This 'moderating the business cycle' is called 'counter-cyclical fiscal policy'. A primary goal of fiscal policy, in the Keynesian sense, is to mitigate the pleasure and pain of the business cycle, to stabilize the economy. Keynes provided a tool to allow Government to stabilize the business cycle without having to do it all itself. It is 'the multiplier'. He also spawned a generation of economists who searched for new tools to foster growth in potential GDP, i.e. how to make the economy grow.
Growth theory is now a recognized sub-discipline of the economics profession.
An increase or decrease will change aggregate expenditure more than the initial change via three fiscal policy multipliers assuming the very short-run and a constant price level. MPC, however, is assumed to be constant. There are essentially two kinds of taxes - induced and autonomous. Induced taxes rise and fall as real GDP varies. Taxes increase or decrease as real GDP changes.