ECON 345 TOPIC 7: THREE SECTOR MODEL WITH A
BANKING SYSTEM AND A POSITIVELY SLOPED
MONEY SUPPLY CURVE
This section extends the 3-sector model to incorporate a banking system with
commercial banks and a central bank that can control liquidity through the
overnight bank rate ibank. The principal analytical result is that the money supply
curve is now upward sloping with respect to the interest rate and is shifted by the
overnight bank rate.
1. Banking is a profitable form of financial ‘intermediation’ insofar as a bank can
offer the public a short-term liquid asset, ‘deposits’, that pays more than cash but
less than what the bank can reap from longer-term investments (loans, mortgages)
funded by the deposits. Thus, bank profits essentially emerge from an arbitrage
process over different asset maturities. Commercial banks also serve a valuable
matching function in bringing together borrowers and lenders in financing
investment and capital projects. In traditional treatments, the most basic feature of
banks is that, since they are not constrained to hold all their deposits, they can lend
out much of the money deposited by the public, and create new money – bankcreated money. Thus, the activities of the banking sector give rise to a ‘money
multiplier’.
Measuring what the aggregate money supply actually can prove complicated when
the money stock consists of both government-denominated fiat money and bankcreated money. But, even more important, there are currently so many financial
assets available that are almost perfect substitutes for liquid cash, it is difficult to
know which near-monies should be included in an aggregate measure. This has led
to a proliferation of measures of aggregate ‘liquidity’, so many that the US Federal
Reserve ultimately gave up on keeping statistics on them. Accordingly, the new
tradition has emerged where the Central Bank does not attempt control the
aggregate money supply directly; it uses the lever of the overnight bank rate to
hopefully push aggregate liquidity (however measured) in the right direction.
In the following exposition, we start from with the simplest traditional measure of
the money supply to illustrate the basic concepts involved. The ‘basics’ of
Canada’s system of banking and monetary control are covered later.
2. It is useful to define the Monetary Base (MB) and Money Supply in the simplest
way, utilizing the narrowest measure of the money supply, M1. C is currency, D is
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checkable deposits in banks and R is desired reserves by banks. Let the
currency/deposit ratio c = C/D and the desired reserve ratio rd = R/D. Here,
Monetary Base: MB = C + R
[1]
Money Supply: M1 = C + D
[2]
3. The money multiplier: Now consider the general relation:
M1 = m MB
[3]
where the objective is to find ‘m’, the money multiplier. The money multiplier
indicates the extent to which bank-created money (‘inside money’) expands
liquidity beyond the monetary base (‘outside money’).
It follows that if there are no banks, D = R= 0: hence, M1 = MB = C, so m = 1.
If there are banks, but no currency: M1 = D and MB = R, thus m = D/R, which by
the above definition implies:
m = 1/rd
[4]
Here the money multiplier ‘m’ is in general greater than 1, but approaches unity as
rd approaches 1. If the reserve ratio was 0.1, then m = 10: $100 base money would
yield a total money stock of $1000 once the base money was subjected to the
continual redepositing and relending processes of the banking system.
It follows that the lower the reserve ratio rd, the more money banks can lend out/
create, so the multiplier will be higher. At one extreme, if rd = 1, no new money
can be created by banks, since every dollar of deposit must be fully held in
reserves. The money multiplier is 1. On the other extreme, if rd = 0, then in
principle even a monetary base of a penny can expand the money supply infinitely.
4. If there are both banks and currency, dividing all terms by D in the equations [1]
and [2] for M1 and MB yields a more realistic multiplier:
m = (1+c)/(c+rd)
[5]
As above, the currency/deposit ratio is c = C/D and the desired reserve ratio is rd =
R/D.
2
The big difference here is that agents now have the option to hold currency, rather
than having to fully redeposit any cheque in the bank. This must limit the extent to
which money is subject to continuing expansion within the banking system, and
lowers the money multiplier. This phenomenon is often referred to as ‘currency
drain’. Here an agent can receive a $1000 cheque drawn on one bank but might
only deposit $500 in their own checking account, keeping $500 in their pocket.
Thus, only half the money continues to remain in the banking system for relending.
Both increases in rd and c must lower the money multiplier. Increases in the former
limit the extent to which banks can lend out money, while increases in the latter
limit the extent to which money so created returns to the banking system.
It is useful to note how strongly even a small amount of ‘currency drain’ can lower
the money multiplier. Before, with rd = 0.1 and c = 0, the multiplier was 10. If c =
0.1 now, the multiplier drops to 1.1/0.2 = 5.5. If c = 0.2, it would drop further to
1.2/0.3 = 4. If an agent aimed to hold currency and deposits 50/50, then c = 1, and
the multiplier is 2/1.1 = 1.82. The higher c is, the closer the multiplier is to 1.
