## Answer Outline for Econ 302 Exercise Set 3

Course Instructor: Leigh Tesfatsion
Date: 25 February 1996

```                          ANSWER KEY

THIRD TAKE-HOME EXERCISE SET [13 Points Total]      L. Tesfatsion
DUE DATE: Thursday, Feb 29, 9:30 A.M.          Econ 302/Spring 96

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

EXERCISE 3.1 [3 Points]:   Hall and Taylor, Chapter 4,
NUMERICAL Exercise Number 1, page 118. Be sure to justify your
assertions and to label graphs carefully.

----------

#Part a)#: The labor supply function is given by

(1)              N  =  1000  +  12[W/P]

and labor demand is

(2)              N  =  2000  -  8[W/P] .

Draw a diagram showing these schedules.  Find the equilibrium
level of employment and the real wage.

----------

#Answer Outline for Ex 3.1 (a) [One Point]#:

W/P
|
|
|
250 *
|
200 |
|
150 |
|
100 |
|
(W/P)*=50 |-------------------------------. E* = (1600,50)
|                               .
0 |-------------------*-------------------*-----    N
|        500      1000         1600    2000
-50 |
|
-100 |
|

For the labor supply curve, the N-intercept is 1000, the
W/P-intercept is -83.33, and the slope dN/d(W/P) is 12.  For the
labor demand curve, the N-intercept is 2000, the (W/P)-intercept
is 250, and the slope dN/d(W/P) is -8.  When these linear
functions are graphed in the N-(W/P) plane (see above), they
intersect at the point E* = (N*,[W/P]*) = (1600,50).

Economically, E* = (1600,50) represents the unique equilibrium
for the labor market, in the sense that the supply of labor
supply equals the demand for labor at E* and only at E*.

Algebraically, the intersection point E* can be found by solving
the two equations (1) and (2) for the two unknowns N and (W/P).
First, use (1) to substitute out for N in (2), obtaining

(3)     1000  +  12[W/P]  =   2000  -  8[W/P]  .

Equation (3) represents one equation in the one unknown, W/P.
Collecting terms in [W/P], one obtains

(4)     (12  +  8)[W/P]  =  2000 - 1000  =  1000.

Solving for W/P, one obtains the solution value

(5)     [W/P]*  =  1000/20  =  50.

Replacing [W/P] in (1) by 50, and solving for N, gives the
solution value for N:

(6)     N*   =   1000  +  (12 x 50)   =   1,600.

---------

#Part b)#: Given existing technology and the capital stock,
output is given by the function Y = 100 x [square root of N],
Graph the production function.  Does the production function
exhibit diminishing marginal product of labor?

----------
#Answer Outline for Ex 3.1 (b) [One Point]#:

By assumption, the production function is given by

(7)       Y   =   F(N)  =  100 x [square root of N],

where, for notational simplicity, dependence on the current
capital stock (captured in the coefficient 100) has been omitted.
To graph the production function as a function of N, one needs to
calculate a table of values for N and Y = F(N).  The production
function exhibits diminishing marginal product of labor if and
only if the marginal product of labor, i.e., the derivative
F_N(N) = dF(N)/dN of F(N) with respect to N---decreases with
increases in N for all positive N.  Also listed, then, are values
for F_N(N), which are seen to decrease as N increases.

Production Function
-------------------------------------
N              Y            F_N(N)
-------------------------------------

0              0               -

1            100.0           50.0

2            141.2           35.4

3            173.2           28.9

4            200.0           25.0

9            300.0           16.7

16            400.0           12.5

Y
|
400|
|
|
|
300|
|
|                                  Production Function
|                                       Y = F(N)
200|
|
|
|
100|
|
|
|
---------------------------------------------- N
0         5        10        15        20

For any positive N, the derivative F_N(N) of F(N) with respect to
N---that is, the marginal product of labor evaluated at N---is
given by

(8)   F_N(N)  =

(1/2) x 100            (1/2) x 100 x 100         5000
----------------  =  ------------------------  =  ------  .
square root of N     100 x [square root of N]      F(N)

It follows from (8) that F_N(N) is #positive# for all positive N.
The fact that F(N) exhibits diminishing marginal product of
labor---that is, that F_N(N) decreases with increases in N for
all positive N---follows directly from (8).  For (8) shows that
F_N(N) is positive for all positive N, which implies F(N)
#increases# with increases in N and hence F_N(N) = 5000/F(N)
#decreases# with increases in N.

The fact that F_N(N) decreases with increases in N can also be
determined more formally by taking the derivative of F_N(N) with
respect to N, which yields the second derivative of F(N):

- 5000 x F_N
(9)     F_N_N(N)     =   -------------  .
