Original Sin and Redemption: Rebalancing the Currency Structure

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Original Sin and Redemption:
Rebalancing the Currency Structure of
Uruguayan Public Debt
Umberto Della Mea
[email protected]
Antonio Juambeltz
[email protected]
1
Debt Management Unit
Ministry of Economy and Finance
Colonia 1089, 3rd Floor, 11100 Montevideo, URUGUAY
Abstract
This document discusses the optimal structure of the Uruguayan public debt given a set of current
parameters of cost and risk. In a traditional portfolio setting, the model decomposes and develops
alternative ways to reestimate the covariance matrix in order to internalize a changing environment.
Policy and institutional innovations, as well as the countercyclical properties of nominal debt, are thus
reflected in the parameters of the model on a forward-looking basis.
32
Resumen
Este documento discute la estructura óptima del endeudamiento público uruguayo en función de un
conjunto de parámetros de costo y riesgo. En el marco de un modelo tradicional de portafolios, esta
aproximación innova formas alternativas de reestimar la matriz de covarianzas, de modo de internalizar cambios e innovaciones institucionales y de política. Así, las propiedades contracíclicas de la
deuda nominal se reflejan en los parámetros del modelo sobre una base forward-looking.
Clasificación JEL: H63
Palabras clave: Gestión de Deuda Pública, Pecado Original, Deuda Indexada
1. The opinions here expressed are those of the authors and do not necessarily represent those of the Debt Management Unit.
Revista de Ciencias Empresariales y Economía
Revista de Ciencias Empresariales y Economía
JEL Classification: H63
Keywords: Public Debt Management, Original Sin, Indexed Debt
33
ORIGINAL SIN AND REDEMPTION: REBALANCING THE CURRENCY STRUCTURE OF URUGUAYAN PUBLIC DEBT
UMBERTO DELLA MEA / ANTONIO JUANBELTZ
1. Introduction
There seems to be a wide consensus in the literature about the main drivers of sovereign debt creditworthiness. These drivers, which often tend to feedback, are in general related to the overall quality
of the country, characterized by its degree of economic, social and institutional development. However, there is also a particularly important role played by the public debt management. In this regard,
aspects like the maturity profile, the currency composition or the intrinsic quality of the asset-liability
management are key factors to determine the current levels of sovereign risk.
Fig. 2: Changes in
Non Financial Public Sector Debt
70%
In Uruguay, the currency composition of the public debt has been the main explanation to the strong
variations observed in the overall debt service and debt stock, relative to the domestic level of activity. A highly dollarized economy, with a high share of foreign-currency denominated public debt,
is sensitive to real exchange rate fluctuations and -in particular- to the level of the exchange rate as
compared to the level of domestic prices. Any appreciation in the currency of denomination of the
public debt, relative to domestic inflation, will increase the debt burden relative to GDP, deteriorating
the solvency indicators and producing an increase in the risk premium that might in turn trigger a
debt crisis.
Since the early 70s, when Uruguay started to open the capital account, the currency choice in favor
of US Dollars covered a wide range of domestic financial instruments in the banking sector and the
capital markets, reaching even the means of payment of the economy. Della Mea (2007) pointed out
some of the reasons that explained this trend over the last decades: the status of legal tender granted
to the foreign currency, setting an equal treatment relative to the domestic currency; the small size
and degree of openness of the economy, where most durable goods were imported and their prices
were quoted in foreign currency to provide a natural hedge in absence of a developed financial market; a long history of medium to high inflation and lack of institutional commitment to price stability;
and finally, the absence of alternatives providing inflation protection, like CPI-indexed instruments.
The public debt was no exception to this process. Still at the beginning of this decade, almost 100%
of the Non-Financial Public Sector debt was foreign currency denominated. This factor was determinant to fuel the financial crisis of 2002, whose aftermath -after a strong real depreciation of the
domestic currency and a deep recession- was a soaring Debt/GDP ratio which more than doubled in
one year, surpassing 100%. A similar consequence is observed over the debt service 2. Fig. 1 illustrates
the pattern of the debt stock and the debt service, relative to GDP, in the current decade:
Fig. 1: Debt Stock and Debt Service
110%
9%
100%
In this respect, the objective of this paper is to set up a model on the optimal currency composition
of the Uruguayan public debt. This model brings into consideration not only the expected costs and
risks associated to a set of possible units of denomination, but also the way these factors are correlated as well as their macroeconomic properties.
8%
90%
34
80%
5%
70%
4%
Debt/GDP
Interest/GDP
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7%
6%
60%
3%
50%
2%
40%
1%
0%
30%
2000
2001
Interest/GDP
2002
2003
2004
2005
The crucial question is how debt managers should behave in order to avoid this kind of exposure,
in particular when the country is likely to be subject to periodical real exchange rate shocks. Foreign
currency debt might be more risky, but at the same time it might also be cheaper on average, especially when domestic capital markets are not developed or when there is uncertainty about future
development of inflation and other key domestic variables. The past choice in favor of foreign currency may have had sensible foundations. The question is how to assess this choice at present and
balance foreign currency liabilities with other available domestic alternatives, in particular nominal
and CPI-indexed debt.
2006
Debt/GDP (RHS)
2. Foreign-currency denominated debt and debt service are valued at end-of-period exchange rates and compared to nominal GDP, valued at
market prices.
The rest of the document is organized as follows: Section 2 surveys some recent economic literature
about optimal composition of public debt. A model of portfolio selection based on the traditional
mean-variance approach is presented in Section 3. This model, in spite of working on a somewhat
standard setting, introduces significant changes in the way of estimation of the covariance matrix
of expected returns, in order to produce more forward-looking parameters. Section 4 discusses the
results obtained in different scenarios and finally, Section 5 concludes.
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(Gross Non Financial Public Sector Debt)
10%
In turn, Fig. 2 discloses the factors of variation of the Non-Financial Public Sector debt into a series of
basic components. The major source of variation is the depreciation of the nominal effective exchange rate, which more than compensated a very low pass-through to the GDP deflator in a context of
strong capital outflows. Moreover, real GDP behavior tends to be negatively correlated with the real
exchange rate, feeding back the process and worsening the overall scenario. Strange as it may seem,
fiscal factors did not seem to contribute in a significant way to explain changes in this ratio over the
period under consideration.
