On September 9, 2026, I attended Professor Pierpaolo Benigno’s keynote, “Currency, Money and Monetary Policy,” at the 57th Annual Conference of the Money, Macro and Finance Society in Lancaster. His presentation connected the central bank’s responsibility for the value of the currency with the provision of liquidity by public and private institutions.
A useful starting point is the distinction between currency and money. Money includes instruments accepted for payments, including privately issued deposits. In the keynote’s terminology, public currency supplies the settlement asset in which private monetary promises are discharged. The unit of account is the denomination, such as the pound. These three ideas—denomination, public settlement asset, and private payment claim—need to be kept distinct.
| Concept | Example | Economic meaning |
|---|---|---|
| Unit of account | The pound (£) | The unit in which prices and obligations are stated. |
| Public money | A central-bank-issued £5 note | A public monetary liability denominated in pounds. |
| Private money | A £5 bank deposit | A claim on a commercial bank, denominated in pounds and usable for payments. Its issuer must honor the promised conversion into public money. |
The £5 deposit and the £5 note can have the same nominal purchasing capacity when conversion at face value is credible, although they are liabilities of different issuers. Money can therefore be private while the currency remains publicly anchored.
The inscription reads: I promise to pay the bearer on demand the sum of five pounds.
What does that promise mean today? The Bank of England explains that notes can be exchanged for other Bank of England notes of the same face value. They are no longer convertible into gold.
A £5 commercial-bank deposit is a private claim payable in public money. A £5 Bank of England note is itself public money and a liability of the central bank. Thus, both involve a monetary liability; the issuer and the means of settlement differ. Neither instrument guarantees a fixed quantity of goods. If prices double, the same £5 buys half as much. This is why honoring a nominal promise and maintaining the purchasing power of the currency are separate questions.
Complete notation guide: quantities, rates, preferences, and frictions
The time index \(t\) denotes the current period; \(t+1\) denotes the next. Interest rates apply from \(t\) to \(t+1\). A net rate of 2% is written as \(i_t=0.02\); its gross return is \(1+i_t=1.02\). Superscripts such as \(X\), \(Q\), \(a\), \(s\), and \(A\) identify an instrument or sector; they are labels, not exponents.
| Symbol | Definition and units | Interpretation |
|---|---|---|
| \(C_t\) | Real household consumption at \(t\), in units of the consumption good. | The quantity consumed, rather than expenditure in currency. |
| \(Y_t\) | Real output at \(t\), in the same goods units. | The displayed model imposes \(C_t=Y_t\). |
| \(P_t\) | Price of one unit of the consumption good, in currency units. | \(1/P_t\) is the goods purchasing power of one currency unit. |
| \(X_t\) | Nominal stock of public liquid liabilities. | Held by banks as reserves in the collateral example; supplies household liquidity directly in the public/private-money setup. It is not every item on an actual central-bank balance sheet. |
| \(x_t=X_t/P_t\) | Real public liquidity. | The public stock measured in goods purchasing power. |
| \(Q_t\) | Nominal liquid bank claims in the collateral-multiplier example. | Liabilities of banks and assets of households, backed by \(X_t+B_t\). |
| \(B_t\) | Nominal value of private assets held by banks in that example. | Only the fraction \(\gamma_t\) contributes to pledgeable collateral. |
| \(A_t\) | Nominal bank-issued deposit-like private money outstanding in the later setup. | A liability of the issuer and a liquid claim of its holder. |
| \(S_t\) | Nominal stablecoin claims outstanding. | Private token promises denominated in the public currency. |
| \(q_t\) | Total real liquidity services in the additive setup: \((X_t+A_t)/P_t\), or \((X_t+A_t+S_t)/P_t\) with stablecoins. | Equal weights encode the model’s substitution assumption. In the public-only starting point, real liquidity is simply \(x_t\). |
| \(L(Y,s)\) | Real-liquidity demand as a function of output \(Y\) and the net interest-rate spread \(s=i-i^X\). | The amount of liquidity desired at a given activity level and opportunity cost. |
| \(S(Y,q)\) | Inverse liquidity demand: the spread associated with output \(Y\) and real liquidity \(q\). | A function, distinct from the stablecoin stock \(S_t\). |
| \(S_Y,\ S_q\) | Partial derivatives of \(S\) with respect to output and real liquidity, respectively. | Each derivative holds the other argument fixed; the displayed strict signs concern the region of liquidity scarcity. |
| Symbol | Explicit definition |
|---|---|
| \(i_t\) | Net nominal return on the benchmark illiquid security; this market rate enters the consumption Euler equation. |
| \(i_t^X\) | Net nominal return administered by the central bank on public liquidity \(X_t\). |
| \(i_t^Q\) | Net nominal return paid on bank claims \(Q_t\) in the multiplier example. |
| \(i_t^a\) | Net nominal return paid on deposit-like private money \(A_t\). |
| \(U(C),\ U_c(C),\ U_{cc}(C)\) | Utility from consumption; its first derivative \(dU/dC>0\); and its second derivative \(d^2U/dC^2<0\). Diminishing marginal utility makes \(U_c\) fall as consumption increases. |
| \(V(q),\ V_q(q)\) | Utility from real liquidity services and its derivative \(dV/dq\). The derivative measures the extra benefit of another unit of liquidity. |
| \(\beta\) | Subjective discount factor, with \(0<\beta<1\), measuring the weight on next-period utility. |
