The optimum quantity of money is most famously associated with Milton Friedman (1969). The optimum is a normative policy conclusion drawn from the long-run properties of a theoretical model. Friedman posited an environment that abstracts from all exogenous shocks and nominal price and wage sluggishness. The basic logic is then straightforward. One criterion for Pareto efficiency is that the private cost of a good or service should be equated to the social cost of this good or service. The service in question is the transactions role of money. The social cost of producing fiat money is essentially zero. Since fiat money pays no interest, the private cost of using money is the nominal interest rate. Hence, one criterion for Pareto efficiency is that the nominal interest rate should equal zero. Since long-run real rates are positive, this implies that monetary policy should bring about a steady deflation in the general price level. This famous policy prescription is now commonly called the Friedman rule.

Although most closely associated with Friedman’s (1969) bold statement of the policy conclusion, the basic idea of the optimum quantity can be found in Tolley (1957), who argues, on similar efficiency grounds, for paying interest on currency. Friedman (1960) credits Tolley with this suggestion, and further notes that an alternative policy would be a steady deflation. It is curious that Friedman (1960) dismisses the ‘Friedman rule’ deflation as not feasible for practical purposes. Finally, the optimum-quantity result is implicit, but never noted, in Bailey (1956) who examines the welfare cost of inflation but does not consider the welfare gain of deflations.

In practice, the optimum-quantity result has had remarkably little influence on monetary policy implementation. Although many central banks pursue low inflation rates with an eventual goal of price stability, no central bank has advocated a policy that would bring about a steady price deflation. There are likely several reasons, both judgemental and theoretical, that have led to this lack of influence. I will briefly review both types of objections.

One of the first theoretical objections to the optimum-quantity results was made by Phelps (1973), who argued that Friedman’s first-best argument ignored the second-best fact that money growth produces seigniorage revenues for a government, and that all forms of taxation produce distortions of some kind. If ‘money’ or ‘liquidity’ is a good like any other, then familiar optimal taxation arguments would suggest that it should be taxed via a steady inflation. This argument seems all the more persuasive given empirical estimates of a fairly low money demand elasticity.

This public finance approach spawned a very large literature. Important contributions include Kimbrough (1986), Guidotti and Vegh (1993), Correia and Teles (1996, 1999), Chari et al. (1996), and Mulligan and Sala-i-Martin (1997). These analyses were much more explicit than Friedman (1969) and considered a fully dynamic theoretical environment with no nominal rigidities. A key relationship in all these models is the transactions or shopping function. The time spent by households shopping (s_t) is a function of the form: s_t = ϕ(c_t, m_t), where c_t denotes real consumption and m_t denotes real cash balances. The function ϕ is assumed to be homogenous of degree k, increasing in consumption, and decreasing in real cash balances, the latter effect motivated by the transactions function of money. Money can be thought of as an intermediate good that facilitates consumption purchases. Now suppose a central government needs to finance an exogenous level of spending and can do so only with distortionary taxes on, say, labour income, or the inflation tax on money balances. In this case, is the Friedman rule still optimal?

Most of these papers were supportive of the Friedman rule, concluding that in such a second-best environment the optimal monetary policy is a zero nominal rate. Mulligan and Sala-i-Martin (1997) argued that the result was fragile as it depended on the degree of homogeneity in ϕ and the alternative tax instruments available to the government, for example, income taxes against consumption taxes. These conflicting results have been usefully explained in DeFiore and Teles (2003), who demonstrated that the reason for the divergent conclusions is an inappropriate specification of how consumption taxes are entered in the transactions cost function. They consider a more general environment in which the government has access to both consumption and income taxes. They also consider the case where money is costly to produce at a constant marginal cost of a. Further, they demonstrate that if ϕ is linearly homogenous (k = 1) then the optimal interest rate is equal to α. This is a modified Friedman rule in that the private cost and social cost of money are set equal to each other, and is analogous to the Diamond and Mirrlees (1971) optimal taxation result: intermediate goods should not be taxed when consumption taxes are available and the technology is constant returns to scale (k = 1). If ϕ is not linearly homogeneous, then the optimal policy involves a tax (or subsidy) on money proportional to α. Since money is essentially costless to produce (α = 0) the optimal nominal interest rate is zero. DeFiore and Teles (2003) thus conclude that the Friedman rule is the optimal second-best policy for all homogeneous transactions technologies. Hence, the Phelps (1973) objection appears to be settled in Friedman’s favour.