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ORIGINAL SIN AND REDEMPTION: REBALANCING THE CURRENCY STRUCTURE OF URUGUAYAN PUBLIC DEBT
Following the International Monetary Fund and The World Bank (2001), the main goal of public debt
management is to ensure that the government’s financing needs and its payment obligations are
met at the lowest possible cost over the medium to long run, consistently with a prudent degree of
risk. To achieve this target, debt managers have increasingly begun to rely on Asset and Liability Management (ALM) approaches, which seeks to minimize adverse changes in future net cash flows of
the government. Originally, ALM approaches were implemented by private companies, in particular
financial intermediaries, to hedge their risks by matching the financial characteristic of their liabilities
and assets. According to the definition of IMF and WB, the objective of an ALM model is “selecting,
to the extent possible, liabilities with matching characteristics in order to help smooth the budgetary
impact of shocks on debt servicing costs. If full matching is not possible, or is too costly, the analysis
of cash flows also provides a basis for measuring the risks of the liability portfolio and measuring
cost/risk tradeoffs”.
Thus, an ALM approach focuses on the net cash flows generated by assets and liabilities, instead of
the asset stocks themselves. Basically, an ALM framework involves the following steps:
a. Identifying an objective function;
b. Identifying and measuring risk exposure in relation to the objective function (or measuring the
sensitivity of performance to unexpected changes in prices);
c. Deciding on an acceptable degree of risk; and
d. Choosing and executing hedging transactions.
Focusing on foreign currencies, Masuoka (1990) argued that countries could improve asset and
liability risk management by holding an adequate level of foreign exchange reserves and borrowing
in appropriate currency denominations. As a corollary, diversification of debt portfolio by currencies
–including local ones– is one of the instruments that allow debt managers to accommodate their
hedging strategies.
Revista de Ciencias Empresariales y Economía
A particular interest has been granted to the corner solution where a country is virtually incapable of
issuing domestic currency denominated debt. Eichengreen and Hausmann (1999) coined the expression original sin, referring to a situation in which the domestic currency cannot be used to borrow
abroad or to borrow long term, even domestically, generating risks of currency mismatches. However,
this problem did not arise because of lack of prudence in emerging market debt managers, but rather
because they were unable to do so, since domestic or foreign investors were unwilling to take the
other side of a hedge contract, as Goldstein and Turner (2004) remark. Eichengreen and Hausmann
suggested that the original sin hypothesis could be present as well in emerging countries without
recent history of high inflation, reflecting problems of incomplete information and sovereign risk.
Investors might have been reluctant to invest in domestic currency denominated assets -including
public bonds- because they were not able to know to what extent borrowers would manipulate local
currency in order to minimize real debt servicing.
36
Borensztein, Levy Yeyati and Panizza (2007) summarize a number of findings on the original sin hypothesis. In particular, studies by Eichengreen, Hausmann and Panizza note that approximately 85%,
out of the USD1.3 trillion placed in international markets during 1999-2001, were issued by countries in different currencies than their own. Since international issues require a reasonably good track
record in terms of creditworthiness, it stems from here that good policies are a necessary though not
sufficient condition to overcome the original sin.
On the domestic side, despite of the difficulties to obtain extensive data on the currency and maturity
composition of domestic debts, Hausmann and Panizza proposed a Main Index of domestic original
sin as the ratio of non domestic currency denominated, fixed rate debt, to total domestic debt. The
risks of a high foreign currency ratio are straightforward in two aspects: the liquidity risk, defined
as the availability of foreign exchange and the balance sheet risk, defined as sensitivity of the net
wealth or net income to changes in the exchange rate. These two effects are normally interlaced,
since countries running a weak balance sheet often end up in capital flights and liquidity problems.
Despite there is no ideal foreign currency share of debt, the Bank of International Settlements (2007)
suggests that a good rule of thumb would be not to exceed the proportion of tradable goods to GDP
(e.g., approximated by the share of exports). In the spirit of ALM modeling, foreign currency debt is
considered as more sustainable to the extent that it matches the production of tradable goods.
Uruguay, as well as many other emerging markets, has started a clear-cut policy of dedollarization of
the public debt portfolio and development of domestic markets, in a context of reduction of financial
fragilities. Onshorization, following the term adopted by Borensztein, Levy Yeyati and Panizza (2007),
implies the substitution of external by domestic debt. In order to make that objective feasible, an
active domestic market should be in place. Jeanneau and Tovar (2006) also believe that there are
good reasons to believe that there is room for the expansion on local bond markets in Latin America.
These reasons are basically: a secular process of integration between mature and emerging markets;
an increasing demand for domestic debt from international investors as a consequence of the improvement in macroeconomic fundamentals obtained by many emerging markets; and quite low
correlations of Latin America local currency bonds with other more developed countries, allowing to
improve the risk/return trade-off embedded in the efficient frontier.
After some years, the first evaluations of this strategy began to appear. Acevedo, Alberola and Broto
(2007), in a study on foreign debt reduction over a group of six emerging markets –including Uruguay- noticed a clear progress in reduction in the overall vulnerability of public debt. After decomposing the factors of reduction into an exchange rate effect and a volume effect, they concluded
that the second was the main driver of this behavior, denoting a proactive debt management in this
direction. Paradoxically, this proactivity limited the effective reduction in the Debt/GDP ratio of these
countries by not having allowed them to take full advantage of the real exchange appreciation enjoyed by their economies. Once again, the trade-off between the expected cost and risk arises.
Nominal, CPI-Indexed and Foreign-Currency denominated debt
Certainly, a foreign currency denomination is nothing but a non- sophisticated form of indexation.
Another common way is to issue directly CPI-indexed debt, so all of these alternatives should be confronted to the more traditional issuance of nominal debt.
The question between nominal and indexed debt was addressed -among others- by Bohn (1988),
Calvo and Guidotti (1990) and Barro (1997). They all pointed out that if there is uncertainty about
levels of public outlays, the tax base and asset prices, this definition matters. Bohn justifies the
existence of some nominal public debt because it naturally hedges against the budgetary effects of
economic fluctuations, especially when there are no lump-sum taxes. Through a model with two different stochastic shocks -aggregate demand and money demand-, inflation is likely to be higher and
the real value of nominal debt is likely to be lower in the negative state of nature, reducing required
tax rates and offsetting the consequences of the shocks. By issuing nominal debt, the government
can generate a smoother path of taxes and therefore, avoid a higher welfare cost.
Calvo and Guidotti proposed a model where the government seeks to minimize a function of social
loss of welfare subject to a budget constraint of exogenous government spending. The rationale of
this approach relies on the fact that the covariance between inflation and government expenditure
is positive. If the government can commit to a predetermined level of inflation, indexation is optimal
and maturity is not crucial. If there is nominal debt and the government cannot fully commit to a
level of inflation, then there is a temptation to inflate prices and decrease the real value of nominal
debt. Barro also analyzed public debt management under the assumption that the government seeks
to smooth out taxes when confronted by stochastic levels of public outlays. Once again, he comes to
the conclusion that there is little space for nominal government bonds, in favor of indexed bonds.
