Problem 1: Monetary Policy and Macroeconomic Dynamics (35 Points)
All variables in the model below are measured in percentage deviations from their long-run levels
(percentage deviations of gross rates in the cases of in�ation and the interest rate).
Suppose that the dynamics of output (yt) are determined by the following intertemporal con-
dition:
yt = Et (yt+1) + �yt�1 � � [it � Et (�t+1)] + ut; (1)
where 0 � � < 1 captures the impact of past output dynamics on current output (for instance,
because of habits in consumption behavior), � > 0 is the elasticity of intertemporal substitution,
it is the nominal interest rate between t and t + 1 decided by the central bank at time t, Et (:) is
the expectation operator, �t+1 is in�ation at t+ 1, and ut is an exogenous output demand shock.
In�ation is determined by the following modi�ed New Keynesian Phillips curve:
�t = ��t�1 + �yt + �Et (�t+1) + zt; (2)
where 0 � � < 1 captures persistence in in�ation implied, for instance, by indexation mechanisms,
� > 0, � 2 (0; 1) is the representative household�s subjective discount factor, and zt is an exogenous
supply-side shock.
Finally, the central bank sets the interest rate to respond to in�ation and output according to
the Taylor rule:
it = �1�t + �2yt + �t; (3)
where the policy response coe¢ cients are such that �1 > 1 and �2 � 0, and �t is an exogenous
interest rate shock.
Assume that the shocks ut, zt, and �t are such that:
ut = �uut�1 + “u;t; (4)
zt = �zzt�1 + “z;t; (5)
and
�t = ���t�1 + “�;t; (6)
where the persistence parameters �u, �z, and �� are all strictly between 0 and 1, and “u;t, “z;t, and
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“�;t are identically and independently distributed innovations with zero mean and variance �2″u , �2″z ,
and �2″� , respectively.
We can guess that the solution of the system (1)-(3) can be written as:
yt = �yyyt�1 + �y��t�1 + �yuut + �yzzt + �y��t; (7)
�t = ��yyt�1 + ����t�1 + ��uut + ��zzt + ����t; (8)
and
it = �iyyt�1 + �i��t�1 + �iuut + �izzt + �i��t; (9)
where the ��s are elasticities that can be obtained with the method of undetermined coe¢ cients.
� Explain why we make this guess based on the information I have given you.
� Now assume that the parameters � and � are such that � = � = 0. What do you
think happens to �yy, �y�, ��y, ���, �iy, and �i� in this case? Why?
� Use the method of undetermined coe¢ cients to solve for the elasticities �yu, �yz,
�y�, ��u, ��z, ���, �iu, �iz, and �i� when � = � = 0. (Suggestion: Start by focusing
only on �y�, ���, and �i�. They are the only ones you will actually need to complete
the rest of the problem. Find these three, work on the rest of the exam, and
come back to compute the other elasticities if you have time.)
� Suppose that an innovation (any of the three) happens at time 0, after which
no other innovation happens. Can the responses of any of the three endogenous
variables (output, in�ation, the interest rate) be more persistent than the shock
that is a¤ected by the innovation? Why?
� Think of periods as quarters and assume the following values for the remaining
parameters of the model: � = 0:5, � = 0:05, � = 0:99, �1 = 1:5, �2 = 0:5, and �� = 0:5
(you do not need the values of the other persistence parameters to answer this
question). Set up an Excel spreadsheet to plot the responses of output, in�ation,
and the interest rate to an innovation “�;t = �1 to the interest rate at time t = 0,
followed by no other innovation. Plot the responses and include the Excel �le
with the answers you will submit.
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� Explain your intuition for the responses you �nd.
� Why do the responses of output and in�ation that you plotted display no hump
(i.e., the peak responses happen immediately)? What do you think we would
need to have a hump in, for instance, the response of in�ation?
� Would you consider the model (1)-(6) to be a good model to describe monetary
policy conduct and macroeconomic dynamics in the aftermath of the 2007-08
Global Financial Crisis or during the current crisis? Why?
Problem 2: House Prices and the Macroeconomy (35 Points)
Consider the following version of the �nancial accelerator framework. The representative individual
lives for two periods and not only makes decisions about consumption and savings, but also housing
purchases. We interpret �period 1�as the �young period�of the individual�s life, and �period 2�
as the �old period.� In the young period of an individual�s life, utility depends only on period-
1 consumption, c1. In the old period of an individual�s life, utility depends both on period-2
consumption, c2, as well as her/his �quantity�of housing (denoted h). (For concreteness, you can
think of �quantity�of housing as the square footage and/or the �quality� of the housing space.)
From the perspective of the beginning of period 1, the individual�s lifetime utility function is:
ln c1 + ln c2 + lnh;
in which ln(:) stands for the natural log function, the term lnh indicates that people directly obtain
happiness from their housing, and, for simplicity, we abstract from discounting of the future.
