Optimal Denominations

Transcription

1 Optimal Denominations Manjong Lee, Neil Wallace, and Tao Zhu February 16, 2004 Abstract Previous work on the denomination structure of currency treats as exogenous the distribution of transactions and the denominations held by people. Here, by way of a matching model, both are endogenous. In the model, trades in pairwise meetings alternate in time with the opportunity to freely choose a portfolio of denominations and people trade off the benefits of small-denomination money for transacting against the costliness of carrying a large quantity of smalldenomination money. For a given denomination structure, a monetary steady state is shown to exist. Powers-of-two and powers-of-three denomination structures are compared numerically for an example. JEL classification #: E40. Key words: currency, denominations, matching-model. 1 Introduction The small literature on the optimal denomination structure of a currency treats as exogenous both the distribution of transactions and the portfolios of denominations held by transactors. Telser [6] starts from a problem in number theory. The problem of Bâchet seeks the smallest number of weights capable of weighing any unknown integer.... The version pertinent Department of Economics, The Pennsylvania State University Department of Economics, The Pennsylvania State University Department of Economics, Cornell University 1

2 to this paper allows the unknown weights to be placed in either pan....the solution is weights that are powers of three, namely, 1, 3, 9, 27 and so on... [For the currency version,] the unknown amount to be weighed corresponds to a... transaction....allowing weights to go in either pan corresponds to making change. Finally, that the weights must be capable of weighing any quantity between one and [some] upper bound corresponds to the assumption that a transaction is equally likely to be anywhere between one and some finite upper bound (Telser [6] pages, ). Bâchet s problem is a one-person problem and the application of the solution to multiple transactors, at least a buyer and a seller, is far from straightforward; in particular, who owns which denominations? Van Hove and Heyndels [8] also assume a uniform distribution of transactions. They assume that the buyer and the seller each hold at least one unit of each denomination and judge denomination structures by the number of monetary objects needed to be exchanged to accomplish the transactions. They show that a powersof-two structure, 1, 2, 4,.., can result in a lower average number of monetary objects used in transactions. The main limitation of all of this work is that too much is taken to be exogenous. Here, we formulate a model in which the distribution of transactions and the portfolios of denominations held are endogenous. We use a matching model of money a model in which trades occur in pairwise meetings, between a buyer and seller. Those meetings alternate in time with the opportunity to freely choose a portfolio of denominations subject, of course, to a wealth constraint. The two-person trade is natural for the study of denomination structure because it brings to the fore the limitations imposed on a buyer and seller if neither has small change. Our model focuses on the trade-off between the benefits for trade of small-denomination money and the costliness for people of carrying a large quantity of small-denomination money a trade-off which is in the spirit of the objectives used in the previous literature. For a given a denomination structure, we show that there exists a nice monetary steady state, nice in that the implied value function for (monetary) wealth is strictly increasing and concave. 1 Then we report on numerical 1 We often apply the term concave to functions defined on discrete subsets of R n. Suppose X I n, where I is an interval. We say that f : X R is (strictly) concave if 2

3 work in which we compare expected utility across steady states with different denomination structures expected utility where the expectation is taken with respect to the steady-state distribution of wealth. Although we could, in principle, search numerically for an optimal denomination structure, we limit ourselves to comparing two alternatives addressed in the previous literature: one is the powers-of-three structure; the other is the powers-of-two structure. The latter is the solution to a version of Bâchet s problem in which the object to be weighed is in one pan and weights are placed only in the other pan, a no-change version. Although the making of change is permitted in our model, in our example the powers-of-two structure is better than the powers-of-three structure. 2 The Model The background environment is borrowed from Shi [4] and Trejos and Wright [7]. Time is discrete. There is a [0, 1] continuum of each of N 3 types of infinitely lived agents and there are N distinct produced and perishable types of divisible goods at each date. A type n agent, n {1, 2,...,N}, produces type n good and consumes only type n+1 goods (modulo N). Each person maximizes expected discounted utility with discount factor β (0, 1). Trading histories are private so that any kind of borrowing and lending in pairwise meetings is impossible. The rest of the model is best described in terms of the sequence of actions. Very briefly, the sequence is as follows. Each person starts a period with a portfolio consisting of integer quantities of various denominations for example, 3 ones and 7 fives. Then the person is allowed to freely adjust denominations for example, exchange 2 fives for 10 ones. Then the person meets one other person at random. They trade, as described below, and then go on to the next period. To facilitate a more detailed description, it is helpful to introduce some notation. Let s = (s 1,s 2,...,s K ) be a given denominational structure of the currency. Here s k is the size of the k-th smallest denomination. We assume that 1/s 1 is a positive integer, that s k /s 1 is an integer, and that s k > s k 1. (For example, in the powers-of-three structure, s k = 3s k 1.) The generic symbol for a person s portfolio at the start of a date is y = (y 1,y 2,...,y K ), where y k 0 is the integer quantity of size-s k money held. there exists g : I n R such that f is the restriction of g to X and g is (strictly) concave. 3

