Consumption=2000+0.9y-100000r-g
Investiment=1000-4500r
Real money demand=y-6000i
Inflation=0.03
G=500
Full employment=1000
M=2100
Find equilibrium values of the real interest rate, consumption,investment,price level
To find the equilibrium values of the real interest rate, consumption, investment, and price level, we need to set the supply of savings equal to the demand for savings.
First, we'll find the supply of savings:
S = Y - C
Given that Consumption (C) is equal to 2000 + 0.9Y - 100000r - G, we have:
S = Y - (2000 + 0.9Y - 100000r - G)
S = Y - 2000 - 0.9Y + 100000r + G
S = 8000 + 99900r
Now, we'll find the demand for savings, which is equal to investment (I):
I = 1000 - 4500r
Equating the supply and demand for savings:
8000 + 99900r = 1000 - 4500r
Rearranging the equation:
104400r = -7000
Dividing by 104400:
r = -7000 / 104400
r ≈ -0.067
Since the real interest rate cannot be negative, this result is not feasible. Therefore, there is no equilibrium in the real interest rate for this model.
Next, we'll find the equilibrium values for consumption, investment, and the price level:
Given the equation for consumption:
C = 2000 + 0.9Y - 100000r - G
Substituting the given values for G and r:
C = 2000 + 0.9Y - 100000(-0.067) - 500
C ≈ 2000 + 0.9Y + 6700 - 500
C ≈ 9200 + 0.9Y
To find the equilibrium value for investment, we need to plug in the equilibrium value of the real interest rate (r ≈ -0.067) into the investment equation:
I = 1000 - 4500(-0.067)
I ≈ 1000 + 301.5
I ≈ 1301.5
Finally, to find the equilibrium price level, we can use the inflation equation:
π = 0.03 = (P - P_0) / P_0
Where P_0 is the initial price level. We're given that the initial money supply (M) is equal to 2100, which is also equal to P_0 * Y.
2100 = P_0 * Y
Therefore, the initial price level (P_0) is:
P_0 = 2100 / Y
Substituting this into the inflation equation:
0.03 = (P - (2100 / Y)) / (2100 / Y)
0.03 = (P * Y - 2100) / 2100
Rearranging the equation and substituting the given full employment value (Y = 1000):
P * 1000 = 0.03 * 2100 + 2100
P * 1000 = 63 + 2100
P * 1000 = 2163
P ≈ 2.163
Therefore, the equilibrium values are approximately:
Real interest rate (r): Not feasible (no equilibrium)
Consumption (C): 9200 + 0.9Y
Investment (I): 1301.5
Price level (P): 2.163
To find the equilibrium values of the real interest rate (r), consumption (C), investment (I), and price level (P), we need to set the aggregate demand equal to aggregate supply in the goods and money markets.
1. Aggregate demand:
AD = C + I + G
2. Aggregate supply:
AS = Y (output/income)
3. Equilibrium in the goods market:
AS = AD
Since AS = Y, we can rewrite aggregate demand as follows:
Y = C + I + G ...(1)
To solve for the equilibrium values, we need to equate aggregate supply and demand after finding the values of consumption (C) and investment (I).
Let's proceed step-by-step to find the values.
Step 1: Finding Consumption (C):
C = 2000 + 0.9Y - 100000r - G
Step 2: Finding Investment (I):
I = 1000 - 4500r
Step 3: Equating aggregate supply and demand by substituting equations (1), (2), and (3):
Y = C + I + G
Y = (2000 + 0.9Y - 100000r - G) + (1000 - 4500r) + G
Y = 2000 + 0.9Y - 100000r + 1000 - 4500r + G + G
Simplifying the equation:
0.1Y = 3000 - 9500r
Step 4: Solving for the equilibrium output/income (Y):
0.1Y - 0.9Y = 3000 - 9500r
-0.8Y = 3000 - 9500r
Y = (3000 - 9500r) / -0.8
Y = (-3000 + 9500r) / 0.8
Step 5: Setting the equilibrium output/income (Y) to full employment level (1000):
(-3000 + 9500r) / 0.8 = 1000
Solving for r:
-3000 + 9500r = 0.8 * 1000
-3000 + 9500r = 800
9500r = 3800
r = 3800 / 9500
r ≈ 0.4
So, the equilibrium real interest rate (r) is approximately 0.4.
Step 6: Substituting the value of r into the consumption equation to find the equilibrium consumption (C):
C = 2000 + 0.9Y - 100000r - G
C = 2000 + 0.9(1000) - 100000(0.4) - 500
C ≈ 2000 + 900 - 40000 - 500
C ≈ 2400 - 40000 - 500
C ≈ -37300
The equilibrium consumption (C) is approximately -37300.
Step 7: Substituting the value of r into the investment equation to find the equilibrium investment (I):
I = 1000 - 4500r
I = 1000 - 4500(0.4)
I ≈ 1000 - 1800
I ≈ -800
The equilibrium investment (I) is approximately -800.
Step 8: Finding the equilibrium price level (P):
We can use the quantity theory of money equation to find the equilibrium price level.
MV = PY
Given M = 2100 and Y = 1000 (full employment), we can rearrange the equation to solve for P:
P = (MV) / Y
P = (2100 * 1000) / 1000
P = 2100
The equilibrium price level (P) is 2100.
To summarize, the equilibrium values are:
Real interest rate (r) ≈ 0.4
Consumption (C) ≈ -37300
Investment (I) ≈ -800
Price level (P) = 2100