| \(E_t[\cdot]\) | Expectation conditional on information available at date \(t\). |
| \(R_{t+1}\) | Nominal stochastic discount factor: \(\beta[U_c(C_{t+1})/U_c(C_t)]P_t/P_{t+1}\). It converts a future nominal payoff into its present valuation; it is not an interest rate. |
| \(\Pi_{t+1}\) | Gross inflation, \(P_{t+1}/P_t\). For example, \(\Pi_{t+1}=1.02\) means prices increase by 2%. |
| \(\pi_{t+1},\ \pi\) | In the linearized model, \(\pi_{t+1}=\ln\Pi_{t+1}\) and \(\pi=\ln\bar\Pi\), the reference log inflation rate. |
| Symbol | Explicit definition |
|---|---|
| \(\rho_t\) | Required collateral coverage per unit of liquid bank claims \(Q_t\); a dimensionless ratio. |
| \(\gamma_t\) | Pledgeable fraction of private assets \(B_t\); also dimensionless. |
| \(\rho_{\gamma,t}\) | Effective public-reserve share \((\rho_t-\gamma_t)/(1-\gamma_t)\), equal to \(X_t/Q_t\) when the constraint binds. |
| \(D_t^A\) | Bank holdings of private securities, in nominal value. |
| \(B_t^A\) | Bank holdings of Treasury securities, in nominal value. This differs from \(B_t\), which denotes private assets in the separate multiplier example. |
| \(N_t^A\) | Bank equity or net worth: assets minus deposit-like liabilities. A positive equity buffer means asset backing exceeds \(A_t\). |
| \(\Psi_{t+1}^A\) | Nominal residual payoff after the displayed payments to depositors and equity holders. |
| \(\mathcal R_{t+1}^A\) | Gross return paid on bank equity. The calligraphic symbol is different from household discount factor \(R_{t+1}\). |
| \(\delta_{t+1}^a\) | Asset-payoff loss or intermediation-cost fraction realized over the period in the bank payoff equation. |
| \(\Delta_t^a\) | Effective banking wedge in the simplified competitive return equation. It measures the proportional gap between the gross market return and the gross return passed to depositors. |
| \(\Delta_t^s\) | Corresponding effective issuance/intermediation wedge for stablecoins. |
The realized loss \(\delta_{t+1}^a\) and the effective pricing wedge \(\Delta_t^a\) play related but distinct roles. Their relation depends on risk pricing, competition, and the treatment of equity; no unconditional equality between them is assumed here. The linearization’s hats, bars, \(d_y\), \(d_i\), and \(\sigma\) are defined explicitly in the mathematical appendix.
The starting point is that a liquid asset offers two benefits: interest income and the ability to make payments. Its holder may accept a lower interest rate because payment services are valuable. This separates the market interest rate, \(i_t\), from the return set by the central bank on public liquidity, \(i_t^X\). The difference is the opportunity cost of holding liquidity.
Let \(X_t\) be nominal public liquidity, \(P_t\) the price level, and \(Y_t\) output. Dividing liquidity by prices gives its purchasing power. Liquidity demand and its inverse are
The function \(L\) tells us how much liquidity people want. The function \(S\) tells us the spread consistent with the available supply. Its subscript \(q\) refers to real liquidity, equal to \(X_t/P_t\) in this initial setup. Thus \(S_Y>0\) means that more activity raises the spread, while \(S_q<0\) means that more real liquidity lowers it, holding the other argument fixed and while liquidity remains scarce. The central bank chooses nominal supply \(X_t\); its real value also depends on prices.
Why is \(S_q\) negative? The derivative \(S_q(Y,q)=\partial S(Y,q)/\partial q\) measures how the interest-rate spread changes when real liquidity increases, holding output \(Y\) fixed. Here, the second argument \(q\) is real public liquidity, \(X_t/P_t\). The negative sign reflects the model’s assumption that additional liquidity provides diminishing marginal payment benefits.
A liquid asset provides payment services as well as interest income. When liquidity is scarce, those payment services are valuable: people are willing to accept a substantially lower interest rate to hold the liquid asset. The gap between the benchmark illiquid return, \(i_t\), and the public-liquidity return, \(i_t^X\), can therefore be large.
As liquidity becomes more abundant at the same level of activity, an additional unit provides less extra help in making payments. People become willing to sacrifice less interest income for that additional unit. The difference between the two assets’ returns narrows. This is the economic intuition behind \(S_q<0\): more real liquidity means a lower marginal payment benefit and a smaller scarcity premium.
For example, suppose the central bank keeps \(i_t^X=2\%\). A reduction in the scarcity premium from 2 percentage points to 1 would lower the market interest rate from 4% to 3%. These numbers illustrate the mechanism, holding the administered return fixed.
The mathematical link to liquidity demand. Write the spread as \(s=i-i^X\). Demand and inverse demand satisfy
A higher spread makes liquidity more costly to hold: holders give up more interest income relative to the benchmark asset. The model therefore assumes \(L_s<0\), where \(L_s=\partial L(Y,s)/\partial s\) is the response of liquidity demand to the spread at fixed output. Substituting inverse demand into demand and differentiating with respect to \(q\), while holding \(Y\) fixed, gives
Consequently, wherever this differentiable inverse exists,
The strict inequality concerns the region where liquidity is scarce. At full liquidity satiation, the marginal payment benefit and the spread can reach zero; further liquidity need not reduce the spread further. The strictly decreasing, invertible demand relationship used above need not continue into that satiated region.
