Ambiguity, Money Transfers, and Endogenous Fluctuations

This article reexamines the implications of stochastic lump-sum money transfers in a monetary overlapping generations model, wherein agents are endowed with maxmin expected utility preferences. It is shown that: 1) there exists a continuum of stationary monetary equilibria, wherein a unique optimal one might exist, and 2) there exist innumerable nonstationary monetary equilibria, wherein the real money balance can move freely within a certain range.


Introduction
In endogenous business cycle theory à la Grandmont (1985Grandmont ( , 1986, it is well-known that lump-sum money transfer plays an important role in the stabilization of business cycles. However, it is also well-known that the introduction of lump-sum money transfer causes a decrease in economic welfare (Wallace, 1980). One might find in Brock and Scheinkman (1980), a way to overcome this trade-off between the stabilization of cycles and optimality. They considered lump-sum money transfers with the stochastic growth of money supply and showed that an increase in the mean of the rate of growth of the money supply induces a welfare loss and that an increase in the variance of the rate of growth of the money supply may cause an increase in welfare. By combining these results, it might be possible to observe that random lump-sum money transfers with a sufficiently high variance can attain simultaneously the stabilization of cycles and the mitigation of welfare loss.
The aim of this article is to reexamine such a role of random lump-sum money transfers when the class of preferences is expanded beyond the standard subjective expected utility. Since Savage's (1954) pioneering work, most studies on economics under uncertainty have considered agents who choose their actions as if they maximize the expected utility with a single prior. Ellsberg's (1961) example, however, presented a situation in which agents might not assign a unique prior to uncertainty because of imprecise information. A representative model describing such a decision-making when information is imprecise is the class of maxmin expected utility (MEU) preferences, axiomatized by Schmeidler (1989) and Gilboa and Schmeidler (1989). Agents endowed with MEU preferences choose their actions as if they maximize the expected utility minimized over a set of priors. Now, a situation that agent's belief is summarized by a set of priors is called ambiguity.
Following Brock and Scheinkman (1980), this study considers an overlapping generations (OLG) model with random lump-sum money transfers, although agents are endowed with MEU preferences. This article contributes to the literature by making three main achievements. First, it is shown that a continuum of stationary monetary equilibria can exist, whereas the economy described by Brock and Scheinkman (1980) has a unique stationary monetary equilibrium. Second, this study finds that there might exist a unique golden rule optimal equilibrium in the continuum of stationary monetary equilibria. This is in sharp contrast to the well-known fact that the introduction of lump-sum money transfers induces a welfare loss 1 . Finally, it is shown that there exist innumerable nonstationary monetary equilibria, wherein the real money balance can move freely within an appropriate positive range. This might represent a remarkable difference from the existing results of Grandmont (1985Grandmont ( , 1986. He argued that chaotic behaviors of equilibria can be observed when the nonlinearity derived from preferences is extremely high, and the introduction of lump-sum money transfers helps to stabilize cycles. In contrast, our result does not require high nonlinearity and the introduction of random lump-sum money transfers causes complex equilibrium dynamics. This article also contributes to the ever-growing literature on the applications of ambiguity to various economic issues. In the last three decades, the implications of ambiguity have been explored in several economic environments. For example, Dow and Werlang (1992) and Epstein and Wang (1994) found asset price indeterminacy under ambiguity. Rigotti and Shannon (2012) found that ambiguity has less role in a canonical general equilibrium setting. Ozaki (2004, 2007) applied ambiguity to the search problem and the irreversible investments. Fukuda (2008) found a poverty trap in an OLG model with ambiguity. Guo and Yannelis (2021) applied ambiguity to implementation theory. However, few studies have applied ambiguity to monetary issues. In addition to the study by Ohtaki and Ozaki (2015), this study is one of the few scarce works 1 See Wallace (1980), whereas his model is under certainty. E. Ohtaki DOI: 10.4236/tel.2021.112015 211 Theoretical Economics Letters that applied ambiguity to monetary theory. The remainder of this paper is as follows. Section 2 presents our model, which is a variant of Brock and Scheinkman (1980). Section 3 provides main results. Section 4 contains some concluding remarks. The proofs of main results are relegated to the Appendix.

The Model
This paper considers a stationary, two-period, monetary overlapping generations model, wherein agents are endowed with the class of maxmin expected utility preferences. Uncertainty enters into the model through monetary shocks, described by stochastic money transfers.

