The Welfare Effects of Bubble Policy: Risk Aversion in a Rational Greater Fool Bubble Model

Transcription

1 The Welfare Effects of Bubble Policy: Risk Aversion in a Rational Greater Fool Bubble Model Harlan Holt March 26, 2017 Abstract This paper presents a greater fool bubble model with risk averse agents and examines the welfare consequences of anti-bubble policy by a central bank. The central bank attempts to protect investors by making a public announcement whenever an agent observes that assets are overpriced. This general deflation of overpriced assets policy makes the value of the asset common knowledge and prevents speculative bubbles from forming. We show that this policy is welfare diminishing for buyers in asset markets if utility exhibits constant absolute risk aversion. This happens because the central bank s information revelation prevents buyers from fully sharing risk with sellers, as in Hirshleifer (1971). While buyers gain from the policy by avoiding purchases of overpriced assets, the welfare loss from the Hirshleifer effect always dominates this gain. The policy can also be welfare diminishing if agents have increasing or decreasing absolute risk aversion, but only under specific assumptions about the distribution of wealth. Visiting Assistant Professor, Union College, Department of Economics, 211A Olin Hall, Schenectady, NY, 12866, holth@union.edu The author thanks John Conlon for his many helpful suggestions and discussions on bubbles, and also thanks Josh Hendrickson, Mark van Boening, Eshragh Motahar, Alecia Cassidy, and participants at seminar presentations at Northern Illinois University and Union College for their comments on previous drafts. 1

2 1 Introduction The severity of the 2008 financial crisis has renewed arguments that central banks should include the deflation of overpriced assets as part of their mandate. Many have argued 1 that wild swings in asset prices can cause large misallocations of capital and harm investors, particularly those unlucky enough to buy into the market at inflated prices. 2 In general, these arguments suppose that, if central banks can identify overpriced assets, then central bank policy may have the potential to improve the welfare of investors by protecting them from states of the world where assets are overpriced. Opposing arguments have suggested (i) that it is not possible to eliminate bubbles because accurately predicting whether assets are overpriced is not possible, (ii) that even if it was possible to eliminate bubbles, the outcome might be destabilizing, and (iii) that the available tools for this policy, such as interest rates, are blunt instruments that might impact other areas of the economy adversely. 3 These counterarguments do not concern themselves with the direct welfare effects of the policy on the individuals trading in the asset market. Rather, they focus on the feasibility of the policy - whether it can be implemented without severe, unintended external consequences (such as causing macroeconomic destabilization), or whether it can even be implemented at all (e.g. whether central banks can identify overpriced assets). Largely missing from the debate is a careful analysis of the microeconomic impacts of antibubble policy on the individual welfare of agents in asset markets. With this in mind, this paper investigates the welfare effects of anti-bubble policy. We build on a previous study of bubble policy in Conlon (2015), which in turn is based on the seminal rational agent bubble model of Allen, Morris, and Postlewaite (1993). 4 Previous attempts to identify welfare effects on investors in asset markets, such as Conlon 1 See, e.g., Cecchetti et al. (2000) and Roubini (2006) 2 There are many historical examples of these wild swings, such as the Dutch Tulipmania in the 17th century, the South Sea bubble of 1720, the Japanese asset bubble of the 1980s, and the US housing bubble in the 2000 s. In each of these examples, the price of an asset, or a collection of assets, rose a great deal, and then quickly collapsed. Kindleberger and Aliber (2011) provides a detailed narrative of many of these events. It has also been argued that many of these historical episodes were not examples of bubbles. See Garber (1990, 2001). 3 See Bernanke and Gertler (1999, 2001), Posen (2006), and Mishkin (2007) 4 The Allen, et al. (1993) model is greatly simplified by Conlon (2004). This makes analysis of rational bubbles like the one in this paper far easier to conduct. 2

3 (2015), have assumed essentially risk neutral agents. This is a serious shortcoming in the prior literature because a major question for bubble policy is whether bubbles impose unnecessary risks on investors, and whether anti-bubble policy can protect investors from these risks. Therefore, it is impossible to present a complete analysis of the effects of these risks on the agents welfare without allowing these agents to be genuinely risk averse. This paper considers a model like the one in Conlon (2015), but with risk averse agents, and shows that anti-bubble policy is likely to be harmful for both buyers and sellers under broad conditions. This occurs because bubble bursting policy can disrupt beneficial risk sharing among the agents in the model. Specifically, buyers and sellers use asset markets to share the risks of uncertain future states of the world. By informing the agents about whether certain states of the world will occur, anti-bubble policy reduces the agents ability to share these risks. Thus, the agents can be made worse off by the policy, even if it is successful in preventing investors from buying assets known by sellers to be overpriced. Once again, this finding has eluded prior research because of inadequate attention to the underlying assumptions about risk. We cannot understand how bubble policy affects agents if those agents This analysis builds an argument in Hirshleifer (1971). If better information about the future state of the world disrupts risk sharing, then the agents might be made worse off by that information. Heretofore, this argument has not appeared in the literature on bubble policy because no prior study has ever attempted to include risk aversion in a welfare analysis of an Allen, et al. (1993) type bubble model. Even though prior research (namely, Conlon (2015)) has attempted to examine the welfare effects of bubble policy, this prior research assumed risk neutral agents. The current paper shows that the assumption of risk neutrality prevents us from learning the full effects of bubble policy on investors, by obscuring the role of risk sharing. 1.1 Allen, Morris, Postlewaite (1993) and Conlon (2004) greater fool bubbles Allen, Morris, and Postlewiate (1993), along with a subsequent refinement by Conlon (2004), present a simple model where heterogeneous, perfectly rational agents are randomly endowed with a risky 3

