Macroeconomics — Unit 4: Internal and External Injections

🔁 Internal injection — definition:

An internal injection is an action by the public that changes the composition between cash (CA) and deposits (D), without changing the monetary base B.

Examples:
• A deposit into the bank: CA decreases, D increases
• A withdrawal from the bank: CA increases, D decreases

Effect on M:
The money supply M does change because the multiplier changes — but B stays constant.

🌐 External injection — definition:

An external injection is an action by the central bank that changes the monetary base B.

Main tools:
• Buying/selling government bonds
• Buying/selling foreign currency
• Monetary loans to commercial banks
• Changing the reserve ratio R

Effect on M:
ΔM = injection × multiplier (when paid through reserves)
ΔM = injection only (when paid through currency in circulation)

📈 Positive internal injection:

When the public deposits cash into the bank:
• CA (cash held by public) decreases
• D (deposits) increases
• B = CA + RZ stays the same (money just moves between sectors)

Effect on M:
M increases because the multiplier rises (more money is held as deposits, which can be lent out).

This is "positive" because M grows.

📉 Negative internal injection:

When the public withdraws cash from the bank:
• CA increases
• D decreases
• B stays the same

Effect on M:
M decreases because the multiplier falls (more money is held as cash, which the bank cannot lend out).

This is "negative" because M shrinks.

🧮 Solution — positive internal injection:

Step 1: Identify the action
The public deposits $100 → CA decreases by 100, D increases by 100

Step 2: Compute the multiplier
multiplier = 1/R = 1/0.2 = 5

Step 3: Compute ΔM
The deposited $100 becomes new lending capacity:
ΔM = 100 × (1/R − 1) = 100 × (5 − 1) = +$400

💡 Note:
The $100 itself was already part of M (as cash). What's new is the additional $400 the bank can lend out due to the multiplier effect on the deposit.

📊 Solution — negative internal injection:

Step 1: Identify the action
The public withdraws $200 → CA increases by 200, D decreases by 200

Step 2: Compute the multiplier
multiplier = 1/R = 1/0.25 = 4

Step 3: Compute ΔM
ΔM = −200 × (1/R − 1) = −200 × (4 − 1) = −$600

💡 Note:
The $200 itself stays in M (as cash). What's lost is the $600 of multiplied lending capacity that the bank can no longer extend.

🌐 Solution — positive external injection through reserves:

Step 1: Identify the injection
Bond purchase by central bank = positive external injection of +150
Payment through reserves → multiplier applies

Step 2: Compute the multiplier
multiplier = 1/R = 1/0.3 ≈ 3.33

Step 3: Compute ΔM
ΔM = injection × multiplier
ΔM = 150 × (1/0.3) = 150 × 3.33 ≈ +$500

💡 Key: When payment goes through bank reserves, the full multiplier effect kicks in.

📉 Solution — negative external injection through reserves:

Step 1: Identify the injection
Bond sale by central bank = negative external injection of −80
Payment through reserves → multiplier applies

Step 2: Compute the multiplier
multiplier = 1/R = 1/0.2 = 5

Step 3: Compute ΔM
ΔM = injection × multiplier
ΔM = −80 × 5 = −$400

💡 Key: Selling bonds drains liquidity from the system, reduced by the full multiplier effect.

💡 The key distinction:

Through reserves (RZ):
The funds reach commercial banks, which can then lend them out.
→ The full multiplier kicks in: ΔM = injection × (1/R)

Through currency in circulation (CA):
The funds go directly into the public's pockets as cash.
→ No multiplier, because the money never enters the banking system.
→ ΔM = injection (one-to-one)

Why it matters:
For a $100 injection at R = 0.2:
• Via reserves: ΔM = 100 × 5 = $500
• Via currency: ΔM = 100

That's a 5× difference!

🏃 The bank-run problem:

Banks operate on fractional reserve banking — they hold only a fraction of deposits as reserves (RZ = R × D), and lend the rest out.

Example: If R = 0.2 and total deposits D = $1,000:
• Reserves on hand: RZ = $200
• Loaned out: $800

If all depositors demand their money at once, the bank only has $200 in cash on hand. It would have to:
• Call in loans (often impossible quickly)
• Borrow from the central bank
• Or fail

This is why deposit insurance exists — to prevent panic withdrawals.

📊 Internal injection — does B change?

Definition:
B (monetary base) = CA + RZ

What happens in a deposit:
• CA decreases by 100
• D increases by 100
• Of that, RZ increases by R × 100, but the rest stays as new lending
• Net: when CA falls by 100 and RZ rises by 20 (at R=0.2), B appears to fall by 80 — but this is only the immediate impact before the multiplier resolves.

Important: Once the system reaches equilibrium, the central bank has not changed B. Only the public has shifted its holdings.

Conclusion: B remains controlled by the central bank only. Internal injections change M through the multiplier, not through B.

