程序代写案例-JANUARY 2020
时间:2021-01-17
MFE ECONOMICS
MID-YEAR ASSESSMENT, JANUARY 2020
Please answer all questions in Part A and exactly one question from Part B.
Part A is worth 60% of the marks (each question’s weight is 15%), Part B is worth 40%.
Please, start the answer to each question on a separate page. Time allowed: 2 hours.
PART A: Please answer each question.
Question A1. In the context of the NK model studied in class, what is the natural real rate
of interest? What might cause it to fall and why?
Answer:
• Flexible prices
– They should refer to it being the real rate of interest that prevails in equilibrium
in the flexible price (θ = 0) special case of the NK model. They may refer to the
flexible price case indirectly by saying that all firms have their prices at the ‘desired’
markup but in this case I would dock a small amount of credit as it’s ambiguous
what they mean.
• Causes of a decline
– The best students will hopefully distinguish shock stories from longer run structural
change
– Shocks: An increase in ‘patience’/‘precaution’ (preference shock discussed in class
- though extra marks if they mention that a similar effect might arise from tighter
bank lending - which isn’t explicitly modeled in the basic NK setup) or anticipation
of weaker growth in the short/medium run (smoothing)
– Long run influences: An ‘increase in β’ (if they get the sign wrong but clearly
understand the intuition then only a small deduction) which may capture, say, a
decline in the trend growth rate of the economy (great if they mention slower tech
growth, some of the secular stagnation debates), longer retirements, Asian glut of
savings, regulatory-induced demand for safe assets. . . (they don’t need to name them
all)
– Would be great to see the student refer to the decline in r∗ is recent decades and
even some of the research on it
• Why?
– Generally all these influences act by increasing desired saving at a given real rate (in
the flex price economy). The reduced saving in equilibrium at the ‘initial’ rate may
not be consistent with consumption and output/labor supply decisions etc. such
that for the households to be optimizing and for the other equilibrium requirements
to be satisfied, a lower r is required to disincentivize savings (i.e. market terms of
trade must reach different equilibrium values to reflect a change in ex ante ‘desire
to save’).
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Question A2. Forward guidance is always at play in monetary policy, but especially at the
zero lower bound. Discuss.
Answer:
• Dynamic IS Curve
– They should likely agree with this statement - and the ‘always’ bit hopefully should
elicit references to the dynamic IS curve and how it can be iterated forwards to
obtain a connection between the output gap (I don’t really mind if they are a
bit loose about the output gap concept or simply think of it as real activity) and
expected future paths of nominal interest rates (tool of policy) the natural real rate
and inflation
– Also, if they have most of what I have written below but then simply seem to believe
that forward guidance is just as important away from ZLB as at the ZLB then they
shouldn’t be docked marks - it’s ok to disagree with the statement as long as they
understand the issues and (a matter of taste really) weigh them up.
– If they write out equations they should be something like (if they explain in words
equivalently, that’s fine too)
y˜t = Et [y˜t+1]− 1
σ
(it − Et [pit+1]− rnt )
= − 1
σ
∞∑
k=0
Et[rt+k − rnt+k]
= − 1
σ
∞∑
k=0
Et[it+k − (rnt+k + pit+k+1)]
• Conventional monetary policy
– They should note that forward guidance most typically is thought of in terms of
guidance about nominal interest rates - additionally though, they can refer to trying
to influence inflation expectations via some credible method (extra marks if they
note that influencing expectations in a rational expectations context is a subtle
concept - double extra marks if they then allude to specific reasons why rational
expectations might not hold)
– Once they have established that setting up a credible commitment to pursue some
form of interest rate policy in the future can be an important element of conventional
monetary policy - even away from the ZLB
– Great if they allude to Williams/Rudebusch ‘secrets of the temple’ intuition too
(where central banks’ aims and preferences may be somewhat latent to the private
sector - justifying extra communication)
• ZLB/Unconventional monetary policy
– Should note that the DIS shows that if one can lower expectations of future short
rates (great if they refer to ‘lower for longer’ and slow tightening after liftoff from
2
ZLB) then even if the short rates / short maturity yields are stuck at the ZLB, real
activity can be influenced. Would be nice to see references to Williams and Swanson
AER paper, but not necessary.
– May also note (likely these are the students who wrote out the equations) that
something like price level targeting or some other nominal commitment (such as
debated in recent times) could also be interpreted as a form of forward guidance
- and extra marks if they connect this to the fact that the ZLB is particularly
damaging/constraining when inflation and inflation expectations are low, since it
limits how low short real rates can go. . .
