## Answer Outline for Final Exam

Course Instructor: Leigh Tesfatsion
Date: 7 May 1996

```
FINAL EXAM:  100 POINTS TOTAL

QUESTION 1: SHORT ANSWERS [30 Points Total]

-----------------

Part A [15 Points]:  Give brief but careful #verbal# definitions
for the #economic# meaning of an IS curve, an LM curve, and an
is derived from the IS and LM curves.

-----------------

Part B [15 Points]: Define what is meant by the expectations-augmented
Phillips curve.  Explain briefly how this Phillips curve can give
rise to episodes of "stagflation" in which both unemployment and
inflation are increasing at the same time.

------------------------------------------------------------------------

QUESTION 2:  [40 Points Total]  Consider an economy described by
the following five relations in some period T:
(1)  Y(T)   =  C(T)   +   I(T)    +   G   +   NE(T)
(2)  C(T)   =  a   +   b[1 - t]Y(T)
(3)  I(T)   =  e   -   dR(T)
(4)  NE(T)  =  g  -  mY(T) -  nR(T)
(5)  M/P(T) =  kY(T) - hR(T)
Endogenous Variables: Y(T), C(T), I(T), NE(T), R(T)
Exogenous Variables:  Policy variables G,t,M;
constants a,b,e,d,g,m,n,k,h;
all exog var's positive with b and t less than 1

------------------

Part A [15 Points]: Derive #algebraic# relations for the IS Curve
and the LM Curve for the economy described by relations (1)-(5).
Carefully #graph# these IS and LM relations, and explain what is
meant by an IS-LM equilibrium.

------------------

Part B [5 Points]: Now suppose the economy is described by
relations (1)-(5) #plus# the following #additional# relations in
each period T=1,2,...:
(6)  K(T+1) =  K(T) + I(T)
(7)  Y*(T)  =  F(N*(T),K(T))
(8)  N*(T)  =  h(w*(T))
(9)  w*(T)  =  dF(N*(T),K(T))/dN
(10) INF(T,T+1)  =  zINF(T-1,T) +  f[Y(T) - Y*(T)]/Y*(T)
(11) INF(T,T+1)  =  [P(T+1)-P(T)]/P(T)
where: Y=F(N,K(T))=period T aggregate production function with
higher capital stock K(T) resulting in higher Y and in higher
marginal product of labor dF(N,K(T))/dN for each N; w=real wage;
N=h(w) is the labor supply function with higher w resulting in
higher labor supply; K(T+1), Y*(T), N*(T), w*(T), INF(T,T+1), and
P(T,T+1) are period T endogenous; K(T), INF(T-1,T), and P(T) are
period T predetermined; and z and f are positive constants.
For the economy described by (1)-(11), explain carefully #in
words# what is meant by internal balance in period T.  Graphically
depict a situation in which the economy in period T is in internal
balance.

---------------------

Part C [20 Points]: Assume the economy described by (1)-(11) is in
internal balance in period 1 with K(1)=100, INF(0,1) = 0 and
P(1)=1.  Suppose the central bank unexpectedly #increases# the
money supply M in period 1 and keeps the money supply at this
higher level in period 2.  Using graphs to illustrate and support
your assertions, explain carefully whether this period 1 increase
in M increases, decreases, or leaves unchanged each of the
following variables:
Y(1), R(1), I(1), P(1), Y*(1), P(2), and Y*(2).

----------------------------------------------------------------------

QUESTION 3:  [30 POINTS TOTAL] The year is 1996, and Democrats and
Republicans are arguing over the most appropriate course of action
to take to achieve the following agreed upon goal: maximum
sustainable economic growth consistent with a stable price level
and a low level of unemployment.  The following proposal is under
discussion:

PROPOSAL: Government should #cut# way back both on government
spending and on taxes in such a way that the government
budget is balanced.  The Federal Reserve Board should
keep tight control of the money supply to convince
the private sector that government is serious about
controlling inflation.

Using the economic model described by relations (1)-(11) in
Question 2 (above) as your basic starting point, analyze the
potential strengths and weaknesses of this proposal for achieving
the stated economic goal.

