Econ 302 Exercise Set 4: Answer Outline

Course Instructor: Leigh Tesfatsion
Due Date: Thursday, 28 March, 9:30 A.M.


                              ANSWER OUTLINE


FOURTH TAKE-HOME EXERCISE SET [17 Points Total]      L. Tesfatsion
DUE DATE: Thursday, March 28, 9:30 A.M.         Econ 302/Spring 96

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EXERCISE 4.1 [4 Points]:   Hall and Taylor, Chapter 7,
NUMERICAL Exercise Number 6, page 205. Be sure to justify your
assertions and to label your graphs carefully.

The following relationships describe the imaginary economy of
Nineland:

   (1)  Y  =  C + I           (income identity)

   (2)  C  =  90 + .9Y        (consumption)

   (3)  I  =  900 - 900R      (investment)

   (4)  M  =  [.9Y - 900R]P   (money market)

Y is output, C is consumption, I is investment, R is the interest
rate, M is the money supply, and P is the price level.  There are
no taxes, government spending, or foreign trade in Nineland.  The
year is 1990 in Nineland.  The price level is 1.  The money
supply is 900.

Part a:  Sketch the IS and LM curves for the year 1999, and
show the point where interest rate and output are determined.
Show what happens in the diagram if the money supply is increased
above 900 in 1999.

Part b: Sketch the aggregate demand curve.  show what happens
in the diagram if the money supply is **decreased** below 900 in
1999.

Part c: Derive an algebraic expression for the aggregate demand
curve in which P is on the left-hand side and Y is on the
right-hand side.

Part d: What are the values of output and the interest rate in
1999 when the money supply is 900?

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     PRELIMINARY REMARKS:  As detailed in HT7, equation (1) is the
national income accounting identity and equations (2) and (3) are
reduced form equations that combine the consumption function and
investment function with the equilibrium requirements that planned
quantities (consumption demand and investment demand) equal realized
quantities (actual consumption C and actual investment I) and all
expectations are correct.  Consequently, equations (1)-(3) express
the requirement that the product market for the economy is in equilibrium.
     By definition, the IS curve is the collection of all
combinations of Y and R for which the economy is in a product
market equilibrium.  Thus, to find the IS curve, one needs to
reduce down equations (1)-(3) to one equation in Y and R.  In
particular, use equations (2) and (3) to substitute out for C and
I in equation (1), leaving one equation in two unknowns, Y and R.
By construction, this is the IS curve.  To graph this IS curve in
the Y-R plane, it helps to put the curve into slope-intercept
form by solving for R as a function of Y.
     Similarly, equation (4) is a reduced form equation that
combines the money demand function with the equilibrium
requirements that planned quantities (money demand) equal
realized quantities (money supply M) and all expectations are
correct.  Consequently, equation (4) expresses the requirement
that the money market for the economy is in equilibrium.
     By definition, the LM curve is the collection of all
combinations of Y and R for which the economy is in a money
market equilibrium.  Thus, equation (4) is the LM curve.  To
graph this curve in the Y-R plane, it helps to put the curve into
slope-intercept form by solving for R as a function of Y.

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Answer Outline for Parts a and d together:

IS CURVE:

Substituting out for C and I in equation (1), one obtains

    Y  =   C  +  I

       =   (90 + .9Y)  +  (900 - 900R)  .

Now collect terms in Y and in R:

    Y[ 1 - .9]  =  990   -  900R

or

    Y  =  990/0.1  -  [900/0.1] x R   =  9900  -  9000R .

Solving for R,

      9000R  =  9900  -  Y

or

      R  =   9900/9000   -   Y/9000

or

(IS)  R  =   1.10  -  Y/9000  .


LM CURVE FOR M = 900:

Equation (4) is the LM curve.  For easier graphing, this curve can
be put into slope-intercept form by solving for R as a function of Y:

        M  =  [.9Y - 900R]P

or, using M = 900 and P = 1,

        900  =  [.9Y - 900R]  .

