B) The model in Chapter 18 (section 18.4) uses the simple Keynesian cross model

Question

B) The model in Chapter 18 (section 18.4) uses the simple Keynesian cross model in Chapter 5 to illustrate the main idea about the concept of automatic fiscal stabilizers. Suppose we instead use the IS-LM model in Chapters 6-7, and assume that the government's income taxation policy is also given by equation (18.4) on p. 349, answer the following questions: (a) Derive the autonomous expenditure multiplier within this IS-LM model. (b) Can we still say in this model that the income tax policy equation (18.4) works as an automatic stabilizer? Explain. (c) What's the effect of an increase in t(0) on income Y? (d) What's the effect of an increase in t(1) on Y?

Explanation / Answer

a) Since the taxation policy equation is not specified here, but question d) gives a hint about the same equation, the taxation policy equation is written as: T = t0 + t1Y where t0 is autonomous tax independent of the level of income and t1 is the marginal tax rate, dependent on income.

The components of aggregate demand (AD) are as follows: AD or Y= C+I+G+NX where C: Consumption, I: Investment, G: Government Spending and NX: Net Exports.

‘C’ depends on disposable income(Yd) which is the difference between Y and tax T. The equation of C is given as: C = a+bYd where a is autonomous consumption and bYd is the induced consumption such that the constant ‘b’ is the marginal propensity to consume (MPC) of the consumption function.

Thus, the equation for Y can be written as:

Y= C(Y-t0-t1Y) + I + G + NX

Differentiating totally both sides w.r.t Y we get,

dY = (delta C/ delta Y) (dY-dt0-t1dY) + 0+ dG+0

This is because, since nothing is mentioned about I or NX, they are assumed to be constant, which becomes 0 once differentiated. Additionally, the partial derivative of consumption function delta C/ delta Y is replaced by the MPC (b)

Therefore , dY = bdY-bdt0-bt1dY + dG

Now, t0 is autonomous tax which does not change with change in Y as it doesn’t depend on Y, the term bdt0 disappears. Therefore the equation becomes:

dY = bdY-bt1dY + dG

Or dY-bdY+bt1dY = (1-b)dG

Or dy( 1-b+bt1) = (1-b)dG

Therefore the value of the autonomous expenditure multiplier is given as dY/dG = (1-b)/(1-b+bt1)

Or autonomous expenditure multiplier dY/dG= (1-MPC)/ [(1-MPC) *(1-t1)]

This is the derivation of the autonomous expenditure multiplier.

b) Automatic stabilizers are tools that reduce fluctuations in the GDP or stabilize the GDP automatically, without direct interference from policymakers. For example, in the case of income taxes such as the one in the question, whenever Y fluctuates, the tax liability, which is the amount that a tax payer is liable to pay, automatically adjusts according to the fluctuations in Y, in order to balance or stabilize Y. In case of a sharp rise in Y, the tax liability automatically rises. Thus, this is an example of an automatic stabilizer.

c) When understanding the impact of change in one parameter, the other variables must remain constant. Thus in this case, I, G and NX are constant.

To understand the impact of t0 on Y, the equation of Y is differentiated with respect to (w.r.t) t0

Since t0 does not depend on Y but an increase in autonomous tax liability reduces the income, Y will have a parallel downward shift without any change in slope of Y and Y will reduce by the amount of rise in t0.

d) ) As explained earlier, the other variables must remain constant. To understand the impact of t1 on Y, the equation of Y is differentiated with respect to (w.r.t) t1.

Since t1 depends on the amount of income Y, a rise in t1 will directly affect the slope of Y. This will result in a non-parallel downward shift of the Y curve. The change in t1 will affect Y through the multiplier effect. Whenever t1 rises, the marginal net tax rate rises the value of the multiplier falls. The larger the marginal net tax rate, the smaller is the multiplier.

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