Idiosyncratic Risk, Aggregate Risk, and the Welfare Effects of Social Security

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1 Disussion Paper No Idiosynrati Risk, Aggregate Risk, and the Welfare Effets of Soial Seurity Daniel Harenberg and Alexander Ludwig

2 Disussion Paper No Idiosynrati Risk, Aggregate Risk, and the Welfare Effets of Soial Seurity Daniel Harenberg and Alexander Ludwig Download this ZEW Disussion Paper from our ftp server: Die Disussion Papers dienen einer möglihst shnellen Verbreitung von neueren Forshungsarbeiten des ZEW. Die Beiträge liegen in alleiniger Verantwortung der Autoren und stellen niht notwendigerweise die Meinung des ZEW dar. Disussion Papers are intended to make results of ZEW researh promptly available to other eonomists in order to enourage disussion and suggestions for revisions. The authors are solely responsible for the ontents whih do not neessarily represent the opinion of the ZEW.

3 Idiosynrati Risk, Aggregate Risk, and the Welfare Effets of Soial Seurity Daniel Harenberg Alexander Ludwig This Version: Deember 1, 017 Abstrat We ask whether a pay-as-you-go finaned soial seurity system is welfare improving in an eonomy with idiosynrati produtivity and aggregate business yle risk. We show analytially that the whole welfare benefit from joint insurane against both risks is greater than the sum of benefits from insurane against the isolated risk omponents. One reason is the onvexity of the welfare gain in total risk. The other reason is a diret risk interation whih amplifies the utility losses from onsumption risk. We proeed with a quantitative evaluation of soial seurity s welfare effets. We find that introduing an unonditional minimum pension leads to substantial welfare gains in expetation, even net of the welfare losses from rowding out. About 60% of the welfare gains would be missing when simply summing up the isolated benefits. JEL lassifiation: C68; E7; E6; G1; H55 Keywords: soial seurity; idiosynrati risk; aggregate risk; welfare An earlier version of this paper irulated under the title Soial Seurity and the Interations Between Aggregate and Idiosynrati Risk. We thank the editor Harold Cole and three anonymous referees as well as Klaus Adam, Alan Auerbah, Martin Barbie, Tino Berger, Antoine Bommier, Johannes Brumm, Georg Dürneker, Ayşe Imrohoroğlu, David Jaeger, Philip Jung, Tom Krebs, Dirk Krueger, Per Krusell, Felix Kubler, Marten Hillebrand, Mihèle Tertilt, Fabrizio Zilibotti, and various seminar partiipants at several plaes for helpful disussions. Daniel Harenberg gratefully aknowledges finanial support by the German National Researh Foundation (SFB 884), by Swiss Re Foundation, and by ETH Zürih Foundation. Alex Ludwig gratefully aknowledges finanial support by the German National Researh Foundation (SPP 1578) and by the Researh Center SAFE, funded by the State of Hessen initiative for researh LOEWE. Chair of Integrative Risk Management and Eonomis, ETH Zürih; dharenberg@ethz.h. SAFE, Goethe University Frankfurt; ludwig@safe.uni-frankfurt.de. 1

4 1 Introdution Many ountries operate large soial seurity systems. One reason is that soial seurity an provide insurane against risks for whih there are no private markets. However, these systems also impose osts by distorting pries and deisions. The question arises whether the benefits of soial seurity outweigh the osts. We address this question in a model eonomy featuring two types of risk, aggregate business yle risk in form of aggregate wage and asset return risk as well as idiosynrati produtivity risk. We follow the literature and assume that insurane markets for both types of risk are inomplete. In suh a setting, soial seurity an inrease eonomi effiieny by providing partial insurane. However, it also distorts deisions leading to welfare losses from rowding out of apital formation. Our analysis differs from the previous literature in that prior studies haraterized soial seurity s welfare effets in models with only one type of risk. One strand of the literature examined soial seurity when only aggregate risk is present, e.g., Krueger and Kubler (006). In that setting, soial seurity by pooling aggregate wage and asset return risks aross generations an improve intergenerational risk sharing. The other strand only onsidered idiosynrati risk, f., e.g., İmrohoroğlu, İmrohoroğlu, and Joines (1995, 1998) and Conesa and Krueger (1999). There, soial seurity provides intragenerational insurane by redistributing ex-post from high to low produtivity households. Broadly speaking, both strands of this literature onlude that the osts of introduing soial seurity outweigh the benefits. Suh a segregated view is inomplete beause households fae both types of risk over the life-yle and beause soial seurity, when appropriately designed, an (partially) insure both types of risk. We also argue that simply ombining the findings from previous studies leads to severe biases in the overall welfare assessment. Our theoretial ontribution is to show analytially why the whole insurane benefit exeeds the sum of the benefits from insurane against isolated risk omponents. Our quantitative ontribution is to establish that joint insurane against both types of risk leads to large net welfare gains, thereby turning previous findings in the literature upside down: soial seurity is welfare improving from the ex-ante perspetive. We emphasize that two important biases emanate from simply ombining previous findings. The first arises even when the two types of risk are statistially independent. This bias is a onsequene of the onvexity of the welfare gain (CW G) in total risk. The welfare gain is onvex in the amount of total risk beause the marginal utility of insurane inreases disproportionately as risk inreases. Joint presene of idiosynrati produtivity and aggregate business yle risk strongly fans out the earnings and onsumption distributions. If soial seurity is designed as a Beveridgean system with flat pension benefits, it provides partial insurane against this total

5 life-yle risk exposure. Beause of CW G, the whole benefit from insurane is therefore greater than the sum of benefits from insurane against the single risk omponents. We show that this differene in welfare assessments, the CW G bias, inreases in the total amount of risk. Sine total life-yle risk is large, we an expet this bias to be large. The seond bias stems from a diret interation of risks in form of a ounter-ylial ross-setional variane (CCV ) of idiosynrati produtivity risk: the variane of persistent idiosynrati shoks is higher in a downturn than in a boom. The CCV has been doumented in the data (Storesletten, Telmer, and Yaron 004b) and analyzed with respet to its asset priing impliations (Mankiw 1986; Constantinides and Duffie 1996; Storesletten, Telmer, and Yaron 007). 1 It leads to a high variane of the idiosynrati inome omponent when the aggregate inome omponent is low. Due to onavity of the utility funtion this amplifies the welfare gains from insurane against both risks. To expose these biases we start our analysis by employing an analytially tratable twoperiod life-yle model in whih a household faes idiosynrati and aggregate wage risk in the first period of life. In absene of soial seurity, retirement onsumption is finaned by private savings whih bear aggregate return risk. We study the welfare onsequene of introduing a payas-you-go (PAYG) finaned soial seurity system with flat, unonditional pension payments. By pooling idiosynrati wage risks within and aggregate risks aross generations, this Beveridgean system jointly provides partial insurane against idiosynrati and aggregate risks. We measure welfare gains by a onsumption equivalent variation. Abstrating from CCV, we derive a term apturing the welfare differene between the whole insurane benefit and the sum of the benefits from insurane against the isolated risk omponents. This differene reflets the CW G bias. We subsequently modify the two-period model to aount for the CCV mehanism and show how an additional welfare differene emerges. Our arguments so far ignore behavioral reations, i.e., the redution of savings aused by soial seurity. In general equilibrium, this savings reation leads to rowding out of aggregate apital, whih entails orresponding hanges in relative pries. Therefore, both the sign and the size of the welfare effets of introduing soial seurity in a model with both risks have to be determined in a quantitative general equilibrium analysis. To ondut suh a quantitative analysis we build a large-sale overlapping generations model in the tradition of Auerbah and Kotlikoff (1987), extended by idiosynrati produtivity risk and aggregate wage and asset return risk. Households an save privately by investing in a 1 Based on Guvenen, Ozkan, and Song (014) a reent paper by Bush and Ludwig (017) finds that in addition to the variane the skewness of persistent idiosynrati shoks is ounterylial. Adding suh a ounterylial left-skewness would strengthen our results, beause more households would find themselves in situations with high marginal utility. 3