4 asset and participate in a pure exchange economy with short-sale constraints. 5 In the model, a bubble arises if a state exists such that each agent knows the asset is overpriced, but believes he or she can sell the asset to an uninformed agent (a greater fool ) before its true value becomes common knowledge. 6 The bubble therefore forms because of a specific information structure. At least one state exists such that each agent knows the asset is overpriced. However, if these agents don t know that everyone else knows the asset is overpriced, this gives them an incentive to hold an overpriced asset in hopes of selling it to a greater fool. Allen, et al. (1993) say that an asset in such a state is in a strong bubble. Conlon (2015) provides the first attempt to analyze the welfare effects of anti-bubble policy in an Allen, Morris, Postlewaite (AMP) type of bubble model. This paper augments the simple AMP bubble model with risk neutral sellers who can produce risky assets according to some production technology and then trade the assets in a market with risk neutral buyers. Conlon (2015) shows that an important factor in AMP bubbles is the existence of a lemons problem. Buyers do not know whether the sellers know the asset is overpriced (i.e. whether they are bad sellers ), or whether they are selling the asset for legitimate reasons, such as risk sharing (i.e. good sellers ). If buyers know there are bad sellers in the market, then these buyers will be less confident about buying the asset, or may avoid buying the asset entirely. Therefore, asymmetric information can make buyers and sellers in asset markets worse off. The lemons problem suggests the possibility that an informed authority (a government regulator, for instance) could step in and reveal the existence of bad sellers to protect buyers. Conlon therefore assumes a central bank that announces to the public when at least one agent knows that the asset is overpriced. The policy then causes prices to fall to their true value in states where the announcement is made, but also causes prices to rise in non-announcement states, because the non-announcement provides an implicit endorsement of the asset s value. This unintended price effect has at least one helpful quality. Sellers who produce the 5 Other important papers in this field have also modeled greater fool bubbles, but often with imperfectly rational agents. Abreu and Brunnermeier (2003) models bubble riding behavior by including irrational noise traders as well as rational traders. Doblas-Madrid (2013) uses a similar model, but with perfectly rational agents and noisy prices. Harrison and Kreps (1973), and a later generalization by Scheinkman and Xiong (2003) assume irrationally overconfident agents to generate a bubble. For an excellent summary and overview of these and other papers, see Barlevy (2015). 6 The use of the term greater fool here is used to distinguish these models from other, largely separate, infinitehorizon bubble models such as Tirole (1985) and Santos and Woodford (1997). In these models, overpricing exists in equilibrium because the overpricing is expected to persist forever. These models may therefore describe assets like money that can perpetually maintain a value above their intrinsic expected dividends. 4

5 asset shift production from the bad-seller states of the world, where the asset is worthless, to the good-seller states, where the asset is valuable. Conlon s (2015) analysis is, however, incomplete. Since Conlon assumes risk neutral agents, he cannot analyze the effects of risk on the expected utility of investors in the model. In order to make confident predictions about the effect of risk on expected utility, this paper incorporates risk aversion into an AMP type bubble model. 7 Assuming that bubbles pose risk to the agents, it then seems plausible that bubble policy can protect investors from that risk. However, it turns out that, if agents are risk averse, then the information provided by the asset-deflation announcements actually tends to make these agents worse off. Risk averse agents participate in asset markets to spread their risks across states of the world by using risky assets as hedges against the risks that nature imposes on them. Financial markets allow these agents to move wealth from states where they value it least (e.g. high wealth states) to states where they value it most (low wealth states). If, in the attempt to protect agents from the risk of buying overpriced assets, information is revealed to the agents about the state of the world, then the policy may actually harm the agents it is designed to protect, if that information prevents them from sharing risk. This basic point about risk sharing was first described in an influential paper by Hirschleifer (1971), and has been subsequently built upon by Schlee (2001), among others. 1.2 The Hirshleifer (1971) Effect, risk sharing and bubble policy Hirshleifer s original statement of the problem is approximately this. Suppose two risk averse agents, with differing random endowments, participate in a pure exchange economy with two possible future states of the world: a state that s good for one of the agents, and a state that s good for the other agent. In the absence of any market imperfections, the agents will trade contingent claims with each other to insure themselves against the loss of wealth in the state of the world that s bad for them. By trading with each other, the agents effectively reallocate their endowments to evenly distribute their wealth across the two states of the world. That is, the agents share risk by essentially selling insurance to each other. This allows each agent to move wealth from their good state to their bad 7 Allen, et al. (1993) briefly discuss downward sloping marginal utilities, but perform no formal welfare analysis. 5

6 state, which makes risk averse agents better off. However, suppose that the agents are told the future state of the world before they can trade. Insurance contracts now become irrelevant since the future state of the world is known with certainty. Mutually beneficial gains from trade are no longer available, so no trade occurs. The agents, instead of trading contingent claims to smooth out wealth across the two states of the world, are now forced to consume their own random endowment in each state, and beneficial risk sharing between the agents is denied. From an ex-ante perspective, risk averse agents are now worse off than if they had never been told the true state of the world, because they cannot trade away the risk that they end up with a poor endowment. Since disruption to risk sharing only matters if agents are risk averse, this means that previous attempts to analyze the microeconomic welfare effects of bubble policy cannot present a complete analysis without incorporating risk averse agents. 2 Generating a bubble There exists a competitive market where rational, heterogeneous agents trade money for an asset that pays a stochastic dividend (as in Allen, et al. (1993) and Conlon (2004, 2015)). We construct a market in which a strong bubble state exists, where each of the agents knows the asset is worthless (pays a future dividend of zero), yet the market price is above zero. This is possible because, even though each agent knows for sure that the asset is overpriced, they do not know that the other agents also know the asset is overpriced. The sellers then rationally try to sell the overpriced asset to a buyer they believe might be ignorant of its true value. Since constant marginal utilities (as in Conlon (2015)) imply the agents are risk neutral, any changes in welfare arising from the policy s effect on risk are lost entirely. The desire to share risk among the agents is central to the existence of asset markets (see Hirshleifer (1971) and Schlee (2001)), so no discussion of the welfare effects of bubble policy is complete until the agents risk attitudes have been included in the analysis. To remedy this shortcoming, we present an Allen, et al. (1993) bubble model with concave utility. This will allow us to generate welfare results for risk 6