Household optimality and the liquidity return
Let \(U_c(C_t)\) denote marginal utility of consumption, \(V_q(X_t/P_t)\) marginal utility of real liquidity, and \(R_{t+1}\) the nominal stochastic discount factor. The household optimality conditions for an illiquid security and a liquid public asset are:
Read the first condition as follows: the discounted financial payoff just compensates for purchasing the asset. For the liquid asset, compensation includes both the financial payoff and the payment benefit, measured by \(V_q/U_c\). The expectation \(E_t\) uses information available at date \(t\).
To make discounting explicit, the household valuation factor in this setup is
The first term discounts future utility. The marginal-utility ratio makes a payoff more valuable when future consumption is scarce. The price ratio converts future currency into purchasing power. This explains why \(R_{t+1}\) is a valuation factor rather than a financial return.
To compare the two assets, note that the one-period rates \(i_t\) and \(i_t^X\) are known at date \(t\), so they can be taken outside the conditional expectation. The first condition gives \(E_tR_{t+1}=1/(1+i_t)\). Substituting this into the second condition gives
This equation follows algebraically from the two household optimality conditions. A positive liquidity benefit supports a positive opportunity-cost spread, often called a liquidity or convenience yield. It is the return a holder gives up in exchange for payment services. As liquidity becomes abundant, its extra payment benefit falls, and the spread can shrink. This benefit is not a backing asset on the issuer’s balance sheet.
How does an increase in liquidity affect consumption? An increase in liquidity can reduce the reward for postponing consumption. Households then have an incentive to consume more today. In this model, that mechanism works through the interest-rate spread. Consumption equals output, \(C_t=Y_t\), so this is also an aggregate-demand channel.
Start with the household’s Euler equation:
Here, \(C_t\) is consumption today and \(C_{t+1}\) is consumption next period. The function \(U_c\) measures marginal utility: the additional utility from another unit of consumption. The discount factor \(\beta\) gives the weight placed on next-period utility, and \(E_t\) denotes expectations conditional on today’s information. The market interest rate is \(i_t\), while \(\Pi_{t+1}\) is gross inflation.
The equation compares consuming one additional unit today with saving its value and consuming the proceeds tomorrow:
- The left-hand side is the additional utility from consuming today.
- The right-hand side is the expected, discounted additional utility from saving instead. The ratio \((1+i_t)/\Pi_{t+1}\) converts today’s forgone consumption into tomorrow’s purchasing power.
At the household’s optimum, these two marginal benefits are equal. If consuming today offered more marginal utility, the household would want to bring consumption forward. If saving offered more, it would want to postpone consumption.
What does the real-return fraction measure? Forgoing one unit of consumption today frees \(P_t\) currency units to invest. Saving that amount in the benchmark asset pays \(P_t(1+i_t)\) next period. Dividing by the future price \(P_{t+1}\) gives the number of goods the proceeds can buy:
This is the gross real return. Future inflation can be uncertain, so the return remains inside the expectation alongside future marginal utility.
The market interest rate has two components: the administered return on public liquidity, \(i_t^X\), and the liquidity spread:
Substituting this relationship into the Euler equation and using \(C_t=Y_t\) gives
Now suppose public liquidity increases.
First, the liquidity spread falls. With real liquidity \(q_t=X_t/P_t\), the market rate satisfies
At a given price level and level of activity, more public liquidity raises \(q_t\). Because \(S_q<0\), the spread falls. If the central bank keeps its administered rate \(i_t^X\) unchanged, the market rate \(i_t\) falls.
Second, saving becomes less attractive. Holding the outlook for future consumption and inflation fixed, the lower market rate reduces the Euler equation’s right-hand side. At the original consumption level, consuming today now provides more marginal utility than saving for tomorrow.
Third, households increase current consumption. This lowers the marginal utility of current consumption because of diminishing marginal utility:
The left-hand side therefore moves toward the lower right-hand side, restoring the balance between consuming and saving.
For example, with inflation fixed at 2%, a fall in the market interest rate from 4% to 3% reduces the net real return from approximately 1.96% to 0.98%. Giving up consumption today buys less additional consumption tomorrow.
This is the spending incentive isolated by the Euler equation. In full equilibrium, consumption, output, prices, and expectations adjust together. The mechanism also depends on liquidity remaining scarce: once the liquidity spread has reached zero, additional liquidity need not lower the market rate further.
Expectations about future policy affect spending today, but the resulting increase in current activity also raises liquidity demand. At a fixed real liquidity supply, the spread rises and partly offsets the stimulus. The mathematical appendix on the Euler equation and muted forward guidance derives this feedback step by step and illustrates how its strength depends on the timing of an announced rate cut.
