Course Handouts: Pages 1-20 ASSET PRICE BUBBLES AND SPECULATION. Jan Werner

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1 Course Handouts: Pages 1-20 ASSET PRICE BUBBLES AND SPECULATION Jan Werner European University Institute May

2 I. Price Bubbles: An Example Example I.1 Time is infinite; so dates are t = 0,1,2,...,. There is no uncertainty. There is one security that pays zero dividend at every date. Two agents (i = 1,2) with the same utility function u i (c) = β t ln(c t ), (1) t=0 where 0 < β < 1. Agents maximize utility u i (c) subject to budget constraints c 0 + p 0 h 0 = w i 0 + p 0 α i 0, (2) c t + p t h t = w i t + p t h t 1, for all t 1, (3) h t 1, for all t (4) Constraint (4) is a short-sales constraint. Clearly, there is a trivial equilibrium with zero prices, p t = 0 for all t, and no trade. Claim: There can be an equilibrium with p t > 0 for every t. If p t > 0, then the price exceeds the fundamental value and there is price bubble. The fundamental value of zero dividends must be zero. 2

3 Suppose that endowments are w 1 0 = B + η, w 2 0 = A η, w 1 t = B, w 2 t = A for t even, w 1 t = A, w 2 t = B for t odd, where βa > B > 0. Initial security holdings are α 1 0 = 1 and α 2 0 = 0. The total supply is 1. Let η = βa B 3(1 + β). (5) There is an equilibrium with strictly positive price p t = η, for all t 0, (6) consumption plans c 1 t = (B + 3η) for t even, c 1 t = (A 3η) for t odd c 2 t = (A 3η) for t even, c 2 t = (B + 3η) for t odd, and security holdings h 1 t = 1 for t even, h 1 t = 2 for t odd h 2 t = 2 for t even, h 2 t = 1 for t odd 3

4 Verifying the equilibrium: (i) markets clear at every date, (ii) budget and short-sales constraints are all satisfied. (iii) verifying that consumption plans and security holdings are optimal: At each odd t, agent s 1 first-order condition is It holds. At each even date t, the first-order condition is β t c 1 t β t c 1 t p t = βt+1 c 1 p t+1, (7) t+1 p t βt+1 c 1 p t+1, (8) t+1 since short-sales constraint is binding. It is satisfied. A suitable transversality condition (see Kocherlakota (1992), Proposition 2) can also be verified to hold. The same for agent 2. So there is equilibrium price bubble. This security is fiat money. 4

5 II. Equilibrium with Portfolio Constraints II.1 The Model. Time: t = 0,1,... Uncertainty: Infinite set of states S. Information about the state at date t is described by a finite partition F t of S. F t+1 is finer than F t (nondecreasing information). F 0 = {S}. This is an infinite event tree. ξ t F t denotes an event at date t; ξt F t 1 is the predecessor of ξ t at date t 1, that is, ξ t ξt. Securities are traded at each date - infinitely-lived securities. Security j pays dividend x j (ξ t ) at date-t event ξ t for every t 1. Price of security j at date t in event ξ t is p j (ξ t ). A portfolio in event ξ t is h(ξ t ); h = {h t } is a portfolio strategy. Assumption: Dividends are positive, i.e., x(ξ t ) 0 for every ξ t. 5

6 Agents. Consumption plans: c(ξ t ) in event ξ t at date t; c t event-contingent consumption plan at date t; c = (c 0,c 1,...,). Agent i s utility function u i : C i R, where C i R +. u i is assumed increasing. Endowment is w i, and initial portfolio is α i R J. Consumption-Portfolio Choice. maxu(c) (10) c,h subject to c(ξ 0 ) + p(ξ 0 )h(ξ 0 ) = w(ξ 0 ) + p(ξ 0 )α, c(ξ t ) = w(ξ t ) + (p(ξ t ) + x(ξ t ))h(ξ t ) p(ξ t )h(ξ t ), ξ t t 1, (11) c C, h H (12) H is the set of feasible portfolio strategies. The agent is restricted in her choices of portfolio strategies to those that are in H. Assumption: H is closed, convex, and 0 H. If H = R, then there is no restriction. This would make no sense. Observation: If u is strictly increasing and p(ξ t ) 0 for every ξ t, then there is no solution to the portfolio-consumption choice problem with H = R. 6

