Government borrowing using bonds with randomly ...

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heavily through the use of tontines, where a group of subscribers purchased bonds with fixed total payments that were divided among 'survivors'. With.
Journal of Monetary Economics 41 (1998) 351—370

Government borrowing using bonds with randomly determined returns: Welfare improving randomization in the context of deficit finance Bruce D. Smith1!, Anne P. Villamil",* ! Department of Economics, University of Texas, Austin, TX 78712, USA " Department of Economics, University of Illinois, 1206 S. Sixth Street, Champaign, IL 60820, USA Received 19 September 1996; received in revised form 12 August 1997; accepted 26 August 1997

Abstract We study the problem of a government that wishes to share optimally the burden of deficit finance among agents with differential access to investment opportunities. In the presence of private information, it is Pareto efficient for the government to borrow in a way that amounts to non-linear taxation, and it must treat agents with access to the best investment opportunities preferentially to keep them in the bond market. In addition, with private information about access to assets, it is often desirable to randomize extraneously the return on the highest yielding government liabilities. The optimal government policy is shown to accord well with historical observations and provides insight into why explicit randomization is not often observed in private contracts. ( 1998 Elsevier Science B.V. All rights reserved. JEL classification: E61; E42; H63 Keywords: Deficit finance; Randomization

1. Introduction The explicit and extraneous randomization of returns on government bonds is a surprisingly common feature of historical government borrowing schemes. * Corresponding author. Tel.: 217 244 6330; e-mail: [email protected]. 1 Also at: Federal Reserve Bank of Minneapolis, Minneapolis, MN 55480, USA. 0304-3932/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 0 4 - 3 9 3 2 ( 9 7 ) 0 0 0 8 0 - 9

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For example, in the 18th century England and France issued bonds where, in exchange for a capital payment, an investor received a title to a bond plus a lottery ticket for a drawing of additional bonds. The payoff from a ‘winning ticket’ often provided an annual income greater than the total capital contributed. Moreover, the quantity of capital contributed was not generally small, and sometimes exceeded average per capita income (see Weir and Velde, 1992). Thus, governments often issued bonds in large denominations with extraneously randomized returns. At this same time England and France also borrowed heavily through the use of tontines, where a group of subscribers purchased bonds with fixed total payments that were divided among ‘survivors’. With a large group of subscribers, the government’s payments on such instruments displayed little randomness, while for any individual the returns were random. Interestingly, these were not simple annuity schemes since subscribers could make the payment contingent on the survival of someone other than themselves. The expected returns on this type of liability were generally favorable (see, Weir, 1989). During the American Revolution the Continental government also attempted to borrow, in Europe, through the use of so-called lottery bonds with randomly determined returns (see Anderson, 1982). This was viewed as a device for making American debt instruments more attractive to European investors. The use of such debt instruments has often been viewed as puzzling because the necessity of paying a risk premium to compensate (presumably) risk averse borrowers for randomized returns makes this an apparently expensive way to borrow (see Weir, 1989). In the episodes described, financial and insurance markets were at best weakly developed. Therefore, one might have expected the government to be relatively better able to bear risk than individuals, so the intentional introduction of extrinsic uncertainty by governments seems even more puzzling, especially since ‘lottery bonds’ and bonds with certain returns were often used simultaneously. Finally, in some of the historical examples, bonds with randomized returns were clearly intended to be sold to relatively wealthy investors. This contrasts with the socio-economic characteristics of participants in recent state sponsored lottery ‘games’ (see Clotfelter and Cook, 1990). These observations merit explanation. This paper is an attempt to understand why a government with a deficit to finance might utilize the apparently expensive instrument of bonds with extraneously randomized returns. Our vehicle for addressing this question is a two-period model in which a government with a utilitarian social welfare function must finance a fixed deficit of a given size. It does this by borrowing from two types of agents, denoted H and ¸, that are identical in all respects but one: agents of type H have access to investment opportunities other than government bonds, and agents of type ¸ do not. This captures situations where wealthier investors have access to investment opportunities not open to poorer investors, or where a government seeks to borrow both at home and abroad,

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and foreign investors have opportunities not open to domestic investors. We assume that agent type and investment activity are private information. If agents did not have differential investment opportunities the government would raise revenue from all types equally. When the deficit is sufficiently large, doing so drives agents with the best investment opportunities (type H agents) out of the bond market. This, in turn, requires all revenue to be raised from type ¸ agents, which a utilitarian government regards as undesirable. Thus, the government raises as much revenue as it can from type H agents without driving them out of the market, and this requires that they be treated preferentially. When type is private information, preferential treatment of type H agents creates an adverse selection problem which optimal government policy must address. Under conditions we describe, the government treats type H agents preferentially by designing an asset for them with a high expected return, but one which is extraneously randomized. In contrast, the asset designed for type ¸ agents has a lower expected (but certain) return. All agents have identical preferences, endowments, and equal access to the government’s liabilities, but the access of type H agents to an outside alternative allows them to partially insure against the possibility that they will receive a low return on their bond-holdings. Thus, type H agents have non-trivially diversified portfolios. Type ¸ agents, having no access to the outside investments, prefer the certain return. The government issues as many types of government bonds, bearing different returns and in different minimum denominations, as there are agent types, and intermediation is prohibited. This policy is constrained Pareto efficient if absolute risk aversion decreases at a rapid enough rate because it is then the optimal way to keep type H agents in the bond market given the adverse selection problem. The potential desirability of extraneous randomization in environments with private information has, of course, been previously noted by Prescott and Townsend (1984) and Arnott and Stiglitz (1988). Our model differs from theirs in that randomization is desirable here only because agents have differential access to alternative (non-government) investment opportunities. This causes agents who are (otherwise) intrinsically identical to have indirect utility functions that differ in such a way that randomized allocations are Pareto superior to ones with no randomization. This insight is similar to that in Mirrlees (1975). However, he assumes that agents differ directly with respect to their endowments and demand functions. Of course, the basic nature of the problem we analyze is similar to that considered by Mirrlees (1971) in his seminal principalagent paper with hidden information. In that model, a government with limited information must distinguish among agents with differential ‘ability’ in order to devise an optimal direct taxation schedule. The paper proceeds as follows. Section 2 describes the model. Section 3 considers non-stochastic planning problems from which some candidate Pareto efficient consumption allocations can be derived, and Section 4 establishes conditions under which randomized allocations are desirable. Section 5

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discusses historical public debt policies and shows that they are consistent with the predictions of the model. Section 6 concludes.