6 risk-free bond and a risky stok. Inluding this portfolio hoie is important. It allows us to appropriately alibrate the risk-return struture of the private savings tehnologies, whih diretly affets the value of soial seurity. The possibility to save in two assets also implies that households have additional means of self-insurane. In our omputational experiment, we onsider a stylized soial seurity reform by introduing a pure Beveridgean PAYG soial seurity system like in our analyti two-period model with a ontribution rate of %. This is the size of the U.S. system when first introdued in We hene study the welfare impliations of introduing a flat rate minimum pension. By alibrating the model to the U.S. eonomy we find that suh a marginal introdution of soial seurity leads to a strong welfare gain of.6% in terms of a onsumption equivalent variation. This welfare improvement is obtained beause strong partial equilibrium insurane gains of 5.% outweigh the substantial welfare losses from rowding out of apital of.6% in general equilibrium. Our key finding of net welfare gains stands in stark ontrast to the previous literature. When instead repliating the earlier literature by onsidering eonomies with only one type of risk we indeed observe net welfare losses. We therefore onlude that it is of ruial quantitative importane to jointly onsider both risks. To unover the soures of the partial equilibrium welfare gain of 5.%, we deompose it into the omponents that are attributable to insurane against the isolated risks as well as the two bias terms, CCV and CW G. We find that the ombined effet of the two bias terms sales up the partial equilibrium welfare gains by 60%. This strong effet reemphasizes our key finding on the quantitative importane of jointly onsidering both risks. Finally, we investigate how muh of the general equilibrium welfare effets stem from hanges in mean onsumption and from hanges in the intra- and intergenerational distribution of onsumption. The notion that soial seurity an insure against aggregate risks dates bak to Diamond (1977) and Merton (1983). They demonstrate how it an partially omplete finanial markets, thereby inreasing eonomi effiieny. Building on these insights, Shiller (1999) and Bohn (001, 009) show that soial seurity an redue onsumption risk of all generations by pooling labor inome and apital inome risks aross generations. Gordon and Varian (1988), Matsen and Thogersen (004), Krueger and Kubler (006), and Ball and Mankiw (007) use a two-period partial equilibrium model in whih households only onsume in the seond period of life, i.e., during retirement. For our analytial results, we extend this model by adding idiosynrati risk. Among the few quantitative papers with aggregate risk and soial seurity, Krueger and Kubler (006) is the most similar to our work. They onlude that the introdution Nearly all OECD ountries feature either a minimum or a basi pension that is independent of previous inome, see OECD (015). 4

7 of a small PAYG system does generally not onstitute a Pareto-improvement. 3 The onept of a Pareto-improvement requires that they take an ex-interim welfare perspetive, whereas we alulate welfare from an ex-ante perspetive. Our analysis also differs substantially beause we inlude idiosynrati risk and analyze interations between the risks. 4 Many quantitative papers onsider idiosynrati risk and soial seurity, e.g., Conesa and Krueger (1999), İmrohoroğlu, İmrohoroğlu, and Joines (1995, 1998), Huggett and Ventura (1999), and Storesletten, Telmer, and Yaron (1999). One general onlusion from this literature is that welfare in a stationary eonomy without soial seurity is higher than in one with a PAYG system. More reently Nishiyama and Smetters (007), Fehr and Habermann (008), and Golosov, Shourideh, Troshkin, and Tsyvinski (013) fous on modeling the institutional features of existing soial seurity systems in detail, whih we abstrat from. Our results demonstrate the benefits of a flat minimum pension. Like all these papers, we ondut a limited poliy design experiment by restriting attention to insurane through the soial seurity system, taking insurane through taxes and transfers during the working period as given. 5 We derive our analytial results in Setion. Setion 3 desribes the quantitative model, Setion 4 presents the alibration and Setion 5 the main results of our quantitative analysis. We onlude in Setion 6. Proofs as well as omputational and alibration details are relegated to separate appendies. A Two-Generations Model.1 Model In eah period t, a ontinuum of households is born who live for two periods only. A household has preferenes over onsumption in the seond period. In the first period of life, a household experienes an idiosynrati produtivity shok, denoted by η. This shok indues ex-post heterogeneity and we denote ex-post different households by i. Age is indexed by j, with j = 1 being working age and j = being retirement. Denoting by i,,t+1 onsumption in retirement, 3 The reent work by Hasanhodzi and Kotlikoff (015) mirrors these findings. 4 Other related papers are Ludwig and Reiter (010) who assess how pension systems should optimally adjust to demographi shoks, Olovsson (010) who ontends that pension payments should be highly risky beause this inreases preautionary savings and thereby apital formation, Peterman and Sommer (015, 016) who disuss the insurane benefits of soial seurity in the Great Depression and the Great Reession, respetively, modeling eah event as a one-time maroeonomi shok, and, finally, Gomes, Mihaelides, and Polkovnihenko (01) who use a model similar to ours to study how hanges in fisal poliy and government debt affet asset pries and the wealth distribution. 5 Huggett and Parra (010) point out the importane of analyzing the optimal design of soial seurity and the inome tax system jointly. 5

8 the expeted utility of a household born in period t is E t [u( i,,t+1 )]. We assume a CRRA per period utility funtion with oeffiient of relative risk aversion θ, u( i,,t+1 ) = 1 θ i,,t θ Gross wage inome is given by η i,1,t w t, where w t is the aggregate and η i,1,t is the idiosynrati, stohasti wage omponent. Wage inome is subjet to soial seurity ontributions at rate τ. During retirement, the household reeives a flat pension inome, y ss t+1. Aordingly, the budget onstraints are given by s i,,t+1 = (1 τ)η i,1,t w t and i,,t+1 s i,,t+1 R t+1 + y ss t+1, (1) where s i,,t+1 denotes gross savings and R t+1 = 1 + r t+1 is the risky gross interest fator. While ontributions depend on the idiosynrati shok η i,1,t, retirees reeive the same flat pension payment, y ss t+1. Thus, soial seurity provides partial intragenerational insurane against idiosynrati risk. We denote by ζ t the shok to aggregate wages and by ϱ t the shok to returns. 6 We further assume that wages grow deterministially at rate λ. Denoting by R and w t the deterministi omponents of returns and wages we aordingly get: w t = w t ζ t = w t 1 (1 + λ)ζ t and R t = Rϱ t. () Abstrating from population growth, 7 the balaned budget of the pure PAYG system reads τw t = y ss t. (3) From equations () and (3) one an see that soial seurity provides partial intergenerational insurane against aggregate risk if ζ t and ϱ t are imperfetly orrelated.. Analysis The CWG Bias. We analyze the welfare effets of introduing a marginal soial seurity system of size dτ > 0 under the following assumptions: Assumption 1. All shoks η i,1,t, ζ t, ϱ t : (a) are distributed log-normal with means µ ln η, µ ln ζ, µ ln ϱ and varianes σ ln η, σ ln ζ, σ ln ϱ, (b) have a mean of one: Eζ = Eϱ = Eη = 1, () are unorrelated over time, and (d) are statistially independent from eah other. 6 In this setion, we limit the analysis to a partial equilibrium, and hene wages and returns are exogenous. 7 Our quantitative model also features population growth. 6