7 averse buyers in addition to the results for sellers from Conlon (2015). As in Conlon (2015) we assume the existence of a central bank that perfectly observes states of the world where at least one agent knows that the asset is worthless. 8 In this model, the central bank will follow the general deflation policy from Conlon (2015) resolving information asymmetry for buyers in the market by announcing whenever it observes this state, which makes the knowledge of overpriced assets common amongst the agents. 2.1 Assumptions and setup There are two heterogeneous representative agents in the model, each of which can be thought of as representing an infinite collection of identical traders in a competitive market. These agents maximize their expected utility across all states of the world. We will call the two agents Ellen (E) and Frank (F ), denoted by i {E, F }. Ellen and Frank have different preferences which motivates them to trade in the market. The market lasts for three periods t = 1, 2, 3. After period 3, the asset pays a random liquifying dividend, the market closes, and agents consume their wealth. There are a finite number of states of the world ω, where ω is an element of a state space Ω. The agents have a common prior probability distribution over Ω, given by π(ω), which is known to everyone. Agents are endowed with an initial amount of both a risky asset, a i,0 (ω) for i {E, F }, and a riskless asset (money) that can be used to acquire the risky asset or can be consumed. Each agent faces background risk, B i (ω), that represents uncertain future states of the world. B i (ω) can be thought of as the wealth each agent would have if the asset did not exist. Initial money holdings are assumed to be constant across all states of the world and included in the background risk. Agent i s holdings of the risky asset at time t are given by a i,t (ω). The risky asset can be sold for money, or held in hopes of recieving the dividend d(ω) at time t = 3. There are no short-sales of the risky asset, so a i,t (ω) 0. As in Allen, et al. (1993) and Conlon (2004, 2015) we have to rule out short-sales of the asset because in the presence of an overpriced asset, sellers will short sell until the asset is no longer overpriced. 8 It is clearly implausible that a real central bank could ever observe these conditions with anything approaching perfect accuracy. However, we assume a perfectly informed central bank in order to establish an upper limit on the amount of information a central bank could possibly have. If a central bank cannot improve buyer welfare even if it is perfectly informed, it could not hope to do so if it were less than perfectly informed. 7

8 2.2 Asset demand under risk aversion We assume each agent has a general utility function which depends both the state of the world and on wealth U = U i,ω (W ) (1) where U i,ω (W ) > 0 and U i,ω (W ) 0 allowing for risk averse utility for each agent.9 Each agent s objective is to maximize the expected utility of their final wealth, W i,3, which is state-dependent. We therefore have the following maximization problem max E [U i,ω] = π(ω)u i,ω (W i,3 (ω)) (2) {X i,t (ω)} ω Ω where the X i,t (ω) are agent i s trades in state ω, at time t, and U i,ω (W i,3 (ω)) is utility as a function of agent i s wealth in period 3, W i,3 (ω), which is in turn a function of the state of the world, ω. We need to maximize (2) cell by cell in the information partition. Since agents cannot tell which state occurs within a given cell, P i,t, of their information partition, their trades, X i,t (ω), are the same over all ω in the cell P i,t, so we ll call it X i,t. Carrying out the maximization problem in (2) for a given information set P i,t gives the relevant first order condition gives us where X is the same for all states in P i,t. [ ( )] π(ω) U i,ω(w Wi,3 (ω) i,3 ) = 0, (3) X i,t ω P i,t Because risk aversion implies downward sloping utility, it is now necessary to keep track of wealth. Final wealth can be calculated as 9 The slightly messy notation is intended to reflect the two dimensions on which utility will change. In Conlon (2015), part of which will be replicated below, marginal utility is constant, so the only way to give the traders different preferences is to force the utility curve to be state-dependent with different marginal utilities in different states of the world. Later, once we assume concave utility, this direct state-dependence will no longer be needed. Marginal utility will change with wealth (which is state-dependent), so it will no longer be necessary to assume that the utility function depends directly on the state. 8

9 ( ) 3 3 W i,3 (ω) = B i (ω) + p τ (ω)x i,τ (ω) + a i,0 (ω) X i,τ (ω) d(ω), (4) τ=1 τ=1 where B i is the background risk (or the agent s wealth at the end of the market if the asset did not exist). 10 Equation (4) simply gives the agent s final wealth as the sum of his/her background risk, B i (ω), plus the sum of revenues/remittances from any sales/purchases of the asset, ( 3 τ=1 p τ (ω)x i,τ (ω), plus the dividend value of final asset holdings a i,0 (ω) ) 3 τ=1 X i,τ (ω) d(ω). Using (4), we can evaluate the derivative inside the brackets of (3). Doing this gives us the agent s sensitivity of final wealth to his/her sales: W i,3 (ω) X i,t = p t d(ω), (5) where p t (ω) = p t because the price is the same for all states in a given cell of their information partition. Notice that (5) is intuitive. If an agent sells one more unit of the asset, s/he recieves the market price for the asset, but loses the dividend paid in state ω. Substituting (5) into the first order condition: [ π(ω) U i,ω (W i,3 ) (p t d(ω)) ] = 0. ω P i,t Distributing the marginal utiltiy and the probabilities gives π(ω)u i,ω(w i,3 )p t π(ω)u i,ω(w i,3 )d(ω) = 0. ω P i,t ω P i,t Finally, adding the second term to both sides and dividing by the expected marginal utility gives us a pricing equation: p t = π(ω)u i,ω (W i,3 )d(ω) π(ω)u i,ω (W i,3 ) = [ ] E U i,ω d(ω) ω P i,t [ ] = W T P i (p t P i,t ) (6) E U i,ω ω P i,t 10 Later, we ll simplify things such that trade occurs only in period 2. 9