Banks introduce another channel: public reserves and pledgeable private assets together support bank-issued money. The multiplier measures how much liquid bank debt can be supported by a given stock of reserves in this particular banking configuration. It does not measure an increase in output or wealth.
| Symbol | Definition | Concrete interpretation |
|---|---|---|
| \(X_t\) | Public reserves held by banks at date \(t\). | An asset for banks, issued by the central bank; each unit counts fully as collateral. |
| \(B_t\) | Private assets held by banks. | These supply additional backing, but only part of their value can be pledged. |
| \(Q_t\) | Liquid claims issued by banks and held by households. | Bank liabilities that households can use for payments, such as deposits. |
| \(\rho_t\) | Required pledgeable collateral per unit of \(Q_t\). | If \(\rho_t=0.4\), each £1 of claims requires £0.40 of eligible collateral. This is a collateral requirement, not a requirement to hold £0.40 in reserves alone. |
| \(\gamma_t\) | The pledgeable fraction of \(B_t\). | If \(\gamma_t=0.2\), £1 of private assets contributes £0.20 of eligible collateral. The other £0.80 does not count toward this constraint; that does not mean it is worthless. |
The quantities \(X_t\), \(B_t\), and \(Q_t\) are measured in the same nominal currency units; \(\rho_t\) and \(\gamma_t\) are dimensionless ratios. The simplified bank balance sheet and its collateral constraint are
The first equation abstracts from bank equity. All claims are matched by assets in this accounting identity, even if only part of those assets qualifies as collateral. The second condition limits how many claims those assets can support. A binding constraint means eligible collateral exactly equals the required amount: the bank has no spare collateral capacity. In that case, the keynote’s multiplier is
The notation \(\rho_{\gamma,t}\) denotes a single effective reserve share, calculated from the two parameters \(\rho_t\) and \(\gamma_t\). It is not their product. With binding collateral,
Thus, if \(\rho_{\gamma,t}=0.25\), reserves must provide £0.25 for every £1 of liquid claims, with private assets providing the remaining £0.75 on the balance sheet. Taking the inverse, \(Q_t/X_t=4\): £1 of reserves can support £4 of claims alongside the necessary private assets. The multiplier does not mean £1 of reserves alone backs £4, nor does it create net wealth: the issued claims are liabilities for banks and assets for households.
Higher required collateral coverage, \(\rho_t\), increases the reserve share. Lower private-asset pledgeability, \(\gamma_t\), also increases it when \(\rho_t<1\). In either case, the same reserve stock supports fewer claims. The equality describes issuance when the constraint binds; otherwise it gives an upper bound, and actual issuance may be lower.
Collateral backing, the multiplier, and deposit spreads
Derivation, one step at a time. The balance sheet gives \(B_t=Q_t-X_t\). Substitute this into the collateral constraint:
For \(\rho_t>\gamma_t\), division by \(\rho_t-\gamma_t\) preserves the inequality:
This is the maximum stock of claims compatible with this constraint at the given reserve stock and parameters, assuming the corresponding private assets can be held. When the constraint binds, equality holds:
With nonnegative asset positions, the binding case considered here is \(0\leq\gamma_t<\rho_t\leq1\), with \(\gamma_t<1\). When \(\rho_t<1\), the multiplier exceeds one; when \(\rho_t=1\), \(Q_t=X_t\). If \(\gamma_t\geq\rho_t\), this constraint no longer imposes this finite upper bound on claims: the displayed binding formula is not applicable. Other constraints or demand would then matter.
For a given price level and reserve stock, a larger \(\rho_t\), or a lower \(\gamma_t\) when \(\rho_t<1\), reduces real private liquidity. The deposit-pricing equations shown in the keynote are
The return on bank claims is \(i_t^Q\). Holding the market–public-liquidity spread fixed, tighter collateral requirements widen the deposit spread. Taking the preceding pricing equation as exact also makes the second relation exact by rearrangement.
A numerical example. Suppose reserves equal 20, required coverage is 40%, and 20% of private assets are pledgeable:
Interpreting the amounts as £ million, the bank holds £20 million in reserves and £60 million in private assets, against £80 million in liquid claims. Verify both conditions:
Total assets equal liabilities, and eligible collateral exactly meets the requirement. The reserve share is \(20/80=0.25\), and the multiplier is \(80/20=4\).
Now suppose private assets become unpledgeable: \(\gamma_t\) falls from 0.2 to zero, while \(\rho_t=0.4\). The effective reserve share rises to 0.4 and the multiplier falls to \(1/0.4=2.5\). With reserves unchanged at £20 million, the maximum claims supported by this constraint fall to \(20/0.4=50\) million. This is a fall in feasible issuance capacity; it does not describe how outstanding deposits adjust during the crisis.
To support the original £80 million of claims when private assets cannot count as collateral, reserves must instead reach \(0.4\times80=32\) million. With claims held at 80 and no equity, the corresponding balance sheet is \(32+48=80\): £12 million more reserves and £12 million fewer private assets. This shows how public liquidity can replace lost private collateral capacity, subject to the model’s other conditions.
The crisis mechanism. A financial crisis can reduce the amount of private collateral that lenders accept. Here that means a fall in \(\gamma_t\), possibly combined with a rise in required coverage \(\rho_t\). Both reduce the claims supported by each reserve unit. The central bank can respond by supplying more \(X_t\). The example is a balance-sheet comparison at fixed prices and target claims, not a complete transition path for output or inflation.
A further setup lets households obtain liquidity services directly from both public money and bank deposits, \(A_t\). Here, \(A_t\) is private money: the nominal value of deposits issued by banks, which are liabilities for their issuers and assets for their holders. Their combined real liquidity is
When both are held and provide identical services, their returns coincide: otherwise holders would prefer the higher-paying instrument. The effective banking wedge \(\Delta_t^a\) measures the proportional share of the gross market return that is not passed to depositors because of intermediation frictions. In the keynote’s simplified competitive representation, it links the rates:
The deposit return is \(i_t^a\). For \(0\leq\Delta_t^a<1\) and a positive gross administered return, a larger wedge means a smaller denominator, hence a higher market rate. For given expected consumption and inflation, saving becomes more attractive and current spending weakens. Banking frictions can thus tighten monetary conditions even without a policy-rate change.
