THE IMPORTANCE OF THE NUMBER OF DIFFERENT AGENTS IN A HETEROGENEOUS ASSET-PRICING MODEL WOUTER J. DEN HAAN

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1 THE IMPORTANCE OF THE NUMBER OF DIFFERENT AGENTS IN A HETEROGENEOUS ASSET-PRICING MODEL WOUTER J. DEN HAAN Department of Economcs, Unversty of Calforna at San Dego and Natonal Bureau of Economc Research July 1999 Models wth heterogeneous agents and ncomplete markets used n the lterature often only have two types of agents to lmt the computatonal complexty. The queston arses whether equlbrum models wth a realstc number of dfferent types have the same mplcatons as models wth a small number of types. In the asset-prcng model consdered n ths paper, several propertes depend crucally on the number of types. For example, n the economy wth only two types nterest rates respond to dosyncratc ncome shocks whch makes t easer to smooth consumpton. Moreover these effects can be so strong that t s possble that a relaxaton of the borrowng constrant reduces an agent s utlty n the economy wth two types. Average nterest rates on the other hand are not very senstve to the number of types. The author would lke to thank Jonas Fsher, Lars Hansen, Ken Judd, Martn Lettau, Per Krusell, Valere Ramey, Tony Smth, Harold Uhlg and an anonymous referee for useful comments.

2 1 1. INTRODUCTION. Dynamc equlbrum models wth heterogeneous agents and ncomplete markets have become popular tools n the macro asset-prcng lterature. Heterogenety usually arses because an agent s ncome s affected not only by aggregate but also by dosyncratc ncome shocks, but other forms of heterogenety have also been consdered. 1 These models have been shown to be an mprovement over standard representatve agent models n several dmensons. For example, they provde an explanaton for the low level of real nterest rates observed n the US data. 2 The reason s that the presence of dosyncratc rsk provdes an addtonal ncentve to save whch lowers real nterest rates. Many models analyzed n the lterature only have two dfferent types of agents. 3 In these models, ndvdual specfc shocks are not truly dosyncratc snce all agents of the same type receve the same shock. One could nterpret an agent s dosyncratc ncome shock n an economy wth two types as a sector specfc shock. Ths has two dsadvantages. Frst, although moral hazard can be used to explan why contracts that are contngent on an ndvdual s ncome realzaton do not exst, t cannot be used to rule out contracts contngent on the performance of the sector. The second dsadvantage of nterpretng the dosyncratc shocks as sector specfc shocks s that the varablty of the average ncome n a sector s a lot smaller than the varablty of ndvdual ncome. Wthout havng a substantal amount of dosyncratc uncertanty the predctons of heterogeneous agent models are not that dfferent from models wth a contnuum of agents. Papers that have analyzed models wth two agents and ncomplete markets deserve a lot of credt for mprovng the predctons of general equlbrum models. Usng recently developed algorthms to accurately solve models wth a large number of dfferent agents 4 we can now answer the queston whether models wth a large number of types also mply low real nterest rates and whether the qualtatve and quanttatve propertes are dfferent from the models wth two types of agents that are studed n the lterature. In ths paper, I analyze an nfnte-horzon equlbrum model of the short-term nterest rate smlar to that used n the lterature except that t has a contnuum of types nstead of two types. Markets are assumed to be ncomplete, but agents can smooth ther consumpton by tradng n rsk-free bonds. For the parameter values consdered n ths paper, average nterest rates n the model wth a large number of dfferent agents are very smlar to average nterest rates n the correspondng model wth only two types of agents. In the presence of borrowng constrants, the model wth a contnuum of types, thus, also predcts low average real nterest rates. Several other propertes of the model studed n ths paper, however, depend crucally on the number of types. 1 For example, Judd, Kubler, and Schmedders (1998) and Krusell and Smth (1998) consder heterogeneous preferences. 2 Cf. Wel (1992). 3 Examples are Heaton and Lucas (1992, 1996), Lucas (1994), Marcet and Sngleton (1999), Telmer (1993), and Zhang (1993). Alternatvely, t s assumed that there s no aggregate uncertanty and a contnuum of agents. In ths case asset prces are constant over tme. See, for example, Ayagar (1994) and Ayagar and Gertler (1991). 4 Cf. Den Haan (1996,1997), Gaspar and Judd (1997), Krusell and Smth (1997, 1998), and Ros-Rull (1996). See Ros-Rull (1997) for an overvew.