4 The wealth implied by y is the inner product of s and y, which we denote sy. Because individuals can freely choose portfolios, the state of an individual with portfolio y at the start of a date is simply sy. The state of the economy at that time is the distribution of wealth. We assume that each specialization type has the same distribution. To achieve compactness, we assume that an individual s wealth z {0,s 1, 2s 1,...,Z} Z, where Z is an integer. We let π denote a probability measure on Z, where π(z) is the fraction of each specialization type with wealth z. As a normalization, we impose zπ(z) = 1. With this normalization, 1/s 1 is the ratio of mean wealth to the size of the smallest unit of money and Z is the ratio of the bound on wealth to mean wealth. We let Y = {y = (y 1,y 2,...,y K ) Z K + : sy Z} be the set of feasible individual portfolios. (Here, Z + denotes the set of non negative integers.) And we let θ denote a probability measure on Y, where θ(y) is the fraction of each specialization type with portfolio y. A person with wealth z chooses y Y with which to enter trade subject to sy z. For a type n person with portfolio y just prior to pairwise meetings, realized utility in a period is u(q n+1 ) q n γ k y k where q n+1 R + is consumption of good n + 1 and q n R + is production of good n, and γ > 0 is the utility cost of carrying money per piece of money. (The linear carrying-of-money cost function could be replaced by a more general class of convex functions of k y k.) The utility function u : R + R is strictly concave, strictly increasing, continuously differentiable and satisfies u(0) = 0 and u ( ) = 0. In addition, u (0) is sufficiently large, but need not be infinite. After the portfolio is chosen, a person meets another person at random. We assume that trading partners see each other s portfolios and types. There are no-coincidence meetings during which nothing happens. In singlecoincidence meetings, we assume that the buyer makes a take-it-or-leave-it offer. We formally define and prove existence for a version in which people can choose lotteries both at the portfolio choice stage and in meetings. The definition is similar and the existence result holds for a version without lotteries (by allowing randomization over multiple deterministic optima for the individual problems). To save space, they are not presented. However, both versions are studied in our numerical work. 4

5 3 Existence of a Steady State We begin by defining a steady state. For a given denomination structure, s, a steady state is a collection of functions (w s,π s,θ s ) that satisfies the conditions described below. The functions w s and π s pertain to the start of a period, prior to the choice of portfolios: w s : Z R, where w s (z) is the expected discounted value of having wealth z, and π s : Z [0, 1], where π s (z) is the fraction of each specialization type with wealth z. The function θ s : Y [0, 1] is a measure where θ s (y) is the fraction of each specialization type with portfolio y. We suppress the dependence of a steady state on the denomination structure s whenever the context is clear. We begin with the choice of a portfolio. We let a person with wealth z choose any lottery over portfolios in Y whose value does not exceed z. That is, we let Γ 1 (z), a subset of probability measures defined on Y, be defined by Γ 1 (z) = {σ : σ(y) = 0 if sy > z}. (1) We let h : Y R denote expected discounted utility after the portfolio is chosen and before pairwise meetings. In terms of h, the portfolio-choice problem is g(z,h) = max σ(y)h(y). (2) σ Γ 1 (z) We denote the set of maximizers in (2) by 1 (z,h). That is, 1 (z,h) is a subset of probability measures on Y, where δ 1 (z,h) is an optimal lottery and δ(y) is the implied probability of holding portfolio y. We now turn to trade in meetings. Let S(y) = {v Z K + : v k y k }. (3) That is, S(y) is the set portfolios that a person with y could surrender. Now, consider a meeting between a buyer with y and a seller with y. We let X(y,y ) = {x [0,Z sy ] : x = s(v v ),v S(y),v S(y )}. (4) That is, X(y,y ) is the set of feasible wealth transfers from the buyer to the seller taking into account the denominations held and the possibility of making change. Let W be an upper bound on w that is defined in the existence proof. We define Γ 2 (y,y,w), a set of probability measures on [0,W] X(y,y ), by 5

