Money and crime in a cash-in-advance model.

AuthorCho, Hyung Sun
  1. Introduction

    The main purpose of this article is to develop a model to explore the coexistence of credit and money and the effects of monetary policy when holding money is risky. Historically, currency has been a resilient medium of exchange because everyone accepts it without the user's identity being revealed. However, given its usefulness in anonymous transactions, money has been a target for theft. Credit involves no such theft problem, ignoring the possibility of identity theft. (1) However, using credit incurs transactions costs, for example, the time waiting to check credit history or making a signature on a receipt. Therefore, people carry both cash and credit and choose a different means of payment to purchase goods as there are circumstances during which either can be advantageous.

    A few models have been constructed to study the coexistence of multiple means of payment, such as those of Lucas and Stokey (1987), Prescott (1987), Ireland (1994), Lacker and Schreft (1996), and Aiyagari, Braun, and Eckstein (1998). In Lucas and Stokey (1987), there are two markets, a market for cash goods and one for credit goods. Only cash is accepted as a medium of exchange in the cash-good market, and only credit is accepted in the credit-good market. The choice of means of payment is exogenously given, and there are no transaction costs of credit. Thus, the model cannot provide any implications for the choice of means of payment.

    In Prescott (1987), Ireland (1994), Lacker and Schreft (1996), and Aiyagari, Braun, and Eckstein (1998), there are alternative means of payment (for example, bank draft, private securities, and credit) other than cash, and an economic agent can choose different means of payment when acquiring consumption goods. In addition to the opportunity cost of holding cash (i.e., the nominal interest rate), the transactions costs of alternative means of payment are introduced as a tradeoff. In equilibrium, cash and other means of payment coexist since the substitution of money for other means of payment is beneficial if the nominal interest rate decreases or transactions costs decrease. Also, other means of payment are useful for larger purchases, and cash is useful for smaller ones. In Prescott (1987), Ireland (1994), and Lacker and Schreft (1996), however, all the results are valid only if the return on nominal assets is positive. If the return on nominal assets is 0, then only money is in use in the economy.

    In my model, money and alternative means of payment coexist with a nonpositive return on money if theft is introduced as another opportunity cost of money, in addition to the nominal interest rate. The model is built on the Ireland (1994) framework. Each household consists of a shopper and a worker. A worker decides how much time to spend on stealing or working. A worker can steal shoppers' money from other households before a shopper starts to purchase consumption goods. Contrary to the findings of He, Huang, and Wright (2005, 2008), there is no failure rate on stealing, but a worker has to give up the opportunity to sell goods to steal cash from others. In the goods market, a shopper purchases consumption goods while a worker produces and sells them. A shopper holds cash and credit, and he can choose one of them to purchase consumption goods.

    In steady-state equilibrium, the marginal rate of substitution of cash goods for credit goods depends not only on the nominal interest rate but also on the risk of holding money, contrary to the findings of Prescott (1987), Ireland (1994), and Lacker and Schreft (1996). Credit is preferred if either the nominal interest rate or the crime rate is higher. Thus, an economic agent may spend on credit even though the net nominal interest rate is not positive.

    With theft, a change in the money grOwth rate affects the nominal interest rate and the crime rate as well as the cash-credit choice. First, a worker steals less if the money growth rate increases. Inflation taxes stealing and increases output. Next, a shopper prefers to use cash to purchase consumption goods in more markets, although the nominal interest rate increases. Lower transactions costs follow. The marginal benefit of using money is greater than the marginal cost of using cash. Finally, as long as the crime rate is positive, welfare improves with the money growth rate, contrary to the findings of Dotsey and Ireland (1996).

    Once the money growth rate is large enough, then the crime rate is driven to 0 and the economy becomes like that described in Ireland (1994). Without theft, the nominal interest rate remains as the only opportunity cost of money, which increases with the money growth rate. A shopper prefers to use credit to purchase consumption goods in more markets if the money growth rate increases. Transactions costs increase since shoppers use credit more often. Welfare decreases with the money growth rate, and there exists the welfare cost of inflation, as described in Dotsey and Ireland (1996). Therefore, the government wants to increase the money growth rate in order to eliminate crime, but once crime disappears the government does not want to increase the money growth rate anymore. In general, the Friedman rule is not optimal.