3 2 The frst mportant dfference between the two models s that the amount of cross-sectonal wealth dsperson s a lot more volatle n the economy wth only two types because n economes wth only two types dosyncratc shocks affect the amount of cross-sectonal dsperson. Ths paper documents how and why these dfferences n the tme-seres propertes of the amount of cross-sectonal wealth dsperson can affect the tme-seres propertes of nterest rates, consumpton, and bond purchases n asset-prcng models wth heterogeneous agents. Not surprsngly, the hgher volatlty of crosssectonal dsperson n the economes wth two types mples a much hgher volatlty of nterest rates as well. Heaton and Lucas (1996) menton that Typcally n models that ft the equty premum, the resultng volatlty of the bond return s too hgh. Ths ssue has only been addressed n economes wth two types. In ths paper I show that economes wth a large number of dfferent agents can have very low average real nterest rates wthout havng an excessvely hgh volatlty for bond returns. Another mportant dfference between the economy wth two types and the economy wth a contnuum of dfferent types s the ablty of the agent to nsure hmself aganst dosyncratc shocks. In the economy studed n ths paper, t s easer to smooth consumpton when there are only two types then when there are a contnuum of types because when there are only two types the nterest rate responds to dosyncratc shocks. The explanaton s the followng. Consder the case where an agent receves a low realzaton of the dosyncratc shock for several perods. In the economy wth two types ths means that half the populaton receves a low realzaton for several perods. Ths ncreases the amount of crosssectonal dsperson and ths causes the nterest rate to drop. Ths reducton n the nterest rate due to dosyncratc shocks works lke a transfer from the rch agents (the lenders) to the poor agents (the borrowers). In an economy wth a contnuum of types, an agent who faces the same sequence of low ncome realzatons s not so lucky to see the nterest rate drop at the same tme. Consequently hs consumpton decreases by more than the consumpton of the agent n the economy wth only two types. These effects are magnfed when one of the two agents s at the borrowng constrant. Moreover, they can be so strong that a relaxaton of the borrowng constrant can actually reduce an agent s welfare n an economy wth two types. The organzaton of ths paper s as follows. In the next secton, an nfnte-horzon endowment economy wth ncomplete markets wll be dscussed. In Secton 3, I analyze the propertes of the economy wth a contnuum of types and those of the economy wth two types. The last secton concludes. 2. AN EQUILIBRIUM MODEL WITH HETEROGENEOUS AGENTS. In ths secton, I develop an nfnte horzon model of the short-term nterest rate smlar to the models n Deaton (1991) and Pschke (1995). In contrast to Deaton (1991) and Pschke (1995), the nterest rate s not constant but vares to ensure that the bond market s n equlbrum. In Secton 2.1 I dscuss the optmzaton problem of the ndvdual agent. In Secton 2.2, I dscuss the equlbrum condton, and n Secton 2.3 I dscuss the parameter values used.

4 THE INDIVIDUAL AGENT S PROBLEM. Ex ante agents are exactly the same, but ex post they dffer due to the presence of dosyncratc shocks. In partcular, the endowment of agent relatve to the per capta endowment, y t, can take on a low value, y L, and a hgh value, y H. The (gross) growth rate of the aggregate endowment, a t = A t /A t-1, can also take on a low (or recesson) value, a R, and a hgh (or boom) value, a B. 5 Both processes are assumed to be frst-order Markov processes. In an economy n whch all shocks are observed wthout costs, t would be optmal to wrte contracts contngent on the realzaton of the dosyncratc shock. If t s mpossble for the lender to verfy the realzaton of the borrower s ncome, the borrower would always report that he receved the lowest possble realzaton. In ths case, the optmal one-perod contract s a bond wth a fxed payment. 6 It s assumed here that agents can only smooth ther consumpton by tradng n a one-perod rsk-free bond. Markets are, thus, ncomplete. Agent s maxmzaton problem s as follows: max { Ct, Bt } t= 0 t = 0 β s.t. t 1 γ t C 1 1 γ (2.1) C + q B = y A + B 1, t t t t t t where C t s the amount of consumpton of agent n perod t, B t s the demand for one-perod bonds that pay one unt of the consumpton commodty n the next perod, q t s the prce of ths one-perod bond, and B 1 s gven. The agent takes the bond prce as gven. The nterest rate, r t, s defned as (1-q t )/q t. The queston arses whether a lender would want to restrct the amount he s wllng to lend to a borrower. It would make sense to lmt the amount of debt by the net present value of the borrower s endowment stream. In ths economy, n whch agents lve and receve an endowment stream for ever, ths constrant s unlkely to serously restrct the demand for loans. In realty, however, consumers and frms do seem to face borrowng constrants. 7 More restrctve borrowng constrants arse n our model f one makes the assumpton that the borrower faces lmted costs of defaultng on the loan. Suppose that the default costs are equal to B t. In ths case, borrowers wll default whenever -B t B t. Ths, of course, mples that the lender wll never lend more than B t. I assume that B t (scaled by the per capta endowment) s a constant. Thus, the followng borrowng constrant s added to the maxmzaton problem: (2.2) B b A. t t In ths paper, statonary varables are denoted by small characters and non-statonary varables are denoted by captal characters. Wang (1995) shows that contracts contngent on the realzaton of the shock are possble when mult-perod contracts are enforceable. See Jaffee and Stgltz (1990, Secton 5.3) and the references theren.

5 4 Note that B t ncludes the nterest payments on the bond. Let B ~ ( = q B ) denote the amount t t t of savngs net of nterest payments. Then Equaton (2.2) can be rewrtten as follows (2.3) ~ B t b A 1 + r t t. Equaton (2.3) s a typcal formulaton of a borrowng constrant. 8 It has the property that the maxmum amount an agent can borrow s decreasng wth the nterest rate. The frst-order condtons for the maxmzaton problem of the agent are the two-part Kuhn-Tucker condtons: (2.4) t t γ t t γ + 1 q [ C ] β E [ C ] and t t t t γ t t γ ( B + b A ) ( q [ C ] β E [ C ] ) =. The equatons of ths model can easly be transformed to a system of equatons that contans only statonary varables. To see ths, defne z t = Z t /A t for any non-statonary varables Z t. Equatons (2.1), (2.2) and (2.4) can then be wrtten as (2.5) ct + qt bt = yt + bt 1 / at, (2.6) bt b, (2.7) t t γ t t γ + 1 t+ 1 q [ c ] β E [ c a ], and γ γ ( β E + + ) ( b + b ) q [ c ] [ c a ] =. t t t t t 1 t EQUILIBRIUM CONDITION. Two dfferent versons of the model are consdered. In both versons there are a contnuum of agents wth unt mass. In the frst verson, the dosyncratc draws, y t, are dstrbuted ndependently across agents and across tme. In the second verson, there are only two types of agents and and each agent always receve the same realzaton of the dosyncratc shock as the other agents of the same type. Note that n the economy wth two types y 1 2 t t 1 2 t t and y are perfectly negatvely correlated snce y + y = 2. Ths restrcton wll be dscussed n detal n the next secton. Let F L H t and F t be the cumulatve dstrbuton functon of the cross-sectonal begnnng-ofperod bond holdngs of the agents who receve the low ncome shock and the hgh ncome shock, respectvely. In the economy wth only two types, all agents who receve the low realzaton for the dosyncratc shock have the same amount of begnnng-of-perod bond holdngs snce they all have the same hstory of dosyncratc shocks. The same s true for the agents who receve the hgh value. Consequently, df L and df H have mass at only one level of b t-1 n the economy wth two agents. In contrast, n the economy wth a contnuum of types, there wll be a wde varety of bond holdngs at each pont n tme. 8 See, for example, Bernanke, Gertler, and Glchrst (1996).

