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This paper studies a Diamond–Dybvig model of providing insurance against unobservable liquidity shocks in the presence of unobservable trades. We show that competitive equilibria are inefficient. A social planner finds it beneficial to introduce a wedge between the interest rate implicit in optimal allocations and the economy's marginal rate of transformation. This improves risk sharing by reducing the attractiveness of joint deviations where agents simultaneously misrepresent their type and engage in trades on private markets. We propose a simple implementation of the optimum that imposes a constraint on the portfolio share that financial intermediaries invest in short-term assets.
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