9 Assumptions 1a-b are frequently employed for analytial tratability. Assumption 1 an be justified by the long periodiity of eah period in a two-period overlapping generations model of approximately years. Assumption 1d is important to illustrate the CW G. Below, we relax it to extend the model by the CCV. To evaluate welfare, we adopt an ex-ante perspetive. The soial welfare funtion of a ohort born in period t is the unonditional expeted utility of a generation, E [u( i,,t+1 )]. We study the onsumption equivalent variation (CEV) from a marginal introdution of soial seurity, whih is the perentage inrease in onsumption, g, required to make the household indifferent between being born into an eonomy without soial seurity (τ = 0) and with a small soial seurity system (τ = dτ > 0). We inlude a supersript PE for partial equilibrium to remain onsistent with the subsequent quantitative analysis, whih onsiders a general equilibrium. We also index the CEV by AR and IR to indiate presene of aggregate and idiosynrati risk, respetively. We an now state our first proposition, whih we prove in Appendix A: Proposition 1. Under Assumption 1, the onsumption equivalent variation from a marginal introdution of soial seurity is given by g P E (AR, IR) = 1 + λ R where σ ln AR σln ζ + σ ln ϱ. Therefore, g P E inequality being strit for σln η > 0 σln AR > 0. ( 1+λ R exp ( θ ( )) σln AR + σln η 1 dτ (4) }{{} Ψ(σln AR,σ ln η ) (AR, IR) g P E (AR, 0) + g P E (0, IR) with the To interpret this proposition, first onsider a deterministi eonomy, where g P E (0, 0) = 1) dτ. This reflets the well-known Aaron (1966) ondition, i.e., soial seurity inreases welfare in a deterministi eonomy if (and only if) its impliit return exeeds the market rate of return, (1 + λ) > R. In the non-degenerate stohasti ase where σ ln η > 0 σ ln AR > 0, term Ψ aptures the welfare benefits from intergenerational and intragenerational (partial) insurane provided by the system. Ψ is (i) inreasing in risk aversion θ, refleting the standard intuition that more risk-averse households value insurane more; (ii) inreasing in σ η beause soial seurity pools histories of idiosynrati earnings risk; (iii) inreasing in σ ϱ beause pension payments are not affeted by return risk; (iv) inreasing in σ ζ beause soial seurity redues exposure to the wage shok, ζ, when young and inreases it when old; 8 (v) onvex in 8 Sine ζ is unorrelated over time, mixing ζ t and ζ t+1 by having τ (0, 1) is welfare improving. 7

10 total risk, σ ln AR + σ ln η. This last finding is entral to our analysis. 9 As a onsequene of the onvexity, the whole welfare gain is greater than the sum of the gains from insurane against individual risk omponents. We denote the welfare differene attributable to the onvexity of the welfare gain by CW G. To further haraterize it, we provide the following formal definition: Definition 1 (Components of CEV). The ontributions to g P E (AR, IR) attributable to idiosynrati and aggregate risk are defined as dg P E dg P E (IR) = g P E (0, IR) g P E (0, 0), (AR) = g P E (AR, 0) g P E (0, 0), so that the CEV an be written as g P E (AR, IR) = g P E (0, 0) + dg P E (AR) + dg P E (IR) + CW G. Under Assumption 1 we an express Ψ in terms of varianes of levels instead of varianes of logs: Ψ(σ AR, σ η ) ( 1 + σ η + σ AR + σ ησ AR) θ, where σar σ ζ + σ ϱ + σ ζ σ ϱ. Employing Definition 1 for logarithmi utility (θ = 1), the CEV writes as g P E (AR, IR) ( ) 1 + λ θ=1 = R 1 dτ λ R σ ARdτ λ }{{}}{{} R σ ηdτ }{{} g P E (0,0) dg P E (AR) θ=1 dg P E (IR) θ=1 θ= λ R σ ARσηdτ } {{ } CW G θ=1 For logarithmi utility, the CW G is aordingly diretly related to the produt of varianes of aggregate and idiosynrati risk. By providing a flat, unonditional transfer, soial seurity redues the variane of retirement onsumption, thereby reduing exposure to eah risk omponent as well as their multipliative interation. 10 As we show formally in Appendix B, dg P E (AR), (IR), and CW G are inreasing in risk aversion θ, so that for θ > 1, the ontribution of dg P E eah omponent in the equation above onstitutes a lower bound on welfare gains.. Modifiation: The CCV Bias. We alter Assumption 1 by onditioning the variane of idiosynrati produtivity risk on the aggregate state of the eonomy while keeping its unonditional 9 The finding mirrors an important result from the literature on the welfare osts of aggregate flutuations, namely that the welfare gain of insuring against aggregate risk is a onvex funtion of risk, f. Luas (1978), De Santis (007), and Krebs (007). Relative to this literature we study the effets of joint insurane and therefore total risk is the sum of the risk omponents. 10 Retirement onsumption in the absene of soial seurity is given by w t Rηi,1,t ζ t ϱ t+1. Its variane is ( w t R) var(η i,1,t ζ t ϱ t+1 ) = ( w t R) (ση + σar + σ ησar ), beause the shoks are independent and have a mean of one, f. Goodman (1960). 8

11 variane equal to σ ln η. Fousing on logarithmi utility we extend Definition 1 by the CCV : Assumption. (a) Let ζ t {ζ, ζ + } for all t, with ζ ± = χ exp(1 ± σ lnζ ) > 0 and probabilities π(ζ t = ζ + ) = π(ζ t = ζ ) = 1, where χ is a normalizing onstant. Let η i,1,t be distributed as log-normal with onditional variane σln η(ζ t = ζ + ) = σln η + η, and σ ln η(ζ t = ζ ) = σ ln η η. The rest of Assumption 1 ontinues to hold. (b) Utility is logarithmi, i.e., θ = 1. Definition (Components of CEV with CCV ). The ontribution to the CEV with CCV, g P E (AR, IR, CCV ), that is attributable to CCV is defined as CCV = g P E (AR, IR, CCV ) (AR, IR). Hene, the total CEV with CCV an be written as g P E g P E g P E (0, 0) + dg P E (AR) + dg P E (IR) + CW G + CCV. We an now state our next result. The proof is provided in Appendix A. Proposition. Under Assumption and using Definition we get (AR, IR, CCV ) = g P E (AR, IR, CCV ) = ( 1 + λ R exp ( ) ( σln 1 ϱ exp ( ) σ 1 ln η ζ l + exp ( ) ) ) σln η ζ h 1 dτ + (5a) and CCV = 1 + λ R exp ( ( ) 1 σln ϱ η 1 ) dτ > 0. ζ ζ + (5b) Equation (5a) is the analogue to equation (4) for disrete ζ and inluding CCV. Equation (5b) shows the inrease of welfare gains through the CCV mehanism. This is due to the fat that the CCV raises (redues) the variane of idiosynrati produtivity risk in states where average onsumption already tends to be low (high). Sine utility is onave, this mehanism inreases the value of soial seurity. The amplifiation of welfare is stronger the larger aggregate risk (σln ζ and σln ϱ) and the larger the variane shifter, η..3 Extensions Harenberg and Ludwig (015) provide an extension of the simple model with utility from first period onsumption in general equilibrium to analytially derive a number of additional important insights, whih we summarize and extend in Appendix B.. This shows, first, how life-yle and preautionary savings are redued in response to the soial seurity reform, leading to rowding out of apital. Seond, it shows that the biases in the welfare assessment 9

