SCHOOL OF ECONOMICS
FACULTY OF ARTS AND SOCIAL SCIENCES
THE UNIVERSITY OF SYDNEY
ECOS 3021: BUSINESS CYCLES AND ASSET MARKETS
Semester 1 2021
Final Exam
Duration: 2 hours
Total points: 100
This is an open book exam. Handwrite your answers clearly on plain, white paper and
then either scan or photograph your papers. Submit your exam answers via the dedicated
upload box in the Canvas exam webpage. Note that an extra 30 minutes are provided for
scanning and uploading. No exams will be accepted via email. No late exams will
be accepted.
There are 3 questions in this exam. You are required to answer all parts. Please read
the questions carefully before writing your answer. You may combine mathematics, hand-
drawn figures or graphs, and explanations in words. In answering the questions, make sure
you show your working, explain your steps, and list any assumptions that you make while
deriving results.
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1. In this question, we will consider the effects of fluctuations in the business cycle (i.e.
consumption) on the price of risk-free bonds. A bond can be purchased at time t for
price Pt, and repays a value of 1 in the following period (t+ 1).
Study the following figures describing the path of consumption over time. Over time, an
investor’s beliefs about the path of future consumption vary. Specifically, the investor’s
beliefs differ between the shaded regions and the non-shaded regions. In the shaded
regions (i.e. the area up until the thick red vertical line) an investor believes that
consumption will be constant forever at the level C. In the non-shaded regions (i.e. in
the area from the red line onward) the investor learns about the evolution of the entire
future consumption path.
(A) In Figure 1, a recession occurs at time t2. At time t1, the investor learns that that
the recession will occur. How does the price of the bond, Pt, evolve over time? Is
the bond price pro-cyclical or counter-cyclical? You may use figures, equations, and
words to explain your answer. (10 Points)
Figure 1:
Time (t)
Ct
C
t0 t1
C
t2
Ct
t3
(B) In Figure 2, a recession occurs at time t2. At time t1 the investor does not yet know
that the recession will occur. At time t2, the investor learns that the recession has
already begun. How does the price of the bond, Pt, evolve over time? Is the bond
price pro-cyclical or counter-cyclical? You may use figures, equations, and words to
explain your answer. (10 Points)
2
Figure 2:
Time (t)
Ct
C
t0 t1
C
t2
Ct
t3
(C) In Figure 3, at time t1 the investor believes there is an increase in uncertainty over
possible outcomes for consumption at time t2. The investor believes that with equal
probabilities, any one of the consumption outcomes C1, C2, C3, C4, C5 may occur at
time t2. At time t2, the uncertainty is resolved and consumption is revealed to
be Ct2 = C5. How does the price of the bond, Pt, evolve over time? Is the bond
price pro-cyclical or counter-cyclical? You may use figures, equations, and words to
explain your answer. (15 Points)
Figure 3:
Time (t)
Ct
C3
t0 t1
C1
C2
C4
Ct2 = C5
t2
Ct
t3
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2. (A) Consider a homeowner that would like to use a mortgage to finance the purchase
of a house, and a bank that is deciding whether to lend to this homeowner. Why
might a bank charge higher interest rates on mortgages with larger loan-to-value
ratios? Provide a brief explanation in words. (7 Points)
(B) A potential home-buyer in Sydney is considering buying a house for $1 million.
This buyer will require a mortgage to purchase the house. Suppose a bank offers
a maximum loan-to-value ratio of 80%. How much can the home-buyer borrow
from the bank? Suppose the house price rises by 10% - how much can the home-
buyer borrow now? Finally, how much does the bank need to reduce the maximum
loan-to-value ratio to ensure the home-buyer borrows no more than they could have
borrowed at the original house price of $1 million? (9 Points)
(C) A potential home-buyer in Sydney is considering buying a house for $1 million. This
buyer would like to use an interest-only mortgage to purchase the house. In this
case, when the home-buyer borrows a mortgage of size B at net interest rate r, then
the home-buyer only repays r × B each period. Suppose a bank offers a maximum
payment-to-income ratio of 30% for this mortgage. Let the interest rate on the loan
be r = 0.05, and the home-buyer’s income is Y = $100, 000 per year. How much
can the home-buyer borrow from the bank? Suppose the home-buyer’s income rises
by 10% - how much can the home-buyer borrow now? Finally, how much does the
bank need to reduce the maximum payment-to-income ratio to ensure the home-
buyer borrows no more than they could have borrowed at their original income of
$100,000? (9 Points)
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3. Consider the problem of a household purchasing a house and borrowing to finance the
cost of purchase:
max
C1,C2,B
log(C1) + β log(C2)
s.t. C1 + P1H1 = Y1 +B
C2 + (1 + r)B = Y2 + P2H
The household pays for the house in period 1 at price P1 and sells the house in period 2
and price P2. The household borrows B in period 1 to finance the purchase of the house
and repays the mortgage with interest in period 2, (1 + r)B.
(A) Write down the inter-temporal budget constraint for this household. (3 Points)
(B) Write down the household’s first order conditions, and solve for the optimal consumption
choices C1 and C2. (7 Points)
(C) Now suppose that the household’s borrowing is constrained by a Payment-to-Income
Ratio constraint. Define the payment-to-income ratio as a function of borrowing B,
net interest rate r, and period 2 income Y2:
PTI ≡ rB
Y2
This ratio is restricted by a maximum payment-to-income constraint, so that the
household problem is now:
max
C1,C2,B
log(C1) + β log(C2)
s.t. C1 + P1H1 = Y1 +B
C2 + (1 + r)B = Y2 + P2H
rB
Y2
≤ θ [Maximum PTI Ratio Constraint]
Rewrite this household problem in terms of C1, C2, and PTI. To do this, rewrite
the budget constraints to eliminate B so that the constraints are instead functions
of the PTI decision variable. (5 Points)
(D) Draw the period 2 budget constraint on a figure with PTI on the X-axis and C2 on
the Y-axis. Now suppose that the household would like to borrow so much that it
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is constrained by the maximum PTI ratio. That is, the household’s optimal choice
of PTI is:
PTI∗ = θ
To illustrate the household’s choice, draw an indifference curve that intersects the
period 2 budget constraint at PTI = θ. Label the optimal choice of C2 in your
figure.
Hint: Be sure to use the period 2 budget constraint from your answer to Part (C).
From this budget constraint, what is the maximum possible choice of C2? What is
the maximum possible choice of PTI? Label these points on your figure. Assume
that the PTI constraint θ is less than the maximum possible PTI ratio derived from
the budget constraint PTImax. (7 Points)
(E) On a new figure, show what happens when there is a reduction in the maximum
payment-to-income ratio, θ. Assume that the household is now constrained at the
new maximum payment-to-income ratio. Be sure to draw the new indifference curve.
What is the effect on B? What is the effect on C1? What is the effect of this change
on C2? Provide brief explanations for your answers. (8 Points)
(F) On a new figure, show what happens when there is a reduction in the price of houses
in period 2, P2. Assume that the household is again constrained at new maximum
payment-to-income ratio. Be sure to draw the new indifference curve. What is the
effect on consumption in period 2, C2? What is the effect of the fall in period 2
house prices on consumption C2 for an unconstrained household? Is the constrained
or unconstrained household more sensitive to the change in house prices? Provide
brief economic intuition for your answer.
Hint: You already solved the decision problem for the unconstrained household in
Question (B). (10 Points)
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