E42 | Macro Paper Warehouse

Cash or card? A structural model of payment choices

Mon, 01 Jan 0001 00:00:00 +0000

Lippi and Moracci (2026) ask how euro area households choose between cash and card payments, and whether existing theoretical models can explain observed behavior. They draw on ECB payment diary surveys (SUCH and SPACE waves I–III, 2015–2024) covering transaction-level records that include purchase size, payment method chosen, cash on hand before each transaction, and merchant acceptance of cards. This granular data allows the authors to isolate unforced payment choices — transactions in which the consumer had sufficient cash, the merchant accepted cards, and the consumer held a card — from mechanically constrained ones.

The authors document three empirical patterns. First, roughly 39% of individuals in the sample violate the simple transaction-size threshold rule of Whitesell (1989): their largest unforced cash payment exceeds their smallest unforced card payment. Second, between 27% and 49% of unforced transactions are settled by card across survey waves, contradicting the “cash burns” policy of Alvarez and Lippi (2017) under which cards are used only when cash is exhausted. Third, and most novel, the probability of card use rises sharply as implied residual cash holdings (m′ = m − s) approach zero — that is, when a cash payment would nearly deplete the wallet. This suggests a precautionary motive: consumers maintain a cash buffer to cover purchases at merchants who do not accept cards.

To rationalize these facts, the authors build an inventory-theoretic model with a compound Poisson expenditure flow (random arrival times and random transaction sizes drawn from a lognormal distribution), imperfect card acceptance (fraction ϕ of merchants accept cards, set at 0.89 for 2023–24), a fixed cost b per cash withdrawal, a fixed cost κ per card transaction (sign unrestricted), and a utility penalty u per missed purchase. The optimal policy takes an (s,S) form for withdrawals and a state-dependent threshold for payment choice. When 0 < κ < b, the agent uses cards for purchases large enough that paying cash would push balances below a threshold m̃, thereby avoiding a costly withdrawal or the risk of missing a future purchase. The critical transaction size above which cards are used, s(m), rises with cash on hand, generating the interaction the data reveals.

The model is calibrated by minimum distance to four moments from the 2023–24 SPACE wave: average cash balances relative to daily expenditure, annual withdrawal frequency, the unforced card expenditure share, and realized purchase frequency. The estimated annual cost of managing consumption transactions for the average euro area household is approximately 15 euros — a remarkably small burden. Three counterfactual experiments quantify welfare implications. Removing card access raises the annual cost from 15 to about 50 euros, implying a card ownership value of roughly 35 euros per year. Near-universal card acceptance (ϕ = 0.99) reduces the annual cost by nearly 75%, from 15 to about 4 euros, while average cash holdings fall from 130% to about 20% of daily expenditure. A complete ban on cash would cost the average consumer approximately 60 euros per year more than the current mixed system. A cashless equilibrium requires both near-universal acceptance (ϕ above 99%) and card costs at or below zero (κ ≤ 0); neither condition alone is sufficient given the estimated magnitude of the missed-purchase cost u.

Q: What is the central empirical puzzle the paper addresses? A: Existing models predict either a pure transaction-size threshold (Whitesell 1989) or a pure cash-burns rule (Alvarez and Lippi 2017). The data shows both rules are violated: 39% of individuals with observed unforced transactions of both types violate the threshold rule, and 27–49% of unforced transactions are paid by card despite available cash. Neither model alone accounts for the novel finding that card usage spikes precisely when a cash payment would nearly exhaust the wallet.

Q: What data does the paper use and what is its key advantage? A: The authors use ECB payment diaries from four survey waves: SUCH (2015–16) and SPACE I, II, III (2019, 2021–22, 2023–24). For each transaction the diary records payment method, purchase size, and cash on hand, along with merchant acceptance of each payment method. Critically, the combined information on cash holdings and acceptance allows the authors to distinguish forced from unforced payment choices, which is essential for identifying the behavioral determinants of payment method selection.

Q: What is the novel empirical fact the paper contributes? A: The paper documents that the probability of card use increases sharply as implied residual cash (m′ = m − s) approaches zero. This pattern holds across all survey waves. It is consistent with a precautionary motive: consumers use cards to avoid depleting a cash buffer that provides insurance for encounters with merchants who do not accept cards.

