ECOS3021 Business Cycles and Asset Markets University of Sydney
2021 Semester 1
Tutorial #4
1. Denote the price of a nominal bond that pays one dollar next period as St1. Now denote the price of a nominal bond that pays one dollar two periods from now as St2. Both bonds can be purchased today, but the first pays off next period while the second pays off two periods from now.
1.a) Write down an expression for the nominal rate of return on each bond as a function of the bond price. Rearrange this expression to express the bond prices as functions of the nominal rates of return.
ANSWER:
The nominal return on each bond are:
n,1 1−St1 n,2 1−St2
rt = St1 , rt = St2 The bond prices are then expressed as:
S1= 1 , S2= 1
t 1+rn,1 t 1+rn,2
tt
1.b) Now write down expressions for the real rates of return on each bond as a function of the bond
prices and current and future price levels. Use these expressions to find the Fisher Equation for each
of the bonds. Hint 1: We can write the gross inflation rate between today and some date h periods in
the future as: Pt+h = Pt+h Pt+h−1 · · · Pt−1 . Hint 2: You may assume that all higher order terms are Pt Pt+h−1 Pt+h−2 Pt
approximately zero (i.e. for any variables xt and yt, the higher order terms xt × yt ≈ 0). ANSWER:
The real rate of return on the first bond is: r1=1/Pt+1−St1/Pt=1 Pt −1
t St1/Pt St1 Pt+1 1+r1=1 Pt
t St1 Pt+1
= (1 + rn,1) 1
1
t 1+πt+1
The real rate of return on the second bond is: r2=1/Pt+2−St2/Pt=1 Pt −1
St2 Pt+2
t St2/Pt 1+r2=1 Pt
t St2 Pt+2
= 1 Pt Pt+1
St2 Pt+1 Pt+2 = (1 + rn,2) 1
1
t 1+πt+1 1+πt+2
To find the Fisher equation for the first bond, rewrite with the nominal rate of return on the left hand side, and then expand the expression on the right hand side:
(1+rn,1)=(1+r1)(1+π ) t t t+1
=1+rt1 +πt+1 +rt1πt+1 ≈ 1 + rt1 + πt+1
where the final equality follows from the approximation that higher order terms are equal to zero i.e. rt1πt+1 ≈ 0.
To find the Fisher equation for the second bond, follow the same steps as above: (1+rn,2)=(1+r2)(1+π )(1+π )
t t t+1 t+2
= 1 + rt2 + πt+1 + πt+2 + rt2πt+1 + rt2πt+2 + πt+1πt+2 + rt2πt+1πt+2 ≈1+rt2 +πt+1 +πt+2
2
2. Consider the model from Lecture 4 for household choices over consumption, money holdings, and nominal bonds:
max logC1 +ωlog M +βlogC2 C1,C2,M,B P1
s.t. P1C1+M+B=P1Y1
P2C2 = P2Y2 + M + B(1 + rn)
In class, we showed that the Euler equation is:
1 =β(1+rn)P1 1 (1)
C1 P2 C2 And the money demand equation is given by:
M rn−1
P =ωC1 1+rn (2)
1
2.a) Rewrite the money demand equation so that the nominal interest rate (i.e. the price) is expressed as a function of money demand (i.e. the quantity). Plot this money demand function on a graph with the interest rate on the y-axis, and money demand on the x-axis.
ANSWER:
The money demand equation is:
M rn−1 P =ωC1 1+rn
n M 1
1
⇒ M rn = ωC1(1 + rn)
P1
⇒r P −ωC1 =ωC1
n ωC1 ⇒r =M−ωC
P1 1
which, on the final line, is expressed as the interest rate in terms of money demand.
We can plot the money demand function in (rn,M/P1)-space as:
3
rn
MD
M P
2.b) On a new graph, again plot the money demand curve from the previous question. Now add a money supply curve, such that the quantity of money supplied is constant/inelastic with respect to the interest rate. Be sure to label the equilibrium interest rate. On your graph, show what happen if there is a positive money demand shock, i.e. an increase in the parameter ω. Label the new equilibrium interest rate and quantity of money.
ANSWER:
If the money supply curve is inelastic with respect to the interest rate, it must be vertical in (rn, M/P1)-space.
4
rn
rn
MS
MD
M1 M P1 P
An increase in money demand, ω, leads to a shift up in the money demand curve. We can see this by taking the derivative of the money demand equation with respect to ω:
2 >0
Since the money supply curve is constant, the increase in the money demand curve leads to an increase in the real interest rate: Since households want to hold more money, but the money supply has not changed, the nominal interest rate on bonds must increase in order to induce households to hold bonds and keep their money holdings at their original quantity.
∂ r n C 1 ω C 12 ∂ω =M −ωC+M
P1 1 P1 −ωC1
5
rn
rn′
rn
MS
M1 M P1 P
MD′ MD
2.c) Combine budget constraints and the money demand equation (2) to find an expression for the bond- demand equation as a function of C1. As above, rewrite the equation to express the bond demand equation with the interest rate as a function of real bond holdings, B/P1. Why is the bond demand curve upward sloping? Use both math and economic intuition in your answer.
ANSWER:
The first period budget constraint after dividing through by P1, is: C1 + M + B = Y1
P1 P1
Now substitute in the money demand equation (2), and rearrange for real bond holdings:
rn −1 B
C1+ωC1 1+rn +P =Y1 1
1 B
⇒C1 1+ω 1+rn +P =Y1
1
B 1 ⇒P =Y1−C1 1+ω 1+rn
1
Now rearrange the equation so that rn is on the left and bond holdings are on the right: rn = −ωC1
B −Y1 +C1(1+ω) P1
6
Note that B − Y1 + C1(1 + ω) < 0, so the right hand side is positive. Additionally, P1
∂rn B 2
B =fracωC1 −Y1 +C1(1+ω) >0
∂P1 P1
so the bond demand curve is upward sloping. Why is this the case? In order to induce households
to hold more bonds, the nominal rate of return on those bonds has to increase.
2.d) Draw the bond demand curve, with B on the x-axis, and rn on the y-axis. Now suppose a central bank P1
determineshowmanybondsareavailableinthebondmarket.1 Thesupplyofbondsisconstant/inelastic with respect to the interest. Illustrate equilibrium in the bond market, being sure to label the equilibrium interest rate.
ANSWER:
The bond demand curve is upward sloping, while the bond supply curve is vertical: rn
BS BD
rn
BB P1 P1
2.e) Suppose the central bank increases the supply of bonds (i.e. by selling bonds into the market). What
is the effect on the equilibrium nominal interest rate? Illustrate your answer. What is the effect of this
1In practice, many fiscal, banking, and corporate institutions issue bonds. But central banks buy and sell large numbers of these bonds in order to influence the number available in the market for bondholders to purchase.
7
policy on real money holdings? What does this suggest about the central bank’s ability to conduct monetary policy using either the money or bond supply?
ANSWER:
An increase in the supply of bonds results in an increase in the nominal interest rate: rn
BS BS′ BD
rn′ rn
BB′ B P1P1 P1
Because the nominal interest rate rises, real money balances held by households decline. This can be seen from the money demand equation.
Because the central bank can manipulate nominal interest rates by either changing the money supply or changing the bond suppl, the ability to conduct monetary policy is unaffected by the choice of instrument. That is, the central bank can achieve the same monetary goals by either choosing the money supply or by choosing the bond supply.
8