Keynesian consumption function question?

Question

Given the Keynesian consumption function, a cut in income tax rates would affect consumption by.............???

Answer 1

Tax-cut positively affects income and higher income leads to higher disposable income which consequently increases consumption taking into account marginal propensity to consume and multiplier effect.

C - consumption

C0 - autonomous consumption

MPC - Marginal propensity to consume

Yd - disposable income

t - tax rate

Assuming there is no crowding-out effect:

C=C0 + MPC*Yd

Yd=Y-tY = Y(1-t)

C=C0 + MPC * Y * (1-t) = C0 + MPC * (Y+ΔY) * (1-(t-Δt)

Tax multiplier:

ΔY/ΔT= -MPC/(1-MPC)

ΔT=Y*(t+Δt) - Yt = Y (t+Δt - t) = YΔt

ΔY= -Y*Δt*MPC/(1-MPC)

ΔC= (MPC * (Y+ΔY) * (1-(t-Δt))) - (MPC * Y * (1-t)) =

= Y*MPC*Δt * (MPC*Δt + MPC*t - 1) / (1-MPC)

P.S. Though not for sure - it is straight method, may be there already is simpler one in theory or one using calculus and more advanced with more assumptions taken into account (for instance another consumption models reactions, more different marginal propensities/import/export/withdraw, etc.).

Answer 2

Cutting income taxes will increase consumption. Money in people's pockets will get spent.