Sharing Asymmetric Tail Risk: Smoothing, Asset Prices and Terms of Trade

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 1324 August 2021 Sharing Asymmetric Tail Risk: Smoothing, Asset Prices and Terms of Trade Giancarlo Corsetti, Anna Lipinska, Giovanni Lombardo Please cite this paper as: Corsetti, Giancarlo, Anna Lipinska, Giovanni Lombardo (2021). “Sharing Asymmetric Tail Risk: Smoothing, Asset Prices and Terms of Trade,” International Finance Discussion Papers 1324. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/IFDP.2021.1324. NOTE: International Finance Discussion Papers (IFDPs) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the International Finance Discussion Papers Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers. Recent IFDPs are available on the Web at www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at www.ssrn.com.

Sharing Asymmetric Tail Risk: Smoothing, Asset Pricing and Terms of Trade ∗ † ‡§ Giancarlo Corsetti Anna Lipin´ska Giovanni Lombardo August 5, 2021 Abstract Crises and tail events have asymmetric effects across borders, raising the value of arrangements improving insurance of macroeconomic risk. Using a twocountry DSGE model, we provide an analytical and quantitative analysis of the channels through which countries gain from sharing (tail) risk. Riskier countries gain in smoother consumption but lose in relative wealth and average consumption. Safer countries benefit from higher wealth and better average terms of trade. Calibrated using the empirical distribution of moments of GDP-growth across countries, the model suggests non-negligible quantitative effects. We offer an algorithm for the correct solution of the equilibrium using DSGE models under complete markets, at higher order of approximation. Keywords: International risk sharing, asymmetry, fat tails, welfare. JEL Classification: F15, F41, G15. ∗University of Cambridge and CEPR. †International Finance Division, Federal Reserve Board. ‡Bank for International Settlements and University of Basel. §The views expressed in this paper do not necessarily reflect the views of the BIS and of the US Federal Reserve System. We thank Fernando Alvarez for an insightful discussion. Giancarlo Corsetti gratefully acknowledges the generous support and hospitality of the BIS when working on the first draft of this paper. Corsetti’s work on this paper is part of the project “Disaster Risk, Asset Prices and the Macroeconomy” (JHUX) sponsored by the Keynes Fund at Cambridge University. 1

1 Introduction The Global Financial Crisis, the sovereign risk crisis in the euro-area, the early effects of the looming Climate Change and more recently the COVID-19 pandemic have progressively exposed the lack of resilience of the global economy to large financial and macroeconomic distress and disasters. Policymakers around the world are bracing for a new global environment with heightened tail risk—the risk of rare but disruptive events. Agents’ perceptions of tail risk may hinder economic recovery from large disturbances (Baker et al., 2020), or even weigh on long-term growth prospects (Kozlowski et al., 2020). While the recent large crises have a strong global component, it has become increasingly clear that regions and countries have a very different exposure to disaster shocks. Global crises and tail events transmit quite asymmetrically across borders, widening the international divide in wealth and welfare. Hence tail risk, even when associated to global disturbances, raises the value of international risk sharing, achievable either through capital market development and integration, or through institutional arrangements. In the context of heightened perception of tail risk, risk sharing arrangements at global level are seen as highly desirable as they give regions and countries opportunities for smoothing consumption and moderate the costs of adjustment to adverse shocks. However, insuring disaster risk has potentially significant macroeconomic and financial implications. For any given distribution of fundamentals, going from a low to a high degree of risk insurance changes the equilibrium valuation of national assets. Any change in asset pricing in turn translates into an equilibrium adjustment in relative wealth and demand, possibly leading to a re-allocation of labor and production across regions and countries. This means that insurance may also affect trade and the international prices of goods. In this paper, we study the joint financial, macroeconomic and welfare implications of enhancing risk sharing in the presence of disaster risk. In the tradition of open macroeconomics, we focus on GDP fluctuations as the fundamental source of macroeconomic risk. Drawing on financial theory and asset pricing, we bring forward the analysis of kurtosis and skewness, in addition to variance, in the distribution of the variable underlying macroeconomic risk. Relative to the literature, we explicitly account for cross-border heterogeneity in both second and higher moments of the distribution of national GDP. For analytical clarity and tractability, we focus our analysis by contrasting the extreme cases of financial autarky (no role for insurance via financial markets) to complete markets (perfect insurance).1 We carry out our analysis 1This guarantees that, in the absence of economic distortions, trade in financial assets will unam- 2

both analytically and numerically, setting parameters based on evidence on the GDP distribution across countries. To motivate our analysis, we present a set of stylized facts on the variance, kurtosis and the skewness of output for a large sample of countries. We show that, first, there is substantial heterogeneity in these moments across countries. Second, the variance of output is positively correlated with kurtosis but negatively correlated to skewness. As one would expect, especially in light of the past decades of data, higher volatility of output is associated with higher frequency of large downturns. Ourtheoreticalcontributionisthreefold. First, weofferanoveldecomposition of the gains from risk sharing into a “smoothing effect” (SE) and a “level effect” (LE). The former captures welfare gains from risk diversification in terms of the distribution of marginal utility growth. The latter synthesizes welfare gains or losses through the average consumption of goods and leisure. These in turn materialize via interrelated channels. The relative wealth channel works via the revaluation of a country assets, including physical, financial and human capital, at the equilibrium prices with perfect insurance (relative to imperfect insurance). Asset prices reflect any adjustment not only in the equilibrium discount factor, but also in the (average) international price of a country’s output —the good price channel— associated to the equilibrium reallocation of production and demand. We specifically highlight the terms of trade as a novel channelbywhichperfectinsurancemayaffectsocialwelfare: foragivenrelativewealth, the country experiencing an improvement in its average terms of trade gains in higher consumption and lower labor effort. An important advantage of our decomposition consists of clarifying how varying the relative riskiness of national GDPs may move the smoothingandleveleffects(SEandLE)inoppositedirections. Riskiercountriesgainin termsofsmootherconsumptionandlabor,butloseoutintermsofaverageconsumption and labor effort. Instead, safer countries benefit from higher average consumption and lower labor effort in excess of possible smoothing losses. Our decomposition maps these movements into aggregate quantities and prices. Intuitively, these movements are the general equilibrium analog of a premium paid or received by a country to benefit from or offer macro insurance. Second, we derive analytically and numerically the independent and joint contribution of different moments of the GDP distribution to a country’s gains from risk sharing, by different channels. We show that smoothing and level effects tend to compensate each other with asymmetries in second moments (volatility of income)– with substantial macroeconomic adjustment to risk sharing via relative wealth and biguously bring about positive welfare gains. 3

terms of trade movements. With fat tails asymmetric to the left (third and fourth moments)inthedistributionofGDP,thegainsfromrisksharingareinsteaddominated by improvement in consumption smoothing of the riskier country. Relatively safe countries gain in terms of higher average asset prices and better average terms of trade. Model simulations accounting for the correlation across moments based on country-pair comparison corroborate the empirical relevance of both the smoothing and the level components of the gains from risk sharing. Overall, tail risk enhances the relative gains from capital market integration of countries that are more exposed to it. We quantify this effect by drawing country pairs from the empirical distribution of countries according to their GDP-growth variance, skewness and kurtosis, and using the resulting moments in our theoretical model. Our simulation suggests that riskier countries in the 95-th percentile of the distribution of outcomes can have a relative gain advantage of about 10% of the total gains from risk sharing (total gains are defined as sum of Home and Foreign country gains form risk sharing). This relative advantage resultsfromdifferentcontributionsfromtheSE andLE componentsofthegainsacross countries. For example, the advantage for riskier countries in the 95-th percentile of the distribution, the consumption smoothing component is about 15%, in part offset by a negative LE. The opposite mechanism is at work for the safer countries, who are gaining mainly from the level effect. Our third and last contribution is related to the way we carry out our study, both analytically and numerically, using perturbation methods.2 For the purpose of studying tail risk, these methods are preferable to alternative (global) methods, as they naturally yield a decomposition of the solution in the higher moments of the data generating process. As a methodological contribution to the literature, we spell out a theoretically consistent algorithm applicable in general equilibrium models. The algorithm yields an efficient solution to the problem of solving for the initial distribution of wealth under complete markets, using perturbation. The usefulness of this contribution is best appreciated in light of a widespread practice in the literature, consisting of omitting the specification of the budget constraint under the assumption of complete markets. This practice is not necessarily consequential when countries are assumed to be sufficiently symmetrical in economic structure and distribution of disturbances. It becomes problematic in exercises realistically allowing for large asymmetries in structure and risk across borders, as omitting the budget constraint from the solution algorithm rules out by construction the level effect component driving the gains from risk sharing.3 2Seee.g. Holmes(1995),Judd(1998),Schmitt-Groh´eandUribe(2007),LombardoandSutherland (2007), Kim et al. (2008), Andreasen et al. (2018), and Lombardo and Uhlig (2018). 3It is easy to produce examples where a model solved using this practice erroneously predicts welfare losses for a country when moving from autarky to complete markets. 4

Analytical tractability allows us to explore in detail how the structure of the economytranslatethepropertiesofthefundamentalprocessdrivingoutputintoincome risk. We provide insight on non-loglinearities in the economic structure that drive the transmission of tail risk, and specifically discuss how income risk varies with the intraand intertemporal elasticity of substitution. For some parameterizations of the model, standardintheliterature, weareabletoderivesharpandinstructivepropositions. One instance is the proposition of “equal gains” from risk sharing, applied to symmetric countries differing only in the volatility of output: asymmetries in risk do not translate into differences in welfare gains, but only affect the composition of these gains. The riskier country gains more in consumption smoothing, the other benefits from higher average consumption. Our decomposition of the gains from risk sharing into smoothing and level effectshaspolicy relevance. Inthepolicy literature, thegainsfromrisk sharing aretypically assessed in terms of consumption smoothing only (e.g. Vin˜als (2015), Constˆancio (2016)), i.e. it focuses only on the first leg of our decomposition. Consistently, most of the empirical evidence and indicators of risk sharing published in policy reports are based on measures of consumption volatility and cross country correlations of consumption.4 Relying on these indicators to assess international risk diversification raises a number of logical issues. As shown in our analysis, the SE is only one component of the total gains from diversification: an assessment of these gains based on these indicators is at best incomplete. When a relatively safe and a relatively risky country integrate their capital markets, it is plausible that the safe country mainly gains in term of higher average wealth and consumption, and despite diversification may even lose out in terms of consumption volatility. By no means this implies that the safe country derives no gain from asset trade, as the wealth and consumption level effects of asset revaluation at the new equilibrium price would more than compensate losses in smoothing, if any. Vice versa, the risky country is likely to lose out in terms of relative wealth and consumption. The “implicit transfer” via asset revaluation and terms of trade adjustment is typically disregarded in policy debate, arguably because it is difficult to quantify. Yet, it is a key channel through which countries benefit from the integration of frictionless financial markets.5 This channel cannot be ignored in policy 4The empirical literature is large. See for example Obstfeld (1994) and the literature review in Kose et al. (2009). One popular approach consists of testing directly the consumption risk-sharing condition, which predicts a perfect correlation of consumption growth of two economies trading in complete financial markets. Another popular approach measures the correlation between domestic consumptiongrowthanddomesticoutputgrowth: themoreacountryisgloballyfinanciallyintegrated, the less domestic consumption depends on idiosyncratic domestic disturbances. 5This point is apparent in the theoretical literature, starting from textbook treatments (e.g. Ljungqvist and Sargent, 2012), and including recent research (e.g. Engel, 2016; Coeurdacier et al., 2019). 5