Note also that, in the extreme case where rd = 0, the money multiplier remains
finite and is equal to (1 + c)/c.
5. Central bank lending and the bank rate: Traditionally, central banks operated as
‘lenders of last resort’ to deal with specific liquidity crises, etc. Since the 1980s,
central bank lending has become a more normal and sometimes dominant part of
daily banking activities. Open Market Operations has served as the traditional
central bank instrument for adjusting the monetary base – buying government
bonds to expand liquidity, and selling government bonds to reduce it. We designate
this as the non-borrowed portion of the monetary base MBn.
Now adding in direct central bank lending to the commercial banks, the monetary
base becomes the sum of two components: MBn, plus the new item A (‘Advances’),
where A is the level of central bank lending (commercial bank borrowing)
undertaken. A may be regarded as the borrowed portion of the monetary base.
The incentive for commercial banks to borrow from the central bank critically
depends on the cost of borrowing, and this is the bank rate ibank; the lower is ibank,
everything else constant, the more borrowing that takes place. The bank’s
decisions also depend on what return they can get on their use of the borrowed
funds in the market place (approximated by ib). Bank profits thus depend on the
spread (ib – ibank), noting that the market return ib would typically be an expected
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return, adjusted for risk. An increase in ib increases the above spread and
encourages commercial bank borrowing; an increase in ibank lowers it.
The bank rate has become the big-ticket monetary policy instrument for controlling
the business cycle and signaling the future state of the economy: it can only be
changed eight times a year. MBn has no such restriction and may be changed daily
for smaller adjustments. Both instruments have been used together in ‘crisis’
situations: increases in MBn typically supplement previous reductions in ibank, and
are typically referred to as ‘quantitative easing’. They involve large-scale
government bond (asset) purchases. If one insists that ibank cannot be negative, then
lowering the bank rate to zero must end its potential effectiveness as a monetary
policy instrument and require that additional liquidity be provided through MBn.
With both these components considered, the monetary base can now be written:
MB = MBn + A (ib, ibank)
[6]
+ Substituting into M1 = m MB, the money supply equation becomes:
M1 = mMBn + mA (ib, ibank)
+ where m = 1+c/c+rd as above.
[7]
This formulation immediately implies a money supply curve that is upward-sloping
in the market bond yield ib, since M1 is positive in ib through the Advances function
A(ib …). As a different variable than ib in this equation, ibank can only shift this
curve. The money supply curve is shifted right if ibank falls, and left if ibank rises.
Application: Suppose a banking crisis situation where agents flee from banks,
raising the currency/deposit ratio c. This immediately lowers the money multiplier
m and lowers M1. Suppose the central bank wants to keep ‘liquidity’ up – to keep
M1 roughly constant. What would it do? Answer: lower ibank or increase MBn or
both.
6. Further dependencies of M1 on ib: Applying the idea that the opportunity cost of
holding currency increases with ib allows the additional argument that both c and rd
are negative in ib. The opportunity cost to holding cash for an individual, or
holding reserves for a bank, increases with a higher return to market payoffs.
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Since these two variables comprise the money multiplier, the money multiplier m
also varies positively with ib. More formally, let c = c(ib) and rd = rd(ib), where
both are negative relationships. This implies the positive relationship:
m = m(ib)
[8]
+
It follows that there are three reasons why the money supply is positively-sloped –
because of A first, and now because of c and rd. If we removed all of these
dependencies on ib, the money supply curve would be vertical as before.
The additional relationships allow much of the money supply expression to be
written in terms of ib alone and the two active policy parameters, MBn and ibank.
Letting M1 be now be denoted as Ms:
Ms = Ms (ib, MBn, ibank)
[9]
+ +
s
M changes endogenously with the interest rate, while central bank monetary
policies to expand (contract) the money supply can either be achieved through the
two policy instruments: raising (lowering) MBn and/or lowering (raising) ibank.
Linking to the business cycle: Note the importance of endogenous variations in
‘liquidity’ over the business cycle. In Phase I, ib is increasing, which raises the
money multiplier, and expands Ms by inducing individuals to hold more bank
deposits and banks to hold less reserves. This adds fuel to the early boom, and
possibly explains why a central bank might want to raise ibank to offset this effect.
In Phases II and III, i is falling, so we get the reverse effect. It is well documented
that the money multiplier increases in expansions and declines in recessions.
7. The expanded money market equation: Equilibrium in the money market now
requires:
Ms (ib, MBn, ibank) = P x Md(ib, W,…)
+ +
- +
[10]
8. Changes in the 3-sector model results:
In many ways, the proceeds of the above analysis do not lead to a major change in
thinking. All effects are contained in the money market, and all we have changed is
the previous vertical slope of the money supply curve to an intermediate slope. For
any 3-sector shock, it is therefore easy to refer to the results we got with the
vertical Ms by itself, and add a small adjustment. Under flexible prices, the
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adjustment will only affect the size of the change in the price level P. The new
baseline rules are as follows:
Any time a real shock raises ib, it now expands the money supply, and puts
additional upward pressure on P.
