Question

Algebraically derive the government spending and tax multiplier for the case in which tax revenues depend on income level. Verbally explain what you do in each step of the derivation.

please do not copy on Chegg i want another type answer thanks and please do not write handwriting

Answer 1

Suppose,

1.Consumption = C = Autonomous Consumption (A) + Marginal Propensity to Consume(c) * Disposable income,

Disposable income = Yd = Real Income(Y) - tax rate (t) * Y

Thus, C = A + c*(Y - tY) = A + c(1-t)*Y

2.Governmentt expenditure, Planned Investment and Net exports are a exogenous variables represented by G, I and NX.

3. So, Aggregate Expenditure (AE) = C + I + G + NX

=> AE = A + c(1-t)*Y + G + I + NX

4.At equilibrium, the aggregate expenditure equals the real income,

So, AE = Y = A + c(1-t)*Y + G + I + NX

=> Y = 1/[1-c(1-t)]*(A + G + I + NX)

5.For government spending multiplier,  we will differentiate Y with respect to G,

So, dY/dG = 1/[1-c(1-t)]
As government spending multiplier is the change in real income due to change in government spending,

Thus, government spending multiplier = 1/[1-c(1-t)]

6.For Tax multiplier, differentiate Y with respect to t, we get,

dY/dt = -c/[1-c(1-t)]^2 * (A + G + I + NX)

(We know Y = 1/[1-c(1-t)]*(A+G+I+NX) => A + G + I + NX = Y * [1-c(1-t)] )

=> dY/dt = -c/[1-c(1-t)] * Y

So, changes in tax rate changes the real income by -c/[1-c(1-t)]*Y. Thus, change in real income due to change in tax revenue is -c/[1-c(1-t)] because tax revenue is tY. So, tax multiplier = -c/[1-c(1-t)]

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