Consider an economy with a constant nominal money supply, a constant level of real output Y = 100, and a constant real interest rate, r = 0.10. Suppose that the income elasticity of demand is 0.5 and the interest elasticity of money demand is –0.1.
By what percentage does the equilibrium price level differ from its initial value if output increases to Y = 106 (and r remains at 0.10).
I don't know where to start.
To solve this problem, we can use the equation of exchange and the equation for money demand to determine the equilibrium price level.
First, let's start with the equation of exchange:
MV = PY
Where:
M is the nominal money supply
V is the velocity of money (assumed constant in this case)
P is the price level
Y is the real output
Since the nominal money supply is constant, let's assume it is denoted as M0.
Now, we can rearrange the equation to solve for the price level (P):
P = MV / Y
Next, let's consider the equation for money demand:
M/P = kY^e r^h
Where:
M/P is the real money supply
k is a constant
Y^e is the income level (real output)
r is the real interest rate
h is the interest elasticity of money demand
Given that the income elasticity of demand is 0.5 and the interest elasticity of money demand is -0.1, we can substitute these values into the equation:
M/P = kY^0.5 r^-0.1
Now, let's solve for Y:
Y^0.5 = (M/P) / (k r^-0.1)
Y = [(M/P) / (k r^-0.1)]^2
Given that the real output increases to Y = 106, we can substitute this value into the equation to find the initial price level. Let's call this P0:
106 = [(M0 / P0) / (k r^-0.1)]^2
Now, we need to calculate the new price level when the output increases to Y = 106. Let's call this P1:
P1 = MV / Y
Substituting the values into the equation:
P1 = M0 / (Y = 106)
Now, let's calculate the percentage difference in the equilibrium price level:
ΔP = (P1 - P0) / P0 * 100
Substituting the values into the equation:
ΔP = (P1 - P0) / P0 * 100
= (M0 / (Y = 106) - P0) / P0 * 100
This will give us the percentage by which the equilibrium price level differs from its initial value.
To find the percentage change in the equilibrium price level, you need to understand the relationship between price level, money demand, and the factors affecting them. Here are the steps to solve this problem:
Step 1: Understand the concept
The equilibrium price level in an economy is determined by the equilibrium condition of money demand and money supply. Money demand depends on real income (Y) and the interest rate (r), while money supply is given as constant in this case.
Step 2: Analyze the given information
In this scenario, we are given that:
- The level of real output (Y) is initially 100 and increases to 106.
- The real interest rate (r) remains unchanged at 0.10.
- The income elasticity of demand is 0.5, indicating that a 1% change in real income leads to a 0.5% change in money demand.
- The interest elasticity of money demand is -0.1, indicating that a 1% change in the interest rate leads to a -0.1% change in money demand.
Step 3: Determine the changes in money demand
Using the income elasticity of demand, we can calculate the percentage change in money demand resulting from the change in real output:
ΔY = Y_new - Y_initial = 106 - 100 = 6
Percentage change in money demand = Income elasticity * Percentage change in real income
= 0.5 * (ΔY / Y_initial) * 100
= 0.5 * (6/100) * 100
= 3%
Using the interest elasticity of money demand, we can calculate the percentage change in money demand resulting from the change in the interest rate:
Percentage change in money demand = Interest elasticity * Percentage change in interest rate
= -0.1 * 0
= 0%
Step 4: Determine the change in the equilibrium price level
Since the nominal money supply remains constant, any change in money demand will be reflected in the price level. Therefore, the change in the equilibrium price level is equal to the change in money demand.
Change in the equilibrium price level = Percentage change in money demand = 3%
Step 5: Determine the percentage difference in the equilibrium price level
To find the percentage difference, divide the change in the equilibrium price level by the initial equilibrium price level and multiply by 100.
Percentage difference in the equilibrium price level = (Change in the equilibrium price level / Initial equilibrium price level) * 100
Since we are given the initial value of the equilibrium price level, we can substitute it into the formula:
Percentage difference in the equilibrium price level = (3 / P_initial) * 100
Where P_initial is the initial equilibrium price level.
This will give you the percentage by which the equilibrium price level differs from its initial value.
Note: We don't have a specific value for the initial equilibrium price level, so you need to substitute the appropriate value into the formula to get the exact answer.