Tài liệu Money and Bonds: An Equivalence Theorem doc

Federal Reserve Ba nk o f Minneapolis Research Department St aff Report 393 July 2007 Mon ey and Bonds: An Equivalen ce Theorem Narayana R. Kocherlakota ∗ University of Minnesota, Federal Reserve Bank of Minneapolis, and NBER ABSTRACT This paper considers four models in which immortal agents face idiosyncratic shocks and trade only a single risk-free a sset over time. The four models specify this single asset to be private bonds, public bonds, public money, or private money respectively. I pro ve that, given an equilibrium in one of these economies, it is possible to pick the exogenous elements in the other three economies so that there is an outcome-equivalent equilibrium in each of them. (The term "exogenous variables" refers to the limits on private issue of money or bonds, or the supplies of publicly issued bonds or money.) ∗ I thank Shouyong Shi and Neil Wallace for great conversations about this paper; I thank Ed Nosal, Chris Phelan, Adam Slawski, Ha kki Yazici and participants in SED 2007 ses sion 44 for their comments. I ac k nowl- edge the support of NSF 06-06695. The views expressed herein are mine and not necessarily those of the Federal Reserve B ank of Minneapolis or the Federal Reserve System. 1. Introduction In this paper, I examine four different models of asset trade. In all of them , immortal agents face idiosy n cratic shock s to ta ste s an d /or productivities. They can trade a single risk-free asset over time. Preferences and risks are the same in all four models. The models differ in t heir specification of w h at this single asset is. In the first t wo models, a gen ts trade interest-bearing bonds. In the first model, agents can trade one period risk-free bonds ava ilable in zero net supply, subject to person- independen t borro wing res trictions. I n the second m odel, a gents can trade one period risk-free bonds available in positive net supply, but they cannot s hort-sell the asset. A go v ernm en t pa ys the interest on these bonds, and regulates their supply, by using time-dependent taxes that are the same for all a gents. In th e other two models, agents can trade money. Money is an asset t hat lasts forever, but pays no dividend. It pla ys no special role in transactions. In the third model, money is in positive supply. A go vernmen t regulates its supply using lump-sum taxes. In the fourth model, there i s no g ov ernment. Agen ts c an issue and redeem pri vate money, subject t o a period-by-period constraint on the difference between past issue a nd past redemption. These models are designed to be closely related to ones already in the literature. The first model is ess entially t he famous A iyagari-Bewley m odel o f self-insurance. The s econd model is motivated by Aiy ag ari and McGrattan’s (1998) study of the optimal quantity of go vernm ent debt. The third model is a version of Lucas’ (1980) pure currency economy. It is used by Imroh orog lu (1992) in her study of the welfare costs of inflation. The fourth model is more nov el, a ltho ug h of course many autho rs hav e been interested in comparin g the consequences of using inside instead of outside money (see, for example, Cavalcanti and Wallace (1999)). The basic lesson of this prior literature is that the exact nature of the traded asset has important e ffects on model out com e s. In A iyagari a n d McGra ttan (1998) (and later Shin (2006)), publ ic debt issue generates welfare costs that do not occur in models with private debt. Lucas (1980) argues that agents cannot achieve as much with mon ey as w ith private debt, sa ying explicitly, "There is a sense in which money is a second-rate asset." Cavalcanti and Wallace (1999) argue that u s ing insid e (privately issued) m one y allows a gents to ach iev e more than outside money. In c ontras t, I prove th e follo win g equivalence the orem. Take an equ ilibriu m in any of the four econo mies. Then, it is possible to specify the exogenous elements of the other three econo m ies so tha t there is an outcome-equivalen t equ ilibrium in eac h . Her e, by "exogenous elements," I mean specifically: 1. borrowing limits in the first m odel 2. bond supplies in the second model 3. money supplies in the third model 4. money issue limits in the fourth model. In fact, the equiva len c e is actually even stron ge r : in all of these outcom e -e qu ivalent equilib ria, agents ha ve ident ica l choic e sets a s in th e original equilibriu m. Why is m y r esult so differen t from the lesson of the prior literature? In the earlier analyses, the m odels with different assets also impose differen t assumptions on the n ature of what might