Answer :
Certainly! Let's tackle the problem step-by-step:
### Part A: Graphing the Supply and Demand for Real Balances
We have two functions:
- Real money supply (M / P)
- Money demand [tex]\( (M / P)^d = 1.000 - 100r \)[/tex]
Given:
- [tex]\( M = 1000 \)[/tex]
- [tex]\( P = 2 \)[/tex]
Therefore, the real money supply [tex]\( M / P \)[/tex] is [tex]\( 1000 / 2 = 500 \)[/tex].
Graph Construction:
1. Plot the real money supply as a horizontal line at the value of 500, since it is constant.
2. The money demand [tex]\( (M / P)^d = 1.000 - 100r \)[/tex] is a downward-sloping line in the [tex]\( (M / P) - r \)[/tex] space.
To get the points for the money demand curve:
- When [tex]\( r = 0 \)[/tex]: [tex]\( (M / P)^d = 1.000 \)[/tex]
- When [tex]\( (M / P)^d = 0 \)[/tex]: [tex]\( 0 = 1.000 - 100r \)[/tex] -> [tex]\( r = 0.01 \)[/tex]
Thus, the money demand line intersects the y-axis (real balances axis) at 1.000 and slopes downward to 0.01 when [tex]\( (M / P)^d = 0 \)[/tex].
### Part B: Equilibrium Interest Rate
To find the equilibrium interest rate, we set the real money supply equal to real money demand:
[tex]\[ \frac{M}{P} = (M / P)^d \][/tex]
[tex]\[ 500 = 1.000 - 100r \][/tex]
Solve for [tex]\( r \)[/tex]:
[tex]\[ 100r = 1.000 - 500 \][/tex]
[tex]\[ 100r = -499 \][/tex]
[tex]\[ r = -4.99 \][/tex]
The equilibrium interest rate is [tex]\(-4.99\)[/tex]. In real-world terms, this negative rate would imply a substantial distortion.
### Part C: Change in Equilibrium Interest Rate with Increased Money Supply
Now, suppose the money supply increases to [tex]\( M_{new} = 1200 \)[/tex]. Again, considering the fixed price level [tex]\( P = 2 \)[/tex]:
[tex]\[ \frac{M_{new}}{P} = 1200 / 2 = 600 \][/tex]
Set the new real money supply equal to real money demand:
[tex]\[ 600 = 1.000 - 100r \][/tex]
Solve for [tex]\( r \)[/tex]:
[tex]\[ 100r = 1.000 - 600 \][/tex]
[tex]\[ 100r = -599 \][/tex]
[tex]\[ r = -5.99 \][/tex]
The new equilibrium interest rate is [tex]\(-5.99\)[/tex].
### Part D: Money Supply for Target Interest Rate
To achieve a target interest rate [tex]\( r = 0.07 \)[/tex]:
Using the money demand equation:
[tex]\[ (M / P)^d = 1.000 - 100 \cdot 0.07 \][/tex]
[tex]\[ (M / P)^d = 1.000 - 7 \][/tex]
[tex]\[ (M / P)^d = -6 \][/tex]
We need to find the money supply (M) corresponding to this:
[tex]\[ \frac{M}{P} = -6 \][/tex]
[tex]\[ M = -6P \][/tex]
[tex]\[ M = -6(2) \][/tex]
[tex]\[ M = -12 \][/tex]
Thus, the central bank should set the money supply to [tex]\(-12\)[/tex].
In the context of this theoretical problem, such a negative money supply is conceptually unrealistic and indicates a fundamental impracticality in the given functional relationships. However, based purely on the algebraic results, this would be the calculated value.
### Part A: Graphing the Supply and Demand for Real Balances
We have two functions:
- Real money supply (M / P)
- Money demand [tex]\( (M / P)^d = 1.000 - 100r \)[/tex]
Given:
- [tex]\( M = 1000 \)[/tex]
- [tex]\( P = 2 \)[/tex]
Therefore, the real money supply [tex]\( M / P \)[/tex] is [tex]\( 1000 / 2 = 500 \)[/tex].
Graph Construction:
1. Plot the real money supply as a horizontal line at the value of 500, since it is constant.
2. The money demand [tex]\( (M / P)^d = 1.000 - 100r \)[/tex] is a downward-sloping line in the [tex]\( (M / P) - r \)[/tex] space.
To get the points for the money demand curve:
- When [tex]\( r = 0 \)[/tex]: [tex]\( (M / P)^d = 1.000 \)[/tex]
- When [tex]\( (M / P)^d = 0 \)[/tex]: [tex]\( 0 = 1.000 - 100r \)[/tex] -> [tex]\( r = 0.01 \)[/tex]
Thus, the money demand line intersects the y-axis (real balances axis) at 1.000 and slopes downward to 0.01 when [tex]\( (M / P)^d = 0 \)[/tex].
### Part B: Equilibrium Interest Rate
To find the equilibrium interest rate, we set the real money supply equal to real money demand:
[tex]\[ \frac{M}{P} = (M / P)^d \][/tex]
[tex]\[ 500 = 1.000 - 100r \][/tex]
Solve for [tex]\( r \)[/tex]:
[tex]\[ 100r = 1.000 - 500 \][/tex]
[tex]\[ 100r = -499 \][/tex]
[tex]\[ r = -4.99 \][/tex]
The equilibrium interest rate is [tex]\(-4.99\)[/tex]. In real-world terms, this negative rate would imply a substantial distortion.
### Part C: Change in Equilibrium Interest Rate with Increased Money Supply
Now, suppose the money supply increases to [tex]\( M_{new} = 1200 \)[/tex]. Again, considering the fixed price level [tex]\( P = 2 \)[/tex]:
[tex]\[ \frac{M_{new}}{P} = 1200 / 2 = 600 \][/tex]
Set the new real money supply equal to real money demand:
[tex]\[ 600 = 1.000 - 100r \][/tex]
Solve for [tex]\( r \)[/tex]:
[tex]\[ 100r = 1.000 - 600 \][/tex]
[tex]\[ 100r = -599 \][/tex]
[tex]\[ r = -5.99 \][/tex]
The new equilibrium interest rate is [tex]\(-5.99\)[/tex].
### Part D: Money Supply for Target Interest Rate
To achieve a target interest rate [tex]\( r = 0.07 \)[/tex]:
Using the money demand equation:
[tex]\[ (M / P)^d = 1.000 - 100 \cdot 0.07 \][/tex]
[tex]\[ (M / P)^d = 1.000 - 7 \][/tex]
[tex]\[ (M / P)^d = -6 \][/tex]
We need to find the money supply (M) corresponding to this:
[tex]\[ \frac{M}{P} = -6 \][/tex]
[tex]\[ M = -6P \][/tex]
[tex]\[ M = -6(2) \][/tex]
[tex]\[ M = -12 \][/tex]
Thus, the central bank should set the money supply to [tex]\(-12\)[/tex].
In the context of this theoretical problem, such a negative money supply is conceptually unrealistic and indicates a fundamental impracticality in the given functional relationships. However, based purely on the algebraic results, this would be the calculated value.