debates on the pros and cons of capital market integration among countries differing in their risk profile and economic features. Our analysis draws on a long standing body of literature highlighting the effects of higher moments on asset prices and risk premia. Early on, Samuelson (1970) already warned against limiting the analysis of optimal portfolio choices to first and second moments (mean-variance models), which in his view can be justified only in limiting cases. In general, higher moments co-determine the values of assets and the degree of hedging they can provide. Ingersoll (1975) emphasizes that skewness plays a role in determining efficient portfolio frontiers, in a way conceptually similar to the role of second moments. Indeed, Kraus and Litzenberger (1976) extend the CAPM to include the third moment, showing that such an extension substantially improves the empirical fit of the model. Harvey and Siddique (2000) and Smith (2007) argue that conditional systematic co-skewness of returns helps addressing the empirical puzzle of the failure of the market beta to explain the cross section of expected returns. More recently, Bekaert and Engstrom (2017) build a consumption-based asset pricing model that can generate skewness of consumption growth as well as the negative correlation of such skewness with the option-implied volatility, as observed in the data.6 Hence, from an international finance perspective, differences in higher moments of cross-country returns can have important implications for relative asset prices. Finally Fang and Lai (1997) extend the analysis of asset pricing to the fourth moment. They show that investors seek compensation for higher variance and kurtosis, while willing to forgo compensation for higher skewness. At the same time, our paper follows the tradition of international macroeconomics, stressing that not only relative asset prices, but also relative good prices are key to understanding cross-border aggregate risk (eg Cole and Obstfeld, 1991).7 A direction of research combining insight from finance and open macro to study tail risk is still relatively unexplored in the international macroeconomics literature,8, but can be expected to become dominant in light of recent crisis (see the growing body of contributions on “growth at risk”, e.g. Adrian et al. (2019).) Our paper speaks directly to the large theoretical and empirical literature on 6Guvenenetal.(2018)showthatidiosyncraticincomefluctuationsdisplaynon-Gaussianfeatures. They show that skewness and kurtosis affect the welfare costs of incomplete insurance. 7In our analysis, asset and good prices together drive the gains from risk sharing at different “orders of risk”. We will specifically detail how GDP volatility translates into income risk, as a function of risk aversion and trade elasticity, i.e. the degree of substitution between domestic and foreign goods. 8The international macroeconomics literature has limited the analysis of risk to first and second moments (e.g. Obstfeld and Rogoff, 2000). This is particularly the case for the vast literature that uses second-order approximations to evaluate optimal policies. 6

the gains and extent of risk sharing across countries, e.g. van Wincoop (1994), Lewis (1996), van Wincoop (1999), Athanasoulis and van Wincoop (2000) and Lewis and Liu (2015). Our analysis is also in line with recent literature, stressing that, to the extent that cross-border insurance allows countries to reduce their reliance on precautionary saving, it may also lead to significant reallocation of capital across borders (a point discussed by Gourinchas and Jeanne (2006), Coeurdacier et al. (2019)). The rest of the paper is organized as follows. Section 2 presents stylized facts on the distribution of output growth, showing that variance, skewness and kurtosis (of GDP) are substantially heterogeneous across borders. Section 3 specifies the model. Section4presentshigherordersolutiontothemodelandSection5discussesallocations under complete markets. Section 6 carries out an analytical decomposition of the gains from risk sharing by moment and channels. Section 7 presents and discusses our numerical results. Section 8 concludes. 2 Volatility and fat tails in the distribution of output: cross country evidence International macroeconomic and finance has long focused on output volatility as the main source of macroeconomic risk driving asset prices and motivating portfolio diversification and risk sharing arrangement across borders (see for example Uribe and Schmitt-Groh´e (2017)). The global financial crisis (GFC) and more recently the COVID-19 pandemic together with rising concerns about climate change have brought forward the need to improve our understanding of fat-tail and especially left-tail risks. To set the stage of our study, we present basic stylized facts about the crosscountry joint distribution of output. Figure 1 shows the cross-section density function of the three moments of the first difference of log per-capita real GDP (PWT, 9.1). A first notable result is heterogeneity across countries: all the moments appear to be quite disperse and with a pronounced long tail. A relatively small fraction of countries displays an exorbitant volatility. Moreover, for skewness, very large negative values are much more likely than large positive ones. Table 1 shows percentiles of the distributions. Differences in moments are considerable, even leaving out the extreme tails of the distribution. A country in the 95% of the distribution would have about 97% of the total variance if combined with a country in the lowest 5%, and about 85% of the total variance if combined with the median country. For skewness and kurtosis 7

Figure 1: Distribution of moments across countries Standard Deviation Skewness Kurtosis Source: Penn World Table (9.1). GDP per capita at 2011 constant prices; 1960-2017; 156 countries. the 90% intervals are (−2.53,1.06) and (−0.43,10.28) respectively. Appendix D shows the three moments of interest for the full list of countries. Thecross-countryheterogeneityhighlightedbyFigure1andTable1motivates our key question. Holding constant the global economy exposure to the risk of extreme adverse realization of output, how would countries with a more negatively skewed distribution (in the data, skewness for Mexico is -2.289) benefit from sharing aggregate output risk with countries with a lower or no exposure to tail risk (in the data, some countriesevenhaveapositivelyskeweddistributionofoutput, asisthecaseforIreland, with a skewness of 1.876)? The second and higher moments of the distribution are not uncorrelated. We illustrate a second important empirical regularity using the simple cross-sectional scatter plot displayed in Figure 2. The two panels report the (log) standard deviation of per-capita GDP growth (first difference of log GDP per capita) on the horizontal axis and the skewness (top panel) and excess kurtosis (bottom panel) of per-capita GDP growth. As shown in the figure, the skewness and kurtosis of per-capita GDP growth are correlated with the standard deviation. In related work (Corsetti et al., 8

Table 1: Distribution of higher moments of GDP growth Percentile Standard Deviation Skewness Excess Kurtosis 5% 2.71 -2.53 -0.43 25% 4.70 -1.00 0.43 50% 6.45 -0.42 1.66 75% 9.03 0.09 4.03 95% 15.59 1.06 10.28 Source: Penn World Table (9.1). GDP per capita at 2011 constant prices; 1960-2017; 156 countries. 2021), we derive formal results, using both cross section analysis and panel regressions (See also Bekaert and Popov, 2019). The key takeaway from this section is straightforward. First, there is considerable heterogeneity in all moments of output distribution. Second, in the data, adverse macroeconomic tail risk tends to be associated with higher volatility of output. A conjecture, recently revived by the literature (Kozlowski et al., 2020), is that large shocks, such as COVID-19, may alter the way firms, households and government perceive the risk distribution when taking decisions, possibly exacerbating both heterogeneity and correlation across moments. The observed distribution of moments and their correlation are of course the result of structural factors and policies. Throughout our analysis, however, we will take them as exogenous, consistent with the specific aim of this paper, to inspect the determinants and channels of the gains from sharing macroeconomic risk. 9

Figure 2: GDP-growth: Skewness and kurtosis against standard deviation. 10

3 Model Our baseline model is a canonical, discrete time, two-country, frictionless model with differentiated goods and home bias in consumption. In each period the state of the economy consists of a realization of the stochastic total-factor-productivity (TFP) processes and a distribution of financial assets. The two countries, Home (H) and Foreign (F), are heterogeneous in size and distribution of their TFP processes—while symmetric in all other parameters. Foreign variables are denoted by a superscript ∗. World population size is normalized to unity, with country H population set to n ∈ (0,1). 3.1 Firms Each country produces a differentiated good. In each country an infinite number of firms operate in perfectly competitive markets using a Cobb-Douglas technology in capital (K ) and labor (L ), subject to an exogenous stochastic process for TFP. t t To account for common and country specific component of TFP, we posit that each country’s TFP is a geometric average of two underlying stochastic processes, D and t D∗ (discussed further below), with different weights. We write the overall production t functions as follows: Y =DιD∗1−ιKαL1−α (3.1) t t t t t Y∗ =D1−ι∗D∗ι∗ K∗αL∗1−α, (3.2) t t t t t where ι,ι∗,α ∈ (0,1). Note that we write D and D∗ for convenience. One can think t t of these processes as sector- or technology-specific, i.e. they don’t have a “national” connotation. National TFP nonetheless depends on the country-specific mix adopted by domestic firms.9 Goods markets are competitive. For tractability, we also posit that the aggregate capital stock is constant throughout the analysis. In each period the representative firm rents capital at rate r and hires K,t workers at the real wage w (in units of the consumption basket) from households to t 9The sectoral composition of output can imply strong cross-country commonalities in TFP disturbances. For example shocks hitting the IT sector (new processors, microchips shortages etc.), the financial sector (new financial products, new payment systems, etc.) or the automobile sector (new engines, new pollution standards, etc.) can simultaneously hit different countries, albeit to different extent. Thisdomesticdiversification,generatingcross-countrycommonalities,hasstrongimplications for international risk sharing. Our model of TFP is meant to capture this considerations if only in a stylized way. 11

solve the following problem: min r K +w L (3.3) K,t t t t Kt,Lt subject to the production function (3.1). The associated factor demands satisfy the following conditions: Y Y t t K = p α ; L = p (1−α) . (3.4) t H,t t H,t r w K,t t where p is the price of domestic output relative to the consumer price index. H,t 3.2 Consumer Problem TherepresentativeagentincountryHconsumesabundleofdomesticandforeigngoods C , trades in units of capital at price P , supplies labor L at a wage w and capital K t K,t t t t at rate r , to firms, and trades in Arrow-Debreu securities A , at the price Λ , K,t t+1 t+1|t that pay one unit of consumption in period t + 1. Capital must be purchased one period in advance. All prices are in units of the consumption basket. Consumers thus solve the following problem: ∞ (cid:88) max E δtU (C ,L ) (3.5) 0 t t Ct,At+1,Kt+1,Lt t=0 subject to the individual budget constraint: C +E Λ A +P K = r K +w L +A +P K . (3.6) t t t+1|t t+1 K,t t+1 K,t t t t t K,t t We consider two specifications of preferences, the standard CRRA form, where C1−ρ −1 L1+ϕ U (C ,L ) := t −χ t (3.7) t t 1−ρ 1+ϕ and GHH preferences (Greenwood et al., 1988), where (cid:18) L1+ϕ (cid:19)1−ρ U (C ,L ) := C −χ t (1−ρ)−1. (3.8) t t t 1+ϕ 12

In either case, ρ > 0 is the degree of risk aversion, and ϕ > 0 is the inverse Frisch elasticity of labor supply. One reason for using these specifications is comparability: bothareubiquitousintheliterature. Anotherreasonisthat, unlikeCRRApreferences, GHH preferences eliminate the wealth effect on labor supply. The comparison will be useful in contrasting demand and supply effects driving the terms of trade.10 Total consumption is a CES function of domestic and foreign goods, i.e. (cid:16) 1 θ−1 1 θ−1(cid:17)θ− θ 1 C t = ν θ C H θ ,t +(1−ν) θ C F, θ t (3.9) where C C denote consumption of Home and Foreign goods, respectively, θ > 0 H,t F,t is the trade elasticity and ν ∈ (0,1) is a function of relative size of countries, and the degree of openness, λ ∈ (0,1), such that (1−ν) = (1−n)λ. λ thus measures the degree of home bias in consumption. Similarly, for the foreign representative agent we have the following preferences: (cid:16) (cid:17)θ−1 C∗ = ν∗ θ 1 C∗ θ− θ 1 +(1−ν∗) θ 1 C∗ θ− θ 1 θ (3.10) t H,t F,t with ν∗ = nλ. The relative prices associated with the above preferences obey the following relationships: 1 = νp1−θ +(1−ν)p 1−θ (3.11) H,t F,t Q1−θ = ν∗p 1−θ +(1−ν∗)p 1−θ (3.12) t H,t F,t with Q = StP t ∗ the real exchange rate. Total demand for home and foreign goods are: t Pt 1−n Y = p−θ(νC +ν∗ QθC∗) (3.13) t H,t t n t t (1−ν)n Y∗ = p−θ( C +(1−ν∗)QθC∗). (3.14) t F,t 1−n t t t 10It should be noted that he empirical literature finds significant income effects in labor supply estimations (e.g. Attanasio et al., 2018) 13