Solve for R as a function of Y:

        900R  =  .9Y  -  900

or

        R  =  [.9/900]Y  -  1.0   =  Y/1000   -  1.0 ,

or

(LM)    R  =  -1.0   +   Y/1000  .


IS-LM EQUILIBRIUM POINT FOR M = 900 (ANSWER TO PART D):

The intersection point (Y^o,R^o) of the IS and LM curves can be
found as follows.  First, use (LM) to substitute out for R in IS,
leaving one equation in Y:

        -1.0  +  Y/1000  =  1.10   -   Y/9000  .

Collect terms in Y and solve for Y^o:

         Y/1000 +  Y/9000  =  1.10  +  1.0

or

         9Y  +  Y  =  9900  +  9000

or

         Y^o  =  9900/10  +  9000/10   =  990 +  900  =  1890 .


Then substitute this solution value Y^o for Y in (LM) and solve
for R^o.  

  R^o   =   -1.0  +  Y^o/1000   =   -1.0  +  1890/1000   =  .89.

By construction, (Y^o, R^o) = (1890,.89) is the only (Y,R) point
that lies on both the IS and LM curves, so it is the unique
intersection point of the IS and LM curves.


SKETCH OF IS AND LM CURVES TOGETHER (NOT TO SCALE):

          R

          |                         LM Curve
          |                      .  [Slope = 1/1000]
          |                     .
      1.10|.                   .
          |          .        .
       .89|                  . .                   IS Curve
          |                 .             .        [Slope = -1/900]
          |                .                        .
          |               .                                    .
          |              .
          |             .
          |            .
          |           .
          |----------------------------------------------  Y
          |         .      1890
          |        .
          |       .
          |      .
          |     .
          |    .
          |   .
          |  .
          | .
          |.
     -1.0 |





EFFECT ON IS-LM EQUILIBRIUM POINT WHEN M IS INCREASED:

Since M does not enter into the IS curve, the IS curve is not
affected by an increase in M.  The form of the LM curve for an
arbitrary value of M is

        M/P  =  .9Y  -  900R

or
        900R  =  -M/P  +  .9Y

or
        R     =  -M/900P  +  Y/1000  .


If the money supply M is increased above 900, then it is clear
from the latter equation that, for each value of Y, R must be
**less** than it was for the original value M=900 because the
intercept term is smaller (more negative) than before.  Moreover,
only the intercept is affected by the change in M, not the slope
of the LM curve.  Thus, the LM curve must undergo a **parallel
downward shift** when M increases above 900.

This shift should be plotted on the previously drawn IS-LM
diagram.  The downward shift of the LM curve (with no change in
the IS curve) results in a new intersection point (Y',R') with Y'
less than 1890 and R' less than .89.


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Answer Outline for Parts b and c Together:

The AD curve is the plot of the IS-LM equilibrium value Y^o as a
function of the price level P.  Consequently, one needs to go
back and express the LM curve for an arbitrary price level P
instead of for the specific value P=1---note that P does not
enter into the IS curve in any case.  Using the same steps used
above, one then needs to solve for the IS-LM equilibrium value
for Y as a function of this arbitrary price level.  For later
purposes, it is useful to leave M at an arbitrary value as well.

LM Curve for Arbitrary P and M:

     R  =  - M/900P   +  [9/9000] Y


IS Curve (neither M nor P enter):

     R  =  1.10  -  Y/9000


IS-LM Equilibrium Value for Y:  Substituting out for R and solving
for Y^o, one obtains


    -M/900P  +  [9/9000] Y   =  1.10    -  Y/9000

or

    [9/9000  +  1/9000] Y  =  1.10  +  M/900P

or

    [1/900]Y  =  1.10  +  M/900P

or

(AD CURVE IN DIRECT FORM)     Y  =  990  +  M/P  .

or

(AD CURVE IN INVERSE FORM)    P  =  M/[Y - 990]  .