Q: How does the theoretical model generate the precautionary motive for cash? A: Cards are accepted in only fraction ϕ of stores; when a merchant does not accept cards and the consumer lacks cash, the purchase is missed at utility cost u. This creates an incentive to maintain positive cash balances. Combined with a fixed withdrawal cost b and a fixed card cost κ, the agent optimally targets a cash level m* and withdraws before the wallet empties (trigger m̄ > 0), holding a buffer against card-rejection events.

Q: What is the key proposition characterizing the optimal payment policy? A: Proposition 1 establishes three regimes. When κ ≤ 0, the card always dominates and is used for all purchases. When κ ≥ b, cash always dominates and cards are used only for forced transactions. In the intermediate case 0 < κ < b, a threshold m̃ ∈ (m̄, m*) divides behavior: for m < m̃ the agent uses cash for all transactions; for m ≥ m̃ the agent uses a card for any purchase exceeding a size threshold s(m), where s(m) is increasing in m. The threshold s(m) distinguishes this policy from Whitesell (1989)’s fixed threshold.

Q: How does the payment threshold s(m) vary with cash on hand, and why? A: s(m) is the purchase size above which the value loss from paying cash — pushing the agent closer to m̄ and raising the probability of a missed purchase or costly withdrawal — exceeds the fixed card cost κ. As m rises, a larger cash payment is needed to trigger this concern, so s(m) increases. This means card use becomes less frequent as cash balances grow for most of the state space, consistent with the empirical finding that cash probability rises with cash on hand.

Q: What are the calibrated parameter values and what do they imply? A: The withdrawal cost b is estimated at 0.003 EUR — very small. The per-transaction card cost κ is about 60% of b, meaning cards are cheaper to use per transaction than visiting an ATM. The cost of a missed purchase u is approximately 1 EUR. The arrival rate λ is calibrated so that about 2% of purchase opportunities are missed under the estimated card acceptance rate of 0.89. These values imply that the payment system imposes a small but non-trivial welfare burden, concentrated in the precautionary costs of maintaining cash.

Q: What is the estimated annual cost of managing consumption transactions? A: Under the optimal policy for 2023–24 parameters, the annual cost C is approximately 15 euros per household. This decomposes into opportunity costs of holding cash (RM), withdrawal costs (bn), card usage costs, and the disutility from missed purchases. The authors characterize this as “remarkably small,” suggesting the current payment system is relatively efficient from the household’s perspective.

Q: How does this cost compare across demographic groups and over time? A: Until 2019 the estimated annual cost was around 20 euros; it stabilized around 15 euros from 2021–22 onward, with the decline driven primarily by households holding less cash in the post-pandemic period. Across age groups, education levels, income brackets, and gender, each subgroup faces a very similar cost as a proportion of their expenditure, indicating limited distributional variation in payment system costs.

Q: What is the welfare value of owning a payment card? A: Setting ϕ = 0 (cash-only economy), the annual cost rises from 15 to approximately 50 euros. The value of card ownership is therefore approximately 35 euros per year. The savings come primarily from lower opportunity costs of holding cash (since card access reduces the precautionary motive) and lower disutility from missed purchases; withdrawal cost reductions play a negligible role.

Q: What happens under near-universal card acceptance (ϕ = 0.99)? A: Average cash holdings fall from about 130% of daily expenditure to about 20% of daily expenditure, a reduction of approximately 110 percentage points. The unconditional card expenditure share rises by 17 percentage points to about 93%, mostly through an increase in forced card transactions (agents more often lack cash). Unforced card expenditure falls by about 10 percentage points because the precautionary motive for using cards — preserving a cash buffer — weakens when acceptance is near-universal. The annual management cost falls by nearly 75%, from 15 to approximately 4 euros.

Q: Under what conditions does a cashless economy emerge? A: The model identifies two jointly necessary conditions: card acceptance near universal (ϕ above 99%) and card costs at or below zero (κ ≤ 0). Raising ϕ alone from the estimated 0.89 to 0.99 reduces cash use substantially but does not eliminate it, because the estimated cost of missed purchases u is large enough that consumers still maintain a small cash buffer. For κ ≤ 0, cash holdings M/e are insensitive to κ and depend only on ϕ. With current card usage costs, even near-universal acceptance would not produce a cashless economy.