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2. Issues on the currency denomination of public debt
UMBERTO DELLA MEA / ANTONIO JUANBELTZ
37
ORIGINAL SIN AND REDEMPTION: REBALANCING THE CURRENCY STRUCTURE OF URUGUAYAN PUBLIC DEBT
Calvo, Izquierdo and Talvi (2002), in a study of the influence of sudden stops in emerging markets,
also pointed out that debt should be issued under terms that eliminate incentives to inflate it away
through money creation, such a CPI-indexed debt. Also, this type of issuance should be reserved for
long maturities, in order to mitigate the effects of volatility in real exchange rates but also to make
explicit ex-ante the redistributive effects of a sudden stop. Emerging markets often have problems for
issuing long term debt, due to the lack of monetary and fiscal policy credibility. However, long-term
CPI-indexed bonds are necessary to develop the domestic capital market, in particular through the
activity of private pension funds. In line with the general ALM framework, these institutions tend to
create a natural base of investors seeking the protection against changes in purchasing power that
CPI indexation provides, as Borensztein, Chamon, Jeanne, Mauro and Zettelmeyer (2004) remark.
This debate started to be increasingly relevant in Uruguay only by the end of the last decade. Licandro and Masoller (2000) addressed the issue of a multicurrency portfolio where public liabilities can
be either nominal, CPI-indexed or US Dollar denominated, in a tax-smoothing framework. In this
setting, they showed the importance of issuing CPI- indexed debt as a way to smooth out the tax
burden over time by solving an implicit trade-off between financial and budgetary risks. The government can predict with more precision the real value of fiscal expenditures while at the same time,
CPI-indexed bonds would contribute to complete the financial markets. In their study on Uruguay,
they also define that there seems to be no room for nominal domestic currency debt due to a higher
cost of issuance, mainly because the private sector is not insured against unpredicted high inflation
levels. The moral hazard generated by the temptation to default on the real value of debt through
higher inflation implies a significant risk premium in the interest rate parity. Undesirable stochastic
properties, arising from absence of significant correlation with other variables affecting the budget,
are also qualified as an empirical drawback for this type of debt.
3. A forward-looking portfolio model
Our problem consists in determining the optimal currency composition of the public debt portfolio,
minimizing its overall expected cost subject to a predetermined level of risk. This design should have
to bring into consideration the expected costs, risks and correlations within the set of different available instruments. However, from a macroeconomic standpoint, it should also have to integrate the
countercyclical properties of these currencies or units of denomination in terms of tax smoothing, as
well as any policy innovation that might affect the future stochastic properties of these variables.
UMBERTO DELLA MEA / ANTONIO JUANBELTZ
Expected Costs
The basic problem is to select the optimal shares of the available debt instruments that minimize the
expected cost of a portfolio, subject to certain restrictions. This cost is represented by:
(3.1) Minω ω' r
where ω is a (Nx1) vector of percentages allocated to the different debt instruments, each one of
them being denominated in a different currency or unit of account, while r is the (Nx1) vector of
expected costs. These costs are measured in units of GDP, to capture the full real effort that the economy has to make in order to meet the obligations on its debt. It also allows the model to take into
account the stochastic properties of the correlation matrix between GDP and other variables like the
real exchange rate or the rate of inflation.
In principle, the ω vector of shares satisfies the restriction:
(3.2) ω’I= 1
where I is a (Nx1) vector of ones 3.
Consider for example the case of a (4x1) portfolio, where the available debt instruments are: a nominal fixed rate domestic currency bond (rNOM), a domestic CPI-indexed fixed rate bond (rCPI) and
any two foreign-currency denominated, fixed rate bonds (rFC , i=1, 2). The expected costs for these
instruments are approximated by the following expressions:
(3.3)
E (rNOM ) =
(3.4)
E (rCPI ) =
(3.5)
E (rFC1 ) =
1+ i
− 1 ≅ i − E [γ ]≅ i − E [π γ ]− E [γ ]
1 + E [γ ]
(1 + r )(1 + E[π]) − 1 ≅ r + E[π]− E[γ ]≅ r + (E[π]− E [π ])− E [γ ]
γ
1 + E [γ ]
(1 + iFC )(1 + E[δ1 ]) − 1 = (1 + iFC )(1 + E[q&1 ])(1 + E[π]) − 1 ≅
(1 + E[π1* ])(1 + E[γ ])
1 + E [γ ]
iFC + E [q&1 ]− E [π1* ]+ (E [π]− E [π γ ])− E [γ ]
(1 + iFC )(1 + E[δ2 ]) − 1 = (1 + iFC )(1 + E[q&2 ])(1 + E[π]) − 1 ≅
(1 + E[π*2 ])(1 + E[γ ])
1 + E [γ ]
iFC + E [q&2 ]− E [π*2 ]+ (E [π]− E [π γ ])− E [γ ]
1
1
38
In this regard, we have chosen to approach the portfolio selection problem under the general meanvariance optimization framework first proposed by Markowitz (1952), including a number of adjustments in the way that expected costs and risks are computed in a forward-looking environment.
(3.6)
E (rFC 2 ) =
2
2
2
For practical purposes, we assume that these instruments are plain vanilla bonds paying an annual
coupon and the forecasting horizon of the debt manager is 10 years, about the average maturity of
Uruguayan market debt. In this notation, π stands for the rate of inflation measured in terms of CPI,
π represents the CPI inflation rate in any other country i, πγ is the GDP deflator, δi the rate of nominal
depreciation with respect to currency i, qi is the bilateral real exchange rate with respect to country i,
γ is the nominal rate of GDP growth and γ stands for the real rate of GDP growth.
The bonds are denominated in the following currencies or units of account: Uruguayan nominal Pesos (NOM), Indexed Units (CPI) 4 , US Dollars (USD), Euros (EUR) 5 , Japanese Yens (JPY), Chilean Pesos
(CHP), Colombian Pesos (COP), Singapore Dollars (SGD) and Hong Kong Dollars (HKD). This menu
reflects not only the choice between the available domestic alternatives (nominal or CPI-indexed) and
the financial markets, but also regional and alternative capital markets that might be useful to explore
3. In fact, given that the analysis introduces a stock of foreign assets as the n-th instrument, this condition is redefined to apply to the first n-1 bonds, which are the
pure liabilities.
4. A unit of account indexed to the Uruguayan domestic CPI.
5. The Euro and the European CPI were linked to the Deutsche Mark and the German CPI in order to complete the sample period, previous to the creation of the
single European currency.
