Some simple Bitcoin Economics

Some simple Bitcoin Economics Linda Schilling 1 and Harald Uhlig 2 1 École Polytechnique - CREST Department of Economics lin.schilling@gmail.com 2 University of Chicago Department of Economics huhlig@uchicago.edu June 2018 Schilling-Uhlig Some simple Bitcoin Economics June 2018 1 / 43

Outline 1 Introduction. 2 The Model 3 Analysis 4 Bitcoins and Monetary Policy 5 Examples 6 Conclusions Schilling-Uhlig Some simple Bitcoin Economics June 2018 2 / 43

Introduction. Outline 1 Introduction. 2 The Model 3 Analysis 4 Bitcoins and Monetary Policy 5 Examples 6 Conclusions Schilling-Uhlig Some simple Bitcoin Economics June 2018 3 / 43

Introduction. Questions Bitcoin, cybercurrencies: increasingly hard to ignore. Satoshi Nakamoto, Bitcoin: A Peer-to-Peer Electronic Cash System, www.bitcoin.org. Increasing number of cybercurrencies. Regulatory concerns. Blockchain technology. (Not a topic today) Literature: growing. Increasingly: serious academics. See paper. Imagine a world, where Bitcoin (or cybercurrencies) are important. Key questions: How do Bitcoin prices evolve? What are the consequences for monetary policy? Schilling-Uhlig Some simple Bitcoin Economics June 2018 4 / 43

Introduction. Bitcoin Price, 2011-09-13 to 2018-02-07 20000 Weighted Price 18000 16000 14000 12000 10000 8000 6000 4000 2000 0 9/13/2011 9/13/2012 9/13/2013 9/13/2014 9/13/2015 9/13/2016 9/13/2017 Data: quandl.com Schilling-Uhlig Some simple Bitcoin Economics June 2018 5 / 43

Introduction. Bitcoin Price, 2017-01-01 to 2018-02-07 20000 Weighted Price 18000 16000 14000 12000 10000 8000 6000 4000 2000 0 Data: quandl.com Schilling-Uhlig Some simple Bitcoin Economics June 2018 6 / 43

This paper Introduction. Approach: a simple model, with money as a medium of exchange. Results: A novel, yet simple endowment economy: two types of agents keep trading. Two types of money: Bitcoins and Dollars. A central bank keeps real value of Dollars constant...... while Bitcoin production is private and decentralized. Fundamental condition : a version of Kareken-Wallace (1981) Speculative condition. Under some conditions: no speculation. Under some conditions: Bitcoin price converges. Implications for monetary policy: two scenarios. Construction of equilibria. Schilling-Uhlig Some simple Bitcoin Economics June 2018 7 / 43

Introduction. 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) Schilling-Uhlig Some simple Bitcoin Economics June 2018 8 / 43

The Model Outline 1 Introduction. 2 The Model 3 Analysis 4 Bitcoins and Monetary Policy 5 Examples 6 Conclusions Schilling-Uhlig Some simple Bitcoin Economics June 2018 9 / 43

The Model The model t = 0, 1, 2,... Randomness: θ t, at beg. of per.. History: θ t. Two types of money: Bitcoins B t and Dollars D t (aggregates). Assume: Central Bank keeps Dollar price constant, P t 1. Goods ( = Dollar) price of Bitcoins: Q t = Q(θ t ). Two types of infinitely lived agents: green and red. Green agent j in even periods t: receives lump sum Dollar transfer ( tax, if < 0 ) from Central Bank. purchases goods from red agents, with Bitcoins or Dollars. enjoys consumption ct,j, utility β t u(c t,j ). Green agents in odd periods t: mines new Bitcoins At,j = f(e t,j ; B t ) at effort e t,j 0, disutil. β t e t,j. receives goods endowment yt,j. Not storable. can sell goods to red agents, against Bitcoins or Dollars. Red agents: flip even and odd periods. Assume: whoever consumes first has all the money. Schilling-Uhlig Some simple Bitcoin Economics June 2018 10 / 43

The Model Timeline c p c o e o p c p Schilling-Uhlig Some simple Bitcoin Economics June 2018 11 / 43

