi equation (71) using the solver PATH
(Dirkse and Ferris, 1995):
x(si ) â‰¤ xi â‰¤ x(si ) âŠ¥ f si , xi , wl h (si , xi , el , g (si , xi , el ) , Î¦ (g (si , xi , el ) , Î¸n )) .
l
(72)
3. Update the spline approximation using the new values of response variables, x = Î¦ (s, Î¸n+1 ).
4. If Î¸n+1 âˆ’ Î¸n 2 â‰¥ 10âˆ’8 then increment n to n + 1 and go to step 2.
Once the rational expectations equilibrium is identiï¬?ed, the spline approximation of the decision
rules is used to simulate the model.
33
The storage model representing the world market is solved using this algorithm and a 50-node spline
approximation. This spline approximation deï¬?nes the continuous Markov chain (11) representing the
world price dynamics. It is subsequently used in the storage-trade model as a transition equation,
which means that it is a part of equation (66).
For the optimal trade policy, the method requires a few modiï¬?cations, since equation (54) includes a
derivative of Ïˆ, Ïˆ being an approximation of the behavior of price with respect to state variables. The
calculation of this approximation is actually done in the above method in step 3. Since derivatives of
Ïˆ appears only in expectations terms, to apply the method we have to make sure that the deï¬?nition
of the function h includes these terms, which leads to a new equation (69):
Ë™ Ë™ Ë™
z = Ee [h(s, x, e, s, x, Î¦s (s, Î¸))] ,
Ë™ (73)
where Î¦s (s, Î¸) is the derivative of Î¦ with respect to state variables.
This is only a sketch of the solution method. In fact, several methods are used in this paper to
solve the models, depending on which one is the most eï¬ƒcient. For example, instead of using the
simple updating rule in step 3, a Newton, or inexact Newton, updating is used when feasible. For
more precisions, see the RECS solver (https://github.com/christophe-gouel/recs), with which
the models are solved.
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