– . . . which is another reason why effective policy at the ZLB is especially important
- they should show they understand that in theory - and in recent times in the
real world - the economy may perform very badly if unconventional policy is not
introduced in a low r∗ and low pi environment
– Extra marks if they say something like - since quantitative easing is an alternative -
maybe forward guidance isn’t so vital at ZLB (that may or may not be correct but
that’s an issue for AER papers. . . ) and double extra marks if they note that some
people (e.g. Rudebusch and Bauer) have argued that LSAPs can enhance forward
guidance on rates through the signaling channel
Question A3. Bank liquidity and solvency are completely distinct concepts. Discuss.
Answer:
• Ways in which they are distinct
– I want them to understand the concept of solvency: value of assets > value of
external liabilities. Capital is a loss absorbing buffer etc.
– I want them to understand the concept of liquidity: can sell or borrow against
assets quickly and easily at a reasonable price (not firesale, or at a price below
fundamentals)
– Formally, you can be solvent and have illiquid assets or be illiquid (hard to raise
money quickly) and you can be liquid and insolvent - all your assets are in cash but
you don’t have enough to cover your obligations
– So they should convey the idea that they are somewhat distinct in theory
• Ways in which they are related
– Would be nice to see some Diamond-Dybvig allusions here - especially since it makes
explicit how banks can create liquidity (promising demand deposits or offering lia-
bilities to, say, money market funds, that can be swiftly liquidated) and how beliefs
about solvency can make it optimal to demand liquidity ‘prematurely’ (i.e. to run)
if those beliefs are negative (in DD those beliefs are not irrational but they may
also make the point about irrational beliefs). Here the illiquidity of the long term
asset means that early sale does impair solvency. So this simple model shows so-
phisticated interactions between liquidity and solvency so even if they are formally
distinct, they are often correlated in practice.
3
– Taking a less formal approach, the students may alternatively (or, for the best marks,
additionally) describe some of the events of the recent financial crisis - especially
the run on repo, how that might induce firesales prices that could (from a mark to
market perspective) impair solvency, and how doubts about solvency arising from
opaque derivatives etc. on the balance sheet made it more difficult to raise funds
(liquidity) via haircuts on collateral. Liquidity in the sense of ‘fair price’ asset sales
and liquidity in the sense of ‘fair rate’ ability to borrow are very closely related in
the world of collateralized lending.
– On the last point, there is a lot of irrelevant stuff they could write about the financial
crisis - but the point is for them to focus on the nub of the question (if they talk
about run on repo, the size of the positions being rolled overnight and the losses
incurred by banks through bad performance of loan pools or through mark to market
of opaque products then those are indications they’ve understood the question)
Question A4. Can markets overcome the problems associated with polluting firms, or is there
a need for government intervention?
Answer: A good answer might begin by laying out the First Welfare Theorem, and ex-
plaining why we expect complete, competitive markets to allocate resources (Pareto) efficiently.
Unchecked pollution by firms is a violation of the conditions underlying the First Welfare the-
orem; there is a failure of the complete markets assumption. Hence it is possible that markets
are inefficient. Good answers will set up the very simple quasilinear model of a polluting firm
and a consumer seen in lectures, illustrating that the firm does indeed pollute more than the
efficient amount.
To discuss whether there is a need for government intervention, students should explain
the insight of the Coase Theorem. The best answers will walk through the model to show
that when the parties can write contracts over pollution (and there is a property right defining
outside options), we should expect them to bargain to an efficient outcome.
A discussion of the extent to which the Coase Theorem implies no need for government
intervention might include (i) the need for low transactions costs (including no significant
asymmetries of information, and the ability of all affected parties to enter the bargain); (ii)
why there was not a market for the pollution in place to start with (potential free-rider problems
with many consumers); (ii) the need for at least a minimal role of government in supporting
enforcement of contracts and defining property rights.
PART B: Please answer exactly one question.
Question B1.
A region (‘the North’) is soon to hold a referendum over whether it should leave or remain
in the ‘Seven Kingdoms’. Refer to the outcome that the North remains as ‘state 1’ and the
outcome that it leaves as ‘state 2’. In the Seven Kingdoms, there is only one good (consumption)
and two agents: agent a, who has a wealth of ωa, and agent b , who has a wealth of ωb (measured
in units of the consumption good). Let agents’ aggregate wealth be ω¯. Agent a believes that
the probability of the North leaving is pia2 . Agent b believes the probability of ‘leave’ is pib2.