A sharp rise in \(\Delta_t^a\) during a crisis shows how impaired private intermediation can itself worsen monetary conditions. The central bank’s response can involve the remuneration of public liquidity, its quantity, or both, depending on which constraints and instruments remain active.
Private-money backing and the banking wedge
This setup differs from the preceding multiplier example. There, reserves sit on bank balance sheets and households hold \(Q_t\). Here, \(X_t+A_t\) measures public and private instruments supplying liquidity services directly to holders. Keeping the configurations distinct prevents double-counting reserves. The private-money bank balance sheet is
Here \(D_t^A\) denotes bank holdings of private securities and \(B_t^A\) holdings of Treasury securities, as in Section 4.2 of Stablecoins and Central Bank Digital Currencies: Who Supplies Liquidity?. Their sum is the bank’s asset portfolio. Including bank equity \(N_t^A\) means that assets equal deposit liabilities plus net worth. Notice that \(B_t^A\) denotes Treasury securities here, whereas \(B_t\) denotes private assets in the earlier multiplier example. The bank’s payoff after remunerating deposits and equity is
Read the payoff in three parts: asset income after intermediation losses; minus the payment to depositors; minus the payment to equity holders. The loss wedge is \(\delta_{t+1}^a\), while \(\mathcal{R}_{t+1}^A\) is the gross return paid on equity. Abstracting from equity returns for the marginal deposit gives the approximate competitive condition
With perfect substitutability in liquidity services and both instruments held, \(i_t^a=i_t^X\). The corresponding effective-wedge representation is
Substitution into the Euler equation produces
The equality defines the simplified effective-wedge representation. Its relationship to the fuller bank payoff equation depends on the assumptions governing intermediation costs, risk, and equity remuneration.
For example, an administered return of 2% and a 1% effective wedge imply \(i_t=1.02/0.99-1\approx3.03\%\). The wedge adjusts the gross payoff; it is not simply added as a percentage-point spread.
What a crisis shock does. In this simplified representation, the derivative at a fixed administered return is
Moving the effective wedge from 1% to 3%, with \(i_t^X=2\%\), raises the implied market rate from about 3.03% to \(1.02/0.97-1\approx5.15\%\). These numerical examples use illustrative rates for a single model period.
How quantity intervention differs. The fixed-wedge equation \(1+i_t=(1+i_t^X)/(1-\Delta_t^a)\) contains no \(X_t\). Within that particular equilibrium, increasing public liquidity alone cannot algebraically remove a fixed wedge. Quantity intervention works through the collateral mechanism above, or by replacing private claims and changing which instruments are held. Public supply can support payments even as some private issuance retreats.
For example, in the additive specification \(q_t=(X_t+A_t)/P_t\), replacing a loss of 10 units of \(A_t\) with 10 units of \(X_t\) preserves total nominal liquidity at unchanged prices. This is an accounting illustration conditional on those quantity changes; private issuance and prices generally respond in equilibrium. It does not establish that the banking wedge disappears.
The frictionless benchmark delivers the striking possibility discussed in the keynote:
Why does this relate to full liquidity satiation? With equal financial returns, the household optimality condition implies a zero marginal liquidity benefit. In the private-money setup, this is \(V_q(q_t)=0\): liquidity needs are fully met at the margin. Reaching this point requires enough appropriately backed supply. Private provision can help achieve it without a large central-bank balance sheet.
Why equal returns imply a zero marginal liquidity benefit
Apply the earlier household calculation to total real liquidity \(q_t=(X_t+A_t)/P_t\). Under the maintained assumption that both instruments are held and supply the same liquidity services,
When \(i_t=i_t^X\), the numerator on the right is zero. With positive marginal utility of consumption and a positive gross market return, the implication is
This is the economic meaning of satiation: an additional unit no longer improves payment or liquidity services at the margin. The utility specification must permit \(V_q(q)=0\) at an attainable quantity; diminishing marginal benefits alone need not imply satiation at a finite quantity. It does not mean unlimited credit or the disappearance of every financial friction. The equality follows from the household conditions in an equilibrium with sufficient backing and liquidity supply.
A further issue is backing and real value. A private promise to pay must be supported by assets and, where needed, loss-absorbing equity. For public liabilities, the monetary and fiscal arrangements matter for purchasing power. The liquidity benefit explains why a claim is useful for payments; it is not an extra asset that can repay creditors.
