Competitive Equilibrium of Central Bank Digital Currency and Private Cryptocurrency: A Perspective of Regulatory

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(School of Economics, Fudan University, Shanghai 200433, China)

Abstract: This paper constructs a searching model to study the impact of monetary policy, fiscal policy, transaction costs of private cryptocurrency and regulatory uncertainty on the competitive equilibrium between Central Bank Digital Currency and private cryptocurrency.The results show: (1) Central Bank Digital Currency and private cryptocurrency can conditionally coexist.(2)The level of economic welfare is related to monetary policy and fiscal policy but independent from the issuance and transaction of private cryptocurrency.(3) Under complete competition of private cryptocurrency mining, the mining cost becomes deadweight loss for whole society.

Keywords: Central Bank Digital Currency; Private cryptocurrency; Regulatory uncertainty

§1.Introduction

As many countries’ central banks issue Central Bank Digital Currency, the competition between Central Bank Digital Currency and private cryptocurrency, such as Bitcoin, will exist for a long time.At present, there are no clear laws and regulations that restrict the issuance and trading of private cryptocurrency.There exists huge regulatory uncertainty in private cryptocurrency ( [7,11,16,18–20]).

At present, it’s a controversial topic whether Central Bank Digital Currency and private cryptocurrency coexist in economy in the lack of framework to guide the research on the regulatory uncertainty of private cryptocurrency ( [4–6,12]).The purpose of this paper is to introduce regulatory uncertainty into searching-based model to exploit following questions: Can private cryptocurrency and Central Bank Digital Currency coexist in competitive equilibrium?How do central bank monetary policy, government fiscal policy, transaction costs and regulatory uncertainty of private cryptocurrency affect their coexistence and competitiveness? Will the economy benefit from the coexistence of private cryptocurrency and Central Bank Digital Currency?

In this paper,the model describes both parties of the transaction can freely choose the Central Bank Digital Currency and private cryptocurrency(hereafter,called Bitcoin for simplicity1Bitcoin is most representative private cryptocurrency and has the largest transaction scale around the world.The technical terms of private cryptocurrency are mainly derived from Bitcoin.So, Bitcoin widely represents private cryptocurrency on the basis of block chain technology.)as the medium of exchange for commodity transaction.But, the buyer can only get back a proportion of the commodity suppose they pay by Bitcoin.The proportion of commodity that buyer obtained is a random variable, which describes the regulatory uncertainty for Bitcoin.Besides, the model also introduces the growth rate of Central Bank Digital Currency, the sales tax on the transaction by Central Bank Digital Currency, and the fee rate of Bitcoin transaction to separately describe the monetary policy of central bank, the government’s fiscal policy, and the transaction cost of private cryptocurrency.

From the perspective of regulatory uncertainty,this paper builds an alternative framework to study the competitive equilibrium of private cryptocurrency and Central Bank Digital Currency,which develops the related literature.The other contribution of this paper is to provide some suggestions on how China’s central banks promotes the circulation of China’s Central Bank Digital Currency under the competition with private cryptocurrency in future.

The remainder of this paper is outlined as follows.The model setting is described in section 2.The section 3 solves the model and conducts welfare analysis.Finally, the section 4 concludes.

§2.Model setting

2.1.Basic assumptions

Following Lagos and Wright[10], the model is built on infinite period discrete frameworkt=0,1,....Each period is divided into two sub-periods.The commodity market and currency market open during the first sub-period and second sub-period, respectively.The commodity market is a free market while the currency market is a government-regulated market.Commodities in the model are infinitely divisible and can be consumed but not stored over different periods.In each sub-period, each unit of labor supply can produce a unit of commodity.Assuming that the two parties of transaction cannot trade by credit.In other word, the commodity market needs a medium of exchange to facilitate the transaction.

Suppose there are two perfectly separable mediums of exchange that can be stored over different periods, Central Bank Digital Currency (hereafter, called CBDC) and Bitcoin.At timet, the outstanding of CBDC and Bitcoin areMtandBt, respectively.The growth rate of CBDC supply is exogenously set by government while the growth rate of Bitcoin supply is exogenously set by its issuance mechanism, that is,

For simplicity, the model assumes that the discount rate is zero between different periods and between two sub-periods of the same period.Without loss of generality, assumesθm>1 andθb>1.