2. The model Consider an economy with a continuum of two-period lived agents with unit mass. Agents are divided into two types, indexed by i"H, ¸. Let h denote the i fraction of agents of type i, with h , h '0 and h #h "1. All agents are H L H L assumed to be endowed with w 50 units of the single good in period t of their t lives, with t"1, 2. In addition, there is a government that must finance an exogenously given real per capita expenditure level of g'0 in each period. Assume that the government levies no taxes, so it must finance the per capita deficit g by borrowing from agents of both types. Agent types are differentiated by their access to a storage technology. Type H agents alone have access to a constant returns to scale technology for storing the good, where one unit stored when young returns x3(0, 1) units when old. Assume that each agent can store only his or her own good, that agent type is private information (ex-ante), and that the activity of storing the good (or the quantity stored) is unobservable. All agents have the additively separable utility function u(ci )#v(ci ), where 1 2 ci3R denotes the consumption of a type i agent at age t. Assume that u and t ` v are strictly increasing, strictly concave, and thrice continuously differentiable, and define R(c),!v@@(c)/v@(c) to be the coefficient of absolute risk aversion. Assume that agents derive no utility from government expenditure (or it affects utility in an additively separable way). The assumption that type H agents can store the good while type ¸ agents cannot is meant to capture the problem facing a government which has a deficit to finance, and must borrow from agents with differential access to alternative investment opportunities. The access of some agents to relatively high return investments limits the government’s ability to extract resources from them. Here type H agents’ ability to store the good gives them access to an asset not available to type ¸ agents (the simplest form the problem can take), and proxies for several different scenarios. For example, wealthier agents might have access to investments not available to poorer agents. Alternatively, it could represent the situation of a government which seeks to borrow from foreign investors, who have investment options (bearing the gross rate of return x) not available to domestic investors. For future reference, it will be useful to have a notation for the storage behavior of an agent who pays a lump sum tax q at age t, and faces a certain t gross rate of return on savings of r. Such an agent chooses a savings level, s, to maximize u(w !q !s)#v(w !q #rs) subject to non-negativity con1 1 2 2 straints. The solution to this problem is given by the savings function

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s,f (w !q , w !q , r). We assume that agents facing no taxation will wish to 1 1 2 2 save at any rate of return greater than or equal to the rate of return on storage, x. This will be the case if and only if xv@(w )'u@(w ) holds. In addition, the 2 1 assumption that utility is additively separable implies that f '0'f . Savings 1 2 will be non-decreasing in rates of return if young and old consumption are gross substitutes, although this is not necessary to our results. Finally, we define the indirect utility function » in the standard way: »(w , w , r),u(w !f ( ) ))# 1 2 1 v(w #rf ( ) )). 2 3. Non-random Pareto efficient allocations We begin by describing the allocations which solve a utilitarian social welfare maximization problem under alternative sets of assumptions about constraints faced by a social planner. We then consider how these allocations can be decentralized by a government which sells bonds competitively, but can impose legal restrictions on bond trades. We begin by constraining the planner to choose non-random consumption allocations. Then, after having derived the optimal non-stochastic allocations in this section, we go on in Section 4 to consider whether explicit extraneous randomization by the government might be desirable. Throughout we assume that the government always honors its obligations. 3.1. Full information As a benchmark, consider the problem of a social planner that knows each agent’s type, and can observe and (if desired) prohibit storage of the good. The planner’s objective is to find an allocation that maximizes an equally weighted sum of the agents’ utilities subject to a resource feasibility constraint. Let k denote the amount of storage by a type H agent. The full information Pareto problem can be written as follows: h [u(ci )#v(ci )] subject to Problem 3.1. Choose ci , ci , and k to maximize + 2 1 i/H,L i 1 2 (1) + h (ci #ci )#h k4w #w !g#h xk. 2 H 1 2 H i 1 i/H,L At an interior optimum, the solution to this problem sets u@(ci )"v@(ci ), for 1 2 i"H,¸, cH"cL, for j"1,2, and k"0. Thus, all agents are treated identically, j j the marginal rates of substitution of different agents are equated, and no (inefficient) goods storage occurs. Moreover, ci "w !f (w , w !g, 1), and 1 1 1 2 ci "w !g#f (w , w !g, 1). In effect the allocation mimics that attainable 2 2 1 2 under an equal lump-sum tax imposed on all agents, and the utility of all agents under this allocation is given by »(w , w !g, 1). 1 2

