UCL ECON7003 Money and Banking

Topic 6a

The Money Multiplier

Stylized history of banking.

Deposit creation.

Actors in the Money Supply process.

Multiple deposit creation.

Simple deposit multiplier and assumptions underlying it.

Deposit creation and money supply.

Stylized history of banking.

Owner of safe house storing gold takes in £100 in gold coins:

Safe house

Assets Liabilities .

Gold coins + £100 Deposit notes + £100

Deposit notes become accepted in transactions:

Substitute for payment in gold.

i.e. Gold (‘commodity money’) →‘customary money’ – a financial instrument (‘bank note’)

Safe house owner now makes loans to the value of £900.

Safe house

Assets Liabilities .

Gold coins £ 100 Deposit notes £1000

Loans issued £ 900

£1000

Note: Trade-off in deciding gold:deposit-note ratio:

Profitability of loans and liquidity.

Liquidity insufficient to meet demands for withdrawal of gold coins may → ‘run’, or panic.

Deposit creation.

Bank of England makes open market purchase of £100 from Bank I.

Bank I uses proceeds to make loan of £100, paid into borrower’s current account.

Bank I

Assets Liabilities .

Securities - £100 Deposits + £100

Reserves + £100

Loans + £100 .

+ £100 + £100

Bank I has expanded balance sheet through issuing credit / creating deposit.

Bank now has ER of £100.

BUT: new deposit created for purpose of loan.

→ assumed the sum will soon be withdrawn.

→ bank cannot safely make further loans / the £100 ER short term only.

→ Likely net outcome :

ΔD=0, ΔR=0, ΔL=100.

Bank I’s customer uses loan to make purchase by cheque.

Firm receiving cheque pays it into its account at Bank A.

Bank A takes full advantage of corresponding ER↑ to make loan to another customer.

Bank A

Assets Liabilities .

Reserves + £ 10 Deposits + £100

Loans + £ 90 .

+ £100 + £100

Actors in money supply process:

(a) The expansion of Bank I’s deposit creation capacity

Bank I’s deposit creation capacity was outcome of decision by CB to expand MB.

Controls MB with certainty through OMOs.

Commercial banks, their borrowers, their depositors:

No part in this decision.

Actors in money supply process:

(b) Bank I’s loan to its customer

Bank I’s own independent decision.

Its borrowers may have affected decision through their level of demand for loans → rL.

CB:

no direct part in decision,

though influence or control over interest rates may have affected it indirectly.

Actors in money supply process:

(c) Use the customer makes of loan.

Customer’s action involved two decisions relevant to MS process:

(α) use entire loan for purchase > hold as deposit

(β) make purchase by cheque > cash

Multiple deposit creation.

Increase in deposits in banking system as a whole so far:

Deposits created

Bk A receives £100 from customer of Bk I £100

Bk B receives £90 from customer of Bk A £ 90

Total deposits created £190

Bank A’s customer uses loan to make purchase by cheque from a further firm.

Firm pays this into its account at Bank B.

Bank B takes full advantage of ER↑ to make loan to another customer.

Bank B

Assets Liabilities .

Reserves + £ 9 Deposits + £ 90

Loans + £ 81 .

+ £ 90 + £ 90

Increase in deposits in banking system as a whole so far:

Deposits created

Bk A receives £100 from customer of Bk I £100

Bk B receives £90 from customer of Bk A £ 90

Bk C receives £81 from customer of Bk B £ 81

Total deposits created £271

Series of further transactions involving loans by Banks C, D, E, F. Assumption: Banks make loans to full amount of ER.

Bank ΔD ΔR ΔLoans I £ 0.00 £ 0.00 £100.00 A £100.00 £10.00 £ 90.00 B £ 90.00 £ 9.00 £ 81.00 C £ 81.00 £ 8.10 £ 72.90 D £ 72.90 £ 7.29 £ 65.61 E £ 65.61 £ 6.56 £ 59.05 F £ 59.05 £ 5.91 £ 53.14 … … … … → ∞ → £0 → £0 → £0 £1000.00 £100.00 £1000.00Or:

ΔD = 100 { 1 + 0.9 + (0.9)2 + (0.9)3 + (0.9)4 + . . . (0.9)n }

‘Simple deposit multiplier’.

R = RR + ER, and RR = r.D, where r ≡ RRR

ER = 0 → R = RR = r.D

R = r.D → D = (1/r).R

→

ΔD = (1/r).ΔR Simple deposit multiplier

Series of further transactions involving loans by Banks C, D, E,F. Assumption: Banks make loans to full amount of ER.