7 Portfolio Constraints. Examples of constraints that may be used to define feasible portfolio set H : short sales constraint h j (ξ t ) b j (ξ t ), ξ t, (13) borrowing constraint p(ξ t )h(ξ t ) B(ξ t ), ξ t, (14) for some sequence of positive bounds {B(ξ t )}. debt constraint [p(ξ t ) + x(ξ t )]h(ξ t ) D(ξ t ), ξ t,,t 1, (15) for some sequence of positive bounds {D(ξ t )}. There are other possible constraints such as transversality constraint, wealth constraint, solvency constraint, collateral constraint, etc. 7

8 First-Order Conditions. First-order conditions for a solution (c,h) such that (i) c(ξ t ) > 0, ξ t, and (ii) portfolio constraint H is not binding at h, are p j (ξ t ) = ξ t+1 ξ t [p j (ξ t+1 ) + x j (ξ t+1 )] ξt+1u ξt u, j, ξ t (16) The price of security j in event ξ t equals the sum over immediate successor events ξ t+1 of cum-dividend payoffs of security j multiplied by the marginal rate of substitution between consumption in event ξ t+1 and consumption in event ξ t. II.2 Equilibrium in Security Markets with Portfolio Constraints. An equilibrium is an allocation {(c i,h i )} and a price system p such that (i) portfolio strategy h i and consumption plan c i are a solution to agent i s choice problem (10) subject to constraints (11), (12), (ii) markets clear, that is h i (ξ t ) = ᾱ, ξ t (17a) i and c i (ξ t ) = w(ξ t ) + x(ξ t )ᾱ, ξ t. (17b) i 8

9 III.1 Definitions. III. Arbitrage and Ponzi Schemes. Let z(h,p) denote the (net) payoff of portfolio strategy h in event ξ t : z(h,p)(ξ t ) = (p(ξ t ) + x(ξ t ))h(ξ t ) p(ξ t )h(ξ t ). (18) An arbitrage is a portfolio strategy ĥ such that z(ĥ,p)(ξ t) 0, ξ t t 1, and p 0 ĥ 0 0, (19) with at least one strict inequality. A finite-time arbitrage is an arbitrage portfolio strategy ĥ such that ĥt = 0 for all t τ for some τ. An infinite-time arbitrage is an arbitrage that is not a finite-time arbitrage. A Ponzi scheme is an infinite-time arbitrage portfolio strategy ĥ such that z(ĥ,p)(ξ t) = 0, ξ t t 1, and p 0 ĥ 0 < 0. (20) This is a strategy of rolling over the debt forever. Arbitrage ĥ is unlimited arbitrage for portfolio constraint H if h + λĥ H, λ 0, h H. (21) Condition (21) says that adding strategy ĥ, or an arbitrary multiple thereof, to any portfolio position h does not violate constraint H. Formally, (21) requires 9

10 that ĥ lies in the asymptotic (or recession) cone of H. If H is a cone, then (21) holds if and only if ĥ H Theorem 1: If agents utility functions are strictly increasing, then, in equilibrium, there is no unlimited arbitrage for H. III.2 Arbitrage and Event Prices. Let H be defined by the borrowing constraint (14). Portfolio strategy ĥ is an unlimited arbitrage under borrowing constraint iff ĥ is an arbitrage and p(ξ t )ĥ(ξ t) 0, ξ t. (22) Clearly, there is no unlimited Ponzi scheme under borrowing constraint at any p. Theorem 2: Assume that p(ξ t ) > 0 for every ξ t. Then there is no unlimited arbitrage under borrowing constraint iff there exist strictly positive numbers q(ξ t ) for all ξ t such that q(ξ t )p j (ξ t ) = q(ξ t+1 )[p j (ξ t+1 ) + x j (ξ t+1 )] ξ t j. (23) ξ t+1 ξ t Strictly positive numbers q(ξ t ) satisfying (23) and normalized so that q(ξ 0 ) = 1 are called event prices. 10