Revista de Ciencias Empresariales y Economía
Revista de Ciencias Empresariales y Economía
1
For example, if an institutional change affects the conduction of monetary policy because the country is seeking to implement a full-fledged inflation targeting regime, this will likely affect the past
parameters of inflation and consequently, it will also change the perception about the real cost and
risks of issuing and holding nominal debt. A stronger commitment to a low and stable rate of inflation will likely lower the risk premium on nominal debt and therefore, the past observed first and
second moments of the inflation rate might not be a good indicator of the future parameters. On
the other hand, if the policymaker decided to follow a new policy rule, in terms of linking the level
of activity and the inflation rate (e.g., increasing the rate of inflation in periods of recession and vice
versa), this rule has to be internalized in the model because it affects the countercyclical properties
of nominal debt. This policy rule affects the correlation coefficient between these two variables and
consequently, may require another intervention in the matrix of variances and covariances in order to
update its information.
39
ORIGINAL SIN AND REDEMPTION: REBALANCING THE CURRENCY STRUCTURE OF URUGUAYAN PUBLIC DEBT
UMBERTO DELLA MEA / ANTONIO JUANBELTZ
in search of further diversification of currencies and sources of funding. In addition, the introduction
of other less common currencies like the CHP, COP, SGD or the HKD is motivated by the relative low
variation of the bilateral real exchange rate. Fig. 3 shows how these currencies do not display a significant higher dispersion from historical averages than other mainstream currencies and therefore,
they could be reasonable candidates to join the debt portfolio. This is also valid in terms of the stability of nominal GDP, measured in currencies other than the Uruguayan Peso, as depicted by Fig. 4.
1982=100
Fig. 3: Bilateral Real Exchange Rates
relative to the benchmark. This yield is adjusted, in the case of foreign currencies, in order to reflect
the expected evolution of the bilateral real exchange rate and the expected rates of inflation (see Annex I). Two scenarios are considered: in the first case (q=H), all the bilateral real exchange rates tend
to converge to their historical averages over the forecasting period of 10 years. No model was able
to detect any significant long run trend or deviation from the mean in these variables. In the second
case (q=F), the bilateral real exchange rates are assumed to remain constant at current levels, with
the exception of a total expected 5% appreciation relative to the US Dollar and a total expected 10%
depreciation relative to southeast Asian currencies, over the same span.
250
The domestic inflation is set at the middle of the current target range of the Central Bank of Uruguay
and in the case of foreign countries, a similar approach was taken based on the type of commitment
that monetary authorities have with the inflation level.
200
Risk and the covariance matrix
The objective function (3.1) is minimized subject to a given level of risk:
150
(3.7)
2
σ = ω'Vω
100
where V is a semidefinite positive matrix of variances and covariances.
50
2006
SGD
2005
2004
2003
2002
HKD
Historical values of the second moments have been computed on an asymptotic, unconditional basis.
They were calculated in terms of % logs of their deviations with respect to their long run trends over
the period 1982-2006. Many historical variances and correlations have been retained, to the extent
they are believed to provide a reasonable approach to the expected values, while others have been
discarded and directly replaced by new values. This is the case for the variance of domestic inflation,
which has been replaced by a lower value than the historical average, in line with the reinforcement
of the commitment of the monetary authority with an inflation target 6. A similar approach was
taken in the case of foreign currencies, where the result obtained by their monetary policies in controlling inflation over the last five years was used to calculate the expected variance. In particular, it is
also assumed that in the future, the domestic CPI and the GDP deflator will evolve in a similar fashion
and therefore, their differences are not expected to be a significant source of variance.
Fig. 4: Real and FX-denominated GDP
60%
% log deviations from trend
40%
20%
0%
-20%
-40%
-60%
-80%
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2006
2005
SGD
2004
2003
2002
COP
2001
2000
1999
CLP
1998
1997
1996
JPY
1995
1994
1993
EUR
1992
1991
1990
USD
1989
1988
1987
40
1986
1985
1984
1983
1982
realGDP
HKD
In addition to these nine units of denomination, a new instrument was introduced. Given that the
country needs to hold international reserves which in practice are mainly US Dollar denominated,
this tenth instrument (RES) was introduced in a fixed proportion of the overall portfolio, bearing a
negative rate of cost (that is, a positive rate of return) and a similar correlation structure to the US
Dollar denominated debt. These reserves are estimated to remain fixed at 7.5% of the total debt
portfolio.
The yields on these instruments are based on the average current benchmark rates prevailing in the
respective domestic markets as of June, 2007, plus an estimation of the Uruguayan sovereign spread
Two empirical cases of the covariance matrix were generated: one set of simulations was performed
over the historical values of the correlation coefficient between the rate of domestic inflation and the
level of activity (rho=H). Another set of simulations was performed on the alternative policy scenario
where this correlation is set at -1 (rho=-1). This is to reflect the countercyclical properties of nominal
debt and the capacity of the policymaker to smooth out variations in the tax burden in periods of
recession by raising the rate of inflation and therefore, generating a better case for nominal debt. It
is important to stress the fact that even when correlation is -1, it does not necessarily mean that all
of the GDP movements are offset by inflation in order to keep nominal GDP constant. It only refers to
the direction of the policy response, but their scope is limited by the exogenous variance parameters
of the inflation. The variance of inflation is required to meet the boundaries defined by the monetary
policy. Finally, a negative unit correlation between the level of activity and the level of inflation should
affect the perception of the market about the risks of holding nominal debt and, consequently, also
the risk premium embedded in the nominal yield. However, this feature was not incorporated in our
analysis given the difficulties to estimate a proper value for this premium under the new rule, making
the case more benevolent for the nominal debt in this state of nature.
6. The value of this variance assumes that the inflation is normally distributed around a policy target and that approximately 50% of the time it
will be within a range of ±1%.
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COP
2001
2000
1999
CLP
1998
1997
1996
JPY
1995
1994
1993
EUR
1992
1991
1990
1989
1988
1987
1986
1985
1984
1983
1982
USD
As previously suggested, a forward-looking covariance matrix does not necessarily correspond to the
historical estimation of the second moments of the observed costs. Instead, our approach consists of
dissecting this matrix, in order to disclose its basic components and determine which of them should
be adequately replaced in order to provide a better estimation on a forward-looking basis.
