xuebaunion@vip.163.com

3551 Trousdale Rkwy, University Park, Los Angeles, CA

留学生论文指导和课程辅导

无忧GPA：https://www.essaygpa.com

工作时间：全年无休-早上8点到凌晨3点

扫码添加客服微信

扫描添加客服微信

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’).

1

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

6

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.

10

END

11

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’).

1

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

6

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.

10

END

11

- 留学生代写
- Python代写
- Java代写
- c/c++代写
- 数据库代写
- 算法代写
- 机器学习代写
- 数据挖掘代写
- 数据分析代写
- Android代写
- html代写
- 计算机网络代写
- 操作系统代写
- 计算机体系结构代写
- R代写
- 数学代写
- 金融作业代写
- 微观经济学代写
- 会计代写
- 统计代写
- 生物代写
- 物理代写
- 机械代写
- Assignment代写
- sql数据库代写
- analysis代写
- Haskell代写
- Linux代写
- Shell代写
- Diode Ideality Factor代写
- 宏观经济学代写
- 经济代写
- 计量经济代写
- math代写
- 金融统计代写
- 经济统计代写
- 概率论代写
- 代数代写
- 工程作业代写
- Databases代写
- 逻辑代写
- JavaScript代写
- Matlab代写
- Unity代写
- BigDate大数据代写
- 汇编代写
- stat代写
- scala代写
- OpenGL代写
- CS代写
- 程序代写
- 简答代写
- Excel代写
- Logisim代写
- 代码代写
- 手写题代写
- 电子工程代写
- 判断代写
- 论文代写
- stata代写
- witness代写
- statscloud代写
- 证明代写
- 非欧几何代写
- 理论代写
- http代写
- MySQL代写
- PHP代写
- 计算代写
- 考试代写
- 博弈论代写
- 英语代写
- essay代写
- 不限代写
- lingo代写
- 线性代数代写
- 文本处理代写
- 商科代写
- visual studio代写
- 光谱分析代写
- report代写
- GCP代写
- 无代写
- 电力系统代写
- refinitiv eikon代写
- 运筹学代写
- simulink代写
- 单片机代写
- GAMS代写
- 人力资源代写
- 报告代写
- SQLAlchemy代写
- Stufio代写
- sklearn代写
- 计算机架构代写
- 贝叶斯代写
- 以太坊代写
- 计算证明代写
- prolog代写
- 交互设计代写
- mips代写
- css代写
- 云计算代写
- dafny代写
- quiz考试代写
- js代写
- 密码学代写
- ml代写
- 水利工程基础代写
- 经济管理代写
- Rmarkdown代写
- 电路代写
- 质量管理画图代写
- sas代写
- 金融数学代写
- processing代写
- 预测分析代写
- 机械力学代写
- vhdl代写
- solidworks代写
- 不涉及代写
- 计算分析代写
- Netlogo代写
- openbugs代写
- 土木代写
- 国际金融专题代写
- 离散数学代写
- openssl代写
- 化学材料代写
- eview代写
- nlp代写
- Assembly language代写
- gproms代写
- studio代写
- robot analyse代写
- pytorch代写
- 证明题代写
- latex代写
- coq代写
- 市场营销论文代写
- 人力资论文代写
- weka代写
- 英文代写
- Minitab代写
- 航空代写
- webots代写
- Advanced Management Accounting代写
- Lunix代写
- 云基础代写
- 有限状态过程代写
- aws代写
- AI代写
- 图灵机代写
- Sociology代写
- 分析代写
- 经济开发代写
- Data代写
- jupyter代写
- 通信考试代写
- 网络安全代写
- 固体力学代写
- spss代写
- 无编程代写
- react代写
- Ocaml代写
- 期货期权代写
- Scheme代写
- 数学统计代写
- 信息安全代写
- Bloomberg代写
- 残疾与创新设计代写
- 历史代写
- 理论题代写
- cpu代写
- 计量代写
- Xpress-IVE代写
- 微积分代写