Physical Environment
Time is discrete and runs from 1 t = to infinity. In each period, there is a single perishable physical good, called the consumption good, and a single agent is born and lives for two periods. Thus, agents might be indexed by t, which is the period in which they are born. All agents are endowed with 1 w units of the consumption good in her first period of life and 2 w units in the second period, where 1 0 w > and 2 0 w ≥ . Agent t consumes 1 t c and 2 1 t c + units of the consumption good in her first and second periods, respectively. Agent t is assumed to rank her consumption plans ( ) 1 2 1 , t t c c + according to a lifetime utility function 2 : u + → R R. We assume throughout, unless specified otherwise, that u is timeseparable and there exist strictly increasing, strictly concave, and continuously differentiable real-valued functions 1 v and 2 v on In period 1, there also exists a one-period-lived agent, called the initial old.
The agent aims to maximize her consumption, 2 1 0 c ≥ , in period 1.

Stochastic Money Transfer
This study considers stochastic money transfers to old agents as monetary shocks. Let be a nonempty finite set of ++ R . We regard each element z Z ∈ as a gross rate of growth of money stock realized in each single period and also call it a state. It is assumed that the state in each period realizes before the new agent in that period enters the economy. Let 0 z Z ∈ be a given initial state (in period 0, which is implicitly defined). The date-event tree, Γ, is defined as follows: 1) the root of the tree is 0 z ; 2) the set of nodes at date Note that, for any date-event 2 We assume strict concavity and the boundary conditions on v1 and v2 in order to provide a sharper argument. To tell the truth, however, we can replace strict concavity of v1 with concavity and can remove the boundary conditions on v1 and v2, provided that the possible money growth rates are given appropriately.

DOI
is held by the initial old. We will denote by ( ) p σ and ( ) q σ the real price of money and the (per-capita) real money balance at a date-event σ ∈ Σ , respectively. Of course, ( ) ( ) ( ) so that, the real price of money has a one-to-one relation to the real money balance.
Remark that states in our model are extrinsic in the sense that they do not affect the initial endowment streams ( )

MEU and Equilibrium
Throughout the remainder of this paper, we assume that each agent's belief on the realization of states in the second period of her life is represented by a nonempty, compact, and convex subset of Z ∆ , denoted by  , which is independent of time and current and past realizations of states. Further, for any ( ) U c is strictly concave because of strict concavity of 1 v and 2 v .
Also assume throughout, unless specified otherwise, that z z π π π π − − ∈ ∈ ∈ ∈ > ∑ ∑   . Now, we are ready to define an equilibrium. Because the real price of money and the real money balance at each date-event have the one-to-one relation, this study defines an equilibrium in terms of real money balances, not real prices of money. An equilibrium is a process of real money balances,  Sunspot money transfers as in our model is also considered in the previous studies such as Brock and Scheinkman (1980 (1) is the MEU-maximizing problem with sequential budget constraints and condition (2) is the market-clearing condition for money. An equilibrium e q is  Stationary if there is some se for any σ ∈ Γ and any z Z ∈ ; q q σ = for any 1 t ≥ and any t t σ ∈ Σ ; and  Monetary if it is positive-valued.
In order to close this section, we provide two remarks. Remark first that our e q is real money balances, not real prices of money. If one wish to obtain real prices of money, let At an equilibrium e q , we can obtain that, for each σ ∈ Σ and each z Z as the real rate of return of money, which is not necessarily equal to one due to stochastic growth of the money stock, and as the real amount of money transfers. Also remark that, at an equilibrium e q , the market-clearing conditions for the consumption good holds at each dateevent σ ∈ Σ . In fact, given an equilibrium e q , it holds that where the first equality follows from the sequential budget constraints and the second equality follows from the money market clearing condition. This is no doubt the market clearing condition for the consumption good at date-event σ ∈ Σ 4 . In other words, each allocation associated with an equilibrium is always feasible. One might note that lump-sum money transfers in our model do not affect the feasibility condition on equilibrium allocation.

