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 from my notes 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. This explains the observation in my notes that money can 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 explanatory linearization, \(\pi_{t+1}=\ln\Pi_{t+1}\) 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. The related paper clarifies this asset label. |
| \(B_t^A\) | 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 forward-guidance note below.
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.
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 photographed slides give the optimality conditions for an illiquid security and a liquid public asset:
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 last equation is an algebraic consequence of the slide 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 this affect spending? In the model, consumption equals output, \(C_t=Y_t\). The Euler equation describes how households divide consumption between today and tomorrow. Substituting the liquidity spread into that equation gives
What does the fraction measure? The market interest rate is \(i_t=i_t^X+S(Y_t,X_t/P_t)\). Its two components are the administered return on public liquidity and the liquidity spread. Consequently,
The numerator \(1+i_t\) is the nominal payoff from saving one currency unit in the benchmark asset for one period. The denominator \(\Pi_{t+1}=P_{t+1}/P_t\) measures the change in prices. Dividing by gross inflation gives the gross real return: how many units of next-period consumption can be obtained by forgoing one unit of consumption today. This return can be uncertain because next-period inflation can be uncertain; that is why it appears inside the expectation in the Euler equation.
Why can more public liquidity encourage spending today? The mechanism has three steps:
- Additional liquidity reduces its scarcity. An increase in \(X_t\) raises real liquidity \(X_t/P_t\) at a given current price level. Holding activity fixed to isolate this initial effect, the spread \(S(Y_t,X_t/P_t)\) falls because \(S_q<0\).
- The market return on saving falls. If the central bank keeps \(i_t^X\) unchanged, a lower spread means a lower \(i_t\). For an unchanged outlook for inflation, postponing consumption yields less additional purchasing power.
- Households have an incentive to consume more now. With the same outlook for future consumption and prices, the reward for giving up consumption today is smaller. Current consumption rises relative to that unchanged future outlook. Since this model has \(C_t=Y_t\), this is an aggregate-demand channel.
How the Euler equation expresses this choice. The symbol \(U_c(C_t)\) is the extra utility from another unit of consumption today, and \(\beta\) is the weight placed on next-period utility. Because the market rate is known at date \(t\), the equation can also be written as
Keep the joint outlook for future consumption and inflation fixed. The expected term then stays unchanged. A fall in \(i_t\) lowers the right-hand side, so \(U_c(C_t)\) must fall as well. With diminishing marginal utility, \(U_{cc}(C_t)<0\), marginal utility falls when consumption increases. This is the precise link from a lower saving return to higher current consumption.
A numerical illustration. Suppose the administered rate remains at 2%, while additional liquidity reduces the spread from 2 percentage points to 1. The market rate falls from 4% to 3%. If next-period inflation is known to be 2%, the gross real return falls from \(1.04/1.02\approx1.0196\) to \(1.03/1.02\approx1.0098\): the net real reward for postponing consumption falls from about 1.96% to 0.98%.
This isolates the spending incentive. In full equilibrium, output, prices, and expectations can also change. Higher current activity itself raises liquidity demand and partly offsets the initial fall in the spread, as the forward-guidance calculation below shows. The two policy instruments act through distinct entries in \(i_t=i_t^X+S(Y_t,X_t/P_t)\): remuneration changes \(i_t^X\), while liquidity supply changes the spread.
The Euler equation and muted forward guidance
Before substituting the liquidity spread, the Euler equation is
The slide’s log-linear representation is
What the hats and coefficients mean. To reproduce the slide’s algebra explicitly, use the following common normalization. It is an explanatory reconstruction; the photographed slide does not state its normalization. A bar denotes a fixed reference equilibrium, with \(\bar C=\bar Y\), \(\bar x=\bar X/\bar P\), and gross reference inflation \(\bar\Pi\):
Both interest-rate deviations use the same denominator \(1+\bar i\). This makes their difference a consistently scaled spread deviation even when the two reference rates differ. The quantity hats are log deviations: for a small change, 0.01 is approximately 1% above the reference value.
Write the opportunity-cost spread as \(s_t=i_t-i_t^X\), with reference \(\bar s=\bar i-\bar i^X\). Evaluating liquidity-demand derivatives at \((\bar Y,\bar s)\), define
Here \(L_Y\) is the partial derivative of liquidity demand with respect to output, and \(L_s<0\) its derivative with respect to the spread. Thus \(d_y\) is an output elasticity, \(d_i\) measures sensitivity to the normalized spread, and \(\sigma\) is the local elasticity of intertemporal substitution. A larger \(\sigma\) means households shift consumption more strongly across time when the real return changes. These definitions give, to first order,
The coefficient becomes easier to understand by solving the local liquidity relation for the market rate and inserting it into the linearized Euler equation:
Substitution places \(-(\sigma d_y/d_i)\widehat{Y}_t\) on the right-hand side. Moving that term to the left and dividing by \(1+\sigma d_y/d_i\) yields the displayed keynote equation. The same term that represents demand-induced spread increases creates the attenuation factor.
For positive sensitivities, the coefficient on expected future output is below one. Suppose news of lower future rates encourages spending today. Higher current activity also increases liquidity demand. At a given real supply, the spread rises and partly offsets the stimulus. This feedback explains the muted forward-guidance effect emphasized in the keynote’s specification.
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. The approximation sign follows the slide; within the preceding pricing equation taken literally, the rearrangement is exact.
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 insight in my notes. 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, \(A_t\). As emphasized in my notes, \(A_t\) 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.
This makes the sharp rise in \(\Delta_t^a\) recorded in my crisis notes meaningful: 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 clarified in Section 4.2 of the related paper. Their sum is the bank’s asset portfolio. The slide adds bank equity \(N_t^A\), so 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, the slide states the competitive condition
With perfect substitutability in liquidity services and both instruments held, \(i_t^a=i_t^X\). The slides then use the effective-wedge representation
Substitution into the Euler equation produces
The initial approximate condition and subsequent equality retain the notation of the keynote. The latter is interpreted within the effective-wedge representation, rather than as an exact derivation from the fuller bank payoff equation.
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 are illustrative rates for a single model period, not estimates from the keynote.
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 is an explanatory consequence of the household conditions, conditional on an equilibrium with sufficient backing and liquidity supply.
My notes also stress 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
The reference to safe assets, fiscal resources, and liquidity premia in my notes brings together three mechanisms. Keeping them separate 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 \(\Delta_t^s<\Delta_t^a\) in my notes 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 leads directly to the question on the final page of my notes: 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 keynote motivates the question; the algebra here does not supply a universal numerical answer.
Sources and interpretation: Pierpaolo Benigno, “Currency, Money and Monetary Policy,” MMF keynote, Lancaster, September 9, 2026, and my notes taken during the presentation. The equations retain the slide notation; additional algebra, numerical examples, and the linearization convention provide explanatory detail. The question about the desirable balance-sheet size is my reflection. Related reading, also used to clarify the private-bank asset labels and backing arrangements: Stablecoins and Central Bank Digital Currencies: Who Supplies Liquidity?