Assume that the volume of commodities paid by CBDC and Bitcoin areqm,tandqb,tat timet.Denote the price of one unit of commodity in CBDC and Bitcoin bypm,tandpb,t, in the commodity market, respectively.Assume one unit of CBDC and Bitcoin is paid forϕtunits andψtunits of commodities, respective.So, we havepm,t=1/ϕtandpb,t=1/ψt.

Following Rocheteau and Wright [15] and Lagos and Rocheteau [9], miners, buyers, sellers and government are introduced into the model.

2.2.Miners’ behavior

A block chain consists of blocks.The blocks contain transaction information for a set of digital currency users over a given period.Miners compete with each other for eligibility(blocks)to record new transactions on the block chain.More technical details can be seen in [1], [17]and [13].Assuming that there are a positive integer number of minersη, andith miner makesei,teffort to compete for the qualification to create a new block in periodt, but the miner’s effort will not produce any commodity.According to the mining success probability theory of Chiu and Koeppl [2], it is assumed that the probability of a miner winning the competition is proportional to its effort and the mining competition of each block is independent, so the probability ofith miner winning a block is.Miners obtain the new Bitcoin,(θb-1)Bt-1by successful mining in the currency market in periodt, and obtain transaction fee when buyers pay with Bitcoin in the commodity market in future.The fee rate of Bitcoin transaction is exogenously given byf>02Kang and Lee [8] finds that the endogenous setting of Bitcoin transaction fee rate does not affect the main conclusions.So, this paper study the exogenous case for simplicity..

In periodt, miners successfully dig outnt ≥1 blocks.As the competition of mining block is independent, each miner repeats the mining competitionnttimes by taking the same strategy.Sinceqb,tunit of commodity transactions are settled in Bitcoin.So, the mineri’s maximization problem of expected profitπi,tis

Following Chiu and Koeppl [2], assume that all homogeneous miners make the same optimal effortetand equally share the new Bitcoin from mining competition such that

The total expected profit Πtand the total effortEtof all miners are

and

2.3.Buyer’s behavior

There is a continuum of buyers of total “mass” equal to one.In the commodity market of periodt(first sub-period), buyers only consume but can’t produce commodities.In the currency market of periodt(second sub-period), buyers supply labor to get the mediums of exchange,CBDC or Bitcoin, for their consumption in the commodity market of periodt+1.Especially,assume the mediums of exchange they get from the currency market of periodtequals to their consumption in the commodity market of periodt+1.

After the currency market is closed, the newly generated CBDC and Bitcoin are transferred to the currency holder directly3Rocheteau and Wright [15] found that whether the government injects or withdraws Central Bank Digital Currency to buyers or sellers does not affect the final equilibrium outcome., that is, the buyer’s CBDC balance and Bitcoin balance grow atθmandθb, respectively.Define the quantity of goods purchased by buyer with CBDC and Bitcoin in the commodity market of periodtareqbm,t ≥0 andqbb,t ≥0, respectively.Their labor supply in the currency market isSt.

Assume that the actual proportion of commodity that the buyer can get back after the Bitcoin transaction is a random variable,ξbecause of the government intervention into the transaction settled by Bitcoin transactions.The proportionξsubjects to a normal distribution with the mean,µand the variance,σ2.When the government take more strict policy on Bitcoin transaction, the regulation risk on Bitcoin increases, that is, lowerµand higherσ2.

Assumed that buyers have exponential utility function such that

whereγis the risk aversion coefficient.