Q: What is the cost of a complete cash ban? A: Under a cashless policy, the annual cost is approximately 75 euros — about 5 times the 15-euro baseline and about 25 euros more than the cash-only cost of 50 euros. A complete ban on cash would increase transaction management costs by approximately 60 euros per year for the average consumer. This is because at ϕ = 0.89, nearly 11% of purchase encounters would result in missed transactions.

Q: How does card acceptance affect cash management in the model and data? A: As ϕ falls, the precautionary motive for holding cash strengthens: the withdrawal trigger m̄ rises, average cash holdings increase, and withdrawals occur when the wallet is still substantially full. This prediction is qualitatively consistent with the empirical finding that in areas with lower card acceptance, individuals hold higher cash balances and withdraw at higher residual cash levels.

Q: What are the main limitations the authors acknowledge? A: Three caveats are identified. First, the model has no exogenous cash inflows (wage payments, gifts); incorporating Miller-Orr-style inflows could affect cash resilience estimates. Second, the card cost κ is fixed and independent of transaction size s; allowing κ(s) = κ₀ + κₛ·s would better capture reward-program economies relevant for the US. Third, merchant card acceptance is treated as exogenous; endogenizing it as a game between merchants would allow a joint welfare evaluation of acceptance decisions, payment choices, and cash management.

Unforced transactions: Transactions in which both cash and card payments are feasible — specifically, cash holdings exceed the purchase size, the merchant accepts cards, and the consumer holds a card. Isolating unforced transactions is necessary to identify behavioral determinants of payment choice, stripping out mechanical constraints imposed by cash insufficiency or merchant non-acceptance.
Precautionary cash buffer: A positive cash balance maintained above the withdrawal trigger (m̄ > 0) to insure against purchases at merchants who do not accept cards. In the model, this buffer arises because card non-acceptance combined with insufficient cash results in a missed purchase at utility cost u; the precautionary motive is stronger when ϕ is lower.
Transaction-size threshold s(m): The purchase size above which a consumer with cash holdings m optimally pays by card (when cards are available and 0 < κ < b). Unlike the fixed threshold of Whitesell (1989), s(m) is increasing in m, generating a novel interaction between cash on hand and payment method choice that the ECB diary data confirms.
Cash burns policy: The policy of Alvarez and Lippi (2017) in which cards are used only when cash is fully exhausted (m = 0). The paper documents that 27–49% of unforced transactions are settled by card across survey waves, constituting a systematic violation of this rule that the model resolves by introducing transaction-size heterogeneity and a precautionary motive.
Imperfect card acceptance (ϕ): The exogenous fraction of merchants willing to accept card payments, set at 0.89 for 2023–24 in the calibration. Imperfect acceptance is the primary driver of the precautionary demand for cash; it also determines the frequency of missed purchases under a cashless policy and is the key parameter governing whether a cashless economy can emerge.
Annual transaction management cost (C): The total yearly household cost of operating within the payment system, defined as C = RM + bn + κ·(number of card purchases) + u·(number of missed purchases). Estimated at approximately 15 euros for the average euro area household in 2023–24, decomposed across opportunity costs of cash holdings, withdrawal costs, card usage costs, and missed-purchase disutility.
Ss withdrawal policy: The optimal cash replenishment rule characterized by a trigger level m̄ and a target level m*. The agent withdraws whenever cash falls to m̄, resetting balances to m*. A strictly positive trigger (m̄ > 0) reflects the precautionary motive: the agent refills before cash is exhausted in order to maintain insurance against card non-acceptance events.

Central Bank Digital Currency with Collateral-Constrained Banks

Mon, 01 Jan 0001 00:00:00 +0000

The paper analyzes the implications of introducing a retail central bank digital currency (CBDC) that competes with commercial bank deposits for household liquidity, in a model where banks must post government bonds as collateral to access central bank lending. The authors revisit Niepelt’s (2022) “equivalence of payment systems” result and find that equivalence survives even under a collateral constraint: the central bank can still offer loans to banks that replicate the no-CBDC equilibrium allocation, but at a lending rate lower than Niepelt’s unconstrained rate, because tighter terms are needed to incentivize sufficient loan uptake when banks must redirect portfolio holdings toward government bonds to qualify. A structural cost remains: banks must hold government bonds as collateral at the expense of extending credit to firms, so equivalence in allocation does not imply full neutrality — banks’ business models and the government’s intermediation role change even when aggregate output and prices are unchanged. In the dynamic extension where the central bank does not sterilize the CBDC introduction, banks respond by narrowing deposit spreads to attract inflows, with the result that a CBDC ramp-up to 5 percent of steady-state output expands rather than contracts bank credit to firms.

