Suppose we have a closed economy with a consumption function of C = .75Y, and an
ID: 1202777 • Letter: S
Question
Suppose we have a closed economy with a consumption function of C = .75Y, and an income tax of 20% imposed at each level of income. Assume that investment and government spending are autonomously given to be I = 60 and G = 100 respectively. a. What is the aggregate expenditures function? Identify its slope and vertical-intercept. b. What is the equilibrium value of income, Y*? Show your derivation. c. Is the government running a surplus, deficit, or balance budget? Prove it! d. What is the value of the multiplier? Must show/derive the multiplier to receive credit.Explanation / Answer
The aggregate expenditure function is defined by the summation of Consumption Expenditure (C), Investment Expenditure (I) and the Government Spending (G); where the consumption function is calculated based on the disposable income DY = (Y - T).
Now it is given that 20% amount of tax is imposed on each level of income. So, the tax amount could be written as,
T = 20% * Y = 0.2Y
So, the disposable income would be, DY = Y - 0.2Y = 0.8Y
Therefore, the aggregate expenditure function would be,
AE = C(DY) + I + G
AE = 0.75 * 0.8Y + 60 + 100
AE = 0.6Y + 160
So, the vertical intercept of the aggregate expenditure function is 160 and the slope is 0.6, where we measure the AE on the vertical axis and Y on horizontal axis.
b. At the equilibrium level, we get the Aggregate Expenditure (AE) being same as the Aggregate Supply (AS). Therefore, we could write,
Y = 0.6Y + 160
0.4Y = 160
Y* = 400
c. The government budget is defined by the difference between the tax revenue and the government expenditure. If we denote 'B' as the budget of the government, then we could write,
B = T - G
Now, at the equilibrium level of income Y* = 400, the tx collected by the government would be, T = 0.2 * 400 = 80.
So, the budget of the government would be,
B = 80 - 100 = - 20.
Thus, the government is running with budget deficit since its expenditure is more than its tax revenue.
d. Now, according to the equilibrium condition, in general form, we could write,
Y = C (Y - tY) + I + G; where t = tax rate, and I is constant according to the model and G is autonomous.
So, by total derivative we could write,
dY = C' * (1 - t) * dY + dI + dG Where C' = Marginal Propensity to Consume (MPC)
{1 - C' * (1 - t)} dY = dG Since dI = 0 as Investment is constant.
dY/dG = 1/{1 - C' * (1 - t)}
This is the value of the multiplier respect to the government expenditure, which depends upon the MPC and the tax rate.
Now from the consumption function, we know that the MPC is 0.75 and the tax rate (t) is 0.20.
So, the value of the multiplier would be,
dY/dG = 1/{1 - 0.75 * (1 - 0.20)}
dY/dG = 1/{1 - 0.6}
dY/dG = 1/0.4
dY/dG = 2.5