Question

suppose an economy is defined by C= 100+ 0.9 (Y-T), I=220-10r, G=300,T=300

a. find equation for IS curve and what is the slope

b. if r=6%. what is equilibrium level of income

c. if the government switches from lump-sum taxation to an income tax with a constant tax rate what happens to the slope of IS

Answer 1

a) C= 100+ 0.9 (Y-T), I=220-10r, G=300,T=300

At equilibrium,

Y = Aggregate Demand(AD)

AD = C+I+G

AD = 100+ 0.9 (Y - 300) + 220-10r + 300

AD = 100 + 0.9Y - 270 + 220 - 10r + 300

AD = 350 + 0.9Y - 10r

Now equating Y with AD we get,

Y = 350 + 0.9Y - 10r

0.1Y + 10r = 350

r = 35 - 0.01Y --------------------------------> This is the equation of the IS curve

Slope of the IS curve is 0.01

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b) When r = 6%

0.06 = 35 - 0.01Y

0.01Y = 34.94

Y = 3494 ---------------------> Equilibrium level of income

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c) With income dependent tax, the consumption function becomes,

C = 100 + 0.9(Y - tY)

where t is the tax rate.

Again, Y = AD

Y = C+I+G

Y = 100 + 0.9(Y - tY) + 220-10r + 300

Y = 100 + 0.9Y - 0.9tY + 220 - 10r + 300

Y = 620 + 0.9Y - 0.9tY - 10r

10r = 620 - Y + 0.9Y - 0.9tY

10r = 620 - Y + 0.9Y(1 - t)

10r = 620 - Y(1 - 0.9(1 - t))

r = 62 -[ Y(0.1 + 0.9t)] / 10

r = 62 - {Y(0.01 + 0.09t)

Thus in this case, the slope of the IS curve increases since t>0. Previously the slope of the IS curve was 0.01. But in this case the slope of the IS curve is 0.01 + 0.9t, which implies that it is greater than the previous slope. Thus in this case, the slope of the IS curve increases and the IS curve becomes steeper.