midterm07-answers

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Name

Midterm Exam

180.604Spring, 2007

Answers

You are expected to answer all parts of all questions. If you cannot solve part of aquestion, do not give up. The exam is written so that you should be able to answer laterparts even if you are stumped by earlier parts.

Write all answers on the exam itself; if you run out of room, use the back of the previouspage.


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Part I. Short Discussion Question.

1. Effects ofG in the Tractable Buffer Stock Model.

Consider the behavior of an employed consumer in the tractable buffer stock modeldiscussed in class. Leading up to time t this consumer has been at the steady statetarget level of market resources m. Before the consumption decision is made in periodt, the consumer learns that G has permanently decreased. Show how the phasediagram for the problem changes, show how the Et[ log Ct+1] diagram changes, anddraw a graph showing the time paths ofc and m before and after the change in G.Explainwhether the new target m is higher, lower, or the same as before.

Answer:

An increase in G has similar effects on the phase diagram to the decrease in

analyzed in TractableBufferStock: For any given m the consumptionfunction shifts down. This reflects the fact that human wealth has declined,because expected future income is lower.

The effect in the Et[log Ct+1] diagram is to shift down the g locus, andtherefore to increase the target level ofm to m > m.

The explanation is as follows. The lower expected consumption growthmeans that consumers have less impatience in the sense of less desireto borrow against future income (which is lower) to finance current con-sumption. With less impatience, they consume less at any given m. A

change in expected future G has no effect on the ratio of current Mtocurrent P, but the target level ofm is higher. To achieve this higher m theconsumer must cut the level of consumption, and experience higher-than-before growth rates ofc asymptoting back into the new steady-state growthrate, at which the growth rate ofCmatches the new lower steady-stategrowth rate of income.


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Part II. Medium Analytical Question.

Horizon-Invariance of Portfolio Shares (Samuelson (1969))

Personal finance advisors and websites often recommend that the proportion of a per-sons wealth invested in risky assets should decline with age. This question examineswhether that advice is explained by the benchmark CRRA Merton-Samuelson portfoliochoice model.

Consider a consumer with CRRA utility function u(c) = c1 with risk aversionparameter > 1. This consumer has assets at the end of period t equal to at and is tryingto decide how much to invest in a risky asset that earns stochastic return log t+1 = t+1 N(, 2) compared to a safe asset that earns a log return r = 0 (so that R = 1). (Thenotational convention is that when an object like does not have a subscript, it is thetime-invariant mean of the time-varying realizations of that object, e.g. Et[t+1] = .)

1. Suppose that for a consumer ending period t with assets at, expected value is afunction of the form

vt(at) = a1t Et

et+1(1)

(1)

where = is the equity premium (because r = 0) and the optimal portfolio sharein the risky asset is

=

( 1)2. (2)

Use the fact [NormTimes] ([NormTimes]: Ifz N(z, 2z), then z N(z, 22z))

to show that

Et

et+1(/

2)

= e2/22. (3)

Answer:

t+1(1 ) = t+1/2 (4)

t+1 N(, 2) (5)t+1(/

2) N((/

2), (/

2)

22) (6)

= N(2/2, 2/2) (7)

which implies that

Et

et+1(/

2)

= e2/2+

2/22 (8)

= e2/22 (9)


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2. Show that the value function as of the time when the consumption decision is madeis

vt(mt) = m1

t (10)

for some constant . (Hint: Solve for the consumption function using the FOC,defining = Et

et+1(1)

, to show that optimal consumption is a constant fraction

ofmt.)

Answer:

Defining = Et

et+1(1)

, decision-period value is

vt(mt) = maxct

c1t a1t (11)

with FOC

ct = (mt ct) (12)

ct = (mt ct)()1/ (13)

(1 + ()1/)ct = mt()1/ (14)

ct =

()1/

(1 + ()1/)

mt (15)

so

vt(mt) = (mt)1 ((1 )mt)

1 (16)

= m1t (1 + (1 )1)

(17)


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It is a standard result in portfolio theory that if the value function in period t +1 hasthe generic form vt+1(mt+1) = m

1t+1 and the consumer faces a portfolio investment

choice like the one outlined above, and the consumer will receive no income in period

t+1 except the income on his capital (that is, no labor or pension or transfer income),then the optimal portfolio share to invest in the risky asset at the end of period t isgiven by (2). (This is the Merton-Samuelson model).

3. Given this information, does the Merton-Samuelson model support or contradict theadvice provided by financial advisors? Explain your conclusions.

Answer:

What this result means is that the share of your total wealth allocated torisky assets is always constant and equal to (2), at every age no matter howfar you are from T. So this contradicts the advice from financial advisors,

unless there is some other kind of wealth not encompassed in the model asspecified above.

4. One key feature of reality that is omitted from the Merton-Samuelson model is thefact that people derive income from sources other than their portfolio of risky andriskless financial assets. Considering human wealth as a riskless asset (and ignoringthe possibility of liquidity constraints), does the pattern of human wealth over thelifetime suggest any pattern of portfolio shares in risky versus riskless financial assets?

Answer:

The model as described does not include human wealth. However, if hu-man wealth were perfectly certain and there were no liquidity constraints,it would be equivalent to current market resources, and would thereforeoperate like an unobserved part ofmt. Since the proportion of total wealthaccounted for by human wealth declines as people get older, if we imaginea component ht that varies with age and is incorporated in a measure oftotal wealth, the constant portfolio share in risky assets (including humanwealth) implies that as the perfectly certain human wealth declines, theshare of observed wealth invested in the risky asset may decline.

