Answer :
a. Substituting the given values into the PAE function, we get:
PAE = (500 + 0.8(Y - T) - 300r) + IP + 200 + 10
b. The short-run equilibrium real interest rate is 0.0714 or approximately 7.14%.
c. If the Federal Reserve wants to address a positive output gap and prevent inflation, they can sell government securities. This reduces the money supply, increases interest rates, and discourages borrowing and spending. This decrease in spending helps reduce aggregate demand and prevent excessive inflation.
d. The long-run equilibrium real interest rate should adjust to the level that was previously calculated, which is approximately 7.14%.
a. The PAE function represents the total spending in the economy and is given by the equation:
PAE = C + I + G + NX
Where:
C = Consumption
I = Investment
G = Government spending
NX = Net exports
From the provided equations, we can determine the values for each component of the PAE function:
C = 500 + 0.8(Y - T) - 300r (Consumption function)
I = IP (Investment)
G = 200 (Government spending)
NX = 10 (Net exports)
Substituting these values into the PAE function, we get:
PAE = (500 + 0.8(Y - T) - 300r) + IP + 200 + 10
b. To calculate the short-run equilibrium, we need to find the point where Aggregate Demand (AD) equals Potential Output (Y*). In other words, we need to find the value of Y where PAE = Y*.
Setting PAE equal to Y*, we have:
(500 + 0.8(Y - T) - 300r) + IP + 200 + 10 = 3700
Simplifying the equation, we get:
500 + 0.8(Y - 150) - 300r + IP + 200 + 10 = 3700
Combine like terms:
0.8(Y - 150) - 300r + IP + 710 = 3700
0.8(Y - 150) - 300r + IP = 2990
Next, substitute the value of IP from the given equation:
0.8(Y - 150) - 300r + (200 - 400r) = 2990
Combine like terms:
0.8(Y - 150) - 700r + 200 = 2990
0.8(Y - 150) - 700r = 2790
Simplify further:
0.8Y - 120 - 700r = 2790
0.8Y - 700r = 2910
Now, substitute the value of Y*:
0.8(3700) - 700r = 2910
2960 - 700r = 2910
-700r = 2910 - 2960
-700r = -50
Dividing both sides by -700:
r = -50 / -700
r = 0.0714
Therefore, the short-run equilibrium real interest rate is 0.0714 or approximately 7.14%.
c. To determine the output gap, we compare the actual output (Y) with the potential output (Y*). If the actual output is below the potential output, it indicates a negative output gap, meaning the economy is producing less than its full capacity. Conversely, if the actual output is above the potential output, it indicates a positive output gap, meaning the economy is producing more than its full capacity.
In this case, the potential output (Y*) is given as 3700. To calculate the output gap, we subtract the actual output (Y) from the potential output (Y*) and divide by the potential output (Y*):
Output gap = (Y - Y*) / Y* * 100
Since we don't have the value for Y, we cannot determine the exact output gap at potential output.
d. To correct the output gap, the Federal Reserve can implement monetary policy actions. One tool they can adopt is the Open Market Operations (OMO). This involves buying or selling government securities (bonds) in the open market to control the money supply.
For example, if the Federal Reserve wants to stimulate the economy and close a negative output gap, they can buy government securities from banks and individuals. This increases the money supply, lowers interest rates, and encourages borrowing and spending. This increase in spending helps boost aggregate demand and supports economic growth.
On the other hand, if the Federal Reserve wants to address a positive output gap and prevent inflation, they can sell government securities. This reduces the money supply, increases interest rates, and discourages borrowing and spending. This decrease in spending helps reduce aggregate demand and prevent excessive inflation.
d. Given that potential output is 3700 units, to ensure long-run equilibrium, the real interest rate (r) should adjust to a level where the actual output (Y) equals the potential output (Y*). In this case, the long-run equilibrium real interest rate should adjust to the level that was previously calculated, which is approximately 7.14%. This supports our response in parts c and d, where the short-run equilibrium was also calculated at the same real interest rate to ensure equilibrium.
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