The first order conditions are: C :U (C ,L )−ζ = 0 (3.15a) t C t t t L :U (C ,L )+w ζ = 0 (3.15b) t L t t t t A :δζ −ζ Λ = 0 (3.15c) t+1 t t−1 t|t−1 ζ t+1 K :P = E δ (r +P ), (3.15d) t+1 K,t t k,t+1 K,t+1 ζ t whereζ istheLagrangemultiplieronthebudgetconstraint, andΛ isthestochastic t t|t−1 discount factor (SDF). International trade in Arrow-Debreu securities implies the following risksharing condition: Q Λ∗ = t Λ . (3.16) t|t−1 Q t|t−1 t−1 The definition of equilibrium is standard and omitted to save space. 3.3 Productivity and Income Risk Countries are heterogenous in terms of the aggregate TFP process, depending on the country-specific mix of technologies, ι and ι∗. These differences ultimately drive the relative income risk faced by residents in each country. For each technology, the stochastic process driving TFP follow an AR(1) process in logs:11 ¯ lnD = (1−ϕ )lnD+ϕ lnD +ωσ ε ; (3.17) t D D t−1 D D,t where the parameter ϕ ∈ (0,1) measures the persistence of the TFP process, σ is D D the standard deviation of the serially-uncorrelated exogenous innovation ε , ω is the D,t 11The specification of the stochastic process in the log of TFP has implications that we discuss laterinthepaper. Thisassumptionisubiquitousinthemacroliteratureandweadoptitasitgreatly simplifies the analysis. 14

perturbation parameter (identical across countries) such that if ω = 0 the model is deterministic, and a bar over variables indicate the deterministic steady state value. An analogous process drives D∗.12 The probability distribution of ε plays a central role in the analysis. For D,t analytical clarity, we will carry out most of our analysis under the assumption that the moments of log(D ) and log(D∗) are mutually independent. t t Assumption 1 (Probability distribution). The probability distribution of the innovation ε is characterized by the fol- D,t lowing moments: E(ε ) = 0 (Mean) D,t E (cid:0) εi εj (cid:1) = 0; i,j ∈ N (Cross-Moments) D,t D∗,t (cid:0) (cid:1) E ε2 = Γγ (Variance) D,t (cid:0) (cid:1) E ε3 = φ (Skewness) D,t (cid:0) (cid:1) E ε4 = η (Kurtosis) D,t  m (cid:0) (cid:1) (m−1)!!(Γγ)2 if m is even AND m > 4 E εm = (m-th Moment) D,t  0 if m is odd AND m > 4 (cid:0) (cid:1) (cid:0) (cid:1) Γ := E ε2 +E ε2 D,t D∗,t The m-th moment assumption coincides with the moments of a Gaussian distribution. We will carry out our analysis up to the fourth order, allowing for departures from Gaussianity. Note that, by our assumption above, the distributions of log(D ) and log(D∗) t t are mean-preserving. If this were not the case higher moments would affect the mean. In this sense, our assumption will help us clarify the specific role of each moment in the distribution of TFP in determining asset prices and the way a country gains from risk sharing.13 12As we discuss further below, in this paper we use perturbation methods as described in Holmes (1995) and Lombardo and Uhlig (2018). 13It will still be possible to gauge the effect of non mean-preserving spreads, i.e. effects of higher moments on the first moment, by modelling how higher moments of TFP impinge on its mean value. Specifically, one could draw on the empirical evidence in Section 2 to specify the interdependence among moments parametrically. 15

The uncertainty generated by unexpected realizations of ε is the source D,t of risk for the households. There are various measures of risk in the literature. A common approach consists of distinguishing between the quantity of risk, depending on the covariance between the SDF and the return on the risky asset, and the price of risk, depending on the mean of the SDF, e.g. Cochrane (2009). This approach may not be ideal when higher moments play an important role, as pointed out by Harvey and Siddique (2000). We prefer to adopt an intuitive and simple measure, using the relative price of risky domestic assets under complete markets. Namely, weassessrelativeriskacrossborderscomparingtheequilibriumvalue of each country’s income stream at the complete-markets state price. Definition 1 ((Inverse) Measure of Relative Risk). Relative risk is defined as a difference between home and foreign asset prices under complete markets: (cid:0) (cid:1) RR := Pcm −Pcm . (3.18) K,t K∗,t If RR > 0, then the Home country is safer than the Foreign. This measure is obviouslynotcountry-specific(i.e., itisthesameforHomeandForeignresidents), since under complete markets the stochastic discount factor is equalized across all agents, independently of where they reside. We should note here that, in equilibrium, uncertainty about productivity translates into uncertainty about income depending on the equilibrium realization of goods prices. This in turn depends on structural parameters of the model, in particular on trade elasticity θ and risk aversion ρ. In what follows we will discuss how relative risk depends on the combination of these parameters. 4 Higher Order Solution of the Model We solve our model, both analytically and numerically, using perturbation methods.14 Indoingso,wecontributetotheliteratureatheoreticallyconsistentalgorithmgenerally applicable in general equilibrium models. In this section, we first motivate and lay out oursolutionmethod. Nextwediscussthenon-loglinearitiesthat,inourmodeleconomy, are key in determining the equilibrium allocation of risk. 14Seee.g. Holmes(1995),Judd(1998),Schmitt-Groh´eandUribe(2007),LombardoandSutherland (2007), Kim et al. (2008), Andreasen et al. (2018), and Lombardo and Uhlig (2018). 16

4.1 Solution Method We adopt a perturbation method in alternative to global methods, used by other contributions in the literature—see e.g., the related paper by Coeurdacier et al. (2019). There are two reasons for our choice. First, perturbation methods naturally yield a decomposition of the solution in the various higher moments of the data generating process. This allows us to relate our solution to variance, kurtosis and skewness in the distribution of fundamentals in a clean way. Second, as already mentioned, a novel contribution of our paper consists of showing how to address the problem of solving for the initial distribution of wealth under complete markets, using perturbation methods at any order of approximation. Following Lombardo and Uhlig (2018), we represent all the variables in our model as functions of the loading parameter for the exogenous stochastic process (ω), e.g. C = C(t;ω). We then take higher order series expansions of the model with t respect to ω around the risk-less equilibrium (ω = 0). The resulting system of stochastic difference equations is recursively linear. Standard solution methods can then be applied recursively to solve for the rational-expectation equilibrium. As customary in the literature, we begin by observing that the risk-sharing condition (3.16) can be solved backward to yield: logζ −logζ∗ +logQ = logζ −logζ∗ +logQ := ρlogκ. (4.1) t t t 0 0 0 where the subscript 0 indicates the time in which the risk-sharing agreement is decided and implemented for the first time. The time-invariant variable κ is the risk-sharing “constant”, ubiquitous in the open-macroeconomics literature under complete markets (e.g. Chari et al., 2002).15 This constant reflects the endogenous initial distribution of wealth under complete markets, which, as we work out in the rest of the paper, is a function of equilibrium assets and goods prices, as well as of the quantities produced. The initial distribution of wealth under a full set of period-by-period state contingent (Arrow) securities is pinned down by a condition on the initial distribution of Arrow securities across borders. Usually this is set equal to zero, de facto imposing thattherisk-sharingagreementisfirststartedfromapositionofzeronetforeignassets, see (see, e.g. Ljungqvist and Sargent, 2012, Ch 8)). This implies that solving forward 15For non-separable preferences or recursive preferences, e.g. `a la Bansal and Yaron (2004), a similar decomposition can be obtained. Details can be obtained from the authors on request. 17

the budget constraint (3.6) and imposing the transversality conditions must satisfy: ∞ (cid:88) A = E Λ (C −p Y ) = 0. (4.2) 0 0 t|0 t H,t t t=0 Upon adding the initial condition constraint (4.2) to the system of equations representing our model, we can solve for all the endogenous variables of the model, including κ. In general this requires solving for a fixed-point. Below we show how to approach the problem using perturbation methods. Take the m−order series-expansion of κ around ω = 0 1 logκ(ω) ≈ κ(0) +κ(1)ω +...+κ(m) ωm (4.3) m! (cid:12) ∂mlogκ(cid:12) where κ(m) := (cid:12) . At each order m, solve for κ(m) and proceed recursively, ∂ωm (cid:12) ω=0 starting from κ(0) = κ¯ and κ(1) = 0 (as certainty equivalence holds at first order). By way of example, a second order expansion implies that 1 κ := logκ(ω)−logκ(0) ≈ κ(2) (4.4) (cid:101) 2 where wlog we set ω = 1.16 Oursolutionalgorithm(whetherappliedanalyticallyornumerically)proceeds as follows 1. Expand to the order of interest the system of equations constituting the model; 2. Find the RE solution for all variables as a function of κ; (cid:101) 3. Use the appropriate series expansion of condition (4.2) to solve for κ. (cid:101) For higher orders of approximation this algorithm can be used recursively starting from lower orders to build the solution for higher orders, i.e. to construct a solution for each of the variables of the model with the same structure as in equation (4.3). In Appendix A, we detail an additional step which is particularly useful for numerical solutions of DSGE models of any size, e.g. using Dynare (Juillard, 1996). 16The accuracy of the approximation clearly depends on the size of ω. Nevertheless, we can normalize this to 1 and scale appropriately the standard deviation of the underlying shocks, wlog. 18

4.2 Economic Structure and the Transmission of Output Risk In economic models, higher order moments (in the stochastic processes of productivity) would matter for risk sharing, even if the economy could be reduced to a system of log-linear equation—this is because the welfare function itself is typically not loglinear. In general, however, higher moments can and do play a key role in determining the equilibrium prices and allocation via different mechanisms. It is instructive to map and briefly review the non-loglinearities in our model, to gain insight on which part of the economy actively shapes the transmission of asymmetric risk. Throughout this section, to ease exposition, we assume that the TFP process has no common component across countries, i.e., ι = ι∗ = 1. To start with, we note that the only equations in our model that are loglinear for any parameter values are the labor supply equations and the production function. In particular, we take the ratio of Home and Foreign labor supply (equation 3.15b and Foreign counterpart), using the production function (equation 3.1 and Foreign p F,t counterpart). Denoting the terms of trade as τ := , we obtain an expression t p H,t linkingrelativeconsumptiontoaweightedgeometricaverageofrelativeoutput,relative productivity and relative prices: (cid:18) C∗(cid:19)jρ (cid:18) Y (cid:19) ( ϕ 1− + α α ) (cid:18) D∗(cid:19) (1 ϕ − + α 1 ) Q t = t t τ , (4.5) C Y∗ D t t t t where j = 1 with CRRA preferences and j = 0 with GHH preferences. All other equations are generally non-loglinear, although some become loglinearinspecialcases. Weorganizeourdiscussiondistinguishingbetweennon-loglinearities that are/are not independent of the international financial arrangements. The first (common to financial autarky and complete markets) arise from the demand aggregators ((3.13) and (3.14)) and the price aggregators ((3.11) and (3.12)). The second pertain to the financial structure, and are thus embedded in the budget constraints (3.6). 4.2.1 Non-loglinearities in Price Indexes and Production Functions Among the loglinearities that are independent of the financial regime, a first one arises from the price indexes (equations 3.11 and 3.12). In general, taking the ratio of equations (3.11) and (3.12) gives a nonlinear relation between the real exchange rate and 19

terms of trade: ντθ−1 +1−ν Qθ−1 = (4.6) t ν∗τθ−1 +1−ν∗ Focusing on equal-size economies, a fourth order approximation of equation (4.6) (n = 1) can be written as: 2 (cid:18) (cid:19) 1 λ Q(cid:101) = (1−λ)τ − (1−λ)(θ−1)2λ 1− τ3 (4.7) t (cid:101)t 6 2 (cid:101)t This shows that, for equal-size economies, up to fourth order of approximation the relationship between the mean log of the real exchange rate and the mean log of the terms of trade is affected by the skewness, but not by the kurtosis, of the terms of trade. For economies of unequal size, also the variance and kurtosis of terms of trade come into play. One may nonetheless note that the expressions above become loglinear (only) when θ = 1, that is, under Cobb-Douglas consumption aggregator, and when λ = 1, that is, under Purchasing Power Parity (PPP).17 Turning to aggregate demand, the ratio of equations (3.13) and (3.14) yields: C∗ ν +(1−ν)Qθ t Y t C t = τθ t , (4.8) Y∗ t C∗ t (1−ν)n +(1− n (1−ν))Qθ t 1−n 1−n t C t which becomes exactly loglinear in the special case of PPP—in which case relative output is proportional to the terms of trade. In the general case (failing PPP), a fourth order approximation of equation (4.8) (again) for the case of equal-size economies, yields: ∗ Y(cid:101) −Y(cid:101) = θτ +X(cid:101) , (4.9) t t (cid:101)t t where (cid:18) (cid:19) 1 λ (cid:16) (cid:17)3 X(cid:101) = −(1−λ)(r +Q(cid:101) (θ−1))+ (1−λ)λ 1− r +Q(cid:101) (θ−1) (4.10) t (cid:101)t t (cid:101)t t 6 2 and where r is relative consumption expressed in equivalent units, that is, (cid:101)t r = C(cid:101) ∗ −C(cid:101) +Q(cid:101) . (4.11) (cid:101)t t t t 17In the cafse of PPP, the real exchange rate is constant and equation 3.12 becomes redundant. 20

