final-sol.pdf

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Econ 202a Spring 2000

Suggested Final Exam Solutions

1 Rational and Near Rational Wage Setting

Firms

Each firm faces a commodity demand ofqd = 1nMp

pp

, where >1. Labor

productivity is given by T(w) = A+B wwR

+Cu. So, the production

function is qs = T(w)L and the cost function must take the form C(qs) =w

T(w)qs

. Hence, the profit function can be written as

i = piqsi C(q

si ) =

pi

wiT(wi)

qdi (1)

for a monopolist. There are two types of firms, rational and near-rationalones. Label them i = r, nr, respectively.

1.1 Price choice of firm i

Firmi chooses it price by maximizing (1) with respect to pi. The first order

condition for this problem is

qdi

pi

wiT(wi)

qdipi

= 0,

which is a sufficient condition since revenues are concave in pi. Rearrangingyields

pi =

1

wiT(wi)

. (2)

1.2 Wage choice of firm i

Similarly, maximizing (1) with respect towi yields the first order condition

T(wi) T(wi)wi

T(wi)2

= 0

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so that, in optimum, the Solow condition

wi = T(wi)

T(wi) (3)

must be satisfied. By the definition ofT(), this implies that

w=A+B

wiwRi

+Cu

BwiwRi

1w

.

Equivalently,

wi = A Cu

(1 )B

1

w

R

i . (4)

1.3 Relative profits

Given the facts that wRr = wR = w1(1 +e) for the rational firms and

wRnr = w1 for the near rational firms, the two wage choices are equal for= e = 0:

wRr =wRnr =

A Cu

(1 )B

1

wR if = e = 0.

Hence, the productivity levels must be equal, too: T(wr) = T(wnr) for =

e = 0. Then, however, price choices must be equal: pr = pnr. Therefore,we can conclude that profits r and nr must be identical if =

e = 0 andrelative profits (the ratio of profits) must be equal to one: r/nr = 1.

Suppose these relative profits could be expressed as a function of inflation. Then, by the arguments above, at zero inflation =e = 0 the optimalchoice ofpi and wi are the same for both firms. Therefore, the change inprofits is the same by the envelope theorem so that the ratio of profits remainsat one after the change. This means, however, that the derivative of the ratior/nr with respect to must be zero.

1 This argument suffices to answerthe question.

1In general,

r

nr

=r

nr nr

r

(nr)2 = 0

for r = nr and r/ = nr/.

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The same results could, of course, also be derived explicitly. This was not

required. For completeness, relative profits could be derived by noting that

i =

pi

wiT(wi)

qdi

=

1

wiT(wi)

wi

T(wi)

1

n

M

p

pip

= 1

1

1

Mn

1

p

1 wiT(wi)

1,

and hence

rnr =

wrT(wr)wnr

T(wnr)

1= w

Rr

wRnrA+B(wr/wR)+CuA+B(wnr/wR)

+Cu

1

by (4) and the fact that true productivity is given byT(wi) = A +B

wi/wR

+

Cu for both firms (where wR = w1(1 +e)). Setting = e = 0 immediately

yields r/nr = 1. Taking the derivative with respect to and evaluating the

resulting term at = e = 0 yields zero.

1.4 Short-run wage Phillips curve

If we suppose that denotes the share of near-rational firms, we can obtain

the wage level at period 0 as

w0 = wnr+ (1 )wr

=

A Cu

(1 )B

1

[w1+ (1 )w1(1 +e)] . (5)

Subtractingw1/w1 from both sides of (5) yields

w0 w1w1

=

A Cu

(1 )B

1

[1 + (1 )e] 1. (6)

1.5 Long-run Phillips curve

Since = e = 0 and = (w0 w1)/w1 in the long-run, (6) implies

1 +LR = a [1 + (1 )LR] (7)

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Figure 1: Phase Diagram for Tobins q

fora [(A Cu)/(1 )B] 1 . Thus,

LR= 1 a

1 a(1 ). (8)

2 Temporary Technological Progress and To-

bins q

A temporary rise in the technology parameterA shifts the qt= 0-scheduletemporarily to the East. Thus, between time t0, when productivity unexpect-

edly rises, and time T > t0, when it expectedly falls back to its initial level,the new dynamic system with qt = 0 and Kt = 0 temporarily governsthe dynamics of the capital stock and Tobins q.

The capital stockKt0 cannot adjust immediately and thus remains at itsinitial steady-state level for one period. However, the shadow price of capital,

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Tobinsq, has to jump to a new level which is such that the subsequent levels

of bothqs andKs (s t0) will be back at a point on the initial saddle path.So, att0Tobinsqjumps discretely to point 2 in figure 1 and it will reach point3 exactly at date T. Between timet0 andT, both variables qs andKs mustfollow the dynamics of the shifted system which take them first to the southeast and then (once the Kt = 0-schedule is surpassed) to the south westof the system. Note that the shadow price must not change discontinuously(jump) at any time after t0, otherwise the capital stock cannot have beenchosen optimally. Between time t0 and T, the capital stock first increasesand then falls while Tobins qkeeps falling until T. Once timeT is reached,the system is again governed by the old qt = 0 and Kt = 0-schedulesand the capital stock continues to fall smoothly while the shadow price of

capitalqs now continuously rises.

