ECON8026代写、Java/C++编程代写

日期：2023-03-26 09:35

ECON8026 Advanced Macroeconomic Analysis

Assignment 4

Semester 1, 2023

Question 1

Preference Shocks in the Consumption-Savings Model. In the two-period consumption-

savings model (in which the representative consumer has no control over his real labor in-

come y1 and y2), suppose the representative consumer’s utility function is u(c1, Bc2), where,

as usual, c1 denotes consumption in period 1, c2 denotes consumption in period 2, and B is

a preference parameter.

a. Use an indifference-curve/budget-constraint diagram to illustrate the effect of an

increase in B on the consumer’s optimal choice of period-1 consumption.

b. Illustrate the effect of an increase in B on the private savings function. Provide

economic interpretation for the result you find.

c. In the months preceding the U.S. invasion of Iraq, data shows that consumers de-

creased their consumption and increased their savings. Is an increase in B and the effects

you analyzed in parts a and b above consistent with the idea that consumption fell and

savings increased because of a looming war? If so, explain why; if not, explain why not.

d. Using a Lagrangian and assuming the utility function is u(c1, Bc2) = ln c1 + ln(Bc2)

, show how the representative consumer’s MRS (and hence optimal choices of consumption

over time) depends on B.

e. How would your analysis in parts a and b change if the consumer’s utility function

were u(Dc1, c2) (instead of u(c1, Bc2)) and you were told that the value D decreased? ( D

is simply some other measure of preference shocks.)

Question 2

Impulse Response Function and Labor Supply: Part 1. Suppose a one-time TFP

shock occurs, as shown below.

As we have studied, an increase in TFP leads to an outwards shift in labor demand (recall

this from our firm analysis unit), which, as long as the upward-sloping labor supply function

does not shift, leads to an increase in the real wage. Using an infinite-horizon (which,

recall, is a heuristic for a “many, many, many time-period” framework) of the combined

consumption-savings and consumption-labor framework (which is an extension of the brief

two-period framework of Chapter 5), qualitatively plot an impulse response function for

the representative consumer’s optimal labor supply that lines up with the impulse response

1

profile for TFP drawn above. Use the lifetime utility function

in which the utility parameters ψ > 0 (the Greek lower-case letter “psi”) and ν > 0 (the

Greek lower-case letter “nu”) are exogenous to the representative consumer.

(You should be able to set up the appropriate budget constraints yourself, although you

don’t need to display them if you don’t think you need to.)

Provide brief justification for the impulse response you have sketched.

Assignment 3

Semester 1, 2023

Question 1

Government and Credit Constraints in the Two-Period Economy. Consider again

our usual two-period consumption-savings model, augmented with a government sector.

Each consumer has preferences described by the utility function

u(c1, c2) = ln c1 + ln c2

where c1 is consumption in period one, and c2 is consumption in period two. The associated

marginal utility functions are

u1(c1, c2) =

1

c1

and u2(c1, c2) =

1

c2

Suppose that both households and the government start with zero initial assets (i.e., A0 = 0

and b0 = 0), and that the real interest rate is always 10 percent. Assume that government

purchases in the first period are one (g1 = 1) and in the second period are 9.9 (g1 = 9.9). In

the first period, the government levies lump-sum taxes in the amount of 8 (t1 = 8). Finally,

the real incomes of the consumer in the two periods are y1 = 9 and y2 = 23.1.

a. What are lump-sum taxes in period two (t2), given the above information?

b. Compute the optimal level of consumption in periods one and two, as well as national

savings in period one.

c. Consider a tax cut in the first period of 1 unit, with government purchases left

unchanged. What is the change in national savings in period one? Provide intuition for the

result you obtain.

d. Now suppose again that t1 = 8 and also that credit constraints on the consumer are

in place, with lenders stipulating that consumers cannot be in debt at the end of period one

(i.e., the credit constraint again takes the form a1 ≥ 0 ). Will this credit constraint affect

consumers’ optimal decisions? Explain why or why not. Is this credit constraint welfare

enhancing, welfare-diminishing, or welfare-neutral?

e. Now with the credit constraint described above in place, consider again the tax cut

of 1 unit in the first period, with no change in government purchases. (That is, t1 falls from

8 units to 7 units.) What is the change in national savings in period one that arises due to

the tax cut? Provide economic intuition for the result you obtain.

Question 2

Habit Persistence in Consumption. An increasingly common utility function used in

macroeconomic applications is one in which period-t utility depends not only on period-t

1

consumption but also on consumption in periods earlier than period t. This idea is known

as “habit persistence,” which is meant to indicate that consumers become “habituated”

to previous levels of consumption. To simplify things, let’s suppose only period-(t ? 1)

consumption enters the period-t utility function. Thus, we can write the instantaneous

utility function as u(c1, c2). When a consumer arrives in period t, ct?1 of course cannot be

changed (because it happened in the past).

a. In a model in which stocks (modeled in the way we introduced them in class) can be

traded every period, how is the pricing equation for St (the nominal stock price) altered due

to the assumption of habit persistence? Consumption in which periods affects the period-t

stock price under habit persistence? To answer this, derive the pricing equation using a

Lagrangian and compare its properties to the standard model’s pricing equation developed

in class. Without habit persistence (i.e., our baseline model in class), consumption in which

periods affects the stock price in period t?

b. Based on your solution in part a and the pattern you notice there, if the instantaneous

utility function were u(ct, ct?1, ct?2) (that is, two lags of consumption appear, meaning that

period t utility depends on consumption in periods t, t?1, and t?2), consumption in which

periods would affect the period-t stock price? No need to derive the result very formally

here, just draw an analogy with what you found above.

