Should Bank Reserves Earn Interest?

Home , Bank reserves

The case for payments of interest on re- Should Bank serves applies not only to the 100% reserve system, but equally to our present fractional Reserves Earn reserve system. Accordingly, even if reserves are not raised to 100%, Reserve Banks should be required to pay interest on their Interest? deposit liabilities. —Milton Friedman A Program for Monetary Stability Scott Freeman Professor University of Texas at Austin As the introductory quote indicates, Milton Friedman (1959), among others, has advocated Joseph H. Haslag 1 Senior Economist paying interest on reserves. In the United States Federal Reserve Bank of Dallas and many other countries, banks and other financial intermediaries are required to hold a fraction of their assets as fiat money—unbacked, interest-free bills of the central bank. In the absence of interest on these reserves, the aver- age return to assets held by banks must lie below the market rate of return. This implies that banks must pay their depositors a return below the market rate of interest, unnecessarily discouraging the holding of bank deposits. Be- Paying interest on reserves cause such intervention into the business of banking is so common, basic questions about would increase the demand for the desirability of such requirements may easily be overlooked. For instance, by forcing banks to deposits and thus for reserves. hold unbacked assets paying no interest, might the central bank be discouraging banking and This, in turn, would raise the the accumulation of capital? But where would the interest come from? value of existing reserves, As with any government expenditure, interest paid on reserves must (at least eventually) come increasing the wealth of those from taxes, raising two questions: Wouldn’t wealth be reduced by the rise in taxes? Wouldn’t who own bank deposits at taxation introduce its own economic distortions, possibly worse than those that result from the the time that interest absence of interest? Paying interest on reserves would increase payments are initiated. the demand for deposits and thus for reserves. This, in turn, would raise the value of existing reserves, increasing the wealth of those who own bank deposits at the time that interest payments are initiated. Bruce Smith (1991) shows that this windfall gain to those holding deposits at the time the policy is enacted comes at the expense of future generations; that is, future generations must pay the taxes to finance the interest payments but do not receive all of the resulting benefits. Thus, Smith shows that the transfer of wealth created by the payment of interest makes future generations worse off. In this article, we propose a means of eliminating this transfer. We begin by discussing the role reserve requirements play in a simple economy. People finance the next period’s consumption by holding deposits. The key feature

FEDERAL RESERVE BANK OF DALLAS 25 ECONOMIC REVIEW FOURTH QUARTER 1995 of the model is that reserve requirements force nothing when old, but wishes to consume in banks to hold fiat money as fractional backing both periods of life. The problem facing these for deposits. The merits of paying interest on people is the means of financing consump- reserves will be clear if the government offsets tion in the second period of life. There is also the wealth transfer identified by Smith. a generation that lives and consumes only in Our idea for an offsetting transfer is adapted the initial period, hereafter referred to as the from a policy proposed by Leonardo Auernheimer “initial old.” (1974).2 When interest on reserves is initiated, In the first period of this model economy, the central bank should expand the stock of there is a fixed stock of M (divisible) pieces of nominal reserves to keep the price level from paper called fiat money. In addition to money, decreasing. If the central bank uses this increase there are also two forms of capital. The first form in the money stock to purchase interest-bearing is available to any individual in isolation. An assets (an open market operation), the interest investment of kt goods in period t will produce generated by these assets can help pay for the f (kt ) consumption goods in period t + 1. The interest paid on reserves, lowering the tax bur- marginal product of capital, which we express den on future generations. We argue that paying as f ′(k), is positive but decreasing. The second interest on reserves, when accompanied by the form of capital produces a constant x consump- appropriate open market operation, can make tion goods (x > 1) in period t + 1 for each good every future generation better off without hurt- invested at t. This latter form of capital can be ing initial deposit holders. made only in amounts greater than y so that no We also take up the second of our nettle- individual alone has the resources to finance some questions: Would taxation introduce its capital. Both forms of capital produce consump- own economic distortions? The taxes available tion goods only once. to government in the real world are generally ad Note that the second form of capital is valorem taxes; the amount of tax collected is set illiquid in this economy because it cannot be at some fraction of an economic variable, such divided into small units. It is easy to see how an as income or sales. Ad valorem taxes artificially intermediary can overcome this illiquidity by discourage the taxed activity, just as the absence simply pooling the deposits of many individuals of interest on reserves discourages deposits at to an amount greater than y. We assume for banks. We show that despite this tax-induced simplicity that the intermediation services are distortion, we can make people unambiguously costlessly and competitively provided by entities better off by paying interest on reserves. This referred to as “banks.” 3 improvement occurs even if the interest must In this economy, we assume that a reserve be funded by a tax used in the real world— requirement is imposed: for each good de- a distorting ad valorem tax on capital—if this posited, a bank must hold fiat money worth γ tax is accompanied by a price-stabilizing open goods but is free to invest the remaining 1 – γ market purchase. goods in the illiquid, or intermediated, capital In sum, our questions about the costs of good.4 (We assume throughout this analysis paying interest on reserves are fairly straight- that the initial old hold positive quantities of forward to resolve. Both capital taxation and both unintermediated capital and deposits.) If open market operations are widely used real- fiat money’s rate of return is less than that of world policy options. Therefore, there exists a capital, banks will hold no more than the re- way to finance the payment of interest on required balances of fiat money. Suppose, for serves that will make the public unambiguously now, that banks do not hold any excess re- better off. serves. (We will verify shortly that this is a wise