Uptofourthorderofapproximation,forequal-sizeeconomies,therelationship between the mean log of relative output and terms of trade is affected only by the mean and skewness of log relative consumption adjusted for the real exchange rate. Variance and kurtosis come into play if the economies differ in size. The set of equations considered so far characterizes the solution for relative pricesandcross-countryratios—therealexchangerateandthetermsoftrade(equation 4.6) as well as relative aggregate demand (equation 4.8). When solving for the level of variables, further non-loglinearities arise from equation (3.13) (or 3.14), even under θ = 1 and PPP. 4.2.2 Non-loglinearities in the Budget Constraint The non-loglinearity that is arguably most consequential for welfare and allocation, however,istheonearisingfromalternativespecificationsoffinancialmarkets—specifically, from the budget constraint (3.6). To fully appreciate this point, note that under autarky, theratioofhomeandforeignhousehold’sbudgetconstraintsimpliesthatrelative consumption at each point in time is determined by relative output: Q C∗ Y∗ t t = τ t . (4.12) t C Y t t This expression is loglinear, meaning that it will not come into play in the way order moments of the distribution will contribute to allocations. This is in sharp contrast to the case of complete markets, in which case the budget constraint (3.6) cannot be reduced to a simple loglinear equation. As discussed in detail below, this difference is crucial in understanding how risk-sharing, or lack thereof, impinges on allocations. It should be stressed that failing to account for the budget constraint causes models to yield a number of puzzling results (i.e., welfare being higher in autarky than under complete markets) that become apparent in the presence of significant heterogeneity across regions and countries. Our analysis thus warns against the common practice of writing quantitative models under complete markets omitting the budget constraint. 21

5 Asset Prices, Terms of Trade and Consumption Under complete markets, because of the non-linearities arising from the budget constraint, risk drives relative wealth and consumption by impinging on assets and goods prices. In this section, we provide analytical insight on the transmission mechanism. We start by showing how higher (co-)moments affect asset prices. Imposing the transversality condition lim Λ P = 0, the price of domestic productive i→∞ t+i|t K,t+i asset (3.15d) can be written as follows: ∞ (cid:88) P = E Λ r . (5.1) K,t t t+i|t K,t+i i=1 where this price is increasing in the comovement between the SDF and the net return. Then,bytakingthefourth-orderseriesexpansionofthisexpressionaroundD ¯ = D ¯∗ = 1 and Λ ¯ = δ = R ¯−1 = R ¯∗−1 we can further write k k ∞ 1−δ (cid:88) (cid:104) (cid:105) P ˙ = E δi r˙ +Λ ˙ +P . (5.2) K,t t K,t+i t+i|t H,t+i δ i=1 where P ˙ = 1+P(cid:101) + 1P(cid:101)2 + 1P(cid:101)3 + 1 P(cid:101)4 .18 In line with this, we can express the K,t K,t 2 K,t 6 K,t 24 K,t home household’s “premium” on home returns as follows: P :=E r Λ(cid:101) + H,t+i t(cid:101)k,t+i t+i|t 1 (cid:104) (cid:105) +E r2 Λ(cid:101) +r Λ(cid:101) 2 + t 2 (cid:101)k,t+i t+i|t (cid:101)k,t+i t+i|t 1 1 (cid:104) (cid:105) +E r2 Λ(cid:101) 2 +E r3 Λ(cid:101) +r Λ(cid:101) 3 (5.3) t 4 (cid:101)k,t+i t+i|t t 6 (cid:101)k,t+i t+i|t (cid:101)k,t+i t+i|t This premium characterizes the “riskiness” of the asset, depending on how its return comoves with its valuation (the SDF)—P is indeed related to the “beta” measure H,t+i of risk discussed in the financial literature (e.g. Cochrane, 2009). The “beta” measure depends on the covariance between the SDF and the gross return on the risky asset: our premium spells out how the riskiness of an asset depends on the various co-moments between the SDF and its net return. Specifically, the first line of equation (5.3) corresponds to the second-order premium and involves the “covariance” between the SDF and the net return. The second line corresponds to the third-order premium and involves the “coskewness” between these variables. The third line corresponds to 18We collect these terms together as the higher orders are due to the log-expansion and not to intrinsic non-loglinearities of the model. 22

the fourth-order premium and involves the “cokurtosis” of the same variables.19 Asymmetriesinriskandsizeaffectassetprices,averagetermsoftrade(reflecting relative demand and supply of national outputs), average real exchange rates and average consumption (reflecting relative wealth). The equilibrium link between these variable is complex, but becomes tractable in some special cases, e.g., under symmetric preferences, implying PPP, or log utility in consumption. To build intuition, the following proposition states a useful didactic result under PPP.20 Proposition 1. Under complete markets and power utility in consumption, holding PPP, relative consumption is equal to the relative value of a country’s current and future output—assessed at the equilibrium good and asset prices: p Y∗ +P∗ F,0 0 K∗,0 κ = . (5.4) p Y +P H,0 0 K,0 Proof. With complete markets and power utility in consumption, it is easy to see that, holding PPP, per equations (4.1) and the aggregate resource constraint, equation (4.2) implies 1 E (cid:80)∞ Λ p Y p Y +P µ := = 0 t=0 t|0 H,t t = H,0 0 K,0 (5.5) n+(1−n)κ E (cid:80)∞ Λ Y E (cid:80)∞ Λ Y 0 t=0 t|0 w,t 0 t=0 t|0 w,t where µ is the share of the value of Home output in world output, current and future, the latter expressed in Home consumption units: Y := np Y + (1 − n)p Y∗, w,t H,t t F,t t whereas we have used equation (5.1) to express the numerator in terms of asset prices and current (time 0) income (equivalent current output evaluated at its time 0 good price).21 A similar expression (as (5.5)) characterizes the Foreign country, making use 1−nµ of the fact that C∗ = Y . (5.4), directly follows taking the ratio of equation t 1−n w,t (5.5) and its foreign counterpart. Thepropositionestablishesthat,underPPP,relativeconsumptionmoveswith 19It may be noted that these are not central moments. We use this terminology for simplicity and directness. 20The proposition generalizes the result in Obstfeld and Rogoff (1996) assuming one world homogeneous good. 21This equivalence follows in the version of the model holding the aggregate capital stock fixed. More in general the numerator of equation (5.5) would be related to the price of a claim to the Home country income. The denominator would be the price of a claim to global income. 23

relative financial wealth, reflecting the relative valuation of country-specific assets— which are claims to the income generated by the domestic (current ad future) production of national goods valued at their equilibrium prices (terms of trade). Note that, from equation (4.8), we also know that the output stream will be endogenous, since in equilibrium prices and wealth differences will impinge on relative labor supply across border. Relaxing PPP, a closed form solution for κ can only be derived for the special case of log utility, ρ = 1. In general, the exchange rate will drive an optimal wedge between marginal utility across borders. The risk-sharing condition (4.1) together with the aggregate resource constraint will imply: (cid:18) (cid:19)−1 1−1 C = n+(1−n)Q ρκ Y = µ Y (5.6) t t w,t Q,t w,t The share of a country consumption in global output depends not only on κ, but also on the real exchange rate. For the Home country, the time-zero condition of the budget constraint (4.2) becomes p Y +P H,0 0 K,0 = 1, (5.7) E (cid:80)∞ µ Λ Y 0 t=0 Q,t t|0 w,t while its foreign counterpart (converted in Home consumption units) is p Y∗ +Q P∗ F,0 0 0 K∗,0 = 1, (5.8) 1−nµ Q E (cid:80)∞ Q,t Λ Q−1Y 0 0 t=0 1−n t|0 t w,t This expression calls attention to the fact that, when capital market integration moves the economy from financial autarky to perfect risk sharing, repricing of goods and assets can be expected to result in significant changes in relative wealth, impinging on relative consumption and labor supply. In the next sections we will articulate this point, resorting to higher-order perturbation methods to fully characterize the model solution in terms of higher moments. 24

6 The Relative Gains from Risk Sharing (RGRS): a decomposition into smoothing and level effects Having described how higher moments drive the valuation effects of capital market integration, in this section we focus on how these effects concur to determine the relative welfare gains across countries. One question often asked in the literature is whether “riskier” countries gain more from integrating their financial markets with safer ones. This question provides a good angle to analyze which specific moments and structural parameters determine the relative riskiness of a country, and drive its relative gains from risk sharing. We start by defining the relative gains from perfect risk sharing, RGRS, in terms of relative welfare changes from financial autarky. Definition 2 (RGRS). ∞ (cid:88) (cid:104)(cid:16) (cid:17) (cid:16) (cid:17)(cid:105) (cid:0) (cid:1) RGRS := E δt U(cid:98) cm −U(cid:98) au − U(cid:98) ∗cm −U(cid:98) ∗au +O ω5 , 0 t t t t t=0 ¯ where U(cid:98) = U −U, cm denotes complete markets and au denotes autarky. t t The RGRS can be decomposed into a “level effect” (LE) and a “smoothing effect” (SE) of moving from autarky to complete markets. These effects are defined as follows: Definition 3 (Level and level effect). LE corresponds to the linear term of the 4th order approximation of RGRS: ∞ LE := E (cid:88) δtC ¯ U ¯ (cid:104)(cid:16) C(cid:98) cm −C(cid:98) au (cid:17) − (cid:16) C(cid:98) ∗cm −C(cid:98) ∗au (cid:17)(cid:105) +O (cid:0) ω5 (cid:1) , 0 C t t t t t=0 SE summarizes all higher moments of the 4th order approximation of RGRS: SE := RGRS −LE. Forthesakeofanalyticaltractability, wewillfocusontwoequal-sizecountries (n = 0.5) with identical preferences, so that PPP holds. We will show that our main conclusions go through also when PPP doesn’t hold. 25

Recall that, under the simplifying assumptions of PPP, mean-preserving distributions and symmetry of initial steady state endowments, mean consumption under autarky is identical across countries. Therefore, LE is simply the difference between home and foreign consumption levels under complete markets. We now state two propositions that establish, for equal-size countries under PPP (a) a simple mapping of RGRS into asset prices under complete markets and autarky, and (b) how the overall RGRS are driven by asymmetries in the distribution of output. Proposition 2 (RGRS and asset prices). Assume that: aggregate capital is fixed (and normalized to 1); α = 1 ; Y ¯ = Y ¯∗; consumption preferences are identical across countries (λ = 1 and PPP holds) and that countries have equal size (n = 1). Then, 2 we have that ∞ (cid:88) (cid:104) (cid:105) E δt U(cid:98) cm −U(cid:98) ∗cm = E(Pcm −P∗,cm) = RR 0 t t K,t K∗,t t=0 and E (cid:88) ∞ δt (cid:104) U(cid:98) au −U(cid:98) ∗au (cid:105) = E(P K au ,t t −P K ∗, ∗ a , u t t) 0 t t (1−ρ) t=0 Proof. The proof follows from direct calculation. Remarkably, for equal-size countries under PPP, our measure of relative risk RR is also a measure of the relative welfare under complete markets. Safer countries invariably gain more from perfect risk sharing. Relative risk translates partly into asymmetries in consumption levels, partly into asymmetries in the volatility and thickness of the tails of the distribution of consumption. While we will analyse how these two effects shape the gains below, we can anticipate here that the relative gains to the safer country accrue mostly in terms of average consumption. The second part of the proposition shows that an analogous condition holds under autarky. Barring trade in assets, relative welfare is proportional to relative asset prices. Since under our assumptions the level component of consumption is identical under autarky, the difference in asset prices reflects exclusively the volatility and thickness of the distribution of consumption. Not surprisingly, in this case safer countries have higher welfare than riskier countries because residents enjoy a smoother consumption. The following proposition shows how RGRS depends on different moments of 26