10 Equation (6) is the certainty equivalent price for the asset and is analogous to the certainty equivalent pricing equation in Conlon (2015). 11 This generalizes the result in Conlon (2015) for any concave utility function. Since the right hand side of (6) is a function of p t (through wealth), an explicit solution for price, p t, can be found at the fixed point W T P (p t P i,t ) = p t, for whichever agents are not short sale constrained. 2.3 The information structure The initial setup follows Conlon (2015). There are nine possible states of the world, e G 1, eg 2, eg 3, e B, f1 G, f 2 G, f 3 G, f B, b. Each state is equally likely to occur, so π(ω) = 1/9 for all ω. The risky asset pays a dividend, d, only in states e G 3 and f 3 G. In states e G 1, eg 2, and eg 3, Ellen is a good seller. That is, she will wish to sell but thinks the asset might be worth something. In states b and e B, Ellen is a bad seller. In these states, Ellen knows the asset will pay no dividend, so she clearly wants to sell to Frank. Finally, Ellen is a buyer in states f1 G, f 2 G, f 3 G, and f B. Symmetrically, there are states of the world in which Frank is a good seller (f1 G, f 2 G, and f 3 G), where he is a bad seller (f B and b), and where he is a buyer (e G 1, e G 2, eg 3, eb ). Note that in state b, both agents are bad sellers who know the asset is worthless. Accordingly, we can write Ellen s period 1 information partition as: E BS 1 = {b, e B }, E GS 1 = {e G 1, e G 2, e G 3 }, and E Buyer 1 = {f B, f G 1, f G 2, f G 3 } Frank s period 1 information partition is likewise: F BS 1 = {b, f B }, F GS 1 = {f G 1, f G 2, f G 3 }, and F Buyer 1 = {e B, e G 1, e G 2, e G 3 } The information partitions for period 1 are represented graphically in Figure 1. In period 2, the agents learn whether the state is b, e G 1, or f G 1. Thus the period 2 information 11 Equation (6) holds only if the agent s short sale constraint does not bind. In the binding case for the short sale constraint, the first order condition in (3) does not hold, and equation (6) is therefore invalid. 10

11 Figure 1: Period 1 information partitions. Solid ovals - Ellen, Dashed ovals - Frank, Dotted ovals - dividend paying states partition for Ellen is E 1 2 = {b}, E 2 2 = {e G 1 }, E 3 2 = {f G 1 }, E BS 2 = {e B }, E GS 2 = {e G 2, e G 3 }, and E Buyer 2 = {f B, f G 2, f G 3 } and likewise for Frank. The nontrivial part of the new information structure is displayed in Figure 2 with states b, e G 1 and f G 1 omitted. Figure 2: Nontrivial parts of the period 2 information partitions. Solid ovals - Ellen, Dashed ovals - Frank, Dotted ovals - dividend paying states Lastly, we assume that agents are endowed with a single unit of the asset when they are sellers. The buyers are endowed with none of the asset. In period 3, all information is revealed. The agents learn the true state of the world, the dividend is paid, and agents consume their wealth. 2.4 Bubble policy and the baseline scenario with constant marginal utility Now that we have a well defined equilibrium that generates a bubble, we can examine what happens when a market authority, e.g. a central bank, uses policy to prevent overpricing. We assume the 11

12 existence of a central bank that can identify when the asset is overpriced (i.e. whether the true state of the world is e B, f B, or b). Since the central bank knows the true state of the world when there are bad sellers in the market, it can pursue a policy of general deflation of overpriced assets by announcing to the public when it observes these states. We assume that the central bank observes bad sellers with complete accuracy. This has two important functions. First, it greatly simplifies the analysis. Second, it allows us to study what the effects of the central bank s policy would be in the most perfect of all worlds. In other words, if the central bank cannot improve welfare even in the case when its information is perfect, it surely cannot improve welfare in the vastly more realistic case where the central bank has limited information. Following Conlon (2015), we examine what happens to the market when the central bank follows the general deflation policy as described above. In period 1, the central bank announces whether the world is in states b, e B, or f B. The policy forces the price to zero in these states because it is now common knowledge among the agents that the asset is worthless. 12 Ellen s new information partition in period 1 is E 1P 1 = {b}, E 2P 1 = {e B }, E 3P 1 = {f B }, E BSP 1 =, E GSP 1 = {e G 1, e G 2, e G 3 }, and E BuyerP 1 = {f G 1, f G 2, f G 3 } and similarly for Frank, where the superscript P indicates the information sets under the general deflation policy. 13 As before, assume that in period 2 the agents learn whether the state is e G 1 or f G 1. Ellen s period 2 information partition is then 12 I.e. each agent knows the asset is worthless, each agent knows that the other agent knows the asset is worthless, and each agent knows that the other agent knows that they know the asset is worthless, and so on. See Allen, Postlewaite, and Shin (1995), Conlon (2004), and Allen, Morris, and Shin (2006) for more discussion of these interesting types of recursive information structures involving depth of knowledge problems. 13 Technically, E BSP = is not actually a cell in the information partition, but we include it to emphasize that therea re no states in which Ellen will successfully sell an asset she knows to be worthless. 12

13 E 1P 2 = {b}, E 2P 2 = {e B }, E 3P 2 = {f B }, E 4P 2 = {e G 1 }, E 5P 2 = {f G 1 } E BSP 2 =, E GSP 1 = {e G 2, e G 3 }, and E BuyerP 1 = {f G 2, f G 3 } and similarly for Frank. 3 The model and results for buyer welfare under the general deflation policy For simplicity, we assume Frank is the buyer, and Ellen the seller for the remainder of the paper. At this point it bears reminding the reader that there are two parts to the ultimate change in the price of the asset. The first part occurs as a direct effect of the policy on the information partition of the buyer, holding the marginal utilities constant. Recall that the relevant period 2 information cell in the bubble equilibrium is F Buyer 2 = {e B, e G 2, eg 3 }. The policy moves state eb into its own cell in the buyer s period 2 information partition where the price falls to zero, so the price in the other relevant information cell, F BuyerP 2 = {e G 2, eg 3 }, rises because a state where the dividend is definitely zero has been removed from this cell. This is the familiar result from Conlon (2015). The second part of the price change occurs due to risk aversion. Downward sloping marginal utility functions mean that when the purchase price rises as an effect of changing the buyer s information partition, the buyer s wealth falls, so his or her ultimate marginal utilities (of wealth) will rise. This will cause a change in the marginal rates of substitution, and, therefore, a further change in the price of the asset. Thus, the behavior of the price of the asset in the cell {e G 2, eg 3 } rests on the marginal rate of substitution of wealth between states e G 2 and eg 3, which we ll denote as MRS 2,3 = U (W 3 ) U (W 2 ). To simplify the notation, let W B = W F,3 (e B ) be Frank s final (period 3) wealth in the bad seller state, e B, and W 2 = W F,3 (e G 2 ), and W 3 = W F,3 (e G 3 ) be Frank s final wealth in the good seller 13