Safe assets, fiscal resources, and liquidity premia: three distinct roles
Safe assets, fiscal resources, and liquidity premia play three distinct roles. Distinguishing them helps explain what issuing money can achieve.
| Question | Relevant mechanism | Explicit interpretation |
|---|---|---|
| Will the private issuer deliver the promised currency amount? | Asset backing and equity | Assets provide resources for repayment. Equity can absorb losses before they impair the liquid claim. A label such as “deposit” or “stablecoin” does not itself supply those resources. |
| What goods will that currency amount purchase? | The nominal anchor and public backing arrangements | The price level \(P_t\) determines purchasing power. Public asset income and, in regimes allowing it, fiscal support help sustain the monetary arrangement. Fiscal resources mean, for example, future primary surpluses: tax revenue less non-interest public spending. |
| Why hold a liquid claim when another asset pays more interest? | The liquidity premium | Payment services create a nonfinancial benefit, \(V_q/U_c\), allowing a lower financial yield. That service value is distinct from the asset portfolio used to honor redemption. |
Nominal repayment and real purchasing power differ. Suppose a £5 deposit is repaid in full. If a consumption good costs £2, the payment buys 2.5 units. If the good’s price rises to £4, the same £5 buys only 1.25 units. The bank has honored the nominal promise, yet the currency’s purchasing power has fallen. This is why liquidity provision and price-level control must be analyzed together.
The related paper distinguishes backing through assets from backing through fiscal capacity. This does not mean every private claim is guaranteed by the state. It identifies alternative institutional arrangements for supporting public liquidity and its value. Nor does expanding \(X_t\) create real goods: it changes the supply and composition of nominal claims, subject to the relevant balance-sheet and policy constraints.
A useful way to read the full-satiation result is therefore: efficient private issuance can reduce the central bank’s direct role in supplying payment instruments, while the backing requirement remains somewhere in the financial system.
Stablecoins add a third source:
The first equation adds stablecoins, \(S_t\), to real liquidity under the model’s additive specification. The inequality describes a case in which their wedge, \(\Delta_t^s\), is lower than banks’ wedge. With safe backing, they can then supply liquidity more efficiently. They remain promises to deliver currency. The quantity \(S_t\) is distinct from the spread function \(S(\cdot)\).
What the stablecoin inequality means
The inequality \(\Delta_t^s<\Delta_t^a\) compares proportional intermediation wedges, not the quantities of stablecoins and bank deposits. It says that the stablecoin issuer loses or absorbs a smaller share of the benchmark gross return in providing the liquid claim. With sufficient safe backing and the relevant competitive conditions, this can make private liquidity provision more efficient.
It is a conditional model comparison, not a claim that all stablecoins are safer or cheaper than bank deposits. Safety of backing, convertibility, and operating costs still matter. Likewise, \(X_t+A_t+S_t\) lists the possible liquidity sources; positive holdings of all three require compatible returns and substitution conditions. The identity alone does not guarantee their coexistence.
The institutional idea is that private issuers can provide payment instruments while the central bank anchors the currency’s value. A small normal-times balance sheet can coexist with a capacity to expand public liquidity during stress. This raises a central policy question: how small can the central-bank balance sheet be, and how small should it be?
Three different balance-sheet questions
| Question | What would need to be established? |
|---|---|
| How small can it be? | The smallest feasible public-liquidity supply consistent with the specified equilibrium, payment needs, backing, and nominal anchor. A result permitting a small positive supply does not by itself identify a unique positive minimum. |
| How small should it be in normal times? | The size that maximizes welfare after accounting for liquidity services, intermediation costs, balance-sheet risks, and the institutional costs of public or private provision. |
| How much should it be able to expand in a crisis? | The capacity to supply additional public liquidity when private money becomes costly or collateral loses pledgeability, together with the assets or fiscal arrangements supporting that response. |
These are questions about the central bank’s balance sheet. The variable \(X_t\) captures public liquidity liabilities in the model; it is not a complete accounting measure of every central-bank asset and liability. The displayed liquidity and Euler equations establish useful mechanisms, but a numerical minimum or a welfare-optimal balance-sheet size requires the remaining policy, backing, and welfare conditions. The answer depends on the policy regime, financial frictions, and backing arrangements.
Mathematical appendix: the Euler equation and muted forward guidance
This appendix explains why a given increase in expected future output can produce a smaller increase in current output when liquidity is scarce. The mechanism is a feedback through interest rates: higher current spending raises payment needs; with real liquidity held fixed, the liquidity spread rises; the higher market interest rate partly offsets the initial incentive to spend.
The derivation proceeds from two equations: the household’s Euler equation and the demand for real liquidity. We first approximate each equation around a reference equilibrium, then combine them. Each approximation and normalization is stated explicitly.
A.1. The two starting equations and their variables
| Symbol | Meaning |
|---|---|
| \(t\), \(t+1\) | The current period and the following period. |
| \(C_t\), \(Y_t\) | Real consumption and real output. The model imposes \(C_t=Y_t\), so the Euler equation can be written using output. |
| \(U_c\), \(U_{cc}\) | The first and second derivatives of consumption utility. We assume \(U_c>0\) and \(U_{cc}<0\): additional consumption is valuable, with diminishing marginal utility. |
| \(\beta\), \(E_t\) | The subjective discount factor, with \(0<\beta<1\), and expectation conditional on information available at date \(t\). |
| \(i_t\), \(i_t^X\) | The net nominal market interest rate and the net nominal rate administered on public liquidity. A 4% rate is written as 0.04. |
| \(s_t\) | The interest-rate spread \(i_t-i_t^X\): the interest income forgone by holding public liquidity instead of the benchmark asset. |
| \(X_t\), \(P_t\), \(x_t\) | Nominal public liquidity, the price level, and real public liquidity \(X_t/P_t\). The quantity \(x_t\) is the same public-liquidity quantity denoted by \(q_t\) in the preceding Euler-equation intuition. |
| \(\Pi_{t+1}\) | Gross inflation, \(P_{t+1}/P_t\). Gross inflation of 1.02 means prices rise by 2%. |
| \(L(Y,s)\) | Demand for real liquidity, with partial derivatives \(L_Y=\partial L/\partial Y\) and \(L_s=\partial L/\partial s\). In the scarcity region, \(L_Y>0\) and \(L_s<0\): output raises payment needs, while a higher opportunity-cost spread reduces liquidity demand. |
The exercise uses a first-order approximation around a deterministic steady state. “First order” means retaining terms proportional to small deviations and dropping products of those deviations. The reference state has constant real output, consumption, and real liquidity. Nominal prices can still grow at the reference inflation rate.
