Rational Bubbles and Middlemen Yu Awaya Kohei Iwasaki Makoto Watanabe University of Rochester University of Wisconsin VU Amsterdam/ TI December 25, 2019 1 / 65
Motivation Bubbles: ◮ Continuous price increases, interrupted by a sudden market crash ◮ Chains of intermediaries engaged in flipping Examples: Dutch tulip mania (1634-7); Mississippi Bubble (1719-20); South Sea Bubble (1720); Roaring Twenties followed by the 1920 crash; Housing bubble preceded the 2008 financial crisis = ⇒ Explore for a (simple) framework of bubbles that features the above 2 / 65
Our Approach ◮ Why would a smart person hold an asset they know is overpriced? ◮ they’re hoping to sell it to another person just before the bubble bursts ◮ Why would that other smart person buy an asset that’s about to collapse? ◮ Bubbles are impossible ◮ They expect the overpricing to grow forever ◮ Our answer: finite horizon, identifying exactly the timing of bubble burst 3 / 65
Our Approach Implications: ◮ The intuition of market participants, “if they want to ride a bubble, they must carefully time the point at which they sell to a “greater fool”, and so, get out of the bubble” ◮ Booms turn into euphoria as “rational exuberance morphs into irrational exuberance” Charles P. Kindleberger (1978) “Manias, Panics, and Crashes: A History of Financial Crises” 4 / 65
Illustrative example ◮ Suppose there are two agents, A 1 and A 2 A 1 A 2 ◮ And two goods—goods x and y 5 / 65
Illustrative example x A 1 A 2 y ◮ Good y can be produced (at a certain cost) and consumed by both agents ◮ Good x is owned by agent A 1 , but consumed only by A 2 6 / 65
Illustrative example ◮ The consumption value of good x is stochastic ◮ Specifically, the value � with some probability v V = 0 with the remaining probability where v > 0 7 / 65
Illustrative example ◮ Obviously, bubble never occur ◮ That is, consider a case where ◮ V = 0, that is the value of object x is 0 ◮ And all agents know this ◮ In this case, trade doesn’t occur ◮ A 2 rejects to produce any positive amount of good y to get good x 8 / 65
Illustrative example ◮ Now suppose the trade can be done through a middleman (flipper) ◮ In particular, there are three agents, A 1 , A 2 and A 3 A 1 A 2 A 3 9 / 65
Illustrative example As before, two goods, x and y ◮ Good x is now owned by A 1 and can be consumed only by A 3 ◮ Good y can be produced and consumed by all agents ◮ The consumption value of good x � v with some probability V = 0 with the remaining probability 10 / 65
Illustrative example Trading protocol is similar as before: ◮ First A 1 and A 2 can trade goods x and y x A 1 A 2 A 3 y ◮ If the trade occurs, then A 2 and A 3 can trade goods x and y x A 1 A 2 A 3 y 11 / 65
Illustrative example ◮ Now suppose as before ◮ V = 0, that is the value of object x is 0 ◮ And all agents know this ◮ Can good x ever be traded with good y ? ◮ Can bubble occur? 12 / 65
Illustrative example ◮ Yes! ◮ There are certain cases in which good x is traded for good y , although everyone knows the consumption value of x is 0 ◮ Specifically suppose A 2 is a fool who (mistakenly) believes that A 3 is a greater fool than he is ◮ That is, A 2 puts high probability on the event than A 3 does on the event that x has value ◮ Consistent with all agents knowing the value of x is 0 ◮ In this case... 13 / 65
Illustrative example Then A 2 is still willing to trade with A 1 x A 1 A 2 A 3 y 14 / 65
Illustrative example Hoping to trade with A 3 x A 1 A 2 A 3 y ◮ Recall A 2 does NOT know that A 3 knows V = 0 15 / 65
Illustrative example Unfortunately for A 2 , A 3 refuses the trade A 1 A 2 A 3 ◮ A 3 knows good x has no value ◮ A 2 turns out to be the greatest fool who cannot find a greater fool 16 / 65
Bubble Middlemen (flippers) are a source of bubbles ◮ End users care about the quality of an asset ◮ Middlemen don’t ◮ Downstream middlemen only care about how end users think about the asset ◮ Upstream middlemen only care about how down stream middlemen think about the asset 17 / 65
Paper Based upon this observation ◮ We construct a tractable model of bubbles in an economy with flippers ◮ An object with no value is traded although everyone knows that it has no value ◮ A fool buys the object, hoping to find a greater fool who buys the object from him ◮ Bubble occurs in the unique equilibrium ◮ The model describes the life of a bubble 18 / 65