6 Γ 2 (y,y,w) = {σ : E σ [ q + βw(x + sy )] βw(sy )}, (5) where E σ denotes the expectation with respect to σ and where the arguments of σ are (q,x). Then we let f(y,y,w) = max E σ [u(q) + βw(sy x)]. (6) σ Γ 2 (y,y ;w) This description implies the payoffs from trade. That is, the buyer s payoff is f(y,y,w) and, because the inequality in (5) will be binding, the seller s is βw(sy ). We denote the set of maximizers in (6) by 2 (y,y,w). Because it can be shown that each maximizer is degenerate in q, in what follows, for a maximizer, we denote the maximizing q by ˆq. To describe the law of motion, it is convenient to define z 1 (y,y ) = {x : sy x = z} for z Z, the set of asset trades that leave the buyer with wealth z. Then we define 2 (y,y,w), a subset of probability measures on Z, by 2 (y,y,w) = {δ : δ(z) = δ(ˆq,x), for δ 2 (y,y,w)}. (7) z 1 (y,y ) That is, for δ 2 (y,y,w), δ(z) is the probability that the buyer with y leaves with wealth z after meeting a seller with y. Now we can complete the conditions for a steady state. The value function w must satisfy w(z) = g(z,h), (8) where the function h is defined by h(y) = γ y k + N 1 N βw(sy) + 1 θ(y )f(y,y,w). (9) N And the measures π and θ must satisfy where π T π (w,θ) and θ T θ (w,π,θ), (10) T π (w,θ) = {ω : ω(z) = (y,y ) θ(y )θ(y )[δ(z) + δ(sy z + sy )], for δ 2 (y,y,w)}, (11) 6

7 and T θ (w,π,θ) = {ω : ω(y) = z π(z)δ(y) for δ 1(z,h)}. (12) (Notice that T π (w,θ) is a set of probability measures on Z and T θ (w,π,θ) is a set of probability measures on Y. Also, notice that the dependence of T θ (w,π,θ) on (w,θ) is through the dependence of h on (w,θ). Finally, in (11), as a convention, δ(x) = 0 if x / Z.) Definition 1 Given a denomination structure s, a steady state is (w,π,θ) that satisfies (1)-(12). Now we prove that if u (0), 1/s 1, and Z are large enough and if γ is small enough, then there exists a nice steady state. Here nice, as above, means a steady state with a strictly increasing and concave value function. We summarize the result formally as follows. Proposition 1 Given a denomination structure s, if u (0), 1/s 1, and Z are sufficiently large and if γ is sufficiently small, then there exists a steady state (w,π,θ) with w strictly increasing and strictly concave. Proof. See Appendix 1. As is well known, the challenge in models like the one above is to show that there is a reasonable monetary steady state. The monetary challenge is met by the conclusion that w is strictly increasing; the reasonableness challenge is met by the concavity of w. The proof shows that there is a neighborhood of γ = 0 with such a steady state. Given earlier results in [10] and [11], the proof is very simple. The proof establishes properties of the mapping used to define a steady state. First, it uses the fact previously established in [11], which, in turn, rests on the results in [10] that for γ = 0 the mapping has fixed-point index 1. The result in [11] can be applied because if γ = 0, then portfolios that consist entirely of the smallest denomination money are equilibrium portfolios. But given such portfolios, the model is the same as the money-only case of the model in [11]. (For such portfolios, the non lottery version is the model in [10].) Second, the mapping is shown to be upper hemicontinuous in γ at γ = 0. Those two facts give the result. Notice, by the way, that the mapping is not continuous in γ at γ = 0; that is, a portfolio that is an equilibrium portfolio when γ = 0 is not, in 7