4
The outcome of the referendum will not affect any individual’s endowment. If agent i’s
consumption in state k as cik, for i = a, b, her final utility from consumption in state k is
ui(c
i
k) = ln c
i
k
Assume no individual is important enough to influence the outcome of the referendum –
i.e. they take their beliefs about the outcome as given.
a.) Are the agents risk averse, risk neutral or risk loving? If pia2 = pib2 =
1
2
, explain
intuitively (no maths) whether there will be any economic gains to introducing a
market for speculating on the outcome of the referendum? [5%]
b.) Write down an expression for the expected utility of agent i as a function of cik, k =
1, 2, and pii2. Show that agent i’s marginal rate of substitution between consumption
in state 1 and state 2 is
MRSi21 = −
1− pii2
pii2
ci2
ci1
.
[10%]
Now suppose that pia2 =
2
3
, pib2 =
1
3
.
c.) Find an expression for the contract curve (i.e., ca2 in terms of ω¯ and ca1). Illustrate,
using an Edgeworth box with state-2 consumption on the vertical axis. [25%]
Before the referendum, there are two markets for state-contingent consumption. In market 1,
agents can buy rights to state-1 consumption. In market 2, agents can buy the right to state-2
consumption. Let p be the price of a unit of state-2 consumption and normalize the price of
(a unit of) state-1 consumption to 1.
d.) Derive a’s demand for consumption in each state as a function of p and ωa. Simi-
larly, find b’s demands. (Hint: are utilities Cobb-Douglas?) [25%]
e.) Solve for Walrasian equilibrium prices and consumption as a function of ωa and ωb.
Is the equilibrium efficient? [20%]
f.) Suppose the ‘true’ probability of leave is 1
2
. What distribution of wealth is required
for the relative market ‘implied probability’ of state 2, p
1−p , to equal
1
2
? [10%]
g.) Briefly discuss the validity of using market ‘implied probabilities’ for predicting
risky events. [5%]
Solution to B1.
(a) ui(c) is concave. Hence, both a and b are risk-averse. Since both are risk-averse and
initially face no risk to their own consumption as a result of the referendum, if both agents also
had the same beliefs about the outcome of the referendum there would be no gains to trade
(i.e. the no trade outcome is already on the contract curve and hence Pareto efficient).
(b) From definition in lecture notes, expected utility: U(ci1, ci2) = pii1ui(ci1) + pii2ui(ci2) =
pii1 ln c
i
1 + (1− pii1) ln ci2. Given this, we can apply the definition of i’s MRS to find:
MRSi21 = −
MU1
MU2
= − pi
i
1/c
i
1
(1− pii1)/ci2
= − pi
i
1
(1− pii1)
ci2
ci1
.
5
(c) Contract curve is all the set of all Pareto efficient allocations. Can be found by equating
MRS across agents:
MRSa21 = MRS
b
21,
which from (b) gives for the case pia1 =
1
3
, pib1 =
2
3
:
1
2
ca2
ca1
= 2
cb2
cb1
.
We also know that in an efficient allocation, aggregate allocations are exhausted in each state:
cbk = ω¯ − cak, for k = 1, 2. Plugging in to the MRS condition and rearranging gives:
ca2 =
4ω¯ca1
ω¯ + 3ca1
.
Since ω¯ ≥ ca1, the contract curve always lies above the 45 degree line in the Edgeworth box
(and touches it only at ca1 = 0, ω¯). It is increasing and concave (check:
dca2
dca1
is positive, but
decreasing). The endowment by contrast is on the 45 degree line. With the two extreme
exceptions, consuming the endowment is no longer efficient (in contrast to part (a.)).
(d) Given prices, individual i’s income is (1 + p)ωi. Since utility is Cobb-Douglas, can use
trick from lectures:
ca1 =
(1 + p)ωa
3p
; ca2 =
2
3
(1 + p)ωa
cb1 =
2
3
(1 + p)ωb
p
; cb2 =
1
3
(1 + p)ωb
(e) Equilibrium: prices clear both markets. By Walras’ law, need only find a p that clears
one of the markets. Clearing market 2:
2
3
(1 + p)ωa +
1
3
(1 + p)ωb = ω¯.
Using ω¯ ≡ ωa + ωb and rearranging for p:
p =
ωa + 2ωb
2ωa + ωb
.
