Banks, Non-Banks and the Interest Rate Effect

In my last post, I used the balance sheet framework of a simple imaginary economy to look at the relationship between bank lending, non-bank lending and payments money.  I thought it would be useful to use this framework to construct a little model.  I particularly wanted to look at the way non-bank lending and bank lending differ, even though both are expansionary.

A full list of the equations, variables and parameters for what follows is set out at the end of this post.

Many of the equations simply set up the accounting relationships shown in the balance sheet matrix.  This is the same as before and set out below:

	Households	Firms	Banks	NBFIs	Net
Money	Mh		-M	Mf	0
Time deposits	D		-D		0
Savings accounts	S			-S	0
Loans		-L	Lb	Lf	0
Total	V	-L	0	0	0

The financial sector is represented by two types of lender: banks and non-bank financial institutions (NBFIs).  Both entities make loans to firms (households are assumed not to borrow).  Banks offer time deposits to households and NBFIs offer something similar which we will call savings accounts.  The key difference between banks and NBFIs is that only banks can offer checking accounts.  As we are going to assume that only checking accounts can be used for payments, we're going to define money as the balance of such accounts.  Because they cannot offer checking accounts, NBFIs have to attract funds into their savings accounts before they can make loans.  As the timing of new loans does not necessarily coincide with the timing of new savings, they also hold some money balances as a float.

I am assuming that households behave as follows:

- they hold money in a fixed ratio to income;

- they have a long term target ratio of total financial assets to income, but move towards this incrementally; and

- they allocate their non-money assets between deposits and savings accounts on the basis of the rates paid on each.

Investment is driven by loans.  I am assuming the demand for loans exceeds supply, so the quantity of loans is determined by credit constraints imposed by banks and NBFIs.  The amount of lending by each can therefore be treated as exogenous.  As we are assuming firms do not retain any money, their spending must be equal to the change in loans.  

I am assuming here that neither consumer spending, nor investment depends on interest rates.  This makes it easier to isolate the effects we want to observe.  We could easily relax this assumption if we wanted.

This set up now means we can compare the effect of an increase in bank lending to an increase in NBFI lending.  The charts below show this for the same absolute increase in the value of loans made by each.

The first thing to note is that the effect on GDP is the same for bank lending as for NBFI lending (this depends on my assumption that spending is not affected by the change in interest rates described below).  New loans have a temporary effect on investment when they are made and a permanent, but lesser impact on consumption due to the increase in overall financial assets.

The difference between the two is in what happens to interest rates.  As we can see from the charts below, an increase in NBFI lending tends to push up rates offered to savers, compared with an increase in bank lending.

If they want to increase lending, NBFIs need to attract more savings.  To do this they will have to raise their savings rates.  This will tend to pull funds away from banks.  The total funds placed with banks cannot fall because it must always be equal to the total of bank loans, but what will happen is that banks will find the ratio of deposits to money falling.  Banks will react to this by raising their own deposit rates to try to attract more longer term funds.

Bank lending will tend to have the reverse effect.  Initially, bank lending adds to checking account balances and banks will need to raise their rates to try to attract savers into longer term deposits.  NBFIs will need to follow suit if they are not to lose funds themselves.  However,  once the initial boost from the new loans has worn off and income falls, household demand for money holdings also falls and they try to invest more into term deposits and savings accounts.  However, banks and NBFIs do not at this stage need more funds and so they can cut their savings rates.

To illustrate further what is going on, the graphs below show the deviations in the balances of household money holdings, deposits and savings accounts.

So we can see that even though bank lending and non-bank lending might both be expansionary, they are not equivalent.  The difference reflects their relative position in the hierarchy of money.  NBFIs need bank money to make payments.  Banks do not need NBFI claims.  In addition to facilitating debt funded expenditure, bank lending will also increase liquidity in a way that NBFI lending will not.  Banks and NBFIs are therefore different.  But this does not mean that NBFI lending does not matter.  

Equations

GDP is consumer spending plus investment by firms.

Y = C + I

Consumer spending and investment are defined by accounting identities.  Consumer spending is income less increase in assets (i.e. saving).

C = Y - ( V - V_t-1 )

Investment is equal to the increase in loans.

I = L - L_t-1

Households holdings of money are in a fixed ratio to GDP.

Mh = α_m . Y

Households also wish to hold total financial assets in an fixed ratio to GDP, but this is achieved by a gradual adjustment based on the difference between a target and actual value.

V = V_t-1 + ε_v . ( V* - V_t-1 )

V* = α_v . Y

The proportion of non-money assets that households wish to hold as savings accounts is a function of the interest rate differential between deposits and savings accounts.

S = ( V - M) . ( λ_h0 + λ_h1 . ( rs - rd ) )

Deposits are then the balancing item.

D = V - Mh - S

The amount of loans by banks and NBFIs adjust incrementally towards a target level.

Lf = Lf_t-1 + ε_f . ( Lf* - Lf_t-1 )

Lb = Lb_t-1 + ε_b . ( Lb* - Lb_t-1 )

Banks and NBFIs each have to make a portfolio decision based on rates.  The proportion of loans that banks would prefer to finance with money, as opposed to deposits depends on the deposit rate. 

M = Lb . ( λ_b0 + λ_b1 . rd )

The amount of money that NBFIs wish to hold depends inversely on the savings rate.

Mf = Lf . ( λ_f0 - λ_f1 . rs )

The above two equations are re-arranged as equations in the interest rate.  In addition to enabling solution, this reflects the idea that banks and NBFIs set the rates they offer and then take all money at those rates.

Lastly, some accounting identities.

M = Mf + Mh

L = Lf + Lb

Mf = S -Lf

Variables

Name	Description	Start Value
C	Consumer spending	100
D	Time deposits	120
I	Investment by firms	0
L	Loans	300
Lb	Loans by banks	200
Lb*	Target loans by banks	200
Lf	Loans by NBFIs	100
Lf*	Target loans by NBFIs	100
M	Total balance in checking accounts	80
Mf	NBFIs' balance in checking accounts	20
Mh	Households' balance in checking accounts	60
S	Savings accounts with NBFIs	120
V	Total household financial assets	300
V*	Target household financial assets	300
rd	Rate on deposit accounts	3.00%
rs	Rate on savings accounts	3.00%

Parameters

Name	Description	Value
αm	Household money holding ratio	0.6
αv	Household wealth ratio	3.0
εb	Bank loan adjustment rate	0.25
εf	NBFI loan adjustment rate	0.25
εv	Household wealth adjustment rate	0.1
λb0	Bank portfolio parameter	0.34
λb1	Bank portfolio parameter	2.0
λf0	NBFI portfolio parameter	0.26
λf1	NBFI portfolio parameter	2.0
λh0	Household portfolio parameter	0.5
λh1	Household portfolio parameter	10.0

The graphs show the result of an increase of 10 in the value of either Lf* or Lb*.