Some Simple Bitcoin Economics Linda M. Schilling* and Harald Uhlig** - PowerPoint PPT Presentation

Some Simple Bitcoin Economics Linda M. Schilling* and Harald Uhlig** *Ecole Polytechnique CREST and **University of Chicago May 24, 2018

Motivation Coexistence of fiat moneys: Dollar versus Bitcoin (any cryptocurrency) CB controlled supply versus uncontrolled production Question ◮ How do Bitcoin prices evolve (Speculation)? ◮ How do Bitcoin prices affect Monetary Policy and vice versa? 2 / 18

Literature Bitcoin Pricing ◮ Athey et al ◮ Garratt and Wallace (2017) ◮ Huberman, Leshno, Moallemi (2017) Currency Competition ◮ Kareken and Wallace (1981) (Monetary) Theory ◮ Bewley (1977) ◮ Townsend (1980) ◮ Kyotaki and Wright (1989) ◮ Lagos and Wright (2005) 3 / 18

The Model I - Economy ◮ Discrete time ◮ Randomness θ t per period ◮ 2 types of agents: ◮ red j ∈ [0 , 1), green j ∈ [1 , 2]: each mass 1 ◮ utility from consuming: u ( · ) strictly increasing, concave ◮ no money in utility function (money intrinsically worthless) ◮ 3 goods: ◮ consumption good perishable/not storable random production y t ∈ [ y , y ] , y > 0 (Bitcoin, Dollar) 2 Fiat Moneys ◮ storable equally adopted as means of payment � 0 , ∞ t odd � U g = β t ( ξ t u ( c t ) − e t ) , ξ t = 1 , t even t =0 4 / 18

Timing - Alternation c c p o o e p c p 5 / 18

Timing - Transfers CB CB MINING MINING c c p e o e o MINING MINING p c p e e CB 6 / 18

The Model II - Moneys ◮ P t ( θ t ) price of consumption good in Dollar ◮ Q t ( θ t ) price of Bitcoin in terms of consumption good Dollars D t : ◮ CB D t = D t − 1 + τ t , τ t : ! P t ≡ 1 ! Bitcoins B t : ◮ B t +1 = B t + f ( e t ) , A t +1 = f ( e t +1 , B t +1 ) ≥ 0 7 / 18

Equilibrium An equilibrium is a stochastic sequence ( A t , [ B t , B t , g , B t , r ] , [ D t , D t , g , D t , r ] , τ t , ( P t , z t , d t ) , ( Q t , x t , b t ) , e t ) t ≥ 0 ◮ Utility is maximized to green and red agents. ◮ Prices clear market for consumption good, Dollars and Bitcoin � 2 ◮ y t = 0 c t , j dj ◮ � 2 � 2 0 z t , j dj = 0 d t , j dj y t = x t , j + z t , j ◮ � 2 � 2 0 x t , j dj = 0 b t , j dj c t , j = b t , j + d t , j ◮ D t = D t , g + D t , r ◮ B t = B t , g + B t , r ◮ Central Bank control P t = 1 ◮ Budget constraints Evolution money stock ◮ 0 ≤ b t , j ≤ B t , j Q t B t +1 , j = B t , j − b t , j / Q t ≥ 0 ◮ 0 ≤ P t d t , j ≤ D t , j D t +1 , j = D t , j − P t d t , j ≥ 0 B t +1 , j = B t , j + x t , j / Q t + A t , j ( e t , j ) ◮ B t +1 , j = B t , j + x t , j / Q t + A t , j ( e t , j ) ◮ 8 / 18

Results Proposition (Fundamental Condition) Assume agents use both Dollars and Bitcoins to buy goods at t and t + 1 , (i.e. x t , x t +1 , z t , z t +1 > 0 ). Then E t [ u ′ ( c t +1 )] · Q t = E t [ u ′ ( c t +1 ) Q t +1 ] If production (consumption) is constant, Q t = E t [ Q t +1 ] Proposition (Speculative Condition) Assume that Bitcoin and Dollar prices are positive. Assume agents do not spend all Bitcoins b t < B t Q t . Then it has to hold � � u ′ ( c t +2 ) Q t +2 u ′ ( c t ) ≤ β 2 E t Q t where this inequality holds with equality if x t > 0 and x t +2 > 0 . 9 / 18

