Answer to Question #214369 in Macroeconomics for geoffrey

Question

a)     A simple closed economy with an mpc equal to 0.5. Investment spending has suddenly fallen, reducing aggregate demand and output to a level that is 100 million below Y*.

               

                 ii.           If the government, instead, decided to try to get the economy to full employment using only a lump-sum tax cut how big of a tax cut would be needed?

               iii.           If the government decided to try to get the economy back to full employment using only an increase in transfers, how large would this increase need to be?

Answer 1

ii)For a simple closed 3-sector economy like the above, the following Keynesian framework can represent it.

$Y = C + I + G,$

where Y = national income, C = consumption expenditure, I = investment expenditure and G = government expenditure.

Now,$C = a + bY' and Y' = Y - T + R,$

where a = autonomous consumption, b = marginal propensity to consume, Y' = disposable income, T = taxes, and R = transfers.

So we have $C = a + b(Y - T + R) = a + bY - bT + bR.$

Substituting C in the original national income equation.

$Y = a + bY - bT + bR + I + G,$

or

$Y - bY = a - bT + bR + I + G,$

or

$Y = \frac{1}{(1-b)} (a - bT + bR + I + G) \Rightarrow \textup{ Equation 1}$

Using this equation, we can calculate the impacts of changes in various variables on the changes in national income.

To see how Y changes concerning change in G or government spending, we will differentiate Equation 1 wrt G; such as

$\frac{\mathrm{d} Y}{\mathrm{d} G} = \frac{\mathrm{d} (\frac{1}{(1-b)} (a - bT + bR + I + G)) }{\mathrm{d} G} = \frac{1}{(1-b)}$

Thus for a unit change in G, Y changes by 1/(1-b), where b is mpc like stated previously. This is the government expenditure or spending multiplier. We can state that,

$\textup{Government Spending Multiplier} = \textup{GM} = \frac{\mathrm{d} Y}{\mathrm{d} G} = \frac{1}{(1-b)}$

Similarly, we can calculate the tax multiplier and transfers multiplier.

ii) Tax multiplier:

$\frac{\mathrm{d} Y}{\mathrm{d} T} = \frac{\mathrm{d} (\frac{1}{(1-b)} (a - bT + bR + I + G)) }{\mathrm{d} T} = \frac{-b}{(1-b)}$ }

$\textup{Tax Multiplier} = \textup{TM} = \frac{\mathrm{d} Y}{\mathrm{d} T} = \frac{-b}{(1-b)}$

Transfers multiplier:

$\frac{\mathrm{d} Y}{\mathrm{d} R} = \frac{\mathrm{d} (\frac{1}{(1-b)} (a - bT + bR + I + G)) }{\mathrm{d} R} = \frac{b}{(1-b)}$

$\textup{Transfers Multiplier} = \textup{RM} = \frac{\mathrm{d} Y}{\mathrm{d} R} = \frac{b}{(1-b)}$

Now we are given that mpc = b = 0.5. So

$\textup{GM} = \frac{\mathrm{d} Y}{\mathrm{d} G} = \frac{1}{(1-b)} = \frac{1}{(1-0.5)} = 2$

$\textup{TM} = \frac{\mathrm{d} Y}{\mathrm{d} T} = \frac{-b}{(1-b)} = \frac{-0.5}{(1-0.5)} = -1$

$\textup{RM} = \frac{\mathrm{d} Y}{\mathrm{d} R} = \frac{b}{(1-b)} = \frac{0.5}{(1-0.5)} = 1$

Now the needed change in national income is 100 million to bring it to full employment equilibrium. We are given three cases. Let's approach each individually.

Case 1: When G has to vary

Earlier, we have calculated GM = 2. And GM = change in Y / change in G = 2. When change in Y = 100, change in G = 50. Thus, government spending must increase by 50 million to increase national income by 100 million.

Case 2: When T has to vary

TM = change in Y / change in T = -1. When change in Y = 100, change in T = -100. Thus, taxes must be decreased by 100 million to increase national income by 100 million.

Case 3: When R has to vary

RM = change in Y / change in R = 1. When change in Y = 100, change in R = 100. Thus, transfers must be increased by 100 million to increase national income by 100 million.