14 states, e G 2 and eg 3. Lemma 1. The period 2 price of the asset, under the deflation policy, is increasing in the marginal rate of substitution of wealth between states e G 2 and eg 3. Proof. Recall that the price of the asset in period 2 after the policy is given by (??) p 2 = π(eg 2 )U (W 2 )d(e G 2 ) + π(eg 3 )U (W 3 )d(e G 3 ) π(e G 2 )U (W 2 ) + π(e G 3 )U (W 3 ) Dividing top and bottom by π(e G 2 )U (W 2 ), noting that d(e G 2 ) = 0 by assumption, and letting MRS 2,3 = U (W 3 ) U (W 2 ) gives p 2 = [ ] π(e G 3 ) U (W 3 ) π(e G 2 ) U (W 2 ) d(e G 3 ) [ 1 + π(eg 3 ) U (W 3 ) π(e G 2 ) U (W 2 ) ] = π(e G 3 ) π(e G)MRS 2,3 d(e G 3 ) 2, 1 + π(eg 3 ) π(e G)MRS 2,3 2 which, on the relevant domain, is increasing in MRS 2,3. Given this result, we now determine how the marginal rate of substitution behaves under risk aversion. Since e G 2 and eg 3 are in the same information set, the changes in purchase price, and therefore the changes in wealth, induced by the policy must be the same in these two states. As it turns out, the change in the marginal rate of substitution, caused by this wealth change, depends on how the coefficient of absolute risk aversion responds to wealth. This is because, even though the policy-induced changes in wealth are equal in each state of the buyer s information set, the proportional changes in marginal utilities are not necessarily equal. How, exactly, the marginal rate of substitution will change is shown in Lemma 2. Lemma 2. If wealth decreases by the same amount in states e G 2 and eg 3, the change in the marginal rate of substitution of wealth between states e G 2 and eg 3 depends on the coefficient of absolute risk aversion in the following ways: (a) If utility has Constant Absolute Risk Aversion (CARA), the change in the MRS is zero. (b) If utility has Decreasing Absolute Risk Aversion (DARA), the change in the MRS will be positive 14

15 if W 2 > W 3, and negative if W 3 > W 2. (c) If utility has Increasing Absolute Risk Aversion (IARA), the change in the MRS will be negative if W 2 > W 3, and positive if W 3 > W 2. Proof. Denote the Arrow-Pratt coefficient of absolute risk aversion as a function of wealth as ARA(W ), defined as ARA(W ) = U (W ) U (W ). (7) The marginal rate of substitution between wealth in states e G 2 and eg 3 is, again, MRS 2,3 = U (W 3 ) U (W 2 ). If we suppose that wealth changes by the same amount, W, in the two states, then the new marginal rate of substitution is MRS2,3 P = U (W 3 + W ) U (W 2 + W ). Differentiating the new marginal rate of substitution with respect to the change in wealth gives MRS P 2,3 W = U (W 2 + W )U (W 3 + W ) U (W 3 + W )U (W 2 + W ) [U (W 2 + W )] 2 = K [ARA(W 2 + W ) ARA(W 3 + W )], (8) where K = U (W 3 + W ) U (W 2 + W ) = MRSP 2,3 > 0. Since K > 0, the sign of (8) is controlled by the expression ARA(W 2 + W ) ARA(W 3 + W ). (9) Thus, the sign of (9) will determine the effect of the policy on the MRS. If utility is CARA, (9) is always zero. If utility is DARA (or IARA) the sign will depend on whether W 2 or W 3 is bigger, that is, on the distribution of wealth between states e G 2 and eg 3. This leads us to the following 15

16 conditions for the sign of MRSP 2,3 W. (a) Expression (9) will always be zero if utility is CARA, so MRSP 2,3 W = 0 via (8). (b) When utility is DARA, Expression (9) will be negative if W 2 > W 3, so MRSP 2,3 W < 0. This is because DARA implies that, if W 2 > W 3, then ARA(W 2 + W ) < ARA(W 3 + W ), so (9) is negative. If W 3 > W 2, under DARA utility, Expression (9 will be positive, so MRSP 2,3 W > 0. (c) When utility is IARA, Expression (9) will be negative if W 3 > W 2, and positive if W 2 > W 3. Lemma 2 shows that the sensitivity of the coefficient of absolute risk aversion to wealth, together with the prior distribution of final wealth, controls how the marginal rate of substitution responds to the general deflation policy. Therefore, by Lemma 1, the price of the asset, and so the ultimate effect of the policy on expected utility, depends on these factors as well. Let a superscript of N P denote a no-policy (bubble) equilibrium variable, and let a superscript of P denote a policy equilibrium variable. Then, also let the change in the marginal rate of substitution caused by the policy be denoted by δ, that is U (W3 P ) U (W2 P ) = U (W NP U (W NP 3 ) 2 ) + δ. (10) The sign of δ is controlled by the distribution of wealth and the sensitivity of absolute risk aversion to changes in wealth, as shown in Lemma 2. Since the policy increases the buyer s purchase price, the resulting W is negative, Lemma 2 implies that = 0 if utility is CARA. δ > 0 if utility is DARA and W 2 > W 3 OR utility is IARA and W 3 > W 2. (11) < 0 if utility is DARA and W 3 > W 2 OR utility is IARA and W 2 > W 3. Thus, the marginal rate of substitution, and therefore the price of the asset, can rise, fall, or remain 16