5. Another possible feature of reality omitted from the usual economic analysis is that

the coefficient of relative risk aversion might vary by age. Suppose that intrinsicrisk aversion increases with age. Would this make the model more consistent or lessconsistent with the advice given by financial advisors?

Answer:

This is another potential explanation for the advice of the personal financegurus. The equation for the portfolio share invested in risky assets declinesdirectly as increases (cf. (2)), so if increases with age it would be naturalto expect to decline.


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Part III. Long Analytical Question.

Productivity Growth and Dynamic Inefficiency in the OLG Model.

Consider a Diamond (1965) OLG economy like the one in the handout OLGModel,assuming logarithmic utility and a Cobb-Douglas aggregate production function,

Yt = F(Kt, PtLt) (18)

where Pt is a measure of labor productivity that grows by

Pt+1 = GPt (19)

from period to period. Assume that population growth is zero (N = 1) and for conveniencenormalize the population at Ls = 1 s, and assume that productivity growth has occurredat the rate g = G 1 forever.

One unit of the quantity P L is called an efficiency unit of labor: It reflects a unit oflabor input to the production process.

1. Assume that F(K , PL) is a CRS function, and show how to rewrite the capitalaccumulation equation

Kt+1 = A1,t (20)

in per-efficiency-unit terms as

kt+1 = a1,t/G (21)

Answer:

In an OLG economy, aggregate capital in period t+1 is equal to the savingsfrom period t:

Kt+1 = A1,t (22)

Kt+1 = a1,tPt (23)Kt+1

Pt

= a1,t (24)

Kt+1Pt+1 Pt+1Pt = a1,t (25)kt+1 = a1,t/Gt+1 (26)

2. Show that under these assumptions, the process for aggregate k dynamics is

kt+1 =

(1 )

Gt+1(1 + )

kt (27)

Answer:


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k

k

kt

kt1

45 Line

k

t1kt

Figure 1: Convergence of OLG Economy After Increase in G

Since (1 ) > 0, equation (37) implies that a larger value ofG impliesa lower steady-state capital stock per efficiency unit. The reason is that

with faster productivity growth, the efficiency units of labor provided bythe young generation are larger relative to the size of the capital stocksaved by the previous generation, so the ratio of capital to efficiency unitsof labor is smaller.

5. Next, use a diagram to show how the kt+1(kt) curve changes when the new growthrate takes effect, and show the dynamic adjustment process for the capital stocktoward its new steady-state, assuming that the economy was at its original steadystate leading up to period t.

Answer:

Defining the original steady-state capital stock as k and the new steady-

state capital stock ask, the convergence process looks as indicated in fig-

ure 1.

6. Define an index of aggregate consumption per efficiency unit of labor in period t ast = c1,t + c2,t/G, and derive a formula for the sustainable level of associated witha given level ofk.

Answer:


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Kt+1 = Kt + Kt P

1t C1,t C2,t (38)

Kt+1Pt

= kt + kt c1,t c2,tPt1Pt (39)Kt+1Pt+1

Pt+1Pt

= kt + k

t c1,t c2,t/G (40)

kt+1G = kt + kt + t (41)

The sustainable level of is the level such that kt+1 = kt = k:

(1 + g)k = k + k (42)

= k gk. (43)

7. Derive the conditions under which a marginal increase in the productivity growthrate g will result in an increase in the steady-state level of, and explain in wordswhy this result holds. (You can leave the term k/g unevaluated in your answer,using only what we know about this term from above).

Answer:

There are two effects of an increase in g. First, for a given k, the sustainableamount of(k) declines, because the faster productivity growth meansthat to keep capital per efficiency unit constant the economy must save

more (each efficiency unit of labor must be supplied with its own capital;faster growth of efficiency units therefore requires faster growth of capital).Second, with a faster g the endogenous saving rate and steady-state capital-per-capita k will change. Whether steady-state consumption per capitarises or falls depends on the balance between these two things.

Steady-state can be written as (k). We are interested ind

dg

=

g

+

k

k

g

(44)

=

k +

k

k

g (45)But we know (from above) that k/g is negative; since k < 0, (45) canpossiblybe positive only if

k

< 0 (46)

k1 g < 0 (47)

r < g , (48)


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where r = f(k) = k1. This is just the dynamic efficiency condition.

In words: We know from above that a higher value ofg will decrease thesteady-state capital stock per efficiency unit. We know from our analysis inclass that the only circumstance in which a decrease in capital per efficiencyunit will directly result in an increase in consumption per efficiency unitis if the dynamic efficiency condition fails to hold. So in order for there tobe any hope of an increase in g increasing , the economy must start outas being dynamically inefficient.

However, dynamic inefficiency is not enough - the second term in (45) mustbe larger than k in order to offset the negative effect of faster g on . Theeconomy must be sufficientlydynamically inefficient that the increase inthe raw marginal product of capital that comes from lower k more than

offsets the capital-dilution effect from the requirement to equip the newefficiency units of labor with capital. In math, faster growth increasesconsumption per efficiency unit when

d

dg

> 0 (49)

k + (r g)

k

g

> 0 (50)

(r g)

k

g

> k. (51)


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