41
ORIGINAL SIN AND REDEMPTION: REBALANCING THE CURRENCY STRUCTURE OF URUGUAYAN PUBLIC DEBT
UMBERTO DELLA MEA / ANTONIO JUANBELTZ
The algebraic form of this matrix is captured in the following way:
(3.7) Covariance Matrix
V (rNOM ) ≅
COV (rNOM , rCPI ) ≅
V (π γ ) + V ( γ ) + 2ρπ γ , γ V (π γ ) V ( γ )
ρ γ ,γ −π V (rNOM ) V (rCPI )
V (rCPI ) ≅ V ( γ − π)
COV (rNOM , rFC1 ) ≅
COV (rNOM , rFC2 ) ≅
ρ γ , γ −δ1 V (rNOM ) V (rFC1 )
ρ γ ,γ −δ2 V (rNOM ) V (rFC2 )
COV (rCPI , rFC1 ) ≅
COV (rCPI , rFC1 ) ≅
ρ γ −π,γ −δ1 V (rCPI ) V (rFC1 )
ρ γ −π,γ −δ1 V (rCPI ) V (rFC1 )
V (rFC1 ) ≅
COV (rFC1 , rFC 2 ) ≅
(
*
1
)
*
1
V ( γ − δ1 ) = V ( γ + π − π − q&1 ) ≅ V γ + π − q&1 =
( )
1
1
ργ − δ1 , γ − δ 2 V (rFC1 ) V (rFC 2 )
( )
V (γ − q&1 )+ V π + 2ρ γ − q& ,π* V (γ − q&1 ) V π
*
1
*
1
V (rFC 2 ) ≅ V ( γ − δ 2 ) = V ( γ + π *2 − π − q& 2 ) ≅
(
)
V γ + π *2 − q& 2 =
( )
V (γ − q& 2 )+ V π *2 + 2ρ γ − q&
*
2 ,π2
Eight different scenarios were generated in total. Over every combination of the two bilateral real
exchange rate scenarios and the two correlations between inflation and the level of activity, two different alternatives are built, generating a total of eight cases under consideration.
As it can be seen, these different assumptions do not generate significantly different results. In fact,
point ‘A’ in Fig. 5 corresponds to a breaking point where the portfolio is mostly composed by CPIindexed debt. At this point, CPI-Indexed debt is approximately 93% of the total portfolio, while the
rest corresponds to foreign currency denominated debt, as a hedge to the external reserve assets. To
the upper left, portfolios are mostly combinations of nominal and indexed debt while, to the lower
right, portfolios are combinations of foreign currency denominated and CPI-indexed debt. This is the
reason why -to the upper left- assumptions on the bilateral real exchange rates are almost irrelevant.
Same happens to the assumptions on the inflation correlation coefficient, to the lower right.
As expected, when this coefficient is set to -1, we obtain lower levels of risk for unit of cost, but this is
only operational in a short and steep interval, left of point ‘A’. A more horizontal slope would require
more variance in the level of inflation and therefore, more time for the monetary authority to be outside the inflation target range, in order to accommodate GDP fluctuations. However, in this scenario,
it is difficult to expect that the yield on nominal debt would remain the same 7 .
4. Cost and risk trade-offs under different scenarios
Revista de Ciencias Empresariales y Economía
Revista de Ciencias Empresariales y Economía
As a baseline scenario, the vector of shares ω is unrestricted and can take any value between 0 and
100%, subject to the adding-up restriction (3.2). However, a second set of parameters was derived in
a restricted case, where the optimal shares cannot differ in more than a predetermined percentage
of the actual portfolio composition. This is to reflect the lack of capacity of the debt manager to shift
too rapidly the full currency denomination of its portfolio and therefore to constrain the space of
the feasible solutions over the short term. In this second set of simulations the key restrictions are:
domestic nominal fixed rate debt can grow up to 10% of total debt, CPI indexed debt up to 35% and
at the same time, USD-denominated debt can decrease from current levels, but not being lower than
60% of the total portfolio. EUR and JPY denominated debt are marginal and the reduction of their
exposure is not an issue for the policymaker.
42
( )
V (γ − q& 2 ) V π *2
The baseline scenario (unrestricted case or ‘uR’), is depicted in Fig. 5. Here, four efficient frontiers are
generated. Two of them correspond to the case where the correlation between the inflation rate and
the real GDP are based on actual historical values, while the other two assume that the correlation
coefficient between these variables (ρπγ,γ or ‘rho’) is set to -1 by the monetary policy in order to offset
variations in the nominal GDP. Within these cases, two alternatives are open: in the first one, bilateral
real exchange rates (‘bRER’) converge to their historical averages over the sample period (case ‘H’),
while in the second (case ‘F’), the Uruguayan currency appreciates a further 5% in real terms against
the US Dollar over the forecasting horizon and appreciates 10% face to the Singapore and Hong
Kong Dollars, keeping the other rates constant.
7. In fact, an alternative way to reduce risk would be to strengthen the overall macroeconomic management in order to reduce the variance
of GDP around its trend, but this would also benefit the other debt instruments. This trade-off would also improve through a reduction in
the risk premium, per unit of risk.
43
ORIGINAL SIN AND REDEMPTION: REBALANCING THE CURRENCY STRUCTURE OF URUGUAYAN PUBLIC DEBT
Simulations performed in the restricted case are not significantly different from a conceptual point of
view, but as expected, they considerably reduce the margin of maneuver for the debt manager. These
results are depicted in Fig. 6:
UMBERTO DELLA MEA / ANTONIO JUANBELTZ
A particularly striking feature of the efficient frontier is the sharp change in the slope (see point ‘A’ in
Fig. 5), corresponding to the situation where domestic debt starts to be replaced by foreign currency
denominated debt. Left of point ‘A’, the slope is almost vertical, meaning that large reductions in
the expected cost can be bought with small additional amounts of risk. In this regard, by analogy
with the Sharpe Ratio, we have defined a quasi-Sharpe index (‘qSharpe’), as the ratio between the
variations in expected cost, divided by the variations in risk. This metrics is calculated from individual
simulation points and depicted in Fig. 8 8 :
(-∆% Expected Cost / ∆% Risk)
These two alternatives in the ‘bRER=H, rho=-1’ case, restricted and unrestricted, appear summarized
in Fig. 7, where it can be easily checked that the unrestricted case dominates the restricted one. In
this figure, we also plot the parameters for the current portfolio as of June, 2007. Even when the
result is suboptimal in terms of what could be achieved, there doesn’t seem to exist a large scope to
improve the overall performance of the debt portfolio related to the restricted case, because of the
high participation of foreign currency. Only a massive shift towards domestic currency instruments
can decrease risk in a significant way, given levels of expected cost.
Clearly, this measure is higher for lower levels of risk, decreasing rapidly and falling below one for
levels of risk higher than 6%. This characteristic, which seems quite independent from the set of assumptions, is related to the share of nominal debt in the portfolio.
While nominal debt provides protection against risk given the negative correlation assumed between
inflation and the level of activity, this gain is relatively small compared to the cost over the CPI-indexed
alternative. Substitution of nominal debt per CPI-indexed debt significantly reduces the funding cost
at a very small price in terms of risk.