12 of rowding out are ambiguous. While rowding out beomes stronger with more risks, this does not only redue the deterministi omponent of wages, it also redues exposure to wage risk beause wage risk positively depends on the size of the apital stok. 11 Third, we unover the importane of disounting: the lower the disount rate, the more relevant are the welfare benefits from insurane against seond period onsumption risk and the lower are the welfare osts of rowding out. In addition to the insights we worked out in our simple two period model, these aspets will play ruial roles in our quantitative analysis to whih we turn next. 3 The Quantitative Model 3.1 Time, Risk, and Demographis Time is disrete and runs from t = 0,...,. At the beginning of eah period t, an aggregate shok z t hits the eonomy. For a given initial z 0, a date-event is uniquely identified by the history of shoks z t = (z 0, z 1,..., z t ) where the z t follow a Markov hain with finite support Z and nonnegative transition matrix π z. Thus, π z (z t+1 z t ) represents the probability of z t+1 given z t. At every point in time t, the eonomy is populated by J overlapping generations indexed by j = 1,..., J. We denote the size of a generation by N j (z t ). Eah generation onsists of a ontinuum of households. We normalize the initial population size to unity, i.e., J j=1 N j (z 0 ) = 1. Population grows at the exogenous rate of n. To keep the analysis foused we abstrat from survival risk. 1 Total population at t is therefore N(z t ) = (1 + n) t. Households within a ohort are ex-ante idential but reeive an idiosynrati shok e j eah period so that there is ex-post intragenerational heterogeneity. We denote by e j the history of idiosynrati shoks with probability π e (e j ). We assume that e j follows a Markov hain with finite support E and stritly positive transition matrix π e. The transition probabilities are π e (e j+1 e j ) In general equilibrium, wages inrease in apital and shoks are multipliative in wages. Suh peuniary effets play key roles for welfare in heterogeneous agent eonomies, f. Davila, Hong, Krusell, and Ríos-Rull (01) and Krueger and Ludwig (017). Also, Harenberg and Ludwig (015) fous on log utility and therefore the saving rate does not reat to hanges of the apital stok. Furthermore, for analytial tratability, the model features a degenerate distribution of households who are all ex-ante idential. Both aspets play additional important roles for the welfare effets of rowding out, see our disussion of the quantitative results in Setion In presene of survival risk, soial seurity an be benefiial if it partially substitutes for missing annuity markets. Caliendo, Guo, and Hosseini (014) demonstrate that this may not hold beause soial seurity rowds out aidental bequests. Also, it is not straightforward to jointly model survival risk and finanial risk with Epstein-Zin preferenes, see Bommier, Harenberg, and Le Grand (017). 13 By a Law of Large Numbers π e (e j ) represents both the individual probability for e j and the fration of the population with that shok history. Likewise, π e (e j+1 e j ) represents both the individual transition probability and its population ounterpart. 10

13 3. Households At any date-event z t, a household is fully haraterized by its age j and its history of idiosynrati shoks e j. Denote by u j (, e j, z t ) the expeted remaining life-time utility from onsumption alloation at age j, history e j, and date-event z t. Preferenes are represented by a reursive utility funtion u j (, ) of the Epstein-Zin-Weil kind (Epstein and Zin 1989, 1991; Weil 1989): 14 u j (, e j, z t ) = 1 [ j (e j, z t ) ] 1 θ γ +β π z (z t+1 z t )π e (e j+1 e j ) [ u j+1 (, e j+1, z t+1 ) ] γ 1 θ z t+1 e j+1 γ 1 θ, u J (, e J, z t ) = J (e J, z t ), > 0, where β is the disount fator and θ ontrols risk aversion. Parameter γ is defined as γ 1 θ 1 1 ψ with ψ denoting the intertemporal elastiity of substitution. Households inelastially supply one unit of labor until they retire at the fixed retirement age j r. They are endowed with a deterministi life-yle produtivity profile ɛ j. The idiosynrati, stohasti omponent of inome, η(e j, z t ), depends on the realization of idiosynrati and aggregate shoks. The dependene of η(e j, z t ) on the aggregate shok is neessary to model the CCV. We assume that E (η(e j, z t ) z t ) = 1. Labor inome is y j (e j, z t ) = w(z t )ɛ j η(e j, z t ), where w(z t ) is the real aggregate wage in terms of the onsumption good at z t. Insurane markets for labor inome risk are losed by assumption. Households an transfer wealth between periods by holding stoks and bonds in amounts s j+1 (e j, z t ) and b j+1 (e j, z t ), respetively. The stok has a risky return r s (z t+1 ) that depends on the realization of the aggregate shok in the following period, whereas the bond pays a one-period risk-free interest rate r b (z t ). The sequential budget onstraint is standard: j (e j, z t ) + s j+1 (e j, z t ) + b j+1 (e j, z t ) = (1 + r s (z t ))s j (e j, z t ) + (1 + r b (z t 1 ))b j (e j, z t ) + (1 τ)y j (e j, z t )I(j) + y ss (z t )(1 I(j)), where τ is a fixed soial seurity ontribution rate, y ss (z t ) is pension inome, and I(j) is an indiator funtion that takes the value 1 if j < j r and 0 otherwise. 15 Households annot die in 14 In a slight abuse of notation, we use letter u to denote remaining lifetime utility in this reursive formulation, whih was used in Setion to denote the per-period utility funtion. 15 We do not onsider an exogenous borrowing onstraint. This may bias results in favor of soial seurity beause inome (and asset) poor households an relax their budget onstraint. With an exogenous borrowing 11