🧮 Detailed solution — positive internal injection:

Initial state:
Given: B = 600, CA = 100, R = 0.25
RZ = B − CA = 600 − 100 = 500
D = RZ / R = 500 / 0.25 = 2,000
M_initial = CA + D = 100 + 2,000 = 2,100

The action — a deposit of 50:
The public deposits $50 into the bank:
• new CA = 100 − 50 = 50
• the amount enters bank reserves: RZ_new = 500 + 50 = 550

Compute new D:
D_new = RZ_new / R = 550 / 0.25 = 2,200

Compute new M:
M_new = CA_new + D_new = 50 + 2,200 = $2,250

Verification:
ΔM = deposit × (1/R − 1) = 50 × (4 − 1) = 50 × 3 = 150
M_new = M_initial + ΔM = 2,100 + 150 = 2,250 ✓

💡 Insight:
In a positive internal injection, B stays the same (600) but M rises by 150 due to the multiplier effect on the deposit.

💱 Buying foreign currency — effect on M:

When the central bank buys foreign currency from the public:
• The public hands over dollars (or euros, etc.)
• The central bank pays in local currency
• That local currency enters the economy → positive external injection

Effect on M:
• If paid through reserves: ΔM = injection × (1/R)
• If paid through currency in circulation: ΔM = injection

Why central banks do this:
• To weaken the local currency (boost exports)
• To accumulate foreign reserves
• To inject liquidity into the economy

📋 Identifying an external injection:

An external injection = an action by the central bank that changes the monetary base B.

Reviewing each option:
✅ Central bank buys bonds from the public: The central bank pays out new money → B rises → external injection.
❌ Public deposits cash: Internal injection — only the composition (CA vs D) changes.
❌ Public withdraws cash: Internal injection — same reason in reverse.
❌ Transfer between accounts: Neither — no change in CA, D, or B.

Only option (a) is an external injection.

🧮 Solution — external injection through currency in circulation:

Given:
• Sale of foreign currency = 200 (negative injection)
• Payment through currency in circulation (explicitly stated!)

What happens?
• The public hands over local currency (from their cash holdings)
• The public receives dollars (foreign currency)
• CA decreases by 200
• RZ does not change (the payment did not pass through banks)

Calculation:
ΔM = ΔCA = −200

ΔM = −200 (a decrease of 200)

💡 When payment is through currency in circulation — there is no multiplier!
If it had been through reserves with R = 0.2: ΔM = −1,000.

📊 Solution — bank balance sheet after a withdrawal:

Initial state:
Assets = RZ + LO = 400 + 1,600 = 2,000
Liabilities = D = 2,000 ✓ (balance sheet balances)

The withdrawal of 100:
The customer takes 100 in cash from the bank.
• The bank pays out 100 from its reserves: RZ goes from 400 to 300
• The customer's deposit shrinks by 100: D goes from 2,000 to 1,900
• Loans don't change: LO = 1,600

Verification:
Assets = 300 + 1,600 = 1,900
Liabilities = 1,900 ✓

RZ = 300, LO = 1,600, D = 1,900

⚠️ Why the claim is incorrect:

The claim confuses two things — B (monetary base) and M (money supply).

Truth in the claim: B really does not change in an internal injection.

Where the claim fails: M is determined by both B and the multiplier:

M = B × multiplier(CA-to-D ratio)

When the public deposits cash:
• B stays the same
• But the multiplier rises, because more of B is held as RZ (which the bank can lend)
• Therefore M rises

Numerical example:
Before: B = 100, all CA → M = 100
After deposit at R = 0.2: B = 100, all D → M = 100/0.2 = 500
ΔM = +400, even though ΔB = 0!

🏦 Monetary loan to commercial banks:

What is a monetary loan?
The central bank lends money directly to commercial banks. The funds enter the banks' reserves (RZ) immediately.

Step 1: Identify the injection
The 100 enters RZ → positive external injection through reserves

Step 2: Compute the multiplier
multiplier = 1/R = 1/0.25 = 4

Step 3: Compute ΔM
ΔM = injection × multiplier = 100 × 4 = +400

💡 Intuition: The loan increases bank reserves by 100, which lets the banking system create 400 in additional money supply through repeated lending.

📋 Formula for external injection through reserves:

The formula:
ΔM = injection × (1/R)

Why?
When the injection enters bank reserves, the banking system can create new money through fractional reserve lending:

1. Bank receives the injection (say 100) — RZ rises by 100
2. Bank holds R × 100 as new reserves; lends out (1−R) × 100
3. The borrower spends it; the recipient deposits it
4. The receiving bank holds R of that, lends out the rest
5. ... and so on

The infinite series sums to: ΔM = 100 × (1/R)

Note the difference from internal injections:
• External through reserves: ΔM = injection × (1/R)
• External through currency in circulation: ΔM = injection
• Internal injection (deposit): ΔM = injection × (1/R − 1)

🔄 Solution — internal injection with given balances:

Step 1: Compute initial M
D = RZ / R = 300 / 0.2 = 1,500
M_initial = CA + D = 200 + 1,500 = $1,700

Step 2: The deposit (positive internal injection)
CA: 200 → 150 (decrease by 50)
The 50 enters bank reserves: RZ rises from 300 to 350

Step 3: Compute new D
D_new = RZ_new / R = 350 / 0.2 = 1,750

Step 4: Compute new M
M_new = CA_new + D_new = 150 + 1,750 = $1,900

Verification: ΔM = +200
Using the formula: ΔM = 50 × (1/R − 1) = 50 × (5 − 1) = 200 ✓