Substitution under Transaction Costs

By Linda M. Schilling and Harald Uhlig

A major selling point and feature of cryp- Wallace [1981], Obstfeld and Rogoff [1995], tocurrencies is that they allow anonymous and [Casas et al., 2016]. Time is discrete, payments around the globe without a third t 0; 1; 2; : : :. The economy is determinis- party watching. For payments of certain tic.D There exists a continuum Œ0; 1 of differ- goods, this censorship resistance feature entiated consumption goods k Œ0; 1 which makes more suitable or are all non-storable across time.2 less costly a medium of exchange than tra- : Trade is carried out, using ditional fiat moneys such as Dollars or Eu- two kinds of . The first shall be called ros. On the other hand, there exist goods , and its aggregate stock is fixed at 1 which are easier to acquire using traditional Bt B. There exists no designated crypto means of payments. The costs of em- centralÁ bank. The second money shall be ploying cryptocurrencies may involve fees called Dollar, and we denote its aggregate to miners, while traditional money might be stock at time t with Dt . There is a des- subject to taxation. In this paper, we there- ignated Dollar central bank, which governs fore explore how asymmetry in transaction the aggregate stock of Dollars Dt per lump costs as well as exchange fees drive currency sum transfers in each period. The central substitution. We build on Schilling and Uh- bank can produce Dollars at zero cost. lig [2018]. Agents alternate in their role as Agents: There are two types of in- buyers and sellers, necessitating currency. finitely lived agents, called ’red’ and ’green’. We assume a continuum of differentiated Each type is given by a unit interval. In odd goods, which can be strictly ordered accord- periods, a green agent receives an exoge- ing to the costs agents incur when purchas- nously given goods endowment ykt;j yt ing these goods with as opposed for all k Œ0; 1, which he can sellD to to Dollars. Exchanging Dollars for Bitcoins red agents,2 against Bitcoins or Dollars, but may be subject to a fee. In a market equi- does not enjoy consuming herself. We as- librium, agents endogenously decide which sume that yt is a deterministic function of goods to acquire using Dollars and which time. In even periods t, the green agent j goods to purchase using Bitcoins. We char- has zero endowment but enjoys consump- acterize the nonstochastic equilibrium and tion ckt;j . To purchase goods k Œ0; 1 the resulting dynamics. The from red agents, she employs her individ-2 marginal good at which agents are indif- ual Bitcoin and Dollar stock she accumu- ferent between purchasing with Bitcoins or lated from previous periods when she was a Dollars depends on and varies in the size of seller. Additionally, she receives a lump- the value-added-tax and transaction fees to sum Dollar transfer t from the central miners. bank to purchase goods. She then en- t joys utility ˇ u.ckt;j /, ˇ .0; 1/. Since I. The Model agents are infinitely lived, they2 may choose to save in Bitcoins or Dollars across time. The model builds on Schilling and Uh- For red agents, flip even and odd peri- lig [2018] and is related to Kareken and 1This assumption corresponds to the fact that the  Schilling: Ecole Polytechnique CREST, 5 maximum quantity of Bitcoins in circulation is bounded Avenue le Chatelier, 91120 Palaiseau, France, from above by 21 million coins. We employ the assump- [email protected]. Uhlig: Depart- tion of a fixed Bitcoin stock to abstract from coin mining ment of Economics, University of Chicago, 1126 and the associated effort in this paper. Schilling-Uhlig East 59th Street, Chicago, IL 60637, U.S.A., huh- (2018) provide details, when allowing for mining and an [email protected]. increase in the Bitcoin stock. 