Any time a real shock lowers ib, it now contracts the money supply, and puts
additional downward pressure on P.
This will be true for all the 3-sector cases you have done. Here are two examples:
(1) Consider the standard Y1 increase, Y2 constant case where r (ib) necessarily
falls. The interest rate decrease lowers Advances, raises c and raises rd – all of
which lower Ms. By the above rule, there is less money out there, so prices should
fall more than before. In Diagram 7.1 below, the movement away from point A is
no longer to B; one now moves down the positively-sloped Ms-curve to settle at C.
Ms (ib, MBn, ibank) = P x Md(ib, W,…)
Before:
0%
-5%.
+5%
Now:
-1%.
-6%.
+5%
P decreases by 6% to match the 1% fall in Ms. Suppose P was fixed in the short
run. Then you would have to raise MBn or lower ibank to raise Ms by a total of 6%.
This would offset the 1% fall in endogenous money, so the increase nets to 5%.
Diagram 7.1
Price adjustment with upward-sloping Ms-curve
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In the diagram, we initially feed the implied lower interest rate from the
goods/bonds market into the money market. Money demand shifts out by the
standard wealth effect. The equilibrium adjustment in the money market with the
vertical Ms- curve requires a fall in P to take the market from A to B. Now the
equilibrium movement is from A to C: P has to fall by more since it has to offset
both the increased demand for money and its falling supply. In the case where P is
assumed fixed, we would be stuck at E, and the only option for maintaining
equilibrium would be to shift the Ms-curve right by raising MBn or lowering ibank.
(2) Now consider the standard Y1 constant, Y2 increase case where interest rates
necessarily increase. The ib increase raises Advances, lowers c and lowers rd, all of
which raise Ms. By the above rule, there is more money out there, so prices should
rise more than before. Again, the equilibrium movement is not simply a vertical
displacement from A to B; one moves up the Ms curve to C. Now, prices are in
principle ambiguous in this backloaded case, but in all of the possible movements,
there is new price pressure upwards. If P rises, it will rise more than in the vertical
Ms case; if P fell before, it now falls by less: if P was constant before, it will now
rise. To illustrate, consider this last case where the net change in Md (…) is 0.
Ms (ib, MBn, ibank) = P x Md(ib, W,…)
Before:
0%
0%
0%
Now:
+1%
+1%
0%
P has to increase by 1% to match the 1% increase in Ms. If P was fixed here, the
change in Ms must be 0 if P doesn’t change, so one would have to lower MBn or
raise ibank to offset the 1% endogenous increase in Ms.
Diagram 7.2 Backloaded Case
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In the above diagram, we feed the higher interest rate determined in the
goods/bonds market into the money market. We consider the simplest case where
the wealth effect on money demand is just big enough to offset the negative
interest rate effect, so the net change in Md (…) = 0. The equilibrium adjustment in
the money market with the vertical Ms curve would be from A to B, with no
change in P. Now, the equilibrium movement is from A to C; P has to rise to
absorb the increased money supply induced by higher interest rates. If P was
assumed fixed for the short-run problem, then we would have to achieve B again,
and the only option would be to shift the full Ms-curve left (from D through B) by
lowering MBn or raising ibank.
The generics of all the cases are fairly simple once you get the basic logic down.
You should work out a variety of 3-sector cases with both Y1 and Y2 changing.
**While bank rate changes get all the headline attention these days (and are presumed to be the most important
monetary instrument from an implementation, efficiency and signaling standpoint), it should be noted that openmarket policies altering MBn could potentially achieve the same effects through the bond market rather than a bank
lending channel. If the objective was a monetary ‘tightening’, raising ibank would achieve this, but so would a ‘large’
open market sale of government bonds. The latter would lead to a rightward shift in the Bs-curve, creating excess
supply of bonds, lowering bond prices and raising bond yields. In the money market, the Ms-curve would shift left,
as MBn and currency in circulation have declined. This is the same qualitative effect on the Ms-curve as the increase
in ibank.
9. New approaches to understanding the banking system have emerged from
detailed studies of the 2007/8 financial crisis in the US. The ‘traditional’ banking
model presented above would suggest that the funds that financed the lending/real
estate boom before 2007 naturally came from large increases in commercial bank
deposits. Except that the data shows that deposits (of the big US banks in
particular) did not increase in the years of the lending boom. So where did all the
funds come from? From transient financiers – which comprise a ‘shadow’ banking
system – who entered the market to realize quick arbitrage profits.
Shadow banks are organizations that (selectively) perform the same borrowing/
lending functions as standard banks, but are not subject to banking regulations.