Question 3

“Hyperbolic Impatience” and Stock Prices. In this problem you will study a slight

extension of the infinite-period economy from Chapter 8. Specifically, suppose the repre-

sentative consumer has a lifetime utility function given by

u(ct) + γβu(ct+1) + γβ

2u(ct+2) + γβ

3u(ct+3) + ...

in which, as usual, u(.) is the consumer’s utility function in any period and β is a number

between zero and one that measures the “normal” degree of consumer impatience. The

number γ (the Greek letter “gamma,” which is the new feature of the analysis

here) is also a number between zero and one, and it measures an “additional”

degree of consumer impatience, but one that ONLY applies between period t

and period t+1.1 This latter aspect is reflected in the fact that the factor γ is

NOT successively raised to higher and higher powers as the summation grows.

The rest of the framework is exactly as studied in Chapter 8: at?1 is the representative

consumer’s holdings of stock at the beginning of period t, the nominal price of each unit of

stock during period t is St , and the nominal dividend payment (per unit of stock) during

period t is Dt. Finally, the representative consumer’s consumption during period t is ct

and the nominal price of consumption during period t is Pt . As usual, analogous notation

describes all these variables in periods t+ 1, t+ 2, etc.

1The idea here, which goes under the name “hyperbolic impatience”, is that in the “very short run” (i.e.,

between period t and period t + 1), individuals’ degree of impatience may be different from their degree of

impatience in the “slightly longer short run” (i.e., between period t + 1 and period t + 2, say). “Hyperbolic

impatience” is a phenomenon that routinely recurs in laboratory experiments in experimental economics

and psychology, and has many farreaching economic, financial, policy, and societal implications.

2

The Lagrangian for the representative consumer’s utility-maximization problem (start-

ing from the perspective of the beginning of period t) is

u(ct) + γβu(ct+1) + γβ

2u(ct+2) + γβ

3u(ct+3) + ...

+ λt

[

Yt + (St +Dt)at?1 ? Ptct ? Stat

]

+ γβλt+1

[

Yt+1 + (St+1 +Dt+1)at ? Pt+1ct+1 ? St+1at+1

]

+ γβ2λt+2

[

Yt+2 + (St+2 +Dt+2)at+1 ? Pt+2ct+2 ? St+2at+2

]

+ γβ3λt+3

[

Yt+3 + (St+3 +Dt+3)at+2 ? Pt+3ct+3 ? St+3at+3

]

+ ...

a. Compute the first-order conditions of the Lagrangian above with respect to both at

and at+1 . (Note: There is no need to compute first-order conditions with respect to any

other variables.)

b. Using the first-order conditions you computed in part a, construct two distinct stock-

pricing equations, one for the price of stock in period t , and one for the price of stock in

period t + 1 . Your final expressions should be of the form St = ... and St+1 = ... (Note:

It’s fine if your expressions here contain Lagrange multipliers in them.)

For the remainder of this problem, suppose it is known that Dt+1 = Dt+2, and

that St+1 = St+2, and that λt = λt+1 = λt+2

c. Does the above information necessarily imply that the economy is in a steady-state?

Briefly and carefully explain why or why not; your response should make clear what the

definition of a “steady state” is. (Note: To address this question, it’s possible, though not

necessary, that you may need to compute other first-order conditions besides the ones you

have already computed above.)

d. Based on the above information and your stock-price expressions from part b, can

you conclude that the period-t stock price (St) is higher than St+1, lower than St+1, equal

to St+1, or is it impossible to determine? Briefly and carefully explain the economics (i.e.,

the economic reasoning, not simply the mathematics) of your finding.

Now also suppose that the utility function in every period is u(c) = ln c, and also that the

real interest rate is zero in every period.

e. Based on the utility function given, the fact that r = 0, and the basic setup of

the problem described above, construct two marginal rates of substitution (MRS): the

MRS between period-t consumption and period-t+ 1 consumption, and the MRS between

period-t+ 1 consumption and period-t+ 2 consumption.

f. Based on the two MRS functions you computed in part e and on the fact that r = 0

in every period, determine which of the following two consumption growth rates

ct+1

ct

OR

ct+2

ct+1

3

is larger. That is, is the consumption growth rate between period t and period t + 1 (the

fraction on the left) expected to be larger than, smaller than, or equal to the consumption

growth rate between period t + 1 and period t + 2 (the fraction on the right), or is it

impossible to determine? Carefully explain your logic, and briefly explain the economics

(i.e., the economic reasoning, not simply the mathematics) of your finding.