While most countries around the world rely on debt to finance their government and economy, keeping this debt under control is a financial imperative. Large government debt negatively impacts long-term economic growth. Increase in a nation's debt results in lower private investment, which leads to diminishing growth and wages in the long term. A high amount of debt can be detrimental even in the absence of a financial crisis.
The debt-to-gross domestic product (GDP) ratio is an important indicator of an economy's financial leverage. "During the last financial crisis, the debt-to-GDP ratio--also called debt ratio--exploded in many countries," says Ferrari, who proposes a mathematical model for the control of this ratio in a paper to publish next week in the SIAM Journal on Control and Optimization .
"It is unclear if governments plan their debt reduction policies according to an optimization criterion, such as the maximization of social welfare or the minimization of social costs," Ferrari says. "In this sense, mathematical modeling can provide a theoretical background for such choices, and might give insights on the policies to follow."
"In my model, governments face two opposing costs," Ferrari explains. "On the one hand, they aim to minimize the total expected opportunity cost due to debt. This may result, for instance, from private investment crowding out public investments, leaving less room for public ventures, and from a tendency to suffer low subsequent growth. On the other hand, by reducing the debt through, say, fiscal policies, the government incurs a cost that is proportional to the amplitude of its action. It is important for governments to properly counterbalance these two costs, and such a problem can be modeled mathematically through a so-called singular stochastic control problem."
"A government's need to counterbalance the cost of having debt and reducing it suggests that it should follow a threshold strategy - that is, it should intervene so as to reduce debt-to-GDP ratio only when the latter is sufficiently large," Ferrari points out. "In my model, in its planning, the government also takes into account the current level of inflation in the country, which is not under the government's control, but managed by an autonomous central bank. As a result, the critical level at which the government should act in order to halt the growth of public debt is inflation-dependent, and this optimal threshold is endogenously determined as part of the solution to the problem."
"Clearly, in reality, when managing the public debt, the government should also consider macroeconomic variables other than inflation, for instance, interest rates, GDP growth rate, and exchange rates," Ferrari says. "However, in order to have a tractable mathematical problem, I have decided to focus only on the role of inflation in the debt's reduction problem faced by the government."
In his work he demonstrates that the solution of the control problem is related to that of an auxiliary optimal stopping problem developed in terms of the marginal cost of having debt and the marginal cost of intervention on the debt ratio. In the optimal stopping problem, the government determines the optimal time to reduce the debt ratio level by one additional unit with the goal of minimizing the associated total expected marginal cost. Solving the optimal stopping problem can then effectively solve the control problem.
"With collaborators, my research group at the Center for Mathematical Economics at Bielefeld University is currently trying to investigate how strategic issues might enter into the picture, how a government can optimally reduce the debt ratio when it only has partial information about the involved macroeconomic quantities, or can optimally increase or decrease the debt level when the interest rate on debt is stochastic and is affected by economic shocks that are not under its control."
"I find problems of optimal management of macroeconomic quantities--like public debt, inflation, or exchange rates--very interesting both from an everyday-life and a mathematical perspective," Ferrari says. "They lead to very challenging mathematical problems in which one needs to consider the interaction between several variables, including macroeconomic and financial quantities and multiple agents, such as government, central banks, and financial agents. I believe that there is still a lot to do in the mathematical analysis/modeling of such problems."
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SIAM Journal on Control and Optimization