Consider a central bank that can adjust the inflation rate by increasing and decreasing

the level of the key interest rate. Each intervention gives rise to proportional costs,

and the central bank faces also a running penalty, e.g., due to misaligned levels of inflation

and interest rate. We model the resulting minimization problem as a Markovian degenerate

two-dimensional bounded-variation stochastic control problem. Its characteristic is that the

mean-reversion level of the diffusive inflation rate is an affine function of the purely controlled

interest rate's current value. By relying on a combination of techniques from viscosity theory

and free-boundary analysis, we provide the structure of the value function and we show that

it satisfies a second-order smooth-fit principle. Such a regularity is then exploited in order to

determine a system of functional equations solved by the two monotone curves that split the

control problem's state space in three connected regions.