The Model Timeline CB CB MINING c p e MINING c o e o p e MINING c p e MINING CB Schilling-Uhlig Some simple Bitcoin Economics June 2018 12 / 43

The Model Timeline CB CB MINING c p e MINING c o e o p e MININGI c p e MININGN CB Schilling-Uhlig Some simple Bitcoin Economics June 2018 13 / 43

The Model Optimization problem of green agents: (drop j ) [ Maximize U = E β t ( ) ] ξ t,g u(c t ) e t t=0 where ξ t,g = 1 in even periods, ξ t,g = 0 in odd periods, s.t. in even periods t: 0 b t Q t B t,g (1) 0 P t d t D t,g (2) 0 c t = b t + d t (3) 0 B t+1,g = B t,g b t /Q t (4) 0 D t+1,g = D t,g P t d t (5) in odd periods t: A t = f(e t ; B t ), with e t 0 (6) y t = x t + z t, with x t 0, z t 0 (7) 0 B t+1,g = A t + B t,g + x t /Q t (8) 0 D t+1,g = D t,g + P t z t +τ t+1 (9) Schilling-Uhlig Some simple Bitcoin Economics June 2018 14 / 43

The Model Monetary Policy and Market clearing The Central Bank achieves P t 1, per suitable transfers τ t. Markets clear: Bitcoin market: B t = B t,r + B t,g (10) Dollar market: D t = D t,r + D t,g (11) Bitcoin denom. cons. market: b t = x t (12) Dollar denom. cons. market: d t = z t (13) Schilling-Uhlig Some simple Bitcoin Economics June 2018 15 / 43

The Model 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 Given prices, choices maximize utility for green and red agents. Budget constraints Evolution money stock 0 bt,j B t,j Q t B t+1,j = B t,j b t,j /Q t 0 0 Pt 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 ) Markets clear (for goods, Bitcoin, Dollars): yt = 2 0 c t,j dj 2 0 z t,j dj = 2 0 d t,j dj y t = x t,j + z t,j 2 0 x t,j dj = 2 0 b t,j dj c t,j = b t,j + d t,j Dt = D t,g + D t,r Bt = B t,g + B t,r Dollar monetary policy: P t = 1 Schilling-Uhlig Some simple Bitcoin Economics June 2018 16 / 43

Analysis Outline 1 Introduction. 2 The Model 3 Analysis 4 Bitcoins and Monetary Policy 5 Examples 6 Conclusions Schilling-Uhlig Some simple Bitcoin Economics June 2018 17 / 43

Analysis Consolidate: B t+1 = B t + f(e t ; B t ) D t c t = D t 1 +τ t = y t Schilling-Uhlig Some simple Bitcoin Economics June 2018 18 / 43

Analysis Avoid speculation with Dollars Assumption A. Assume throughout: for all t, Proposition u (y t ) β 2 E t [u (y t+2 )] > 0 (14) (All Dollars are spent:) Agents will always spend all Dollars. Thus, D t = D t,g and D t,r = 0 in even periods and D t = D t,r and D t,g = 0 in odd periods. This is a consequence of assumption 14 and P t 1. Proposition (Dollar Injections:) In equilibrium, D t = z t and τ t = z t z t 1 Schilling-Uhlig Some simple Bitcoin Economics June 2018 19 / 43

Analysis Bitcoin Production Proposition (Bitcoin Production Condition:) Suppose that Dollar sales are nonzero, z t > 0 in period t. Then [ 1 βe t u (c t+1 ) f(e ] t; B t ) Q t+1 e t (15) This inequality is an equality, if there is positive production A t > 0 of Bitcoins and associated positive effort e t > 0 at time t as well as positive spending of Bitcoins b t+1 > 0 in t + 1. Schilling-Uhlig Some simple Bitcoin Economics June 2018 20 / 43

Analysis The Fundamental Condition The following is a version of Kareken-Wallace (1981). Proposition (Fundamental Condition:) Suppose that sales happen both in the Bitcoin-denom. cons. market as well as the Dollar-denom. cons. market at time t as well as at time t + 1, i.e. suppose that x t > 0, z t > 0, x t+1 > 0 and z t+1 > 0. Then [ E t u (c t+1 ) ] [ = E t u (c t+1 ) Q ] t+1 (16) Q t In particular, if consumption and production is constant at t + 1, c t+1 = y t+1 ȳ, then Q t = E t [Q t+1 ] (17) i.e., the price of a Bitcoin in Dollar is a martingale. Schilling-Uhlig Some simple Bitcoin Economics June 2018 21 / 43