the TFP distribution, under the same assumptions as for the previous propositions. Proposition 3 (Distribution of Gains). Assume that: aggregate capital is fixed (and normalized to 1); α = 1 ; Y ¯ = Y ¯∗; consumption preferences are identical across countries (λ = 1 and PPP holds) and that countries have equal size (n = 1). Then, 2 for given total variance Γ := γ +γ∗ and total kurtosis N := η +η∗ we have that (cid:18) (cid:19) ∂RGRS sign = sign((θ−1)(ρ−1)(ι−ι∗)(ι−(1−ι∗))) (6.1a) ∂(γ −γ∗)2 (cid:18) (cid:19) ∂RGRS sign = −sign((θ−1)(ι−(1−ι∗))), (6.1b) ∂(φ−φ∗) (cid:18) (cid:19) ∂RGRS sign = sign((θ−1)(ρ−1)(ι−(1−ι∗))). (6.1c) ∂(η −η∗) Proof. Propositions 2 and 3 are proved by direct calculation of the solution of the model. In particular, solving the model to fourth-order of accuracy under the assumptions of Proposition 3 yields δ(θ−1)3ρ(ρ+1)(ι−(1−ι∗))3 RGRS =− × 96(1−δ)θ3 (cid:8) 3(ι−ι∗)(ρ−1)Γ2 (cid:0) (γ −γ∗)2 −1 (cid:1) +4(φ−φ∗)−2(ρ−1)((η −η∗)+(ι−ι∗)N)} (6.2) where N := η +η∗. We start by noting a remarkable “equal-gain” result established by the proposition. As long as ι = ι∗, the first equation in Proposition 6.1 simplifies to (cid:18) (cid:19) ∂RGRS = 0 ∂(γ −γ∗)2 The benefits from risk sharing are always symmetric for countries with symmetric size, preferences and technology, but differing in the volatility of their output.22 The relevance of this result lies in the fact that, if only for a special case, it clearly illustrates 22RGRS are also zero, trivially, when technologies are identical, i.e. ι=1−ι∗. In this case, shocks are global and there are no gains from risk sharing. 27

the interplay of the level and smoothing channels in determining the RGRS. Specifically, up to a fourth order of approximation, differences in the volatility of output do not impinge at all on the relative welfare gains from complete markets relative to autarky: the welfare improvement is independent of the relative riskiness of the national income process. It follows that, varying relative output volatility, any gains in terms of consumption smoothing are exactly offset by losses in average consumption (and vice-versa). The equal gains result above provides a useful benchmark against which to assess the general case away from symmetry. Henceforth, for clarity of exposition, we assume that ι > 1−ι∗, that is, each country has a higher intensity in one technology. Whenthestochasticpropertiesofthetwotechnologiesdiffer,theresultdifferdepending on ρ and θ. Provided that ρ > 1 and θ > 1, the gains from risk sharing are larger for the country whose production has a relatively stronger bias in ‘own’ technology, proportionately to the square of the aggregate variance Γ. This effect is partially compensated by the square of relative variances. Note that, since γ ∈ (0,1), then (γ −γ∗)2 ≤ 1, so that this compensation is typically only partial. For θ > 1, when countries differ in terms of the frequency of very negative or positive realization of TFP (skewness), the country whose income is more exposed to negative events will tend to benefit more from risk sharing. Because of the benefit of insuring (low-probability but) large realizations of income, the gains in consumption smoothing exceed the equilibrium losses in terms of the level effect. By the same token, the country whose income distribution is characterized by fatter tails (excess kurtosis) and thus prone to extreme events will tend to benefit more from risk sharing, unless consumption preferences are sub-logarithmic. Note that, for aggregate kurtosis (N) there is a further effect on the RGRS: the country with a relatively stronger technological ‘bias’ will gain relatively more, the larger aggregate kurtosis is. Proposition 3 highlights that the gains from risk sharing cannot be correctly understood without a comprehensive analysis of the different channels through which they materialize. The gains accruing from relative consumption levels are usually inversely related to the gains accruing from smoothing. Which channel prevails, smoothing vs level, varies with the distribution of the shock, and structural parameters of the economy. The proposition also establishes the way higher skewness impinges on the relative gains from risk sharing depends exclusively on θ, a parameter that also matters at the fourth order. Conversely, ρ (together with θ) drives the relative gains from risk sharing only at even orders. Below we expand on the analysis of the different role of 28

the (static) elasticity of substitution θ between goods and risk aversion ρ in shaping tail risk sharing at different orders of accuracy. 6.1 Trade Elasticity The key to understand the role of θ is that higher relative output volatility, skewness and kurtosis do not mechanically translate into higher income volatility, skewness and kurtosis—the mapping depends on the equilibrium response of relative good prices to output shock. This in turn depends on the elasticity of substitution between domestic and foreign goods, θ. This point is best appreciated in light of the literature pioneered by Cole and Obstfeld (1991), Corsetti and Pesenti (2001) and Corsetti et al. (2008). In their well known contribution, Cole and Obstfeld (1991) show that in the limiting case of a unit elasticity of substitution, relative prices and relative output move opposite to each other in the same proportion—hence output fluctuations cannot cause any variation in relative incomes. Under PPP, using equation (4.8), relative non-financial income can be written as p Y H,t t = τθ−1. (6.3) p Y∗ F,t t If θ = 1 non-financial income is always identical across countries, independently of shocks to the supply of output. Indeed, under autarky, per equation (4.12) and using again equation (4.8) we have that au ∗,au θ−1 (cid:16) ∗ (cid:17) C(cid:101) −C(cid:101) = Y(cid:101) −Y(cid:101) . (6.4) t t t t θ With θ = 1, consumption is equalized across countries. Introducing complete markets in this environment would be irrelevant because domestic income and wealth are alreadyperfectlycorrelatedacrosscountries. Relativepriceadjustmentmakescontingent transfers (providing insurance) redundant: relative welfare is identical across countries whether or not markets are complete. In light of our definition 1, risk is also identical, independently of the different stochastic properties of the TFP processes. Conversely, a θ above or below 1 affects how the relative moments of output translate into the relative moments of income via the equilibrium adjustment in the relative price of goods. Intuitively, an elasticity above unity implies that prices still 29

move opposite relative to a country output, but less than proportionally relative to quantities. Hence, higher output volatility translates into higher income volatility. On the contrary, with an elasticity below 1, prices move more than proportionally to any change in relative output. With perfect insurance, higher quantity volatility translates into lower income volatility.23 In general equilibrium in turn, the relative price adjustment will affect relative wealth, the ratio of the present discounted value of income, and therefore the relative adjustment in the relative risk (Definition 1). 6.2 Risk Aversion In our Proposition 3 above, risk aversion plays a role in determining the RGRS from asymmetries in variance and kurtosis, but not from asymmetries in skewness. To gain insight into this result, and more in general on how risk aversion influences the different channelsthroughwhichcountriesbenefitfrommutualinsuranceofmacroeconomicrisk, it is instructive to focus on the case θ → ∞, so that home and foreign goods are perfect substitutes and there is no adjustment in the relative price of goods. With a homogeneous good, replacing the net return on domestic capital with domestic output, the asset price equation (5.1) simplifies as follows: (cid:88) ∞ (cid:88) ∞ (cid:18) nY +(1−n)Y∗ (cid:19)−ρ P = E Λ r = E δi t+i t+i Y (6.5) K,t t t+i|t K,t+i t nY +(1−n)Y∗ t+i i=1 i=1 t t where the expression in parenthesis in the discount factor is the growth rate of world consumption. An analogous expression holds for the foreign economy. Now,inthelimitingcaseofρ = 1(utilityfromconsumptionislogarithmic)the log-expansion of the utility function has no higher order terms, that is, U(C ) = log(C ) t t toanyorderofaccuracy. Hence, underourassumptions(equal-sizecountriesproducing a homogeneous good normalized such that Y ¯ = Y ¯∗ = 1), expected utility in autarky is identical for the two countries, EU(Cau) = EU(C∗,au). By proposition 2, then, t t the difference in utility under complete markets coincides with RR. In other words, it only depends on changes in asset prices. It follows that, moving from autarky to perfect insurance has no smoothing effect. Relative gains from risk sharing are driven exclusively by asset valuation. 23Withhomebiasinconsumption,theresponseofthetermsoftradetoagivenoutputshockisnot necessarily monotonic, see Corsetti et al. (2008) for a detailed analysis. This observation is relevant in numerical analysis for elasticity of substitution below 1/2. 30

Under log preference, (6.5) simplifies to: ∞ (cid:88) P =2E δi(1+x )−1 (6.6) K,t t t+i i=1 where x = Y t ∗ . Combining this with the symmetric expression for the other country t Yt ∞ P∗ =2E (cid:88) δi (cid:0) 1+x−1 (cid:1)−1 (6.7) K,t t t+i i=1 we obtain ∞ (cid:88) (1−x ) P −P∗ =2E δi t+i (6.8) K,t K,t t (1+x ) t+i i=1 Our key result for the limiting case of log preferences in our structurally symmetric economy follows from taking a log expansion of order m of the ratio on the right-hand side of this expression (1−x ) x x3 x5 t+i ≈ − t+i + t+i − t+i −...+O (cid:0) xm+1 (cid:1) , (6.9) (1+x ) 2 24 240 t+i t+i Relative asset prices are affected only by odd moments of the distribution of relative income x : only differences in the degree of asymmetry of the distributions matter t for asset price revaluation and hence welfare under log preferences. Gains from risk sharing would be missed by focusing only on second moments. This main conclusion is strengthened when we move away from the logpreference case. Specifically, it can be shown that the even derivatives of (6.5) (evaluated at the symmetric deterministic steady state) are multiplied by the term (1−ρ), while the odd derivatives – those capturing the asymmetries in the distribution – are not. Even moments (e.g., variance or kurtosis) have an extra, direct effect on relative asset prices and welfare as long as ρ (cid:54)= 1—and will thus affect the RGRS. Comparing two countries with asymmetric kurtosis in the distribution of output, higher values of ρ amplify the range of variation in the discount factor, making the country with relatively fatter tail relatively riskier. Thus, the difference in asset prices will be increasing in ρ. It is worth reiterating that all these results would go unnoticed if we restricted our attention to Gaussian stochastic processes, widely used in the literature, as the odd moments of Gaussian distributions are always zero. That 31

said, we also note that the above results are derived under the Assumption 1, which implies independence of the two income processes. 7 Quantitative Analysis In this section, we reconsider and generalize our main results using quantitative analysis. In particular we allow for endogenous labor and cross-country differences in preferences over consumption goods (home bias). In a richer specification of the model, we canassessthe(SEandLE)componentsofrelativewelfaregainsandthemacroeconomic effects of risk sharing, as function of key structural parameters. In addition to expanding on the consumption smoothing and relative wealth channels highlighted in the previous section, our quantitative analysis will highlight a novel one. Gains from risk sharing may also materialize via a terms of trade channel— reflecting any adjustment in relative labor supply and demand for home and foreign goods across borders. A country that experiences an improvement in its terms of trade can enjoy a higher consumption-to-labor ratio for any given asset price revaluation. To bring forward the implications of risk sharing for the terms of trade, we assess the model for different degrees of home bias, as this is the key parameter determining the terms-of-trade effect, and contrast standard CRRA preferences with GHH preferences, since the latter rule out wealth effect on labor supply. In our exercises we will restrict attention to specific values for the trade elasticity (θ = 1.5) and risk aversion (ρ = 4), bearing in mind their role in determining our results as explained above.24 7.1 Parametrization We set the frequency of the model to annual (reflecting the frequency of our main PWT database). The share of labor in production is 1−α = 0.7, the risk aversion parameter is ρ = 4, the inverse Frisch elasticity of labor supply is ϕ = 1.75 (see Attanasio et al., 2018), the weight of disutility from working is normalized to χ = 20,25 and the 24We chose a relatively high value for risk aversion following Coeurdacier and Gourinchas (2016) in a related analysis. 25This value yields approximately hours worked equal to 20% of total time, in line with US data. That said, the value of χ does not affect the results. 32