- 材料学代写
- 代写
- 会计信息系统代写
- 凸优化代写
- 投资代写
- F#代写
- C#代写
- arm代写
- 伪代码代写
- 白话代写
- IC集成电路代写
- reasoning代写
- agents代写
- 精算代写
- opencl代写
- Perl代写
- 图像处理代写
- 工程电磁场代写
- 时间序列代写
- 数据结构算法代写
- 网络基础代写
- 画图代写
- Marie代写
- ASP代写
- EViews代写
- Interval Temporal Logic代写
- ccgarch代写
- rmgarch代写
- jmp代写
- 选择填空代写
- mathematics代写
- winbugs代写
- maya代写
- Directx代写
- PPT代写
- 可视化代写
- 工程材料代写
- 环境代写
- abaqus代写
- 投资组合代写
- 选择题代写
- openmp.c代写
- cuda.cu代写
- 传感器基础代写
- 区块链比特币代写
- 土壤固结代写
- 电气代写
- 电子设计代写
- 主观题代写
- 金融微积代写
- ajax代写
- Risk theory代写
- tcp代写
- tableau代写
- mylab代写
- research paper代写
- 手写代写
- 管理代写
- paper代写
- 毕设代写
- 衍生品代写
- 学术论文代写
- 计算画图代写
- SPIM汇编代写
- 演讲稿代写
- 金融实证代写
- 环境化学代写
- 通信代写
- 股权市场代写
- 计算机逻辑代写
- Microsoft Visio代写
- 业务流程管理代写
- Spark代写
- USYD代写
- 数值分析代写
- 有限元代写
- 抽代代写
- 不限定代写
- IOS代写
- scikit-learn代写
- ts angular代写
- sml代写
- 管理决策分析代写
- vba代写
- 墨大代写
- erlang代写
- Azure代写
- 粒子物理代写
- 编译器代写
- socket代写
- 商业分析代写
- 财务报表分析代写
- Machine Learning代写
- 国际贸易代写
- code代写
- 流体力学代写
- 辅导代写
- 设计代写
- marketing代写
- web代写
- 计算机代写
- verilog代写
- 心理学代写
- 线性回归代写
- 高级数据分析代写
- clingo代写
- Mplab代写
- coventorware代写
- creo代写
- nosql代写
- 供应链代写
- uml代写
- 数字业务技术代写
- 数字业务管理代写
- 结构分析代写
- tf-idf代写
- 地理代写
- financial modeling代写
- quantlib代写
- 电力电子元件代写
- atenda 2D代写
- 宏观代写
- 媒体代写
- 政治代写
- 化学代写
- 随机过程代写
- self attension算法代写
- arm assembly代写
- wireshark代写
- openCV代写
- Uncertainty Quantificatio代写
- prolong代写
- IPYthon代写
- Digital system design 代写
- julia代写
- Advanced Geotechnical Engineering代写
- 回答问题代写
- junit代写
- solidty代写
- maple代写
- 光电技术代写
- 网页代写
- 网络分析代写
- ENVI代写
- gimp代写
- sfml代写
- 社会学代写
- simulationX solidwork代写
- unity 3D代写
- ansys代写
- react native代写
- Alloy代写
- Applied Matrix代写
- JMP PRO代写
- 微观代写
- 人类健康代写
- 市场代写
- proposal代写
- 软件代写
- 信息检索代写
- 商法代写
- 信号代写
- pycharm代写
- 金融风险管理代写
- 数据可视化代写
- fashion代写
- 加拿大代写
- 经济学代写
- Behavioural Finance代写
- cytoscape代写
- 推荐代写
- 金融经济代写
- optimization代写
- alteryxy代写
- tabluea代写
- sas viya代写
- ads代写
- 实时系统代写
- 药剂学代写
- os代写
- Mathematica代写
- Xcode代写
- Swift代写
- rattle代写
- 人工智能代写
- 流体代写
- 结构力学代写
- Communications代写
- 动物学代写
- 问答代写
- MiKTEX代写
- 图论代写
- 数据科学代写
- 计算机安全代写
- 日本历史代写
- gis代写
- rs代写
- 语言代写
- 电学代写
- flutter代写
- drat代写
- 澳洲代写
- 医药代写
- ox代写
- 营销代写
- pddl代写
- 工程项目代写
- archi代写
- Propositional Logic代写
- 国际财务管理代写
- 高宏代写
- 模型代写
- 润色代写
- 营养学论文代写
- 热力学代写
- Acct代写
- Data Synthesis代写