Due to the �time to build�nature of housing (that is, it takes time to build a housing unit),
the representative individual has to incur expenses in her/his young period to purchase housing for
her/his old period. The real price in period 1 (i.e., measured in terms of period-1 consumption) of
a �unit�of housing (again, think of a unit of housing as square footage) is pH1 , and the real price
in period 2 (i.e., measured in terms of period-2 consumption) of a unit of housing is pH2 .
In addition to housing decisions, the representative individual also makes stock purchase deci-
sions. The individual begins period 1 with zero stock holdings (a0 = 0), and ends period 2 with
zero stock holdings (a2 = 0). How many shares of stock the individual ends period 1 with, and
hence begins period 2 with, is to be optimally chosen. The real price in period 1 (i.e., measured
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in terms of period-1 consumption) of each share of stock is S1, and the real price in period 2 (i.e.,
measured in terms of period-2 consumption) of each share of stock is S2. For simplicity, suppose
that stock never pays any dividends (that is, dividends = 0 always).
Because housing is a big-ticket item, the representative individual has to accumulate �nancial
assets (stock) while young to overcome the informational asymmetry problem associated being able
to purchase housing. Suppose the �nancing constraint that governs the purchase of housing is:
pH1 h = RHS2a1:
In this �nancing constraint, RH > 0 is a government-controlled �leverage ratio�for housing. Note
well the subscripts on variables that appear in the �nancing constraint.
Finally, the real quantities of income in the young period and the old period are y1 and y2, and
they are exogenously given.
The sequential Lagrangian for the representative individual�s problem lifetime utility maximiza-
tion problem is:
L = ln c1 + ln c2 + lnh+ �1�y1 � c1 � S1a1 � pH1 h
�+�2
�y2 + S2a1 + p
H2 h� c2
�+ �
�RHS2a1 � pH1 h
�;
in which � is the Lagrange multiplier on the �nancing constraint, and �1 and �2 are, respectively,
the Lagrange multipliers on the period-1 and period-2 budget constraints.
� Brie�y describe the informational asymmetry problem that motivates the exis-
tence of �nancing constraints, and why it can be a serious problem in �nancial
transactions.
� Brie�y describe the role that the leverage ratio RH plays in the �housing �nance�
market. In particular, brie�y discuss what higher leverage ratios imply for the
individual�s ability to �nance a house purchase (i.e., �obtain a mortgage�).
� Based on the sequential Lagrangian presented above, compute the two �rst-order
conditions with respect to a1 and h. (You can safely ignore any other �rst-order
conditions for the purposes of this problem.)
� Based on the �rst-order condition with respect to h obtained above, solve for
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the period-1 real price of housing pH1 (that is, your �nal expression should be
of the form pH1 = :::, where the term on the right hand side is for you to deter-
mine). Note: You do NOT have to eliminate Lagrange multipliers from the �nal
expression.
� Based on the expression for pH1 obtained above, assuming that � > 0 and RH > 0,
answer the following: Is the period-1 price of housing larger than or smaller than
what it would be if �nancing constraints for housing were not at all an issue? Or
is it impossible to determine? Carefully explain the logic of your analysis, and
provide brief economic interpretation of your conclusion.
For the remainder of this problem , suppose that �1 = �2 = 1.
� Consider the period-1 housing market, with the quantity h of housing drawn on
the horizontal axis and the period-1 price, pH1 , of housing drawn on the vertical
axis. Using the house-price expression you obtained above, qualitatively sketch
the relationship between h and pH1 that it implies. Your sketch should it make
clear whether the relationship is upward-sloping, downward-sloping, perfectly
horizontal, or perfectly vertical. Clearly present the algebraic/logical steps that
lead to your sketch, and clearly label your sketch.
� In the same type of sketch as above, clearly show and label what happens if pH2
rises. (Examples of what �could happen�are that the relationship you sketched
rotates, or shifts, or both rotates and shifts, etc.) Explain the logic behind your
conclusion, and provide brief economic interpretation of your conclusion.
� In the same type of sketch as above, clearly show and label what happens if RH
rises. (Examples of what �could happen�are that the relationship you sketched
rotates, or shifts, or both rotates and shifts, etc.) Explain the logic behind your
conclusion, and provide brief economic interpretation of your conclusion.
Essay Question: Pandemic Dynamics and Policy Response (30 Points)
Based on the material you studied in this class and any other source you may have
read, how do economic dynamics interact with those of a pandemics? How does
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containment policy contribute to shaping pandemic and macroeconomic dynamics?
And what roles do/should monetary and �scal policy play? Using at most three
(letter-size) pages of text and three �gures (not counted toward the page limit) write
an essay that tackles these questions. You will be rewarded for not simply copying
material from the slides, but you should make sure to indicate explicitly the material
you refer to in building your analysis (examples, �Eichenbaum-Rebelo-Trabandt model
slides,���nancial accelerator slides,��paper X that I read for my own interest,�and
so on).
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