6 5 The state varables of agent are bt 1, yt, and the aggregate state varables a t. Agent s demand for bond holdngs s assumed to be a functon of the state varables. 9 The equlbrum condton for the bond market s gven by (2.8) b / at b / at L t 1 t t L t 1 b( b, y, a ) d F ( b ) H t 1 t t H t 1 = 0 b( b, y, a ) d F ( b ) + In the economy wth two types, nformaton about the dosyncratc shock and bond holdngs of one type dsclose the correspondng values of the other type. The set of aggregate state varables ( a t ) s, thus, equal to the growth rate of the aggregate endowment, a t. In the economy wth a contnuum of dfferent agents, the aggregate state varables are a t, the cross-sectonal dstrbuton of the bond holdngs of the low-ncome agents, and the cross-sectonal dstrbuton of the bond holdngs of the hgh-ncome agents. A soluton to the model, n ths case, conssts of a consumpton functon, c( bt 1, yt, at, Ft L, Ft H ), an nvestment functon, b( bt 1, yt, at, Ft L, Ft H ), a bond prce functon, q( at, Ft L, Ft H ), and the functonals F L ( a, a 1, F 1, F 1 ) and F H ( a, a 1, F 1, F 1 ) that descrbe the law of moton of F L t and t t t L t H t t t L t H F t H, respectvely. The only reason why the current value of a t s an argument of the transton functonals s that the begnnng-of-perod bond holdngs are scaled relatve to the per capta endowment. For example, suppose that n perod t-1 an agent borrows the maxmum amount b A t-1. Thus, b t-1 = - b. Ths means that n perod t, begnnng-of-perod bond holdngs, relatve to the aggregate endowment, are equal to - b /a t. Several papers n the lterature dscuss the exstence of equlbrum. Exstence of equlbrum wth borrowng constrants n an economy lke ours but wth a fnte number of agents s dscussed n Magll and Qunz (1994). Den Haan (1997) compares the numercal soluton to the model wth a contnuum of agents wth the smulated results of an economy wth 100,000 agents and fnds that the results are very smlar. Ths suggests that there are no dfferences between an economy wth a contnuum of agents and an economy wth a large but fnte number of agents. Related papers are Duffe, Geanokoplos, Mas-Colell, and McLennan (1994), Levne (1989), and Levne and Zame (1993) PARAMETER VALUES. The tme perod n the model corresponds to a year and the dscount rate s set equal to Asset prces and consumpton behavor crucally depend on the assumed values for the degree of relatve rsk averson, γ, and the borrowng constrant parameter, b. I, therefore, consder values for γequal to 1, 3, and 5 and values for b rangng from 0.2 to See Judd, Kubler, and Schmedders (1998) for a dscusson on state varables n ths type of model.

7 6 The parameter values of the stochastc drvng processes for A t and y t are dentcal to those used n Heaton and Lucas (1996). Heaton and Lucas (1996) obtaned estmates for these processes usng the ncome seres from the Panel Studes of Income Dynamcs (PSID) under the assumpton that both processes are two-state frst-order Markov processes. The two values for the dosyncratc shock 10, y t, are and and the two values for the aggregate (gross) growth rate, a t, are and The probablty of recevng n ths perod the same dosyncratc shock as was receved n the last perod s equal to and the probablty of recevng the same aggregate shock s equal to Note that the tme-seres specfcatons for A t and y t are the same n both economes. The amount of dosyncratc uncertanty as well as the amount of aggregate uncertanty, therefore, does not depend on the number of types n the model. In both economes they match the correspondng emprcal estmates. It s mportant to understand that matchng the emprcal value for the amount of dosyncratc uncertanty as well as the emprcal value for the amount of aggregate uncertanty mposes restrctons on the jont dstrbuton of Y 1 2 t and Yt n the economy wth two agents. To understand ths consder the case where there s no aggregate uncertanty and A t = A n every perod. 11 To ensure that A t = A, t must be the case n the economy wth two agents that Y t 1 2 t and Yt j t are perfectly negatvely correlated. In contrast, n the economy wth a contnuum of agents Y and Y can be ndependent for j. A law of large numbers guarantees that A t = A. If one would assume that Y 1 2 t and Yt are ndependent n the economy wth only two agents as well then the amount of aggregate uncertanty would depend on the number agents. In ths case, the propertes of the model would depend on the number of agents even when markets are complete because the amount of aggregate uncertanty depends on the number of agents. 3. THE PROPERTIES OF DIVERSE AND LESS DIVERSE ECONOMIES. In ths secton, I analyze the propertes of the two equlbrum models descrbed n Secton 2. Agents n the two economes face exactly the same amount of dosyncratc and aggregate endowment rsk. The jont tme-seres propertes dffer consderably, however. In the economy wth two types, an agent s dosyncratc shock s perfectly postvely correlated wth the shocks of half of the populaton. In the economy wth a contnuum of dfferent agents, an agent s dosyncratc shock s dstrbuted as an ndependent random varable. In Secton 3.1, I analyze the tme-seres propertes of the nterest rate. In Secton 3.2, I analyze the tme-seres propertes of consumpton and bond purchases n the two economes. In Secton 3.3, I analyze the ablty to nsure aganst dosyncratc shocks n the two economes. 10 The reported values are two tmes those reported n Heaton and Lucas (1996) snce Heaton and Lucas (1996) defne y t as the share of the aggregate endowment one agent receves, whle ths paper defnes y t as the endowment relatve to the per capta endowment to ensure that the defnton also makes sense n the case wth a contnuum of types. 11 Snce the amount of aggregate rsk s small relatve to the amount of dosyncratc rsk ths example s not that dfferent from the actual model used.