---------------------------------------------------------------
---------------------------------------------------------------

QUESTION 1: SHORT ANSWERS [30 Points Total]

-----------------

Part A [15 Points]:  Give brief but careful #verbal# definitions
for the #economic# meaning of an IS curve, an LM curve, and an
is derived from the IS and LM curves.

Answer Outline for Part A: The IS Curve shows all
combinations of the real interest rate R and real GDP Y that are
consistent with product market equilibrium, in the sense that
demand for newly produced goods and services equals supply and
all expectations are fulfilled.

The LM Curve shows all combinations of the real interest
rate R and real GDP Y that are consistent with money market
equilibrium, in the sense that the demand for money equals the
supply of money and all expectations are fulfilled.

The AD Curve is a schedule showing the aggregate spending Y
of the economy at each general price level P, assuming the
economy is in an IS-LM equilibrium (or short-run equilibrium) at
this price level.

The AD curve is derived from the succession of intersection
points (Y^o,R^o) of the IS and LM Curves as the general price
level P is varied.  Specifically, it is the graph of the resulting
solution values Y^o against the corresponding P values.  Note that
P enters only into the LM curve; the IS Curve is not affected by
changes in P.
The graphical derivation of the AD Curve from the IS and LM
curves, given in Hall and Taylor Chapter 7 (see also the HT7
lecture notes), is *crudely* reproduced below.

GRAPHICAL DERIVATION OF THE AGGREGATE DEMAND CURVE:

(LM given P=P_1)
R                                 lm
|
| is                        lm
|
|         is         lm
|                                   (LM given P=P_2)
|              lmis                       lm
|              .
|         lm   .         is           lm
|              .
|    lm        .                 is
|              .             lm                (LM given P=P_3)
|              .             .           is    lm
|              .             .            lm
|              .      lm     .        lm  .      is
|              .             .    lm      .
------------------------------------------------------  Y
0          Y^o(P_1)      Y^o(P_2)        Y^o(P_3)

P
|
|          .
|          .
|          .             .
|          .             .
-------------------------------------------------  Y
0          Y^o(P_1)      Y^o(P_2)        Y^o(P_3)

The Aggregate Demand Curve in the Y-P Plane

-----------------
Part B [15 Points]: Define what is meant by the
expectations-augmented Phillips curve.  Explain briefly how this
Phillips curve can give rise to episodes of "stagflation" in which
both unemployment and inflation are increasing at the same time.

Answer Outline for Part B:  The expectations-augmented
Phillips Curve is a relation that postulates that the actual
inflation rate from any period T to period T+1 depends positively
on the expected inflation rate from T to T+1 and positively on the
GDP gap in period T.  The simple algebraic form of this relation
used by HT is as follows:

(*)  INF(T,T+1)      =   INF^e(T,T+1)      +    f[Y(T) - Y*(T)]/Y*(T) ,

actual inflation     expected inflation        GDP gap in period T
rate from T to T+1   rate from T to T+1

where f is a positive exogenously given constant.  Unemployment is
increasing if the GDP gap is becoming more negative over time,
implying that the right-hand-side term in (*) is becoming more
negative over time.  The actual inflation rate on the left-hand-side
of (*) can still be increasing over time, however, as long as the
*expected* inflation rate is increasing over time at a sufficiently
fast rate.  In this case the effect of the increasing expected
inflation rate outweighs the effect of the decreasing GDP gap.

------------------------------------------------------------------------

QUESTION 2:  [40 Points Total]  Consider an economy described by
the following five relations in some period T:
(1)  Y(T)   =  C(T)   +   I(T)    +   G   +   NE(T)
(2)  C(T)   =  a   +   b[1 - t]Y(T)
(3)  I(T)   =  e   -   dR(T)
(4)  NE(T)  =  g  -  mY(T) -  nR(T)
(5)  M/P(T) =  kY(T) - hR(T)
Endogenous Variables: Y(T), C(T), I(T), NE(T), R(T)
Exogenous Variables:  Policy variables G,t,M;
constants a,b,e,d,g,m,n,k,h;
all exog var's positive with b and t less than 1
------------------------------------
Part A [15 Points]: Derive #algebraic# relations for the IS Curve
and the LM Curve for the economy described by relations (1)-(5).