Results Assumption 1: For y ∈ [ y , y ] u ′ ( y ) > β 2 E t [ u ′ ( y )] Proposition Under Ass. 1, agents spend all Dollars in each period. Assumption 2: u ′ ( y ) > β E t [ u ′ ( y )] Theorem (No Bitcoin Speculation) Given Ass 2 holds, assume Dollar and Bitcoin prices are positive. Then all Bitcoins are spent in each period. 10 / 18

Application: Bitcoin price evolution Given Ass 2, (Version of Kareken-Wallace) E t [ u ′ ( c t +1 )] · 1 = E t [ u ′ ( c t +1 ) Q t +1 ] , for all t Q t We know cov ( X , Y ) = E [ X · Y ] − E [ X ] · E [ Y ] Rewrite Q t = σ u ′ ( c ) | t σ Q t +1 | t cov t [ u ′ ( c t +1 ) Q t +1 ] + E t [ Q t +1 ] E t [ u ′ ( c t +1 )] σ u ′ ( c ) | t σ Q t +1 | t � �� � � �� � = κ t > 0 = corr t ( u ′ ( c t +1 ) , Q t +1 ) 11 / 18

Application: Bitcoin price evolution II Q t = E t [ Q t +1 ] + κ t · corr t ( u ′ ( c t +1 ) , Q t +1 ) , Corollary Under assumption (2), the Bitcoin price process is a (i) martingale (Q t = E t [ Q t +1 ] , for all t) ⇔ corr t ( u ′ ( c t +1 ) , Q t +1 ) = 0 , (ii) supermartingale (Q t ≥ E t [ Q t +1 ] , for all t) ⇔ corr t ( u ′ ( c t +1 ) , Q t +1 ) > 0 (iii) submartingale (Q t ≥ E t [ Q t +1 ] , for all t) ⇔ corr t ( u ′ ( c t +1 ) , Q t +1 ) < 0 12 / 18

Application: Price Convergence Corollary (Bitcoin Price Bound) Under Ass 2, there exists an upper bound for the Bitcoin price. Q t = b t ≤ b t + d t = y t ≤ y B t B t B t B 0 Theorem (Bitcoin Price Convergence) Under assumption (2), assume the Bitcoin price is a sub- or a super martingale (correlation between the price and marginal utility does not switch). Then the Bitcoin price converges almost surely point wise and in L 1 . Q t → Q ∞ a . s . and E [ | Q t − Q ∞ | ] → 0 13 / 18

Monetary Policy Market clearing : D t = y t − Q t B t , for all t Conventional : Bitcoin prices independent of central bank policies D t = D t ( Q t ) Unconventional : Consider an equilibrium: ◮ CB maintains P t = 1 independently of D t and ◮ D t set independently of production ⇒ CB impacts Bitcoin price Q t = y t − D t B t ◮ Implication: If D t independent of production: E [ Q t +1 ] ≥ Q t ◮ y t iid: P ( Q t +1 < s ) = P ( y t < B t s + D t ) 14 / 18

Conclusion We analyze a model of currency competition in which we derive sufficient conditions such that in equilibrium ◮ there is no Bitcoin speculation ◮ evolution of Bitcoin price process is determined by its correlation with marginal utility ◮ we can characterize central bank policy as function of Bitcoin price evolution or vice versa 15 / 18

Back Up - Monetary Policy - Unconventional Proposition (Dollar Stock evolution) If the Dollar quantity is set independently of production, the Bitcoin price process is a submartingale, E t [ Q t +1 ] ≥ Q t . Proposition (Bitcoin Price Distribution) As the Bitcoin or Dollar quantity rises, high Bitcoin price realizations become less likely (FOSD). Proposition As productivity increases (in terms of FOSD), the Bitcoin price is higher in expectation. 16 / 18

Back-Up - Inflation Given Ass 2, (Version of Kareken-Wallace) � � � P t � u ′ ( c t +1 ) Q t +1 u ′ ( c t +1 ) · = E t , for all t E t P t +1 Q t � �� � 1 π t +1 With inflation π t +1 > 1 17 / 18

Back-Up - Inflation π t +1 > 1 deterministic Q t = π t +1 E t [ Q t +1 ] + π t +1 κ t · corr t ( u ′ ( c t +1 ) , Q t +1 ) , (1) If inflation high, Q t can be supermartingale despite negative correlation between marginal consumption and Bitcoin price. 18 / 18