Analysis The Speculative Condition Proposition (Speculative Condition:) Suppose that B t > 0, Q t > 0, z t > 0 and that b t < Q t B t. Then, [ u (c t ) β 2 E t u (c t+2 ) Q ] t+2 Q t (18) where this equation furthermore holds with equality, if x t > 0 and x t+2 > 0. Schilling-Uhlig Some simple Bitcoin Economics June 2018 22 / 43

Analysis Seller Participation Condition Proposition (Seller Participation Condition:) Suppose that B t > 0, Q t > 0, z t > 0. Then [ E t u (c t+1 ) ] [ E t u (c t+1 ) Q ] t+1 Q t (19) Schilling-Uhlig Some simple Bitcoin Economics June 2018 23 / 43

Analysis The Sharpened No-Speculation Assumption Assumption A. For all t, u (y t ) βe t [u (y t+1 )] > 0 (20) This is a slightly sharper version of assumption 1, which only required u (y t ) β 2 E t [u (y t+2 )] > 0 Schilling-Uhlig Some simple Bitcoin Economics June 2018 24 / 43

Analysis The No-Bitcoin-Speculation Theorem Theorem (No-Bitcoin-Speculation Theorem.) Suppose that B t > 0 and Q t > 0 for all t. Impose assumption 2. Then in every period, all Bitcoins are spent. Proof. β 2 E t [u (c t+2 )Q t+2 ] = β 2 E t [E t+1 [u (c t+2 )Q t+2 ]] (law of iter. expect.) β 2 E t [E t+1 [u (c t+2 )] Q t+1 ] (equ. (19) at t + 1) < βe t [u (c t+1 )Q t+1 ] (ass. 2 at t + 1) βe t [u (c t+1 )]Q t (equ. (19) at t) < u (c t )Q t (ass. 2 at t) Thus, the specul. cond. (18) cannot hold in t. Hence b t = Q t B t. Schilling-Uhlig Some simple Bitcoin Economics June 2018 25 / 43

Analysis A (very high) bound for Bitcoin Prices Corollary (Bitcoin price bound) Suppose that B t > 0 and Q t > 0 for all t. The Bitcoin price is bounded by 0 Q t Q where Q = ȳ B 0 (21) Schilling-Uhlig Some simple Bitcoin Economics June 2018 26 / 43

Analysis Bitcoin Correlation-Pricing Rewrite (16) as Corollary Q t = cov t(u (c t+1 ), Q t+1 ) E t [u (c t+1 )] (Bitcoin Correlation Pricing Formula:) +E t [Q t+1 ] (22) Suppose that B t > 0 and Q t > 0 for all t. Impose assumption 2. In equilibrium, where Q t = κ t corr t (u (c t+1 ), Q t+1 )+E t [Q t+1 ] (23) κ t = σ u (c) t σ Qt+1 t E t [u (c t+1 )] > 0 (24) where σ u (c) t is the standard deviation of marginal utility of consumption, conditional on date-t information, etc.. Schilling-Uhlig Some simple Bitcoin Economics June 2018 27 / 43

Analysis Martingale Properties Corollary (Martingale Properties of Equilibrium Bitcoin Prices:) Suppose B t > 0 and Q t > 0 for all t. Impose ass. 2. If and only if for all t, marg. util. of cons. and Bitcoin price are positively correlated at t + 1, given t info, the Bitcoin price is a supermartingale and strictly falls in expectation, Q t > E t [Q t+1 ] (25) If and only if marginal utility and the Bitcoin price are always neg. corr., Q t < E t [Q t+1 ] (26) If and only if marginal utility and the Bitcoin price are always uncorr., the Bitcoin price is a martingale, Q t = E t [Q t+1 ] (27) Schilling-Uhlig Some simple Bitcoin Economics June 2018 28 / 43