Carefully #graph# these IS and LM relations, and explain what is
meant by an IS-LM equilibrium.

NOTE FOR GRADING:  STUDENTS SHOULD SHOW THE DERIVATION OF THE
ASKED FOR ALGEBRAIC RELATIONS FOR THE IS AND LM CURVES.  ONLY HALF
CREDIT WILL BE GIVEN IF STUDENTS SIMPLY GIVE THE FINAL FORM
WITHOUT SHOWING THEIR WORK.

Answer Outline for Part A:  As noted in Part A of Question 1,
the IS Curve shows all combinations of the real interest rate R
and real GDP Y that are consistent with product market
equilibrium.  For the economy described by relations (1)-(5), the
relations dealing with the product market are relations (1), (2),
(3), (4)---that is, all relations except the money market relation
(5).
The IS curve is obtained from the product market relations
(1)-(4) as follows:

#Step 1#:  Reduce down relations (1)-(4) to one equation in the
two unknowns R(T) and Y(T) by substituting out for the other
endogenous variables C(T), I(T), and NE(T) that appear in these
relations.

Y(T) = (a + b[1-t]Y(T)) + (e - dR(T)) + G + (g - mY(T) - nR(T))

#Step 2#: Collect terms in Y(T) and in R(T)

(1 - b[1-t] + m) x Y(T)  =  (a + e + G + g)   -  (d+n) x R(T)

or equivalently

(d+n) x R(T)   =   (a + e + G + g)  -  (1-b[1-t] + m) x Y(T)

#Step 3#: Solve for R as a function of Y (slope-intercept form)

(a + e + G + g)          (1 - [1-b] + m)
R(T)  =  ---------------     -   -----------------  Y(T)
(d+n)                   (d+n)

...............    .....................
R-Intercept (+)       Slope dR/dY (-)

---------------

#Graphical Illustration#:

R
|
(a+e+G+g)|
---------.is
(d+n)  |    is
|         is
|              is <-------------IS Curve
|                   is
|                        is
|                            is
|                                  is
|-----------------------------------------------   Y
0

IS Curve with positive R-intercept and negative slope dR/dY.

As noted in Part A of Question 1, the LM Curve shows all
combinations of the real interest rate R and real GDP Y that are
consistent with money market equilibrium.  For the economy at
hand, the only relation dealing with the money market is relation
(5).  The LM curve is obtained from the money market relation (5)
as follows.

#Step 1#:  Relation (5) is already one relation in the two endogenous
variables of interest, Y(T) and R(T); no other endogenous variables
appear in relation (5).

#Step 2#: Collect terms in Y(T) and in R(T)

hR(T)  =  - M/P(T)  +  kY(T)

#Step 3#: Solve for R(T) as a function of Y(T) (slope-intercept form)

R(T)   =    - M/hP(T)         +   (k/h)    x   Y(T)
...............   ...............
R-Intercept (-)   Slope dR/dY (+)

---------------

#Graphical Illustration#:

R
|
|                                LM Curve given M/P(T)
|                                      lm
|                                   lm
|                                lm
|                             lm
|                          lm
|                       lm
|                    lm
|                 lm
|--------------lm---------------------------------   Y
0          lm
|       lm
|    lm
-M/hP(T) . lm
|

LM Curve with negative R-intercept and positive slope dR/dY.

------------------
Part B [5 Points]: Now suppose the economy is described by
relations (1)-(5) #plus# the following #additional# relations in
each period T=1,2,...:
(6)  K(T+1) =  K(T) + I(T)
(7)  Y*(T)  =  F(N*(T),K(T))
(8)  N*(T)  =  h(w*(T))
(9)  w*(T)  =  dF(N*(T),K(T))/dN
(10) INF(T,T+1)  =  zINF(T-1,T) +  f[Y(T) - Y*(T)]/Y*(T)
(11) INF(T,T+1)  =  [P(T+1)-P(T)]/P(T)
where: Y=F(N,K(T))=period T aggregate production function with
higher capital stock K(T) resulting in higher Y and in higher
marginal product of labor dF(N,K(T))/dN for each N; w=real wage;
N=h(w) is the labor supply function with higher w resulting in
higher labor supply; K(T+1), Y*(T), N*(T), w*(T), INF(T,T+1), and
P(T,T+1) are period T endogenous; K(T), INF(T-1,T), and P(T) are
period T predetermined; and z and f are positive constants.