A.2. Define the reference state and the deviations
A bar denotes a reference value. Assume \(\bar Y>0\), \(\bar x>0\), \(\bar\Pi>0\), and \(1+\bar i>0\), so the logarithms and return normalizations below are well defined. Constant reference consumption implies the steady-state Euler condition:
The reference liquidity condition is \(\bar x=L(\bar Y,\bar s)\), where \(\bar s=\bar i-\bar i^X\) and \(\bar C=\bar Y\). Equation (A.3) follows by substituting constant consumption into (A.1) and cancelling the same marginal utility on both sides.
Output and liquidity: define their proportional deviations using logarithms:
A hat of 0.01 represents approximately 1% above the reference value. More precisely, the percentage change is \(100(e^{0.01}-1)\), about 1.005%. For the small changes considered here, the approximation is convenient.
Interest rates: use changes in net rates divided by the same reference gross market return:
The common denominator is important. It makes the difference between the two normalized rates equal to the normalized change in the spread:
These rate hats are defined as scaled changes, rather than exact logarithms. In particular, the administered-rate hat uses \(1+\bar i\), even if \(\bar i^X\) differs from \(\bar i\). If \(\bar i=0.04\), a one-percentage-point rate increase gives a normalized change of \(0.01/1.04\approx0.009615\).
Inflation: define log inflation and its deviation from the reference rate:
The tilde is a convenient label for the inflation deviation. Thus \(E_t\widetilde\pi_{t+1}=E_t(\pi_{t+1}-\pi)\), the expected log inflation rate relative to the reference rate.
A.3. Approximate the Euler equation, one component at a time
Define the positive coefficient
This is the local elasticity of intertemporal substitution. It measures how strongly households shift consumption between periods when the real return changes. A larger \(\sigma\) gives a stronger response. For logarithmic utility, \(U(C)=\ln C\), we have \(U_c=1/C\), \(U_{cc}=-1/C^2\), and therefore \(\sigma=1\).
First, approximate marginal utility. A first-order Taylor expansion around \(\bar Y\) gives
For small changes, \(Y_t-\bar Y\approx\bar Y\widehat Y_t\). Divide by \(U_c(\bar Y)\) and use (A.8):
The negative sign has an economic meaning: higher output means higher consumption and therefore lower marginal utility. The same expansion applies to next-period marginal utility.
Second, express the nominal return. The definition in (A.5) gives the exact identity
Third, approximate the inflation adjustment. From (A.7),
Inflation above its reference rate reduces the purchasing power of a given nominal payoff.
Combine the three components. Divide (A.1) by \(U_c(\bar Y)\), substitute (A.9)–(A.11), and use \(\beta(1+\bar i)/\bar\Pi=1\):
Expanding the product and retaining only first-order terms gives
For example, the product \(\widehat i_t\widehat Y_{t+1}\) is dropped because it multiplies two small deviations. The market rate is known at date \(t\), so \(E_t\widehat i_t=\widehat i_t\). The expectation is taken after expanding the payoff: this calculation does not replace an expected logarithm with the logarithm of an expectation.
Cancel the constant 1 on each side, then multiply by \(-\sigma\). From this point, an equality sign denotes equality within the first-order linearized system. The result is
Read (A.14) in two parts. Higher expected future output encourages consumption today. A higher market interest rate relative to expected inflation encourages postponing consumption, reducing current output relative to that future outlook. Risk remains present in the exact equation (A.1); products of deviations and their covariance effects are beyond this first-order expansion around the deterministic reference.
A.4. Approximate liquidity demand
Expand \(x_t=L(Y_t,s_t)\) around \((\bar Y,\bar s)\):
Divide by \(\bar x\). Use \((x_t-\bar x)/\bar x\approx\widehat x_t\), \((Y_t-\bar Y)/\bar Y\approx\widehat Y_t\), and the exact spread identity (A.6):
Define two positive coefficients:
- \(d_y\) is the elasticity of liquidity demand with respect to output, holding the spread fixed. For example, \(d_y=1\) means a 1% rise in output raises desired real liquidity by approximately 1%.
- \(d_i\) measures the sensitivity of liquidity demand to the normalized interest-rate spread. It is positive because the definition includes a minus sign and \(L_s<0\). It is not an elasticity with respect to the logarithm of the spread.
The linearized liquidity equation is therefore
A.5. Solve for the market interest rate
Move the interest-rate term in (A.17) to the left:
Divide by \(d_i\), then add \(\widehat i_t^X\):
This equation separates three influences on the market rate. The administered rate contributes directly. Higher output raises payment needs and the spread. More real liquidity reduces scarcity and the spread. In particular, holding real liquidity and its administered return fixed, a rise in current output raises the market interest rate. The coefficient \(d_y/d_i\) measures the strength of that response in the normalized units.