Bank ΔD ΔR ΔLoansI £ 0.00 £ 0.00 £100.00A £100.00 £10.00 £ 90.00B £ 90.00 £ 9.00 £ 81.00C £ 81.00 £ 8.10 £ 72.90D £ 72.90 £ 7.29 £ 65.61E £ 65.61 £ 6.56 £ 59.05F £ 59.05 £ 5.91 £ 53.14… … … …

→ ∞ → £0 → £0 → £0£1000.00 £100.00 £1000.00

Or:

ΔD = 100 { 1 + 0.9 + (0.9)2 + (0.9)3 + (0.9)4 + . . . (0.9)n }

Equilibrium: Point at which process ends,

i.e. deposit creation ceases:

In terms of number of transactions in ‘chain’:

Reached at limit as n → ∞i.e. as (0.9)n → 0.

In terms of level of ER in banking system:

Reached at limit as all ER exhaustedi.e. as ER → 0.

In terms of total R in banking system: R = RR + ER

ER → 0→ R → RR = r.D

i.e. ER exhausted, R reaches min. level consistent with RRR.

OMP of £100 → £900 expansion of deposits in banking system:

Banking System Assets Liabilities .Securities - £100 Deposits + £900Reserves + £100Loans + £900 .

+ £900 + £900

ER not yet exhausted: ER = £10 > 0.

Equivalently R not yet → equilibrium level: R = 100 > r.D = 90.

i.e. Simple deposit multiplication process has not yet reached equilibrium.

Once process has reached equilibrium, we have.

Banking System

Assets Liabilities .

Securities - £ 100 Deposits + £1000

Reserves + £ 100

Loans + £1000 .

+ £1000 + £1000

ΔD = (1/r).ΔR

r = 0.10 and ΔR = 100

→ ΔD = (1/0.10).100 = 10.(100) = 1000

Now banks use 50% of ER to purchase secs. > make loans. No difference to deposit creation process / equilibrium!

Banking System

Assets Liabilities .Securities - £ 100 Deposits + £1000Reserves + £ 100Securities + £ 500Loans + £ 500 .

+ £1000 + £1000

Or, simplifying: Banking System Assets Liabilities .Reserves + £ 100 Deposits + £1000Securities + £ 400Loans + £ 500 .

+ £1000 + £1000

The assumptions underlying the simple deposit multiplier.Note: Money here defined as M1 ≡ C + bank deposits

(1) All banks exhaust ER in making loans.i.e. till ER = 0, or equivalently to maximum extent consistent with RRR (i.e. R = r.D).

(2) All transactions are carried out by cheque.→ Amount of currency in circulation unaffected (ΔC = 0).

Lifting these assumptions:Deposit creation process is interrupted:

(1) Banks retain ER > lend to full amount possible consistent with RRR(2) Customers shift into cash; money ‘leaks’ out of the banking system.

Deposit creation and money supply:

M ≡ M1 = C + D

→ deposit creation is supply of money.

Extent to which the two processes proceed:

MS process is interrupted before deposit creation → equilibrium of SDM.

Assumptions on which SDM equilibrium rests are unrealistic.

If the two assumptions underlying the SDM held, CB would have full control over MS process:

Assumption (1)

BB continue deposit creation until ER exhausted.

i.e. the process continues till equilibrium implied by SDM.

i.e. with certainty, given the level of R in the economy.

Assumption (2)

CB can control R with certainty, since MB = R + C, and ΔC = 0.

The CB is thus sole ‘actor’ with determining role in MS process.

Lifting the assumptions, we have:

CB no longer has full control of MS process:

Assumption (1)

If BB decide not to issue loans to full extent of ER resulting from OMP, and instead to hold ER > 0:

Some of the ΔR from the OMP ‘leak’ out of the deposit creation process.

Process is interrupted / does not reach equilibrium level implied by SDM.

Assumption (2)

If banks’ borrowers decide to shift from cheque to cash transactions:

ΔC > 0 and some of the ΔR from the OMP ‘leaks’ out of the banking system, → out of deposit creation process.

i.e. Lifting the assumptions:

→ CB no longer sole actor in MS process:

Decisions of banks, depositors, borrowers now also affect extent to which process proceeds.

To allow for lifting of these assumptions, 2 additional ratios:

Lifting of assumption (1), we need:

e ≡ ER / D excess reserve ratio

Lifting of assumption (2), we need:

c ≡ C / D currency ratio

Model that will incorporate these ratios.

The money multiplier ≡ m:

m = (1 + c) / (r + e + c)

ΔD = (1/r).ΔR Simple deposit multiplier

Underlying assumptions:(1) Banks exhaust all ER in making loans.(2) All transactions are carried out by cheque.