Summary of a forthcoming paper, AI-assisted and human-reviewed. See the linked original for the authoritative claims and full conditions.

In depth

Q1. What is the equivalence of payment systems result and how does the collateral constraint change it?

Brunnermeier and Niepelt (2019) and Niepelt (2022) established that the central bank can neutralize the real effects of CBDC introduction by lending to banks at an appropriate rate to replace lost deposit funding, a result the present paper revisits by adding a collateral requirement on central bank lending — specifically, that banks must hold eligible government bonds up to a fraction θb of their central bank loan value. Under this constraint, Proposition 1 shows that equivalence survives: there exists a central bank lending rate that replicates the no-CBDC equilibrium allocation and price system. However, this lending rate is lower than Niepelt’s unconstrained rate by a factor increasing in the restrictiveness of the constraint (lower θb requires a lower lending rate), because when banks are collateral-constrained, cheaper terms are needed to induce them to borrow enough from the central bank to offset deposit outflows.

Q2. What is Corollary 1 and why does “full neutrality” fail?

Corollary 1 states that even when the central bank achieves allocation equivalence by setting the appropriate lending rate, banks must redirect portfolio holdings from firm loans to government bonds to meet the collateral requirement — crowding out bank credit to firms by an amount equal to the bond uptake, with the crowding-out diminishing as the collateral constraint becomes less restrictive (higher θb). This is the sense in which “full neutrality” fails under the collateral constraint: aggregate output and prices are unchanged, but the composition of credit changes — banks extend less to firms and hold more government bonds — and the government or household sector must absorb the gap in firm financing. In the limiting case where CBDC and deposits are equally valuable to households (λ = 1), the government alone compensates for the reduction in bank loans, effectively expanding its own intermediation role.

Q3. What does the dynamic extension show about bank disintermediation?

Simulating a gradual and near-permanent increase in CBDC to 5 percent of steady-state output without central bank sterilization, the paper finds that banks respond by narrowing their deposit interest spread to attract deposit inflows, such that total deposits do not fall and bank loans to firms expand rather than contract — the opposite of the disintermediation hypothesis. The mechanism relies on the assumption that banks have market power in their regional deposit markets (each bank is a monopsonist): in response to CBDC competition, the bank voluntarily reduces the rent it extracts on deposits (the spread between the risk-free rate and the deposit rate), attracting more deposit inflows. This deposit inflow, combined with central bank loan uptake, expands the bank’s balance sheet and increases credit extension to firms. The result stands in contrast to models with competitive deposit markets, where banks cannot respond to CBDC competition through deposit pricing.

Q4. What changes even if credit is not reduced?

Even when the dynamic model shows credit expansion rather than contraction, the paper establishes that CBDC introduction alters banks’ balance sheet composition and business model: banks shift toward holding more government bonds and away from firm loans, the government assumes a larger credit intermediation role, and the aggregate distribution of capital ownership changes — constituting the form of non-neutrality that survives even when total credit is unchanged. This is what Corollary 1 calls the failure of “full neutrality”: the real allocation equivalence holds at the aggregate level, but the sectoral distribution of who provides credit to firms shifts from the banking sector toward the public sector. The paper interprets this as a structural consequence of the collateral requirement on central bank lending that is absent in the frictionless equivalence benchmark.