To the right of point ‘A’, the slope becomes less steep and the qSharpe Ratio is below one. The introduction of foreign-currency denominated debt contributes to lower funding costs but at the price of
assuming a high risk.
44
Fig. 9 depicts, in the unrestricted baseline scenario, how the portfolio composition changes as we
move towards higher levels of risk. This analysis reveals two outstanding characteristics of the debt
portfolio: in the first place, the lowest risk portfolio is basically a combination of nominal and CPIindexed debt, with only a residual amount in JPY. However, the level of risk rapidly decreases as
nominal debt is being substituted by CPI-indexed debt. The level of risk increases more significantly
as foreign-currency denominated debt is included in the portfolio. On the other hand, there doesn’t
seem to be gains from diversification in foreign currency. In the end, foreign currencies are foreign
currencies and their strong covariances tend to bias the choice towards the less costly option, in this
case the EUR.
8. Small discontinuities in the curve might be caused by the discrete simulations.
Revista de Ciencias Empresariales y Economía
Revista de Ciencias Empresariales y Economía
Risk, Exp.Cost
45
ORIGINAL SIN AND REDEMPTION: REBALANCING THE CURRENCY STRUCTURE OF URUGUAYAN PUBLIC DEBT
A conceptually similar story stems from the restricted case, as shown in Figs. 10 and 11. These two
outstanding features, CPI-indexed debt dominating over nominal debt and no foreign currency diversification –except for the restrictions originated in the current levels- still hold.
(-∆% Expected Cost / ∆% Risk)
UMBERTO DELLA MEA / ANTONIO JUANBELTZ
5. Conclusions
In this document, we have analyzed the optimal structure of the Uruguayan public debt given a set
of current parameters of risk and expected cost. Even when the general setting is based on the traditional mean-variance portfolio approach, we introduce strong innovations in the way to estimate
the covariance matrix in a more forward looking environment. Policy and institutional changes are
reflected in the parameters, substituting historical values that have lost their validity. The result is a
hybrid matrix whose components can be taken as a more reasonable approach to future behavior.
46
On the other hand, foreign currencies are intrinsically risky. While in some cases the expected cost can
be lower, the amount of risk involved does not seem to compensate this reduction. No benefits of
currency diversification can be found, either. All the foreign-currency denominated instruments tend
to show a high correlation and therefore, the lower expected cost makes the difference.
Revista de Ciencias Empresariales y Economía
Revista de Ciencias Empresariales y Economía
Taking the prevailing yields on the different debt instruments and the scenarios on the relative behavior of the different currencies, the recommendation is a significant increase in the share of CPIindexed debt. While nominal debt may provide a hedge in the case of recession and loss of tax collection, the price of this hedge seems too high. A significant decrease in the overall expected cost can
be obtained in exchange for only a marginal increase in risk. The way to improve the case in favor of
nominal currency would be decreasing the amount of risk associated to this type of debt. Given the
current parameters, however, this seems only feasible if the central bank allows for a higher variance
in the level of inflation.
47
ORIGINAL SIN AND REDEMPTION: REBALANCING THE CURRENCY STRUCTURE OF URUGUAYAN PUBLIC DEBT
Acevedo, Paloma, Enrique Alberola y Carmen Broto, 2007, Local Debt Expansion... Vulnerability
Reduction? An Assessment for Six Emerging Markets, Banco de España - Asuntos Internacionales,
March 2007.
Bank for International Settlements, 2007, Financial Stability and Local Currency Bond Markets, Committee of Global Financial Stability Publications N°28, June 2007, 30–35.
Barro, Robert, 1997, Optimal Management of Indexed and Nominal Debt, National Bureau of Economic Research Working Paper N°6197, September 1997.
Bohn, Henning, 1988, Why Do We Have Nominal Government Debt?, Journal of Monetary Economics 21, 127-140.
NOM
CPI
USD
EUR
JPY
CLP
COP
SGD
HKD
RES
10.00%
4.00%
6.40%
5.88%
3.19%
7.78%
10.48%
4.18%
6.01%
5.10%
Expected Inflation
2.50%
2.00%
0.00%
3.00%
3.00%
0.90%
2.30%
2.50%
Expected change in bilateral
RER (Option H)
5.00%
(0.11%)
(0.41%)
1.29%
(0.31%)
(0.13%)
0.61%
0.17%
(0.11%)
Expected change in bilateral
RER (Option F)
(0.51%)
0.00%
0.00%
0.00%
0.00%
0.96%
0.96%
(0.51%)
Real GDP Growth
2.75%
Net Expected Cost (Option H)
1.96%
1.22%
0.92%
0.60%
1.72%
1.52%
4.25%
1.11%
1.02%
(0.32%)
Net Expected Cost (Option F)
1.96%
1.22%
0.51%
1.02%
0.43%
1.84%
4.39%
1.45%
1.82%
(0.72%)
Annex 2: Covariance Matrix
Borensztein, Eduardo, Eduardo Levy-Yeyati and Ugo Panizza, 2006, Living With Debt. How to limit
the risks of Sovereign Finance, Inter-American Development Bank.
Option
Burger, John and Frances Warnock, 2003, Diversification, Original Sin and International Bond Portfolios, Federal Reserve International Finance Discussion Papers N°755.
NOM
Calvo, Guillermo, Alejandro Izquierdo and Ernesto Talvi, 2002, Sudden Stops, the Real Exchange Rate
and Fiscal Sustainability: Argentina´s lessons, National Bureau of Economic Research Working Paper
N°9828.
Della Mea, Umberto, 2007, On the Currency Structure of Uruguayan Public Debt, Proceedings of the
Workshop on Designing Government Debt Management Strategies, The World Bank.
Goldstein, Morris and Philip Turner, 2004, Controlling Currency Mismatches in Emerging Markets,
Institute for International Economics, Washington DC.
International Monetary Fund and The World Bank, 2001, Discussion of the Guidelines for Public Debt
Management, Chapter IV, 10–35.
International Monetary Fund, 2006, Structural Changes in Emerging Sovereign Debt and Implications
for Financial Stability, Chapter III, Global Financial Stability Report, April, 85-126.
Revista de Ciencias Empresariales y Economía
Yield
Borensztein, Eduardo, Marcos Chamon, Olivier Jeanne, Paolo Mauro and Jeromin Zettelmeyer, 2004,
Sovereign Debt Structure for Crisis Prevention, International Monetary Fund Occasional Paper N°37.
Calvo, Guillermo and Pablo Guidotti, 1990, Indexation and Maturity of Government Bonds: An Explanatory Model, Capital Markets and Debt Management, Dornbusch and Draghi eds., NYU Press.
48
Annex 1: Assumptions on the Vector of Expected Costs
Jeanneau, Serge and Camilo Tovar, 2006, Domestic bond markets in Latin America: achievements
and challenges, BIS Quarterly Review, June, 51–64.
