Currency 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 cryptocurrencies more suitable or are all non-storable across time.2 less costly a medium of exchange than tra- currencies: Trade is carried out, using ditional fiat moneys such as Dollars or Eu- two kinds of money. The first shall be called ros. On the other hand, there exist goods Bitcoin, 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 Bitcoins 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 exchange rate 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.