F(N)^2

Clearly (9) is #negative# for all positive N because F_N and
F(N)^2 are both positive for all positive N.

**IMPORTANT:***  Note that a graph of the production function
can at best #illustrate# diminishing marginal product of labor
for a limited range of positive N values but cannot #establish#
diminishing product of labor for #all# positive N.  For the
latter, one needs to use an analytical argument such as given in
the previous two paragraphs.

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

#Part c)#: Using the labor market from Part a) and the production
function from Part b), determine the equilibrium level of output
for this economy.

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

#Answer Outline for Ex 3.1 (c) [One Point]#:

From Part a), the equilibrium level of employment is
N* = 1600.  It follows from Part b) that the equilibrium
level of output is given by

Y*  =  F(N*)  =  100 x [square root of 1600]

=  100 x 40

=  4000 .

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

EXERCISE 3.2 [2 Points]:   Hall and Taylor, Chapter 4, NUMERICAL
Exercise Number 2, page 118. Be sure to justify your assertions.

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

#First Part#:

Assume that over a 10 year period the growth rate of capital is 4
percent, the growth rate of employment is 2 percent, and the
growth rate of real output is 5 percent.  Calculate the growth
rate of total factor productivity.

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

#Answer Outline for Exercise 3.2 First Part [One Point]#:

Following HT4, assume the production function takes the form
Y = AF(N,K), where A denotes total factor productivity.  Using
the economic growth formula, the growth rate of total factor
productivity is then given by

DA         DY           DN             DK
----   =   ----  -  .70 ----   -   .30 ----
A          Y            N              K

=    .05  -  .70[.02]   -   .30[.04]

=    .05  -  .0140  -  .0120

=    .0240  .

That is, the growth rate of total factor productivity is 2.4
percent. [Note that round off error can significantly affect your
answer here.  The calculations above are rounded off to four
decimal places.]

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

#Second Part#:

Suppose that a permanent cut in the budget deficit increases
investment, and the growth rate of capital rises by 1 percent.
How much does the growth rate of output increase?  Suppose that a
tax reduction increases the supply of labor by 1 percent in one
year.  What happens to the growth rate of real output?

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

#Answer Outline for Exercise 3.2 Second Part [One Point]#:

Using the econonomic growth formula, if DK/K rises by 0.01 over
some period of time, then DY/Y rises by 0.30[0.01] = 0.003 (i.e.,
by 0.3 percent) over that period of time.  And if DN/N rises by
0.01 over some period of time, then DY/Y increases by 0.70[0.01]
= 0.007 (i.e., by 0.7 percent) over that same period of time.

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

EXERCISE 3.3 [3 Points]: Hall and Taylor, Chapter 6, NUMERICAL
Exercise Number 1, page 173. Be sure to justify your assertions
and to label graphs carefully if you make use of them.

Suppose the model of the economy is given by

(1)   Y  =   C  +  I  +  G   +   X     (NOTE: X = NE)

(2)   C  =  a   +  bY_d

(3)   Y_d  =  [1-t]Y

(4)   X  =   g  -  mY

where I = \$900 billion, G = \$1,200 billion, m = 0.1, b = 0.9,
t = 0.3, a = \$220 billion, and g = \$500 billion.

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

#Part a)#: Show that the value of GDP at the point of spending
balance is \$6,000 billion.  Compared to the example on page 165
with exogenous net exports, is the (investment) multiplier larger
or smaller?

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

#Answer Outline for Exercise 3.3 (a) [One Point}#:

Use equations (2), (3), and (4) to substitute out for C, Y_d, and
X in equation (1).  The result is:

Y[1 - b[1-t] + m]  =   a  +  I  +  G  +   g ,

which yields the spending balance solution for Y:

a  +  I  +  G  +  g
Y^o =  ----------------------  .
1 - b[1-t] + m

Substituting in the given numerical values for the terms on the
right hand side, one obtains Y^o = \$6,000 billion.

It follows from this formula for Y^o that the (investment)
multiplier for the economy at hand is given by

dY^o               1                1
------   =   ---------------   =   ------  =   2.13.
dI           1 - b[1-t] + m        0.47

This is #smaller# than the (investment) multiplier 2.70 obtained
by HT on page 165 with exogenously given net exports X (or NE).

The reason for this finding is that the change in net exports in
response to a change in income Y works to partially offset the
change in Y.

For example, as discussed in HT6, if investment I decreases by
\$100, then the "first round" effect is a decrease in Y by \$100,
which in turn leads to a decrease in consumption C by b[1-t]\$100
and hence a further decrease in Y by b[1-t]\$100in the "second
round."  However, given relation (4) for net exports, these (and
further round) decreases in Y are partially offset by #increases#
in net exports in response to decreases in Y.