17 the same, depending on the sign of δ. Recall that there were two components of the ultimate change in welfare caused by the price change in the asset. The first component is a direct effect of the change in the information partition as the central bank makes the announcement to the agents, holding marginal utilities constant. The second arises due to the effect of risk aversion. We consider each separately. Lemma 3 addresses the part of the change in welfare that does not depend on risk aversion. The part of the price change that does not depend on the marginal utilities changing is just the size of the price change times the no-policy marginal utilities. In the bad state, e B, the price falls to the dividend value which is zero in this case. Thus the price change is just p NP 2 0 = p NP 2. In the good states, e G 2 and eg 3, we know the price rises to pp 2, so the price change is pn 2 P pp 2, which is less than zero. Then, the total change in utility that doesn t depend on risk aversion is given as π(e B )U ( WB NP ) p NP 2 + π(e G 2 )U ( W NP ) 2 (p NP 2 p P 2 ) + π(e G 3 )U ( W NP ) 3 (p NP 2 p P 2 ). (12) }{{}}{{} Expected utility gain Expected utility loss in good states in bad state which conceivably can be positive, negative or zero. Lemma 3 shows that the Expression (12) depends on the sign of δ, which as discussed above, changes sign with the assumptions about the utility function and the distribution of wealth. This is because the change in utility must obviously depend on how the price of the asset changes, because increases or decreases in the price of the asset affect the final wealths of the agents in all of the relevant states of the world. What is not clear is how, precisely, Expression (12) depends on δ. However, from Lemma 2, since CARA utility has the useful property of not affecting the marginal rates of substitution between states e G 2 and eg 3, this implies δ = 0, and the price of the asset does not change relative to the case where marginal utilities are held constant. Since the price of the asset doesn t change, it s clear that the welfare effects, holding the marginal utilities constant, should be zero. We can use this case to logically prove the other cases. When δ < 0, the purchase price in the policy equilibrium, p P 2, increases, and so from Expression (12), the buyer should be worse off since the second two terms fall (rise negatively), and the first term remains unchanged. Similarly, when δ > 0, the purchase 17

18 price falls in the policy equilibrium, and the buyer is better off. This is formally proved in Lemma 3. Lemma 3. Expression(12) has the opposite sign of δ. Proof. Recall that in period t = 2, the price in equilibrium in a given cell, P F,2, of Frank s information partition is given by the certainty equivalence (CE) formula, p 2 = ω P F,2 π(ω)u (W ω )d(ω) ω P i,2 π(ω)u, (13) (W ω ) where, π(ω) is the probability of state ω occurring, and W ω is the final (period 3) wealth of the agent in state ω. Recall that, for the no-policy equilibrium, the relevant period 2 information set for the buyer includes three states: e B, e G 2, and eg 3. Expanding (13) out for this case, 2 = π(eb )U ( WB NP p NP ) d(e B ) + π(e G 2 )U ( W NP 2 π(e B )U ( W NP B )) + π(e G 2 )U ( W NP 2 ) d(e G 2 ) + π(e G 3 )U ( W3 NP ) d(e G 3 ) ) + π(e G 3 )U ( ) W3 NP, (14) Rearranging gives us [ π(e B )U ( W NP B ) + π(e G 2 )U ( W NP 2 = π(e B )U ( W NP B ) + π(e G 3 )U ( W3 NP ) ] p NP 2 ) d(e B ) + π(e G 2 )U ( W NP 2 ) d(e G 2 ) + π(e G 3 )U ( W3 NP ) d(e G 3 ). (15) Note that (15) holds for any assumption about risk preferences. In the policy equilibrium, the general deflation policy reveals whether e B has occurred. This causes the asset price to fall to zero in the bad state, e B, and rise in the good states, e G 2 and e G 3. Since the policy changes both the price of the asset and the wealths in each state, the policy equilibrium analogue for Equation (15) will be a function of the new price and wealths. Thus, p P 2 18

19 is the solution to [ π(e G 2 )U ( W P 2 ) + π(e G 3 )U ( W P 3 )] p P 2 = π(e G 2 )U ( W P 2 ) d(e G 2 ) + π(e G 3 )U ( W P 3 ) d(e G 3 ). (16) Divide both sides by π(e G 2 )U ( W P 2 ) to re-express (16) in terms of the marginal rate of substitution between wealth in states e G 2 and eg 3, giving [ 1 + π(eg 3 )U ( W3 P π(e G 2 )U ( W2 P )] [ ) p P 2 = d(e G 2 ) + π(eg 3 )U ( W3 P π(e G 2 )U ( W2 P ) ] )d(e G 3 ). (17) Use equation (10) to re-express (17) in terms of the no-policy marginal utilities. [ ( π(e G )U ( W3 NP ) )] [ π(e G 2 )U ( ) W2 NP + δ p P 2 = d(e G 2 ) + ( π(e G 3 )U ( W3 NP ) π(e G 2 )U ( W NP 2 ) + δ ) d(e G 3 ) ]. (18) Multiplying both sides by U (W NP 2 ) then gives [ ] π(e G 2 )U (W2 NP )+π(e G 3 )U (W3 NP ) + δπ(e G 2 )U (W2 NP ) p P 2 = π(e G 2 )U (W NP 2 )d(e G 2 ) + π(e G 3 )U (W NP 3 )d(e G 3 ) + δπ(e G 2 )U (W NP 2 )d(e G 3 ) (19) Now, subtract (19) from (15) and recall that d(e G 2 ) = d(eb ) = 0 by assumption. Collecting like terms and simplifying gives π(e B )U (WB NP )p NP 2 + π(e G 2 )U (W2 NP )(p NP 2 p P 2 )+π(e G 3 )U (W3 NP )(p NP 2 p P 2 ) π(e G 2 )U (W NP 2 )p P 2 δ = π(e G 2 )U (W NP 2 )d(e G 3 )δ (20) 19

