Consider a hypothetical economy in which households spend $$0\mathrm{.}80$ of each additional dollar of their after$-$tax income. Suppose that this economy is experiencing a recession. The government would like to stimulate aggregate demand and is deciding whether it should increase its spending by $$1$ billion or reduce income tax by $$1$ billion. Assume other things remain constant, and the marginal propensity to consume remains at $0\mathrm{.}8.$

a$)$ How much would a $$1$ billion increase in government spending increase aggregate demand?

b$)$ How much would a $$1$ billion reduction in income tax increase aggregate demand?

c$)$ Alternatively, suppose the hypothetical economy is $$7\mathrm{.}5$ billion below its long$-$run equilibrium. All else equal, how much should government spending increase to close this gap?