Licandro, Gerardo and Andrés Masoller, 2000, La Composición Óptima de la Deuda Pública Uruguaya, Revista de Economía, Banco Central del Uruguay, Vol.7 N°2, 135-180.
Markowitz, Harry, 1952, Portfolio Selection, The Journal of Finance, Vol.7, N°1, 77-91.
Masuoka, Toshiya, 1990, Asset and Liability Management in the Developing Countries - Modern Financial Techniques, Policy Research Working Paper N°454, The World Bank.
ρ πγ ,γ
= -1
NOM
CPI
USD
EUR
JPY
CLP
COP
SGD
HKD
0.351%
0.232%
1.443%
0.784%
0.559%
1.381%
1.563%
0.934%
1.050%
CPI
0.232%
0.404%
1.399%
0.628%
0.633%
1.074%
1.171%
1.071%
0.994%
USD
1.443%
1.399%
10.597%
8.301%
7.525%
6.119%
7.632%
8.073%
7.042%
EUR
0.784%
0.628%
8.301%
9.342%
7.967%
3.678%
5.160%
7.040%
5.364%
JPY
0.559%
0.633%
7.525%
7.967%
8.148%
2.384%
3.756%
6.257%
4.688%
CLP
1.381%
1.074%
6.119%
3.678%
2.384%
5.897%
6.564%
4.340%
4.453%
COP
1.563%
1.171%
7.632%
5.160%
3.756%
6.564%
7.697%
5.566%
5.393%
SGD
0.934%
1.071%
8.073%
7.040%
6.257%
4.340%
5.566%
6.636%
5.289%
HKD
1.050%
0.994%
7.042%
5.364%
4.688%
4.453%
5.393%
5.289%
4.783%
Option
NOM
ρ πγ ,γ
= Historical
NOM
CPI
USD
EUR
JPY
CLP
COP
SGD
HKD
0.703%
0.328%
2.042%
1.110%
0.792%
1.954%
2.212%
1.322%
1.485%
CPI
0.328%
0.404%
1.399%
0.628%
0.633%
1.074%
1.171%
1.071%
0.994%
USD
2.042%
1.399%
10.597%
8.301%
7.525%
6.119%
7.632%
8.073%
7.042%
EUR
1.110%
0.628%
8.301%
9.342%
7.967%
3.678%
5.160%
7.040%
5.364%
JPY
0.792%
0.633%
7.525%
7.967%
8.148%
2.384%
3.756%
6.257%
4.688%
CLP
1.954%
1.074%
6.119%
3.678%
2.384%
5.897%
6.564%
4.340%
4.453%
COP
2.212%
1.171%
7.632%
5.160%
3.756%
6.564%
7.697%
5.566%
5.393%
SGD
1.322%
1.071%
8.073%
7.040%
6.257%
4.340%
5.566%
6.636%
5.289%
HKD
1.485%
0.994%
7.042%
5.364%
4.688%
4.453%
5.393%
5.289%
4.783%
Revista de Ciencias Empresariales y Economía
6. References
UMBERTO DELLA MEA / ANTONIO JUANBELTZ
49
Asymmetric English Auctions Revisited*
Juan Dubra
.7&5!
Universidad de Montevideo
Abril, 2008
50
Abstract
I introduce a property, the Own Effect Property (OEP) of player´s valuations thatensures the existence
of an ex post efficient equilibrium in asymmetric English auctions.
The use of the OEP has the advantage of yielding an ex post efficient equilibrium without assuming
differentiability of valuations or that signals are drawn from a density.
These technical, non economic, assumptions have been ubiquitous in the study of (potentially) asymmetric English auctions. Therefore, my work highlights the economic content of what it takes to
obtain efficient ex post equilibria.
I generalize prior work by Echenique and Manelli (2006) and by Birulin and Izmalkov (2003). Relative
to Krishna (2003), I weaken his single crossing properties, drop his differentiability assumption, but
I assume that one player´s valuation is weakly increasing in other players´signals, while he uses a
different assumption (neither stronger nor weaker).
I also show that the OEP is necessary for the existence of an ex post efficient equilibrium, when one
* I thank Federico Echenique and Alejandro Manelli for their comments. This paper started as an attempt to weaken some of the assumptions in
their paper Echenique and Manelli (2006) on comparative statics. This paper owes them a lot: the main property of this paper is a weak version
of their Dominant Effect Property, and the method of proof that I use was first used in an earlier version of their paper. I also thank Vijay Krishna
who helped me in substantial issues and in improving the presentation.
.7&5! Correspondence address: Universidad de Montevideo, Prudencio de Pena 2440, Montevideo - CP 11600 - Uruguay. Email: [email protected] and
[email protected]
Revista de Ciencias Empresariales y Economía
Revista de Ciencias Empresariales y Economía
Resumen
Introduzco una propiedad, la Own Effect Property (OEP) de las valuaciones de los jugadores que
asegura la existencia de un equilibrio expost y eficiente en remates ingleses asimétricos. El uso de la
OEP tiene la ventaja de generar tal tipo de equilibrios sin necesidad de asumir diferenciabilidad de las
valuaciones, o que las señales vienen de una densidad. Estos supuestos técnicos y no económicos han
sido omnipresentes en el estudio de remates ingleses asimétricos. Por lo tanto, mi trabajo resalta el
contenido económico de lo que se necesita para obtener equilibrios expost eficientes.
Este trabajo generaliza estudios previos de Echenique y Manelli (2006) y de Birulin e Izmalkov (2003).
En relación a Krishna (2003), uso una propiedad “single crossing” que es más débil que la suya, elimino sus supuestos de diferenciabilidad, pero asumo que la valuación de cada jugador es débilmente
creciente en las señales de los otros jugadores, mientras que él usa un supuesto que no es ni más
fuerte ni más débil que este.
También muestro que la OEP es necesaria para la existencia de un equilibrio expost y eficiente, cuando
las valuaciones satisfacen una cierta propiedad de regularidad.
51
ASYMMETRIC ENGLISH AUCTIONS REVISITED
Journal of Economic Literature Classification Numbers: D44, D82
Keywords: Auctions, Efficiency, Ex-post Equilibrium
1. Introduction
This paper gives a minimal set of assumptions that ensures existence of an efficient ex-post equilibrium (once signals are known, players don´t want to change their behavior), in asymmetric English
auctions. I introduce a new assumption that I call Own Effect Property, and a new method to nd
equilibria, that does not require the ubiquitous assumptions that value functions are differentiable
and that signals are drawn from a density. Moreover, the Own Effect Property is weaker than all the
versions of the single crossing property that have been used in this branch of the literature. Also,
the new assumption is simple, and highlights the economic content of what is needed to obtain the
desired equilibrium.