be t erm ed the repayment or collection technology. For examp le, in m odels with priv ate risk-free debt, the borrow er must mak e a repaym en t that is independent of the borrower’s dec is ions or shocks . In ess en c e, the lender is essentially able to impose a lump- sum tax on th e borrower at the tim e of repayment. In models with public risk- free d eb t, like Aiyagari and McGrattan (1998), the government m ak es a repa ymen t that is independent of any aggregat e shocks. H owever, it is ty p ically assumed that the go vernment must use linear taxes to collect the resources for its repa yments. This restriction to linear taxes means that governmen t repaymen t of public debts must d istort agents’ decisions in a wa y that is not true of private debt repa ym e nt. T he treatment of taxes in models with outside money is often ev en more drastic; thu s, in their m odels, Lucas (1980), Cavalcanti and Wallace (1999) and Kocherlakota (2003) all assume that the government can use no taxes other than inflation taxes. In this paper, I elimin ate these differences across the models in their specification of the repay ment technology. In particular, I assume in the models with public debt issue that the governmen t is able to levy a head tax — that is, a lump-sum tax that is the same for all agents. ( Note that giv en the potential heterogeneity in the model setting, t he go v ernm en t 2 cannot generally implement a first-best outcome using the (uniform) head tax.) Once I endow the go vernmen t with this instrument, I can pro ve the equivalence theorem. The theorem really contains two distinct results. First, I sho w that the public issue and private issue of bonds/mon ey are equivalent to one another. In this equ ivalence, the above head tax plays a key r ole. In the m odels with public issue, the gov ernm ent uses a head tax t hat i s exactly equal to the interest payment made by a borro wer in t he private-issue economy who holds the maximal level of debt in e ach period. It is in this sense that the collection powers of the private sector and public sector are the same. Of course, these collection powe rs may well be limited by enforcemen t problems of various kinds; the crucial assumption is that the enforcemen t problems are the same in the private and public sectors. Second, the theorem sho ws that risk-free bonds and money are equiv alen t to one another. The k ey to this demonstration i s that money can have a positive real rate of r eturn even th ou gh it does not pay dividends. This p ric e rise can occur in equilibrium in the th ird model i f the g overnmen t shrinks the supply o f money using t he head tax. The size of t he needed head tax is exactly the same as in the econom y with public debt issue. It can occur in th e fourth model if the limits on net money issue are sh rin kin g ov e r time. Th e theorem is related t o Wallace’s (19 81) famo u s Modigliani-Miller theorem for open market operations. Wallace p r oves tha t the m o ney/bond c omposition of a government’s debt portfolio does not affec t equilibrium outcomes. Like my theorem, Wallace ’s relies on two crucial assumptions. First, a s noted above, the go vernmen t must ha ve access to lump-sum taxes. Second, money cannot have a transactions a dvantage over bonds. This assumption is not satisfied by cash-in-advance , money-in -th e -u tility function , or transaction cost models. Lik e Wallace’s paper, mine is also closely related to Barro’s (1974) analysis of governm ent debt. Taub (1994) poses the que stion, "Are currency and credit equivalent mech anisms?" that motivates this paper. As I do, he answers this question affirmativ ely. Ho wever, he confine s his analysis to a rather special examp le (linear utilit y ) . Le v ine (19 91) and Green and Zhou (2005) use linear utility exam ples to demonstrate ho w a go vernment using public money issue can ac hieve a first-best outcome in a world in whic h a gents experience shocks to their need for consumption. In their examples, the go vernmen t ac hiev es this good outcome by 3 using an inflationary m onetary policy. My theorem demonstrates that t he gov ernm ent could instead use an appropriate debt po licy, o r that private agents could achiev e this desirable outcome with appropriately set borrowing limits. 