8 INTEREST RATES IN DIVERSE AND LESS DIVERSE ECONOMIES. In ths secton, I analyze the behavor of the nterest rate n the economy wth two types as well as n the economy wth a contnuum of types. Important for the tme-seres propertes of the nterest rate are the tme-seres propertes of the amount of cross-sectonal dsperson and the relatonshp between the amount of cross-sectonal dsperson and the nterest rate. In Secton 3.1.1, I dscuss the relatonshp between the amount of cross-sectonal dsperson and the nterest rate. In Secton 3.1.2, I analyze the behavor of nterest rates n the two models. Fgure 1: Cross-Sectonal Dsperson and the Interest Rate AMOUNT OF CROSS-SECTIONAL DISPERSION NOTE: Ths fgure plots the nterest rate as a functon of the low-ncome agent s begnnng-of-perod debt n the economy wth two types. As the debt of the low-ncome agent ncreases, the amount of cross-sectonal dsperson ncreases. The borrowng constrant parameter s equal to 1.4 and the parameter of relatve rsk-averson s equal to 3. The graph s drawn for the low realzaton of the aggregate growth rate The Amount of Cross-Sectonal Dsperson and the Interest Rate. In ths secton, I dscuss the relatonshp between the amount of cross-sectonal dsperson and the nterest rate. An example of ths relatonshp s gven n Fgure 1 that plots the nterest rate as a functon of the low-ncome agent s begnnng-of-perod debt n the economy wth two types. Note that as the debt of the low-ncome agent ncreases, the amount of cross-sectonal dsperson ncreases. The decreasng concave relatonshp between the nterest rate and the amount of cross-sectonal dsperson depcted n Fgure 1 s robust to changes n the parameter values. Ths relatonshp can be easly understood when one recalls the property that the margnal propensty to save s ncreasng n wealth. 12 Consder a wealth transfer from a poor agent to a rch agent. In response to ths transfer the poor agent would want to save less (or borrow more) and the rch agent would want to save more. Snce the rch agent s margnal propensty to save s hgher than the poor agent s margnal propensty to save, the nterest rate has to decrease to keep the bond market n equlbrum. Moreover, the effect becomes smaller 12 Carroll and Kmball (1996) showed that ths property holds n ntertemporal optmzaton models under very general condtons.

9 8 when the amount of cross-sectonal dsperson decreases,.e., the relatonshp s concave. To understand ths, consder the extreme case when there s no cross-sectonal dsperson. In ths case, the margnal propenstes to save are equal for all agents and a margnal wealth transfer would ncrease the amount of dsperson but would have no effect on the nterest rate. The effects are magnfed n the presence of borrowng constrants snce borrowng constrants reduce the margnal propensty to save of the less wealthy even further. When a poor agent has reached hs debt lmt, a wealth transfer from ths agent to a wealther agent requres a large drop n the nterest rate snce the new nterest rate has to be such that the wealther agent s wllng to consume the total wealth transfer. Ths explans the knk n Fgure 1. At the knk, the poor agent has reached hs borrowng constrant and the slope of the functon sharply decreases Tme-Seres Behavor of the Interest Rate. In ths secton, I analyze the tme-seres behavor of the nterest rate n an economy wth two types of agents and n an economy wth a contnuum of dfferent types. In partcular, I consder the average, the standard devaton, the autocorrelaton coeffcent, and the correlaton wth the growth rate of the aggregate endowment. I start wth the averages of the bond prce and the nterest rate. Fgure 2 plots the average bond prce n both economes as a functon of the borrowng constrant parameter. I choose to plot the average bond prce because the dfferences are bgger for the bond prce than for the nterest rate. 13 Nevertheless, the results are remarkably smlar for most parameter values. Ths s good news for several papers n the asset-prcng lterature that manly focused on average returns. 14 As documented n the graph, some dfferences can be found when the borrowng constrant parameter s low and the parameter of relatve rsk averson s hgh. When the borrowng constrant parameter s equal to 0.2 and the parameter of relatve rsk averson s equal to three, for example, then the average bond prce n the economy wth economy wth two types s 1.4 percent hgher than the average bond prce n the economy wth a contnuum of types. 15 For many purposes these dfferences are of lttle mportance. When one wants to know the effect of a change n the borrowng constrant parameter from 0.2 to 2.0, for example, then the economy wth two types predcts a decrease n the average bond prce of 13.3 percent and the economy wth a contnuum of types predcts a decrease of 12.0 percent. The results from the last subsecton provde an ntutve explanaton for the fndng that the average nterest rate (bond prce) s lower (hgher) n the economy wth economy wth two types. Snce the nterest rate s a concave functon of the amount of cross-sectonal dsperson and the amount of crosssectonal dsperson s consderably more volatle n economes wth two types of agents, Jensen s 13 Jensen s nequalty mples that the dfferences are smaller for the average nterest rates than for average bond prces because the nterest rate s a concave functon of the bond prce. 14 Examples are Heaton and Lucas (1996), Lucas (1994), and Telmer (1993). 15 Not shown n the graph s the result that when the borrowng constrant parameter s equal to 0.2 and the parameter of relatve rsk averson s equal to fve, then the dfference n average bond prces s equal to 2.9 percentage ponts.