Analysis Bitcoin Price Convergence Theorem (Bitcoin Price Convergence Theorem.) Suppose that B t > 0 and Q t > 0 for all t. Impose assumption 2. For all t and conditional on information at date t, suppose that marginal utility u (c t+1 ) and the Bitcoin price Q t+1 are either always nonnegatively correlated or always non-positively correlated. Then the Bitcoin price Q t converges almost surely pointwise as well as in L 1 norm to a (random) limit Q, Q t Q a.s. and E[ Q t Q ] 0 (28) Proof. Q t or Q t is a bounded supermartingale. Apply Doob s martingale convergence theorem. Schilling-Uhlig Some simple Bitcoin Economics June 2018 29 / 43

Bitcoins and Monetary Policy Outline 1 Introduction. 2 The Model 3 Analysis 4 Bitcoins and Monetary Policy 5 Examples 6 Conclusions Schilling-Uhlig Some simple Bitcoin Economics June 2018 30 / 43

Bitcoins and Monetary Policy Scenario 1 - Conventional approach Assume that Bitcoin prices move independently of central bank policies. Impose assumption 2. Then Proposition (Conventional Monetary Policy:) The equilibrium Dollar quantity is given as The central bank s transfers are D t = y t Q t B t (29) τ t = y t Q t B t z t 1 (30) Schilling-Uhlig Some simple Bitcoin Economics June 2018 31 / 43

Bitcoins and Monetary Policy Scenario 1 - Conventional approach Proposition (Dollar Stock Evolution:) Tomorrow s expected Dollar quantity equals today s Dollar quantity corrected for deviation from expected production, purchasing power of newly produced Bitcoin and correlation E t [D t+1 ] = D t (y t E t [y t+1 ]) A t Q t +κ t B t+1 corr t (u (c t+1 ), Q t+1 ) Likewise, the central bank s expected transfers satisfy E t [τ t+1 ] = (y t E t [y t+1 ]) A t Q t +κ t B t+1 corr t (u (c t+1 ), Q t+1 ) If the Bitcoin price is a martingale, then E t [D t+1 ] = D t E t [τ t+1 ] = (y t E t [y t+1 ]) A t Q t (y t E t [y t+1 ]) A t Q t Schilling-Uhlig Some simple Bitcoin Economics June 2018 32 / 43

Bitcoins and Monetary Policy Scenario 2 - Unconventional approach Unconventional view, but compatible with equilibrium: the Central Bank can maintain the price level P t 1 independently of the transfers she sets. Further, assume that she sets transfers independently of production. Note that Q t = y t D t B t (31) Intuitively, the causality is in reverse compared to scenario 1: now central bank policy drives Bitcoin prices. However, the process for the Dollar stock cannot be arbitrary. To see this, suppose that yt ȳ is constant. We already know that Q t must then be a martingale. Suppose B t is constant as well. Equation (31) now implies that D t must be a martingale too. Schilling-Uhlig Some simple Bitcoin Economics June 2018 33 / 43

Bitcoins and Monetary Policy Scenario 2 - Unconventional approach Proposition (Submartingale Implication:) If the Dollar quantity is set independently of production, the Bitcoin price process is a submartingale,e t [Q t+1 ] Q t. Schilling-Uhlig Some simple Bitcoin Economics June 2018 34 / 43

Bitcoins and Monetary Policy Scenario 2 - Unconventional approach Suppose that production y t is iid. Let F denote the distribution of y t, y t F. The distribution G t of the Bitcoin price is then given by G t (s) = P(Q t s) = F(B t s+d t ). (32) Proposition (Bitcoin Price Distribution:) In scenario 2, if Bitcoin quantity or Dollar quantity is higher, high Bitcoin price realizations are less likely in the sense of first order stochastic dominance. Schilling-Uhlig Some simple Bitcoin Economics June 2018 35 / 43

Bitcoins and Monetary Policy Scenario 2 - Unconventional approach Compare two economies with y t F 1 vs y t F 2,iid. Definition Economy 2 is more productive than economy 1, if F 2 first order stochastically dominates F 1. Economy 2 has more predictable production than economy 1, if F 2 second order stochastically dominates F 1. Proposition (Bitcoins and Productivity) Assume scenario 2. In more productive economies or economies with higher predictability of production, the Bitcoin price is higher in expectation. Schilling-Uhlig Some simple Bitcoin Economics June 2018 36 / 43

