The final exam will be 2 hours (plus 10 minutes of reading time) on TBD. Nonprogrammable calculators are allowed and one A4 sheet of hand written notes. Notes
may be double sided. Exam booklets will be provided. You must use a pen to write
your answers.
Examinable Material and Expectations:
1. The exam will cover material from the whole course, although with more content reflecting
material since the midterm. Anything covered in class, in the tutorials, or in the problem
sets is examinable.
a. We will review topics in the last lecture.
2. I will provide relevant formulas such as production functions to be used in answering a question
(see questions below to see examples of what material will be provided and what you will be
assumed to know).
3. You will be expected to understand the “economics” behind any equations provided. Correct
math and incorrect logic may result in zero points earned.
4. In answering questions, be precise, showing all of the steps, and indicate if you are making
any assumptions along the way.
5. I will not post solutions for the questions below because very similar questions may show up
on the exam. You should be able to come up with your own solutions using the lecture slides,
textbook, tutorial solution videos, and solutions to the problem set.
Example Questions:
1. Assume an output production function Y (t) = A(t)(1 − aL)L(t) and a production function
of new ideas A˙(t) = B[aLL]
γAθ
. Also, assume population grows at rate n. For the simple
endogenous growth model, this setup implies ˙gA(t) = γngA(t) + (θ − 1)gA(t)
2
, where gA(t) is
growth rate of knowledge.
(a) Why is either θ < 1 and n > 0 or θ = 1 and n = 0? Explain.
(b) What is the equilibrium growth rate of output (per capita) if θ = 1 and n = 0?
(c) If θ < 1 and n > 0, steady-state growth of knowledge is g
∗
A =
γ
1−θ
n. Why, in economic
terms, does the steady-state knowledge growth rate increase in γ, n, and θ (hint: discuss
in terms of the production functions)?
(d) What is the equilibrium growth rate of output per capita if θ < 1 and n > 0? Explain
why.
2. Consider the role of different frictions in explaining why monetary policy shocks have real
effects.
(a) In the imperfect competition/menu cost model of nominal rigidity, the flex-price equilibrium relative price is Pi/P =
η
η−1
Y
γ−1
, where η > 1 is the elasticity of elasticity of
demand for good i and 1/(γ − 1) is the elasticity of labour supply with γ > 1. Noting
that Yi = Y and Pi = P in equilibrium, derive an expression for yi
in terms of pi − E[p],
γ, and η.
(b) In the Lucas islands model, the Lucas supply curve is y =
1
γ−1
σ
2
z
σ2
z+σ2m
(p − E[p]), where
σ
2
z > 0 is the variance of the good-specific taste shock and σ
2
m > 0 is the variance of the
aggregate demand shock. Noting again that Yi = Y and Pi = P in equilibrium, use the
Lucas supply curve to derive a comparable expression for yi
in terms of pi −E[p], γ, and
the signal-to-noise ratio λ ≡
σ
2
z
σ2
z+σ2m
.
(c) Suppose σm = 0 instead of σm > 0 and consider a one-unit aggregate demand shock such
that pi − E[p] = 1, show that yi
is higher for the Lucas economy than for the imperfect
competition economy. Explain why.
(d) Suppose instead that η → ∞ and, again, consider a one-unit aggregate demand shock
such that pi − E[p] = 1, show that yi
is lower for the Lucas economy than for the
imperfect competition economy. Explain why.
(e) Discuss why it is necessary to have both a nominal friction and a real rigidity for the
nominal rigidity to explain why aggregate demand shocks can have sizeable effects of
real GDP in the imperfect competition/menu cost model of nominal rigidity.
(f) Discuss why the Lucas model suggests aggregate demand cannot have lasting effects on
real GDP. Why is this an example of the Lucas critique?
(g) Which model is better supported by international data on real/nominal GDP growth
correlations and the level and variance of inflation or nominal GDP growth? Explain.
3. Consider a simple Taylor rule with an inflation target of zero: it = ¯r + φππt + φyy˜t
, where it
is the nominal interest rate, ¯r > 0 is the natural real interest rate, πt
is inflation, and ˜yt
is
the output gap. The aggregate demand and supply equations are given by ˜yt = −β(rt−1 −
r¯) + ρy˜t−1 + ε
D
t and πt = πt−1 + αy˜t + ε
S
t
, where ε
D
t and ε
S
t are demand and supply shocks,
respectively. The relationship between it and rt
is given by the Fisher identity (assuming
expected inflation is equal to current inflation): rt = it −πt
. All parameters (¯r, φπ, φy, β, ρ, α)
are > 0. Further, assume that the following parameters (α, β, ρ, φy < 1). Suppose that there
is one-time 10% supply shock at time t = 0 so that ε
S
0 = 0.1. There is no further demand or
supply shock after t = 0. Assume that prior to t = 0, the economy was in steady state with
i = ¯r, π = 0, ˜y = 0, and r = ¯r.
(a) Solve for inflation and the output gap at t = 0 and 1 (i.e., π0, π1, y˜0, y˜1) as functions of
model parameters (or compute the exact values if available).
(b) Suppose that φπ ≤ 1. Show that ˜y1 ≥ y˜0 ≥ 0 and π1 ≥ π0 > 0.
(c) Suppose that φπ > 1. Show that it is possible to stabilize inflation after only one period
(i.e., π1 = 0). At what value of φπ would this occur?
(d) What can you say about the role of the value of φπ for inflation stabilization? But what
is the cost of a higher ratio of φπ/φy?
4. Consider the delegation problem under discretionary policy. Suppose social loss minimization
implies π = π
∗ +
b
a+b
2 (y
∗ − y
f lex) + b
2
a+b
2 (π
e − π
∗
), while loss minimization for a “hawkish”
central banker with a
0 > a implies π = π
∗ +
b
a
0+b
2 (y
∗ − y
f lex) + b
2
a
0+b
2 (π
e − π
∗
). Let π
∗
, π
EQ,
and π
EQ0
be equilibrium inflation under rule-based policy with commitment, discretionary
policy without delegation, and discretionary policy with delegation, respectively.
(a) Show, mathematically, that π
∗ < πEQ0
< πEQ.
(b) What value of a
0 would imply that the equilibrium inflation rate under discretionary
policy with delegation is equal to the inflation rate under rule-based policy with commitment? Hint: What inflation rate can be achieved under perfect commitment in this
model?
5. Consider the following New Keynesian model, where inflation is forward looking but output
is backward looking
xt = ρxt−1 + t
πt = βEtπt+1 + κxt
,
where xt
is the output gap, which is defined as yt−y
f lex
t
, and t
is an i.i.d white noise demand
shock.
(a) Consider the Rational Expectations (RE) minimum state variable solution
πt = a + bxt
.
Use the method of undetermined coefficients to find the RE solution, i.e., find values
of a and b in terms of β, κ, and ρ such that forcasts of inflation made using the above
equation have i.i.d. white noise errors.
(b) What is required to guarantee that your solution is unique? Write down the specific
conditions for this model.