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This solution is identical to that obtained by Bryant and Wallace (1984), and can be decentralized as they describe:2 The government prohibits goods storage, sells bonds with a minimum real value of F and rate of return r, and prohibits agents from intermediating bonds. If F and r are chosen to satisfy F"f (w , w !g, 1) and r"(F!g)/F, each agent will voluntarily purchase 1 2 bonds with a minimum real value of F [when »(w , w !g, 1)5»(w , w , 0)]. 1 2 1 2 This issue of indivisible, large denomination bonds permits the government to raise enough revenue to cover its expenditure, and does so without inducing any distortions or inequities across agent types. 3.2. »oluntary participation and private information Now assume that market participation cannot be compelled and that agent type cannot be directly observed (i.e., the planner cannot prevent type H agents from autarkically storing the good or type ¸ agents from consuming their endowments, nor does the planner know the type of any specific individual). The government must then finance its deficit, g, by selling bonds, which agents must purchase voluntarily. In order to induce type H agents to participate in the bond market the government must treat them preferentially: this preferential treatment then gives rise to an adverse selection problem. Thus, relative to the previous section, both voluntary participation and standard incentive compatibility constraints must be appended to the planner’s problem. The planner now solves Problem 3.2. h [u(ci )#v(ci )] subject Problem 3.2. Choose ci , ci , and k to maximize + 2 1 i/H,L i 1 2 to + h (ci #ci )#h k4w #w !g#h xk, 2 H 1 2 H i 1 i/H,L (2) u(cH)#v(cH)5»(w ,w ,x), 2 1 2 1 (3) u(cL)#v(cL)5u(w )#v(w ), 2 1 2 1 (4) u(cH)#v(cH)5u(cL)#v(cL), 2 1 2 1 (5) u(cL)#v(cL)5u(cH#k)#v(cH!xk). 2 1 2 1 Eqs. (2) and (3) are the voluntary participation constraints for bond markets. Eq. (4) is the incentive compatibility constraint for type H agents, which imposes that type H agents weakly prefer the allocation (cH, cH) to (cL, cL).3 Eq. (5) 1 2 1 2 2 See Cooley and Smith (1993) for a discussion of indeterminacy in decentralization schemes. 3 Formally, Eq. (4) should be written as u(cH)#v(cH)*»(cL, cL, x). However, since 1 2 1 2 u@(cL)"v@(cL)'xv@(cL) holds (see below), »(cL,cL,x)"u(cL)#v(cL). See Villamil (1988) for a dis1 2 2 1 2 1 2 cussion of private information in this context.

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imposes incentive compatibility for type ¸ agents, since a type ¸ agent taking a type H allocation cannot mimic the storage of type H agents and therefore consumes cH#k when young and cH!xk when old. 1 2 Our assumptions on savings behavior imply that the constraint set for Problem 3.2 is non-empty if, for instance »(w , w !g/h ,1)5u(w )#v(w ) 1 2 L 1 2 holds (that is, if the government’s budget deficit is not too large). Under such a condition Problem 3.2 has a solution, and we denote its associated allocation by cJ i, i, t"1, 2, and the investment level of agents by kI . We now describe the t optimal allocation, which can fall into two general categories. Case 1: »(w , w !g, 1)5»(w , w , x). In this case the allocation from Prob1 2 1 2 lem 3.2 satisfies Eqs. (2)—(5), since cH"cL, for t"1,2. t t Case 2: »(w , w , x)'»(w , w !g, 1). In this case, which we focus on, the 1 2 1 2 allocation solving Problem 3.1 violates the voluntary participation constraint Eq. (2). As a result, in order to keep type H agents in the bond market, the government must improve the allocation they receive. In doing so, however, the government also causes the incentive constraint Eq. (5) to bind. It is easy to show that if Eq. (2) holds as an equality the constraint Eq. (4) must be satisfied, and for simplicity, we focus on the situation where Eq. (3) does not bind. But, to emphasize, constraints (1), (2) and (5) must bind in Problem 3.2. The following proposition characterizes the solution to Problem 3.2. The proof is in Appendix A. Proposition 1. ¹he solution to Problem 3.2 satisfies Eqs. (1), (2) and (5) at equality, has u@(cJ H)"xv@(cJ H), u@(cJ L)"v@(cJ L), and f (w , w , x)'kI '0. 1 2 1 2 1 2 The solution to Problem 3.2 has several interesting features. First, Appendix A shows that goods storage occurs. While it is not technologically efficient to store goods here, there is a minimal level of goods storage that is necessary to give type H agents a utility level of »(w , w , x) without having type ¸ agents 1 2 mimic their bond purchases. Second, the two types of agents have different marginal rates of substitution. This also reflects an inefficiency that arises due to the necessity of treating type H agents preferentially in the presence of private information. Third, type H agents hold diversified portfolios (that is, they store goods and hold government bonds), and they are also ‘on their savings functions’. Type ¸ agents, in contrast, are not, since u@(cL)"v@(cL)'rLv@(cL). 1 2 2 In order to decentralize the solution to Problem 3.2, the government issues two types of bonds, and prevents intermediation. Agents who buy type h bonds with gross return x can buy only type h bonds, and are permitted to purchase at most Fh"f (w , w , x)!kI '0 units (in real terms). Type H agents buy these 1 2 bonds. Agents who purchase type l bonds must purchase at least Fl"w !cL 1 1 units (in real terms) which earn the gross return rl"(cL!w )/Fl. Such bonds 2 2 are sold to type ¸ agents.

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4. Pareto efficient randomization Starting from the optimal non-stochastic allocation of Section 3.2, we now describe conditions under which the extrinsic randomization of bond returns by the government can yield a further welfare improvement. Since our objective is only to show that some extrinsic randomization is desirable, we proceed as follows. Assume that the planner chooses, for i"H, ¸, deterministic values ci for young consumption, and values ci (z) for old consumption that may 1 2 depend on an extraneous state z. For simplicity, let z3M1, 2N, and let p3(0, 1) be the exogenous probability (which is the same in all periods) that z"1.4 Assume that realizations of z are independently and identically distributed across agents, and that z is realized at the beginning of old age.5 To simplify notation, let E h(ci (z)),ph(ci (1))#(1!p)h(ci (2)), where h( ) ) is an arbitrary function, and z 2 2 2 E denotes the expectation operator. Consider the solution to the following planning problem. Problem 4.1. Choose ci , ci (z) and k to maximize + h [u(ci )#E v(ci (z))] 1 2 i/H,L i 1 z 2 subject to + h [ci #E ci (z)]#h k4w #w !g#h xk, (6) i 1 z 2 H 1 2 H i/H,L (7) u(cH)#E v(cH(z))5»(w , w , x), 1 2 1 z 2 (8) u(cL)#E v(cL(z))5u(w )#v(w ), 1 2 1 z 2 u(cH)#E v(cH(z))5u(cL)#E v(cL(z)), (9) 1 z 2 1 z 2 (10) u(cL)#E v(cL(z))5u(cH#k)#E v(cH(z)!xk). 1 z 2 1 z 2 Eq. (6) is the resource constraint confronting the planner when bond returns are extraneously randomized, while Eqs. (7) and (8) are the voluntary participation constraints. Eqs. (9) and (10) are the incentive compatibility constraints for type ¸ and type H agents, respectively, in the presence of extraneous randomization. The solution to Problem 4.1 coincides with the solution to Problem 3.1 unless Eq. (7) binds. When Eq. (7) binds, so does Eq. (10), as in the previous section. In