Price path An object without fundamental value is traded at a positive price price time 19 / 65
Price path Price of the object increases—and accelerates—as time passes price time While the fundamental of the economy does NOT grow 20 / 65
Price path And someday, it bursts price time 21 / 65
Paper And ◮ Provide a simple condition for which bubble is detrimental ◮ Show bubble-bursting policy (Conlon, 2015) does not affect welfare ◮ Information increases size of bubble ◮ Not information on fundamentals, but information on knowledge of the other agents 22 / 65
Fools We do NOT assume irrational agents nor heterogeneous priors ◮ Fools are not irrational, but ignorant people 23 / 65
The Model 24 / 65
Objects ◮ Two goods— x and y ◮ Good x is durable and indivisible ◮ Good y is perishable and divisible 25 / 65
Environment N agents, A 1 , A 2 ,..., A N A 1 A 2 A N 26 / 65
Environment ◮ Good x is owned by A 1 and can be consumed only by A N ◮ The consumption value of good x � v > 0 with some probability V = 0 with remaining probability ◮ Good y can be produced and consumed by all agents ◮ The cost of producing ˆ y units of good y is ˆ y ◮ The utility of consuming ˆ y units of good y is κ ˆ y 27 / 65
Environment x x A n − 1 A n + 1 A n y y ◮ Agent A n − 1 and A n + 1 can trade only through A n ◮ First A n − 1 and A n can (if both want) exchange x and some amount of good y ◮ Conditional on the trade between A n − 1 and A n , A n and A n + 1 can exchange x and some amount of good y ◮ The amount of y is determined by Nash bargaining 28 / 65
Knowledge ◮ Introduce type space ◮ Each type describes who knows what ◮ In a way reminiscent of Rubinstein’s Email game ◮ Rather schematic ◮ A way to help illustrating the relevant knowledge structure 29 / 65
Knowledge A N − 2 A N − 1 A N ◮ If V = 0, A N gets a signal s N with some probability ◮ Thus, if A N gets s N , then he knows that V = 0 ◮ If not, A N becomes optimistic about the value of good x 30 / 65
Knowledge A N − 2 A N − 1 A N ◮ If A N gets the signal s N , then he sends a signal (“rumor”) s N − 1 to A N − 1 ◮ The “rumor” reaches A N − 1 with some probability ◮ Thus, if A N − 1 gets s N − 1 , then he knows that A N knows V = 0 31 / 65
Knowledge A N − 2 A N − 1 A N ◮ If A N − 1 gets the signal s N − 1 , then he sends a signal (“rumor”) s N − 2 to A N − 2 ◮ The “rumor” reaches A N − 2 with some probability ◮ Thus, if A N − 2 gets a signal s N − 2 , then he knows that A N − 1 knows that A N knows V = 0 32 / 65
Knowledge In general A n − 1 A n A N ◮ If A n gets the signal s n , then he sends a signal (“rumor”) s n − 1 to A n − 1 ◮ The “rumor” reaches A n − 1 with some probability ◮ Thus, if A n − 1 gets a signal s n − 1 , then he knows that A n knows that ... that A N knows that V = 0 33 / 65
Knowledge A 1 A 2 A N ◮ If A 1 gets the signal s 1 , the process stops 34 / 65
Knowledge ◮ Finally, assume all but A N always know the value of x 35 / 65
Type space Formally, the set of the state of the world Ω = { ω v , ω φ , ω N , ..., ω 1 } where ◮ ω v means V = v ◮ ω φ means V = 0 and no agents get a signal ◮ ω n means V = 0 and agent n is the last one to get a signal 36 / 65
Partition Partition of ◮ A N is {{ ω v , ω φ } , { ω N , .. ω 1 } } � �� � � �� � signal no signal ◮ A n is {{ ω v } , { ω φ , .., ω n + 1 } , { ω n , ..., ω 1 } } � �� � � �� � � �� � V = v V = 0, signal V = 0, no signal 37 / 65
Prior ◮ Prior distribution µ on Ω ◮ Homogeneous prior— µ is common knowledge 38 / 65
Price ◮ Price (the amount of good y ) is determined by Nash barganing ◮ Outside option is 0 ◮ The value of good x is unknown, but the expected value is common knowledge ◮ Can be generalized ◮ Let θ be the bargaining power of A n in trade between A n and A n + 1 ◮ Price of each pair is NOT observed by outsiders ◮ Over-the-couter market 39 / 65
Timing 1. Nature determines V 2. Signals (“rumors”) are send, and a type is determined 3. Actual trades start 40 / 65
Main result Definition We say bubble occurs if ◮ Everyone knows the value of good x is 0 ◮ And yet good x is exchanged with positive amount of good y 41 / 65
Main result Theorem The equilibrium is unique. In the equilibrium, a bubble occurs when ω ∈ { ω N , ω N − 1 , · · · , ω 3 } . Moreover, a bubble bursts for sure. 42 / 65
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