**PS 3 Solutions.**

**1a **This is true: with a higher interest rate,
there is less investment (investment as a function of output shifts down), so
demand shifts down, which gives a smaller level of equilibrium output.

**1b** This is true: a fiscal expansion increases Y
and I, a monetary expansion increases Y and decreases i. So combining these two
may lead to unchanged interest rates and higher output.

**1c** False: Monetary policy means changing M, which
enters only the LM curve (financial markets). The IS curve stays the same.

**2a** The LM curve comes from equating money supply
and demand: 280 = Y/10*i, so

i = Y/2800.

**2b** The IS curve represents equilibrium output as a
function of the interest rate. To derive it, we need to equate the demand and
the supply of goods:

Y = C+I+G = 50+0.5*(Y-10)+20-100*i+10=75-100*i+0.5*Y.

This implies

Y = 150-200*i.

**2c** Solving for the intersection of IS and LM:
2800*i = 150-200*i, so

i = 150/3000 = 0.05 = 5%.

Output is then

Y = 150-200*0.05 = 140.

**2d** If G increases by 15, it does not change the LM
curve but shifts the IS curve up (to the right). It becomes Y = 180-200*i. The
new equilibrium is given by

2800*i = 180-200*i, so

i = 180/3000 = 0.06 = 6%.

Output is

Y = 180-200*0.06=168,

and investment becomes I = 20-100*0.06=14 (it used to be 15). This is what is called the crowding out effect of government spending: higher G means higher interest rates, which then might decrease investment: higher output is good for investment but higher interest rates are bad. In our example, investment did not depend on output, so an increase in G had only the negative effect on investment.

**2e** Now it is the LM curve that shifts out (to the
right), and becomes i = Y/3550. IS stays the same; so the equilibrium interest
rate is given by

3550*i = 150-200*i, so i = 150/3750 = 0.04 = 4%.

Output is Y = 150-200*i = 142, and investment is 20-100*i = 16. A monetary expansion increases output and decreases the interest rate, which both help investment. Here we did not have the positive effect through output, only the boost through lower interest rates.

In general, both fiscal and monetary policy increase output, but they have opposite effects on the interest rate; which means that a fiscal expansion has an ambiguous effect on investment, while a monetary expansion has a clear positive effect.

**3a** R^{D} =1/6*(1-0.1)* M^{D},
since the demand for checks is (1-0.1)* M^{D}, and 1/6 is the amount of
reserves.

**3b **

H^{D} =0.1* M^{D} + R^{D} =
0.1*$Y*L(i)+1/6 *0.9*$Y*L(i)=$Y*L(i)*(6/60+9/60)=0.25*$Y*L(i)

**3c** H= H^{D} =0.25*$Y*L(I); the graph is a
downward sloping line and a vertical supply

M^{D} =M=4*H (Using that H=0.25M)

The intuition for M being 4 times bigger than H is the following: H goes into the hands of consumers and banks. Consumers use it as currency (C); banks will end up keeping it as reserves. How much reserves banks will keep?

Suppose someone gets a dollar then she will deposit 0.9
of it to a bank. The bank puts aside 1/7 of it, and 6/7 is given to someone (as
a loan, or via purchasing bonds with it). That person deposits 90% of the money
to the bank again 0.9*1/7*6/7 is put aside, 0.9*6/7*6/7 is again given to
someone, etc. In the end, the amount of total checkable deposits is
0.9*6/7*(1+0.9*6/7+(0.9*6/7)^{2}+
) = 0.9*6/7*1/(1-0.9*6/7); and
reserves are 0.9*1/7*(1+0.9*6/7+(0.9*6/7)^{2}+
) = 6*R.

Now C = 0.1*M, CD = 0.9*M, so C =1/9*CD = 6/9*R. Hence H = C+R = 15/9*R, so R=9/15*H, CD = 54/15*H and C = 6/15*H. Therefore M = C+CD = 6/15*H+54/15*H=60/15*H=4*H.

**3d** If the 0.1 goes up, then the 0.25 multiplier
(which is 1/6+5/6*cash-to-money ratio) goes up, so H-demand goes up, so the
demand line shifts up in the H diagram, the interest rate increases.

The intuition behind this result is the following: people
want to hold more of their money in currency. This has two effects on the
demand for H: a direct increase (more currency wanted by the public) and an
indirect decrease: less checkable deposits, hence less reserves. The decrease
in checkable deposits is the same as the increase in currency, but reserves are
only 1/6 of deposits, so the net effect is an increase in H^{D}.

In the M-world, the 4 goes down (1/0.25), so money supply goes down while demand stays the same. Again, the same interest rate increase.

The intuition is also similar to the H-case: the
increased desire for currency will lead to smaller reserves, therefore smaller
deposits. Since the multiplication of H into M is achieved through bank
deposits, so if there are smaller deposits around, the multiplication effect is
weaker, hence the same amount of H implies a smaller amount of M^{S}.

The CB should expand the supply of H (hence M) to maintain the same interest rate. In more details: If the interest rate is to be kept constant, then M must also stay constant. So the amount of currency will increase, and the amount of checkable deposits will decrease by the same amount. This means a 1/6 decrease in reserves so H must increase (1-1/6 is positive).

Note that banks might be in temporary problems here: when
people start contracting their deposits, banks have only 1/6 of them available
immediately, the rest is lent to some firms or consumers. After a while, banks
will be able to get all their funds back and then pay all of their depositors
but there might be a period when they need extra cash. This would be an example
of a *bank run.*

**3e** In both diagrams, the demand line shifts up,
thus the equilibrium interest rate increases. With more income, people will
want to make more transactions, so they want to hold more money. At the given
interest rate, however, there is not enough money to satisfy their needs --
interest rates must rise to dissuade people from holding more money.