14 debt, s J+1 (e J, z t ) + b J+1 (e J, z t ) 0. Sine there are no bequests, households are born with zero assets, i.e., s 1 (e 1, z t ) = b 1 (e 1, z t ) = Firms There is a representative firm that produes output, Y (z t ), using apital, K(z t ), and labor, L(z t ). The prodution tehnology is Cobb-Douglas with apital share α and deterministi laboraugmenting produtivity growth λ. At eah date-event there is a multipliative shok to total fator produtivity, ζ(z t ), so that we have Y (z t ) = ζ(z t )K(z t ) α ((1 + λ) t L(z t )) 1 α. Assuming a stohasti depreiation rate δ(z t ), 16 the apital stok evolves aording to K(z t ) = I(z t 1 ) + K(z t 1 )(1 δ(z t 1 )). Beause of perfet ompetition, the firm remunerates the fators of prodution aording to their marginal produtivities. Thus, the aggregate wage, w(z t ), and the return on apital, r(z t ), are given by ( w(z t ) = (1 + λ) t K(z t ) α ) (1 α)ζ(z t ), (6a) (1 + λ) t L(z t ) ( (1 + λ) r(z t t L(z t ) 1 α ) ) = αζ(z t ) δ(z K(z t t ). (6b) ) The apital stok is finaned by issuing stoks and bonds in quantities S and B, so that K(z t ) = S(z t ) + B(z t ) = S(z t )(1 + κ f ). The debt-equity ratio, κ f, is exogenous and onstant. Therefore, the firm only deides on aggregate apital and not on the apital struture. 17 This mehanial leverage allows us to keep the depreiation shoks, whih drive stok return volatility, small in the alibration. This is desirable, beause large depreiation shoks imply unrealistially large flutuations on the real side of the eonomy. As derived in Appendix B.5, the leveraged stok return is ( r s (z t ) = r(z t ) + κ f r(z t ) r b (z t 1 ) ), (7) whih shows that leverage inreases mean and variane of stok returns. 18 onstraint it would be natural to modify the soial seurity system to have a progressive ontribution rate with an exemption for inome poor households. 16 The same assumption is employed by Storesletten, Telmer, and Yaron (007), Gomes and Mihaelides (008), and Krueger and Kubler (006), among others. 17 Leverage is frequently modeled this way in the finane literature to inrease the volatility of stok returns, f., e.g., Boldrin, Christiano, and Fisher (1995) and Croe (014). 18 As Gomes and Mihaelides (008) point out, the empirial equity premium is for levered firms. Our model is onsistent with this target, whereas standard models should rather ompare to an unlevered empirial ounterpart. 1

15 3.4 Soial Seurity Soial seurity is organized as a PAYG system just like in the two-generations model of Setion. Denoting by P (z t ) the number of pensioners, P (z t ) = J j=jr N j (z t ), the budget onstraint aordingly reads as τw(z t )L(z t ) = y ss (z t )P (z t ) Equilibrium We study a ompetitive general equilibrium, where households and firms maximize and all markets lear. The orresponding value funtion of the household is denoted v j ( ). In the omputational solution, we fous on reursive Markov equilibria. We express all aggregate variables in terms of labor effiieny units, i.e., we divide aggregate variables by (1+λ) t L(z t ) = (1 + λ) t j r 1 j=1 ɛ j N j (z t ). The orresponding normalized variable is written in lower ase, e.g., k(z t ) = K(zt ). Individual variables are detrended only by the level of tehnology, and (1+λ) t L(z t ) the orresponding variables are denoted with a tilde, e.g., j ( ) = j( ). Aordingly, the (1+λ) t monotone transformation of the value funtion is denoted by ṽ j ( ). Sine the model has (ex-post) heterogeneous households and aggregate unertainty, the distribution of households beomes part of the state spae. We denote by Φ the distribution of households over age, urrent inome state, stoks, and bonds. The orresponding equilibrium law of motion, Φ = H(Φ, z, z ), is indued by household s optimal hoies and the exogenous shok proesses. 0 Every period there are five markets that lear: onsumption good, apital, labor, stoks, and bonds. A preise definition of the reursive Markov equilibrium is relegated to Appendix B. 3.6 Computational Solution We ompute an equilibrium of our model by applying the Krusell and Smith (1998) method. 1 To approximate the law of motion of the distribution, H(Φ, z, z ), we onsider various foreast funtions, Ĥ, of the average apital stok and the ex-ante equity premium and selet the one with the best fit. The average goodness of fit of the seleted approximate law of motion is in the range of R = 0.99 for all of the alibrations. The state spae is further redued by one dimension by reasting the problem in terms of ash-on-hand. To speed up the solution, we 19 By the balaned budget, intergenerational sharing of aggregate risk is limited to generations alive at the same point in time. It may be desirable to also share this risk with future, unborn generations. This ould be ahieved by adding a soial seurity trust fund to the model. 0 Next period s aggregate shok z is an element of the law of motion beause it determines the distribution of next period s idiosynrati inome states. 1 Also see, e.g., Storesletten, Telmer, and Yaron (007) and Gomes and Mihaelides (008). 13

16 employ a variant of the endogenous grid method (Carroll 006) that allows for two ontinuous hoies. Details of the omputational solution are provided in Appendix C. 3.7 Welfare Criterion and Dynami Effiieny Soial Welfare Funtion. We employ the same welfare onept as in the two-generations eonomy of Setion, namely ex-ante expeted utility of a household at the start of eonomi life. As explained in Davila, Hong, Krusell, and Ríos-Rull (01), in an eonomy with ex-ante idential but ex-post heterogeneous agents, this onept represents a natural objetive for a soial planner who is behind the Rawlsian veil of ignorane. It is a Utilitarian welfare riterion, whih weighs lifetime utilities with their respetive probabilities. In our model with aggregate shoks, this riterion means that we evaluate the expeted life-time utility of many different households that are randomly born into a state of an eonomy and then form the welfare index by taking the unonditional average of these households expeted life-time utility. Formally, a household s ex-ante expeted welfare of being born into an eonomy with poliy A an be written as the unonditional expetation E [ ṽ 1 ( A, e 1, z t ) ], where the expetation is taken over all date-events z t. It is an expetation over all possible equilibrium values of aggregate apital and indued pries. Analogous to Setion, when omparing poliy A to poliy B we express the welfare differene of two suh ex-ante welfare measures in terms of a onsumption equivalent variation, g. As we prove in Appendix B.5, it is given by g = E [ ṽ 1 ( B, e 1, z t ) ] E [ṽ 1 ( A, e 1, z t )] 1. (8) A positive number indiates the perentage of lifetime onsumption a household would be willing to give up under poliy A in order to be born into an eonomy with poliy B. We ompare the long-run welfare effets of suh a reform. While this does not inlude the transition between the two eonomies, it is important to understand that for the experiment desribed below (an introdution of soial seurity), inluding the welfare effets along the transition would inrease g. The reason is that moving from poliy A to poliy B implies a gradual derease in apital. Thus, generations that live through the transition experiene the full benefits from insurane but are spared some of the long-run welfare osts of rowding out. Therefore, by ignoring the transition, we alulate a lower bound on the welfare effets. Dynami Effiieny. In our eonomy, there are two soures for ineffiienies. One is missing insurane markets against aggregate and idiosynrati risk, the other is the possibility of an 14

