Question

Thus far we have assumed that fiscal policy variables (G and T) are independent of the level of income. In

the real world, however, taxes are not independent of income. Typically, when income is higher, the tax

rate is higher as well.

Consider the following equations that govern our economy:

Z = C + I + G

C = c0 + c1YD

T = t0 + t1Y

YD = Y − T

Assume G and I are exogenous. Further, assume t1 is between 0 and 1, c1 + t1 < 1 and t0 is positive.

(a) Solve for equilibrium output in terms of the exogenous variables.

b) What is the multiplier? Is the multiplier higher when t1 is 0 or when t1 is positive? Note) t1 = 0

c) Suppose c0 increases by $300 m. What is the change in equilibrium Y when c1 = 0.5 and t1 = 0.2. What about when c1 = 0.5 and t1 = 0?

.

Answer 1

a) Given the equations of the real sector we find the equilibrium condition where aggregate output/income is equal to aggregate expenditure:

Y = AE

Y = C + I + G

Y = {c0 + c1YD} + I + G

Y = {c0 + c1[Y – T]} + I + G

Y = {c0 + c1[Y – (t0 + t1Y)]} + I + G

Y = {c0 + c1[Y – t0 - t1Y]} + I + G

Y = c0 + c1Y – c1t0 – c1t1Y + I + G

Y – c1Y + c1t1Y = c0 – c1t0 + I + G

Y ( 1 – c1 + c1t1) = c0 – c1t0 + I + G

Y = c0 – c1t0 + I + G / (1 – c1 + c1t1)     {Equilibrium level of income}

b) Let’s assume that autonomous consumption level increases from c0 to c2 (no changes in any other variable) , which leads to an increase in the equilibrium from Y to Y2

Y2 = c2 – c1t0 + I + G / (1 – c1 + c1t1)    

Thus change in equilibrium can be calculated as:

Y2 – Y = [c2 – c1t0 + I + G / (1 – c1 + c1t1)] - [c0 – c1t0 + I + G / (1 – c1Y + c1t1)]

ΔY = [(c2 – c1t0 + I + G) – (c0 – c1t0 + I + G)]/ 1 – c1 + c1t1

ΔY = [c2 – c1t0 + I + G - c0 + c1t0 – I – G] / 1 – c1 + c1t1

ΔY = [(c2 – c0) + (c1t0 - c1t0) + (I – I) + (G – G)] / 1 – c1 + c1t1

ΔY = c2 – c0 / 1 – c1 + c1t1

ΔY = ΔC / 1 – c1 + c1t1

ΔY/ ΔC = 1 / 1 – c1 + c1t1

Spending Multiplier = 1 / 1 – c1 + c1t1    - Equation 1

The multiplier states the change in value of equilibrium income due to change in consumption spending, holding other things constant.

If t1 = 0, Spending Multiplier = 1 / 1 – c1 ------ Equation 2

If t1 is positive, then c1t1 is a positive value and the spending multiplier holds as it is that is given as in equation 1.

Comparing the denominators of the two equations we can see:

(1 – c1 + c1t1 ) > (1 – c1)

Thus, 1 / (1 – c1) > 1 / (1 – c1 + c1t1) i.e. multiplier when t1 = 0 is greater than the multiplier when t1 is positive.   

c) If ΔC = 300, c1 = 0.5 and t1 = 0.2 , then using the multiplier equation 1 we can find change in Y:

ΔY/ ΔC = 1 / 1 – c1 + c1t1

ΔY/ 300 = 1/ 1 – 0.5 + 0.5*0.2

ΔY/ 300 = 1/1 – 0.5 + 0.1

ΔY/ 300 = 1/0.60

ΔY/ 300 = 1.66

ΔY = 1.66* 300

ΔY = 499.99

Thus, when t1 is positive then an increase in consumption by $300m leads to an increase in income by approximately $500 m

If ΔC = 300, c1 = 0.5 and t1 = 0.2 , then using the multiplier equation 2 we can find change in Y:

ΔY/ ΔC = 1 / 1 – c1

ΔY/ 300 = 1 / 1- 0.5

ΔY/ 300 = 1/ 0.5

ΔY/ 300 = 2

ΔY = 2* 300

ΔY = 600

Thus, when t1= 0 then an increase in consumption by $300m leads to an increase in equilibrium income by approximately $600 m