Question

So far, we have been assuming that the fiscal policy variable T is independent of
the level of income (exogenous). In the real world, however, this is not the case.
Taxes typically depend on the level of income, so tax revenue tends to be higher
when income is higher. In this problem, we examine how this automatic response
of taxes can help reduce the impact of changes in autonomous spending on
output.
Consider the following model of the economy:
C = C0 + c1Yd
T = t0 + t1Y
Yd = Y - T
G and I are both constant (exogenous).
a. Is t1 (marginal propensity to tax) greater or less than one? Explain.
b. Solve for equilibrium output.
c. What is the multiplier? Does the economy respond more to changes in
autonomous spending when t1 is zero or when t1 is positive? Demonstrate.

Answer 1

a. Is t1 (marginal propensity to tax) greater or less than one? Explain.

The marginal propensity to tax is always less than $1.$ This is because it represents a fraction of the national income that is taken into taxation.

b. Solve for equilibrium output.

For a closed economy:

$$Y = C + I + G$$

$$Y = c_0 + c_1Y^d + I + G$$

$$Y = c_0 + c_1(Y - T) + I + G$$

$$Y = c_0 + c_1(Y - t_0 + t_1Y) + I + G$$

$$Y - Y(c_1 + t_1) = c_0 - c_1t_0 + I + G$$

$$Y [1 - (c_1 + t_1) ] = c_0 - c_1t_0 + I + G$$

$$Y^* = \dfrac{c_0 - c_1t_0 + I + G}{ [1 - (c_1 + t_1) ]}$$

c. What is the multiplier? Does the economy respond more to changes in

autonomous spending when t1 is zero or when t1 is positive? Demonstrate.

$$\dfrac{\Delta Y^*}{\Delta G} = \dfrac{1}{ [1 - (c_1 + t_1) ]}$$

When $t_1 = 0;$ the multiplier becomes:

$$\dfrac{\Delta Y^*}{\Delta G} = \dfrac{1}{1 - c_1}$$

$1 - c_1>[1 - (c_1 + t_1) ]$ . Therefore:

$$\dfrac{1}{ [1 - (c_1 + t_1) ]}>\dfrac{1}{1 - c_1}$$

Thus, the economy will respond to more changes in autonomous spending when $t>0$