1 2 PAPERS AND PROCEEDINGS JANUARY 2019 ods. Red and green agents alternate in shadow price of a Bitcoin in Dollar, which consuming and producing the consump- we shall denote with Qt , is also the official tion goods. Since goods are perishable, exchange rate, if exchange takes place at the alternation creates the absence of a all. Suppose that the goods buyer gives double-coincidence of wants, and thereby one Bitcoin to the goods seller in period reasons to trade using currency. Denote by t, and that the seller in exchange pays an 1=.1 1=/ R 1 1 1= Á amount of Dollars Qt . We assume, that ct;j c dk the CES ag- Q D 0 kt;j the goods seller receives the Bitcoin in full, gregator with  > 1 denoting the elasticity while the goods buyer only receives Qt of substitution. The consumption-utility Q Dollars, with the difference of .1 /Qt Dol- function u. / is strictly increasing, concave Q  lars collected as transaction cost. Per our and twice differentiable. We assume that convention, we therefore have Q Q . Q t t whoever consumes first has all the money. From the perspective of the goodsD seller, Agents maximize discounted expected life- Qt the exchange rate is Qt . Conversely, Q D  time utility if the goods buyer obtains one Bitcoin in ex- " # change for paying Qt Dollars to the goods X1 t seller, we assume that all Dollars arrive (1) U E ˇ t;j u.ct;j / Qt D t 0 at the seller, while the goods seller needs to D give up 1= Bitcoins for one Bitcoin to ar- Formally, we impose alternation of utility rive at the goods buyer, and where .1 /= from consumption per t;j 1 t is odd for Bitcoins are collected as transaction cost. D f g j Œ0; 1/ and t;j 1 t is even for j The exchange rate from the perspective of 2 D f g 2 Œ1; 2. the goods seller now becomes Qt Qt . Q D Prices and Transaction Costs: We assume that usage of currency comes at a As usual, define the aggregate Dollar 1=.1 / R 1 1  Á cost. Let .1 .k// Œ0; 1; k Œ0; 1 the ex- price indices Pt P dk . 2 2 D 0 k;t ogenous, product-specific transaction cost The central bank implements an exoge- for goods purchased with Dollar, reflecting nously given path Pt for the aggregate Dol- perhaps a goods-specific value-added tax lar price index via appropriate lump sum (VAT). Purchases with Bitcoins cannot be Dollar transfers  paid to the goods buyer taxed, but are nonetheless subject to an ex- t at the beginning of the period. Let t ogenous transaction cost .1 ˛.k// Œ0; 1, Pt denote the resulting inflation. In equi-D perhaps reflecting transaction fees2 which Pt 1 librium, the central bank achieves her price are paid to miners. path target. Assume that ˛.k/= .k/ is strictly increas- ing and continuous in . In equilibrium, k Denote by Dt;i the Dollar stock of indi- there will therefore exist a critical good vidual i when entering time period t. The c , such that all goods c are R kt Œ0; 1 k kt aggregate stock equals Dt Dt;i di 2 Ä c D Œ0;2 purchased with Dollars and all goods k > kt where i Œ0; 1 denote red agents and i will be purchased with Bitcoins. Let Pk;t 2 2 Œ1; 2 denote green agents. Similarly, let Bt;i and P be the period-t price of good k Ok;t be the Bitcoin stock of individual i at time in Dollars and Bitcoins respectively before R t. Note that B Œ0;2 Bt;i di. Exchange of transaction costs, set by goods sellers at currency and purchasesD of goods happens which they are indifferent between accept- simultaneously. Thus, the goods buyer can ing either currency. A buyer, therefore, spend all his Bitcoins on goods, purchase needs to pay P = .k/ Dollars or