11 Let H be defined by the debt constraint (15). Portfolio strategy ĥ is an unlimited arbitrage under debt constraint iff ĥ is an arbitrage and [p(ξ t ) + x(ξ t )]ĥ(ξ t ) 0, ξ t,,t 1 (24) Theorem 3: There is no unlimited arbitrage under debt constraint iff there exist strictly positive event prices. 11

12 IV. Price Bubbles. VI.1 Definitions. Assume that p is such that there exist event prices q >> 0 satisfying (23). The present value at date 0 of security j under q is defined by t=1 ξ t F t q(ξ t )x j (ξ t ). (25) Similarly, we can define present value of security j at any event. Price bubble is the difference between the price and the present value of a security. Price bubble at ξ t is σ j (ξ t ) p j (ξ t ) 1 q(ξ t ) τ>t ξ τ ξ t q(ξ τ )x j (ξ τ ) (26) Theorem 4: Suppose that there exist strictly positive event prices. Then (i) If p(ξ t ) 0 for all ξ t, then 0 σ j (ξ t ) p j (ξ t ), ξ t j. (ii) If security j is of finite maturity (that is, x jt = 0 for t τ for some τ, and that security is not traded after date τ), then σ j (ξ t ) = 0 for all ξ t. (iii) It holds q(ξ t )σ j (ξ t ) = ξ t+1 ξ t q(ξ t+1 )σ j (ξ t+1 ) ξ t j. 12

13 IV.2 Price Bubbles in Equilibrium. The question is whether price bubbles can be nonzero in equilibrium in security markets. We say that security markets are complete at prices p if the one-period payoff matrix [p j (ξ t+1 ) + x j (ξ t+1 )] j,ξt+1 has the rank equal to the number of events ξ t+1 that are successors of ξ t, for every ξ t. Let H be defined by the borrowing constraint (14). Theorem 5: Let p 0 be an equilibrium price system under the borrowing constraint. Suppose that security markets are complete at p and there exist strictly positive event prices q. If and t=1 q(ξ t ) w(ξ t ) <, ξ t F t ᾱ 0 >> 0, then price bubbles are zero. See Santos and Woodford (1997) (also for incomplete markets), Huang and Werner (2000). 13

14 Let H be defined by the debt constraint (15). Theorem 5-d: Let p 0 be an equilibrium price system under the debt constraint. Suppose that security markets are complete at p and there exist strictly positive event prices q. If and t=1 q(ξ t ) w(ξ t ) <, ξ t F t ᾱ 0 >> 0, then price bubbles are zero. [This is a conjecture. I am not sure whether the result holds for incomplete markets.] Remark: It is crucial in Theorem 5 that borrowing limits B(ξ t ) are positive, (so that they do not become minimum saving requirements). Similarly, D(ξ t ) are positive in Theorem 6. 14

15 IV.3 Pareto Optimal Equilibria in Security Markets. A special case of the borrowing constraint is the wealth constraint with B(ξ t ) = 1 q(ξ t ) τ>t where q is a sequence of event prices. ξ τ ξ t q(ξ τ )w(ξ τ ) ξ t, (27) Theorem 6: Let p and {c i,h i } be a security market equilibrium under the wealth constraint. If security markets are complete at p and price bubbles are zero, then {c i } and P given by P(c) ξ t E q(ξ t )c(ξ t ) (28) are an Arrow-Debreu equilibrium. Further, consumption allocation {c i } is Pareto optimal. 15