2. Setup Consider an infinite horizon e nvironm e nt with a u n it m e asu re o f agents in wh ich time is indexed by t he na tural numbers. At t he beginning o f period 1, for each agent, Nature dra ws an infinite sequence (θ t ) ∞ t=1 from the set Θ ∞ , where Θ is finite. The draws are i.i.d. acr o ss agents, with m easure μ. Hence, there is no aggregate risk. At the beginning of period t, a g iven agent ob ser ves his own realization of θ t ; his information at date t consists of the history θ t =(θ 1 , , θ t ). The shocks affect i n d ivid u als as fo llows. Th e typical agent has pre fe ren ces of the f orm ∞ X t=1 β t−1 u(c t ,y t ,θ t ), where c t is the agent’s consum ption in period t, y t is th e agent’s output in period t, and 0 <β<1. The agent’s utility function u is assumed to be strictly increasin g in c t ,strictly decreasing in y t , a n d is a function of the realization o f θ t . I t he n consider four d ifferent (possibly inc omplete markets) trading s tructures embed- ded in this setting. A. Private-Bond Econom y The first market structure is a private-bond economy. It is c om pletely characterized b y a borrowing limit seque nce B priv =(B priv t+1 ) ∞ t=1 , where B priv t+1 ∈ R + . (Note that the borrowing limits are the s am e for all agents in all periods.) At each date, the agents trade one-period risk-free real bonds in zero net supply for consumption. They are initially endowed with zero units of bonds each. Eac h agent’s bond-holdings in period t must be no smaller than −B priv t+1 (as measured in terms of consumption in that period). In this economy, in d ivid ua ls take intere st rates r =(r t ) ∞ t=1 ,r t ∈ R, as giv en and then choose consumption, output, and bond-holdings (c, y, b)=(c t ,y t ,b t+1 ) ∞ t=1 , (c t ,y t ,b t+1 ):Θ t → 4 R 2 + × R. Hence, the agen t’s problem is max (c,y,b) E ∞ X t=1 β t−1 u(c t ,y t ,θ t ) s.t. c t (θ t )+b t+1 (θ t ) ≤ y t (θ t )+b t (θ t−1 )(1 + r t−1 ) ∀(θ t ,t≥ 1) {b t+1 (θ t )+B priv t+1 },c t (θ t ),y t (θ t ) ≥ 0 ∀(θ t ,t≥ 1) b 1 =0 An equilib rium i n a private-bond economy B priv is a specification of (c, y, b, r) such that (c, y, b) solves the agen t ’s problem given r and m arkets clear for all t: Z c t dμ = Z y t dμ Z b t+1 dμ =0 B. Public-Bond Econom y The second is a pub lic-bond economy. A t eac h date, there is a go vernmen t that sells one-period risk-free real bonds. The economy is completely characterized by an exogenously specified bond supply sequence B pub =(B pub t ) ∞ t=1 ,whereB pub t+1 ∈ R + and an in itial period return r 0 . The government raises B pub t+1 units of consumption in period t b y selling one-period risk-free bonds. I t collects τ b t units of consump tion fro m each agent; th e tax is the same for all agents, and is d etermined endogenously in equilibrium. 1 Each agen t is in itially endowed with bonds t hat pay off B pub 1 (1+r 0 ) units of consumption. At eac h date, agen ts trade co nsumption and the gov ernmen t-issued bonds. Agents are not allow ed to short-sell these bonds. In this econom y, the individuals take interest rates r =(r t ) ∞ t=1 as given and then choose consumption, output, and bond-holdings. Hence, the individual’s problem is max (c,y,b) E ∞ X t=1 β t−1 u(c t ,y t ,θ t ) 1 Taxes are endogenously determined in this public-bond economy and in the public-money economy dis- cussed in the next section. It is important to note that the main equivalence theorem is valid even if taxes are exogenously specified. I treat taxes as endogenous so as to ensure that the government flow budget constraint is satisfied for off-equilibrium interest rate/price sequences, as well as in equilibrium. 5 s.t. c t (θ t )+b t+1 (θ t ) ≤ y t (θ t )+b t (θ t )(1 + r t−1 ) − τ bond t ∀(θ t ,t≥ 2) c 1 (θ 1 )+b 2 (θ 1 ) ≤ y 1 (θ 1 )+B pub 1 (1 + r 0 ) − τ bond 1 ∀θ 1 b t+1 (θ t ),c t (θ t ),y t (θ t ) ≥ 0 ∀θ t ,t≥ 1 An equilibrium in a public-bond economy (B pub ,r 0 ) is a s pecification of (c, y, b, r, τ bond ) suc h that (c, y, b) solves the i n div id ual’s problem given (r, τ bond ) and mar kets clear for all t: Z c t dμ = Z y t dμ for all t Z b t+1 dμ = B pub t+1 for all t Together, these imply t hat a gov ernm en t budget constraint holds at eac h date: τ b t = −B pub t+1 + B pub t (1 + r t−1 ) C. Public-Money Economy The third economy is a public-m oney economy. By mo ney, I mean an infinite ly -lived asset that pays no dividends. Each a gen t is initially endowed with M pub 1 units of money. Then, the economy is completely characterized by an exogenously specified money supply sequence M pub =(M pub t+1 ) ∞ t=1 ,whereM pub t+1 ∈ R + .Inperiodt,thegovernmentcollectsτ mon t units of consumption from eac h agent; again, the taxes are the same for all agents, and are determ in ed endogenou sly in equilibrium . At eac h date, agen ts t rad e money a nd consumption ; the go vernment trades so as to ensure that there are M pub t+1 units of mon ey outstandin g. In this econom y, the i ndividuals tak e money prices p as g iv en and then c hoose con- sumption, output, and m oney-holdings. Hence, the individual’s