According to the expectation formula for log-normally distributed random variable4E[exp(Y)=exp(µ+σ2/2)], of which Y ∼N(µ,σ2)., the buyer’s utility maximization problem is

2.4.Seller’s behavior

There is a continuum of sellers of total “mass” equal to one.In the commodity market of periodt(first sub-period), sellers only produce but can’t consume commodities.In the currency market of periodt(second sub-period), sellers spend all mediums of exchange, CBDC or Bitcoin for consumption.Suppose that quantity of commodities sold by the seller with the CBDC and Bitcoin in the commodity market of periodtareqsm,t ≥0 andqsb,t ≥0, respectively.But seller have to pay the sales tax to the government for the commodity transaction in the CBDC and the tax rate isτ ∈[0,1].Suppose that the seller’s preference satisfies the following quasi-linear function in periodt

2.5.Government behavior

Assume that the government budget is break-even and its budget constraint is

whereGtis the government expenditure,δpm,tqm,tis the operating cost of CBDC andδis the ratio of CBDC operating cost to the transaction scale of the CBDC in the commodity market.

It is assumed that the sales tax rate on CBDC transaction is much greater than the operating cost rate, that is,τ ≫δ.Therefore, the scale of government expenditure mainly depends on the scale of commodity transactions by CBDC.

§3.Equilibrium and welfare analysis

Exogenous policy variables of the model include the growth rate of CBDC supplyθm,the sales tax rate on CBDC transaction,τ, the fee rate of Bitcoin transaction,f, the mean and variance of actual proportion of commodity that buyers can get back after the Bitcoin transaction,µandσ2.They measure the monetary policy of central bank, fiscal policy of government, transaction cost of Bitcoin, and regulatory risk of Bitcoin, respectively.

where the equation (3.1) indicates that the price for a unit of commodity in CBDC equals to the product of CBDC price per unit of commodity in the currency market of periodt-1 and the growth rate of CBDC supply.The equation (3.2) indicates that the price for a unit of commodity in Bitcoin equals to the product of CBDC price per unit of commodity in the currency market of periodt-1 and the CBDC growth rate adjusted by sales tax rate on CBDC transaction.The conditions for the coexistence of CBDC and Bitcoin in the economy are derived from the equations (3.3) and (3.4).

Proposition 3.1.In equilibrium,the coexistence of Central Bank Digital Currency and Bitcoin satisfies following conditions:

Proposition 3.1 shows that when the expected proportion of commodity that buyers can get back after the Bitcoin transaction,µis within a certain range, ceteris paribus, CBDC and Bitcoin coexist in equilibrium.Suppose the regulatory risk of Bitcoin is too high, that is,µis below the lower limit, the buyers in the commodity market will no longer use Bitcoin for commodity transactions.Suppose the regulatory risk of Bitcoin is too low, that is,µexceeds the upper limit, the buyers in the commodity market will no longer use CBDC for commodity transactions.

Proposition 3.2.In the equilibrium where CBDC and Bitcoin coexist,CBDC and Bitcoin constitute competition.Suppose the growth rate of CBDC supply increases,the sales tax rate on CBDC transaction decreases,the fee rate of Bitcoin transaction increases,or the regulatory risk of Bitcoin increases(µ↓and σ2↑),then in the commodity market,the volume of commodities paid by CBDC,qm,t will increase and the volume of commodities paid by Bitcoin,qb,t will decrease.

It needs to note that as the growth rate of CBDC supplyθmincreases, the value of the CBDC per unit will decline but the volume of commodities paid by CBDC,qm,tin commodity market will increase intperiod.Based on the assumption that newly generated CBDC and Bitcoin are transferred to the currency holder at one time after the currency market is closed,the CBDC balance and Bitcoin balance of buyers grows at the corresponding supply growth rate of CBDC and Bitcoin, separately.It can be regarded as the “interest” for holding the medium of exchange in digital account.So, the rising growth rate of the CBDC supply has enhanced buyers’ motivation to hold CBDC.It’s similar with the mechanism of contemporary financial system that when the central bank increases the money supply and promote the amount of credit loan into real economy, the price of various asset rises such that asset holders can obtain investment returns and ultimately promote the consumption of the economy.