8 general, close to those that are equilibrium portfolios when γ > 0 and small. That is why the model can have rich implications for portfolios even if γ is small. In accord with such discontinuity, there is a general conclusion about portfolios that is related to Telser s conclusions about optimal denomination structure; namely, portfolios do not have surplus divisibility. Lemma 1 If σ solves the problem in (2) and y is in the support of σ, then ỹ satisfying (i) sỹ = sy, (ii) S(y) S(ỹ), and (iii) k ỹk < k y k. Proof. Suppose, by contradiction, that ỹ exists. It suffices to show that h(ỹ) > h(y). By (9) and properties (i) and (iii), h(ỹ) > h(y) if f(ỹ,y,w) f(y,y,w). To establish the latter, suppose x X(y,y ) (see (4)). Then there exists v and v with x = sv sv with v S(y). But by condition (ii), there exists ṽ with ṽ S(ỹ) with sṽ = sv. Therefore x X(ỹ,y ), which implies that any offer that is feasible when the buyer has y is also feasible when the buyer has ỹ. This and (i) imply f(ỹ,y,w) f(y,y,w). To see that this lemma rules out some portfolios, suppose s is the powersof-two structure and consider y = (3, 0,..., 0) and ỹ = (1, 1, 0,...0). If σ solves the problem in (2), then y cannot be in the support of σ because ỹ satisfies the three conditions in the lemma. And, using strict concavity of w, we can say a bit more about offers in meetings. The following lemma shows how to use the best trade when the buyer s portfolio consists entirely of the smallest denomination to construct the best trade for all portfolios with the same wealth. The best trade is described in terms of a lottery, possibly degenerate, over the amount of wealth transferred. The corresponding implied (deterministic) output is that which satisfies the producer s participation constraint in (5) with equality. Lemma 2 Consider a buyer with y and a seller with y. (i) If the buyer s portfolio is (sy/s 1, 0, 0,..., 0), then the solution for the wealth transfer from the buyer is a lottery over {m,m + s 1 } for some m Z. (ii) Let m be as described in part (i). In general, the solution for the wealth transfer from the buyer is a lottery over {m,m + } = {max x m X(y,y ), min x m+s 1 X(y,y )}, (13) where X(y,y ) is defined in (4) and where the lottery is degenerate on m if m + does not exist. 8

9 Proof. Both (i) and (ii) are obvious consequences of strict concavity of w. Lemma 2 helps simplify the computation of a steady state. 4 Comparing Powers of Two and Three We compare welfare under different denomination structures in terms of πw expected utility prior to the assignment of wealth holdings, where the assignment is made in accord with the steady-state distribution. Here, we compare two denomination structures: powers-of-two and powers-of-three. We make this comparison for two versions, one with and one without lotteries. We assume that both denomination structures have the same s 1, the same smallest denomination. The standard computational procedure involves iterating on the mapping studied to prove existence, the usual mapping studied in heterogeneous-agent models. Even conceptually, this is not straightforward. First, as in many other models, there are no theorems that guarantee that the iterates of such a mapping converge. Second, there are no uniqueness results pertaining to nice steady states. Nevertheless, experience with related one-denomination models (see Molico [2]) suggests that neither of these issues arises in computations. As noted above, we report results for the version with lotteries and the version without lotteries. Lotteries and randomization are identical at the portfolio choice stage. The two differ with regard to what happens in meetings and that difference, of course, produces different steady states. Even in the non lottery version, we always find unique outcomes for both choice problems. 2 For outcomes for which there is a unique portfolio associated with each wealth level, it is well-known that πw satisfies (1 β)πw = πc + 1 N πgπ. (14) 2 This may seem surprising for the portfolio choice. We suspect that it happens because the wealth distribution is smooth and has full support. If the wealth distribution were degenerate, then any steady state would seem to have mixing. After all, if everyone else has the smallest denomination, then why should I? 9