trade elasticity of substitution is θ = 1.5,26 and the discount factor is δ = (1+0.02) implying a real rate of 2% per year. We assume that the persistence of the TFP process is ϕ = 0.7, amounting to the average TFP across OECD countries. We assume that D ι = ι∗ = 1. The baseline moments are set at the median value shown in Table 1. In particular, denoting the 5-th percentile by x and the 95-th by y, for each moment, we set the total variance (Γ := γ + γ∗), total skewness (Φ := φ + φ∗) and total kurtosis (N := η+η∗) at the value corresponding to x+y (see Table 1 for the specific values). y Then we set the range of values for γ, φ and η at {50%,60%,70%, } where j = j {Γ,Φ,N} respectively. We present our analysis at first focusing on GHH preferences, which allow us to purge out the terms-of-trade effect stemming from income effects on laborsupply. WethendiscussourbaselinewithCRRApreferences, usingtheAppendix to elaborate on the comparison of the two cases. 7.2 Relative Gains from Risk Sharing by Channels and Moments The results from our quantitative exercises are shown in Tables 2 through 5 in the text and in Appendix C. To enhance comparison across tables, we report the RGRS as share of the global gains from risk sharing, i.e. the sum of Home and Foreign gains: we denote this share by RGRS . Correspondingly, we scale the LE and SE by total s welfare gains, with notation LE and SE respectively. s s All our tables include five panels. The first three panels report, respectively, RGRS , and its decomposition into the two additive terms SE and LE . The last s s s two panels show the relative adjustment in asset prices and terms of trade. In each panel, rows describe the results for four degrees of home bias (1 − λ), in decreasing order (1=no home bias). Columns show results for different values of each moment, in the ranges defined above. Results assuming GHH preferences are shown in Tables 2 through 4. Focusing on asymmetric variances first, the results shown in table 2 confirms and generalizes our analytical results. The country with the most volatile output gains more. In relative terms, however, thegainsfromimprovedsmoothing(theleveleffectshowninthesecond 26Consistently with our discussion of Proposition 3, the sign of RGRS (and the direction of the channels) change around θ = 1. For reasons of space, we show results only for θ > 1, as this is the typical range considered in the literature. 33

panel of the table) are to a large extent offset by a negative level effect (LE, in the third panel). The opposite is true for the country with a lower volatility of output. This country gains mainly in terms of average utility from consumption and leisure. In the limiting case of no-home-bias (λ = 1, corresponding to PPP), the SE and LE offset each other exactly. This generalizes Proposition 3, stated for endowment economies, to a production economy under GHH preferences. The drivers of the level effect are shown in the bottom part of the Table 2. Observe that both the relative price of assets and the relative price of goods (terms of trade) deteriorate for the country with the more volatile output—improve for the other country. Indeed, the country with a low output volatility enjoys higher relative wealth and purchasing power. Note that, with GHH preferences, the terms of trade adjust only with home bias in consumption. This is because, with home bias, changes in relative income and wealth modify the composition of global demand in favor of the output produced by the safer (and thus richer) country—driving up its relative price. With GHH preferences, when λ approaches 1–the case of PPP–the terms of trade do not move at all. Comparing the results in three tables Tables 2 through 4, it is apparent that tail risk magnifies the RGRS among heterogeneous countries. To appreciate this point, in each table, consider a “risky” (Home) country in the upper decile of the distribution of the corresponding moment, for the intermediate degree of home bias (.75). Relative to the median (Foreign) country, the risky Home country only gains 0.4 percentage points (of total gains) more than the Foreign country, if it falls in the upper decile of the variance distribution (Table 2, first panel). However, it gains 8.7 percentage points more if it falls in the upper decile of the (negative) skewness distribution (Table 3, first panel); and 8.8 percentage points more if it falls in the upper decile of the kurtosis distribution (Table 4, first panel). With tail risk, the relative gains for the riskier country are 20 times larger (8.7 or 8.8 versus 0.4). In the three tables, the SE and the LE have opposite sign, but the relative weight of these two components is different. In the Tables for skewness and kurtosis (Tables 3 and 4), the SE for the risky country (i.e., a country whose output distribution has the fatter tail, or the larger mass on negative realization), is much larger relative to the LE. Insuring against extreme realizations of output yields large gains in terms of consumptionsmoothing. Theleveleffect(themacroeconomic“price”oftheinsurance), while non negligible, plays a smaller role compared to the case of asymmetric variances in Table 2. A notable result is that, everything else equal, the riskier country gains more 34

(the RGRS are higher) when home bias is high. This result is only in part explained by the SE component of the RGRS. As shown in Tables 3 and 4, what matters is that, when home bias is high, the risky country actually suffers smaller losses in terms of LE (LE is less negative in the third panel of the tables). This is so, despite the fact that the fall in asset prices is more pronounced (fourth panel). But while a large fall in asset prices makes the risky country relatively worse-off in financial terms, the loss in real purchasing power of their residents remains contained because of a moderate deterioration in their terms of trade (fifth panel). As is well known, GHH preferences insulate labor supply from wealth effects– for more general preferences, these wealth effects activate an additional mechanism by which risk sharing may cause terms of trade adjustment. If only for this reason, it is particularly instructive to consider the case of CRRA preferences. To save space, we only discuss results for the variance in the main text (Table 5) – we present the results for the other moments in the Appendix C. With CRRA preferences, when moving from autarky to perfect risk sharing (κ), the safer country consumes more and works less on average. This implies that, through the effects of market integration, the terms of trade tend to improve for the safer country. The gains in purchasing power may be strong enough to tilt the RGRS s in its favor under PPP and for an intermediate degree of home bias. Yet home bias interacts with risk. A strong degree of home bias tilts the RGRS back in favor of the s riskier country, in line with the results using GHH preferences. Note that, in Table 5, the RGRS changes sign between the first and the second row. The nonlinear effects s of labor supply and terms of trade adjustment under CRRA preferences mitigate and can even reverse the RGRS for riskier countries. s 7.3 Model Simulations So far we have carried out our analysis focusing on each higher moment in the distribution of GDP by country separately. In this section, we will use the empirical evidence in Section 2 (see also Table 9), to assess the potential gains from risk sharing treating the different moments jointly. In doing so, we will be in a good position to capture the full extent of asymmetries in the distribution of GDP across borders. We conduct our exercise by randomly drawing 1449 non-repeated pairs of countries from Table 9, each draw generating a pair of vectors of moments for Home and Foreign. The results are shown in Figure 3 and Table 6. Starting from Figure 3, 35

Table 2: Welfare and Relative Price Effects of Risk Sharing by Volatility of TFP, GHH preferences Volatility(γ†) Home-Bias 50% 60% 70% 95% Relative Welfare Gains (RGRSs ††) 0.2 0.0000 0.4311 0.8665 2.1008 0.75 0.0000 0.0824 0.1655 0.4005 1. 0.0000 0.0000 0.0000 0.0000 Smoothing Effect (SEs ††) 0.2 0.0000 1.1634 2.3387 5.6798 0.75 0.0000 1.5242 3.0626 7.4155 1. 0.0000 2.3436 4.7085 11.3920 Level Effect (LEs ††) 0.2 0.0000 −0.7322 −1.4722 −3.5790 0.75 0.0000 −1.4418 −2.8970 −7.0150 1. 0.0000 −2.3436 −4.7085 −11.3920 Relative Gains in Asset Prices 0.2 0.0000 −1.1350 −2.2700 −5.3422 0.75 0.0000 −0.3353 −0.6706 −1.5781 1. 0.0000 −0.1983 −0.3965 −0.9332 Home Average Terms of Trade 0.2 0.0000 0.8172 1.6345 3.8465 0.75 0.0000 0.0631 0.1262 0.2970 1. 0.0000 0.0000 0.0000 0.0000 All measures are in percentages. n = 1, θ = 1.5,ρ = 4, φ = φ∗ = 50%, η = η∗ = 50%. 2 Fourth-orderapproximation. † ColumnsrepresentpercentilesoftheempiricaldistributionofGDPvolatilityinTable1. ††RelativerisksharinggainsdividedbysumofHomeandForeigngainsfromrisksharing,both in percent consumption equivalent (pce) units. Gains are then decomposed into the relative smoothingandleveleffect,sothatRGRSs=SEs+LEs. 36

Table 3: Welfare and Relative Price Effects of Risk Sharing by Skewness of TFP, GHH Preferences Skewness(φ†) Home-Bias 50% 60% 70% 95% Relative Welfare Gains (RGRSs ††) 0.2 0.0000 0.8312 1.6624 10.1457 0.75 0.0000 0.7160 1.4320 8.7421 1. 0.0000 0.5380 1.0761 6.5698 Smoothing Effect (SEs ††) 0.2 0.0000 0.9693 1.9386 11.8312 0.75 0.0000 0.9424 1.8848 11.5060 1. 0.0000 0.9040 1.8080 11.0381 Level Effect (LEs ††) 0.2 0.0000 −0.1381 −0.2762 −1.6855 0.75 0.0000 −0.2264 −0.4527 −2.7638 1. 0.0000 −0.3659 −0.7319 −4.4682 Relative Gains in Asset Prices 0.2 0.0000 −0.2697 −0.5394 −3.2931 0.75 0.0000 −0.0734 −0.1468 −0.8965 1. 0.0000 −0.0424 −0.0848 −0.5175 Home Average Terms of Trade 0.2 0.0000 0.1720 0.3441 2.1007 0.75 0.0000 0.0101 0.0203 0.1237 1. 0.0000 0.0000 0.0000 0.0000 All measures are in percentages. n= 1, θ=1.5, ρ=4, γ =50%, φ∗ =1−φ, η=η∗ =50%. 2 Fourth-orderapproximation. †ColumnsrepresentpercentilesoftheempiricaldistributionofGDPskewnessinTable1. ††RelativerisksharinggainsdividedbysumofHomeandForeigngainsfromrisksharing,both in percent consumption equivalent (pce) units. Gains are then decomposed into the relative smoothingandleveleffect,sothatRGRSs=SEs+LEs. 37

Table 4: Welfare and Relative Price Effects of Risk Sharing by Kurtosis of TFP, GHH Preferences Kurtosis(η†) Home-Bias 50% 60% 70% 95% Relative Welfare Gains (RGRSs ††) 0.2 0.0000 3.0267 6.0527 10.2219 0.75 0.0000 2.6073 5.2144 8.8078 1. 0.0000 1.9592 3.9184 6.6192 Smoothing Effect (SEs ††) 0.2 0.0000 3.2188 6.4368 10.8706 0.75 0.0000 2.8238 5.6474 9.5392 1. 0.0000 2.3098 4.6195 7.8035 Level Effect(LEs ††) 0.2 0.0000 −0.1921 −0.3841 −0.6487 0.75 0.0000 −0.2165 −0.4330 −0.7314 1. 0.0000 −0.3505 −0.7011 −1.1843 Relative Gains in Asset Prices 0.2 0.0000 −0.4376 −0.8751 −1.4784 0.75 0.0000 −0.1071 −0.2141 −0.3618 1. 0.0000 −0.0570 −0.1139 −0.1925 Home Average Terms of Trade 0.2 0.0000 0.2185 0.4369 0.7381 0.75 0.0000 0.0095 0.0190 0.0321 1. 0.0000 0.0000 0.0000 0.0000 All measures are in percentages. n= 1, θ=1.5, ρ=4, γ =50%, φ=φ∗ =50%, η∗ =1−η. 2 Fourth-orderapproximation. † ColumnsrepresentpercentilesoftheempiricaldistributionofGDPkurtosisinTable1. ††RelativerisksharinggainsdividedbysumofHomeandForeigngainsfromrisksharing,both in percent consumption equivalent (pce) units. Gains are then decomposed into the relative smoothingandleveleffect,sothatRGRSs=SEs+LEs. 38

Table5: WelfareandRelativePriceEffectsofRiskSharingbyVolatilityofTFP,CRRA Preferences Volatility(γ†) Home-Bias 50% 60% 70% 95% Relative Welfare Gains (RGRSs ††) 0.2 0.0000 0.0959 0.1922 0.4566 0.75 0.0000 −0.0289 −0.0580 −0.1374 1. 0.0000 −0.0545 −0.1090 −0.2584 Smoothing Effect (SEs ††) 0.2 0.0000 1.2578 2.5197 5.9872 0.75 0.0000 1.5613 3.1265 7.4146 1. 0.0000 2.0900 4.1849 9.9189 Level Effect (LEs ††) 0.2 0.0000 −1.1619 −2.3275 −5.5306 0.75 0.0000 −1.5902 −3.1844 −7.5520 1. 0.0000 −2.1444 −4.2939 −10.1773 Relative Gains in Asset Prices 0.2 0.0000 −0.4028 −0.8056 −1.8959 0.75 0.0000 −0.1370 −0.2740 −0.6449 1. 0.0000 −0.0822 −0.1645 −0.3871 Home Average Terms of Trade 0.2 0.0000 0.3983 0.7965 1.8745 0.75 0.0000 0.1347 0.2694 0.6339 1. 0.0000 0.0818 0.1636 0.3851 All measures are in percentages. n = 1, θ = 1.5,ρ = 4, φ = φ∗ = 50%, η = η∗ = 50%. 2 Fourth-orderapproximation. † ColumnsrepresentpercentilesoftheempiricaldistributionofGDPvolatilityinTable1. ††RelativerisksharinggainsdividedbysumofHomeandForeigngainsfromrisksharing,both in percent consumption equivalent (pce) units. Gains are then decomposed into the relative smoothingandleveleffect,sothatRGRSs=SEs+LEs. 39