A.6. Substitute into the Euler equation and isolate current output
Replace \(\widehat i_t\) in (A.14) with the expression in (A.18):
Multiply each term inside the brackets by \(-\sigma\):
The key term is \(-\sigma(d_y/d_i)\widehat Y_t\). Current output appears on the right because higher spending raises the market rate, which in turn restrains spending. Move this term to the left:
Finally, divide by \(1+\sigma d_y/d_i\). Define
The resulting aggregate-demand equation is
Because \(\sigma>0\), \(d_y>0\), and \(d_i>0\), the denominator in (A.22) exceeds one. Hence \(0<\lambda<1\). Recalling \(\widetilde\pi_{t+1}=\pi_{t+1}-\pi\), equation (A.23) is the same expression written using the expected inflation deviation.
A.7. Understand the attenuation coefficient with numbers
Suppose, for illustration, that \(\sigma=1\), \(d_y=1\), and \(d_i=2\). Then
Hold the administered rate, expected inflation, and real liquidity at their reference values, so their deviations are zero. Suppose expected next-period log output rises by 0.01, approximately 1%. Equation (A.20) becomes
Moving the feedback term to the left gives
Current output therefore rises by approximately 0.67%. The direct effect of stronger expected future activity encourages spending, while the resulting increase in the liquidity spread offsets part of it. With the market rate fixed in (A.14), the corresponding conditional response would be approximately 1%.
| Change | Effect on \(\lambda\) | Economic reason |
|---|---|---|
| Higher \(d_y\) | Lower \(\lambda\) | Output creates a larger increase in liquidity demand, requiring a stronger spread adjustment at fixed supply. |
| Higher \(d_i\) | Higher \(\lambda\) | Liquidity demand responds more strongly to the spread, so a smaller rate increase is needed to accommodate the same increase in payment needs. |
| Higher \(\sigma\) | Lower \(\lambda\) | Consumption responds more strongly to the interest-rate increase generated by higher current activity. |
The last row concerns the attenuation coefficient. The effect of \(\sigma\) on the response to a policy-rate change also includes the direct factor \(\sigma\) in (A.23); it cannot be inferred from \(\lambda\) alone.
A.8. Why the timing of forward guidance matters
Forward guidance changes expectations about future policy rates. To isolate how the horizon affects transmission, consider a fully anticipated rate cut lasting one period at date \(t+h\). The integer \(h\geq0\) is the number of periods until the cut; \(h=0\) means a cut today.
For this deterministic conditional experiment, assume there are no other shocks, so the anticipated paths are realized. Keep the real-liquidity and inflation paths at their reference values. Set all other administered-rate deviations to zero, and assume output returns to its reference value after the cut. Let the normalized rate reduction be \(\varepsilon>0\), so \(\widehat i^X_{t+h}=-\varepsilon\). The announcement is already known at date \(t\).
At the date of the cut, next-period output is at its reference value. Equation (A.23) gives
One period before the cut, for \(h\geq1\), there is no current policy-rate deviation. The effect operates through expected next-period output:
Each additional step backward contributes another factor \(\lambda\). After \(h\) such steps, the effect today is
The factor \(\lambda\) appears once at the date of the cut and another \(h\) times as its effect is transmitted back to today. Relative to an otherwise identical cut today, the additional attenuation from announcing the cut \(h\) periods ahead is therefore
For a numerical illustration, retain \(\sigma=1\) and \(\lambda=2/3\), and set the reference market rate to 4%. A one-percentage-point administered-rate cut corresponds to \(\varepsilon=0.01/1.04\). The conditional responses are:
| Timing of the cut | Current log-output response, expressed as an approximate percentage | Response relative to the same cut today |
|---|---|---|
| Today: \(h=0\) | 0.641% | 100% |
| One period ahead: \(h=1\) | 0.427% | 66.7% |
| Four periods ahead: \(h=4\) | 0.127% | 19.8% |
The figures are illustrative calculations from (A.25), with model periods left unspecified. They show how repeated liquidity-demand feedback reduces the current effect of a more distant announced rate cut under the stated assumptions.
A.9. The scope of the result
Real liquidity is held fixed when isolating the feedback. Holding nominal \(X_t\) fixed does not hold \(x_t=X_t/P_t\) fixed if prices change. A policy that expands real liquidity alongside activity can weaken the spread increase. For example, in (A.18), \(\widehat x_t=d_y\widehat Y_t\) cancels the output-induced contribution to the spread.
The approximation applies locally in the region of liquidity scarcity. It requires smooth liquidity demand with finite \(L_s<0\), positive \(d_i\), and a locally invertible demand relationship. Full satiation can change these conditions, so the scarcity-region derivation should not be mechanically extended beyond them.
The horizon calculation isolates one transmission mechanism. In a complete equilibrium, inflation, liquidity supply, and expectations can respond to policy. Determining the overall effect requires the rest of the model and its policy rules. The result established here is the source of the attenuation term: higher current activity raises the liquidity spread, which partly offsets the incentive to spend.
References: Pierpaolo Benigno, “Currency, Money and Monetary Policy,” keynote at the 57th Annual Conference of the Money, Macro and Finance Society, Lancaster, September 9, 2026. Related reading: Stablecoins and Central Bank Digital Currencies: Who Supplies Liquidity?. Banknote illustration and explanation: Bank of England £5 note and Banknote FAQs.