Key concepts

equivalence of payment systems : the theoretical result (from Brunnermeier-Niepelt 2019 and Niepelt 2022) that the central bank can ensure the same equilibrium allocation whether or not CBDC exists, by adjusting its lending terms to banks; this paper revisits and extends the result to environments with a collateral constraint.

collateral constraint (θb) : the requirement in this model that banks hold eligible government bonds as a fraction of the central bank loans they take on; adding this friction to Niepelt’s framework preserves equivalence in allocation but requires a lower central bank lending rate and crowds out bank loans to firms.

disintermediation : the concern that CBDC adoption would cause households to shift en masse from bank deposits to CBDC, reducing bank funding and contracting bank credit; the paper finds this does not occur in either the equivalence analysis or the dynamic extension.

monopsony in deposits : the market structure assumption that each regional bank is the sole deposit provider in its region, giving it pricing power over deposit rates; this is what enables banks in the dynamic model to narrow the deposit spread in response to CBDC competition, generating deposit inflows rather than outflows.

full neutrality : a stronger invariance result requiring that not only the equilibrium allocation but also banks’ balance sheet composition and business model are unchanged by CBDC introduction; the paper shows this fails under the collateral constraint even when allocation equivalence holds.

Debasements and Small Coins: An Untold Story of Commodity Money

Mon, 01 Jan 0001 00:00:00 +0000

This paper applies a multiple-denomination commodity money model — building on Lee, Wallace, and Zhu (2005) — to coinage episodes in late medieval England, and derives two main findings. Shortages of small coins are severely inconvenient because halfpennies and farthings serve not merely as small change but as consumption-smoothing instruments: parameterized to 15th-century England (per-capita silver approximately 35 grams, penny approximately 1 gram), the model shows that adding a halfpenny is highly welfare-improving for poor agents even at infrequent expenditure, and welfare-improving for all agents when monetary transactions occur at least twice weekly. Debasing the penny by 50 percent has approximately the same welfare effect as introducing a halfpenny and replicates the three stylized facts of the debasement puzzle — large minting volumes, cocirculation of old and new coins, and no additional mint inducement — as equilibrium outcomes rather than paradoxes. However, full-bodiedness creates a commitment device against over-issuance that cannot be replicated by sufficiently small coins, since precious metals have a practical lower bound on coin content, so debasement relieves but does not solve the structural small-coin problem, pointing to the historical necessity of a transition to fiat money.

Summary of a forthcoming paper, AI-assisted and human-reviewed. See the linked original for the authoritative claims and full conditions.

In depth

Q1. What is the debasement puzzle and how does the paper resolve it?

The debasement puzzle, documented by Rolnick, Velde, and Weber, consists of three facts: following a debasement, minting volumes rose sharply, old and new coins cocirculated sometimes by weight, and yet people still paid minting fees rather than receiving inducements — all of which are puzzling because the absence of an inducement suggests no straightforward arbitrage. The paper resolves the puzzle by modeling a debasement as equivalent to introducing a new denomination: it draws agents to the mint because it supplies the welfare-improving small denomination that agents wanted, not because of a price arbitrage. Cocirculation by weight emerges naturally along the equilibrium path because agents hold both old and new coins in optimal portfolios, and the counterfactual welfare calculation shows the welfare gain from eliminating the shortage is large, explaining why agents willingly pay minting fees to obtain the new coins.

Q2. How does the paper measure the inconvenience of a coin shortage?

The paper measures inconvenience as the welfare difference between the shortage equilibrium and a hypothetical scenario in which the mint suddenly eliminates the shortage — an unanticipated shock that adds the missing denomination to the coinage structure. This counterfactual is tractably computable in the model and directly mirrors the intuition of a historical agent who compares their constrained experience to the imagined experience of having access to the missing coins. Applied to the penny, the model shows that adding a halfpenny (debasing the penny by 50 percent) yields a welfare gain equivalent to the full shortage inconvenience; the result is large for poor agents even at once-monthly expenditure and extends to all agents when transactions are at least twice weekly.

Q3. Why can debasement not permanently solve the small-coin problem?

Full-bodied coinage — coins whose face value equals their precious-metal content — constrains the minimum viable coin size: very small coins are practically too easy to counterfeit and too difficult to handle, so debasement merely pushes the lower denomination boundary down without eliminating it. The model uses this practical indivisibility of precious metals as the structural constraint that prevents an infinite regress of smaller and smaller coins. This constraint points to why fiat money — which severs the link between value and metallic content — ultimately emerged as the only way to provide arbitrarily small denominations at negligible production cost. The paper frames this as the resolution to the historical “big problem of small change.”