decision.) If st denotes deposits per young per- A model of reserve requirement banking son, then banks will hold fiat money balances γ To address these questions, let us examine worth Nst goods. Those required reserves a simple model adapted from the framework represent the total demand for fiat money shared by David Romer (1985), Thomas Sargent measured in goods. The supply of fiat money and Neil Wallace (1985), Scott Freeman (1987), is M dollars or, when measured in goods, vt M, and Smith (1991) in which financial intermediar- where vt represents the goods that can be pur- ies that mobilize capital are subject to a reserve chased by a single dollar. The goods value of requirement. a dollar is simply the inverse of the dollar price

In each period, starting from some initial (pt ) of one good, or vt = 1/pt. Furthermore, the period 1, N people who live two periods are gross real rate of return from holding fiat money born. Each produces y goods when young and is the ratio of goods purchased by a single dollar

26 Figure 1 Determining Savings, Deposits, and Unintermediated Capital for a Given Reserve Requirement

Desired savings

f ′(k) = x(1 – γ) – γ x(1 – γ) + γ A

f ′(k)

Goods Deposits Unintermediated capital

Total savings

in period t + 1 to the goods purchased by a subsequent generation suffers from this lower single dollar in the current period, or vt +1/vt. rate of return on their deposits. Moreover, with For the demand for fiat money to equal its x > 1, equation 2 indicates that the bank best supply, serves its depositors by not holding reserves in excess of those required. γ (1) Nst = vt M. People will invest in the asset paying the better rate of return. This implies that the people

Notice that when deposits, st, are constant over who hold both deposits and unintermediated time, the demand for fiat money is constant over capital will invest in unintermediated capital time. Therefore, when the stock of fiat money is up to the point that its marginal rate of return also constant over time, the value of a dollar and just equals the rate of return offered by inter- the price level will both be constant over time. It mediaries: follows that the gross real rate of return of a 5 ′ γ γ dollar, vt +1/vt, equals 1. (3) f (k) = (1 – )x + . What, then, will be the rate of return offered by competitive banks? Assuming for sim- Because an increase in the reserve require- plicity that intermediation services are costlessly ment lowers the return on intermediated capital, provided by banks in a competitive market, then people switch from deposits to unintermediated banks will offer depositors the rate of return that capital. This switching occurs until the rate of the banks can earn on the assets they hold. This return on unintermediated capital falls to equal (gross, real) rate of return (call it R) is the new lower rate of return on deposits. Figure 1 illustrates the basic point made (2) R = (1 – γ)x + γ in equation 3. The desired savings curve plots the quantity of savings for next-period con- because for each good deposited, the bank can sumption at different rates of return. The rate of invest (1 – γ) in capital paying the rate of return return on the vertical axis is equal to the rate x and purchase γ in fiat money paying the rate offered by competitive banks and is determined of return 1. Notice in equation 2 that increasing by the returns on intermediated capital and the reserve requirement lowers the rate of return reserves. The horizontal line emanating from on deposits by forcing banks to hold more low- the value x (1 – γ) + γ on the vertical axis in return fiat money per deposit. Clearly, every Figure 1 is the return on deposits. For a given

