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ECOM009 Macroeconomics

创建日期：2024-04-11 17:40:46

所属分类：经济Economics

标签： ECOM009

Macroeconomics

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ECOM009 Macroeconomics

YOU ARE NOT PERMITTED TO START READING THE CONTENTS
OF THIS QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY AN
INVIGILATOR.
Answer TWO question from Section A and TWO questions from Section B.
CALCULATORS ARE PERMITTED IN THIS EXAMINATION. PLEASE
STATE ON YOUR ANSWER BOOK THE NAME AND TYPE OF MA-
CHINE USED.
COMPLETE ALL ROUGH WORKINGS IN THE ANSWER BOOK AND
CROSS THROUGH ANY WORK THAT IS NOT TO BE ASSESSED.
IMPORTANT NOTE: THE ACADEMIC REGULATIONS STATE THAT
POSSESSION OF UNAUTHORISED MATERIAL AT ANY TIME WHEN
A STUDENT IS UNDER EXAMINATION CONDITIONS IS AN ASSESS-
MENT OFFENCE AND CAN LEAD TO EXPULSION FROM QMUL.
PLEASE CHECK NOW TO ENSURE YOU DO NOT HAVE ANY NOTES,
MOBILE PHONES OR UNAUTHORISED ELECTRONIC DEVICES ON
YOUR PERSON. IF YOU HAVE ANY THEN PLEASE RAISE YOUR
HANDANDGIVE THEMTOAN INVIGILATOR IMMEDIATELY. PLEASE
BE AWARE THAT IF YOU ARE FOUND TO HAVE HIDDEN UNAUTHO-
RISED MATERIAL ELSEWHERE, INCLUDING TOILETS AND CLOAK-
ROOMS IT WILL BE TREATED AS BEING FOUND IN YOUR POS-
SESSION. UNAUTHORISED MATERIAL FOUND ON YOUR MOBILE
PHONE OR OTHER ELECTRONIC DEVICE WILL BE CONSIDERED
THE SAME AS BEING IN POSSESSION OF PAPER NOTES. MOBILE
PHONES CAUSING A DISRUPTION IS ALSO AN ASSESSMENT OF-
FENCE.
EXAM PAPERS CANNOT BE REMOVED FROM THE EXAM ROOM.