Consumptions can be found by plugging prices back into the demand function.
(f) From (e), they need an equal distribution of wealth for p = 1. In this case, prices reflect
a fair gamble, exactly what we would expect if people knew the ‘true probability’ of leave were
1
2
.
(g) As part (e) shows, when people have different beliefs prices only reflect a weighted
average of the beliefs of the two agents: notice if a owns all the wealth, then p would equal
1
2
and hence the implied probability would be 1
3
; if b held all the wealth the implied market
probability would be 2
3
.
Some interesting consequences include(unlikely that stufents will go this far):
• Market odds might have very little relationship to true probabilities if people’s beliefs
are not grounded in reality – in this case we cannot learn anything about the world from
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prices;
• Even if people’s beliefs are accurate on average, prices may not reflect this: instead
they reflect a weighted average of beliefs, where the weight on a person’s beliefs depends
positively on their wealth (as aggregate demand becomes relatively sensitive to their
beliefs).
• More generally, it also depends on the relative risk aversion of participants. Prices will
converge towards the beliefs of the less risk averse, again because they are more willing
to trade based on perceived mismatches between prices and their beliefs.
In the latter 2 cases, we need to adjust prices for the influence of differing levels of wealth/risk
aversion before using them to make inferences about the world.
Question B2.
In this question we consider a household choosing its consumption/savings plan in an en-
vironment where they live for 3 periods and in which there is no randomness. The household
receives a stream of income over the three periods, {Yt}3t=1, has initial wealth it brings into
t = 1, denoted, a0 and in each period can either consume or save. Saving is enabled by pur-
chasing at one-period bonds at a price Qt (where we can alternatively think in terms of an
implied riskless gross real interest rate on the bonds from t to t + 1 as Rt where Qt ≡ R−1t ).
The flow budget constraint for period t ≡ {1, 2, 3} is:
at−1 + Yt = Qtat + Ct
=
at
Rt
+ Ct
where Ct is consumption in t. The household has period felicity function:
u(C) = log (C)
and, as of t = 1, chooses its plan to maximize:
U(C1, C2, C3) ≡ u(C1) + βu(C2) + β2u(C3)
a.) Why will a3 ≤ 0 under the optimal plan? [10%]
b.) Why is it sensible to impose the constraint a3 ≥ 0 for any plan available to the
household? [10%]
c.) Thus, show that candidate plans must satisfy:
a0 = C − Y (1)
Y ≡ Y1 + Y2
R1
+
Y3
R1R2
C ≡ C1 + C2
R1
+
C3
R1R2
7
[15%]
d.) What are the interpretations of Y , C and, overall, equation (1)? [10%]
e.) Suppose the household entered t = 1 as a debtor, what does that mean for the
relative size of Y and C? (Very) briefly interpret this. [12%]
f.) Derive the first order condition for optimality (the Euler equation) relating Ct, Ct+1
and Rt. That is, you should show how you obtain:
1
Ct
= βRt
1
Ct+1
[13%]
g.) Now, suppose R1 = 1.05, R2 = 1.10, β = 0.995, Y1 = 10, Y2 = 15, Y3 = 20 and
a0 = 0. What isG2 ≡ C2C1 and G3 ≡ C3C2 under the optimal plan? [15%]
h.) What is C1 under the optimal plan? [15%]
Solution to B2.
a). Because the household will not be alive in period 4 so no optimal plan will imply positive
savings as there exist other plans that give weakly greater consumption in every period
and strictly more in at least one period - implying that a candidate plan with a3 > 0
cannot satisfy the maximization, given non-satiated preferences. Basically the not alive
bit gets them some marks and the better students should explain why.
b). Because this prevents them dying in debt - no free lunch from lenders (a very good stu-
dent might argue that the limit could be a boundedly negative if there is some sort of
charitable element etc. but I ignore that here). If not, the problem becomes unbounded,
consumption would be desired to be arbitrarily large as the agent will finance consump-
tion by taking on debt they know they won’t be around to pay off. Arbitrarily large
consumption cannot feature in an equilibrium with finite production.
c). They need to set a3 = 0 and then rearrange the 3rd period budget constraint to express
a2 = C3 − Y3, substitute that into the 2nd period one and so on....
d). Present value of the stream of lifetime income, present value of the stream of lifetime con-
sumption and an ‘intertemporal budget constraint’ (with some properties of the optimal
solution and no-debt at death imposed) indicating that the PV of consumption must
equal the PV of income plus any initial wealth.
e). They should note that entering as a debtor is equivalent to a0 < 0 implying that consump-
tion henceforth must be lower in present value than income, since a chunk of income must
payoff the initial debt, as well as funding consumption.
f). They may do this in various ways - could be a quasi verbal marginal argument where they
calculate marginal benefit of consumption foregone (today’s MU) and marginal ‘cost’
(tomorrow’s MU discounted and allowing for the bond return) and then equating them,
or they may use a diagram (in which case they need to fully explain everything) or a
Lagrangian etc.