SKETCH OF AD CURVE:

The inverse form for the AD curve reveals that Y must greater
than 990 in value in order for the price level to have a finite
positive value.  The value Y = 990 is an "asymptote" for Y in the
sense that P diverges to infinity as Y decreases towards 990.


             asymptote
      P        .  x
      |        .
      |        .   x
      |        .
      |        .     x
      |        .
      |        .        x
      |        .
      |        .             x
      |        .
      |        .                   x
      |        .                                AD CURVE
      |        .                           x
      |        .                                    x
      |        .
      |-----------------------------------------   Y
              990


If M is now decreased, it is clear from the analytical form of
the AD Curve derived above (in either direct or inverse form)
that, for each Y value, P must be smaller than before in order
for this Y and P pair to lie on the AD curve.  In other words,
the decrease in M results in a **downward** movement of the AD curve.

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EXERCISE 4.2 [3 Points]:   Hall and Taylor, Chapter 7,
ANALYTICAL Exercise Number 6, page 206. Be sure to justify your
assertions and to label your graphs carefully.

Suppose that two administrations, one Democratic and the other
Republican, both use fiscal and monetary policy to keep output
at its potential level, but that the Democratic administration
raises more in taxes and maintains a larger money supply than the
Republican administration.

Part a:  On a single graph, show how the IS and LM curves of
these two administrations differ.

Part b: Indicate whether the following variables will be higher
under the Democratic or Republican administrations, or whether
they will be unchanged:  consumption; investment; net exports;
government saving; and private saving.

Part c: Under which administration will foreign holdings of
U.S. financial assets grow more slowly?

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Answer Outline for Part a:

Using the IS-LM model developed in HT7, the LM curve in any
period T has the general form

(LM)    R(T)  =  - [ M/hP(T)  +  INF^e(T,T+1) ]  +  [k/h] Y(T)

The Democratic administration (DA) maintains a **higher** money
supply than the Republican administration (RA), and the money
supply affects the intercept but not the slope of the LM curve (a
larger M imples a more negative intercept).  Thus, the LM curve
for the DA is **shifted downward** relative to the LM curve for
the RA.  The IS curve for the HT model is not affected by changes
in the money supply M.

The IS curve for the HT model in any period T has the general
form

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

The DA imposes a **higher** tax rate t on income than the RA, and
the tax rate affects the slope but not the intercept of the IS
curve (a larger t implies a more negative slope).  In particular,
then, using (IS), the IS curve for the DA is **rotated downward**
relative to the IS curve for the RA.  The LM curve for the HT
model is not affected by changes in the tax rate t.

Recall that both the DA and the RA are presumed to use fiscal and
monetary policy to achieve the potential GDP level Y*.  Assuming
all exogenous variables other than M and t take on the same values
for both the DA and the RA, and using x's and y's for the DA and
o's and p's for the RA, the relative IS and LM relations for the
DA and the RA can be illustrated as follows:


       R              LM (RA)  p
       |                               y  LM (DA)
       |o                   p
       |  x  o                      y
       |          o      p
R (RA) |     x         o         y
       |              p     o
       |        x             y  o
       |           p                  o
       |            x      y              o
       |        p                               o
R (DA) |                yx                           o      IS (RA)
       |     p                                           o
       |             y        x
       |  p
       |          y                 x
       |                                   IS (DA)
       |       y                         x
       |---------------------------------------------------- Y
                        Y*

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Answer Outline for Part b:

The HT relations for the indicated variables are as follows.

    C  =  a + b[1-t]Y  ;

    I  =  e - dR ;

    NE =  g - mY - nR ;

    S_g (government savings)  =  tY - G  ;

    S_p (private savings) = [1-t]Y - C  =  [1-t]Y - a - b[1-t]Y
                                        =  [1-b][1-t]Y -  a.

From Part a, M is **higher** for the DA than the RA, t is **higher**
for the DA than for the RA, output Y* is the **same** for both
the DA and the RA, and the interest rate R* corresponding to Y*
is **higher** for the RA than for the DA.

