I am facing a problem that may seem simple at first but with which I struggle.
The question relates to economics where I try to see the effect of deficit at time t ($D_t$) onto Output at time t+h ($Y_{t+h}$). That is, I want the multiplier $\Delta Y_{t+h}/\Delta D_t$ where $\Delta$ stands for the first difference operator. In addition to this, $D_t$ is composed of two elements, government consumption $G_t$ and taxes $T_t$ such that $D_t=G_t-T_t$.
What I know is that $\Delta Y_{t+h}/\Delta G_t$ is different from $\Delta Y_{t+h}/\Delta T_t$ in the literature on multipliers. Those results correspond to a regression coefficient. It means that $\Delta Y_{t+h}/\Delta D_t$ depends on the relative size of the changes $\Delta G_t$ and $\Delta T_t$.
What I would like is to have $\Delta Y_{t+h}/\Delta D_t$, not losing from sight that both $G$ and $T$ can vary in $D$. In other words, I want to know if the variation in the deficit is due to an increase in $G$ or a decrease in $T$.
Thank you very much.
Olivier.