FEDERAL RESERVE BANK OF DALLAS 27 ECONOMIC REVIEW FOURTH QUARTER 1995 Figure 2 The Effect of an Increase in the Reserve Requirement from γ to γ′

Desired savings

x(1 – γ) + γ

x(1 – γ′) + γ′

f ′(k)

Goods Deposits Unintermediated capital

Total savings

level of savings, the distribution between inter- reserve requirements has on savings and each mediated and unintermediated capital depends form of capital. on the assumption that the return on uninter- There is, we should note, a group that mediated capital falls with each additional unit benefits from the imposition of a reserve re- of this form of capital. Figure 1 captures this quirement: the initial old. By assumption, this feature by representing the f′(k) curve as a group starts with a portfolio of assets that in- downward sloping line. From equation 3, people clude fiat money. If reserve requirements were add units of unintermediated capital up to removed, this fiat money would have no value. the point at which the return offered by banks Consequently, the value of the initial old’s port- equals the return on unintermediated capital. folio would fall. Alternatively, increasing the This occurs at point A in Figure 1. The horizon- reserve requirement increases the demand for tal distance between the vertical axis and point fiat money, making each dollar more valuable A measures how much unintermediated capital and raising the welfare of the initial old by people will choose. The difference between raising the value of fiat money (see equation 1). desired savings and unintermediated capital— In short, the reserve requirement transfers wealth the horizontal distance between total goods from all future generations to the initial old. saved and point A—measures the quantity of The central bank can increase the rate of deposits. return on deposits if it increases the rate of An additional implication of equation 3 return on fiat money by, for example, paying seen in Figure 1 is that with reserve require- interest on required reserves. This will increase ments, the output produced by one more unit of the rate of return paid to depositors, but will it unintermediated capital, f ′(k), is less than the make them better off? As we demonstrate in the output from a unit of intermediated capital, x.6 next section, the answer depends on how this Therefore, by encouraging people to switch higher rate of return is financed. from intermediated capital to unintermediated capital, a reserve requirement reduces output A case with interest payments on reserves for each good switched. More generally, higher In this section, we consider how different reserve requirements discourage total savings financing schemes affect the desirability of pay- because of the lower rates of return offered on ing interest on reserves. Paying interest on re- both unintermediated capital and deposits. serves will be deemed desirable if at least one Figure 2 illustrates the effects an increase in group is made better off while no other group is