10 9 nequalty suggests that the average nterest rate should be lower n the economy wth economy wth two types. 16 Fgure 2: The Average Bond Prce n Dverse and Less Dverse Economes economy wth two types and hgh rsk averson low rsk averson contnuum of types & hgh rsk averson BORROWING CONSTRAINT PARAMETER NOTE: Ths graph plots the average bond prce as a functon of the borrowng constrant parameter. The borrowng constrant parameter ndcates the maxmum amount an agent s allowed to borrow relatve to the per capta endowment. The parameter of relatve rsk averson s equal to one (three) n the case of low (hgh) rsk-averson. The other parameter values are reported n Secton 2.3. Populaton moments are approxmated usng the moments calculated wth a smulated draw of 100,000 observatons. Table 1 documents the dfferences across the two types of economes for the followng tme seres statstcs of the nterest rate: the standard devaton, the frst-order autocorrelaton coeffcent, and the correlaton wth the growth rate of the aggregate endowment. The followng observatons can be made from the table. Frst, bg dfferences are observed when one compares the standard devaton of nterest rates across the two types of economes. Ths can be explaned by the large dfferences n the volatlty of the amount of cross-sectonal dsperson across the two types of economes. Heaton and Lucas (1996) report that n models wth two types of agents t s not possble to ft the equty premum wthout bond returns beng excessvely volatle. Heaton and Lucas (1996) report a standard devaton for US bond returns equal to As documented n Table 1, for values of γ and b that produce low average nterest rates (.e. hgh values of γ and low values of b ), the volatlty n the economy wth two types s ndeed much hgher than what s observed n the data. In the correspondng economes wth a contnuum of agents, however, the volatlty of nterest rates s never larger than what s observed n the data. Second, the autocorrelaton of the nterest rate s much hgher and much closer to the observed autocorrelaton n the economy wth two types. The reason s that n ths type of economy the nterest rate s nfluenced by realzatons of the dosyncratc shock whch are much more persstent than shocks to the 16 Note that ths s not a formal argument. One reason s that characterzng the amount of cross-sectonal dsperson s much more complex n the economy wth a contnuum of agents. For example, n the economy wth two types the amount of begnnng-of-perod bond holdngs of the low ncome agents completely descrbes the amount of cross-sectonal dsperson. In the economy wth a contnuum of types ths s not the case. Also, even f the same measure of cross-sectonal dsperson s used, the relatonshp between the nterest rate and the amount of cross-sectonal dsperson does not have to be the same n the two types of economes. Fnally, the average amount of crosssectonal dsperson may dffer across the two types of economes.

11 10 aggregate growth rate. Of course, ths doesn t seem lke a plausble explanaton for the observed persstence. Table 1: Tme-Seres Behavor of the Interest Rate. Standard Devaton Autocorrelaton Correlaton wth a t γ b types 2 types types 2 types types 2 types NOTE: Ths table reports tme-seres statstcs of the nterest rate on a one-perod bond. γdenotes the parameter of relatve rsk averson and b ndcates the amount the agent s allowed to borrow relatve to the per capta endowment. The other parameter values are reported n Secton 2.3. Populaton moments are approxmated usng the sample moments of a smulated draw of 100,000 observatons. Thrd, the correlaton wth the growth rate of the aggregate endowment s close to one n the economy wth a contnuum of types because ths random varable s the only shock that affects the nterest rate. In the economy wth only two types ths coeffcent s much lower because the nterest rate s also affected by dosyncratc shocks. Note that n the presence of complete markets, the nterest rates would be exactly the same n both types of economes. Consstent wth ths s the observaton that the dfferences across the two types of economes become smaller when the borrowng constrant parameter ncreases, that s, when the fnancal frctons become smaller CONSUMPTION AND BOND HOLDINGS In ths secton, I analyze the tme-seres behavor of consumpton and bond purchases n the economy wth two types and the behavor of these varables n the economy wth a contnuum of types. The consumpton varable that I focus on s the percentage change n ndvdual consumpton, ln C t = ln (c t A t ). Snce bond purchases can take on negatve values and values close to zero, t does not make sense to use the percentage change. To study the behavor of bond purchases I, therefore, focus on the

12 11 amount of bonds purchased relatve to the per capta endowment, b t = B t /A t. Ths varable has the dsadvantage that ts uncondtonal covarance wth any aggregate varable s by constructon equal to zero because agents are ex ante dentcal. The statstcs consdered for the consumpton varable are the standard devaton, the correlaton wth the ndvdual endowment growth rate, the correlaton wth the aggregate consumpton growth rate, and the correlaton wth the nterest rate. The statstcs consdered for the bond purchases are the standard devaton, the frst-order autocorrelaton coeffcent, the correlaton wth the dosyncratc endowment shock, and the fracton of tmes the agents s at the constrant. The results are reported n Tables 2 and 3 for consumpton and bond purchases, respectvely. The followng observatons can be made. Frst, n both types of economes, the correlaton between ndvdual consumpton growth and ncome growth decreases and the correlaton between ndvdual and aggregate consumpton growth ncreases when the borrowng constrant parameter ncreases. Results not reported here show that the amount of seral correlaton n consumpton growth reduces when the borrowng constrant parameter ncreases. These fndngs, thus, document that as fnancal frctons weaken agents are better able to smooth consumpton. Second, the correlaton of consumpton growth wth the nterest rate n the economy wth two types s much smaller than the correlaton n the economy wth a contnuum of types. Gven the much hgher volatlty of nterest rates n the economy wth two types, t s no surprse that the correlaton coeffcents dffer across economes. Thrd, the qualtatve change n the tme-seres statstcs n response to changes n the parameter values s very smlar. For example, the correlaton between ndvdual consumpton growth and the nterest rate ncreases n both types of economes wth an ncrease n the borrowng constrant parameter, although the amount of correlaton dffers substantally n both types of economes. Table 2: Tme-Seres Behavor of Indvdual Consumpton Growth. Standard Devaton Correlaton wth ln(y t ) Correlaton wth a t Correlaton wth r t γ b types 2 types types 2 types types 2 types types 2 types NOTE: Ths table reports the tme-seres statstc for the percentage change n ndvdual consumpton growth, ln C t. The varable ln Y t denotes the percentage change n the ndvdual endowment. a t s the growth rate of aggregate endowment whch equals the growth rate of aggregate consumpton. r t s the nterest rate. γdenotes the parameter of relatve rsk averson and b ndcates the amount the agent s allowed to borrow relatve to the per capta endowment. The other parameter values are reported n Secton 2.3. Populaton moments are approxmated usng the sample moments of a smulated draw of 100,000 observatons.