10 Here the z th component of the vector c is the carrying cost implied by the portfolio associated with wealth level z, and the element in row z and column z of the matrix G is G zz = u[q(z,z )] q(z,z ), (15) where q(z,z ) is output when the buyer has wealth z and the seller has wealth z. If q maximizes u(q) q, then u(q ) q is an (unattainable) upper bound on N(1 β)πw. This upper bound would be attained if q was produced and consumed in every single-coincidence meeting and no money was held. 4.1 Parameters We need to specify u,n,β,z,s 1, and γ. We let u(q) = q 1/2 /(1/2). This satisfies all our assumptions and has the virtue of implying a linear firstorder condition for the choice of lotteries in meetings. It implies q = 1 and u(q ) q = 1. We let N = 3, the smallest magnitude consistent with our assumption. We let β =.9 (1/12), which corresponds to one meeting per month and an annual discount factor of.9. We choose Z = 4, which is large enough so that almost no one is at the bound in a steady state. We set s 1 = 1. 8 Recalling that 1/s 1 is the ratio of the mean amount of (monetary) wealth to the size of the smallest unit, this choice of s 1 is a large smallest unit. However, it is consistent with France in the 17-th century if the smallest denomination is taken to be the single coin that was being minted. 3 These choices of Z and s 1 imply that there are 33 elements in the set Z, a conveniently small number for computations. For these choices, s 2 = 1 (1, 2, 4, 8, 16) is the 8 powers-of-two denomination structure and s 3 = 1 (1, 3, 9, 27) is the powersof-three denomination structure. 4 As regards γ, we study γ =.001 which 8 makes the carrying cost per unit a tenth of one percent of the best level of output. 3 See Redish [3] for the size of the coins being minted and see Spooner [5] for estimates of the per capita stock of money. 4 Notice that we have omitted from the powers-of-two structure s 6 = 4. Even omitting this, the powers-of-two structure turns out to be better. In any case, omitting it has no discernible effect on welfare because almost no one has enough wealth to hold such a unit. 10

11 4.2 Computational details The mapping used for the iterative scheme is the mapping studied to prove existence in Appendix 1. The mapping uses an initial condition: (w,π,θ) = x. Given an iteration-i outcome, denoted x i, the initial condition for the i+1 iteration, denoted z i, is formed by a weighted average of the components x i and x i 1, where different weights are used for the different components. The convergence criterion is max (z i+1 z i )/z i 10 5, where the maximization is taken over all elements except that (i) w(0), which is zero, is excluded, and (ii) π and θ are replaced by their corresponding cumulative distributions to avoid dividing by numbers near 0. 5 To get initial conditions and as a check on our computations, we did two preliminary computations. Both are for one-denomination structures, ones that are equivalent to forcing people to hold only the smallest denomination money. We do this for γ = 0 and for the assumed γ. We use the former as one initial condition. The other initial condition we use is the latter, but with θ given by the portfolio that minimizes carrying costs. In each case, the resulting sequence converged to the same limit. 4.3 Results The results for welfare and some other summary statistics are in table 1. The first row describes the steady state we find if γ = 0 (and if we impose the restriction, innocuous in that case, that only the smallest denomination is held). The second row describes the steady state if we impose that only the smallest denomination is held. The outcomes for welfare for those cases bracket those for the powers-of-two and three structures. Both for the lottery and non lottery version, the powers of-two structure has higher welfare than the powers-of-three structure, although the differences are small, no matter how they are measured. In the table, Eq is expected output over singlecoincidence meetings. Underlying these welfare results are outcomes for π and w that do not vary much between the two denomination structures. Figure 1 shows w and π for the powers-of-two structure. (In these figures and those below, the domain is 8 times wealth, so that mean wealth corresponds to 8.) Those for the powers-of-three structure could not be distinguished visually. (See table 2 in Appendix 2 for the underlying data for both structures.) Although the 5 A copy of the computer code is available upon request. 11