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

#Part b)#:  What proportion of investment is private saving?
Government saving? Saving by the rest of the world?

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

#Answer Outline for Exercise 3.3 (b) [One Point]#:

We know that

I    =     S_p     +      S_g       +       S_r

\$900
billion

where

S_p  =  Y_d - C  =  [1-t]Y  - (a + b[1-t]Y)

=  Y( [1-t] - b[1-t] ) - a

=  Y[1-b][1-t] - a

=  Y[1-0.9][1-0.3] - \$220

=  Y[0.1][0.7] -  \$220

=  Y[0.07] - \$220

=  \$6000[0.07] - \$220

=  \$200 .

Therefore

S_p/I   =   200/900   =   0.222   (or 22.2 percent).

Also,

S_g  =  tY  -  G

=  0.3[\$6000] - \$1,200

=  \$600.

Therefore

S_g/I  =  600/900  =  0.667   (or 66.7 percent).

Finally,

S_r  =   I  -  S_p  -  S_g   =   \$900 - \$200 - \$600

=   \$100 .

Therefore

S_r/I  =  100/900  =  0.111   (or 11.1 percent) .

Note that 22.2 + 66.7 + 11.1 =  100.

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

#Part c)#: Now suppose that I increases by \$100 billion.  By what
proportion of the increase in investment do each of the three
categories of savings increase?

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

#Answer Outline for Exercise 3.3 (c) [One Point]#:

By Part a), dY^o/dI = 2.13.  Thus, if I increases by \$100, then
the change in Y is approximately given by

Delta(Y)   =    2.13  x  \$100  =  \$213.

As seen in Part b), S_p =  Y[0.07] - \$220, implying that dS_p/dY = 0.07.
Thus, the change in S_p due to the increase in I by \$100 is
approximately given by

Delta(S_p)  =  0.07 x Delta(Y)

=  0.07 x \$213

=  \$15   (rounded off to the nearest dollar) .

Similarly, S_g = tY - G =  [0.30]Y - \$1200 implies that dS_g/dY = 0.30
Thus, the change in S_g due to the increase in I by \$100 is
approximately given by

Delta(S_g)  =  0.30  x  Delta(Y)

=  0.30  x  \$213

=  \$64   (rounded off to the nearest dollar) .

Finally,

S_r  =  I  -  S_p  -  S_g

=  I  -  (Y[0.07] - \$220)  -  ([0.30]Y - \$1200)

=  I + \$1200 + \$220 - Y[0.07 + 0.30]

=  I + \$1420  -  Y[0.37] .

Thus, the change in S_r due to an increase in I by \$100 is
approximately given by

Delta(S_r)  =  \$100  -  Delta(Y)[0.37]

=  \$100 - \$213[0.37]

=  \$100 - \$78.81

=  \$21  (rounded off to the nearest dollar).

Another way to calculate Delta(S_r) is by noting that, for the
economy at hand, V = 0 (because household income prior to income
taxes is assumed in equation (3) to just be given by Y, not by
Y+V+F+N---see Hall and Taylor (2-1), page 51.) Thus, using
equation (4),

S_r  =  - X  =  - [g - mY]  =  - [500 - .1Y]  =  -500 + .1Y,

implying that the change in S_r due to the change in I is
approximately given by

Delta(S_r)  =  .1 x Delta(Y)

=  .1 x \$213

=  \$21  (rounded off to the nearest dollar).

As a simple calculational check, note that 15 + 64 + 21 = 100.

The desired proportions of savings to investement can now be
determined as follows:

Delta(S_p)/Delta(I)  =  15/100  =  .15   (15 percent)

Delta(S_g)/Delta(I)  =  64/100  =  .64   (64 percent)

Delta(S_r)/Delta(I)  =  21/100  =  .21   (21 percent)

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

EXERCISE 3.4 [5 Points Total---1 point for Part (a) and 2 points
each for Parts (b) and (c)]: [Compare Hall and Taylor, Chapter 6,
ANALYTICAL Exercise Number 6, page 175.] Consider an economy
described by the following equations:

(1)  Y  =  C  +  I   +  G  +  NE;

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

(3)  NE =  g - mY ;

(4)  G  =  tY   (balanced budget restriction on government)

Exogenous Variables:  I,  a,  b,  t,  where all terms are
strictly positive and b and t are also strictly less than 1.

Endogenous Variables: Y, C, NE, G

justify your assertions and to label graphs carefully.

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

#Part (a)#: Explain carefully why G is #endogenous# in model (1)-(4).

