In the model that we have studied, consumption is a function of disposable income or income minus taxes C(Y −T), which implies that taxes T are a lump-sum subtraction from income Y. Now suppose a different kind of tax that is closer to what we have in reality. Let τ be the income tax rate in the country so that disposable income is Y −τY or more simply Y(1−τ). So the consumption function is now C(Y[1−τ]), which is read “consumption is a function of total income times one minus the income tax rate.” The new equation for the IS curve (goods market clearing) is the following: IS : Y = C Y[1−τ] +I(r) +G
(a) Derive the expression for the government purchases multiplier dY dG with this new consumption function.
(b) Derive the expression for the government taxes multiplier dY dτ , which means the effect on output from a decrease in the income tax rate τ. [Note that |·| is the absolute value sign.]
(c) If this is a country where MPC∗Y > 1 (which we would expect in rich countries and/or countries with higher consumption rates), which multiplier is bigger?