28 harmed. In the model outlined above, the groups which equals the gross rate of return on capital can be identified using the date at which the regardless of the size of the reserve requirement. policy is implemented as the reference point; For any positive reserve requirement, all future thus, the two groups that come to mind are generations are made better off by this plan to those already holding money when the policy is pay interest on reserves because they are of- implemented (the initial old) and the future fered a higher rate of return on their deposits. generations. Would anyone oppose such a plan to pay The central bank as an intermediary. Con- interest on reserves from central bank capital? sider first a policy that would have the govern- Yes, the initial old would. Notice that this fi- ment pay the interest from interest-bearing assets nancing scheme begins with the central bank of the central bank. Suppose that instead of confiscating the initial old’s money balances with- leaving the initial stock of central bank money in out any compensation. Such a tax collection the hands of the initial old, the central bank scheme reduces the wealth holdings of the ini- takes it and uses it to purchase (intermediated) tial old, reducing their consumption. capital. This gives the central bank ownership of The payment of interest on reserves from a stock of capital. (We focus our attention here central bank capital has the same welfare effects on an equilibrium in which the stocks of re- as abandoning reserve requirements. In both serves, central bank capital, and the value of cases, future generations receive a better rate of money are constant over time.) Formally, the return (x) on their deposits, but the initial old central bank’s balance sheet constraint is lose the value of their initial balances of fiat money. (4) K g = vM, There are three differences between abandoning reserve requirements and confiscating where K g is an interest-bearing asset that repre- the initial old’s money balances. First, when sents the value of capital held by the central reserve requirements are simply abandoned, all bank.7 The central bank will pay interest on its fiat money becomes worthless.8 Under central liability, reserves (central bank money), using bank intermediation, however, there is still a the return on this capital net of its replacement demand for reserves, and the value of a dollar is cost, xKg – K g = (x – 1)K g. If ρ denotes the again determined by the equality of supply and nominal net interest paid on a dollar of reserves, demand for reserves set forth in equation 1. As then the interest paid on reserves equals ρvM, we have shown, banks will hold zero excess implying that each period the central bank’s reserves, so that budget requires (8) γNs = vM. (5) (x – 1)K g = ρvM. Second, the model economy described Because the central bank owns capital exactly above specifies that intermediated capital al- equal to the value of reserves (K g = vM ), the ways returns x units of the consumption good central bank can offer interest on reserves equal for every one unit invested. It is not difficult to to the net return of capital: imagine a situation in which returns are related to the quality of the investment decisions made. (6) ρ = (x – 1), Under central bank intermediation, we entrust a governmental body, the central bank, with in- which implies that the gross rate of return on vestment decisions. The central bank may not reserves, 1 + ρ, is x. Because the central bank be motivated by maximization of profits. Conse- backs its money with capital, reserves pay the quently, if the central bank does not choose as same rate of return as other interest-bearing wisely as private banks, the return offered on assets owned by private banks. Under this plan, reserves may be below the market rate of return. the central bank has become an intermediary Of course, one way to remove the investment paying market interest rates to its depositors decision from the purview of the central bank is (private banks). Therefore, depositors at private to open the discount window. Banks could bor- banks will no longer care what fraction of their row funds at the market rate of return and make deposits is required to go into reserves. The the investment decisions. Then the central bank’s gross rate of return on deposits is now only responsibility would be to restrict its lend- ing to sound banks. (7) R = (1 – γ)x + γ (1 + ρ) Third, we have thus far assumed that inter- = (1 – γ)x + γx = x, mediation services are costlessly provided. A