17 ineffiient intergenerational alloation of mean onsumption aross generations even when insurane markets are omplete. The latter is known as dynami ineffiieny, whih an arise in OLG models (Samuelson 1958; Diamond 1965). In a dynamially ineffiient eonomy an intergenerational realloation of resoures from the young to the old through a PAYG pension system an help to ure this ineffiieny. In our experiments we want to fous on dynamially effiient eonomies to avoid making a normative ase for soial seurity that is not based on a partial ompletion of missing asset markets for insuring idiosynrati and aggregate risk. To this aim, we hek the dynami effiieny riterion of Demange (00), Theorem 1, whih applies to stohasti eonomies suh as ours and very general notions of effiieny (e.g., ex-ante effiieny). Speifially, we ompute the onditions proposed by Krueger and Kubler (006) in Proposition 1, whih are suffiient onditions for Demange s effiieny riterion. While these onditions an be onveniently evaluated numerially, they may be far from neessary onditions. We restate them in the following definition whih is adapted to our notation. Definition 3 (Dynami effiieny, Krueger and Kubler (006)). Suppose that a) whenever the one period risk-free interest rate, r b (z t ), is larger than the impliit average soial seurity return, (1+n)(1+λ) 1, then there exist two next period states z t+1, z t+1 Z suh that (i) next period s bond returns in the orresponding date-events are above the impliit return next period, i.e., r b ( z t+1 ) > (1 + n)(1 + λ) 1 and r b ( z t+1 ) > (1+n)(1+λ) 1, and (ii) the stok return satisfies r s ( z t+1 ) > r b (z t ) and r s ( z t+1 ) < r b (z t ), and b) from any initial equilibrium state, a high interest rate r b (z t ) > (1 + n)(1 + λ) 1 is reahed in finite time. If onditions a) and b) are fulfilled, the eonomy is dynamially effiient. This definition of dynami effiieny implies that eonomies an be dynamially effiient even if the average bond return is less than the average impliit soial soial seurity return. It is ruial to understand that bond returns and impliit soial seurity returns are flutuating in our quantitative model. While the average bond return may be less than the average impliit soial seurity return, ondition (a) states that in equilibrium there need to exist states with high bond returns and the eonomy needs to stay in suh a state with positive probability. Condition (b) in turn says that suh a state with a high bond return must be reahed in finite time. Details on the numerial implementation an be found in Appendix C.6. 15

18 3.8 Experiment and Deomposition Analyses Experiment. In terms of the previous setion, our omputational experiment onsists of omparing poliy A, whih has a soial seurity ontribution rate of τ = 0%, to poliy B, whih has τ = %. It an be interpreted as the introdution of a marginal soial seurity system in form of a minimum pension, as in Setion. We then ompute the welfare gains from this poliy reform by omparing two long-run equilibria. If the introdution of soial seurity leads to a welfare improvement despite the fat that the eonomy is dynamially effiient, then this must be a onsequene of the partial ompletion of markets through soial seurity. As in the simple model of Setion, this partial ompletion of markets dereases the onsumption variane. It also leads to behavioral adjustments through redued savings, f. Setion.3, and inreased stok holdings, both of whih tend to inrease average onsumption. Disentangling these effets is ruial for understanding our quantitative results whih we ahieve with the deomposition analyses desribed next. Gains from Insurane and Losses from Crowding Out. Our first deomposition of the general equilibrium welfare effets aims at disentangling the effets of welfare gains in partial equilibrium from those indued by the rowding out of apital, as in our general equilibrium extension of the simple model, f. Setion.3. We thereby also disentangle the long run (= general equilibrium) from the short run (= partial equilibrium) welfare effets. In our partial equilibrium experiment, the soial seurity system hanges, but pries, i.e., wages and returns, remain unaffeted. Coneptually, this orresponds to a small open eonomy with free movement of the fators of prodution. To formalize this, denote by P A = {{z t, r(z t ), r s (z t ), r b (z t ), w(z t )} t=0 τ = 0%} the sequene of shoks and pries obtained from the general equilibrium of the eonomy without a soial seurity system, i.e., under poliy A (τ = 0%). Likewise, denote by ĤA the approximate law of motion of this equilibrium. We ompute the partial equilibrium under the old prie sequene P A and the old law of motion ĤA, but with poliy B (τ = %). The welfare gains stemming from insurane are then: 3 g P E = E [ṽ1 ( B, e 1, z t ) P A, ĤA, τ = % ] E [ ṽ 1 ( A, e 1, z t ) P A, ĤA, τ = 0% ] 1. (9) 3 We need to take into aount the approximate law of motion ĤA in this definition beause households form their expetations based on the laws of motion. 16

19 Analogously, the orresponding general equilibrium number is g GE = E [ṽ1 ( B, e 1, z t ) P B, ĤB, τ = % ] E [ ṽ 1 ( A, e 1, z t ) P A, ĤA, τ = 0% ] 1, (10) where the ruial differene is that in the new equilibrium with poliy B (τ = %), households optimize given the new general equilibrium pries and laws of motion, P B, ĤB. The welfare osts of rowding out (CO) are given by the differene g CO = g GE g P E. To relate the osts of rowding out to our onept of dynami effiieny in Setion 3.7, notie that dynami (in)effiieny refers to the mean alloation of onsumption aross generations. In our model, however, there is also a dispersion of onsumption around the mean whih is indued by different idiosynrati shok histories. Therefore, from the ex-post perspetive, households may gain or loose from a derease of the apital stok beause depending on eah idiosynrati shok history and resulting asset position either the negative aggregate wage or the positive aggregate return effet dominates, f., e.g., Kuhle (01). From the ex-ante perspetive the question whether there is too muh or too little apital in the eonomy then depends on the weight a household reeives in the respetive welfare riterion, f. Davila, Hong, Krusell, and Ríos-Rull (01). The redution in the apital stok ould therefore by itself lead to an inrease in welfare even in a dynamially effiient eonomy, meaning that g CO > 0. In our results, we never enountered this ase; this is why we speak of welfare osts from rowding out. Welfare Impliations of Changes in the Mean and the Distribution of Consumption. equivalent variations g GE, g P E, g CO enompass two effets. One is the welfare impliation of poliy-indued hanges of mean onsumption alloations, the mean effet, the seond is the welfare impliation from a hange in the intra- and intergenerational distribution of onsumption, the distribution effet. We deompose the total CEV into these effets by omputing the welfare hange due to a hange of the distribution as (see Appendix B.5): g P E,distr g GE,distr = E [ CA P A, ĤA, τ = 0% ] E [ CB P A, ĤA, τ = % ] = E [ CA P A, ĤA, τ = 0% ] E [ CB P B, ĤB, τ = % ] ( ) 1 + g P E 1, The ( ) 1 + g GE 1, (11) where C A ( C B ) is aggregate, growth-adjusted onsumption under poliy regime A (B). The respetive differenes g P E,mean = g P E g P E,distr and g GE,mean = g GE g GE,distr are then the 17