13 12 Table 3: Tme-Seres Behavor of Indvdual Bond Purchases. Standard Devaton Autocorrelaton Correlaton wth y t Fracton at Constrant γ b types 2 types types 2 types types 2 types types 2 types NOTE: Ths table reports the tme-seres statstc for the bond purchases relatve to the current per capta endowment, b t. The varable y t denotes the agent s endowment relatve to the per capta endowment. γdenotes the parameter of relatve rsk averson and b ndcates the amount the agent s allowed to borrow relatve to the per capta endowment. The other parameter values are reported n Secton 2.3. Populaton moments are approxmated usng the sample moments of a smulated draw of 100,000 observatons INSURANCE AGAINST IDIOSYNCRATIC RISK. A crucal feature of the economy wth two types s that the nterest rate drops when an agent receves the low realzaton for several perods because of the ncrease n cross-sectonal dsperson. Ths works lke a transfer from the rch agents (the lender) to the poor agents (the borrower) and suggests that agents n economes wth only two types are better off than agents n economes wth a contnuum of types. There s another reason, however, that makes t harder to smooth consumpton n economes wth two types. It s harder to smooth consumpton n an economy wth two types because an agent wll not lend to another agent of the same type and can never lend more than the agent of the other type s allowed to borrow. Ths wll prevent hm from buldng up a large buffer stock durng good tmes. Agents n an economy wth a contnuum of types can lend to a wde varety of dfferent agents and at tmes accumulate assets well n excess over the maxmum amount that s possble n an economy wth two types. To document the quanttatve mportance of the nterest rate effect, I plot n Fgure 3 the mpulse response functon of the nterest rate when the same agent receves the low-ncome realzaton for several perods. The graph plots the nterest rate for three levels of the borrowng constrant parameter and a parameter of relatve rsk averson equal to three. The agents n the economy wth two types start out wth zero bond holdngs, and wthout loss of generalty, I consder the case where the economy s n a recesson. The graph also plots the nterest rate n the economy wth a contnuum of types that, of course, does not respond to the realzatons of an ndvdual s ncome shock. 17 Consder the ntal perod n the two models. Snce the ntal levels of bond holdngs are equal to zero n the economy wth two types, there s lttle cross-sectonal dsperson when the agent receves hs frst low-ncome realzaton. In the 17 It s assumed that the economy has been n a recesson for several perods so that there are no more changes n the nterest rate n the economy wth a contnuum of agents.

14 13 economy wth a contnuum of types there always s a certan amount of dsperson. Intally, therefore, the nterest rate s lower n the economy wth a contnuum of types, snce hgher cross-sectonal dsperson corresponds to lower nterest rates. When the agent receves addtonal negatve shocks, the amount of cross-sectonal dsperson n the economy wth a contnuum of types s not affected but ncreases n the economy wth two types. As documented n the graph, ths causes the nterest rate to drop. Quanttatvely, these effects are enormous. When the agent reaches hs constrant n the economy wth two types, then the nterest rate s equal to mnus 15%, mnus 3.7%, and mnus 1.8%, for values of the borrowng constrant parameter equal to 0.2, 1.0, and 2.0, respectvely. These negatve nterest rates mply that the agent consumes more than hs ncome level when he reaches the constrant. Not bad, to lve n a world n whch changes n the nterest rate provde ths knd of nsurance! Fgure 3: The Response of the Interest Rate to Idosyncratc Shocks bbar=1, two types bbar=2 bbar=1 bbar=2, two types bbar=0.2 bbar=0.2, two types TIME NOTE: Ths graph plots the realzaton of the nterest rate n the economy wth two types when the agent receves the low-ncome realzaton for several perods. The straght lne wth the same thckness ndcates the nterest rate n the correspondng economy wth a contnuum of types. The parameter of rsk averson s equal to three and the aggregate growth rate s always equal to the low value. The parameter bbar (= b ) ndcates the amount an agent s allowed to borrow relatve to the per capta endowment. The value of the other parameters are reported n Secton 2.3. The graph does not reveal the ablty of an agent n the economy wth a contnuum of types to accumulate more assets durng good tmes. The next two fgures reveal both ths effect and the nterest rate effect dscussed above. In partcular, Fgures 4 and 5 plot for the economy wth a contnuum of types and the economy wth two types a realzaton for consumpton (relatve to per capta ncome) and bond purchases (relatve to per capta ncome), respectvely. The parameter of rsk averson s equal to one and the borrowng constrant parameter s equal to one. Durng the frst 30 perods, the agent s ht several tmes by negatve shocks and s repeatedly at the borrowng constrant. As documented n the graph, the behavor of bond purchases s very smlar n both economes durng ths perod. In contrast, as documented n Fgure 4 the consumpton level n the economy wth two types s hgher than the consumpton level n the economy wth a contnuum of types durng ths perod wth frequent negatve shocks because of reductons n the nterest rate. The behavor of bonds s qute dfferent across the two