For the economy described by (1)-(11), explain carefully #in
words# what is meant by internal balance in period T.  Graphically
depict a situation in which the economy in period T is in internal
balance.

Answer Outline for Part B:  An economy is said to be in
internal balance if all of its domestic markets (money, product,
and labor) are in equilibrium.  In particular, then, an economy is
in internal balance if two conditions hold: (i) it is in an IS-LM
equilibrium (or short-run equilibrium); and (ii) actual GDP is
equal to potential GDP.

GRAPHICAL DEPICTION OF INTERNAL BALANCE IN PERIOD T:

R                                 lm
|
| is                        lm  <------LM Curve given P=P(T)
|
|         is         lm
|
|              lmis  <----------------IS-LM equilibrium
|              .                      holds with Y(T)=Y*(T)
|         lm   .         is
|              .
|    lm        .                 is   <--------IS Curve
|              .
|              .                         is
|              .
|              .                                 is
|              .
------------------------------------------------------  Y
0            Y*(T)

---------------------
Part C [20 Points]: Assume the economy described by (1)-(11) is in
internal balance in period 1 with K(1)=100, INF(0,1) = 0 and
P(1)=1.  Suppose the central bank unexpectedly #increases# the
money supply M in period 1 and keeps the money supply at this
higher level in period 2.  Using graphs to illustrate and support
your assertions, explain carefully whether this period 1 increase
in M increases, decreases, or leaves unchanged each of the
following variables:
Y(1), R(1), I(1), P(1), Y*(1), P(2), and Y*(2).

Answer Outline for Part C:  Suppose the economy is in a
situation of internal balance as graphically depicted in Part B,
with T=1 and P(T)=1.  Recall that the price level P(1) in period 1
is predetermined, and so remains fixed throughout period 1.  Thus,
the sudden increase in the money supply M to some higher level,
say M', means that the real money supply M'/P(1) is suddenly
larger (M is higher and P(1) is unchanged).  This shifts the LM
curve **down** in the sense that a lower level of R corresponds to
each level of Y; but the IS Curve is not affected since M does not
enter into the algebraic relation for the IS Curve.
Consequently, a new IS-LM equilibrium is established at the
intersection point of the shifted-down LM curve and the unchanged
IS Curve, implying that Y(1) is **higher** and R(1) is **lower**.
THIS NEW IS-LM EQUILIBRIUM SHOULD BE GRAPHICALLY DEPICTED.
The lower level of R means that investment I(1) is
**higher**.  The predetermined ("sticky") price level P(1) is
**unchanged**.  Also, the potential GDP level Y*(1) for period 1
is **unchanged**; for nothing has happened that shifts either the
labor supply function (8) or the labor demand function (9) **in
period 1**, hence N*(1) is unchanged.  Since K(1) is
predetermined, it follows that Y*(1)=F(N*(1),K(1)) is the same as
before.
What about P(2)?  Note that a **positive** GDP gap has opened
up in period 1.  This is because, by assumption, Y(1) was
**originally** equal to Y*(1) because the economy was originally
in internal balance; but then the increase in M increased Y(1)
above Y*(1).  It follows from the Phillips Curve relation (10) for
T=1 and the definition (11) for the inflation rate INF(T,T+1) with
T=1, together with the assumption that INF(0,1)=0, that the price
level P(2) must then **increase** relative to the level P(2)=P(1)
that it would have taken on if M had not increased in period 1.
Finally, what about Y*(2)?  It follows from relation (6) for
T=1 that the increase in I(1) in period 1 leads to a period 2
capital stock K(2) that is **higher** than it would have been
without the increase in M in period 1.  Consequently, it follows
from relations (7) and (9) for T=2 (plus the assumptions given for
the production function and the marginal product of labor) that
both output and the marginal product of labor are **higher** for
each level of employment N.  THESE SHIFTS IN THE PRODUCTION
FUNCTION AND THE LABOR DEMAND CURVE IN PERIOD 2 SHOULD BE
GRAPHICALLY DEPICTED.  Consequently, N*(2) will by higher in
period 2 than it otherwise would have been because it is
determined from the intersection of the shifted up labor demand
curve with the upward sloping labor supply function.