FEDERAL RESERVE BANK OF DALLAS 29 ECONOMIC REVIEW FOURTH QUARTER 1995 more realistic assumption recognizes that costs, ply increases help the future generations? Let the such as those of record-keeping, are associated central bank use the increase in the stock of its with creating private intermediary services. With money to buy capital. The central bank’s ex- central bank intermediation, there is a second change of (intermediated) capital for fiat money level of record-keeping; people make deposits is an open market purchase. The additional at banks and then banks make deposits (hold capital can then be used to help finance the reserves) at the central bank. If it is costly to payment of interest on reserves, lessening the keep records and otherwise manage deposits, tax burden of future generations. Freeman and the total of these costs will be higher under this Haslag (forthcoming) show that this tax-financed two-level system of intermediation than under interest on reserves makes the future genera- the one-level setup. tions better off. (A formal proof is also presented Tax-financed interest on reserves. Suppose in the appendix.) The higher rate of return on that the central bank wants to finance interest on deposits encourages savings through banks at its reserves without hurting the initial old. It can do optimal level, without a transfer of wealth from so if each future generation is taxed to pay the the future generations to the initial owners of interest.9 Would the benefits of the increased money. In short, the future generations pay rate of return exceed the cost of the taxation? enough taxes to finance the interest payments Increasing the rate of return on deposits would on reserves but do not pay for a transfer to the increase deposits and thus capital, as desired. initial old. The increased deposits, however, also increase the demand for reserves. Greater demand for A case with distortionary taxes fiat money increases the value of the initial The financing scheme outlined above is reserves owned by the initial old. (Note from based on a lump-sum tax. The desirability of equation 1, the equality of supply and demand taxing to pay interest on reserves may no longer for reserves implies that v = γNs/M. Clearly, an hold if the tax, like many real-world taxes, itself increase in s will increase v.) In effect, the taxes distorts individual incentives. An income tax, for paid by future generations go to pay interest on example, may well reduce incentives to work reserves and to increase the wealth of the initial and invest, therefore causing more economic old. Smith (1991) demonstrates that this transfer distortion than the absence of interest on re- of wealth from the future generations to the serves. To address this concern, we now exam- initial generation lowers the welfare of the fu- ine the payment of interest on reserves financed ture generations despite the greater rate of re- by a tax commonly used in the real world, a tax turn on deposits. To understand this result, note on capital. We show that people are better off that the taxes paid by people in the future with interest paid on reserves, even if it must be generations are exactly equal to the value of the financed by a tax that discourages the holding of interest payments received on reserves. These capital. two changes to lifetime wealth, therefore, ex- Consider, in particular, a tax applied against actly cancel each other out. However, the policy the return from both types of capital; that is, the has a side effect: the reserves that the initial payment of interest on reserves is to be financed generation owns and that subsequent genera- by a tax of α times the return to both intermedi- tions need have been made more expensive. ated and unintermediated capital. As with the This transfers wealth from subsequent genera- lump-sum case described above, we assume tions to the initial old. Therefore, the central that the government conducts an open market bank cannot increase the welfare of the future purchase that keeps the price level constant. generations simply by financing interest on re- Thus, the net interest on the government’s capi- serves through pay-as-you-go taxation. tal goods plus revenue from the capital tax is Auernheimer (1974) suggests a way to fi- equal to the government’s net interest on re- nance the payment of interest on reserves with- serves. out hurting or helping the initial old.10 The initial The question is whether interest-bearing old gain under the tax plan just described be- required reserves are welfare-improving when cause of an increase in the value of their initial financed with a distortionary capital tax. Free- money balances. The value of money can be man and Haslag (forthcoming) and the appen- brought back to its initial level if the central dix to this article show that the total net return to bank prints more money, such that the increased the future generations is increased when the demand for money is exactly matched by an government pays interest on required reserves, increased supply of money. How can a plan even if the interest is financed by a tax on featuring taxes and accommodating money sup- capital.