A second theoretical objection to the optimum-quantity result is that, in a world with nominal rigidities, a steady general price deflation would produce unwanted relative price movements since not all nominal prices would be adjusted simultaneously. Strictly speaking this is not a theoretical objection to Friedman (1969), as he assumed a world with perfectly flexible nominal prices and wages. But if one believes that nominal rigidities are important, and that they matter even in the long run, then this is a relevant objection to the Friedman rule. For example, in the dynamic new Keynesian (DNK) class of models (for example, Woodford 2003) the assumed nominal rigidities have permanent effects so that any departure from price stability causes permanent movements in relative prices. Hence, these models typically suggest that optimal policy is a stable price level, and that a Friedman-rule deflation would be suboptimal. These DNK models typically abstract from the nominal interest rate distortions that are at the heart of the optimum-quantity result.

A model that combined the DNK nominal rigidities with the nominal rate distortion would presumably result in a long-run optimal nominal interest rate somewhere between zero and the steady-state real rate.

The principle judgemental objection to the Friedman rule is historical. The instances in US history in which deflations occurred are associated with severe recessions, most famously in the 1929–1933 period. A related judgemental concern deals with the zero bound. If the central bank’s principal tool to stimulate the economy is a reduction in the nominal rate of interest, then the zero nominal rate prescribed by the Friedman rule apparently leaves no additional ammunition in the monetary policy arsenal (as nominal rates cannot be negative). This nervousness about the Friedman rule was enhanced by the experience of Japan during the 1990s. The Japanese economy performed poorly at a time in which general prices were falling and the short-term nominal rate was zero.

Since central banks have not followed Friedman’s (1969) proposal to set the nominal rate to zero, a natural issue is to quantify the welfare costs of being away from Friedman’s optimum quantity of money. Following in the footsteps of Bailey (1956) and Lucas (2000) uses a theoretical environment similar to that of Correia and Teles (1996, 1999) to address this question. The welfare cost is approximately the area underneath the money demand curve between the optimal zero nominal rate and the interest rate under question. Lucas reports that the welfare cost of a four per cent nominal rate is between 0.2 per cent and one per cent of annual income, the difference depending upon the assumed behaviour of money demand as the nominal rate approaches zero. Since a zero nominal rate has not been observed in the United States in the post-Second World War period, the data cannot determine which estimate is more accurate. But either estimate suggests a fairly modest welfare cost.

Studies analysing the optimality of the Friedman rule have been reignited by the new class of search-theoretic monetary models. These models are micro-based, replacing the function ϕ in DeFiore and Teles (2003) with a search-based trading environment in which money improves the chances of successfully finding a suitable partner with whom to trade. In an innovative paper, Lagos and Wright (2005) use a search-theoretic environment to address the optimality of the Friedman rule and the welfare consequences of deviating from it. In search models of money the buyer and seller engage in a bargaining game to determine the transactions price at a given meeting. The buyer is carrying money and has thus postponed previous consumption. If sellers have some bargaining power, then there is a hold-up problem because part of the gain associated with the holding of money is received by the seller. This bargaining distortion leads the buyers to economize on money holdings so that they are below the socially efficient level. Lagos and Wright (2005) demonstrate that the optimal policy in this search environment is the Friedman rule (a similar conclusion is reached by Shi 1997). But more interestingly, the welfare cost of being away from the Friedman rule, at say a four per cent nominal rate, is significantly higher than calculated by Lucas (2000). This arises because the positive nominal rate exacerbates an already suboptimal level of real balances arising from the hold-up problem.

The search models of money have rekindled interest in the optimality of the Friedman rule at just the time when DeFiore and Teles (2003) appear to have settled the issue in the aggregative monetary models. The coming years will probably see further work on the Friedman rule from this search-theoretic perspective. A key issue is the nature of the bargaining process that arises at trading opportunities. These recent developments testify to the continued prominence of the optimum quantity of money in monetary theory, if not practice. The lasting contribution of the theory is to introduce explicit, utility-based welfare analysis into monetary economics.