Consequently, plugging in this higher level of N*(2) in the
shifted up production function, potential GDP Y*(2) in period 2
will be **higher** than it would have been in the absence of the
period 1 increase in M.  THE GRAPHICAL DEPICTION OF THESE EFFECTS
SHOULD BE GIVEN WITH SOME ACCOMPANYING VERBAL EXPLANATION.

SUMMARY EFFECTS:  Y(1) increased; R(1) lower; I(1) higher;
P(1) unchanged, Y*(1) unchanged;
P(2) increased; Y*(2) increased.

---------------------------------------------------------------------

QUESTION 3:  [30 POINTS TOTAL] The year is 1996, and Democrats and
Republicans are arguing over the most appropriate course of action
to take to achieve the following agreed upon goal: maximum
sustainable economic growth consistent with a stable price level
and a low level of unemployment.  The following proposal is under
discussion:

PROPOSAL: Government should #cut# way back both on government
spending and on taxes in such a way that the government
budget is balanced.  The Federal Reserve Board should
keep tight control of the money supply to convince
the private sector that government is serious about
controlling inflation.

Using the economic model described by relations (1)-(11) in
Question 2 (above) as your basic starting point, analyze the
potential strengths and weaknesses of this proposal for achieving
the stated economic goal.

similar to the simplified Gingrich/Dole presidential platform.
Consequently, the in-class discussion of potential strengths and
weaknesses of the latter platform are relevant here as well.  The
website summary of this discussion is reproduced below, with some
modifications appropriate for the proposal at hand---namely, a
consideration that the current proposal calls for the government
budget to be balanced **now**.  The discussion below is of course
more detailed than would be expected for student answers on this
30 point (approximately 1/2 hour) question.

----------------------

Interpretation of Proposal (Platform):

The basic idea is that government should only do for people
what people cannot do for themselves.  While most political
parties accept this idea, conservatives differ from liberals with
regard to their interpretation of the phrase "what people cannot
do for themselves."

Except for a few basic functions such as national defense and
the regulation of foreign relations (plus the collection of
minimal taxes to finance these functions), conservatives such as
Gingrich (and to a lesser extent Dole) believe that most things
should be left to the voluntary efforts of private citizens.  They
believe that a cut-back in government intervention would unleash a
tide of productive private effort---in particular, large increases
in private investment, including expenditures on research and
development (R and D) that could positively affect the total
factor productivity coefficient A in the aggregate production
function.  These private efforts would thus result in a higher
capital stock and improved technology that would in turn lead to
increases in the potential GDP level Y* over time.  Any imbalance
that might arise between Y and Y* would be corrected for quickly
and automatically by offsetting movements in the price level and
the interest rate.

On the other hand, conservatives worry about the supposed
propensity of the government to pay for increases in current
expenditures by means of debt financing (i.e., the sale of newly
issued government bonds to U.S. private citizens and to ROW) or by
printing press financing (i.e., the sale of newly issued
government bonds directly to the central bank---the Fed Reserve in
the U.S. ---in return for a deposit account or cash).  Either
means of financing runs up the current government budget deficit:

[G+F+N]  -  tY   =   DB/P    +    DM/P  ,

and hence reduces government savings:

S_g  =  tY - [G+F+N],

where F = government net transfers to the private sector and N =
interest payments on the current government debt (outstanding
government bonds held by private U.S. citizens and by ROW).
Conservatives point to the identity

S_p   +    S_g     +     S_r     =     I

to argue that reductions in S_g means that either investment I
will have to be cut or that the reduction in S_g will have to be
made up by increases in private savings S_p or by increases in ROW
savings S_r (HC borrowing from ROW).  Decreasing I or increasing S_r
are not very attractive options ("mortgaging the future of our
children"), and increasing S_p has proved hard to do.