Examples Outline 1 Introduction. 2 The Model 3 Analysis 4 Bitcoins and Monetary Policy 5 Examples 6 Conclusions Schilling-Uhlig Some simple Bitcoin Economics June 2018 37 / 43

Examples Constructing an equilibrium: an example. Suppose θ t {L, H}, each with probability 1/2. Let m t be iid, m t = m(θ t ), with m(l) m(h) and E[m t ] = (m L + m H )/2 = 1. Pick 0 < β < 1 such that m(l) > β. At date t and for ǫ(θ t ) = ǫ t (θ t ), consider two cases Case A: ǫ t (H) = 2 t, ǫ t (L) = 2 t Case B: ǫ t (H) = 2 t, ǫ t (L) = 2 t. Pick Q 0 > ξ +(m(h) m(l))/2. Set Q t+1 = Q t +ǫ t+1 cov t(m t+1,ǫ t+1 ) E t [m t+1 ] Fix some strictly concave u( ). Let y t = (u ) 1 (m t ). Start with some initial B 0. With B t and Q t, equation (15) delivers new Bitcoin mining A t and thus B t+1. The No-Bitcoin-Speculation Theorem now implies the purchases x t = b t = Q t /B t and z t = d t = y t b t. Be careful with B 0, so that b t y t for all t. Or: fix ex post. Schilling-Uhlig Some simple Bitcoin Economics June 2018 38 / 43

Examples Super-, sub-, non-martingale examples Consider three constructions, Always A: Always impose case A, i.e. ǫ t (H) = 2 t, ǫ t (L) = 2 t. Always A results in supermartingale Q t > E t [Q t+1 ]. Always B: Always impose case B, i.e. ǫ t (H) = 2 t, ǫ t (L) = 2 t. Always B results in submartingale Q t < E t [Q t+1 ]. Alternate: In even periods, impose case A, i.e. ǫ t (H) = 2 t, ǫ t (L) = 2 t. In odd periods, impose case B, i.e. ǫ t (H) = 2 t, ǫ t (L) = 2 t. This results in a price process that is neither a supermartingale nor a submartingale, but which one still can show to converge almost surely and in L 1 norm. Schilling-Uhlig Some simple Bitcoin Economics June 2018 39 / 43

Examples Bitcoin Price, 2017-01-01 to 2018-02-07 20000 Weighted Price 18000 16000 14000 12000 10000 8000 6000 4000 2000 0 Data: quandl.com Schilling-Uhlig Some simple Bitcoin Economics June 2018 40 / 43

Examples Bubble and bust examples θ t {L, H}, but now P(θ t = L) = p < 0.5. Suppose that m(l) = m(h) = 1. Pick some Q > 0 as well as some Q > Q. Pick some Q 0 [Q, Q ]. If Q t < Q, let Q t+1 = { Qt pq 1 p Q if θ t = H if θ t = L If Q t Q, let Q t+1 = Q t. Therefore Q t will be a martingale and satisfies (22). If Q 0 is sufficiently far above Q and if p is reasonably small, then typical sample paths will feature a reasonably quickly rising Bitcoin price Q t, which crashes eventually to Q and stays there, unless it reaches the upper bound Q first. Schilling-Uhlig Some simple Bitcoin Economics June 2018 41 / 43

Conclusions Outline 1 Introduction. 2 The Model 3 Analysis 4 Bitcoins and Monetary Policy 5 Examples 6 Conclusions Schilling-Uhlig Some simple Bitcoin Economics June 2018 42 / 43

Conclusions Recap and Conclusions. Approach: a simple model, with money as a medium of exchange. Results: A novel, yet simple endowment economy: two types of agents keep trading. Two types of money: Bitcoins and Dollars. A central bank keeps real value of Dollars constant...... while Bitcoin production is private and decentralized. Fundamental condition : a version of Kareken-Wallace (1981) Speculative condition. Under some conditions: no speculation. Under some conditions: Bitcoin price converges. Implications for monetary policy: two scenarios. Construction of equilibria. Schilling-Uhlig Some simple Bitcoin Economics June 2018 43 / 43