12 Table 1: Comparison between powers-of-two and three Non Lottery Lottery Eq E k y k πw Eq E k y k πw γ = s 1 only Powers-of Powers-of wealth distributions have full support, almost no one has as much as 3 times average wealth. Not surprisingly, the lottery version has both a higher value function and a more concentrated wealth distribution. Figure 1: Powers-of-two: w and π Lottery 60 Lottery Non Lottery Non Lottery The higher ex ante welfare of the lottery version has several sources, including more meetings where money holdings of the buyer and the seller are consistent with output near unity. This is confirmed in figure 2 where we show the distribution of output over single-coincidence meetings for the powers-of-two structure. The distribution for the lottery version is much closer to the best output level, unity for our example, than is that for the non lottery version. 12

13 Figure 2: Powers-of-two: frequency distribution of output over singlecoincidence meetings Lottery Non Lottery As suggested by the results for the average number of monetary units held in table 1, the portfolios differ quite a bit across the two denomination structures. The portfolios in the lottery version are very simple: for each wealth level, there is a unique best portfolio and it is the deterministic portfolio that minimizes the number of units of money held. For example, in the powers-of-three structure, someone with wealth 27/8, approximately three times average wealth, holds a single unit of money, despite the fact that this portfolio and the implied trades produce a risk of losing almost all of it in meetings with sufficiently poor producers. The expected wealth transfers E σ x in (5) and (6) are shown in figure 4 in Appendix 2. The portfolios in the non lottery version do not minimize carrying costs (see table 3 in Appendix 2). In the powers-of-three structure steady state, every person holds at least one unit of the smallest denomination. And with one exception, offers of money are either 0, which happens when the seller is sufficiently wealthy relative to the buyer, or 1 unit of the smallest denomination. The one exception is that a buyer with upper bound wealth offers two units of the smallest denomination to a seller with zero. In the powers-of-two structure, those who are sufficiently wealthy do not hold the smallest denomination. For example, a person with wealth 28/8 holds the 13

14 portfolio (0, 2, 0, 1, 1). And a buyer with that portfolio offers 1/4 to a seller with 0, even though a buyer with wealth 29/8, who necessarily holds a unit of the smallest denomination, offers only 1/8 to such a producer (see figure 3 in appendix 2). Such non monotonicity of offers in the wealth of the buyer cannot happen in a one-denomination structure. Again, however, most offers are either 0 or 1 unit of the smallest denomination. The predominance of offers equal to 0 or 1 unit of the smallest denomination in the non lottery version is, of course, a consequence of the indivisibility implied by s 1 = 1/8. Moreover, that indivisibility almost certainly plays a role in our finding that the powers of-two structure is better than the powersof-three structure. 5 Concluding Remarks We have set out a very simple model suitable for studying the effects of different denomination structures. However, there are many effects of the denomination structure that are not in our model. For example, because we have assumed that money never wears out, replacement costs play no role. And we have not considered illegal activity and the impact of the denomination structure on the ease of carrying out illegal transactions. Even aside from those omissions, it is easy to come up with plausible variations of our model. For example, additional heterogeneity could be introduced by replacing our return function by u(q n+1 ) ηq n γ k y k, where η is an idiosyncratic taste (productivity) shock. Indeed, such heterogeneity would seem to bring the model closer to generating the kind of distribution of transactions assumed in the previous literature. And the assumption that the opportunity to choose a portfolio alternates over time with pairwise meetings could be generalized in many ways. Thus, as hardly needs to be mentioned, our numerical finding is, at best, in the nature of a counter-example to the idea that powers-of-three are best. Does that make our model and ones like it irrelevant? We think not. The model highlights the kinds of features that are needed in order to study the interaction between the denomination structure, on the one hand, and the portfolios held and the distribution of transactions, on the other. At a minimum, having before us a model in which those interactions are modeled and in which the optimum problem is posed in terms of welfare helps us see the limitations of models in which the distribution of transactions is 14