8
g). The point of this and the next part is to see if they can do some simple manipulations
and realize that the Euler equation is a marginal condition but requires a boundary
condition to nail down which of infinitely many ‘parallel’ consumption paths exhausts
the household’s resources exactly.
h). See comment in part (g).
Question B3.
In this question we consider a household with period felicity function
u(Ct, Lt) ≡ (C
ν
t L
1−ν
t )
1−σ − 1
1− σ
where σ > 0, ν ∈ [0, 1], Ct is real consumption and Lt is leisure. The household is a price
taker in all markets and, in particular, takes the real wage, wt as given.
a.) Derive the marginal utility of consumption, denoted uC(C,L) ≡ ∂u(C,L)∂C and the
marginal utility of leisure, uL(C,L) ≡ ∂u(C,L)∂L . [15%]
b.) Show that Λt,t+1 ≡ β uC(Ct+1,Lt+1)uC(Ct,Lt) is given by:
Λt,t+1 = β
(
Ct+1
Ct
)−σ (Lt+1
Ct+1
)(1−σ)(1−ν)
(
Lt
Ct
)(1−σ)(1−ν) .
[15%]
c.) Comment on the form Λt,t+1 takes in the limiting case of ν = 1 and, for general
ν, briefly interpret why Lt and Lt+1 affect the valuations of real payoffs in t + 1
from the perspective of t (assuming that under optimality, Λt,t+1 is the household’s
stochastic discount factor). [15%]
d.) Assume the household earns the real wage, wt, for labor supplied (leisure foregone)
in t. Using a marginal argument (equating marginal benefit of more leisure to the
marginal ‘cost’ of more leisure), show that optimal labor supply implies:
Lt =
1− ν
ν
Ct
wt
.
[15%]
e.) What is the elasticity of leisure in t with respect to the real wage in t under the
optimal plan? [10%]
f.) Show that under optimality:
Λt,t+1 = β
(
Ct+1
Ct
)−σ (
wt
wt+1
)(1−σ)(1−ν)
(2)
9
[15%]
g.) Under optimality, our household uses (in t) Λt,t+1 to value a real payoff in t + 1,
Yt+1 as
P Yt = Et[Λt,t+1Yt+1].
Briefly interpret the role of the wage term in equation (5) in how the household
discounts payoffs in different contingenices in t+1 (refer to your answer in the third
part of the question). [15%]
Solution to B3.
a). Differentiation yields
uC(C,L) = ν(C
νL1−ν)−σ
(
Lt
Ct
)1−ν
(3)
uL(C,L) = (1− ν)(CνL1−ν)−σ
(
Lt
Ct
)−
ν (4)
b). Just divide - using the first part of the question.
Λt,t+1 = β
(
Ct+1
Ct
)−σ (Lt+1
Ct+1
)(1−σ)(1−ν)
(
Lt
Ct
)(1−σ)(1−ν)
c). Back to CRRA in terms solely of consumption. Leisure in the periods is irrelevant for
utility of marginal unit of consumption in either period. When ν 6= 0 the utility derived
from a marginal unit of consumption is influenced by the prevailing amount of leisure (or
equivalently, labor supply) so evaluating a real payoff tomorrow must take into account
tomorrow’s (presumably as yet unknown) leisure/labor decision.
d). Use the marginal utilities calculated in the first part. The aim here is hopefully to filter
students who understand marginal arguments.
Lt =
1− ν
ν
Ct
wt
e). -1 reflecting the CD kernel.
f). This requires them to rearrange the intratempral leisure-consumption condition above and
substitute out the leisure:consumption ratios in each period.
Λt,t+1 = β
(
Ct+1
Ct
)−σ (
wt
wt+1
)(1−σ)(1−ν)
(5)
g). Basically this is the same as in the earlier part of the question but the point I want them
to recognize is that the leisure choice discussed previously, is elicited by whatever the
wage turns out to be.
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END
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