20 Finally, subtracting δu (W NP 2 )p P 2 from both sides, gives π(e B )U ( WB NP ) p NP 2 +π(e G 2 )U ( W2 NP ) (p NP 2 p P 2 )+π(e G 3 )U ( W3 NP ) (p NP 2 p P 2 ) = U (W2 NP ) ( p P 2 d(e G 3 ) ) δ. Since ( p P 2 d(eg 3 )) must be negative, 14 and U ( ) must always be positive, then we have the following three cases corresponding to the three cases in Lemma 2: If δ = 0, then U (W NP 2 ) ( p P 2 d(eg 3 )) δ = 0. If δ > 0, then U (W NP 2 ) ( p P 2 d(eg 3 )) δ < 0. If δ < 0, then U (W NP 2 ) ( p P 2 d(eg 3 )) δ > 0. (21) Since δ is the change in the marginal rate of substitution resulting from the policy, equation (21) implies that the expected change in utility evaluated at the no-policy marginal utilities depends on the sign of δ. Thus, the ultimate changes in welfare, evaluated at the no-policy marginal utilites, depend on the sign of δ. Lemmas 2 and 3 together with Lemma 1 define how welfare changes when we hold the marginal utilities constant at the no-policy level. This is the first of the two components of the price change defined at the beginning of this Section. The second part of the price change, that owing to the effect of risk aversion, is discussed next, and represents the losses to risk sharing as discussed earlier A graphical treatment Here we ll break down the effect of the policy into the two different parts, the effect of changes in wealth, holding marginal utility constant at the no-policy level, as in (12), and the additional effect of allowing marginal utility to change when wealth changes. 14 The price in period 2 could never be larger than the highest possible dividend in period 3, and they could only be equal if there were no uncertainty. 20

21 The first part is given in equation (21) above, and is illustrated by the rectangles in Figures 3 and 4. The horizontal dashed lines in each figure represent the no-policy marginal utilities from (21). Thus, in Figure 3, the box ABCD is the utility gain for the buyer in the bad state of the world, calculated at the no-policy marginal utility, U (WB NP ). This area corresponds to the first term in (21). In figure 4, the boxes F GHI and JKLM represent the utility losses for the buyer in the good states, again calculated at the no-policy marginal utilities, U (W NP 2 ) and U (W NP 3 ), as in (21). These boxes represent the second and third terms in (21) respectively. Equation (21) implies that areas of boxes ABCD, F GHI, and JKLM (where the areas of F GHI and JKLM are negative) must sum to U (W NP 2 ) ( p P 2 d(eg 3 )) δ. In the CARA case, this value will always be zero. In the DARA (IARA) case it could be negative or positive depending on the distribution of wealth between states e G 3 and eg 2. In Figure 3, showing the bad seller state, the marginal utility of wealth falls as a result of the policy because the purchase price falls to zero in this state, so the buyer s wealth rises. Thus, the agent s utility gain in the bad state of the world is not ABCD, but the smaller area, ABED. The darkly shaded area BCE represents the loss in utility in this state because the true marginal utility from this gain in wealth is less than the no-policy marginal utility. In Figure 4, showing the good seller states of the world, the marginal utility of wealth rises as a result of the policy, because the purchase price of the asset rises in these states, so the buyer s wealth falls. This means that the total utility loss to the buyer in these states is not F GHI and JKLM, but the larger areas F NHI and JOLM. That is, the buyer is even worse off in these states, by an amount given by in the darkly shaded areas GHN and KLO. 21

22 Figure 3: Utility change in the bad seller state Figure 4: Utility change in the good seller states 22

23 Theorem 1. The general deflation policy will always reduce welfare for buyers if δ 0. Proof. The total change in expected utility due to the policy is W P π(e B B ) WB NP U (W )dw }{{} EU(e B ) W P + π(e G 2 2 ) W2 NP U (W )dw }{{} EU(e G 2 ) W P + π(e 3 3 G) W3 NP }{{} EU(e G 3 ) U (W )dw, (22) where the second and third integrals are negative since W P k it is obvious that < W NP k for k = 2, 3. From Figure 3, 2 U ( (WB NP ) W P B > U (W )dw, (23) p NP W NP B since p NP 2 = W P B W NP B 4, it is obvious that and U (WB NP ) > U (W ) for (WB NP < W < WB P. Similarly, from Figure and W P (p NP 2 p P 2 )U (W2 NP 2 ) > U (W )dw, (24) W2 NP W P (p NP 2 p P 2 )U (W3 NP 3 ) > U (W )dw, (25) W3 NP where note that both sides of (24) and (25) are negative. Using (23), (24), and (25) to bound (22, gives π(e B )p NP 2 U ( WB NP ) + π(e G 2 )(p NP 2 p P 2 )U ( W2 NP ) + π(e 3 G )(p NP 2 p P 2 )U ( W3 NP ) >π(e B ) W P B ) W NP B U (W )dw + π(e G 2 ) W P 2 W NP 2 U (W )dw + π(e G 3 ) W P 3 W NP 3 U (W )dw (26) 23

24 Using Equation (21), we can replace the left hand side of this equation, so that U (W2 NP ) ( p P 2 d(e G 3 ) ) δ >π(e B ) W P B ) W NP B U (W )dw + π(e G 2 ) W P 2 W NP 2 W P U (W )dw + π(e G 3 3 ) U (W )dw W3 NP (27) The right hand side of this equation is the total change in utility from the policy. The sign of the left hand side of this equation is given by (21), and is controlled by the sign of δ as in Lemma 3. Using the three conditions listed in Lemma 3, we can say the following: If δ = 0, then the policy will reduce welfare since the right hand side of (27 is less than zero. If δ > 0, then the policy will reduce welfare since the right hand side of (27 is smaller than left hand side which is negative. If δ < 0, then the policy s effect on welfare is ambiguous since the right hand side of (?? is less than the left hand side which is positive. Thus the left hand side can be positive, negative, or zero, and so the policy might improve welfare, diminish welfare, or be welfare neutral. Thus it must be the case that the policy diminishes welfare if δ 0, and changes welfare ambiguously if δ < 0. Theorem 1 gives us a definite welfare result for the policy as long as δ 0. This corresponds to two specific cases. First, if utility is CARA (regardless of the distribution of wealth), and second, if utility is DARA and W 2 > W The reason why the wealth distribution matters with DARA utility is that when W 2 > W 3, the asset serves as a type of insurance against low wealth in state e G 3. The dividend paying state is e G 3. So the assumption that W 2 > W 3 makes the dividend paying state Frank s lowest wealth state. Because the policy raises the price of the asset in states e G 2 and eg 3, and because Frank is more risk averse in state e G 3 due to DARA utility, this harms Frank because the price of insurance against the low wealth state of the world is rising. This, combined with the Hirshleifer effect, which we know makes Frank worse off, is sufficient to show that Frank will be 15 The IARA results are clearly opposite to the DARA results. Discussion of IARA is omitted here because of its empirical implausibility. 24