Consequently, using this information in the above formulas, one
sees that:

   C is **lower** for the DA than for the RA (less disposable income) ;

   I is **higher** for the DA than for the RA (lower R);

   NE is **higher** for the DA than for the RA (lower R and same Y);

   S_g is **higher** for the DA than for the RA (higher t, same Y and G);

   S_p is **lower** for the DA than for the RA (less disposable income).

Thus, under the DA, there is relatively **more** private
investment, net exports, and government savings, and relatively
**less** consumption and private savings.

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Answer Outline for Part c:

The rate of change in the foreign holding of financial assets
(e.g., bonds) of the Home Country (HC) constitutes ROW lending to
the HC, i.e., **savings by ROW** vis-a-vis the HC---see HT
Chapter 2, equation (2-7).  From HT Chapter 2, equation (2-3),
ROW savings vis-a-vis the HC can also be expressed as

     S_r  =  - NE  -  V  ,

where V denotes factor income and transfer payments from ROW to
the HC in net terms.  As determined in Part b, net exports are
**greater** for the DA than for the RA.  Thus, assuming V is the
same under both administrations, ROW savings are **less** under
the DA.  Consequently, the foreign holding of HC financial assets
**grows more slowly** under the DA.
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EXERCISE 4.3 [4 Points]: Hall and Taylor, Chapter 7, MACROSOLVE
Exercise Number 1, page 207. Be sure to justify your assertions
and to label graphs carefully.  #All MacroSolve data that you
generate to answer this exercise should be appended to your
answer#.

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EXERCISE 4.4 [6 Points]:  Hall and Taylor, Chapter 8, NUMERICAL
Exercise Number 1, page 225.  Be sure to justify your assertions
and to label your graphs carefully.

     REMARK: Round-off errors can have quite an effect on your
     answers.  Try to avoid unnecessary round-offs and report
     calculations for decimal numbers using at least two decimal
     places.

Suppose the economy has the aggregate demand curve

                               M
          Y  =  3,401 + 2.887 ---
                               P

and the price adjustment schedule

                              Y(T) - 6000
          INF(T,T+1)  =  1.2  ------------  .
                                  6000

The money supply is $900 billion.

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Part a:  Plot the aggregate demand curve and the potential GDP
line.  Explain why the aggregate demand curve is not a straight
line.

Part b: If P_0 = .5, what will Y_0 be?  Will this place
upward or downward pressure on prices?

Part c: Compute the path of the economy---that is, calculate
GDP, the price level, and inflation---for each year until GDP is
within 1 percent of potential.

Part d: Diagram the economy's path on the demand curve plotted
in Part a.  Then draw your own version of Figure 8-7 and 8-8.
(You may assume that inflation was initially zero.)  From these
graphs, does the economy overshoot or converge directly to
equilibrium?

Part e: Assume that inflation is given by

                                      Y(T) - 6000
     INF(T,T+1) = .6INF(T-1,T) + 1.2 -------------  .
                                         6000

Compute the path of the economy for the first five years and
diagram the economy's path as in Part d.  Now is there
overshooting?

Part f: In Part e, what does the .6INF(T-1,T) term in the
price-adjustment equation represent?  Explain the relationship
between this term and overshooting.

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Answer Outline for Part a:

     From the given form of the aggregate demand curve, Y
approaches 3,401 (from above) as P gets very large (because M/P
converges to zero as P approaches infinity).  Also, Y approaches
infinity as P gets close to zero (because M/P diverges to
infinity as P converges to zero).  Finally, for the particular
given value M=$900 billion and potential GDP level Y*=6000, when
Y = 6000 the corresponding value P* for P is given by

                                900
       6000 =  3,401  +  2.887 -----
                                 P*

or

       6000P*  =  3,401P*  +  2.887 x 900

or

       2599P*  =  2598.30

or

       P*  =  1.00  (rounded off to two decimal places)