3 Central Bank with Superior Knowledge

3.1 Linear optimal monetary rule

Given aggregate supply

yt = (pt Et1[pt]) +ut, (9)

and aggregate demand

mdt =pt+ vt, (10)

output can be written as

yt = mst vt Et1[m

st vt] +ut

= mst vt Et1[mst ] +ut1+ t. (11)

The second step follows from the assumptions on the stochastic processes.Since the conditional expectations over the linear monetary rule mst = a+but1+ct+dvt1 are Et1[m

st ] =a+but1+d vt1, it follows that

mst Et1[mst ] =ct.

Hence, output can be written as

yt = (c+ 1)t vt+ut1. (12)

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For c = 1, the disturbance of aggregate supply is completely offset by

the monetary rule so that the variance of output will be minimized. Theparameters a, b, d are irrelevant since their impact is entirely anticipatedby the private sector. This arguments can be made formal by minimizingV(yt) = E[(ct vt+ ut1+ t)

2] with respect to a, b, c, d (a, b, d dontenter the minimization problem).

3.2 Non-linear optimal monetary rule

Allowing for higher order terms such as in a rule like mst = a+ but1 +ct+dvt1+ b(ut1)2 +c(t)2 +d(vt1)2 will result in minimizing V(yt) =E[(ct+ c(t)2 vt+ ut1+ t)2]. Now neithera, b, d nor a, b, d enter,

and settingc = 1 is still optimal. Since the noise t is completely offset bythe choice ofc = 1 already, anything else but c = 0 would add noise backin. The same is true for any higher order polynomial.

3.3 Stochastic process ofut

The process

ut= ut1+t1+ t (13)

is an ARMA(1,1) process. Using lag operators, (13) can be rewritten as

1 L

1 +Lut= t (14)

or

(1 L)(1 L +2L2 3L3 +4L4 . . . ) ut = t (15)

(1 L L +L2 +2L2 2L3 3L3 +. . . ) ut = t

(1 (+)L + (+)L2 (+)2L3 +. . . ) ut = t. (16)

Without going through any further derivations, the first step (15) immedi-

ately shows that we found an AR(

) process with

ut=s=1

ksuts+ t

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for some ks. The (not required) further steps show that

ut = (+)s=1

()s1uts+ t

in fact.

3.4 New optimal monetary rule

The optimal monetary rule does not change at all. Both the Central Bankand the Private Sector know all (infinitely many) past realizations ofuts(s = 0, 1, 2, . . . ). So it is still the case that the Private Sector rationally

foresees that the expected deviation of the money supply from its mean isequal to the unknown part of the money rule, mst Et1[m

st ] =ct+ c

(t)2.Thus, including any past realizations ofuts orvts is entirely irrelevant. Itis still optimal for the central bank to set c = 1 and c = 0.

4 Microfoundations of Money Demand

4.1 Interpretation of budget constraint

The individual can choose to allocate her current income Yt to three different

uses: to savingsAt+1, or to money holding Mt, or to consumption Ct. Themore money Mt she chooses to put aside today (at t), the more of todaysincome Yt becomes useful. In the limit (

MtPt

), income is fully useful.In optimum, the individual will choose some intermediate amount of moneyholdings.

4.2 Choice variables

The individual can choose Ct, At+1, and Mt. Since she is restricted by thebudget constraint, choosing any two of the three implies her choice of thethird. Thus, two intertemporal first-order conditions (Euler equations) will

be enough to pin the optimal consumption, savings and money holdings pathsdown.

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4.3 First-order conditions

Intertemporal utilityUt =

s=tstu(Cs)2 can be rewritten as

Ut = u(Ct) +Ut+1= u(Ct) +u(Ct+1) +2Ut+2. (17)

Then, plugging the consumption choice Ct = (1 +r)At+Mt1Pt

+g(MtPt )Yt

At+1 MtPt

into (17) fort and t + 1, and maximizing (17) with respect toAt+1andMt yields the two first-order conditions

u(Ct) +(1 +r)u(Ct+1) = 0 (18)

and

1Pt

u(Ct)

1 g

MtPt

Yt

+ 1

Pt+1u(Ct+1) = 0. (19)

Using (18) in (19) yields

1 g

MtPt

Yt=

1

1 +r

PtPt+1

. (20)

4.4 Money demand under specific functional form for

g()

Given (20) andYs= Y, and usingg(MtPt ) = 1k e

Mt

Pt so thatg(MtPt ) =k e

Mt

Pt,we find

k Y eMt

Pt = 1 1

1 +r

PtPt+1

.

Observe that the nominal interest rate must be related to the real interestrate through 1 +it+1= (1 +r)

Pt+1Pt

. Then, taking logs of both sides yields

MtPt

= ln kY ln it+11 +it+1

. (21)

This function is increasing in income and decreasing in the nominal interestrate, just as any LM-function should be.

2This relationship is correct for 1. All the following derivations hold for the more common case where

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