Question

Although our development of the Keynesian cross in this chapter assumes that taxes are a fixed amount, most countries levy some taxes that rise automatically with national income. (Examples in the United States include the income tax and the payroll tax.) Let’s represent the tax system by writing tax revenue as

T= ̄ ̄T+tY,

where ̄ ̄T and t are parameters of the tax code. The parameter ̄ ̄T is a lump-sum tax (or, if negative, a lump-sum transfer).
The parameter t is the marginal tax rate: if income rises by $1, taxes rise

by t×$ 1.

1. How does this tax system change the way consumption responds to changes in GDP?

2. In the Keynesian cross, how does this tax system alter the government - purchases multiplier?

3. In the IS–LM model, how does this tax system alter the slope of the IS curve?

Answer 1

(1)

Consumption (C) = C0 + c(Y - T) = C0 + c(Y - tY) where C0: Autonomous consumption & c: MPC

In closed-economy equilibrium, Y = C + I + G

Y = C0 + c(Y - tY) + I0 + G0 where I0, G0: Autonomous investment and government spending

Y = (C0 + I0 + G0) + cY - ctY

[1 - c(1 - t)] x Y = (C0 + I0 + G0)

Y = [(C0 + I0 + G0)] / [1 - c(1 - t)]

So, when Autonomous consumption (C0) increases (decreases) by 1 unit, GDP increases (decreases) by {1 / [1 - c(1 - t)]} units. This increase (decrease) is less than the multiplier when tax is lump-sum, value of which is {1 / (1 - c)}.

(2)

In Keynesian cross model, when Government spending (G0) increases (decreases) by 1 unit, GDP increases (decreases) by {1 / [1 - c(1 - t)]} units. This increase (decrease), which is the government spending multiplier, is less than the multiplier when tax is lump-sum, value of which is {1 / (1 - c)}.

(3)

In IS-LM model, when tax rate depends on income, spending multiplier has a lower value compared to the multiplier when tax was lumps-sum. This decrease in multiplier value will make IS curve flatter.