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

#Answer Outline for Exercise 3.4 (a) [One Point]#:

G is endogenous in model (1)-(4) because there is an equation in
the model, namely equation (4), that determines the value of G
once Y is determined by equations (1) through (3).  That is, the
value of G is determined #within the model#, and this is the
definition of an endogenous variable for a model.

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

#Part (b)#:  Determine an analytical expression for the
#investment# multiplier for model (1)-(4).  Is this investment
multiplier larger or smaller than in the case (covered in HT6)
where G is exogenous? Provide a careful explanation for your
finding.

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

#Answer Outline for Exercise 3.4 (b) [Two Points]#:

The model given by equations (1), (2), and (3) alone, with G
classified as an exogenous variable, is the same model as
developed in the class lecture notes HT6.  As determined in those
notes, the investment multiplier for model (1)-(3) is given by:

#Investment Multiplier with Exogenous G#:

dY^o                 1
(*)             ------    =    ----------------  .
dI             1 - b[1-t] + m

#With# equation (4) included in the model, we need to determine
the solution value for Y as a function of I.  Using equations
(2), (3), and (4) to substitute out for C, NE, and G in equation
(1), one obtains one equation in Y:

Y  =  ( a  +  b[1-t]Y )  +   I   +   tY   +   (g - mY).

Collecting terms in Y, one obtains

Y x (1 - b[1-t] - t + m)   =   a   +   I    +    g

or

Y x ( [1-t][1-b] +  m)   =    a   +   I    +    g .

Dividing through by the term in parentheses on the left, one
obtains:

#The Solution Value for Y as Determined by Model (1)-(4)#:

a + I + g
(**)         Y+  =  ----------------    .
[1-t][1-b] + m

It follows from (**) that the investment multiplier for model
(1)-(4) is given by

dY+                 1
(+)         -----      =   --------------      .
dI            [1-t][1-b] + m

Is the multiplier (+) for endogenous G larger or smaller than the
multiplier (*) with exogenous G?  The multiplier (+) is #larger#
than the multiplier (*) if and only if the denominator of (+) is
#smaller# than the denominator of (*), that is, if and only if

[1-t][1-b] + m   is smaller than   1 - b[1-t] + m  .

or equivalently, writing out terms, if and only if

(++)    1 - b - t + bt + m   is smaller than   1 - b + bt + m

Cancelling identical terms on each side, on sees that (++) holds
as long as -t is smaller than 0; but this is true because t is
assumed to be positive as part of the given specifications for
model (1)-(4).

Therefore, it has been established mathematicaly that the
multiplier (+) with endogenous G is larger than the multiplier
(*) with exogenous G.  But what is the reason for this
difference in more intuitive economic terms?

The reason for the difference is that tax revenues in model
(1)-(4) do not "leak out" of the economy as they do for model
(1)-(3) with exogenous G.  Rather, they enter back in by way of
changes in G.  Consequently, any "first round" increase in Y due
to an increase in I is #amplified# in all future rounds because
it leads to increased tax revenues and hence increases in G which
further increase Y.  And similarly for any first round decrease
in Y due to a decrease in I.

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

#Part (c)#: Suppose t increases.  Determine carefully whether the
solution value for Y determined by model (1)-(4) increases,
decreases, or stays the same in response to this increase in t.
Provide a careful explanation for your finding.

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

#Answer Outline for Exercise 3.4 (c) [Two Points]#:

The solution value for Y as determined by model (1)-(4) is given
in (**).  Recall that the income tax rate t is restricted to lie
between 0 and 1.  As t increases towards 1, the denumerator in
(**) becomes #smaller#, hence the solution value of Y given by
(**) becomes #larger#.

More formally, differentiating (**) with respect to t, one
obtains:

#Tax Multiplier for Model (1)-(4)#:

dY+        - [a + I + g]
-----   =  -----------------  x ( - [1-b] )
dt        ([1-t][1-b] + m)^2

[1-b]Y+
=  -----------------  ,
[1-t][1-b] + m

which is positively valued given the assumed sign restrictions on
b,t, and m.   Thus Y+ #increases# with increases in t.

What is the reason for this in more intuitive economic
terms?  Recall, for example, that Y #decreases# with increases in
t in the HT6 model (1)-(3) with exogenously given G, the reverse
of the result we have just obtained for model (1)-(4).

The reason why Y+ increases when t increases, with government
expenditures G determined endogenously by equation (4), is because
the subsequent increase in G more than offsets the reduction in
household spending caused by the higher income tax rate t.  This
is because part of the increase in the taxes assessed on household
income is paid out of what would have been household savings.  In
other words, part of the income that households would previously
have saved (a leakage out of the circular flow of income) is now
retained in the circular flow because government spends this
income on goods and services.

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

End of Answer Key for Exercise Set 3

```