15 14 types of economes when the agent has receved a seres of hgh realzatons. Ths happens around perods 50, 110, and 130. In the economy wth two types ths means that the agents of the other type have receved a seres of low realzatons and have reached ther borrowng constrants. Consequently, equlbrum on the bond market lmts the amount of savngs that the agent wth the postve shocks can accumulate. In the economy wth a contnuum of types, there s no such lmtaton and the agent accumulates assets well n excess of the borrowng constrant parameter. The graph for consumpton shows that the agent n the economy wth a contnuum of types s better able to smooth hs consumpton n the perods after whch he has bult up a large buffer stock of savngs. Fgure 4: A Realzaton for Indvdual Consumpton Economy wth a Contnuum of Types Economy wth two Types TIME NOTE: Ths graph plots the consumpton levels relatve to the per capta endowment for the ndcated economy. The parameter of relatve rsk averson and the borrowng constrant parameter are equal to one. The values of the other parameters are reported n Secton 2.3. Fgure 5: A Realzaton for Indvdual Savngs Economy wth a Contnuum of Agents Two-Agent Economy TIME NOTE: Ths graph plots the savngs decson relatve to the per capta endowment for the ndcated economy. The parameter of relatve rsk averson and the borrowng constrant parameter are equal to one. The values of the other parameters are reported n Secton 2.3.

16 15 The queston arses how these dfferences n the ablty to nsure aganst dosyncratc shock affect the agents welfare. In Table 4, I report the welfare dfferences between the dfferent economes. To do ths I calculate the (uncondtonal) expected dscounted utlty for the economes consdered. The expected utlty s calculated as the average of the actual utlty across 50,000 replcatons of 250 observatons. 18 In each replcaton, the ntal condtons are drawn from the ergodc dstrbuton. Panel A of Table 4 compares the ncomplete markets economes relatve to the complete markets economes. In partcular, t reports the permanent percentage ncrease n consumpton that would make the agents n the ncomplete markets economes as well off as the agents n the complete markets economy. The table documents that agents n the ncomplete markets economes are substantally worse off at the lower levels of the borrowng constrant parameters, especally for the hgher levels of rsk averson. Table 4: Welfare Comparsons. A. Permanent percentage ncrease of consumpton to become as well off as n complete markets economy. γ = 1 γ = 3 γ = 5 b types 2 types types 2 types types 2 types B. Permanent percentage ncrease of consumpton to become as well off as n the economy wth two types. b γ = 1 γ = 3 γ = NOTE: Panel A reports the permanent ncrease n consumpton n all perods that s requred to make an agent n the ndcated economy as well off as an agent n the complete markets economy. Panel B reports the permanent change n consumpton that s requred to make an agent n the economy wth a contnuum of types as well off as an agent n the economy wth two types. To calculate the (uncondtonal) expected dscounted utlty the average of the actual dscounted utlty levels across 50,000 replcatons of 250 observatons s used. In each replcaton, the ntal condtons are drawn from the ergodc dstrbuton. γs the parameter of relatve rsk averson and b ndcates the amount an agent s allowed to borrow relatve to the per capta endowment. The values of the other parameters are reported n Secton 2.3. Panel B reports the permanent percentage change n consumpton that would make an agent n the economy wth a contnuum of types as well off as an agent n the economy wth two types. For most parameter values, ths number s postve, whch means that an agent s better off n an economy wth only 18 The dscounted utlty of consumpton n perods 251 and hgher s so small that gnorng t does not affect the results.

17 16 two dfferent types of agents. 19 In those cases, the nterest rate effects are more mportant than the ablty to accumulate hgh levels of assets. The results are quanttatvely mportant for some parameter values. When the parameter of relatve rsk averson s equal to fve (three) and the borrowng constrant parameter s equal to 0.2, for example, then an agent n the economy wth two types s wllng to permanently reduce hs consumpton by 0.96 (0.27) percent to avod beng n an economy wth a contnuum of types. An mportant excepton s the case when the parameter of rsk averson s equal to fve and the borrowng constrant parameter s equal to two. In ths case, the agent s better off n the economy wth a contnuum of types. One more nterestng observaton can be made about the economy wth two types. As documented n Panel A, when the parameter of rsk averson s equal to fve, then an ncrease n the borrowng constrant parameter from 0.2 to 1.0 makes the agent better off, but a further ncrease leads to a decrease n the agent s expected utlty. The reason s that for hgher levels of rsk-averson, the nterest rate effects are so helpful n smoothng consumpton that beng n an economy wth a hgher borrowng constrant parameter does not make you necessarly better off. To understand ths result, I plot n Fgure 6 the mpulse response functon of consumpton (relatve to the per capta endowment) when the agent receves the low-ncome realzaton for several perods. Fgure 6: The Response of Consumpton to a Seres of Negatve Idosyncratc Shocks bbar=0.2 bbar = 2 bbar = Endowment Level TIME NOTE: Ths graph plots the realzaton of consumpton n the economy wth two types when the agent receves the low-ncome realzaton for several perods. The parameter b ndcates the amount an agent s allowed to borrow relatve to the per capta endowment. The parameter of rsk averson s equal to fve and the aggregate growth rate s always set at the low value. The graph makes clear how t can be possble that an agent s expected utlty s hgher when the borrowng constrant parameter s equal to 1.0 than when the borrowng constrant parameter s equal to 0.2 or 2.0. Note that the agent s consumpton when b = 1.0 s never far below the two alternatves. But when b = 0.2 an agent s consumpton s consderably lower than the alternatves f he has been ht a few 19 Ths paper, thus, provdes the frst poltcally correct argument aganst dversty.

