Hello yvonne,
These three problems are actually quite similar. Let's see each of them.
Question 1
First of all, we need to define money supply. In this framework, we'll
define money supply (M) as the sum of the total cash held by people
(C) plus the the total sum of deposits at the bank (D):
M = C + D
In this case, C = 0 because holding currency is outlawed.
Now, when you deposits $100, D immediately increases by $100. The bank
will then want to lend that money. Since it's required to hold 7.5% in
reserves, then it will be able to lend:
(1 - 0.075) * 100 = $92.50
Someone will borrow that money, but since holding currency is
outlawed, the money will be deposited into the bank. So, now D will
be:
100 + (1 - 0.075)*100
Now, once again, the bank will want to lend more money since it has
new deposits for $92.50. Given that it must hold 7.5% in reserves, the
amount of money it can lend becomes:
(1 - 0.075)*(1 - 0.075)*100 = (1 - 0.075)*92.50 = $85.5625
Again, the borrower will have to redeposit the full amount. Therefore,
total desposits at this time would be:
D = 100 + (1 - 0.075)*100 + (1 - 0.075)*(1 - 0.075)*100
D = 100 + 100*(1 - 0.075) + 100*(1 - 0.075)^2
The idea should now be clear. The next time, the bank will lend
100*(1 - 0.075)*(1 - 0.075)*(1 - 0.075) = 100*(1 - 0.075)^3 = $79.14...
which will be redeposited by the borrower, taking total deposits to:
D = 100 + 100*(1 - 0.075) + 100*(1 - 0.075)^2 + 100*(1 - 0.075)^3
D = 100 + 92.50 + 85.5625 + 79.14
As you can see, the increase in deposits is smaller in each cycle.
This process is repeated an infinite number of times until the money
available for lending becomes zero. The formula for the result of this
infinite sum is:
D = 100*(1/0.075) = 1333.33...
Total deposits increase by $1,333.33. Therefore, money supply
increases by the same amount. Notice that the general formula is:
(Inital Deposit / Reserve Requirement)
Question 2
The idea here is exactly the same. The bank now only lends a fraction
(1 - 0.075 - 0.0375)
of what it receives in each cycle. Therefore, this would be (for the
purpose of this problem) the same case as if the bank had a required
reserve requirement of 7.5 + 3.75 = 11.25%.
Applying the formula we found above, we get that money supply increases by:
100/0.1125 = $888.88
Question 3
In this case, we have that cash must be equal to 10% of total
deposits. Let's call 'c' to this ratio (so c=0.1). We're told that you
deposit $100 into the bank. However, if you're not holding any money,
this would violate the assumption that you hold 10% of the deposits in
cash. Therefore, let's assume that you have $100 and deposit an amount
that would leave your cash holdings equal to 10% of your deposits. We
have that, at first:
C + D = 100
Also, C = cD (because you hold in cash a fraction c of your deposits). Therefore,
cD + D = 100
(1 + c)D = 100
D = 100/(1+c)
So you start by depositing 100/(1+c) (which is $90.90... in your case)
and holding in cash [c/(1+c)]*100 (which is $9.09... in your case).
When the bank receives that deposit, it will hold a part of it in
reserves and lend the rest. The amount that will be lent is clearly
(1-r)*100/(1+c).
Now, the borrower will hold a fraction of that in cash, and the rest
will be redeposited. We've seen that if you have 100, you must deposit
100/(1+c) and hold [c/(1+c)]*100 in cash in order to keep your cash as
10% of your deposits. Therefore, the borrower that now has
(1-r)*100/(1+c) will deposit:
[(1-r)*100/(1+c)] / (1+c) = 100*(1-r)/(1+c)^2
and hold in cash:
c*100*(1-r)/(1+c)^2 = [c/(1+c)]*100*(1-r)/(1+c)
Therefore, total deposits and cash up to this point are:
D = 100/(1+c) + 100*(1-r)/(1+c)^2
C = [c/(1+c)]*100 + [c/(1+c)]*100*(1-r)/(1+c)
So the bank has received new deposits for 100*(1-r)/(1+c)^2. It will
keep a fraction of that in reserves and lend the rest. Total amount
lent will now be:
100*(1-r)^2/(1+c)^2
Following the same reasoning as before, the borrower will deposit:
100*(1-r)^2/(1+c)^3
and keep in cash:
c*100*(1-r)^2/(1+c)^3 = [c/(1+c)]*100*(1-r)^2/(1+c)^2
Therefore, up to this point, total deposits and currency holding are:
D = 100/(1+c) + 100*(1-r)/(1+c)^2 + 100*(1-r)^2/(1+c)^3
C = [c/(1+c)]*100 + [c/(1+c)]*100*(1-r)/(1+c) + [c/(1+c)]*100*(1-r)^2/(1+c)^2
Once again, this procedure is repeated "infinite" times
(theoretically) until there is no more money left for the banks to
lend. We get the following results:
D = 100/(c+r)
C = 100c/(c+r)
Therefore,
M = 100*(1+c)/(c+r)
Applying your values c = 0.1 and r = 0.1125, we get that the change in
money supply caused by the initial deposit is:
M = 100*(1.1)/(0.1+0.1125) = $517.64
You can find additional information on this subject at the following site
http://www.mhhe.com/economics/mcconnell15e/graphics/mcconnell15eco/common/dothemath/complexmoneymultiplier.html
Google search terms
"money multiplier"
://www.google.com.ar/search?sourceid=navclient&ie=UTF-8&rls=GGLJ,GGLJ:2006-23,GGLJ:en&q=%22money+multiplier%22
I hope this helps! If you have any doubt regarding my answer, please
don't hesitate to request clarification before rating it. Otherwise, I
await your rating and final comments.
Best wishes!
elmarto |