    Introducing the risk of holding money is not a new idea. In 1987, Prescott pointed out that the risk of loss by theft or fire can be an additional feature of money. He, Huang, and Wright (2005, 2008) note that cash is relatively riskier to carry around, compared with other means of payment (for example, checks and debit or credit cards), and they introduce a theft problem in a search framework. A buyer can lose his money while he searches for consumption goods because a seller can be a thief at any time. A bank can endogenously arise where a buyer can choose to deposit his money for safekeeping purposes. In equilibrium, with endogenous theft, money can possibly disappear if the storage fee is less than the amount of money that could be lost. Thus, a buyer can deposit even with a negative nominal interest rate, and optimal inflation can be positive, as in my model. However, in He, Huang, and Wright (2005, 2008), a buyer carries one means of payment, either cash or checks, when purchasing consumption goods. The model cannot show why people use different means of payment to acquire a different variety of consumption goods. On the contrary, my article can show the choice of means of payments in the goods market and the effects of monetary policy on individual consumption and theft in a clear and more concise way.

    The remainder of the article is organized as follows. Section 2 describes the basic environment. Section 3 discusses a money-only economy. Section 4 studies a cash-credit economy and its implications in steady state. Section 5 presents monetary policy and welfare implications, and section 6 concludes.

  2. The Economic Environment and Timing

    Time is discrete and is indexed by t = 0, 1, 2, .... The economy consists of a continuum of infinitely lived households with unit mass. Each household consists of two agents: a worker and a shopper. A continuum of spatially separated markets indexed by i [member of] [0,1] exists at each period. In each market i, a worker produces and sells distinct, perishable consumption goods indexed by i [member of] [0,1]. The household has preferences (2) given by

    [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [beta] is the discount factor; [c.sub.t] (i) represents consumption goods purchased at market i in period t; and [x.sub.t] represents transactions costs.

    At the beginning of period t, each household enters with [M.sub.t] units of currency and [B.sub.t] units of one-period nominal bonds. Then, the household learns the money growth rate, [[theta].sub.t]. The government, the representation of the central bank, controls the nominal money supply through nominal lump-sum transfers of money, [P.sub.t][[tau].sub.t]. The government budget constraint is

    [M.sup.S.sub.t+1] = [M.sup.S.sub.t+1] + [P.sub.t][[tau].sub.t]

    [M.sup.S.sub.t+1] = [[theta].sub.t] [M.sup.S.sub.t], where [P.sub.t] is the average price level of consumption goods, [P.sub.t] = [[integral].sup.1.sub.0] [P.sub.t](i) di, (1)

    and [P.sub.t](i) is the price of consumption goods at market i.

    The asset market opens, and each household exchanges money for interest-bearing one-period government nominal bonds, [B.sub.t]. Each bond sells for qt units of money in period t and is a claim to one unit of money in period t + 1. The asset market closes, and a worker and a shopper at each household leave for a goods market.

    At the goods market, first a worker decides how much time to devote to working, [n.sup.w.sub.t], and to stealing, [n.sub.s.sub.t], given one unit of time:

    [n.sup.w.sub.t] + [n.sup.s.sub.t] = 1. (2)

    A worker steals a proportion of money from shoppers of other households before he starts to produce consumption goods. A worker does not steal his own household shopper's money. The amount of money that a worker steals is denoted by [phi]([n.sup.sub.t])[[bar.M].sub.t], where [[bar.M].sub.t] is the average quantity of money held by other shoppers. The function [phi]([n.sup.s.sub.t]) defines the yield from effort in stealing, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3)

    where [pi] > 1 is the degree of stealing efficiency. (3) A shopper will lose all of his cash if a worker spends more than 1/[pi] unit of time to steal. However, stolen cash is not a free lunch, although no punishment or failure rate for stealing is assumed. If a worker spends more time stealing, then he works less and produces less output. Thus, he has to give up income for stolen cash. A shopper cannot spend the stolen money, [phi]([n.sup.s.sub.t])[[bar.M].sub.t], within the period, but he can use it to pay off credit at the end of the period. (4)

    After a worker steals money, everyone starts to exchange consumption goods. A worker produces consumption goods with a linear...

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