Characterization of Equilibrium
Our first task is to characterize a monetary equilibrium. When agents' preferences are represented by standard expected utility functions, a monetary equilibrium is characterized by a system of difference equations 5 . However, because the class of MEU preferences are not smooth at some points 6 , a monetary equili-4 Note that See for example Ohtaki (2011Ohtaki ( , 2015. 6 To be more precise, indifference hyperplanes have kinks at consumption plans ( ) which is the set of probability measures minimizing the second-period expected utility over  .
As an immediate corollary, we can find that there is no stationary monetary This implies the existence of some π ∈  such that The se q is now characterized by is strictly increasing in q and the right-hand side of the above equation is independent of z, se z q must be independent of z, which contradicts the hypothesis that se

Existence and Indeterminacy of Stationary Monetary Equilibrium
As mentioned in the previous subsection, there is no stationary monetary equi- , which is a solution of the inclusion Examining this inequality, we can obtain the following result.
This proposition says that every element of ( ) Q ζ can construct a stationary deterministic monetary equilibrium, and therefore, the stationary deterministic monetary equilibrium is indeterminate. As argued in Subsection 3.4, agents partially sacrifice consumption smoothing to avoid ambiguity. In the OLG framework, it enlarges the intergenerational transfers supported at monetary equilibrium. This is the main reason of indeterminacy of stationary monetary equilibrium. Note that, if , we can conclude that ( ) Q ζ is the largest set of stationary deterministic monetary equilibria.

Optimality of Stationary Monetary Equilibrium
We then examine the existence of optimal monetary equilibrium. Here, optimality is (ex-post) Pareto optimality. Some of this reason is the economy is deterministic, except for monetary policy 7 . An allocation { } Also note that any golden rule optimal allocation is Pareto optimal, given current assumptions.
Proposition 3. Assume that ( ) ( ) Then, there is a stationary deterministic monetary equilibrium, allocation of which is golden rule optimal, if and only if 1 ζ ζ ≤ ≤ . Furthermore, the equilibrium is unique. 7 For (ex-ante) optimality in stochastic environments, see Chattopadhyay and Gottardi (1999) and Ohtaki (2013) for example. 8 In other words, Pareto optimality cares welfare of all generations including the initial old, whereas golden rule optimality does welfare of all newly born agents only. Interested readers might find a more theoretical relationship between these two criteria on optimality in Ohtaki (2013 If  is a singleton with the unique element π, the last condition degenerates into 1 1 z z Z z π − ∈ = ∑ . On the other hand, in the presence of ambiguity, we do not necessarily require that 1 1 z z Z z π − ∈ = ∑ to obtain the optimal monetary equilibrium.
Note that, as shown in Proposition 2, there might be a continuum of stationary monetary equilibria. By Proposition 3, however, the golden rule optimal equilibrium, if any, is unique. As a corollary of Proposition 3, therefore, we can say that the golden rule optimal equilibrium is measure zero in the space of stationary monetary equilibria.

Complexity of Deterministic Monetary Equilibria
It is well-known that, in a canonical pure-endowment OLG model with a twoperiod-lived agent per generation, stationary monetary equilibrium is unstable and each monetary equilibrium with a sufficiently low initial real money balance converges to the nonmonetary one. Here, we reexamine this observation in the present framework.
As a corollary of Proposition 1, a deterministic monetary equilibrium can be identified with a positive sequence ( ) 1 where ζ and ζ are defined as in Equation (1). This can be interpreted as a requirement that, in a deterministic monetary equilibrium, the marginal rate of substitution at the equilibrium allocation lies on the range of inflation-adjusted deflation rates. Moreover, we might be able to say that this is a trade-off between uncertainty aversion and consumption smoothing. That is, Equation (2)  1 qv w q v w q ′′ ′ − + + ≥ for all q Q * ∈ holds, for example, if the index of relative risk aversion of 2 v is greater than unity and the initial endowment in the second period is sufficiently small. Actually, when 2 0 w = , the condition exactly holds if the index of relative risk aversion of 2 v is greater than of equal to unity.
Propositions 4 and 5 imply that, for any utility index functions 1 v and 2 v , there might exist innumerable nonstationary monetary equilibria, wherein the real money balance can move freely within a certain range of positive numbers. This is far from the existing results in endogenous business cycle theory, in which the business cycles are explained by chaos theory.