P =˛.k/ k;t Ok;t Bitcoins from the goods seller against Dol- Bitcoins to obtain a unit of good k. lar, and spend these Bitcoins again. In addition, buyers and sellers can ex- change currency in either direction, at an We shall focus on symmetric equilib- exchange fee of .1 / Œ0; 1. We ria, which allows us to drop the agent- adopt the convention that the2 goods buyer’s individual subscripts i and j , unless needed VOL. VOL NO. ISSUE CURRENCY SUBSTITUTION 3 for clarification. Denote by lars CD;t and Bitcoins CB;t , her currency stocks increase to Z Pk;t (2) CD;t ck;t dk D k kc .k/ 0 Dt 1;j Dt;j RD;t Qt t;j D;t Ä t Ä C D C Q Q C 0 Bt 1;j Bt;j RB;t t;j B;t the total quantity of Dollars the representa- Ä C D C C Q C tive buyer spends on consumption goods in where  is the quantity of Bitcoins ob- Q t;j Dollars at time t potentially after exchang- tained (“sold”, if negative) by the goods ing some Bitcoins to Dollars. Due to trans- seller j and Q is the price from the per- Q t action costs, the representative goods seller spective of that goods seller. In equilib- only receives rium, all goods sellers choose the same Z currency trade t t;j and the mar- Q D Q RD;t Pk;t ck;t dk ket clears via corresponding currency trades D k kc Ä t by goods buyers and, possibly, transaction fees. If goods buyers sell Bitcoins, the mar- Dollars as revenue. Similarly, let CB;t D ket clearing condition is t t > 0 and R Pk;t Q c O ck;t dk be the quantity of Bitcoins D k>kt ˛.k/ the price from the perspective of goods sell- spent on consumption goods demanded in ers is Qt Qt =. If goods buyers buy R Q D Bitcoins, and let RB;t c Pk;t ck;t dk Bitcoins, the market clearing condition is D k>kt O the number of Bitcoins the seller receives t t < 0 and the price from the D Q through goods purchases. The goods buyer perspective of goods sellers is Qt Qt . Q D faces the time t Dollar budget constraint Market clearing for all goods markets re- quires yk;t yt ct ck;t for all k Œ0; 1 D D D 2 (3) 0 CD;t Dt;i t Qt t;i and t, since buyers cannot produce and Ä Ä C C goods are not storable. where t;i is the quantity of Bitcoins ex- changed for Dollars by the goods buyer. II. Analysis She then carries the difference Dt 1;i be- C tween the right-hand side of (3) and CD;t Consider the goods seller at time t, who over as Dollars into period t 1. If t;i > 0 becomes a goods buyer in t 1. Given her ( ), the goods buyerC is selling (ac- C t;i < 0 shadow price Qt 1 for a Bitcoin tomorrow, quiring) Bitcoins for Dollars to the goods she sets goods pricesC in terms of Bitcoins seller. Likewise, the goods-buying agent and Dollars according to faces the time t Bitcoin budget constraint Pk;t (4) 0 C B  (5) Qt 1 ; for all k Œ0; 1 B;t t;i t;i C D P 2 Ä Ä Ok;t and carries the difference between the right- in order to be indifferent between receiv- hand side and CB;t as Bitcoin balance Bt 1;i ing either currency. Now consider the into the next period. In the symmetricC buyer. With Qt Dollars, a buyer can equilibrium, all choose the same t t;i . purchase Qt .k/=Pk;t units of good k. On Next, consider the goods sellers in periodD t. the other hand, one Bitcoin buys ˛.k/=P We assume that they receive all transaction Ok;t units of good k. The buyer is indifferent costs and Bitcoin-Dollar exchange fees, D;t c c at a critical good k kt ; kt .0; 1/, if and B;t , as lump sum payments. c P c D 2 ˛.kt / Okt ;t c . Equation (5) then implies .k / Qt P c 1  t D kt ;t D;t CD;t RD;t Qt max t ; 0 D C  f g c ˛.kt / Qt 1  (6) c B;t CB;t RB;t min t ; 0 .kt / D Qt 1 D  f g C The left hand side is a strictly increas- As she sells goods to green agents for Dol- c ing function in k. Thus, the good kt 4 