PLOT OF AD CURVE AND POTENTIAL GDP (FULL EMPLOYMENT) LINE:

                         FE Line
        asymptote          fe
          |    .           fe
      P    --> .  x        fe
      |        .           fe
      |        .   x       fe
      |        .           fe
      |        .     x     fe
      |        .           fe
      |        .        x  fe
      |        .           fe
 1.00 |.  .  .  .  .  .    x
      |        .           fe
      |        .           fe     x
      |        .           fe                  AD CURVE (M=900)
      |        .           fe                x
      |        .           fe                          x
      |        .           fe
      |-----------------------------------------   Y
      0       3,401       6000


     By definition, the aggregate demand relation connecting Y
and P would be linear if and only if it had the form

(*)           Y  =   a   +   bP

for some constants a and b.  Note from relation (*) that the rate
of change of Y with respect to P satisfies dY/dP = b, a constant,
and all higher derivatives of Y with respect to P are zero.  The
graph of (*) in the Y-P plane is thus a straight line.  On the
other hand, the aggregate demand curve has the form

(**)          Y  =   a    +   b/P  ,

which differs from the form of relation (*).  In particular, the
rate of change of Y with respect to P is given by dY/dP = -b/P^2,
which varies with P rather than being a constant, and higher
derivatives of Y with respect to P are not zero.  The graph of
relation (**) in the Y-P plane is not a straight line.  If b is
positive, as it is for the aggregate demand curve, the graph of
relation (**) in the Y-P plane is a downward sloping curve which
has a bowl-like shape.  For very large P, the term b/P approaches
zero and Y approaches a.  For very small P, the term b/P approaches
infinity and Y approaches infinity.

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Answer Outline for Part b:

     If the price level in some period 0 is given by P_0 = .5, it
follows from the form of the AD curve with M=900 that the GDP Y_0
for period 0 is given by

                        900
  Y_0  =  3.401 + 2.887 ----  =  3,401 + [2.887 x 1800] = 8597.60
                        .5

Since Y_0 is **greater** than potential GDP Y* = 6000, which is
the "normal" level of output, this will put upward pressure on
the price level.  In particular, it follows from the given price
adjustment equation that the inflation rate INF(0,1) between
periods 0 and 1 is

               P(1) - P_0               8597.60  - 6000
  INF(0,1)  =  -----------   =    1.2  -----------------   =  .52,
                   P_0                       6000

a positive number, implying that the price level P(1) for the
next period 1 will be greater than the price level P_0 in the
current period 0.  In particular, solving this relation for
P(1) with P_0 = .5 gives

     P(1) - .5  =  .52  x  .5

or

     P(1)  =  .76  .

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Answer Outline for Part c:

     As in Part b, assume that the price level in period 0 is
given by P_0 = .5, so that the corresponding GDP level is given
by Y_0 = 8597.60, the inflation rate INF(0,1) = .52, and the
price level for period 1 is given by P(1) = .76. Plugging the
price level P(1) into the AD curve then gives Y(1), and plugging
Y(1) and P(1) into the price adjustment relation gives INF(1,2)
and hence P(2) = [INF(1,2) x P(2) + P(2)], and so forth for
successive periods.  The numerical values, calculated until Y is
within 1 percent ($60 billion) of Y*=6000, are as follows:


   Year T        INF(T,T+1)        P(T)         Y(T)
   ----------------------------------------------------
     0               -             .50         8597.60

     1              .52            .76         6819.82

     2              .16            .88         6353.61

     3              .07            .94         6165.15

     4              .03            .97         6076.77

     5              .02            .98         6039.15


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Answer Outline for Part d:

     Plotting the movement over time of Y(T) and P(T) on the AD
diagram from part a, in analogy to HT figure 8-7, it is seen that
the (Y(T),P(T)) points are moving steadily upward along the AD
curve from T=0 to T=5 (P(T) is steadily rising and Y(T) is steadily
falling as T increases).  Moreover, Y(T) is approaching ever more
closely to the potential GDP level Y*=6000 from above, and P(T)
is approaching ever more closely to the "potential" price level
P*=1.00 from below.  In particular, the convergence of Y(T) to
Y*=6000 and P(T) to P*=1.00 is direct; there is no overshooting.
     These movements are also depicted, below, in figures that
are analogous to HT figure 8-8. [NOTE: The figures below are not
actual plots of the numerical values in the table above; they
simply illustrate the general form such plots would take.]