problem is max (c,y,M ) E ∞ X t=1 β t−1 u(c t ,y t ,θ t ) s.t. c t (θ t )+M t+1 (θ t )p t ≤ y t (θ t )+M t (θ t−1 )p t − τ mon t ∀θ t ,t≥ 2 c 1 (θ 1 )+M 2 (θ 1 )p 1 ≤ y 1 (θ 1 )+M pub 1 p 1 − τ mon 1 ∀θ 1 M t+1 (θ t ),c t (θ t ),y t (θ t ) ≥ 0 ∀(θ t ,t≥ 1) 6 An equilibrium in a public -money ec on omy (M pub ) is a specification of (c, y, M, p, τ mon ) such that (c, y, M) s olves the individual’s problem given (p, τ mon ) and markets clear for all t: Z c t dμ = Z y t dμ Z M t+1 dμ = M pub t+1 Again, a go vernment budget constrain t is implied at eac h date b y market-clearing: τ mon t = M pub t p t − M pub t+1 p t There is no cash-in-advance constrain t or any transaction cost advantage associated with mon ey in this se tting. D. Private Money Econom y The fourth and final econo my is a private-money economy. In this economy, there is no gov ernm en t. A gen ts are able to issue their money in exchange for consumption, and redeem others’ monies in exc hange for consumption. Howev er, in eac h history, they face an ex ogen ous upper bound on the net am ou nt of money issue that they have done in their lifetimes. The econom y is completely characterized b y the exogenous upper bound process M priv =(M priv t+1 ) ∞ t=1 , where M priv t+1 ∈ R + . In this econom y, the individuals take money prices p as given and then c hoose con- sumption, how m u ch money to issue and ho w muc h money to redeem. (I assume that all monies are traded at the same price p; there m ay be other equilibria in which this restriction is n ot satisfied.) H enc e, the individual’s pro ble m is max (c,y,M iss ,M red ) E ∞ X t=1 β t−1 u(c t ,y t ,θ t ) s.t. c t (θ t )+M red t+1 (θ t )p t ≤ y t (θ t )+M iss t+1 (θ t )p t ∀θ t ,t≥ 1 M iss t+1 (θ t ),M red t+1 (θ t ),c t (θ t ),y t (θ t ) ≥ 0 for all θ t ,t t X s=1 [M iss s+1 (θ s ) − M red s+1 (θ s )] ≤ M priv t+1 ∀θ t ,t≥ 1 An equ ilib rium in a private-money e conomy (M priv ) is a specification of (c, y, M red ,M iss ,p) 7 suc h that (c, y, M red ,M iss ,p) solves the individual’s problem and markets clear for all t: Z c t dμ = Z y t dμ Z M red t+1 dμ = Z M iss t+1 dμ Again, there is no cash-in-advance constrain t or transaction cost advantages associated with mon ey in this se tting. 3. Example Economy In this section, I w o rk through an example of the abo ve general structure that il- lustrates the gene ral equivalence theore m th at follows in the next section. I sta r t with an equilibrium in a private-bond econom y. I then construct a public-bond economy, a public- money econom y, and a private-money economy. I show that in eac h of these economies, there is an equilibrium with the same consu mption allocation as the original, p rivate-bond, equilibrium. Even more stron gly, agents have exactly the same budget set s in each of these equilibria. In t he example, output is inelastically su p plied. Half of the agents rece ive an endow - ment stream of the f orm (1 + h, 1, 1 , ) and the ot her half get an endowm ent stream of the form (1 − h, 1, 1, ), where 1 >h>0. I will call the first half "rich" and the second half "poor." The agents have identical preferences of the form ∞ X t=1 β t−1 ln(c t ), where 1 >β>0. A. Private-Bond Econom y Consider first a priva te-bond ec on omy in which th e borrow in g limit B priv t+1 is con stant at βλ(1 − β) −1 , where (1 − β) <λ<1. We can construct an equilibriu m in th is eco n omy as follows. Set the interest rate r t to be constant at 1/β − 1. Ric h agents consume a constant amount c r ,where c r =(1+h)(1 − β)+β =1+h(1 − β) 8 Ric h agents’ bond-holdings b r t+1 equal βh for period t ≥ 1. Poor agents consu me a constant amount c p =2− c r . Poor agents’ bond- holdin gs b p t+1 equal −hβ for period t ≥ 1. Note that the borrowing limit has been chosen so that it nev er binds in equilib rium. It is readily checked tha t the abo ve specification forms an equilibrium. Markets clear. The agents’ flo w budget constrain ts are satisfied because c r + b r 2 − 1 − h =0 c r + b r t+1 − 1 − b r t β −1 =0for all t ≥ 2 and similarly for poor agents. Because the borrow in g limit d oes not bind, the agents’ Euler equations are satisfied. We need only chec k the agents’ transv ersality conditions, whic h are satisfied because the two limits lim t→∞ β t−1 u 0 (c r )(b r t+1 − 2β) lim t→∞ β t−1 u 0 (c p )(b p t+1 − 2β) are both equal to zero. B. Public-Bond Econom y I n ow want to design a public-bond ec on o my with an outcome-equ ivalent equilib rium. In a public-bond econom y, agen ts are not allowed to borrow. Hence, to get a non-autarkic equilib rium, th ere must be a positiv e amou nt of d e b t outstan d in g. I set B pub t = βλ(1 − β) −1 (the private economy borrowing limit) for all t and r 0 =1/β − 1. As above, w e can c onstruct an equilibriu m in this economy in which the equ ilibriu m intere st rate r t is constant at 1/β −1. Rich agents consume c r (as defined above) in eac h period, and poor agents consume c p in 9 [...]