In equilibrium, the money market clears in periodt:

Then we have

In fact, any personal computer installed with a mining program can participate in Bitcoin mining.When the mining work is fully competitive, the number of miners tends to infinity,η →∞, the optimal effort of each mineretand the total expected profit of all miners Πtboth converge to 0.The total effortEtconverges to the total reward for mining work.In other word,the mining cost becomes deadweight loss for the economy, which contradicts the results of Cong et al.[3] and Raskin et.al [14].The equation (2.1) is rewritten as

Proposition 3.3.In the equilibrium where CBDC and Bitcoin coexist,the miner market is fully competitive and the number of miners,η tends to infinity,then

(i)Suppose the growth rate of CBDC supply increases,the sales tax rate on CBDC transactions decreases or the regulatory risk of Bitcoin increases(µ↓and σ2↑),then the total effort of miners Et decreases;

(ii)Suppose the fee rate of Bitcoin transaction increases,then the total effort of miners,Et possibly increases or decreases.

It needs to note that as the fee rate of Bitcoin transaction rises, the volume of commodities paid by Bitcoin in the commodity marketqb,tdecreases.At the same time, the number of commodities that can be exchanged for each unit of Bitcoinψtalso decreases.The total effort of miners,Etconverges to the total reward for mining work, that is, total transaction fee of Bitcoin transaction and the volume of commodities that newly minted Bitcoin can be exchanged for.As a result of the rise in the fee rate of Bitcoin transaction, the total effort of miners possibly goes up or down.On one hand, the total fee of Bitcoin transaction,fqb,tpossibly goes up or down.On the other hand, the newly minted Bitcoin can be exchanged for less volume of commodities.

According to the government budget constraint (2.6), we have

We now conduct a welfare analysis for the coexistence between CBDC and Bitcoin.First,we define the total output of the commodity market as the social welfare level,qm,t+qb,tsuch that

Proposition 3.4.In the equilibrium where CBDC and Bitcoin coexist,the social welfare is positively related with the growth rate of CBDC supply,negatively related with the sales tax rate on CBDC transaction,and independent from the fee rate of Bitcoin transaction and the growth rate of Bitcoin supply5When the seller’s cost function v(·) is a power function whose power exponent exceeds two, it can be proved that the effect of growth rate of CBDC supply, θm and the sales tax rate on CBDC transactions, τ on the total output of commodity market is marginally decreasing.It does not affect the main results of the model..

Finally, this paper discusses how the social welfare changes after the commodities originally settled in Bitcoin in the commodity market are paid by CBDC completely.Given{ϕt-1,ψt-1,θb},Proposition 3.1 shows that there are two situations where only CBDC exists in the economy and the Bitcoin is completely substituted by CBDC.The first situation is that when the fee rate of Bitcoin transaction,fincreases and the regulatory risk of Bitcoin increases (µ↓), the social welfare doesn’t change.The other situation is that when the growth rate of the CBDC supply increases and the sales tax rate on CBDC transaction decreases,then the increase of commodities settled in CBDC in the commodity market is greater than the decrease of commodities settled in Bitcoin.Thus, the social welfare is promoted.

§4.Conclusion

From the perspective of regulatory uncertainty, this paper studies how the monetary policy of central bank, fiscal policy of government, transaction costs and regulatory uncertainty of private cryptocurrency on the competitive equilibrium between private cryptocurrency and Central Bank Digital Currency.It draws the following conclusions and policy suggestions:

First, the Central Bank Digital Currency and private cryptocurrency can coexist under certain conditions.When given monetary and fiscal policy, Central Bank Digital Currency and private cryptocurrency construct complete competition.The government can strengthen the supervision on private cryptocurrency.It cracks down on the transaction of private cryptocurrency to promote the risk expectation and transaction fee rate of private cryptocurrency.It’s beneficial to accelerate the process replacing private cryptocurrency with Central Bank Digital Currency in economy.

Second, the social welfare is related with monetary policy and fiscal policy but unrelated with the issuance and transaction mechanism of private cryptocurrency.In other word, the government should maintain the independence of monetary policy and fiscal policy to realize the macroeconomic development target and optimize social welfare.

Third, under the complete competition of private cryptocurrency mining, the mining profit tends to zero for whole society.The total cost of mining becomes a social deadweight loss.It’s positively related with the volume of commodities paid by private cryptocurrency.The government should crack down on the mining work of private cryptocurrency and reduce the waste of social resource.