25 worse off in this case. While Theorem 1 will hold as long as δ 0, the results are ambiguous if δ < 0. 4 An Example Problem Including Risk Aversion 4.1 The bubble equilibrium with constant marginal utilities We begin by generating an example of a bubble equilibrium (again following Conlon (2015)) where the final, equilibrium, marginal utilities for each state of the world are as in Table 1, and where x, y, and z are all positive. These marginal utilities will generate certainty equivalent prices for the risky asset in each state of the world, as in equation (6). They are also arranged in such a way that changing x, y, and z do not affect the asset s price. They simply control the weight or importance that the agents give to different states of the world. 16 In states {b, e B, e G 1, eg 2, eg 3 } Ellen is endowed with one unit of the risky asset at the beginning of the world. In states {b, f B, f G 1, f G 2, f G 3 }, Frank is endowed with one unit of the asset. Table 1: Assumed marginal utilities in the bubble equilibrium State b e B f B e G 1 e G 2 e G 3 f G 1 f G 2 f G 3 Ellen x x z 2y y y 5z z 2z Frank x z x 5z z 2z 2y y y Working backwards, at t = 3, all information has been revealed and the agents know the true state of the world, so p 3 (ω) = d(ω). Suppose the asset will pay a dividend of d = 4 in states e G 3 and f G 3 Buyer, and a dividend of d = 0 in all other states (as in Figure 1). For F2 = {e B, e G 2, eg 3 }, in period 2 Frank believes the state could be e B, e G 2, or eg 3. Then, utilizing (6), we have 16 At this point, these marginal utilities are simply assumed. Later, we will work backwards from this example to derive the implied wealths in each state necessary to produce this particular bubble equilibrium for a given risk averse utility function. 25

26 [ E W T P F (ω) = E ] U F,ω p 3 ω F Buyer 2 [ ] = U F,ω Buyer ω F2 (z)(0)(1/3) + (z)(0)(1/3) + (2z)(4)(1/3) (z)(1/3) + (z)(1/3) + (2z)(1/3) = 2 Ellen s problem is similar. Within the set F Buyer 2, Ellen has two possible information sets: E2 BS = {e B }, and E2 GS = {e G 2, eg 3 }. For ω EBS 2, W T P E (ω) = (x)(0)(1/9) (1) = 0, and for ω E GS, W T P E (ω) = (y)(0)(1/9) + (y)(4)(1/9) (y) + (y) = 2. The market is competitive with short sale constraints, so the price for any given state ω, p t (ω), is just the greater of the two willingnesses to pay. Therefore, in state e B, Ellen sells to Frank at a price of 2, and in states e G 2, and eg 3, Ellen is indifferent between buying and selling at the price of 2, and so is willing to sell. Similar calculations can be made for ω E Buyer 2, and also for the period 1 information sets for both agents. Applying similar calculations to the remaining states in both periods 1 and 2, we arrive at an equilibrium set of prices and net sales for each state of the world. The prices are shown below in Table 2, and the pattern of trade is shown in Table 3. In period 2, Ellen sells to Frank in information sets E2 BS = {e B } and E2 GS = {e G 2, eg 3 }, while Frank sells to Ellen in information sets F BS 2 = {f B } and F GS 2 = {f G 2, f G 3 }.17 Table 2: Bubble Equilibrium Prices State b e B f B e G 1 e G 2 e G 3 f G 1 f G 2 f G 3 Period Period Period It is not an accident that we have chosen the parameters such that trade occurs only in period 2. This greatly simplifies the analysis. 26

27 Table 3: Sales from Ellen to Frank - Bubble Equilibrium State b e B f B e G 1 e G 2 e G 3 f G 1 f G 2 f G 3 Period Period Period This exercise generates three states of the world where the asset is overpriced. In states b, e B, and f B, at least one agent knows the asset pays a final dividend of zero, yet the price is above zero in period 1. In state b, both agents realize the asset is worthless, but the price is positive. This is the familiar strong bubble result from Allen, et al. (1993). In state b, Ellen knows the asset is worthless, but does not know that Frank also knows the asset is worthless. From her point of view, she may be in state e B, in which case Frank will be in F Buyer 1. This means that she believes that Frank might believe that the true state could be e G 3. If this is the case, Frank will be willing to buy the asset from Ellen for a positive price. Frank, meanwhile, is thinking the exact same thing about Ellen in state b, so in period 1, both agents are hoping to sell a worthless asset to the other in the next period. In period 2, both agents learn that since they re trying to sell to each other, so the true state must be b, and the bubble crashes to a price of zero in period 2. Re-applying the calculations in Equation (6) to the new information structure, holding the marginal utilities from Table 1 constant, we arrive at a new set of prices and trades shown in Tables 4 and 5. Table 4: Risk Neutral Prices Under Bubble Bursting Policy State b e B f B e G 1 e G 2 e G 3 f1 G f2 G f3 G Period Period Period Table 5: Sales from Ellen to Frank - Policy Equilibrium State b e B f B e G 1 e G 2 e G 3 f G 1 f G 2 f G 3 Period Period Period