PAPERS AND PROCEEDINGS JANUARY 2019 is unique for every t if it exists. Other- is the No-Speculation-Theorem in Schilling wise, if ˛.k/ > Qt for all k Œ0; 1, set and Uhlig [2018], assuming the strong im- .k/ Qt 1 C 2 patience condition there holds. kc 0, and if ˛.k/ < Qt for all k, set t .k/ Qt 1 c D C c k 1. All goods k k are payed in P c ; t t D Ä t kt Dollars, and it holds ˛.k/ Qt while seller indiff: Qt 1 .k/ Qt 1 C D Pkc ; t c Ä C O t goods k > kt are paid for in Bitcoins with c ˛.k / Pkc ; t ˛.k/ Qt t t > . To determine prices for goods buyer indiff: Qt .k/ Qt 1 c D .kt / P c ; t purchasedC in Dollars and Bitcoins, consider Okt c two goods k; k0 < kt which a buyer pays for in Dollars. Both goods are provided at the ˛.kc/ same supply and are thus consumed in the combined: Qt t Qt 1 D .kc/ same quantity in equilibrium. After trans- C t P c action costs, they must therefore have the c kt ;t Dollars spent: CD;t kt c yt same Dollar price. The same remark holds D .kt / for the after-transaction Bitcoin prices for P c c c Okt ;t goods k; k0 > kt . This implies Bitcoins spent: CB;t .1 kt / c yt D ˛.kt / c .k/ c CD;t kt (7) Pk P c ; for all k k ratio: Qt D kt .kc/ Ä t CB;t D 1 kc t t ˛.k/ c 1 (8) P P c ; for all k > k exch. arbitr.: Qt 1 Qt Qt 1 k kt c t O D O ˛.kt / C Ä Ä  C seller Dollars: 0 CD;t Qt t Allocation The purchase of yt units Ä c seller Bitcoins: 0 CB;t t of a good k > kt requires yt Pk=˛.k/ Ä C c O D yt P c =˛.k / Bitcoins. Spending is con- Okt t stant across k. Analogous for Dollar pur- c c Note that we only need to keep track of the chases. Therefore, CD;t k yt P c = .k / t kt t prices for the critical good, as the pricing c D c and CB;t .1 k /yt P c =˛.k //. What if D t Okt t for all other goods follows from the indif- the buyer enters the period with amounts ference conditions stated above. The of Dollars and Bitcoins which differ from goods sellers receive the entire monetary the intended spending with each currency? amount resulting from goods spending by Suppose, say, Dt < CD;t and Bt > CB;t . To the goods buyers because they receive the acquire the additional CD;t Dt Dollars, the direct payments and the transaction fees CD;t Dt goods buyer transfers t Bit- and exchange fees as a lump sum payment. D Qt coins to the goods seller, with the exchange The latter two equations say that the goods subject to the Dollar-Bitcoin exchange fee seller cannot transfer more Dollars (Bit- described above, implying Qt 1 Qt =. coins) in the exchange rate market to the C D Equivalently, if Dt > CD;t and Bt < CB;t , goods buyer, than she receives in total (pos- the goods buyer acquires Bitcoins in ex- itive money balances). In both of the fol- change for Dollars and Qt 1 Qt . lowing examples, assume (9) and (10), i.e. C D Collecting equations The following buyers never hold on to Dollars or Bitcoins set of equations characterize the aggregate at any t. evolution of the economy, assuming that buyers never hold on to Dollars or Bitcoins III. Example 1: no currency exchange to the next period. Suppose that Dollar inflation and output (9) CD;t Qt t Dt are constant t  1; yt y, with  D Á  Á (10) CB;t t B and y parameters. We wish to feature an C D example, where no currency exchange takes The justification for this latter assumption place. Thus, t 0, CB;t B, CD;t D D D VOL. VOL NO. ISSUE CURRENCY SUBSTITUTION 5