I will now present the model, the assumptions and the results. I postpone the discussion of the literature until Section 3, because it needs a series of de nitions. It suffices here to say that the paper most
related to this is Krishna (2003). Although he uses differentiability, densities, and two single crossing
conditions that are stronger than the one in this paper, his set of sufficient conditions for the existence of efficient ex-post equilibria in Asymmetric English Auctions does not imply my assumptions,
since I assume that when some player´s valuations are tied, valuations are weakly increasing in other
players´signals. This assumption is neither weaker nor stronger than his assumption that one signal´s
increase causes the sum of valuations to increase.
The contribution of this paper is important because the English auction has had a prominent role
in allocating objects among a potential set of buyers, both in real life, and in economic theory. The
English was the rst auction format, having been used since the times of the Roman and Babylonian
empires, and it is the most commonly used form of auction to sell goods nowadays (see McAfee and
McMillan (1987) and Cassady (1967) who claims that 75% or more of all auctions are English). Its
popularity stems from a variety of reasons: it yields more revenue than the other common auction
format, the sealed bid, in a variety of contexts (see Milgrom (1989), Milgrom and Weber (1982) and
Lopomo, 1998); it allocates the object efficiently in a wider range of environments; and it economizes on information gathering and bid preparation costs (see Milgrom, 1989). Also, relative to other
theoretical constructions of reduced use in the real world, like the second-price auction, it does not
require the winner to reveal his true valuation, thus avoiding renegotiation between the seller and
the highest bidder and also avoiding any conflict that could happen if (for example) in a second-price
auction the price paid is signi cantly less than what the winning bidder stated he was willing to pay.
Finally, in a variety of contexts the English Auction has an ex post equilibrium, one which remains an
equilibrium even if bidders know each others´valuations or signals. If this is the equilibrium actually
played, then players have no ex-post regrets, or reasons to renegotiate, making the auction format
attractive. Also, Perry and Reny (2005) have argued that
2. The Model and Main Results
Revista de Ciencias Empresariales y Economía
“Simultaneous (sealed-bid) auction formats, we contend, often require participants to collect
and provide signi cant amounts of redundant information. For example, the remarkable efficient
auction due to Dasgupta and Maskin (2000) requires a bidder to submit to the auctioneer his
entire preference pro le (over all possible bundles of goods) for all possible vectors of signals of
the other bidders.Consequently, the vast majority of information a bidder is required to express is
redundant since the only relevant preference profile is that corresponding to the actual realized
vector of signals of the others. Perry and Reny´s (2002) auction also requires bidders to submit
large amounts of redundant information (...) However, given the actual vector of bidder signals,
most of these “as if” comparisons are not required to determine the efficient outcome.”
52
This new assumption allows us to work with valuations that are not differentiable, and with types
which are not drawn from densities.1 The method of proof is new and highlights the economic content of the assumptions that ensure efficiency in this kind of auctions. An early version of Echenique
and Manelli (2006) contained a similar proof on which mine is based, but relied on stronger assumptions. This contribution is relevant, because English auctions have played a prominent role in real life,
and in economic theory.
They go on to argue that this excessive amount of information gathering may make bidders collect
less relevant information about the actual state of the world, leading to an inefficient allocation of
the object. They go on to design an auction that, they claim, could be used in practice, even when the
English auction fails to provide an efficient ex-post equilibrium. While this agenda is interesting, my
paper studies under what condition an auction that is actually used in practice, and that economists
have been studying for a long time, yields an efficient ex post equilibrium.
To summarize, I present a simple assumption of payoff functions that is weaker than all the assumptions of its kind, that ensures the existence of an efficient ex-post equilibrium in the Asymmetric
English Auction, when one assumes that valuations are non decreasing.
Let&
B signal
!! """"! #! be the set of players. Each player i observes a signal4 % " #$! &%2 This
!
is only known to player i. Signals are drawn according to some probability
measure
over- #$! &%
)*+- $
' )*+-"?)?%4%&/
that need not posses a density. The signals affect the values that players have for the objects. Player
'""?M !&*2
!
&()&
i´s valuation is a continuous'+#!&%"#
function ( & #$! &% #
that maps profiles of signals (one for each
44 !
player) into real numbers, and that is&()&
strictly
in its own signal, so that for all- $ and all
%* increasing
*&-%!&4/ %#!
+ ! !"
"!! # "! implies $! #"!
(vectors are written in bold) in
For any# $;
$
#
$
#"
$&
'
!
!
!
!
!
that $! # $
let W #( $) be the set of players i such -/*$" # $ for all k (the set of “winners”at# $).
Let IW #( $)I be the cardinality of W #( $). I now introduce the two conditions that are sufficient for the
existence of an efficient equilibrium.
Definition. The set of #-.*"+-)
functions
33 # " ! # $
$ " $ "$43"() % " ! # $
is Increasing at Ties if for every# $such*'1*
that !! # $! " % and all
#
!
!
$! &
!!
"
!
#$! &
!! $ '
The interpretation of the above property is as follows. Suppose# $is a profile of signals for which at
least two players have equal and highest valuations. Then, the property requires that if one of the
winner´s signal increases
to $! & then the effect of the other player´s signals, when they increase from
) *+
to !! *+ !! & does not hurt player- $ . Example 6 of Maskin (2001) shows that even if some sort of
,!+$
(! #;)
0(! %;) signal does not affect player
single crossing property is satisfied, one still needs that player
valuation “very”negatively, if an efficient equilibrium is to exist (an equilibrium is efficient if it always
allocates the object to one of the players with the highest valuation). Valuations in Maskin´s example
are not Increasing at Ties, and that is why he finds that no efficient equilibrium exists. Stronger versions of this property have been standard in the literature. The most common assumption of this kind
is $5($
that $! #is increasing in si and weakly increasing in s_i. The only paper in this literature that has an
assumption that is not stronger than Increasing at Ties is Krishna´s. The assumption in that paper is
neither weaker nor stronger than Increasing at Ties: it states that when i´s signal increases, the sum
of all player´s valuations increases. I now turn to the more substantial assumption on valuations.
Definition. The set of #-.*"+-)
functions
satisfies the Own Effect Property (OEP) if for every# $such that
1 The assumption that types are drawn from densities, and not just any distribution, has been made in the past in this literature, but it is actually not needed (or used!) neither in Krishna (2003), nor in Birulin and Izmalkov (2003). I thank Vijay Krishna for bringing this oddity to my
attention.
Revista de Ciencias Empresariales y Economía
assumes that the valuations satisfy a certain regularity condition.
JUAN DUBRA
53