a) Suppose the economy is characterized by the following behavioral equations:
C = c0 + c1YD
YD = Y – T
I = b0 + b1Y
Government spending is constant. Note that investment now increases with output. Chapter 5 will discuss the reasons for this relation. Solve for equilibrium output.
Y=(I-b0)/b1
To solve for equilibrium output, we need to set aggregate output (Y) equal to aggregate demand (AD).
Aggregate demand can be expressed as the sum of consumption (C) and investment (I).
C = c0 + c1YD (equation 1)
YD = Y - T (equation 2)
I = b0 + b1Y (equation 3)
Since government spending (G) is constant, we can say that T = G.
By substituting equation 2 into equation 1, we can rewrite the consumption equation as:
C = c0 + c1(Y - G)
Now we can express aggregate demand (AD) as:
AD = C + I
AD = (c0 + c1(Y - G)) + (b0 + b1Y)
AD = c0 + c1Y - c1G + b0 + b1Y
To find equilibrium output, we need AD to equal Y. Thus, we have:
Y = c0 + c1Y - c1G + b0 + b1Y
Rearranging the equation, we can group the terms with Y on one side and the constant terms on the other side:
Y - c1Y - b1Y = c0 - c1G + b0
Factoring out Y, we get:
Y(1 - c1 - b1) = c0 - c1G + b0
Dividing both sides by (1 - c1 - b1), we find:
Y = (c0 - c1G + b0) / (1 - c1 - b1)
This equation gives us the equilibrium output.