Help me please! In the model that we have studied, consumption is a function of

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Help me please!

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 tau be the income tax rate in the country so that disposable income is Y - tauY or more simply Y(1 - tau). So the consumption function is now C(Y[1 - tau]), 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 - tau]) + I(r) + G Derive the expression for the government purchases multiplier dY/dG with this new consumption function. Derive the expression for the government taxes multiplier |dY/dtau|, which means the effect on output from a decrease in the income tax rate tau. 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?

Explanation / Answer

a)      Y=C(Y(1-t)) + I(r) +G

Taking the first differenciation on both sides,

(We use the product rule on C)

dY= c[(1-t)dY] + dI + dG (where c is the differenciation with respect to consumption)

Now, dI=0

Thus, dY= c[(1-t)dY] + dG

dY[ 1-c(1-t)] = dG or dY/dG= 1/ [1-c(1-t)]

b)      Y=C(Y(1-t)) + I(r) +G

Taking the first differenciation on both sides,

(We use the product rule on C)

dY= c[(1-t)dY + ((-1)dt)Y] + dI + dG

Now, dI=0, dG=0

Thus, dY= c[ dY- tdY -Ydt] = c.dY – t.c.dY – c.Y.dt

dY (1-c(1-t)) =-Y.c.dt

dY/dt= -Y.c/ (1-c(1-t))

c)      Tax multiplier is bigger in absolute sense.

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