4 Our description treats p as exogenous, but clearly randomization can be no less desirable on welfare grounds if the government is free to choose p. Below we consider a more general maximization problem which treats p as a choice variable, and Proposition 3 (below) states conditions under which the restriction to a two-state distribution is without loss of generality. 5 This captures an important feature of several historical randomization devices employed in government borrowing. In particular, governments have often confronted individuals with random returns on some bonds while facing little or no (as here) randomness with respect to total interest obligations on these bonds.

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this case, Eq. (9) cannot bind, and we restrict attention to the case in which Eq. (8) does not bind. Thus, for the remainder of the section, constraints (6), (7) and (10) bind. It is easy to verify in this case that the solution to Problem 4.1 has cL(1)"cL(2), so that only (or at most) type H agents face extrinsic uncertainty. 2 2 This is consistent with the historical observation that governments with large deficits to finance have made use of bonds involving explicitly and extraneously randomized returns, and that have attractive return distributions. These bonds typically were sold to agents with good alternative investment opportunities. In addition, the solution to Problem 4.1 has u@(cL)"v@(cL), k'0, and u@(cH)" 1 2 1 xE v@(cH(z)). The latter condition implies that type H agents continue to be ‘on z 2 their savings functions’. Proposition 2 states a sufficient condition for cH(1)OcH(2) to hold, so that 2 2 type H agents face extraneous uncertainty. We find that when the elasticity of absolute risk aversion with respect to old age consumption is sufficiently large, extrinsic randomization is welfare improving. The proof is in Appendix A. Proposition 2. Suppose that (1!x)v@@(cJ H!x kI ) v@@(cJ H) 2 2 ! '! v@(cJ H!x kI )!u@(cJ H# kI ) v@(cJ H) 1 2 2 holds, where cJ H, cJ H, and kI denote solutions to Problem 3.2. ¹hen cH(1)OcH(2). 2 2 1 2 Proposition 2 asserts that extraneous randomization improves welfare if

C

D

(1!x)v@(cJ H!xkI ) 2 R(cJ H!xkI )'R(cJ H). 2 2 v@(cJ H!x kI )!u@(cJ H#kI ) 1 2

(11)

Since v@(cJ H!xkI ) 2 (1 v@(cJ H!x kI )!u@(cJ H#kI ) 2 1 holds, Eq. (11) indicates that extraneous randomization is desirable if absolute risk aversion decreases at a rapid enough rate. Indeed, Eq. (11) can only hold if absolute risk aversion is decreasing (this implies that in a choice between a safe and a risky asset, the risky asset is a normal good). This is a common assumption about preferences. Intuitively, the ability of type H agents to store the good (or, more generally, invest in assets other than government liabilities) enables them to transfer resources from youth to old age in a way that is infeasible for type ¸ agents. If agents’ preferences display decreasing absolute risk aversion, type H agents will — when old — behave as if they are less risk averse than type ¸ agents. If they behave as if they are sufficiently less risk averse (i.e., if risk aversion decreases rapidly enough), the extraneous introduction of risk by the government is an efficient way to deter type ¸ agents from mimicking type H agents. (1!x)

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How fast must absolute risk aversion decrease in order for Eq. (11) to be satisfied? The answer depends on the level of relative risk aversion, on the rate of return on storage, and on the volume of storage that must be undertaken to induce incentive compatibility in the solution to Problem 3.2. In Appendix A we derive the following sufficient condition for Eq. (11) to hold: !(1!x)

C

R@(cJ H!xk)(cJ H!xk) 2 2 R(cJ H!xk) 2

D

u@@(cJ H#k)(cJ H#k) 1 1 5xR(cJ H!xk)(cJ H!xk)!(cJ H!xk) . (12) 2 2 2 u@(cJ H#k)(cJ H#k) 1 1 Eq. (12) indicates that low levels of relative risk aversion and low returns on storage are conducive to the desirability of randomized returns. The following example provides a further illustration of the same point. Example. Suppose that u(c )"/c1~o/(1!o) and v(c )"c1~o/(1!o), with 2 1 2 1 /5x and o'0. Then !cR@(c)/R(c),1 for all c. In addition, u@(cJ H)" 1 xv@(cJ H) implies that cJ H"cJ H(x//)1@o4cJ H, and consequently, that cJ H!xk4 2 1 1 2 2 (cJ H#k)(x//)1@o for all k50. This implies that Eq. (12) necessarily holds if 1 1!x5xo#o(x//)1@o is satisfied. It is therefore the case that extraneous randomization is welfare improving if either x or o is sufficiently small. Having described some conditions under which it is desirable to randomize bond returns, we now turn to a consideration of a fully general Pareto problem with extrinsic randomization. We then state conditions under which the restriction to the two state distribution in Problem 4.1 is without any loss of generality.6 Let z3M1, 2,2, ZN be a finite set of states, with Z'2, and let p(z) denote the probability that state z occurs. Let the planner solve the following problem. Problem 4.2. Choose cL, cL, cH, k, cH(z) and p(z), z"1,2,Z, to maximize 1 2 1 2 h [u(cL)#v(cL)]#h [u(cH)#R p(z)v(cH(z))] subject to7 L 1 2 H 1 z 2 h [cL#cL]#h [cH#R p(z)cH(z)]#h k4w #w !g#h xk, (13) 2 H 1 2 H z 2 H 1 L 1 (14) u(cH)#R p(z)v(cH(z))5»(w ,w ,x), 2 1 2 1 z (15) u(cL)#v(cL)5u(w )#v(w ), 2 1 2 1 6 We thank an anonymous referee for drawing our attention to this point. 7 Notice that we now allow the planner to choose the values p(z), as well as the values cH(z). We 2 have also imposed that the allocation received by type ¸ agents is nonstochastic, a result that was established in solving Problem 4.1.