Concluding Remarks
This article has explored the implications of stochastic lump-sum money trans- The last result is a remarkable difference from the existing literature such as Grandmont (1985Grandmont ( , 1986, which have argued that endogenous business cycles require high nonlinearity to the model. In this sense, the last result represents a new mechanism of endogenous business cycles. It is a future work whether our mechanism of endogenous fluctuations can be observed in more general model with, for example, intertemporal production technology. Remark that the stochastic growth of money in this study is as sunspots in the sense that it does not affect endowments nor preferences. Such money creation may be interpreted as a situation in which the monetary authority determines a policy based on false information. Our last result signifies that such decisions, based on false information, cause endogenous fluctuations. Theoretical Economics Letters Shuhei Shiozawa, and Hiroyuki Ozaki for their helpful comments.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper. DOI: 10.4236/tel.2021.112015 220 Theoretical Economics Letters

Appendix: Proof of Main Results
In order to obtain our results, we introduce some notation. In the current setting, some of objective functions derived from the utility function U is not necessarily differentiable. However, we can define the "superdifferential" of those functions. The superdifferential of a concave function : dom and each of its elements is called a supergradient of f at x 9 . One can immediately show that are positive when f is strongly monotone.
Proof of Proposition 1. Let e q be a mapping of Σ to ( ) 1 0, w and, for each σ ∈ Σ , define the function For each , n x y∈ℜ , , x y represents their inner product, i.e., In order to prove Proposition 2, we prepare a lemma. Lemma 1. Let ( ) q ζ be a solution of the equation that Proof of Lemma 1. Given a positive number x such that ( ) ( ) lim 0 x q w f q ↑ = −∞ < , there exists some ( ) ( ) We claim that this ( ) a contradiction. Therefore, ( ) q x is a unique solution of the equation We then claim that ( ) q ⋅ is increasing. To verify this, let ( ) ( )

Proof of Proposition 2. Let
It is easy to verify that this q is a desired stationary deterministic monetary equilibrium.
On the other hand, if there is a golden rule optimal stationary deterministic monetary equilibrium q , it must satisfy that because it is a stationary deterministic monetary equilibrium. Therefore, we obtain that 1 ζ ζ ≤ ≤ .
Uniqueness of a stationary deterministic monetary equilibrium, of which allocation is golden rule optimal follows from uniqueness of q . Q.E.D.
Before proving Propositions 4 and 5, we prepare a lemma. where the first inequality follows from the fact that 2 V is nonincreasing, the second equality follows from the definition of q * , the third and fourth inequalities follow from the facts that t q Q * ∈ and that 1 V is increasing, the fifth equality follows from the definition of q * , and the last inequality follows from the fact that 2 V is nonincreasing. Summarizing this result, we have By Equation (2), this implies that ( ) 1 t t q ∞ = is a deterministic monetary equilibrium. This establishes the proof of Proposition 4. Q.E.D.
In order to prove Proposition 5, we add two lemmas. Suppose now that q q * * * > . Then, we have where the first inequality follows from the fact that 1 V is increasing, the second equality follows from the definition of q * * , the third inequality follows from the fact that ζ ζ < , and the last equality follows from the definition of q * . This is, however, a contradiction. Therefore q q * * * ≤ . On the other hand, suppose that q q * * * > . Then, we have where the first inequality follows from the fact that 1 V is increasing, the second equality follows from the definition of q * * , the third inequality follows from the fact that ζ ζ < , and the last equality follows from the definition of q * . This is, however, a contradiction. Therefore q q * * * ≤ . This completes the proof of Lemma 3. where the first inequality follows from the fact that 1 V is increasing, the second equality follows from the definition of q * * , the third inequality follows from the fact that 2 V is nondecreasing, and the last equality follows from the definition of q * . This is, however, a contradiction. Therefore, q q * * * ≤ .
On the other hand, suppose that q q * * * > . Then, we have where the first inequality follows from the fact that 1 V is increasing, the second equality follows from the definition of q * * , the third inequality follows from the fact that 2 V is nondecreasing, and the last equality follows from the definition of q * . This is, however, a contradiction. Therefore q q * * * ≤ . This completes the proof of Lemma 4. Q.E.D.
Proof of Proposition 5. Recall the functions 1 V and 2 V defined in Lemma 2. Under assumptions in this proposition, 1 V is increasing on its domain and 2 V is nondecreasing on Q * .
By Lemma 4, , q q Q * * * * * ∈ . Note that Equation (3)  where the first inequality follows from the fact that 2 V is nondecreasing, the second equality follows from the definition of q * * , the third and fourth inequalities follows from the facts that , t q q q * * * *   ∈   and that 1 V is increasing, the This establishes the proof of Proposition 5. Q.E.D.