Macroeconomics — Chapter 5: Comprehensive Test on Supply and Equilibrium

📊 The vertical supply curve:

The reason:
The quantity of money (M) is determined by the central bank as an exogenous variable (external to the model).

The meaning:
• The central bank chooses how much money there will be in the economy
• The decision does not depend on the interest rate
• At every interest rate — the same quantity of money

In the graph:
• A vertical line at M/P
• No matter what r is — the quantity is fixed

💡 Simplifying assumption:
In reality, there is some relationship between r and M, but in the course we assume that the central bank has complete control.

📈 Increasing the quantity of money:

Step 1: the effect on the graph
• M rose → M/P rose
• The (vertical) supply curve shifted right

Step 2: finding the new equilibrium
• At the old interest rate there is excess supply of money
• The public holds more money than it wants
• It will try to get rid of the money → it will buy bonds
• The price of bonds will rise → the interest rate will fall

The result:
• M/P ↑
• r ↓

💡 This is expansionary monetary policy!

📉 Selling bonds by the central bank:

The mechanism:
1. The central bank sells bonds to the public
2. The public pays with money
3. The money is “absorbed” from the economy into the central bank
4. The quantity of money (M) falls

In the money market:
• The supply curve shifts left
• At the old interest rate there is excess demand for money
• The public sells bonds in order to obtain money
• The price of bonds falls → the interest rate rises

💡 This is contractionary monetary policy!

💰 The inverse relationship between bond price and interest rate:

Numerical example:
A bond promises to pay 100 dollars in one year.

If the price today is 95:
Yield = (100-95)/95 = 5.26%

If the price today is 90:
Yield = (100-90)/90 = 11.1%

The conclusion:
• Bond price ↑ → yield (interest rate) ↓
• Bond price ↓ → yield (interest rate) ↑

The formula:
P = FV / (1 + r)
When P rises, r must fall!

💡 This explains how actions in the bond market affect the interest rate.

⚖️ Equilibrium condition:

Money supply equals money demand.

Graphically, it is the intersection of the supply and demand curves.

The equilibrium interest rate is determined at this point.

🧮 Equilibrium calculation:

Set demand equal to supply:
400 = 500 - 200r

200r = 100
r = 0.5 (50%)

🧮 New equilibrium:

450 = 500 - 200r

200r = 50
r = 0.25 (25%)

📊 Excess demand for money — adjustment mechanism:

The situation:
The public wants to hold more money than it currently has.

What does the public do?
It tries to obtain money by selling bonds.

The effect:
• The supply of bonds rises
• Bond prices fall
• The interest rate rises

Result:
A higher interest rate reduces the demand for money until a new equilibrium is reached.

📊 Excess supply of money — adjustment mechanism:

The situation:
The public holds more money than it wants to hold.

What does the public do?
It tries to get rid of excess money by buying bonds.

The effect:
• Demand for bonds rises
• Bond prices rise
• The interest rate falls

Result:
A lower interest rate increases the demand for money until a new equilibrium is reached.

🧮 Detailed solution:

Given:
(M/P)^d = 2Y - 1000r
Y = 400
(M/P)^s = 600

Step 1: Substitute Y into the demand function:
(M/P)^d = 2 × 400 - 1000r = 800 - 1000r

Step 2: Equilibrium condition:
600 = 800 - 1000r

Step 3: Solve:
1000r = 800 - 600
1000r = 200
r = 200/1000 = 0.2 (20%)

Check:
800 - 1000 × 0.2 = 600 ✓

📉 Increase in the price level:

Effect on real supply:
• M, the nominal money supply, is fixed
• P rises
• Therefore M/P decreases
• The real money supply curve shifts left

Market result:
At the old interest rate there is excess demand for money. The public sells bonds to obtain money, bond prices fall, and the interest rate rises.

💡 Inflation raises the interest rate if the central bank does not respond.

🏛️ Lowering the interest rate:

To lower the interest rate, the central bank must shift the money supply curve to the right, meaning it must increase M/P.

How?
• Buy bonds from the public
• The public receives money in exchange for bonds
• M increases, so M/P increases
• At the old interest rate there is excess supply of money
• The interest rate falls

This is an expansionary monetary policy.

🧮 Detailed solution:

Step 1: Calculate real money supply:
M/P = 800/2 = 400

Step 2: Equilibrium condition:
400 = 600 - 100r

Step 3: Solve:
100r = 600 - 400
100r = 200
r = 200/100 = 2 = 200%

Check:
600 - 100 × 2 = 400 ✓

The number is not realistic economically, but it is the correct mathematical result.

📋 Expansionary monetary policy:

Goal:
To encourage economic activity by lowering the interest rate.

Tools:
• Buying bonds from the public
• Buying foreign currency
• Reducing reserve requirements
• Lowering the policy interest rate
• Increasing loans to banks

Results in the money market:
• M increases, so M/P increases
• The supply curve shifts right
• The equilibrium interest rate falls

Effect on the economy:
Lower interest rates encourage investment, output, and employment.

📋 Contractionary monetary policy:

Goal:
To cool down the economy and fight inflation by raising the interest rate.

Tools:
• Selling bonds to the public
• Selling foreign currency
• Increasing reserve requirements
• Raising the policy interest rate
• Reducing loans to banks

Results:
M ↓ → M/P ↓, the supply curve shifts left, and the equilibrium interest rate rises.

High interest reduces investment and reduces inflationary pressure.

🧮 Detailed solution:

Initial situation:
800 = 1000 - 500r
r = (1000 - 800) / 500 = 0.4 (40%)

New situation:
Demand increased: (M/P)^d = 1200 - 500r
Supply did not change: M/P = 800

New equilibrium:
800 = 1200 - 500r
500r = 400
r = 400/500 = 0.8 (80%)

Output increased, money demand shifted right, and the interest rate rose.

⚠️ The lower bound of the interest rate:

The claim is incorrect. There is a lower bound for the interest rate.

Liquidity trap:
When the interest rate approaches zero, holding money has almost no opportunity cost. The public becomes nearly indifferent between money and bonds, the demand curve becomes horizontal, and increasing M no longer lowers r.

Zero lower bound:
Usually the interest rate cannot fall below zero. Even with a very large money supply, r cannot keep falling forever.

🔗 The connection between the money market and the goods market:

In the money market, the interest rate r is determined. The interest rate affects investment I in the goods market.

• r ↑ → I ↓
• r ↓ → I ↑

Investment is part of aggregate demand, so it affects output Y.

There is also feedback: Y affects the demand for money. When Y rises, money demand rises.

This is the basis of the IS-LM model.

🧮 Detailed solution:

Step 1: Calculate real money supply:
M/P = 900/2 = 450

Step 2: Substitute Y into demand:
(M/P)^d = 500 - 250r

Step 3: Equilibrium condition:
450 = 500 - 250r

Step 4: Solve:
250r = 50
r = 50/250 = 0.2 (20%)

Check:
500 - 250 × 0.2 = 450 ✓

📊 Summary — determining the interest rate:

The interest rate is determined by two forces:

1. Money supply (M/P)^s:
Determined by the central bank. The supply curve is vertical.

2. Money demand (M/P)^d:
Determined by the public. It depends on r and Y. The demand curve slopes downward.

Equilibrium:
The intersection of supply and demand determines r*.

The interest rate is the “price” of money, determined by supply and demand like any market price.