15 exogenous even if, in the end, we decide to use such a model. Of course, we have to concede that models of the sort we have described may have very limited use in the future. The computing power that permits us to study some of the properties of models like ours is also responsible for the widespread use of plastic in the form of credit, debit, and stored-value cards. If plastic were to completely replace currency, then divisibility would become essentially free and there would be no need to choose a denomination structure. 6 Appendix 1 As a prelude to the proof, we begin with some notation and assumptions. Let γ [0, 1]. Let R [N (N 1)β] 1 < 1. Also, let D be the unique solution to u (D) = [2/(Rβ)] 2. Notice that D > 0 provided that u (0) > [2/(Rβ)] 2, one of our assumptions. Let W be the unique solution of N(1 β)w = u(βw). We assume that 1/s 1 βw/d and that Z 4. Let W be the set of non-decreasing and concave functions w : Z [0,W] with w(4) D/β. Let K W be the set of non-decreasing functions from Z to [0,W]. Notice that the interior of W (relative to K) is non-empty and that an element of the interior is strictly increasing, strictly concave, and satisfies w(4) > D/β. Let Π be the set of probability measures π defined on Z satisfying π(z)z = 1. Let Θ be the set of probability measures θ on Y satisfying θ(y)sy = 1. Now we can formally define the mapping to be studied. Let the mapping T w : W Θ [0, 1] K be defined by T w (w,θ,γ)(z) = g(z,h), (16) where g(z,h) is given in (2) and h is given by (9). (Here, the dependence of T w (w,θ,γ) on (w,θ,γ) is through the dependence of h on (w,θ,γ) which is given by (9) with f given by (6).) We let T : W Π Θ [0, 1] K Π Θ be defined by T(w,π,θ,γ) = (T w (w,θ,γ),t π (w,θ),t θ (w,π,θ)), (17) where T w (w,θ,γ) is given by (16), T π (w,θ) is given by (11), and T θ (w,π,θ) is given by (12). Lemma 3 A fixed point of T is a steady state. 15

16 Proof. Obvious. Our proof of proposition 1 uses a fixed-point index theorem. Definition 2 Let Λ Π Θ and let W denote the boundary of W (with respect to K). Let G denote the set of upper-hemicontinuous (u.h.c.), compact valued, and convex valued mappings g : W Λ K Λ satisfying (w,λ) / g(w,λ) for all (w,λ) W Λ. Two mappings g 0,g 1 G are said to be homotopic on W Λ if there exists a u.h.c., compact valued, and convex valued mapping G : W Λ [0, 1] K Λ such that (i) (w,λ,α) / G(w,λ,α) for all (w,λ,α) W Λ [0, 1] and (ii) G(w,λ,α) = g α (w,λ) for all (w,λ,α) W Λ {0, 1}. The version of the fixed point index theorem we need is the following (see [1, Theorem 36.1, p. 218] and [9, 13.6a, p. 604]). There exists a fixed-point index defined on G, denoted ind, satisfying: (A1) If g is constant on W Λ with the value (w 0,λ 0 ) where w 0 W W, then ind(g) = 1. (A2) If ind(g) 0, then there exist some (w,λ) W W Λ with (w,λ) g(w,λ). (A3) If g 0,g 1 G are homotopic on W Λ, then ind(g 0 ) = ind(g 1 ). The next three lemmas constitute a proof of proposition 1 by appealing the above theorem. Lemma 4 T is u.h.c., compact valued, and convex valued. Proof. This follows from the Theorem of Maximum and convexification by lotteries. (For the non lottery version, we would allow all possible randomizations over the elements of the sets corresponding to 1 and 2.) In what follows, we write T(.,γ) as T γ (.). Lemma 5 ind(t 0 ) = 1. Proof. Notice that T 0 is identical to the mapping Φ studied in ([10]) and it is also identical to the mapping Φ 1 studied in ([11]). Then, by the exact logic to show ind(φ 1 ) = 1 in ([11, Lemma 2]), we have ind(t 0 ) = 1. Lemma 6 There exists γ 0 > 0 such that if γ γ 0, then T γ has a fixed point. Proof. This follows from Lemma 5, Lemma 4, and the fixed point theorem. 16

17 7 Appendix 2 Table 2: Value Functions and Distributions w π Non Lottery Lottery Non Lottery Lottery 8z powers-2 Powers-3 powers-2 Powers-3 powers-2 Powers-3 powers-2 Powers

18 Table 3: Optimal Portfolios: Non Lottery Powers-of-2 Powers-of-3 8z

19 Figure 3: Wealth Transfer: Non Lottery Figure 4: Wealth Transfer: Lottery 19

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