30 The intuition behind this result is fairly finance interest-bearing reserves will avoid some straightforward. If reserves pay no interest, a of the pitfalls associated with either directly reserve requirement directly distorts the return taxing initial required reserves or the lump-sum to intermediated capital. In this way, the reserve tax alone. When open market purchases accom- requirement is like a tax on the return to inter- pany the payment of interest on reserves, mem- mediated capital, while unintermediated capital bers of the future generations are better off is not directly taxed. Paying the market rate of while the initial old are unaffected. Clearly, this interest on reserves means that deposits earn the makes society better off. We further show that same return as unintermediated capital, ending paying interest on reserves is strictly better than the discouragement of deposits resulting from not paying interest, even if the taxes are the lower return from required reserves. Taxing distortionary. This last result underscores the both intermediated and unintermediated capital distortionary effect associated with reserve re- at the same rate spreads the distortion equally, quirements. Spreading the distortion across both and thus efficiently, across the two types of types of capital—in the spirit of the Ramsey rule capital. In short, people do not make investment of efficient taxation—raises welfare. choices between the two forms of capital based A key feature of the welfare improvement on after-tax returns. When taxes are applied is the accommodating open market purchase equally, both the pre- and after-tax returns are suggested by Auernheimer. The payment of in- equalized. The gain from the increased return terest on reserves effects a transfer from future on deposits more than offsets the lower after-tax generations to the initial old. This transfer can return on unintermediated capital. Consequently, be exactly offset by an open market purchase. future generations have a higher total return The assets thus purchased can then be used to than when the return of only one type of capital help finance the payment of interest. Such an is distorted.11 accommodation is not beyond the central bank’s The payment of interest on reserves en- normal operations. Indeed, Haslag and Hein courages people to marginally substitute inter- (1995, 1989) provide evidence that the Federal mediated capital for unintermediated capital. For Reserve systematically accommodates changes each extra unit of intermediated capital, x goods in reserve requirements with open market op- are produced, while an extra unit of uninter- erations. mediated capital produces f ′(k) goods. We have Overall, the main purpose of this article is seen that when intermediated capital is subject to demonstrate that paying interest on reserves to reserve requirements without interest, f ′(k) = improves welfare in a broader class of model (1 – γ)x + γ = x – (x – 1)γ, which is less than x. economies than previously believed. We extend Therefore, when people switch one unit of sav- the class of economies along two distinct lines. ings from unintermediated capital to intermedi- For some time, people have recognized the ated capital, more output is gained (x) from the improvement that is possible in Friedman’s set- increase in intermediated capital than is lost ting with infinitely lived people and lump-sum [f ′(k)] from the drop in unintermediated capital. taxes. Smith raises questions about the desirabil- Therefore, there is more overall output and ity of paying interest on reserves when the initial greater welfare when interest is paid on re- (finite-lived) money holders benefit but are not serves. Output and welfare would be even greater taxed. Our first extension shows that welfare if the interest could be funded by lump-sum improvement is still possible in this economy if taxes, but stuck as we are with distorting taxes, a simple coordinated financing scheme is the payment of interest on reserves is still an adopted. The second extension shows that pay- improvement. ing interest on reserves can improve people’s welfare, even if the interest is funded through Conclusions distortionary taxes. In this article, we demonstrate how alter- native schemes to finance interest payments on Notes required reserves will affect people. We con- 1 George Tolley (1957) also argues that the central bank sider four different schemes: directly taxing should pay interest on reserves. Joshua Feinman initial required reserves, a lump-sum tax on (1993) traces the historical evolution of reserve re- future generations, and two financing schemes quirements in the U.S. banking system. Feinman also that are accompanied by open market pur- notes that the Federal Reserve has explicitly supported chases—lump-sum taxes and capital taxes. legislation authorizing the payment of interest on We show that using lump-sum taxes ac- reserves since the 1970s. commodated by an open market purchase to 2 Auernheimer’s proposal is designed to offset changes