16 V. Examples. Example V.1: Binomial event-tree with Prob(up ξ t ) = Prob(down ξ t ) = 1 2. Two consumers with 0 < β < 1. u i (c) = β t E[ln(c t )] t=0 Two securities with dividends x 1 t 1, t, for security 1, and x 2 (ξ t ) = u, or = d depending on whether ξ t = (ξ t 1,up) or ξ t = (ξ t 1,down) for security 2. Assume u > d. Initial portfolio holdings are α 1 = (0,1) and α 2 = (0,0). Endowments are w 1 (ξ t ) = A u, w 2 (ξ t ) = B, whenever ξ t = (ξ t 1,up), and w 1 (ξ t ) = B d, w 2 (ξ t ) = A, whenever ξ t = (ξ t 1,down) for all t 1, and w 1 0 = w 2 0 = A+B 2. Note that w 1 t + w 2 t + (α 1 + α 2 )x t = A + B. 16

17 This is a no-aggregate risk environment. Portfolio constraint is the wealth constraint (27). Security market equilibrium under the wealth constraint has security prices p 1 (ξ t ) = β τ = β 1 β τ=1 p 2 (ξ t ) = u + d β τ = u + d β β, and consumption allocation τ=1 c 1 (ξ t ) = c 2 (ξ t ) = A + B, ξ t. 2 and some portfolio strategies h 1,h 2. Security markets are complete at p and price bubbles are zero. Event prices are q(ξ t ) = 1 2 t β t. Equilibrium with bubble has security prices p 1 (ξ t ) = β τ + σ(ξ t ) = τ=1 p 2 (ξ t ) = u + d 2 τ=1 β 1 β + σ(ξ t), β τ = u + d 2 β 1 β, where σ(ξ t ) is chosen arbitrarily subject to (iii) of Theorem 4, that is, σ(ξ t ) = β[ 1 2 σ(ξ t,up) σ(ξ t,down)], (30) and σ(ξ t ) 0, ξ t. Equilibrium consumption allocation {c 1,c 2 } is as before. Portfolio strategies are different. At these security prices markets are complete and there is price bubble on security 1 equal to σ. There is no price bubble on security 2. 17

18 Example V.2: Suppose that initial asset holdings are α 1 = (1,1) and α 2 = ( 1,0). Portfolio constraint is the wealth constraint (27). There is an equilibrium security prices and p 1 (ξ t ) = β τ + σ(ξ t ) = τ=1 p 2 (ξ t ) = u + d 2 τ=1 β 1 β + σ(ξ t), β τ = u + d 2 β 1 β, where σ(ξ t ) 0 is chosen arbitrarily subject to (30). At these prices markets are complete and there is price bubble on security 1 equal to σ. There is no price bubble on security 2. Equilibrium allocation is and c 1 (ξ t ) = A + B 2 c 2 (ξ t ) = A + B 2 + σ 0 (1 β), ξ t, σ 0 (1 β), ξ t. This is an Arrow-Debreu equilibrium allocation with transfers: consumer 1 receives date-0 transfer σ 0 from consumer 2. Note that σ 0 has to be small so that agent 2 s consumption be positive. Event prices are q(ξ t ) = 1 2 β t. t 18

19 Example V.3: Kocherlakota (2008) There is one infinitely lived security with risk-free dividend d, the same at every date, and a full set of one-period Arrow securities at every date-event. One (representative) agent with utility function u i (c) = β t E[v(c t )], (31) t=0 where 0 < β < 1. Consumption endowment is constant ω(ξ t ) = ω. Initial security holding is α 0 = 1. Portfolio constraint is the debt constraint (15) with D(ξ t ) = ω (1 β) The no-trade equilibrium is c t = ω + d, h t = 1, with p(ξ t ) = βd (1 β) Event prices are q(ξ t ) = β t Prob(ξ t ). No bubble in this equilibrium. 19

20 Consider perturebed debt constraint with adjusted limits ˆD given by where σ satisfies (30). ˆD(ξ t ) = ω (1 β) σ(ξ t) (31) There is an equilibrium with c t = ω + d, h t = 1, and p(ξ t ) = βd (1 β) + σ(ξ t). There is a bubble! Event prices are q(ξ t ) = β t Prob(ξ t ). Note that if the bubble is non-zero, then it must grow without bound for some state and debt limit ˆD given by (31) will become negative (!?) in that state. 20