In addition, repeated increases in M resulting from a
printing press financing of government expenditures will result in
eventual inflation as Y rises above potential GDP Y*.  An increase
in the expected inflation rate reduces the expected real earnings
on securities such as bonds whose promised payments to holders are
denominated in dollars---for example, \$100 per year.
Consequently, the price P_B for such bonds will drop, which
implies that the nominal rate of return on these bonds (e.g.,
\$100/P_B) will rise.

Conservatives therefore argue that the Federal Reserve Board
should only increase M to keep it in line with the growing
transactions needs of a growing economy.  That is, basically the
money supply should not grow any faster than Y*.

Potential Problems:

In terms of the Hall and Taylor model, a cut in G shifts
#down# the IS curve, and a cut in t rotates the IS curve #upwards#
around the R-intercept, offsetting to some extent the cut-back in
G.  In general, these displacements in the IS curve can result in
two possible types of outcomes.

If the expansionary effects of the decrease in t sufficiently
outweigh the contractionary effects of the cut in G, then Y and R
will both increase.  In this case there will be a reduction in the
current level of unemployment (a reduction in the size of the GDP
gap) but also a reduction in current investment which will result
in a smaller capital stock (and hence a lower potential GDP level
Y*) in the next period.  If Y actually rises above Y*, there will
also be an increase in the inflation rate.

On the other hand, if the contractionary effects of the cut
in G outweigh the expansionary effects of the decrease in t, then
both Y and R will decrease.  In this case there will be an
increase in the amount of unemployment in the current period, but
there will also be more private investment and hence a higher
capital stock and potential GDP level Y* in the next period.

However, the proposal explicitly calls for the cuts in G and
t to be coordinated so that the government budget is balanced,
i.e., the government budget deficit is reduced to zero.  As seen
in exercise 6.1 on the Sixth Exercise Set, an **increase** in the
deficit shifts **up** the IS curve and leads to a higher Y and R;
so conversely a **cut** in the deficit leads to a shift **down**
in the IS Curve and hence a **lower** level of both Y and R.

Suppose, then that a decrease in Y and R result from the
decrease in G and t because the cuts are coordinated so as to
reduce the deficit from some positive amount to zero.  What
potential problems could still arise?

First, there is no guarantee in actuality that the decrease
in R will encourage expenditures on newly produced capital goods
that are the most socially productive from a long-run point of
view.  For example, private firms might respond to the decrease in
R by increasing their borrowing in order to acquire existing real
estate or companies (acquisitions and mergers) rather than to
finance investment in new capital goods or in R and D.  Thus,
large amounts of increased private expenditure in reponse to a
decrease in R might actually result in only minimal increases in
future potential GDP levels Y*.  In this case, will the increase
in private investment induced by the lower R be worth the cost of
increased unemployment, particularly when this increased
unemployment is accompanied by a cut-back in government spending
on social programs such as job re-training, unemployment insurance
benefits, and student loans?

Second, even if the increase in private investment
expenditures does result in increased potential GDP Y* in future
periods, it does not necessarily follow that the actual GDP level
Y will also increase by this same amount.  The latter belief rests
on the assumption that the economy is essentially stable, in the
sense that Y tends to converge to Y*.  As discussed for the
Libertarian presidential platform, this requires a
well-functioning price adjustment mechanism and appropriate
accommodation by the Fed.  For example, excessive fears of
inflation by the Fed could result in too tight a monetary policy,
which keeps Y restricted below Y*.

Finally, the Hall and Taylor model includes only a very
simple tax policy parameter---a single income tax rate t.  In
fact, conservatives are more interested in implementing tax cuts
on "capital gains" (gains made from price appreciation on
financial assets such as stocks and bonds) rather than on income
in general.  Conservatives argue that tax savings on capital
gains---because they first accrue to the richer portion of the
population---have a greater chance of being reinvested in the
economy rather than being consumed.

Liberals counter that this "trickle down" theory of economic
growth, in effect since the early nineteen eighties, has created
an extremely unequal distribution of income in the U.S. over the
past sixteen years that is leading to increased social unrest.
For example, they point to the "excessive" salaries being paid to
chief executive officers (CEOs) of companies relative to the pay
of ordinary workers, and to the angry response of voters when this
situation was brought to their attention by Pat Buchanan in the
presidential primary election.

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