1
Section A
1. [30 marks] Assume consumers have infinite lifetimes and maximize the utility
function
Ut = Et
∞∑
s=t
−βs (c¯− ct+s)
2
2
,
where ct+s is consumption at time t+ s, β > 0 is the subjective discount factor and
Et is the expectation operator conditional on all information available to consumers
at time t. The coefficient c¯ is a positive parameter large enough for the marginal
utility of consumption to be always positive over the relevant range. Consumers
can freely borrow and lend at a constant riskless rate r such that β (1 + r) = 1.
Labour income yt follows the stochastic process
yt = yt−1 + λyt−2 + t,
with λ > 0 and εt a white-noise process distributed normally with zero mean.
(a) This is standard and results in the Euler equation
ct = Et(ct+1).
(b) Guess the linear function ct = α0 + α1at + α2yt + α3yt−1.
Replacing in the Euler equation and taking expectations yields
α1at + α2yt + α3yt−1 = α1at+1 + α2(yt + λyt−1) + α3yt,
which can be rearranged as
at+1 − at = α−11 [−α3yt + (α3 − α2λ)yt−1] (1)
Replacing for ct in the dynamic budget identity yields
at+1 − at = rat + yt − (α0 + α1at + α2yt + α3yt−1). (2)
Equating coefficients on the same variables on the right hand sides of both (1)
and (16) one obtains the consumption function
ct = rat +
r
r(1 + r)− λ [(1 + r)yt + λyt−1]. (3)
The consumption response to an innovation is
ct − ct−1 = ct − Et−1ct = r(1 + r)
r(1 + r)− λεt.
As λ > 0, income is more persistent than a random walk and consumption
responds more than one-to-one to an income innovation. Extra points for
students who mention Deaton’s paradox.
2
(c) The saving function is
st = − λ
r(1 + r)− λ [yt + ryt−1].
Again, saving responds negatively to an income innovation as income is more
persistent than a random walk.
2. The question is rather standard. Only part c) requires some extra thought.
(a)
max
It,Kt
V0 =
∫ ∞
0
[
Kαt − It −
I2t
2
]
e−rtdt (4)
s.t. K0 given (5)
K˙t = It − δKt (6)
The necessary conditions for an optimum are
∂L0
∂It
= − [1 + It] + qt = 0 (7)
∂L0
∂Kt
= αKα−1t − δqt + q˙t − rqt = 0 (8)
lim
t→∞
qtKte
−rt = 0 (9)
This can be reduced to a system of two differential equations
K˙t = (qt − 1) (10)
q˙t = (r + δ)qt − αKα−1t (11)
(b) Saddle path convergence to steady state from the right.
(c) The K˙ locus is expected to shift up in the (Kt, qt) space at time t1. The steady
state capital stock after t1 is going to be higher. At time t0 the economy jumps
onto the unique path which crosses the stable arm of the new saddle path at
time t1. qt jumps up at t0, keeps rising until t1 and falls from t1 onwards.The
capital stock increases from t0 until the new steady state is reached..
3. (a) The individual optimization problem is
max
c1t ,c2t+1
log c1t +
1
1 + ρ
log c2t+1 (12)
s.t. a1t = Wt − τ − c1t (13)
c2t+1 = τ(1 + n) + a1t (1 + rt+1) . (14)
where we have already imposed solvency.
The Euler equation can be written as
c2t+1 =
1 + rt+1
1 + ρ
c1t. (15)
3
Replacing in the IBC and solving for c1t yields
c1t =
1 + ρ
2 + ρ
[
Wt − τ + τ(1 + n)
1 + rt+1
]
. (16)
The associated individual saving function is
s1t = Wt − τ − c1t = 1
2 + ρ
(
Wt − τ − τ(1 + n)(1 + ρ)
1 + rt+1
)
(17)
(b) Given the Cobb-Douglas technology it is Wt = (1− α) k˜αt , with k˜t = Kt/(Lt),
and rt+1 = αk˜
α−1
t+1 − δ. Hence individual saving is a function of the current
stock of capital
s1t =
1
2 + ρ
(
(1− α) k˜αt − τ −
τ(1 + n)(1 + ρ)
1 + αk˜α−1t+1 − δ
)
. (18)
The stock of capital at t+ 1 equals the total saving of the young at time t or
Kt+1 = Lt
1
2 + ρ
(
(1− α) k˜αt − τ −
τ(1 + n)(1 + ρ)
1 + αk˜α−1t+1 − δ
)
(19)
or in per capita terms
k˜t+1 =
1
(2 + ρ)(1 + n)
(
(1− α) k˜αt − τ −
τ(1 + n)(1 + ρ)
1 + αk˜α−1t+1 − δ
)
. (20)
Rearranging yields,
k˜t+1 +
τ(1 + n)(1 + ρ)
1 + αk˜α−1t+1 − δ
=
1
(2 + ρ)(1 + n)
(
(1− α) k˜αt − τ
)
. (21)
Equation (21) implies that, for any level of k˜t, kt+1 is lower and, the function
has a lower slope, than if τ = 0.
Dynamics is standard.
(c) In steady state it has to be k˜t+1 = k˜t. If τ > 0 the solution to (20) is a lower
steady state level of k˜∗ for the stable steady state equilibrium B (see Figure
1).
A pay as you go social security system depresses capital formation. Intuition:
private saving by the young generation falls and the government uses the
contribution not to invest in physical capital but to pay the pensions of the
currently old. Hence aggregate saving, and investment, fall at given kt.
If the marginal product of capital at the decentralized equilibrium with τ = 0
is below the Golden Rule marginal product of capital αk˜α−1GR = δ + n, the
economy is dynamically inefficient. There exist τ > 0 which yields a Pareto
improvement relative to the decentralized equilibrium with τ = 0.
4
kt
kt+1
B
B′
A′
A k∗k∗
′
Figure 1: Steady state with pay-as-you-go pension system.
Section B
4. Usual stuff, driven by concavity of preferences and uneveness of income over time
or states of nature.
5. Marginal cost of adjustment is non-zero in the convex adjustment cost theory. The
two coincide in steady state with zero depreciation rate as there is no adjustment
and therefore the marginal cost of adjustment is zero.
6. The theory predicts that risk premia are driven by the covariance between the
future marginal utility of consumption and asset returns. Empirically the theory,
in its incarnation taught to students, fares pretty badly. Students should discuss
the equity premium and risk-free rate puzzles.
7. Standard aggregation in the life cycle model.

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MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 IE5600 ECOS 3997 EDST2003 FIN 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