FEDERAL RESERVE BANK OF DALLAS 31 ECONOMIC REVIEW FOURTH QUARTER 1995 in the demand for money induced by inflation rate Bacchetta, Phillippe, and Ramon Caminal (forthcoming), changes. Also see Phillippe Bacchetta and Ramon “A Note on Reserve Requirements and Public Finance,” Caminal (forthcoming), who apply the idea to reserve International Review of Economics and Finance. requirement changes. 3 Certainly there are many other services provided by Diamond, Peter A., and James A. Mirrlees (1971), banks, but this one is simple to model and adequate to “Optimal Taxation and Public Production II: Tax Rules,” illustrate the points of this article. Other services of American Economic Review 61 (June): 261–78. banks are implicitly included in x. 4 In the United States, the requirement for checkable Feinman, Joshua N. (1993), “Reserve Requirements: deposits at large banks is currently 10 percent, or History, Current Practice, and Potential Reform,” Federal γ = 0.10. Reserve Bulletin, June, 569–89. 5 More generally, if the economy is growing at the gross

rate n (that is, Nt = nNt –1) and the fiat money stock is Freeman, Scott (1987), “Reserve Requirements and

growing at the gross rate z (that is, Mt = zMt –1), the Optimal Seignorage,” Journal of Monetary Economics 19 gross rate of return on a dollar will be n/z. (March): 307–14. 6 From equation 3, f ′(k) = x – (x – 1) γ < x. 7 The central bank could also buy bonds from private ———, and Joseph H. Haslag (forthcoming), “On the banks, which would then use these funds to invest in Optimality of Interest-Bearing Reserves,” Economic intermediated capital. This scheme is closer to actual Theory. open market purchases but is equivalent in its effects to the direct purchases of capital by the central bank. Friedman, Milton (1959), A Program for Monetary Stability 8 This would not be true if there were an additional (New York: Fordham University Press). demand for fiat money as currency (negotiable notes passed from hand to hand). In most modern econo- Haslag, Joseph H., and Scott E. Hein (1995), “Does It mies, the government retains a monopoly on the Matter How Monetary Policy Is Implemented? Journal of issuance of currency by outlawing its issuance by Monetary Economics 27 (May): 311–26. private banks backed by bank holdings of capital. This is exactly equivalent to a reserve requirement ———, and ——— (1989), “Reserve Requirements, the of 100 percent on currency. Monetary Base, and Economic Activity,” Federal Reserve 9 This is the financing scheme associated with Fried- Bank of Dallas Economic Review, March, 1–16. man’s (1959) proposal and investigated by Smith (1991). Kimbrough, Kent (1989), “Optimal Taxation in a Monetary 10 Auernheimer (1974) describes just such a monetary Economy with Financial Intermediaries,” Journal of policy accommodation scheme in describing the Macroeconomics 11 (Fall): 493 – 511. revenue-maximizing rate of inflation. 11 The idea that taxing all goods improves welfare is Ramsey, Frank P. (1927), “A Contribution to the Theory of discussed in Frank Ramsey’s (1927) rule for efficient Taxation,” Economic Journal 37 (March): 47–61. taxation. According to Ramsey, the government can raise welfare by setting distortionary taxes such that Romer, David (1985), “Financial Intermediation, Reserve the percentage reduction in the quantity demanded of Requirements, and Inside Money,” Journal of Monetary each commodity is the same. In our setting, Ramsey’s Economics 16 (September): 175–94. rule is implemented by taxing both types of capital as opposed to taxing only one type. This result is Sargent, Thomas J., and Neil Wallace (1985), “Interest on demonstrated by Peter Diamond and James Mirrlees Reserves,” Journal of Monetary Economics 15 (May): (1971) in a general setting. Diamond and Mirrlees 279–90. demonstrate that taxing an intermediate input is not part of an optimal policy plan. In a monetary economy, Smith, Bruce, D. (1991), “Interest on Reserves and Kent Kimbrough (1989) shows that the Ramsey tax rule Sunspot Equilibria: Friedman’s Proposal Reconsidered,” applied to final goods improves welfare relative to a Review of Economic Studies 58 (January): 93–105. case in which intermediate goods were taxed. Tolley, George S. (1957), “Providing for the Growth of the References Money Supply,” Journal of Political Economy 65 (Decem- Auernheimer, Leonardo (1974), “The Honest Govern- ber): 477– 84. ment’s Guide to the Revenue from the Creation of Money,” Journal of Political Economy 82 (May/June): 598– 606.

32 Appendix

In this appendix, we show more formally that (A.3) (1 – ␶)x(S – k ) + (1 – ␶)f (k ) paying interest on reserves will make people better > [x (1 – γ) + γ](S – k *) + f (k*). off, even if the interest is financed with a distortionary tax on capital. To do so, we must first calculate We can now use the government budget the tax rate that would be needed to pay the market constraint (equation A.2) to cancel several of the rate of interest on reserves. We let S represent tax terms with terms on the right-hand side of total savings—deposits plus unintermediated equation A.3, leaving us with capital—per young person and use asterisks to in- dicate values of variables in the absence of interest (A.4) –xk + f (k ) > –xk* + f (k*), on reserves. The government must finance net interest on reserves of (x – 1)γ(S – k) from taxes or on the return from savings, ␶x(S – k) + ␶f (k), and from the interest on the capital it acquires from (A.5) x(k * – k ) > f (k *) – f (k). the open market purchase in the initial period, (x – 1)γ [(S – k) – (S* – k*)]. We know that k * > k because unintermediated Altogether, this implies the government capital is taxed when interest is paid on reserves. budget constraint is Because f (.) is a concave function (capital has a diminishing marginal product), (A.1) (x – 1)γ(S – k) = ␶x(S – k) + ␶f (k) + (x – 1)γ[(S – k ) – (S* – k *)], (A.6) f ′(k)(k * – k ) > f (k *) – f (k). or When interest is paid on reserves, we know that the two forms of capital must offer the same (A.2) (x – 1)γ(S* – k *) = ␶x (S – k ) + ␶f (k). marginal rate of return; that is, f ′(k ) = x. It follows that the inequality (equation A.5) is satisfied, Paying interest on reserves makes future proving that future generations are better off generations better off if for any given level of with interest paid on reserves, even if it must be savings, S = S *, the total return net of taxes is financed through a distorting capital tax. greater when interest is paid on reserves:

FEDERAL RESERVE BANK OF DALLAS 33 ECONOMIC REVIEW FOURTH QUARTER 1995