20 equivalent variations apturing the welfare impliations of hanges in mean onsumption. In partial equilibrium, insurane through soial seurity redues both the intragenerational onsumption distribution as well as the onsumption growth rate over the life-yle beause preautionary savings go down (intergenerational distribution). Additional distributional hanges arise in general equilibrium beause the rowding out of apital auses inreasing returns and dereasing wages, the welfare impat of whih we apture by g CO,distr = g GE,distr g P E,distr. Likewise, the mean effet of rowding out, i.e., the hange in mean onsumption due to a hange in equilibrium pries, is omputed as g CO,mean = g GE,mean g P E,mean. Soures of Partial Equilibrium Welfare Gains. Finally, we deompose g P E into the effets attributable to insurane against aggregate risk, idiosynrati risk, as well as the two biases, CW G and CCV, respetively, as in our simple model of Setion. Realling our deomposition of the CEV in Definitions 1 and we have: g P E g P E (AR, IR, CCV ) = g P E (0, 0) + dg (AR) + dg (IR) + CW G + CCV (AR, IR) = g P E (0, 0) + dg (AR) + dg (IR) + CW G g P E (0, IR) = g P E (0, 0) + dg (IR) g P E (AR, 0) = g P E (0, 0) + dg (AR). The right-hand side of the first line shows all of the omponents. To isolate those, we ompute g P E (AR, 0) and g P E (0, 0), as in equation (9), but for an eonomy with only aggregate risk and one without risk, respetively. 4 With those numbers at hand, we an bak out the welfare effet attributable to aggregate risk, dg (AR). Likewise, we ompute g P E (0, IR) for an eonomy featuring only idiosynrati risk to bak out dg (IR). Next, we ompute g P E (AR, IR). As we already know dg (AR) and dg (IR), we an bak out the CW G. In the same manner, we obtain CCV. While we are mainly interested in the overall effet attributable to the respetive risk omponent, we further deompose those into the respetive welfare effets of hanges in the mean and the distribution of onsumption. 4 Calibration The seletion of targets and parameters to be alibrated is informed by our theoretial insights, in partiular Propositions 1 and, as well as Setion.3. Aordingly, the oeffiient of relative 4 As shown in Appendix B.4, g P E (0, 0) an be alulated from the present disounted value of lifetime inome, independent of preferene parameters. 18

21 risk aversion, θ, the varianes of the shoks, the returns on savings and the disount fator are ruial in determining the value of soial seurity. Guided by this, our baseline alibration takes a very onservative approah, in the sense that it features a low θ and small aggregate shoks. In the sensitivity analysis of Setion 5.3, we then first inrease θ to math the Sharpe ratio, ς = E[rs,t r b,t] σ[r s,t r b,t], and then aggregate shoks to math the equity premium, µ = E [r s,t r b,t ]. For the disount fator our target is also onservative and we report results with a less onservative target in Appendix E.3. One set of parameters, the set of first-stage parameters, is determined exogenously by either taking its value from other studies or measuring it in the data. The seond set of parameters is jointly alibrated by mathing the model-simulated moments to their orresponding moments in the data. Aordingly, we refer to those parameters as seond-stage parameters. 5 Table 1 summarizes our onservative baseline alibration, desribed next. 6 Additional information on our empirial approah to measure alibration targets and on the numerial implementation of the proedure is provided in Appendies C and D, respetively. 4.1 Demographis Households begin working at the biologial age of 1, whih orresponds to j = 1. We set J = 58, implying a life expetany at birth of 78 years, whih is omputed from the Human Mortality Database (HMD) for year 007. We set j r = 45, orresponding to a statutory retirement age of 65. Population grows at a rate of 1.1%. 4. Households In our baseline alibration, we treat the oeffiient of risk aversion as a first-stage parameter, setting it to 3, whih is well within the standard range of [, 4]. 7 The intertemporal elastiity of substitution is set to 0.5. This is at the lower end of the range of values used in the literature, as reviewed, e.g., by Bansal and Yaron (004). A higher value of the elastiity of substitution means that households reat more strongly to prie hanges. As a onsequene, welfare losses from rowding out are lower, as shown in our sensitivity analysis in Setion 5.3. In our baseline 5 The seond-stage parameters jointly determine all targeted moments. When we say that we alibrate a parameter to a target, we mean that it has the strongest impat on that target. 6 For lak of better data on the period when the soial seurity system was introdued in the United States in 1935 with a ontribution rate of % (the data analogue to our thought experiment), we take averages for postwar data for alibration. In Setion 5.3 we report results when the ontribution rate is also set to its postwar average of 9.5%. 7 Given this hoie, our model produes a Sharpe ratio of ς = and an equity premium of µ = 0.76%, well below their empirial ounterparts of 0.33 and 5.60%, whih we expliitly target in Setion

22 Table 1: Summary of the Baseline Calibration Parameter Value Target (soure) Stage Demographis Biologial age at j = st Model age at retirement, jr 45 Statutory retirement age of 65 (SSA) 1 st Model age maximum, J 58 Life expetany of 78 years (HMD) 1 st Population growth, n U.S. Soial Se. Admin. (SSA) 1 st Households Disount fator, β Capital output ratio,.65 (NIPA) nd Coeffiient of relative risk aversion, θ st Intertemporal elastiity of substitution, ψ st Age produtivity, {ɛj} Cf. Appendix D Estimates based on PSID data 1 st Autoorrelation of log earnings, ρ Estimates based on PSID data 1 st Variane of persistent shok, σ υ(z) {0.04, 0.008} Estimates based on PSID data 1 st Variane of transitory shok, σ ε Estimates based on PSID data 1 st Firms Capital share, α 0.3 Wage share (NIPA) 1 st Leverage ratio, κf 0.66 Rajan and Zingales (1995) 1 st Tehnology growth, λ TFP growth (NIPA) 1 st Mean depreiation rate of apital, δ Bond return, 0.03 (Shiller) nd Aggregate Risk Standard deviation of depreiation, δ Std. of onsumption growth, (Shiller) nd Aggregate produtivity states, 1 ± ζ 0.09 Std. of TFP, 0.09 (NIPA) 1 st Transition probabilities of produtivity, π ζ Autoorrelation of TFP, 0.88 (NIPA) 1 st Conditional prob. of depreiation shoks, π δ Corr.(TFP, returns), 0.50 (NIPA, Shiller) nd Notes: 1 st stage parameters are set exogenously, nd stage parameters are jointly alibrated to the targets. 0

23 alibration, the disount fator β is alibrated to math the apital-output ratio of.65, whih we alulate from NIPA data, f. Appendix D.. 8 We obtain β = 0.987, whih is a reasonable estimate for a model at an annual frequeny suh as ours. The parametrization of the labor inome proess is based on household earnings data from the PSID applying the proedure of Bush and Ludwig (017). Our earnings measure exludes soial seurity ontributions but inludes all other taxes and transfers. 9 The age speifi produtivity profile ɛ j is extrated from the deterministi omponent of the earnings proess displayed in Appendix D.1. Calibration of the stohasti omponent η(e j, z t ) is derived from the estimates of the proess log(η i,j,t ) = ξ i + ν i,j,t + ε i,j,t, ε i,j,t N ( ) 0, σε, (1a) ν i,j,t = ρν i,j 1,t 1 + υ i,j,t, υ i,j,t N ( 0, συ(z t ) ), (1b) where the variane of the persistent shok, συ(z t ), depends expliitly on the aggregate state. The estimated value of the autoorrelation oeffiient is ρ = The estimated onditional variane of the persistent shok, συ(z t ), is 0.04 in reessions and in booms. 30 The estimated variane of idiosynrati shoks is σε = We approximate the AR(1) proess using the Rouwenhorst method, f. Kopeky and Suen (010), and approximate the transitory omponent ε j,t by Gaussian quadrature (for details see Appendix C). 4.3 Firms We set the value of the apital share parameter to α = 0.3. This is diretly estimated from NIPA data on total ompensation as a fration of GDP. Our estimate of the deterministi trend growth rate is based on data on total fator produtivity. The point estimate is λ = 0.018, whih is in line with other studies. Leverage in the firm setor is set to κ f = 0.66 (Rajan and Zingales 1995). The mean depreiation rate of apital, δ 0, is a seond-stage parameter. We alibrate it to math an average bond return of.3%. 31 In eonomies without aggregate risk we alibrate δ 0 to produe a risk-free return of 4.%, orresponding to the empirial estimate of Siegel (00). 8 Our estimate is in line with the estimates of, e.g., Fernández-Villaverde and Krueger (011). 9 We thank Christopher Bush for providing us with the estimates. 30 See Appendix D.1 for the identifiation of reessions and booms in the data and Setion 4.4 for the orresponding definition in the model. 31 The empirial bond return, equity premium, et., are alulated from the data on Robert Shiller s website, see 1

