Current works and working papers

  • The Envelope Theorem, Euler and Bellman Equations, without Differentiability” (with Jan Werner), January 2021.
  • We extend the standard Bellman’s theory of dynamic programming and the theory of recursive contracts  with forward-looking constraints of Marcet and Marimon (2019) to encompass non-differentiability of the value function resulting from the presence of binding constraints or non-unique solutions. The envelope theorem provides the link between the Bellman equation and the Euler equations, but it may fail to do so if the value function is non-differentiable. We introduce an envelope selection condition which guarantees that solutions generated from the Bellman equation satisfy the Euler equations with or without forward-looking constraints.  In standard dynamic programming, ignoring the envelope selection condition may result in inconsistent multipliers, but not in non-optimal outcomes. In recursive contracts and dynamic planner’s problems with recursive utilities, it can result in inconsistent promises and non-optimal outcomes. A recursive  method of solving dynamic optimization problems with non-differentiable value function involves expanding the co-state and imposing the envelope selection condition. Read more
  • Breaking the Spell with Credit-Easing: Self-Confirming Credit Crises in Competitive Search Economies” (with Gaetano Gaballo), August 2020.

We develop a theory of self-confirming crises in which lenders charge high interest rates because they wrongly believe that lower rates would further increase their losses. In a directed search economy, this misperception can persist because at the equilibrium there is no evidence that can confute it, preventing the constrained-efficient outcome. A policy maker with the same beliefs as lenders will find it optimal to offer a subsidy contingent on losses to induce low interest rates. As a by-product, this policy generates new information for the market that may disprove misperceptions and make superfluous the implementation of any subsidy. We provide new micro-evidence suggesting that the 2009 TALF intervention in the market of newly generated ABS was an example of the optimal policy in our model. Read more

This paper further advances the design of an optimal Financial Stability Fund (Fund) of Abraham et al. (2019) by not having the Fund absorbing all the sovereign debt of a country. The Fund’s long-term contracts are subject to two-sided limited enforcement constraints: at any point in time the borrowing country may breach the contract and exit, while the Fund cannot have expected losses. The country’s constraint therefore represents a sovereignty constraint, whereas the lenders’ constraint can be interpreted as a debt sustainability analysis (DSA). The country can borrow one-period defaultable bonds on the private international market, while having a state-contingent contract with the Fund, which provides insurance and, possibly, credit. The Fund contract has no seniority with respect to the privately held sovereign debt and, therefore, takes this external debt into account. In equilibrium, the Fund contract prevents the country from defaulting on its entire debt position. As a result, the debt in the private international market becomes risk free, although it is constrained when the Fund’s limited enforcement constraint binds. The share of debt held by the Fund might be indeterminate; nevertheless, there is one contract that minimizes the debt absorbed by the Fund. Our model provides a theoretical and quantitative framework to address sovereign debt-overhang problems, in the Euro Area and elsewhere, transforming risky sovereign debts into ‘safe assets’.Read more


Countries which share a common currency potentially have strong incentives to share macroeconomic risks through a system of transfers to compensate for the loss of national monetary policy. However, the option to leave the currency union and regain national monetary policy can place severe limits on the size and persistence of transfers which are feasible inside the union. In this paper, we derive the optimal transfer policy for a currency union as a dynamic contract subject to enforcement constraints, whereby each country has the option to unpeg from the common currency, with or without default on existing obligations.  Our analysis shows that the lack of independent monetary policy, or an equivalent independent policy instrument, limits the extent of risk-sharing within a currency union; nevertheless, in the latter, the optimal state-contingent transfer policy take as given the optimal monetary policy as to implement a constrained efficient allocation that minimises the losses of the monetary union. At the steady state welfare is lower than in a fiscal union with independent monetary policies. Neverthess, in our simulations, the macroeconomic stabilisation effects and the social values achieved, under the two different union regimes, are quantitatively almost the same.Read more

We study the introduction, and possible design, of a European Unemployment Insurance System (EUIS) using a multi-country dynamic general equilibrium model with labour market frictions. Our calibration provides a novel diagnosis of European labour markets, revealing the key parameters – in particular, job-separation and job-arrival rates – that explain their different performance in terms of unemployment (or employment) and its persistence.  We find that there are substantial gains from reforming currently suboptimal unemployment benefit systems. In spite of country differences, it is possible to unanimously agree on a (minimal) EUIS, which countries can complement by additional national benefits. The EUIS features an unlimited duration of eligibility, which eliminates the risk of not finding a job before the receipt of benefits ends, and a low replacement rate of 10%, which stabilizes incentives to work. Country-specific payroll taxes eliminate cross-country persistent transfers. The resulting tax differences across countries may be the best statistic of their structural labour market differences, in terms of job creation and destruction, providing clear incentives for reform. We also find that there are only small additional welfare gains from insuring against country-specific shocks in the proposed harmonized EUIS. Read more

We develop a model of the Financial Stability Fund (Fund), which can be set by a union of sovereign countries. The Fund can improve the countries’ ability to share risks, and borrow and lend, with respect to the standard instrument used to smooth fluctuations: sovereign debt financing. Efficiency gains arise from the ability of the Fund to offer long-term contingent financial contracts, subject to limited enforcement (LE) and moral hazard (MH) constraints.  By contrast, standard sovereign debt contracts are uncontingent and subject to untimely debt roll-overs and default risk. We quantitatively compare the constrained-efficient Fund economy with the incomplete markets economy with default. In particular, we characterize how (implicit) interest rates and asset holdings differ, as well as how both economies react differently to the same productivity and government expenditure shocks. In our economies, calibrated to the euro area ‘stressed countries’, substantial welfare gains are achieved, particularly in times of crisis. Our theory provides a basis for the design of a Fund beyond the current scope of the European Stability Mechanism (ESM), and a theoretical and quantitative framework to assess alternative risk-sharing (shock-absorbing) facilities, as well as, proposals to deal with the euro area ‘debt-overhang problem’. Read more

In an overlapping generations economy with incomplete insurance markets, the introduction of an employment fund – akin to the one introduced in Austria in 2003, also known as ‘Austrian backpack’– can enhance production efficiency and social welfare. It complements the two classical systems of public insurance: pay-as-you-go pensions and unemployment insurance (UI). We show this in a calibrated dynamic general equilibrium model with heterogeneous agents of the Spanish economy (2014). A ‘backpack’ (BP) employment fund is an individual (across jobs) transferable fund, which earns the economy interest rate as a return and is financed with a payroll tax (a BP tax). The worker can use the fund if he or she becomes unemployed or retires. To complement the existing Spanish pension and UI systems with a welfare maximising 19% BP tax would raise welfare by 1.3% of average consumption at the new steady state, and would be preferred to the status quo by most economic and demographic groups. Our model also provides a framework where other reforms – e.g. a partial, or complete, substitution of current unsustainable pension systems – can be quantitatively assessed.Read more

This eBook provides an overview of the findings and proposals of the Horizon 2020 ADEMU research project (June 2015 to May 2018), which aimed at reassessing the fiscal and monetary framework of the European Economic and Monetary Union in the wake of the euro crisis.