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u(cH)#R p(z)v(cH(z))5u(cL)#v(cL), 1 z 2 1 2 u(cL)#v(cL)5u(cH#k)#R p(z)v(cH(z)!xk), 2 1 z 2 1 R p(z)"1, z 04p(z); z"1,2, Z.

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(16) (17) (18) (19)

Let cJ L, t"1, 2, kI , cJ H, cJ H(z), and pJ (z), z"1,2, Z, denote the solution to 1 2 t Problem 4.2. We then have the following result. Proposition 3. Assume that8 lim v@(c)"R, c?0 lim v@@(c)"!R, c?0 v@@(y!xkI )/v@@(y) is a decreasing function of y,

(20) (21) (22)

lim v@@(y!xkI )/v@@(y)"1. (23) y?= Suppose that Eq. (15) does not bind in Problem 4.2, and that there exist three distinct states, z , z , and z , such that pJ (z )pJ (z )pJ (z )'0 holds. ¹hen, at an in1 2 3 1 2 3 terior optimum, [cJ H(z )!cJ H(z )][cJ H(z )!cJ H(z )]"0 must be satisfied. 2 3 2 2 2 2 2 1 The proof of Proposition 3 appears in the Appendix A. The proposition states conditions under which a planner will place positive probability on at most two distinct old age consumption levels for type H agents. Thus restricting consideration to two states in Problem 4.1 had no impact on our results. To conclude this section, we describe how the optimal stochastic allocation can be decentralized. In order to do so, we augment the notation from Section 2. Consider the savings problem of a young agent who now faces a random lump-sum tax of q (z) when old, z"1,2, where the probability that z"1 is p, 2 and who faces a deterministic gross rate of return r. The agent chooses s to maximize u(w !q !s)#pv(w !q (1)#rs)#(1!p)v(w !q (2)#rs), 1 1 2 2 2 2 where the solution is a savings function s,fI (w !q , w !q (1), w !q (2), 1 1 2 2 2 2 r; p). The optimal random consumption allocation can be supported by having the government issue two types of bonds, while it simultaneously prohibits agents from intermediating them. The bonds sold to type ¸ agents are sold in a minimum denomination of Fl and bear a deterministic return rl. The government chooses Fl and rl to satisfy cL"w !Fl and cL"w #rlFl. The bonds 2 2 1 1 sold to type H agents are sold only in the indivisible amount Fh, and bear a gross return rh(1) with probability p, and rh(2) with probability 1!p. The government 8 All of the conditions (20)—(23) are satisfied if v(c)"c1~o/(1!o); o'0.

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chooses Fh, rh(1) and rh(2) to satisfy cH"w !Fh!fI [w !Fh,w #rh(1)Fh,w #rh(2)Fh, x; p], 1 1 1 2 2 cH(1)"w #rh(1)Fh#xfI ( ) ), 2 2 cH(2)"w #rh(2)Fh#xfI ( ) ). 2 2 Thus, the government issues some bonds with a certain return, and — at the same time — issues other bonds bearing an explicitly randomized return.

5. Historical evidence We have considered the problem of a government with a deficit to finance that (a) cannot compel participation in the bond market, and (b) cannot directly observe the characteristics of agents. In addition, (c) some agents have access to outside investment opportunities, and (d) agents have differential access to these investments. Under these conditions — and when the government can limit ‘intermediation’ of its liabilities — we have shown that the extraneous randomization of returns on government liabilities can be desirable from a welfare perspective. The model predicts: (i) Extraneous randomization of returns on government liabilities is desirable only when the deficit is fairly large (Eq. (7) binds). (ii) When extraneous randomization is employed, it occurs on bonds issued in large minimum denominations yielding relatively high expected returns. (iii) Extraneous randomization is observed only if the government can limit intermediation of its liabilities.9 This section delineates historical instances where governments made heavy use of liabilities with extraneously randomized returns, when their revenue needs were large. The use of such instruments ended when the ‘intermediation’ of government bonds became sufficiently efficient. The liabilities were issued in large denominations and offered relatively high expected returns. Thus the predictions of the model seem to be substantiated by historical observation. For example, in the 17th and 18th centuries the British and French governments were frequently at war. During these periods of large sustained deficits, both governments relied heavily on debt instruments with extraneously randomized returns. The instruments fell into three categories. 9 If the government cannot prevent intermediation, it is not possible to decentralize the solution to Problem 4.1. Without limitations on intermediation, agents can enter into pools and purchase a large number of government bonds with extraneously randomized returns. This sheds the risk while preserving the high return on these assets. Such actions then negate the potential use of such bonds as a solution to the adverse selection problem.