18 17 tmes by a low realzaton. Smlarly, when b = 2.0 an agent s consumpton s consderably lower when he has been ht by a large strng of low realzatons. Note that when the agent reaches hs constrant n the economy wth two types, he consumes more than hs endowment level because the nterest rate s negatve. One mght thnk that the agent that s allowed to borrow less would beneft less from the negatve nterest rates. In fact, the opposte s true and agents are consumng more when they face tghter borrowng constrants. The reason s that the nterest rate drops to a much more negatve value n economes wth tght borrowng constrants. 4. CONCLUDING COMMENTS. In ths paper, I analyze the propertes of an asset-prcng model wth a contnuum of types and the propertes of the correspondng model wth two types. Besdes the number of types, the two models are exactly dentcal. In partcular, the unvarate tme-seres specfcatons of the ndvdual and aggregate endowment are the same n the two models. The followng propertes for the tme-seres behavor of nterest rates are found. Average nterest rates are remarkably smlar for the parameters consdered here. For some parameter values nterest rates are on average somewhat lower n economes wth two types. Ths s consstent wth the fndng that the nterest rate s a concave functon of the amount of cross-sectonal dsperson and cross-sectonal dsperson s more volatle n economes wth two types. Interest rates are much more volatle n economes wth only two types because an mportant part of the fluctuaton n nterest rates s due to dosyncratc shocks. Unlke economes wth two types, economes wth a contnuum of types can have low real nterest rates wthout havng excessvely volatle bond returns. The dfferent tme-seres behavor of nterest rates causes the behavor of consumpton and bond purchases to dffer as well. The followng dfferences are found. In an economy wth two types, each agent can never nvest more than agents of the other types are allowed to borrow. In an economy wth a contnuum of types, equlbrum on the bond market does not create such a constrant. Consequently, n the economy wth a contnuum of types, agents accumulate durng good tmes assets well n excess of the amount they are allowed to borrow. Ths makes t easer to smooth consumpton. There s also a reason why t s easer to smooth consumpton n the economy wth two types. If, n an economy wth two types, the same agent s ht by a seres of negatve dosyncratc shocks, the amount of cross-sectonal dsperson ncreases whch leads to a drop n the nterest rate. The reducton n the nterest rate works lke a transfer from the rch agents (the lenders) to the poor agents (the borrowers). For most parameter values, ths effect s more mportant and agent s expected utlty s hgher n economes wth only two types.

19 18 For hgher values of the parameter of relatve rsk averson these nterest rate effects become so strong that the effect of fnancal frctons on agent s welfare has a dfferent sgn n the two types of economes. In economes wth a contnuum of types, an agent s welfare s always ncreasng when the maxmum amount he s allowed to borrow ncreases. In the economy wth two types, however, there are parameter values for whch an agent s welfare s decreasng when the borrowng constrant s relaxed.

20 19 APPENDIX. THE SOLUTION ALGORITHM. In Den Haan (1997) t s shown n detal how the model n Secton 2 wth a contnuum of agents s solved. In ths secton, a bref outlne of the algorthm s gven. Den Haan (1996) and Krusell and Smth (1998) deal wth the nfnte dmensonal set of state varables by approxmatng the cross-sectonal dstrbuton wth a fnte set of moments. Den Haan (1997) follows the same approach but assocates a densty wth these moments. Ths makes t possble to avod the Monte Carlo ntegraton technques used n Den Haan (1996) and Krusell and Smth (1998). A notable paper that dscusses soluton technques for models wth a large (but fnte) number of state varables wthout reducng the dmenson of the set of state varables s Gaspar and Judd (1997). In the model, the bond prce at perod t s a functon of a t and the cross-sectonal dstrbuton of bond holdngs. A key feature of the proposed algorthm s to approxmate the cross-sectonal dstrbuton of wealth and ncome holdngs at perod t usng an (M 1) vector, φ t, contanng moments of the crosssectonal dstrbuton. To approxmate the bond prce functon I, therefore, use the functon Θ(a t,φ t ; δ Θ ), where δ Θ s a vector of parameters and Θ( ) s chosen from a class of functons that can approxmate any functon arbtrarly well. Smlarly, I use the functon Φ(a t+1, a t,φ t ; δ Φ ) to approxmate the transton law of φ t. Snce φ t s a vector, Φ( ) s a vector-valued functon. Solvng the ndvdual problem requres c t+ 1 approxmatng one more functon. I approxmate the condtonal expectaton, E t [ ] γ, and denote the approxmatng functon by Ψ( yt, bt 1, at, φ t ;δ Ψ ). For a partcular functonal form for Θ( ), Φ( ), and Ψ( ), the algorthm solves for the parameter values δ Φ, δ Θ, and δ Ψ wth the followng teraton scheme: Step 1: Gven parameter values for δ Φ and δ Θ solve the ndvdual s problem, that s, obtan parameter values for δ Ψ. The number of state varables for ths problem s equal to sx. Other than the somewhat hgh number of state varables, ths s a straghtforward numercal problem. Step 2: Gven the decson rules for the ndvduals, solve the aggregate problem, that s, obtan parameter values for δ Φ and δ Θ. Snce the cross-sectonal densty s assumed to belong to a certan class of denstes, knowng the moments s the same as knowng the densty. At each pont n the state space one can use numercal ntegraton technques and the ndvdual polcy functons derved n step 1 to calculate the equlbrum nterest rate and the moments of next perod s cross-sectonal dstrbuton. A smple projecton s used to calculate the values for δ Φ and δ Θ. If these parameter values are close to the ones used to solve the ndvdual s problem n step 1, then the algorthm has converged. If not, one has to repeat steps 1 and 2. In ths paper, I use the frst two moments of the bond holdngs of the agents who receve the low ncome shock and the frst two moments of the bond holdngs of the agents who receve the hgh ncome shock to approxmate the cross-sectonal dstrbutons. In ths case M equals 3 snce the means of the two cross-sectonal dstrbutons add up to zero. Other choces of the numercal soluton procedure are the same as those for EXP2Q500 descrbed n Table 2 of Den Haan (1997).