     Y(T)
      |
8597.6|
      | y
      |  y
      |     y
      |          y
      |               y
      |                        y
 6000 |----------------------------------------------
      |
      |
      |
      |
      |
      |................................................   T
      0

     P(T)
      |
      |
  1.0 |----------------------------------------------
      |                       p
      |           p
      |     p
      |  p
  0.5 |p
      |
      |
      |
      |................................................  T
      0


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Answer Outline for Part e:

     The originally given price adjustment relation (a
traditional Phillips curve) is now to be replaced by the
following expectations-augmented Phillips curve:

                                      Y(T) - 6000
     INF(T,T+1) = .6INF(T-1,T) + 1.2 -------------  .
                                         6000

When the part d numerical calculations are redone with
this modified price adjustment relation, assuming the initial
inflation rate INF(-1,0) is 0, the following values are obtained.


   Year T         INF(T,T+1)        P(T)         Y(T)
   ----------------------------------------------------
     0              .00             .50         8597.60

     1              .52             .76         6819.82

     2              .48            1.12         5717.32

     3              .23            1.38         5284.86

     4             -.005           1.37         5293.34

     5             -.144           1.17         5617.48



     As seen from these values, there is now overshooting.  The
GDP level Y(T) starts **above** potential GDP Y*=6000 but falls
**below** this level before it starts to recover and increase
back up to Y*=6000.  Similarly the price level P(T) starts
**below** the "potential" price level P*=1.00 and rises **above**
it before it turns around and starts to fall back toward P*=1.00.

     In particular, the presence of the term .6INF(T-1,T) in the
price adjustment relation allows prices to keep on rising even
when a negative GDP gap opens up---note what happens to the price
level between period T=2 and period T=3, during which the GDP gap
is negative.

     The movements traced out by Y(T) and P(T) from T=0 to T=5
are depicted in the following diagrams, analogous to HT figure
8-8.  [Again, these are not plots of the actual numerically
determined values.]



     Y(T)
      |
8597.6|
      | y
      |  y
      |    y
      |      y
      |         y
      |            y
 6000 |----------------------------------------------
      |                  y                  y
      |                        y      y
      |
      |
      |
      |................................................   T
      0

     P(T)
      |                        p     p
      |                 p                   p
  1.0 |----------------------------------------------
      |          p
      |       p
      |    p
      |  p
  0.5 |p
      |
      |
      |
      |................................................  T
      0


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Answer Outline for Part f:

     The term .6INF(T-1,T) in the price adjustment relation for
part d represents an "adaptive expectation" for the inflation
rate INF(T,T+1) from period T to period T+1 that is formed by
firms at the beginning of period T.  It is "adaptive" because it
is simply a weighted average over past realized inflation rates,
in this case only the most recent realized inflation rate.

     The presence of this expectation term in the price adjustment
relation leads to overshooting in part d because, in period 1
when Y is approaching its potential level, the price level is
still rising; the inflation rate INF(1,2) is 0.52.  Therefore, in
period 2, the inflation rate INF(2,3) is **expected** to be
positive---namely, it is expected to be 0.6 x 0.52---and this
positive expected inflation rate can (and does) outweigh the
negative GDP gap that has opened up in period 2, so that the
actual inflation rate INF(2,3) is indeed positive.  And similarly
for period 3.  Eventually the GDP gap becomes so negative that it
outweighs the positive expected inflation term and the actual
price level begins to drop, which implies that the expected
inflation rate begins to drop as well with a one period lag.

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