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1. ‘¸ottery-loans’ or lottery-bonds. Bond holders purchased a ‘ticket’ which entitled them to a minimum ‘prize’ plus the chance at augmented earnings. In the English lottery-loan of 1694, 100,000 tickets were sold at a price of 10 pounds each.10 ‘Each ticket guaranteed a minimum return ... (of ) one pound annually for sixteen years, an effective interest rate of 6%’, (cf. Jennings and Trout, 1982, p. 28). Prizes made the effective (mean) interest rate paid by the government 11.25%, which Jennings and Trout (1982), p. 29) indicate was high by current standards. 2. ¸ife annuities. The purchaser paid the government a stipulated principal and named a nominee. The annuitant received interest as long as the nominee was alive, which made the return random to the holder while the government faced little or no payment randomness (it benefited from the ability to sell a large number of life annuities). The annuitant could name anyone as nominee, thus arranging a random return not contingent on his own life. 3. ¹ontines. In this case a fixed payment by the government was divided among debt holders whose nominees were still alive. The government faced little randomness in its payments if it had a large set of subscribers. The lottery aspects of government tontine sales (from the point of view of purchasers) were well recognized by the governments that employed them (cf., Jennings and Trout, 1982, pp. 6, 26). Tontines were sold in large minimum denominations and offered high expected returns (Weir, 1989). The price of a tontine share in France was 300 livre when many workers were earning one or two livres a day (cf. Jennings and Trout, 1982, p. 2), and the price of a share in some 18th century English tontines was as high as 100 pounds. Moreover, the importance of the inability to compel bond market participation in the use of these kinds of debt instruments is illustrated by Dickson (1967) and Jennings and Trout (1982), who discuss the offerings of instruments that were under-subscribed.11 The relevance of private information is considered by Dickson (1967, p. 78), who argued (in the English case) that ‘the Exchequer 2was never certain who its creditors were at any one time’. Finally, these liabilities were used primarily in periods of large deficits, since they were mainly issued in wartime. Thus, the use of liabilities with randomized returns accords well with the predictions of our model. Nor were the episodes we have described isolated in nature: Dickson (1967), Jennings et al. (1988), Weir (1989), Weir and Velde (1992), and Calomiris (1992) describe a large number of episodes in which governments issued bonds with extraneously randomized payoffs. 10 This is of the same order of magnitude as per capita income at the time, and substantiates the model’s prediction that these bonds would be issued in large denominations. 11 A further illustration of the inability to compel participation is that ‘the (French) government even made them (its debt instruments) available to residents of nations at war with France’ (cf. Jennings and Trout, 1982, p. 52.)

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Of course, for welfare improvements to result from extraneous randomization in government borrowing, the government must be able to limit agents from ‘intermediating’ or ‘sharing’ its liabilities. Agents did attempt to do exactly this. That governments sought to inhibit such intermediation is indicated by the fact that the English Parliament made sharing of lottery-loan tickets illegal in 1743 (cf. Dickson, 1967, p. 507).12 The French government also made some attempts to prevent life-annuities and tontines from being intermediated (Weir, 1989, p. 111; Weir and Velde, 1992, pp. 32—33). As the intermediation of bonds with randomized returns became more common and harder to prevent,13 the use of these bonds waned and eventually ceased altogether. An example of this intermediation and its consequences occurred in Geneva in the 1760s, where syndicates were formed which mobilized pools of investors. These syndicates compiled a list of nominees for tontines with the nominees consisting of young girls who had survived smallpox and who came from families with a history of longevity. Once this kind of syndicate became sufficiently widespread, the French government ceased to issue tontines altogether. The lack of ability to prevent intermediation of this kind no doubt goes a long way toward explaining why government bonds with randomized returns are not employed today.14 All of these observations are consistent with the predictions of our analysis. An alternative explanation for them is that governments were simply attempting to exploit a taste for gambling by running analogs to modern state lotteries. We reject this explanation for two reasons. First, if participants in the schemes were ‘risk-preferers’, then the relevant debt instruments should have borne no higher expected returns than alternative instruments with deterministic returns. As we have noted, this was not the case. Second, unlike modern state lotteries, ‘lottery-tickets’ in our episodes were generally sold in quite large minimum denominations. As would therefore be expected, Dickson (1967, p. 302) reports that in England these instruments were held by the middle and upper classes. In contrast, Clotfelter and Cook (1990) report that participants in modern state lotteries are primarily from lower socio-economic classes. Thus, the explanation we propose seems a reasonable one for observed historical experience.

12 It is also the case that a 1697 act of Parliament forbade brokers to deal in any government securities without the permission of the treasury (cf. Dickson, 1967, p. 493). 13 The development of this intermediation was clearly stimulated by improvements in recordkeeping and communication (cf. Jennings and Trout, 1982). 14 It is worthy of note that many Japanese banks do offer deposits with extraneously randomized returns. We thank Preston McAfee for bringing this fact to our attention.

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6. Conclusions We have described an environment in which a government must finance a fixed deficit of a given size. When some agents have access to investment opportunities other than government bonds, government borrowing is constrained by the desirability of keeping these agents in the bond market. However, treating some agents preferentially creates an adverse selection problem. The optimal solution to these two problems involves price discrimination by the government, and may involve the simultaneous use of bonds with random and non-random returns. Interestingly, agents with the best outside investment opportunities purchase bonds with random returns, and extraneous randomization of bond returns is observed only when the government’s revenue needs are sufficiently large. These two features accord well with the historical observations cited in Section 5. The constrained Pareto efficient bond policy we consider is a form of price discrimination, thus the existence of secondary markets or insurance would undermine it. For simplicity, we assume that the government can impose legal restrictions which prohibit the intermediation of bonds with randomized returns. However, in this respect it is interesting to note that poorly developed financial and insurance markets are common in many high inflation countries that choose to monetize their deficits, often due to specific forms of government intervention. Bencivenga and Smith (1992) study the optimal degree of financial repression in a developing economy faced with a sustained deficit that must be monetized, and find that a government with a deficit (that is unwilling or unable to decrease spending or increase explicit taxes) may be required by simple feasibility to engage in financial repression to support its monetization program. Such repression is much more difficult in more developed countries, which no doubt explains why explicit ‘lottery bonds’ are not typically observed today in more developed economies. We also believe our analysis sheds light on two other issues of general interest. The first is the question, raised by Arnott and Stiglitz (1988) among others, of why extraneous randomization does not arise as often as theory seems to suggest. Arnott and Stiglitz suggest six potential answers: (1) agents do not understand that randomization is optimal; (2) randomized contracts may be costly to enforce; (3) secondary markets or insurance neutralize the effects of randomization; (4) agents view lotteries as unfair; (5) expected utility theory is deficient; or (6) individuals do not trust randomization mechanisms. As Section 5 indicates, governments have historically made heavy use of bonds with randomized returns — in a way that is consistent with our theory — so long as they could prevent bonds from being intermediated. This suggests that only (3) is persuasive in light of historical observation.

