Question

1. Suppose that the model of the economy is given by

Y = C + I + G + X

C = a + b Yd

Yd = (1 – t)Y

X = g – mY

a. Derive the equilibrium GDP (Y) and the expenditure multiplier (Me ) expressed in general notations.

b. Suppose I = $900 billion, G = $1,200 billion, a = 220, b = 0.9, t = 0.3, g = 500, and m = 0.1. Solve for the equilibrium GDP (Y) and the expenditure multiplier (Me ) using your answers to part a.

c. Is the expenditure multiplier (Me ) with variable import spending (refer the numerical solution of Me from part b) larger or smaller than the expenditure multiplier (Me ) with fixed import spending? In addition to an algebra comparison, provide an intuition with your answer. Hint: The expenditure multiplier (Me ) with fixed import spending (i.e. constant X)) is 1 1−?(1−?) , in the problem, b = 0.9, t = 0.3.

d. Solve for private saving (Sp), government saving (Sg), and the rest of the world saving (Sr) when investment spending (I) is $900 billion.

2. Consider following simple closed economy (X = 0) and all taxes are fixed (a constant T):

Y = C + I + G

C = a + b Yd

Yd = Y – T

a. Derive the equilibrium GDP (Y), the expenditure multiplier (Me ), and the fixed tax multiplier (MT ) expressed in general notation.

b. Suppose government changes government spending G and fixed taxes T by the same amount (G = T). Derive the balanced budget multiplier, Y/G with G = T, using solutions of Me and MT from part a. [Hint] Y = meG + mTT

c. Illustrate the effect on income Y of a balanced budget increase in government spending and taxes (i.e., G=T>0) on the income expenditure (45 degree) diagram. Fully label your diagram.

Answer 1

a. Now, Y=C+I+G+X

or, Y=a+bY_D+I+G+g-mY

or, Y=a+b(1-t)Y+I+G+g-mY

or, Y(1-b+bt+m) = a+I+G+g

or, Y = a+I+G+g/(1-b+bt+m) is the equilibrium GDP

and expenditure multiplier = 1/(1-MPC) = 1/(1-b)

b. Y = a+I+G+g/(1-b+bt+m)

or, Y = (220+900+1,200+500) / (1-0.9+0.9*0.3+0.1)

or, Y = 2,720 / 0.47

or, Y = $5,787.23

and Expenditure multiplier = 1/(1-0.9) = 1/0.1 = 10

c. Expenditure multiplier with fixed import spending = 1/{1-b(1-t)} = 1/(1-0.9+0.9*0.3) = 1/0.37 = 2.7

Thus,  expenditure multiplier with fixed import spending is smaller than  expenditure multiplier with variable import spending.

d. Private saving = Y-T-C = Y-tY-{a+b(1-t)Y} = 5,787.23-(0.3*5,787.23)-{220+0.9(1-0.3)*5,787.23} = 5,787.23-1,736.17-220-3,645.96 = 185.10

Government saving = T-G = tY-G = 0.3*5,787.23 - 1,200 = 1,736.17 - 1,200 = 536.17

National saving = (Private saving + government saving) = (185.10+536.17) = 721.27