Lifting these assumptions: Deposit creation process is interrupted.

To allow for lifting of these assumptions, 2 additional ratios:

e ≡ ER / D excess reserve ratio

c ≡ C / D currency ratio

Model that will incorporate these ratios.

The money multiplier ≡ m:

m = (1 + c) / (r + e + c)

Deriving the money multiplier

MB = R + C

R = RR + ER and RR = r.D , so we can write

R = r.D + ER

→ MB = r.D + ER + C

Note: Parameter of 1 on ER and C:

→ increase in MB that public decides to hold in cash

→ one-for-one increase in MB / not multiplied through creation of additional deposits.

Similarly, increase that goes into ER.

MB = r.D + ER + C

e = ER / D → ER = e.D

c = C /D → C = c.D

→ MB = r.D + e.D + c.D

MB = (r + e + c).D

D = {1 / (r + e + c)} . MB

M ≡ M1 ≡ D + C

Substituting C = c.D:

M = D + c.D = (1 + c ).D

We have D = {1 / (r + e + c)} . MB

→ M = { (1 + c) / (r + e + c) }.MB

We thus have:

m = (1 + c) / (r + e + c) the money multiplier

m = (1 + c) / (r + e + c) the money multiplier

i.e. The change in M (≡ M1) resulting from given change in MB.

In terms of calculus: m = dM / dMB.

Note:

Outside money: Created by CB (i.e. through expansion of MB).

Inside money: created by the commercial banking system.

i.e. through multiple deposit creation.

Money multiplier: numerical example.

We have:r = 0.10C = £400 bnD = £800 bnER = £0.8 bn

c = C / D = 400 / 800 = 0.5

e = ER / D = 0.8 / 800 = 0.001

m = (1 + c) / (r + e + c) = (1 + 0.5) / (0.10 + 0.001 + 0.5) = 1.5 / 0.601 = 2.496

We have:

m = (1 + c) / (r + e + c)

= (1 + 0.5) / (0.10 + 0.001 + 0.5) = 1.5 / 0.601

= 2.496

Note:

If ER = 0 and no leakage into cash transactions (i.e. e = c = 0),

m would reduce to:

(1 + 0) / (r + 0 + 0) = 1/r

i.e. the SDM model would hold / m would be:

1 / r = 1 / 0.1 = 10

i.e. about 4 times greater.

Determinants of money supply (a) required reserve ratio.

Inverse relationship (ceteris paribus) between RRR and MS:

r↑ → supportable D↓

→ Bank must contract loans and consequently deposits.

i.e. A contraction in MS.

Relationship (ceteris paribus) between RRR and m:

Also inverse -- r is in denominator of m !

Recall above example:

m = (1 + c) / (r + e + c)

= (1 + 0.5) / (0.10 + 0.001 + 0.5) = 1.5 / 0.601

= 2.496

Now suppose r ↑ from 0.10 to 0.15:

m = (1 + c) / (r + e + c)

= (1 + 0.5) / (0.15 + 0.001 + 0.5) = 1.5 / 0.651

= 2.304

i.e. m has fallen from 2.496 to 2.304.

Determinants of money supply (b) cash ratio.

Relationship (ceteris paribus) between NBP’s cash to deposits ratio and

money supply

money multiplier

Deposits undergo multiple expansion; currency does not.

→ shift from deposits into cash, i.e. rise in c, is switch from component of M that undergoes multiplication into one that does not.

i.e. both m and M are negatively related to c.

Recall above example:

m = (1 + c) / (r + e + c)

= (1 + 0.5) / (0.10 + 0.001 + 0.5) = 1.5 / 0.601

= 2.496

Now suppose c rises from 0.5 to 0.75 (cet. par.):

m = m = (1 + c) / (r + e + c)

= (1 + 0.75) / (0.10 + 0.001 + 0.75) = 1.75 / 0.851

= 2.056

m falls from 2.496 to 2.009.

Determinants of money supply (c) excess reserve ratio.

Recall above example:

m = (1 + c) / (r + e + c)

= (1 + 0.5) / (0.10 + 0.001 + 0.5) = 1.5 / 0.601

= 2.496

Now suppose fivefold increase in e:

m = (1 + 0.5) / (0.10 + 0.005 + 0.5) = 1.5 / 0.605 = 2.479

i.e. The increase in r has reduced m VERY LITTLE -- from 2.496 to 2.479.

Note: RRR ↑ is equivalent to ER↑.

Both: bank reduces amount of deposits supported by given level of R.

Bank determines e by costs and benefits of holding ER.

Two main factors:

Market interest rates

Expected deposit outflows.