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Acknowledgements We thank Ricardo Caballero, Ed Green and an anonymous referee for comments, and gratefully acknowledge financial support from NSF grant SES 89-09242.

Appendix A. A.1. Proof of Proposition 1 Consider Problem 3.2 with only the binding constraints displayed. h [u(ci )#v(ci )] subject Problem 3.2.@ Choose ci , ci , and k to maximize + 2 1 i/H,L i 1 2 to Eqs. (1), (2) and (5). Let j 50, n"1,2,3, be the Lagrange multiplier associated with Eqs. (1), (2) n and (5), respectively. At an interior optimum, (A.1) u@(cH)(h #j )!j u@(cH#k)"h j , H 1 1 H 2 3 1 (A.2) v@(cH)(h #j )!j v@(cH!xk)"h j , H 1 2 H 2 3 2 (A.3) j [v@(cH!xk)x!u@(cH#k)]!j h (1!x)40, 1 1 H 2 3 (A.5) u@(cL)(h #j )"h j , 1 L 3 L 1 (A.6) v@(cL)(h #j )"h j . 2 L 3 L 1 Eqs. (A.4) and Eq. (A.5) imply that u@(cL)"v@(cL), while Eqs. (A.1), (A.2) and (A.3) 2 1 imply that (h #j )[xv@(cH)!u@(cH)]40, with equality if k'0. 1 2 H 2 We now establish that k'0. To do so suppose, for the purpose of deriving a contradiction, that k"0. Then Eqs. (A.1) and (A.2) imply that u@(cH)"v@(cH). 2 1 Since Eq. (2) binds, it follows that cH'cL, for t"1,2. But then Eq. (5) is t t violated, giving the desired contradiction. Thus k'0. It remains to establish that f (w , w , x)'k holds. Note that Eq. (2) binds and 1 2 (h #j )[xv@(cH)!u@(cH)]"0. These two equations yield a unique solution 1 2 H 2 given by cH"w !f (w , w , x) and cH"w #xf (w , w , x). Since Eq. (2) is 2 2 1 2 1 1 1 2 an equality, Problem 3.2@ reduces to maximizing u(cL)#v(cL), subject to 2 1 Eqs. (1), (2) and (5). From Eq. (1) at equality, h (cL#cL!w !w )" 2 1 2 L 1 h [w #w !cH!cH!(1!x)k]!g"h (1!x)[f (w , w , x)!k]!g. 2 H 1 2 1 H 1 2 Clearly then, it is not optimal to set k5f (w , w , x). This completes the 1 2 proof. h A.2. Proof of Proposition 2 Consider the following augmented version of Problem 4.1:

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Problem 4.1@. Choose ci , ci (z) and k to maximize + h Mu(ci )#pv(ci (1)) 1 2 i/H,L i 1 2 #(1!p)v(ci (2))N subject to: Eqs. (6), (7) and (10) and 2 (A.7) u@(cH)"xpv@(cH(1))#x(1!p)v@(cH(2)). 2 2 1 Since the solution to Problem 4.1 satisfies Eq. (A.7), imposition of this constraint does not alter the optimal choices for the social planner. Eq. (6), Eq. (7), and Eq. (10), which hold as equalities, and Eq. (A.7) constitute four equations involving cH, cH(1), cH(2), k, cL, and cL [since cL(1)"cL(2)"cL]. We now use 2 2 2 2 1 2 1 2 Eq. (6), Eq. (10), and (A.7) to eliminate cH, cH(1), and k from Problem 4.1@. 1 2 First, let Eq. (A.7) define a function a such that cH,a(cH(1), cH(2)). Clearly, 2 2 1 cJ H"a(cJ H, cJ H) holds. Differentiation of Eq. (A.7) yields 2 2 1 v@@(cH(1)) 2 '0. (i) a (cH(1),cH(2))"px 2 1 2 u@@(cH) 1 v@@(cH(2)) 2 '0. (ii) a (cH(1), cH(2))"(1!p)x 2 2 2 u@@(cH) 1 Second, substitute cH"a(cH(1), cH(2)) into Eq. (6) at equality. This gives 2 2 1 a function b which satisfies k,b(cH(1), cH(2); cL, cL). Observe that 1 2 2 2 kI "b(cJ H, cJ H; cJ L, cJ L) holds, and that differentiation of b( ) ) yields: 1 2 1 2 (iii) b "!((a #p)/(1!x))(0. 1 1 (iv) b "!((a #1!p)/(1!x))(0. 2 2 Third, substitute cH"a(cH(1), cH(2)) and k"b( ) ) into Eq. (10) at equality. 2 2 1 This defines a function c such that cH(2),c(cH(1); cL, cL). As before 1 2 2 2 cJ H"c(cJ H; cJ L, cJ L). Moreover, differentiation of c( ) ) yields 2 1 2 2 (v) c (cJ H;cJ L,cJ L)"!p/(1!p). 1 2 1 2 Finally, define a function d as follows: d(cH(1); cL, cL),u(a(cH(1), c( ) )))#pv(cH(1))#(1!p)v(c( ) )), 2 1 2 2 2 where d( ) ) expresses the left-hand side of Eq. (7), the (binding) voluntary participation constraint for type H agents, solely as a function of cH(1) and cL, for t 2 t"1, 2. We now show that d( ) ) is locally convex in cH(1), so local randomization 2 relaxes Eq. (7) and consequently is welfare improving. ¸emma. (a) d (cJ H; cJ L, cJ L)"0 holds, and (b) d (cJ H; cJ L, cJ L)'0 if the inequality in 11 2 1 2 1 2 1 2 Proposition 2 is satisfied. Proof. Differentiation of d yields d (cJ H; cJ L, cJ L)"u@(a( ) ))[a #a c ]#pv@(cJ H)#(1!p)v@(cJ H)c . 1 2 1 2 1 2 1 2 2 1 Expressions (i), (ii), and (v) then imply part (a).