Accordingly, they have a natural tendency to take on greater risk. A key
explanation for their existence in the housing boom prior to 2007 involves credit
market ‘distortions’: namely, if credit markets are imperfect, there can be a wedge
between borrowing and lending rates, so (arbitrage) profits can be made by
borrowing cheaply from one credit source and lending to other individuals at a
higher rate. This becomes particularly relevant in the case of high-risk individuals
who do not qualify for conventional mortgage finance. Here there is a central role
for shadow banks in the creation of high-risk ‘subprime’ mortgages – a defining
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component of the ‘toxic’ mortgage-backed securities that played such a role in the
2007/8 Financial Crisis and which underlay serious instances of bank failure.
There is now strong interest in modeling how a shadow banking system can
interact with a formal banking system, adding to the pool of loanable funds during
boom periods, yet largely disappearing at other points in the business cycle.
10. The Collapse of SVB: In early March 2023, we had our memories taken back
to the bank failures of the 2007/8 Financial Crisis (Lehmann Bros. and Washington
Mutual being among the most significant) as Silicon Valley Bank closed its doors
in 48 hours. It was America’s 16th largest bank. Here the bank’s potential balance
sheet problem emerged from the boom-bust nature of the technology sector itself.
Over the earlier pandemic years, technology startups did very well, and deposited
ample amounts into SVB. However, given a quick deterioration of circumstances
thereafter, the bank was able to find relatively few borrowers for new venture
capital projects or mortgages, and their large excess of funds had to be invested
elsewhere. The funds went mainly into low-yield Treasuries and other long-term
bonds, with the intention of holding them to maturity. The Federal Reserve’s
sequence of interest rate hikes over 2022 systematically lowered the value of these
long-term bonds, which in turn created pressure on the asset side of the bank’s
balance sheet. (If Total Assets – Total Liabilities < 0, the bank is deemed
insolvent.) While the problem was acknowledged by SVB by September 2022, it
did not (or could not) re-position its asset mix to sell the low yield bonds and pick
up higher yield bonds that reflected the new expected inflation. So, the basic
problem remained.
However, more factors were at play here. In the technology downturn of this
period, many startups needed liquidity simply to keep going – to meet payroll, etc.
– so an abnormal number needed to withdraw money from their accounts at SVB.
This was unrelated to any balance sheet problem the bank might have faced. Also
unrelated is the fact that many depositors were corporate depositors that held vastly
bigger accounts than would be covered by the FDIC’s deposit insurance limit of
250k (in Canada, it is 100k per account): in fact, only around 10% of the deposits
were so insured. Clearly, if one combines the large ongoing volume of withdrawals
with even a hint that the bank’s liquidity position was vulnerable, then the many
corporate depositors with large uninsured balances would panic and want to exit
too. This is in fact what created a classic bank run: even if these depositors did not
need the money for immediate purposes, they would run to withdraw their funds
for fear of losing their uninsured balances if the bank actually failed – that they
would end up as ‘residual claimants’. This implied a ‘race’ to withdraw. SVB
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could not successfully sell assets to fend off this bank run. It sold $21 billion in
bonds the week before its closure, taking a $1.8 billion loss, and seeking money
from investors to offset that loss. But the capital call failed, and that is basically the
end of the story. SVB’s demise likely would not have occurred in a precipitate
fashion if more depositors were retail, so perhaps 80% of their deposits were
insured, or in a sector which was less boom/bust in the first place. Nonetheless,
there would be little reason to expect a ‘contagion’ of bank failures outside of the
immediate technology/finance sector. The situation was fairly unique.
© Geoffrey Newman 2023 ALL RIGHTS RESERVED
These notes and study problems are for personal use only, and are not to be
used for a business purpose or uploaded to any external website.
Study Problems:
1. After the financial crisis, regulators put more pressure on banks to hold greater
reserves. What would be the effect on the money multiplier and money supply?
Suppose that, in this regulatory setting, the central bank also wanted to keep the
money supply (liquidity) constant. What policies would the central bank use?
2. In our model with a banking sector, suppose that neither the currency-deposit
ratio nor the reserve ratio are sensitive to i. Would the upward-sloping Mscurve be steeper or flatter as compared to a model where they were sensitive?
3. Suppose c = 0.2 and rd = .05. Let both values increase by 0.1 in the middle of
the financial crisis. Does the money multiplier rise or fall?
4. Consider a Y1 negative shock, Y2 constant. In the endogenous money supply
model, how does the change in P differ from the result in the model without
banks.
5. Consider a Y2 negative shock, Y1 constant in the model with the upward
sloping Ms curve. How does the change in P differ from the change in the
model with the vertical Ms? Suppose the effect of wealth on money demand is
very small for this problem.
6. For #5 above, assume now that P was fixed in the short run? What change in
the overnight bank rate might be advocated by the central bank?
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