t t Dt D0 for all t. Further, assume Pkc ;t  Pkc ;0. From the seller indiffer- WDc D that kt is constant in t. The Bitcoins-spent ence equation, it follows that Bitcoin prices t t equation then implies that P c is constant evolve according to Pkc ;t   Pkc ;0. Ok ;t O D  O in t, while the Dollars-spent equation im- In order for the goods buyer to keep sell- plies that P c and, therefore, the Dollar ing Bitcoins to the goods seller, requires kt ;t price of a Bitcoin must rise at the rate of t B CB;t > 0. Sufficient for this is that t D inflation, Q  Q , P c P c . The P c is not rising in t, i.e. that  1. t 0 k ;t k ;t Ok ;t combined equationD allows toD solve for kc per Consider next the case, that kc 1 Äand .kc/=˛.kc/ . Assume that there is an that no transactions ever take placeD with D interior solution. Given the initial Dollar Bitcoin, t B, because ˛.k/= .k/ <  for D t quantity D0 and exploiting D0 CD;0 as all k. It follows that CD;t P1;0 y= .1/. D D well as B CB;0, the ratio equation de- The Bitcoins-spent equation does not im- D livers Q0 and thus Qt t 0. The Bitcoins- pose a restriction on the rise in prices Pkc ;t f g  O spent equation can be solved for P c , while and therefore no restriction on . A par- Okt P c can be calculated, using the Dollars- ticularly interesting special case is the case kt ;0 spent equation. Two things remain to be of a frictionless exchange, i.e.  1. Then D checked. First, absence of Dollar-Bitcoin Qt 1 Qt Q0, which can take any ini- C D Á exchanges requires that the exchange fee tial value, as long as the seller can afford to 1  is sufficiently high. Via the exchange- purchase all Bitcoins in all periods without arbitrage condition, this requires Qt 1 < encountering a negative dollar balance. Per 1 C the seller Dollar equation, this means we Qt <  Qt 1, i.e.  .; 1=/. Second, the C 2 no-speculation theorem logic in Schilling- need Qt B CD;t or Q0 P1;0 y=.B .1// at date t Ä0, which suffices.Ä Each period Uhlig (2018) indeed implies that all Dollars D and Bitcoins are spent, provided the strong unfolds per the buyer repeatedly spending impatience assumption there holds. all his Dollars and then selling as many Bit- coins to the seller against Dollars as possi- IV. Example 2: currency exchange ble, until all Bitcoins are ultimately sold to the seller. The seller is indifferent between Suppose again that Dollar inflation and keeping Dollars or Bitcoins to the next pe- output are constant t  1; yt y, riod. with  and y parameters.Á We wish toÁ fea- ture an example, where Bitcoins and Dol- REFERENCES lars are traded and where goods buyers always sell some Bitcoins to goods sellers Camila Casas, Federico J D´ıez, Gita Gopinath, and Pierre-Olivier Gourin- against Dollars, t > 0, thus trading more goods with Dollars than they own when en- chas. Dominant currency paradigm. Technical report, National Bureau of tering the period, Dt < CD;t . The exchange arbitrage equation implies that we are at Economic Research, 2016. the bound Qt Qt 1. This allows us John Kareken and Neil Wallace. On the D C to calculate the critical good per the com- indeterminacy of equilibrium exchange c c bined equation,  ˛.kt /= .kt /, where we rates. The Quarterly Journal of Eco- Dc c further note that k kt must be con- nomics, 96(2):207–222, 1981. Á c stant in t. Consider first the case, that kt is interior. At time zero, the goods buyers Maurice Obstfeld and Kenneth Rogoff. Ex- have all the Dollars and Bitcoins. Since the change rate dynamics redux. Journal of buyer is acquiring Dollars for spending pur- political economy, 103(3):624–660, 1995. poses, the ratio equation shows that Q and 0 Linda Schilling and Harald Uhlig. Some D must satisfy D =B < .kc=.1 kc//Q . 0 0 0 simple bitcoin economics. Technical re- Fix some initial Bitcoin price Q and Dol- 0 port, National Bureau of Economic Re- lar quantity D0, satisfying this inequality. Let the Dollar quantity and Dollar prices search, 2018. t rise at the rate of inflation, Dt  and D