Table 6: Summary of Empirical Risk-Sharing Effects Percentiles§ 5% 95% Relative Welfare Gains (RGRS†) -9.504 9.774 s Smoothing Effect (SE†) -14.892 15.387 s Level Effect (LE†) -9.310 9.010 s Relative Gains in Asset Prices †† -1.843 1.916 Home Average Terms of Trade †† -0.291 0.317 Note: §We report 5th and 95th percentiles of the 90% interval of the empirical distributionsforeachvariable. Forthemethodology,seethenotetoFigure3. †Valuesreportedareinpercentagesoftotalgains. Notethatthepercentileoperator (P(x))isnon-linear,sothatP(RGRSs)(cid:54)=P(SEs)+P(LEs). ††Valuesreportedinpercentages. in panel (a) we plot the distribution of the Relative Gains from Risk Sharing RGRS s predicted by our model. The following panels refer the decomposition of RGRS into s smoothing effect SE (panel (b)), and level effect LE (panel (c)), as well as to the s s distribution of the relative change in asset prices (panel (d)) and in the Home terms of trade (panel (e)), all relative to autarky. The summary of these results in Table 6 reports also the 90% interval of each distribution.27 Our new figure highlights at least three remarkable results. To start with, as shown in panel (a), many country pairs gain about the same from risk sharing (corresponding to the mass around zero). Yet, there is a dense tail of countries that draw significantly larger or lower relative gains (right and left tail of distribution). Considering the 90% interval of the distribution, the relative gains/losses from risk sharing (as a percentage of the total gains) are of the order of 10 percentage points on either side. In light of our analysis, relatively riskier countries tend be the winner in terms of improved smoothing, reflecting heterogeneity in exposure to tail risk. The decomposition of RGRS into smoothing and level effects, in panels (b) and (c), corroborates s this analytical insight. A significant group of riskier countries mostly gains in terms of improved smoothing, in excess of the loss from a lower average level of wealth and consumption. A significant group of safer countries mostly gains in terms of relative 27A comment is in order about taking country pairs as our unit of observation, as opposed to, say, pairing a country with regions obtained from aggregating countries. Aggregating countries into regions would be more directly representative of the gains that one individual country could obtain from joining a larger integrated capital market (or aggregate risk sharing institutional arrangement). However, aggregation would also implicitly internalize some of the potential gains from risk sharing, asthesewouldalreadybeachievedthroughwithin-regiondiversification. Wefindourapproachcloser in line with the goal of bringing our evidence to bear on the potential gains from, and effects of, aggregate GDP risk sharing. 40

Figure 3: Empirical Distribution of Risk Sharing Effects (a) Relative Welfare Gains (RGRS ) s (b) Smoothing Effect (SE ) (c) Level Effect (LE ) s s (d) Relative Gains in Asset Prices (e) Home Average Terms of Trade Note: The frequency distributions shown in each panel are derived by randomly drawing 1449 non-repeated pairs of countries using the estimated moments displayed in Table 9 for our sample. Specifically, for each pair of country, we assignmomentstoaHomeandaForeigncountry(eliminatingrepetition),andcomputethemeasuresonthex-axisof eachpanel. Totalmoments(i.e. Γ,ΦandN)changethroughdraws(countrypairs). Inthesimulationsweassumethat λ=0.75, i.e. a considerable degree of trade openness. In the x axis values reported are in percentages of total gains for relative welfare gains, smoothing and level effect, in case of asset prices and terms of trade values are reported in percentages. 41

wealth and consumption–these gains more than offsetting the relative loss in smoothing. The range of relative gains from smoothing corresponding to the 90% interval of the distribution is comprised between −14.8% and 15.5% of total gains from risk sharing; the range for the level effect is between −9.3% and 9.0%.28 The level effect in panel (c), in turn, maps into the distribution of changes in asset prices and the terms of trade, shown in the following two panels. In our analysis, safer countries experience a relative re-valuation of their assets and an appreciation of their terms of trade (right tail of asset gains’ distribution in panel (d) and left tail of terms of trade distribution in panel (e)). Riskier countries experience capital losses in their assets and a depreciation of their terms of trade (left tail of asset gains’ distribution and right tail of terms of trade distributions). The 90% interval for the changes in asset prices ranges approximately from -1.8% to 1.9%; for terms of trade, from -0.3% to 0.3%. The main takeaway is straightforward. Tail risk enhances the relative gains from capital market integration of countries that are more exposed to it. This relative advantage however corresponds to significant differences in the sign, magnitude and combinationsoftheSE andLE componentsofthegainsacrossriskyandsafecountries. Quantitatively, these effects are substantial. A relatively riskier country in the 95-th percentile of the distribution of gains can enjoy an RGRS of about 10% of the global s gains from risk sharing. Similar magnitudes are obtained for the SE and LE . Level s s effects are supported by equally sizeable adjustments of asset prices and terms of trade. 8 Conclusions In this paper, we have reconsidered the welfare and macroeconomic effects of insuring fundamental output risk across borders, offering a decomposition of welfare gains into a smoothing and a level effect, as well as a discussion of these effects by moments of the distribution of output and by transmission channels. After mapping output risk into income risk, highlighting the role of key structural features of the economy, we have brought volatility, kurtosis and skewness in the distribution of output to bear on our decompositionofthewelfaregainsfromrisksharing. Wearticulatethelevelcomponent ofthese gainsdistinguishingbetweentwochannels: arelativewealthchannel, reflecting the revaluation of a country assets at the new equilibrium prices, and a terms of trade 28The percentile operator (P(x)) is non-linear, so that P(RGRS ) (cid:54)= P(SE )+P(LE ), although s s s RGRS =SE +LE . s s s 42

channel, reflecting changes in the relative price of domestically produced goods. These adjustments are the general equilibrium analog of the price of insurance that riskier countries pay, safer countries obtain, when joining a capital market union. Either type of countries gains from insuring aggregate risk, but the composition of these gains differs. While riskier countries tend to gain mostly in terms of consumption smoothing, they may lose out in terms of average consumption and labor effort. Based on the empirical evidence for a large sample of countries, we offer an assessment of the potential gains from macroeconomic risk sharing, decomposing them by smoothing and level effects and by channels. In this exercise, tail risk features prominently in shaping these gains. The distribution of relative gains is bi-modal. A significant group of riskier countries benefit mostly from consumption smoothing, and a significant group of safer countries benefit mostly from the level effects of capital market integration. In the majority of applied and policy assessments of the gains from capital market integration, risk sharing is measured in terms of volatility and correlation of consumption, sometimes in relation to the volatility of output, most often ignoring the real exchange rate in breach of the theoretical condition for smoothing. Our analysis clarifies that these indicators can at best provide an incomplete assessment of capital market integration. By focusing on what we dub the smoothing effect of risk sharing, they miss the main channels through which relatively safer countries gain. Providing macroeconomic insurance is rewarded with a larger share of world output to the residents in the safer regions—corresponding to a rise in relative wealth and an appreciation of their terms of trade. Looking forward, the challenge for the literature is to devise indicators of these level effects, which we show are likely to play a non secondary role in defining why many countries have a clear interest in achieving international risk sharing, especially at times of heightened tail risk. Bibliography Adrian, T., Boyarchenko, N., and Giannone, D. (2019). Vulnerable growth. American Economic Review, 109(4):1263–89. Andreasen, M. M., Ferna´ndez-Villaverde, J., and Rubio-Ram´ırez, J. F. (2018). The pruned state-space system for non-linear DSGE models: Theory and empirical applications. Review of Economic Studies. 43

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Appendix A Higher-order accurate solution of the risk-sharing constant In the simple Lucas’ endowment model discussed in the main text, the risk-sharing constant κ,29 defined as C 2,t κ = (A.1) C 1,t depends recursively on exogenous variables, i.e. assuming ι = ι∗ = 1, 1 E (cid:80)∞ δt(D )−ρD = 0 t=0 w,t 1,t (A.2) n+(1−n)κ E (cid:80)∞ δt(D )−ρD 0 t=0 w,t w,t In general, and specifically when there is production, the right-hand-side of equation (A.2) is endogenous (e.g. the discount factor depends on consumption, and income depends on production). Therefore, to find the risk sharing constant in this more general setting we typically need to use some fixed-point algorithm. We propose a closed-form solution for κ based on perturbation methods. We present a version that is accurate up to second order but that is straightforward to extend to higher orders. In describing our solution we follow a procedure that allows for easy numerical implementation, e.g. using Dynare. Our solution is based on the observation that κ depends on second order terms, and in particular on the variance of the exogenous processes. We can thus represent the risk-sharing constant as log(κ) := κ¯E ε2 (A.3) t κ,t+1 so that κ¯E ε2 = C(cid:101) −C(cid:101) (A.4) t κ,t+1 2,t 1,t 29The term “constant” refers to time invariance. This coefficient is not invariant to risk. 48

where κ¯ is the unknown parameter we want to solve for, and ε , is a mean-zero iid κ,t auxiliary shock with variance denoted by σ2. This implies that E ε2 = σ2. So far κ t κ,t+1 κ we have thus re-scaled the original kappa by σ2. κ The second-order solution of a DSGE model can be written in a second-order VAR form as (e.g. following Dynare notation) 1 1 (cid:126) y = Ay +Bu + [C(y ⊗y )+D(u ⊗u )+2F (y ⊗u )]+ GΣ2 (A.5) t t−1 t t−1 t−1 t t t−1 t 2 2 where y ∈ Rny is the vector of all the n variables (endogenous and exogenous excludt y ing innovations), u ∈ Rni is the vector of all the n (iid) innovations, A, B, C, D, F, t i G are conformable matrices, and for any column vectors x and z, (x⊗z) is the vec- (cid:0) (cid:1) torized outer product of these vectors. Σ2 := E u u(cid:48) , and(cid:126)· is the vectorization t t+1 t+1 operator.30 The key term in equation (A.5) is the last one, which shifts the mean of variables in proportion to the exogenous risk, captured by the variance matrix Σ2 (also referred to as the stochastic steady state in the literature). Using regular perturbations (see e.g. Lombardo and Uhlig, 2018), none of the matrices in (A.5) depends on exogenous risk. This means that the only place where σ κ appears is in Σ2. The vector y contains the variable measuring Arrow-Debreu securities. Ast (cid:126) sume the latter are in position i , and that σ occupies position j in the vector Σ2. AD κ σκ Then we have that 1 y [i ] = A[i ,:]y +B[i ,:]u + [C[i ,:](y ⊗y ) t AD AD t−1 AD t AD t−1 t−1 2 1 (cid:126) +D[i ,:](u ⊗u )+2F[i ,:](y ⊗u )]+ G[i ,:]Σ2 (A.6) AD t t AD t−1 t AD 2 where for a matrix X, X[i,j] denotes the element in row i and column j, and where X[i,:] denotes the row i of matrix X; for a vector z, z[j ] is the j −th element in z. σκ σκ In particular, Σ (cid:126) 2[j ] = σ2. σκ κ Note that if we set κ¯ = 1, we can solve for σ2 that satisfies some restriction on κ y [i ]. In particular we know that under complete markets it must be that y [i ] = 0 t AD 0 AD 30To date, Dynare returns only the product GΣ (cid:126)2 in the variable “oo .dr.ghs2”. In order to implement our algorithm this product must be factorized in the two components. This can be easily done by modifying Dynare function dyn second order solver.m at about line 173, by adding a new variable e.g. dr.G=LHS\(-RHS);, where LHS and RHS are variables defined in the function. 49

