Banking with Inside Money: An Efficiency Analysis

Q1. What is the core model and how does inside money arise?

The paper adds a single departure from the original DD real model: output requires labor from workers and technology from entrepreneurs, which introduces a motive for money to be valued — entrepreneurs borrow units of account (inside money/deposits) from banks at date 0 to pay workers’ wages, and these deposits then circulate as a means of payment for consumption goods at dates 1 and 2. Unlike the outside-money models in Allen and Gale (1998), Skeie (2008), and Allen et al. (2014), inside money is created ex nihilo on the bank’s balance sheet when loans are extended — deposits do not represent a transfer of pre-existing funds but are liabilities created through lending. Banks in this model are price-takers and cannot take direct decisions on real investments or liquidations, which are the responsibility of entrepreneurs. This is the key distinction from the DD and subsequent literature: it is the production of deposits in the provision of loans that generates inside money, and it is the impossibility of making these nominal claims produce state-contingent real payoffs that prevents efficiency.

Q2. Why can’t non-contingent nominal deposits achieve the first best?

In any competitive equilibrium with valued deposits, the no-arbitrage condition requires that the real return on deposits equals the real return on the productive technology R in each period, so the ratio of patient-to-impatient consumption (c₂/c₁) for workers must equal R — but the first-best allocation requires c₁ and c₂ to satisfy the planner’s Euler equation u′(c₁) = Ru′(c₂), which for coefficient of relative risk aversion greater than 1 implies 1 < c₁*/c₂* < R, not c₂/c₁ = R.** This is formalized by comparing the equilibrium allocation (Proposition 1 and the Corollary) — where workers’ consumption satisfies cᵢW(1) = 1/p₁ and cᵢW(2) = R/p₁ with p₁ ∈ (0.5, ∞) — against the efficiency condition (equation 8). Because the real value of deposits is pinned by the price level in the goods market, and competitive banks have no power to engineer the price adjustments needed to create state-contingency, the nominal deposit contract is generically inefficient. This contrasts with Allen and Gale (1998) and Skeie (2008), where central bank control over either prices or real investment liquidation allows efficient outcomes.

Q3. What is the formal result on state-contingent deposit contracts?

Proposition 2 establishes that introducing contingent deposit rates (paying a higher rate to impatient depositors, id₂(1) > id₂(2)) yields an aggregate allocation in which total consumption at date 1 is at most 2 (the liquidation value) and total consumption at date 2 is at least 2R — the same aggregate feasibility constraints as the non-contingent case — and this allocation is incompatible with efficiency and no better in welfare terms than the baseline. The reason is structural: for goods to be supplied at both dates 1 and 2, the rate id₂(2) must satisfy id₂(2)·(P₁/P₂) < R ≤ id₂(1)·(P₁/P₂), but this means only impatient entrepreneurs supply goods at date 1, leaving the aggregate supply schedule identical to the non-contingent case. Even if banks had perfect information about depositor types and could implement contingent contracts without incentive compatibility concerns, the first-best allocation would remain outside the consumption possibility set of the competitive equilibrium.

Q4. What is the result on early loan liquidation?

Proposition 3 shows that allowing entrepreneurs to choose how much of their loan to repay early (at date 1 versus date 2) produces a unique equilibrium in which entrepreneurs are indifferent about when to liquidate, equilibrium deposit and loan rates satisfy id₁ = ib₁ = 0 and (1 + ib₂)(P₁/P₂) = (1 + id₂)(P₁/P₂) = R, and the resulting allocation remains inefficient. The key constraint is unchanged: competition across banks drives both deposit and lending rates to equalize in real terms, so the supply of goods at each date is still not controlled by the bank and cannot reproduce the first-best schedule. Allowing borrowers to prepay their loans does not alter the fundamental tension between fixed nominal contracts and state-contingent real outcomes.

Q5. When can banks be welfare-dominated by bilateral trade?

In the symmetric equilibrium (P₁ = D₁), the banking allocation gives E(uB) = λu(1) + (1−λ)u(R), which is welfare-dominated by the bilateral labor market allocation E(uLM) whenever the coefficient of relative risk aversion and/or the technology return R exceed a threshold — specifically, when agents are risk averse enough that the midpoint consumption available under bilateral bargaining (2R/(R+1)) is preferred to the lottery {1 with probability λ, R with probability 1−λ} — contradicting the presumption that bank intermediation is necessarily superior to direct contracting. This result, formalized by condition (41), implies that the social value of banking as an institution depends on the degree of risk aversion and the illiquidity premium R: the banking allocation is preferred when agents are relatively risk tolerant and/or R is large (so the lottery’s spread is attractive), but bilateral trade may dominate when agents are risk-averse and R is modest.

Q6. What role can central banks play and what is the lender-of-last-resort result?

The paper shows that in the baseline nominal economy, outside money is not demanded by any agent — deposits dominate cash in rate of return and the interbank payment flows net to zero — so the central bank has no leverage to affect real allocations through open-market operations; efficiency is out of reach even for a central bank. However, the paper identifies a limited but important role for central bank intervention: the lender-of-last-resort function can prevent bank runs that would otherwise be self-fulfilling equilibria in the model, even though the central bank cannot restore the first-best allocation. This is because the existence of an emergency liquidity backstop eliminates the coordination failure that makes runs self-fulfilling, without requiring the central bank to replicate the state-contingent real payoffs needed for efficiency. A central bank could potentially be incorporated into an extended model with an uneven distribution of payment flows across banks (creating a demand for reserves), but the paper argues that even then, competition across banks would still prevent contingent deposit rates from achieving efficiency.

Key Concepts

inside money : bank-created deposits that arise ex nihilo when loans are extended to borrowers and circulate as means of payment between agents; the paper’s key departure from the prior banking literature, which modeled deposits as outside money (central-bank-issued fiat money) intermediated by banks rather than money created through lending.

consumption possibility set : the set of feasible allocations achievable by the competitive equilibrium with inside-money banking; the paper’s central result is that the efficient first-best allocation — satisfying u′(c₁*) = Ru′(c₂*) — lies outside this set, so the inefficiency is not correctable by improving incentive design within the existing contract space.

nominal deposit contract : a demandable deposit that specifies a fixed nominal interest rate independent of the realization of individual liquidity preference shocks; the paper’s analysis shows that such contracts cannot produce the state-contingent real payoffs required for efficient risk-sharing in an inside-money economy, even when supplemented with contingent rates or early loan liquidation.

lender of last resort : the central bank’s capacity to provide emergency liquidity to banks facing runs by coordinating expectations away from the bank-run equilibrium; the paper’s limited positive result for central bank policy — it can prevent runs even when it cannot achieve efficiency.

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