... equilibrium rate of return is constant at r > 0, and the value of outstanding public debt is constant at B pub Theorem 1 designs a publicmoney economy in which the equilibrium rate of return is also r, and value of outstanding public obligations (now in the form of money) is B pub In this public -money economy, the price of money must rise at rate r Hence, the quantity of money must fall at this same rate... collection powers of the private and public sector may well differ, and money almost certainly does provide liquidity benefits that bonds do not A great deal of attention has been given to modelling and understanding the latter phenomenon In light of the theorem in this paper, the former issue seems an especially important one for understanding the impact of government financing decisions References [1]... 1) This collection limit is exactly equal to τ bond and τ mon in the equivalent public bond and money economies constructed in Theorem 1 Hence, by assuming that the government can levy taxes equal to τ bond and τ mon , I am assuming that the government can levy the same lump-sum taxes as can a private lender This means, for example, that the private and public sector must face the same limits on enforcement... isomorphism can be generalized 4 An Equivalence Theorem In this section, I prove the main theorem in the paper The theorem starts with an equilibrium (c, y, b, r) in a private-bond economy defined by B priv It then shows how, by translating the bond-holdings upward by B priv and crafting taxes in the right way, we can 13 get an outcome-equivalent equilibrium in the public-bond economy The key to the theorem. .. penalty if they fail to pay those taxes Kocherlakota (2003) and Berentsen and Waller (2006) argue that these penalties could be used to enforce cross-agent transfers of resources, and thereby eliminate the need for money altogether However, to ensure that there is no need for money, the government must be able to impose an arbitrarily large penalty, and know the realization of θ for each agent Neither of... agents’ money- holdings Mt+1 equal β t+1 h + β t+1 λ(1 − β)−1 p and poor agents’ bond-holdings Mt+1 equal −β t+1 h + β t+1 λ(1 − β)−1 To verify the claim that these prices and quantities form an equilibrium, note that markets clear and that individual Euler equations are satisfied Clearly, the transversality conditions are also satisfied, because the money supply converges to zero Finally, we can verify... this kind (in which agents trade only capital and risk-free bonds, and face shocks to their labor productivities) Aiyagari (1995) finds that the optimal capital income tax rate is positive, and the equilibrium interest rate in the economy is less than the rate of time preference Using the logic of Theorem 1, one could redo Aiyagari’s analysis in a public -money economy Aiyagari’s result implies that in... sucks out this money using the same taxes that it used to finance its interest payments in the public-bond economy It is worth pointing out that the proof of Theorem 1 establishes a stronger result than Theorem 1 itself The statement of Theorem 1 is that the equilibrium outcomes across the 3 It is possible to extend Theorem 1 to include model economies in which agents can trade both public and private... section, I discuss several aspects of Theorem 1 A Equivalences Theorem 1 establishes two kinds of equivalences The first is between private issue and public issue (of money or bonds) Consider, for example, a private-bond economy in which agents have a constant borrowing limit B priv In this economy, all agents begin with the same holdings of bonds (zero) They can run down their holdings to −B priv... governments can use a broader range of taxes than is assumed in the above economies More generally, suppose that in all four economies, the government can use any element of a class C of tax schedules ψ = {ψ t }∞ , where an agent who has production history t=1 yt in period t pays a tax ψ t (y t ) We can prove a version of Theorem 1 if C is closed under the addition of a sequence of constants, so that . 393 July 2007 Mon ey and Bonds: An Equivalen ce Theorem Narayana R. Kocherlakota ∗ University of Minnesota, Federal Reserve Bank of Minneapolis, and NBER ABSTRACT This. or money. ) ∗ I thank Shouyong Shi and Neil Wallace for great conversations about this paper; I thank Ed Nosal, Chris Phelan, Adam Slawski, Ha kki Yazici and