24 4.4 Aggregate Risk Aggregate risk is driven by a four-state Markov hain with support Z ={z 1,..., z 4 } and transition matrix π z. Eah aggregate state maps into a ombination of a total fator produtivity (TFP) shok and a depreiation shok, (ζ(z), δ(z)). Both shoks an take a high and a low value, given by ζ(z) = 1 ± ζ and δ(z) = δ 0 ± δ. We define reessions as the low TFP states z {z 1, z }, where ζ(z) = 1 ζ. The transition probability of remaining in a low TFP state is π ζ. To govern the orrelation between TFP and depreiation shoks, we let the probability of the high depreiation state onditional on the low TFP state be π δ. Assuming symmetry of the transition probabilities, the Markov hain of aggregate shoks is haraterized by four parameters, ( ζ, δ, π ζ, π δ ), see Appendix D.3 for details. We set ζ and π ζ to math the standard deviation and autoorrelation of TFP of 0.09 and 0.88, both estimated using NIPA data. The remaining parameters, δ and π δ, are alibrated as seond-stage parameters to jointly math the standard deviation of aggregate onsumption growth of 0.03 and the orrelation of the ylial omponent of TFP with risky returns of 0.5. We get ζ = 0.09, δ = 0.080, π ζ = 0.941, π δ = Results 5.1 Baseline Calibration Dynami Effiieny. We first report the results of heking the two onditions for dynami effiieny of Definition 3 before the introdution of soial seurity. 3 Table shows that about 40% of the simulated periods have a high bond return (larger than the average soial seurity return). Suh a high bond state is reahed from any simulated initial ondition in finite time with a maximum of 10 periods so that ondition (b) is satisfied. Conditional on being in suh a high bond return state, we hek ondition (a) and find that it is violated about 5% of the time. Sine the onditions of Definition 3 are suffiient, but not neessary, we an have at least 95% onfidene that the baseline eonomy is dynamially effiient. Aggregate Effets and Welfare Consequenes. The effets of introduing soial seurity at a ontribution rate of % on apital aumulation, pries and welfare are doumented in Table 3. Our experiment leads, on average, to a long-run redution in the apital stok of 11.61%, whih is aompanied by a 3.8% redution in gross wages, an inrease in the return on stoks of 0.99 perentage points, and an inrease in the return on bonds of 1.01 perentage points. The average 3 We obtain similar results for the baseline eonomy with soial seurity (τ = %), as well as for our other alibrations. Details are provided in Appendix E.1.

25 Table : Dynami Effiieny of Baseline Eonomy, τ = 0% Condition (a) Condition (b) High Bond Returns Conditional violation Max. periods Avg. periods Simulated periods 38.1% 4.7% Notes: Dynami effiieny onditions of Definition 3. High bond returns: fration of high bond return states in whih 1 + r b (z t ) > (1 + n)(1 + λ). Conditional violation: Violation of onditions (a)(i) and (a)(ii), onditional on being in a high bond return state. Avg., resp. max., periods: average, resp. maximum, number of simulation periods to reah a high bond return state. Number of total simulated periods is after disarding a phase-in period. return on bonds inreases to a greater extent, beause the insurane provided through soial seurity leads households to rebalane their portfolios towards stoks. This redues relative demand for bonds, dereasing their prie and inreasing their return. Table 3: Aggregate Effets of The Soial Seurity Experiment Variable Change Aggregate apital, K Aggregate wage, w Stok return, r s Bond return, r b Consumption equivalent variation K/K = 11.61% w/w = 3.8% r s = +0.99pp r b = +1.01pp g GE = +.56% Notes: X/X is the expeted perent hange in variable X between two steady states, i.e., X/X = E(Xt τ=%) E(Xt τ=0%) E(X t τ=0%). x is the hange in variable x expressed in perentage points (pp), i.e., x = E(x t τ = %) E(x t τ = 0%). g GE is the onsumption equivalent variation in general equilibrium, f. Setion 3.8. Table 3 also reports the onsumption equivalent variation, g GE, as defined in equation (10). The reform yields a CEV of.6% despite the sizeable rowding out of apital. This onstitutes a substantial welfare gain from a minimum pension at a ontribution rate of %. Conditional Distribution of Welfare Gains. We now report the distribution of the CEV onditional on the household being born into a reession or a boom. That is, we ompute the CEV for eah history of aggregate shoks, g,t GE 1, thereby E[ṽ 1 ( A,e 1,z t ) z t,p A,ĤA,τ=0%] omparing a household being born into an eonomy with soial seurity to a household being = E[ṽ 1( B,e 1,z t ) z t,p B,ĤB,τ=%] born into an eonomy without soial seurity, before they learn their idiosynrati shoks. 3

26 Figure 1 shows the distribution of g,t GE for reessions (z t {z 1, z }) in Panel (a) and booms (z t {z 3, z 4 }) in Panel (b). First, notie that the CEVs are always positive. Seond, as ontributions to soial seurity imply higher utility osts in reessions when inomes are already low and as aggregate shoks are persistent, CEVs are on average higher in booms (with an average of.83%) than in reessions (with an average of.5%). Furthermore, the distribution of CEVs is left-skewed in reessions and right-skewed in booms. Figure 1: Distribution of CEV: Reessions and Booms (a) Reessions (b) Booms Benefits from Insurane versus Costs from Crowding Out. Where do these substantial welfare gains ome from? To provide an answer, we first deompose the total welfare gain into the benefits from insurane and the losses from rowding out by onduting the partial equilibrium (PE) experiment desribed in Setion 3.8. Aordingly, the sequenes of wages and returns before and after the introdution of soial seurity are idential. As a onsequene, the CEV in this experiment reflets purely the benefits from insurane. Subtrating this number from the overall welfare gain reported in Table 3 yields the losses from rowding out. As Table 4 reveals, the net welfare gains attributable to the total insurane provided by soial seurity amount to +5.% and the losses from rowding out stand at.6%. Welfare Impliations of Changes in the Mean and the Distribution of Consumption. Table 5 reports the results of our welfare deomposition of the numbers in Table 4 into the mean effet, g mean, and the distribution effet, g distr, as desribed in Setion 3.8. Turn first to the distribution effet. The gains from a redution of the dispersion of onsumption are large, 4