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For part (b), further differentiation yields d (cJ H; cJ L ,cJ L)"u@@(cJ H)[a #a c ]2#u@(cJ H)[a #a c #a c 1 11 12 1 21 1 1 1 2 1 11 2 1 2 #a (c )2#a c ]#pv@@(cJ H)#(1!p)v@@(cJ H)(c )2 2 1 2 22 1 2 11 #(1!p)v@(cJ H)c . 2 11 It is straightforward but tedious to show that when evaluated at (cJ H, cJ L, cJ L), 2 1 2 (vi) [a #a c #a c #a (c )2#a c ]" 11 12 1 21 1 22 1 2 11 1!p p v@@@(cJ H) 2 ; a c # p 1 11 (1!p)2 v@@(cJ H) 2 (vii) a #a c "0; 1 2 1 pxa v@@@(cJ H) 2 1 H (viii) c (cJ ; cJ L, cJ L)(p#xa )"! 11 2 1 2 1 (1!p)2v@@(cJ H) 2 p2v@@(cJ H!xkI )(1!x) 2 ! . (1!p)2[v@(cJ H!xkI )!u@(cJ H#kI )] 2 1 Substituting (vi), (vii) and (viii) into d gives 11 pv@@(cJ H) xa v@(cJ H)v@@@(cJ H) 2# 1 2 2 d (cJ H; cJ L, cJ L)" 11 2 1 2 1!p (1!p)v@@(cJ H) 2 xa #(1!p)v@(cJ H)c 1# 1 2 11 p

G

C

D

H

C

D

pv@@(cJ H) v@(cJ H)p(1!x)v@@(cJ H!xkI ) 2 2! 2 " . (1!p)[v@(cJ H!xkI )!u@(cJ H#kI )] 1!p 2 1 Clearly d (cJ H;cJ L,cJ L)'0 if the inequality in Proposition 2 holds. This completes 11 2 1 2 the proof. A.3. A sufficient condition for Pareto improving randomization We now show that Eq. (12) is sufficient to guarantee that there are welfare improvements from the extraneous randomization of returns on government liabilities. In order to do so, we define the following function: R(cJ H!xk)(1!x)v@(cJ H!xk) 2 G(cJ H, cJ H, k), 2 !R(cJ H). 1 2 2 v@(cJ H!xk)!u@(cJ H#k) 2 1 Since (cJ H, cJ H) satisfies u@(cJ H)"xv@(cJ H), it follows that G(cJ H, cJ H, 0)"0. Moreover 1 2 1 2 1 2 u(cJ H)#v(cJ H)"»(w , w , x), so (cJ H, cJ H) is completely determined and indepen1 2 1 2 1 2 dent of k. G is effectively a function of k alone, and if G (cJ H, cJ H, k)'0 for all 3 1 2 k'0, then G(cJ H, cJ H, kI )'0 will hold. This is exactly Eq. (11). Differentiation of 1 2

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G( ) ) establishes that G '0 holds iff 3 xR@(cJ H!xk) 2 ! R(cJ H!xk) 2 u@(cJ H#k) u@@(cJ H#k) 1 1 '! !xR(cJ H!xk) . 2 H H v@(cJ !xk)!u@(cJ #k) u@(cJ H#k) 1 2 1 Since xv@(cJ H!xk)5u@(cJ H#k) for all k50, a sufficient condition for 1 2 G (cJ H, cJ H, k)'0, for all k50, is 3 1 2 (1!x)R@(cJ H!xk) u@@(cJ H#k) 2 1 ! 5xR(cJ H!xk)! . 2 R(cJ H!xk) u@(cJ H#k) 2 1 Multiplying both sides by (cJ H!xk) gives Eq. (12). 2

G

G

HG

H

H

A.4. Proof of Proposition 3 Let j ,2, j denote the Lagrange multipliers associated with constraints 1 7 Eqs. (13)—(19), respectively, in Problem 4.2. In addition, let cJ L, cJ L, kI , cJ H, cJ H(z), and 1 2 1 2 pJ (z) denote the optimal allocation in the same problem, while jI , j"1,2, 7, j denotes the corresponding values of the multipliers. Since Eq. (15) and Eq. (16) do not bind (jI "jI "0), at an interior optimum the following first order 3 4 condition for cH(z) must be satisfied: 2 pJ (z)(h #jI )v@[cJ H(z)]"pJ (z)MjI h #jI v@[cJ H(z)!xkI ]N. (A.8) H 2 2 1 H 5 2 Observe that since pJ (z)'0 must hold for some z, Eq. (A.8) implies the following is satisfied: jI /(h #jI )(1. 5 H 2 We now define the function Q by

(A.9)

Q(y),(h #jI )v@(y)!jI v@(y!xkI )!h jI , H 2 5 H 1 where jI , jI , jI , and kI are fixed at their optimal levels. Then, if pJ (z ), pJ (z ), and 1 2 5 1 2 pJ (z ) are all positive, Eq. (A.8) implies that cJ H(z ), cJ H(z ), and cJ H(z ) are all 3 2 1 2 2 2 3 solutions to the equation Q[cJ H(z)]"0. (A.10) 2 Conditions (20) and (21) imply that Q(xkI )"!R (since kI '0 must hold), and that Q@(xkI )'0. In addition, conditions (22) and (23) allow one to establish the existence and uniqueness of a value yL 3(xkI ,R) such that Q@(y)'(()0 holds iff y((')yL . It is immediate that Eq. (A.10) can have at most one solution in the interval (0, yL ], and similarly it can have at most one solution in the interval [yL , R). It follows that either cJ H(z )"cJ H(z ), or cJ H(z )"cJ H(z ) must hold, 2 2 2 3 2 2 2 1 establishing the result.

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