Key concepts

debasement puzzle : the simultaneous occurrence of unusually large minting volumes and cocirculation of old and new coins following a debasement, without any additional mint inducement; resolved in this paper as the equilibrium response to supplying a welfare-improving small denomination.

full-bodiedness : the property of commodity coins whose face value equals their precious-metal content; acts as a commitment device against over-issuance in the model but creates a practical indivisibility constraint on the minimum coin size.

multiple-denomination model : the Lee-Wallace-Zhu framework extended in this paper; explains the social demand for multiple coin denominations via wide transaction-value heterogeneity and the burden of carrying many coins.

How Banks Create Gridlock in Payment Systems to Save Liquidity: The Case of Canada

Mon, 01 Jan 0001 00:00:00 +0000

This paper uses detailed transaction-level data from Canada’s new high-value payment system (HVPS) to show how participants save liquidity by strategically exploiting the gridlock resolution arrangement built into the system. Observed behaviors are found to be consistent with the equilibrium of a “gridlock game” that captures the key incentives participants face: by withholding outgoing payments to induce gridlock events, participants trigger the system’s bilateral netting algorithm, which settles stuck payment queues at lower liquidity cost than bilateral sequential settlement would require. The findings have implications for the design of high-value payment systems and shed light on financial institutions’ liquidity preference in payment system environments.

Summary based on a working paper version, AI-assisted and human-reviewed. See the linked published article for the authoritative version.

In depth

Q1. What is the gridlock resolution arrangement and why do banks exploit it?

Modern high-value payment systems (HVPSs) include a gridlock resolution mechanism that activates when a set of payments are mutually stuck in queues—each waiting for an incoming payment before it can be sent—and resolves them simultaneously via bilateral netting, which requires less settlement liquidity than sequential settlement; banks strategically withhold outgoing payments to trigger these events and thereby save liquidity. The HVPS studied is Canada’s new large-value transfer system, which replaced the older LVTS. The gridlock game captures the incentive structure: if a bank expects counterparties to send payments that would be netted against its own obligations in a gridlock, it is optimal to withhold and wait rather than settle bilaterally at higher liquidity cost.

Q2. How is the gridlock game formalized?

The “gridlock game” is a formal game-theoretic model that captures the key incentives participants face in the HVPS: players choose whether and when to send payments, and the equilibrium characterizes the strategic withholding behavior as a rational response to the liquidity-saving opportunities created by the gridlock resolution mechanism. The equilibrium of this game is shown to be consistent with the actual patterns observed in the HVPS data: the timing, magnitude, and counterparty structure of strategic withholding are aligned with the game’s equilibrium predictions.

Q3. What are the implications for HVPS design?

The finding that participants strategically exploit the gridlock resolution mechanism has implications for HVPS design: while gridlock resolution was intended as an exception-handling mechanism for unintended payment queue build-ups, participants have adapted to use it as a routine liquidity management tool, changing the system’s effective operation in ways the designers may not have anticipated. System designers must account for the strategic response of sophisticated participants when evaluating the performance of gridlock resolution mechanisms, since the equilibrium behavior changes the frequency, timing, and magnitude of gridlock events relative to the non-strategic benchmark.

Q4. What does the evidence reveal about banks’ liquidity preferences?

The strategic gridlock behavior reveals that financial institutions place significant value on conserving payment system liquidity—enough to coordinate timing of payment submissions in ways that exploit system-level netting opportunities—consistent with liquidity being a scarce and valuable resource in modern payment systems. This preference for liquidity conservation is amplified in environments where central bank reserves are costly and where payment system participants face collateral or reserve constraints.

Key concepts

gridlock in high-value payment systems : a situation in which a set of payments are mutually stuck in queues—each waiting for incoming funds before outgoing payment can be made—requiring the system’s bilateral netting algorithm to simultaneously settle them; exploited strategically by banks to save settlement liquidity. gridlock game : the paper’s game-theoretic model of strategic payment submission timing in an HVPS; captures the incentive to withhold outgoing payments to trigger gridlock resolution events that settle payment queues at lower net liquidity cost. bilateral netting in HVPS : the gridlock resolution mechanism that settles multiple mutually stuck payments by computing net obligations among participants and settling only the differences; requires less total settlement liquidity than sequential bilateral settlement and is the mechanism banks exploit in the gridlock game.