(Ljungqvist andSargent, 2012). Oneway toimplement this condition is to assumethat at time 0 and -1 the economy was at the stochastic steady state, i.e. all elements of equation (A.6) are zero except the last one, i.e.31 1 (cid:126) y [i ] = 0 = G[i ,:]Σ2 (A.7) 0 AD AD 2 Then we can solve for σ2 as κ G[i ,j⊥]Σ (cid:126) 2[j⊥] σ2 = − AD σκ σκ (A.8) κ G[i ,j ] AD σκ where j⊥ denotes all the elements excluding j . σκ σκ Now we simply need to swap values, i.e. κ¯ ← σ2 κ σ2 ← 1. (A.9) κ With this assignment of values, κ is the second-order accurate risk-sharing constant that implements complete markets. Our proposed algorithm, correctly implements complete markets up to second order accuracy. It should be noted also that our approach does not affect the firstorder solution. This solution correctly describes growth rates of variables, since the risk-sharing constant is invariant to time (Ljungqvist and Sargent, 2012). Our approach is reminiscent of the solution algorithm proposed by Devereux and Sutherland (2011) (DS) to solve for portfolio shares up to second order. DS introduce an auxiliary iid shock in the budget constraint of investors as a placeholder for portfolio shares. By knowing the position of this auxiliary shock DS can then use simple linear algebra to derive the shares. Although we solve a different problem, our algorithm shares with DS the idea of using auxiliary iid shocks as placeholders for parameters that would otherwise drop out of the perturbed solution. 31Equally easily implementable is any other condition, e.g. Ey [i ]=0. 0 AD 50

B GHH preferences: Analytics Alargenumberofpapersassumesthathouseholds’preferencesaresuchthatthereisno wealth effect on labor supply, following the seminal work of ?, GHH. GHH preferences are non-separable in consumption and labor, i.e. (for the Home country) (cid:32) (cid:33)1−ρ L1+φ C −χ t t 1+φ U (C ,L ) := , (B.1) t t 1−ρ and an identical expression for the foreign country. Under these preferences the first-order conditions (3.15a) and (3.15b) can be written as (cid:32) (cid:33)−ρ L1+φ C :U (C ,L ) := C −χ t = λ (B.2) t C t t t t 1+φ L : −U (C ,L ) := χLφU (C ,L ) = w λ , (B.3) t L t t t C t t t t so that labor supply does not depend directly on the marginal utility of consumption. Importantly, these preferences imply that the risk-sharing condition (??) does not simply depend on relative consumption and the exchange rate, but on labor too, i.e. logζ∗ −logζ −logQ = logζ∗ −logζ −logQ := κ . (B.4) t t t 0 0 0 (cid:101)GHH The same technique discussed above can be used to solve for κ at any order of GHH approximation. C Further Quantitative Results (CRRA preferences) This appendix shows the effect of asymmetries in skewness and kurtosis under CRRA preferences. 51

Table7: WelfareandRelativePriceEffectsofRiskSharingbySkewnessofTFP,CRRA preferences Skewness(φ†) Home-Bias 50% 60% 70% 95% Relative Welfare Gains (RGRSs ††) 0.2 0.0000 0.6266 1.2531 7.6504 0.75 0.0000 0.8052 1.6103 9.8310 1. 0.0000 0.7160 1.4321 8.7432 Smoothing Effect (SEs ††) 0.2 0.0000 0.7147 1.4294 8.7264 0.75 0.0000 0.8932 1.7865 10.9063 1. 0.0000 0.8299 1.6599 10.1340 Level Effect (LEs ††) 0.2 0.0000 −0.0881 −0.1762 −1.0760 0.75 0.0000 −0.0881 −0.1761 −1.0753 1. 0.0000 −0.1139 −0.2278 −1.3908 Relative Gains in Asset Prices 0.2 0.0000 −0.0425 −0.0850 −0.5190 0.75 0.0000 −0.0096 −0.0193 −0.1177 1. 0.0000 −0.0054 −0.0107 −0.0654 Home Average Terms of Trade 0.2 0.0000 0.0405 0.0811 0.4949 0.75 0.0000 0.0076 0.0151 0.0923 1. 0.0000 0.0043 0.0087 0.0531 All measures are in percentages. n= 1, θ=1.5, ρ=4, γ =50%, φ∗ =1−φ, η=η∗ =50%. 2 Fourth-orderapproximation. †ColumnsrepresentpercentilesoftheempiricaldistributionofGDPskewnessinTable1. ††RelativerisksharinggainsdividedbysumofHomeandForeigngainsfromrisksharing,both in percent consumption equivalent (pce) units. Gains are then decomposed into the relative smoothingandleveleffect,sothatRGRSs=SEs+LEs. 52

Table 8: Welfare and Relative Price Effects of Risk Sharing by Kurtosis of TFP, CRRA preferences Kurtosis(η†) Home-Bias 50% 60% 70% 95% Relative Welfare Gains (RGRSs ††) 0.2 0.0000 0.9644 1.9287 3.2581 0.75 0.0000 1.2393 2.4785 4.1869 1. 0.0000 1.1021 2.2042 3.7235 Smoothing Effect (SEs ††) 0.2 0.0000 1.0375 2.0749 3.5051 0.75 0.0000 1.2601 2.5201 4.2572 1. 0.0000 1.1227 2.2453 3.7930 Level Effect (LEs ††) 0.2 0.0000 −0.0731 −0.1462 −0.2470 0.75 0.0000 −0.0208 −0.0416 −0.0703 1. 0.0000 −0.0206 −0.0411 −0.0695 Relative Gains in Asset Prices 0.2 0.0000 −0.0281 −0.0562 −0.0949 0.75 0.0000 −0.0047 −0.0095 −0.0160 1. 0.0000 −0.0023 −0.0047 −0.0079 Home Average Terms of Trade 0.2 0.0000 0.0253 0.0506 0.0855 0.75 0.0000 0.0018 0.0035 0.0060 1. 0.0000 0.0008 0.0016 0.0027 All measures are in percentages. n= 1, θ=1.5, ρ=4, γ =50%, φ=φ∗ =50%, η∗ =1−η. 2 Fourth-orderapproximation. † ColumnsrepresentpercentilesoftheempiricaldistributionofGDPkurtosisinTable1. ††RelativerisksharinggainsdividedbysumofHomeandForeigngainsfromrisksharing,both in percent consumption equivalent (pce) units. Gains are then decomposed into the relative smoothingandleveleffect,sothatRGRSs=SEs+LEs. 53

D Distribution of moments across countries Table 9 shows the three moments of interest for the PWT9.1 list of countries. Country iso-code to English name conversion is shown in Table 10. 54

stnemoM elpmaS :9 elbaT sisotruK ssenwekS vedtS yrtnuoC sisotruK ssenwekS vedtS yrtnuoC sisotruK ssenwekS vedtS yrtnuoC 464.3 173.1- 624.3 DKM 124.0 471.0- 497.3 NIF 058.0 456.0 577.6 GRA 218.01 362.2 271.5 TLM 463.0- 126.0- 535.2 ARF 962.1 298.0- 333.2 SUA 140.0 495.0- 519.2 DLN 340.0- 876.0- 279.1 TUA 270.3 396.0- 557.2 RBG 402.1 223.0- 689.3 RON 884.1 229.0- 887.2 LEB 537.0 035.0- 647.4 CRG 107.3 117.1- 258.5 RGB 169.0 337.0- 734.3 LZN 682.5 650.2- 893.7 VRH 518.3 338.1- 746.4 LOP 785.4 999.0- 685.3 NUH 624.0- 404.0 662.5 ARB 187.0 442.0- 412.4 TRP 410.0- 850.0- 187.5 NDI 273.2 803.1- 407.2 NAC 107.0 143.0- 740.5 UOR 455.3 446.0- 665.2 EHC 571.0 724.0- 773.4 DNI 446.0- 314.0- 739.9 SUR 643.4 360.1- 267.6 LHC 222.11 678.1 164.5 LRI 173.3 162.1- 714.4 NHC 519.0 564.0 378.21 UAS 981.1 615.0- 175.6 LSI 624.9 235.2- 354.6 KVS 681.0- 840.0- 466.3 RSI 280.0- 730.0 914.3 LOC 890.4 938.1- 627.4 NVS 114.0 982.0 011.3 ATI 919.1 507.0- 795.3 IRC 332.0 725.0- 049.2 EWS 630.2 620.0 187.7 PYC 238.0- 560.0 668.3 NPJ 147.0- 472.0- 159.4 RUT 117.9 106.2- 458.4 EZC 376.1 418.0- 694.5 ROK 910.0 255.0- 384.2 UED 443.0 616.0- 502.2 ASU 945.1 026.1- 982.8 UTL 930.0- 054.0- 819.2 KND 786.1 427.0 421.3 FAZ 219.3 471.1- 093.5 XUL 030.0 985.0- 556.3 PSE 554.0 715.0- 232.11 BMZ 033.5 640.2- 518.9 AVL 924.8 034.2- 341.8 TSE 376.9 982.2- 367.4 XEM 55

Table 10: List of countries ABW=Aruba FRA=France MYS=Malaysia AGO=Angola GAB=Gabon NAM=Namibia AIA=Anguilla GBR=UnitedKingdom NER=Niger ALB=Albania GHA=Ghana NGA=Nigeria ARE=UnitedArabEmirates GIN=Guinea NIC=Nicaragua ARG=Argentina GMB=Gambia NLD=Netherlands ATG=AntiguaandBarbuda GNB=Guinea-Bissau NOR=Norway AUS=Australia GNQ=EquatorialGuinea NPL=Nepal AUT=Austria GRC=Greece NZL=NewZealand BDI=Burundi GRD=Grenada OMN=Oman BEL=Belgium GTM=Guatemala PAK=Pakistan BEN=Benin HKG=China,HongKongSAR PAN=Panama BFA=BurkinaFaso HND=Honduras PER=Peru BGD=Bangladesh HTI=Haiti PHL=Philippines BGR=Bulgaria HUN=Hungary POL=Poland BHR=Bahrain IDN=Indonesia PRT=Portugal BHS=Bahamas IND=India PRY=Paraguay BLZ=Belize IRL=Ireland PSE=StateofPalestine BMU=Bermuda IRN=Iran(IslamicRepublicof) QAT=Qatar BOL=Bolivia(PlurinationalStateof) IRQ=Iraq ROU=Romania BRA=Brazil ISL=Iceland RWA=Rwanda BRB=Barbados ISR=Israel SAU=SaudiArabia BRN=BruneiDarussalam ITA=Italy SDN=Sudan BTN=Bhutan JAM=Jamaica SEN=Senegal BWA=Botswana JOR=Jordan SGP=Singapore CAF=CentralAfricanRepublic JPN=Japan SLE=SierraLeone CAN=Canada KEN=Kenya SLV=ElSalvador CHE=Switzerland KHM=Cambodia STP=SaoTomeandPrincipe CHL=Chile KNA=SaintKittsandNevis SUR=Suriname CHN=China KOR=RepublicofKorea SWE=Sweden CIV=Coted’Ivoire KWT=Kuwait SWZ=Eswatini CMR=Cameroon LAO=LaoPeople’sDR SYC=Seychelles COD=Congo,DemocraticRepublic LBN=Lebanon SYR=SyrianArabRepublic COG=Congo LBR=Liberia TCA=TurksandCaicosIslands COL=Colombia LCA=SaintLucia TCD=Chad COM=Comoros LKA=SriLanka TGO=Togo CPV=CaboVerde LSO=Lesotho THA=Thailand CRI=CostaRica LUX=Luxembourg TTO=TrinidadandTobago CYM=CaymanIslands MAC=China,MacaoSAR TUN=Tunisia CYP=Cyprus MAR=Morocco TUR=Turkey DEU=Germany MDG=Madagascar TWN=Taiwan DJI=Djibouti MDV=Maldives TZA=U.R.ofTanzania: Mainland DMA=Dominica MEX=Mexico UGA=Uganda DNK=Denmark MLI=Mali URY=Uruguay DOM=DominicanRepublic MLT=Malta USA=UnitedStatesofAmerica DZA=Algeria MMR=Myanmar VCT=St. Vincent&Grenadines ECU=Ecuador MNG=Mongolia VEN=Venezuela(BolivarianRepublicof) EGY=Egypt MOZ=Mozambique VGB=BritishVirginIslands ESP=Spain MRT=Mauritania VNM=VietNam ETH=Ethiopia MSR=Montserrat ZAF=SouthAfrica FIN=Finland MUS=Mauritius ZMB=Zambia FJI=Fiji MWI=Malawi ZWE=Zimbabwe 56