Optimal reduction of public debt under partial observation of the economic growth
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Optimal reduction of public debt under partial observation of the economic growth Giorgia Callegaro1 · Claudia Ceci2 · Giorgio Ferrari3
Received: 30 January 2019 / Accepted: 10 June 2020 / Published online: 15 September 2020 © The Author(s) 2020
Abstract We consider a government that aims at reducing the debt-to-(gross domestic product) (GDP) ratio of a country. The government observes the level of the debt-to-GDP ratio and an indicator of the state of the economy, but does not directly observe the development of the underlying macroeconomic conditions. The government’s criterion is to minimise the sum of the total expected costs of holding debt and of debt reduction policies. We model this problem as a singular stochastic control problem under partial observation. The contribution of the paper is twofold. Firstly, we provide a general formulation of the model in which the level of the debt-toGDP ratio and the value of the macroeconomic indicator evolve as a diffusion and a jump-diffusion, respectively, with coefficients depending on the regimes of the economy. The latter are described through a finite-state continuous-time Markov chain. We reduce the original problem via filtering techniques to an equivalent one with full information (the so-called separated problem), and we provide a general verification result in terms of a related optimal stopping problem under full information. Secondly, we specialise to a case study in which the economy faces only two regimes and the macroeconomic indicator has a suitable diffusive dynamics. In this setting, we provide the optimal debt reduction policy. This is given in terms of the contin-
B G. Ferrari
[email protected] G. Callegaro [email protected] C. Ceci [email protected]
1
Department of Mathematics, University of Padova, via Trieste 63, 35121 Padova, Italy
2
Department of Economics, University “G. D’Annunzio” of Chieti-Pescara, Viale Pindaro 42, 65127 Pescara, Italy
3
Center for Mathematical Economics (IMW), Bielefeld University, Universitätsstrasse 25, 33615 Bielefeld, Germany
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G. Callegaro et al.
uous free boundary arising in an auxiliary fully two-dimensional optimal stopping problem. Keywords Singular stochastic control · Partial observation · Optimal stopping · Free boundary · Debt-to-GDP ratio Mathematics Subject Classification (2010) 93E20 · 60G35 · 93E11 · 60G40 · 60J60 · 91B64 JEL Classification C61 · H63 · E32
1 Introduction The question of optimally managing the debt-to-GDP ratio (also called “debt ratio”) of a country has become particularly important in the last years. Indeed, concurrently with the financial crisis started in 2007, debt-to-GDP ratios exploded from an average of 53% to circa 80% in developed countries. Clearly, the debt management policy of a government highly depends on the underlying macroeconomic conditions; indeed, these affect for example the growth rate of the GDP which in turn determines the growth rate of the debt-to-GDP ratio of a country. However, in practice, it is typically neither possibl
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