## Practice Midterm Exam and Answer Key from S96

Course Instructor: Leigh Tesfatsion
Date: 7 March 1996

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QUESTION: [50 Points Total]

Part A: [10 Points]  Given an open economy such as the U.S. in
some period T, carefully define for this economy what is meant
by: (1) #real GDP# Y(T); and (2) #potential real GDP# Y*(T).

Answer Outline for Part A: [See HT2 and HT4.]

The "real GDP" (real gross domestic product) Y(T) for the U.S.
in period T is the value (in base year prices) of all final goods
and services produced within the borders of the U.S. during
period T.

The "potential real GDP" Y*(T) for the U.S. in period T is
the level of real GDP that would result for the U.S. in period T
if period T employment N(T) were equal to the potential employment
level N*(T) and the period T capital stock K(T) were fully in
use.  Potential employment N*(T) is the level of employment that
would result in the U.S. in period T, given existing incentives
such as fringe benefits and the minimum wage, if wages were fully
flexible and responsive to demand and supply pressures.  Also,
the capital stock K(T) is "fully in use" if it is being used at
normal capacity levels.

Part B: [20 Points]  Using carefully labelled graphs for
illustration, explain how Y*(T) is derived from labor market
clearing conditions, assuming that the period T aggregate
production function Y = F(N,K,A) exhibits positive but
diminishing marginal product of labor and the supply of labor is
determined as an increasing function of the real wage W/P.

Answer Outline for Part B: [See HT4.]

By assumption, the labor supply N^S is assumed to increase
with increases in the real wage W/P, say

N^S = h(W/P)

where h(W/P) increases with increases in W/P.  This labor supply
relation thus results in a curve that is upward sloping in the
N-W/P plane (larger values of W/P correpond to larger values of N^S
along this curve).

Assuming profit maximizing firms, period T labor demand N^D
is determined by the condition that the real wage be equal to
the marginal product of labor: that is, by the condition

W/P = F_N(N^D,K(T),A).

The assumption of diminishing marginal product of labor implies
that F_N(N,K(T),A) decreases with increases in N.  Thus, this
labor demand relation results in a curve that is downward sloping
in the N-W/P plane (larger values of N^D correspond to smaller
values of W/P along this curve).

The labor market is said to clear if N^D=N^S, that is, if
labor demand is equal to labor supply.  This corresponds,
graphically, to an intersection point of the demand and supply
curves for labor in the N-W/P plane.

The potential employment level N*(T), interpreted to be the
"full" (labor market clearing) level of employment, is thus
determined (along with the potential real wage) by the intersection
of the supply and demand curves for labor in the N-W/P plane.
The potential output level Y*(T) is then determined as the level
of output generated by the aggregate production function
evaluated at N*(T), given K(T) and A.  That is, the potential
output level Y*(T) is given by Y*(T) = F(N*(T),K(T),A).

The required graphical illustration for this derivation of
Y*(T) is given in Figures 4.3 and 4.4 in the class lecture notes
Hall and Taylor, Chapter 4, p. 97.]

x
W/P
|       o                     x
|
|           o             x
|                o
(W/P)*(T) |.....................LE
|                 x   .  o
|            x        .        o
|      x              .                o
-------------------------------------------------    N
0                   N*(T)

Fig. 4.3: Labor Market Equilibrium at Point LE
[x's depict the labor supply curve and o's depict the labor demand curve]

Y
|
|
|                                               p
|                               p
Y*(T) |.....................p
|             p       .
|       p             .
|                     .
|  p                  .
----------------------------------------------------   N
0                   N*(T)

Fig. 4.4: Determination of potential (full employment) GDP level Y*(T)
[The p's depict the aggregate production function.]

Part C: [20 Points] Suppose the period T capital stock K(T) for
the economy in Part B is subject to a negative shock that
unexpectedly decreases it to some lower value K(T)'.  Using
well-labelled graphs, explain carefully the likely impact on
Y*(T) of this decrease in K(T).  Justify your assertions.

Answer Outline for Part C:  [See HT4.]

Given a fixed technology A, the most plausible assumption is
that the marginal product of labor and output both decrease for
each positive level of N if K(T) decreases to some lower level
K(T)'; for, at any given employment level N, workers have less
capital to work with than before.

It then follows from Part B that both the labor demand
function in the N-W/P plane and the aggregate production function
in the Y-N plane move downward if K(T) decreases to some lower
level K(T)'.  This downward movement in the labor demand function
implies that the potential (labor market clearing) level of
employment N*(T) decreases to some lower level N*(T)' when K(T)
decreases to K(T)'.  This decrease in N*(T), together with the
downward movement in the production function, then implies that
the potential output level Y*(T) decreases to some lower level
Y*(T)'.

[Technical Note, NOT required:  The downward movement in the
production function is a downward **rotation** about the origin
(0,0) if zero output is always obtained with zero employment,
regardless of capital, and the downward movement in the labor
demand function is a downward **rotation** around (0,+infinity)
if the marginal product of labor goes to +infinity as N goes to
zero, regardless of capital.]

Figures such as HT4 4.3 and 4.4, reproduced above, should be
used to depict this decrease in Y*(T) in response to the decrease
in K(T).

QUESTION: [50 Points Total]  Suppose an economy in period T is
described in reduced form by the following three spending balance
equations, a slightly modified version of the Hall and Taylor
model in chapter 6:

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

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

(3)  I  =  e  +  uY

#Endogenous Variables#: Y = real GDP; C = real consumption; I =
real investment

#Exogenous Variables#:  G = real government spending; NE = real
net exports; t = income tax rate lying between 0 and 1; b =
marginal propensity to consume lying between 0 and 1; and a, e, u
are positive coefficients with u strictly less in value than (1 -
b[1-t]).

Part A: [9 Points]  Provide a plausible #economic# interpretation
for each of the model equations (1) through (3).

Answer Outline for Part A:  [See HT 6.]

Equation (1) is the #national income accounting identity#
for the (home country) economy which represents its realized real
GDP Y as the sum of four distinct types of realized spending:
realized HC household consumption spending C; realized investment
spending I by HC firms and HC households; realized HC government
spending; and realized spending on net exports (i.e., realized
spending by ROW on final goods and services newly produced within
the borders of the HC minus realized spending by HC agents on
goods and services produced within the borders of ROW).
As explained in HT6, equation (2) can be viewed as a
combination of one behavioral relation and two equilibrium
relations:

(i)          C^D = a + b[1-t]Y^e ;

(ii)         C^D = C;

(iii)        Y^e = Y,

where (i) is a consumption function giving consumption demand C^D
as a function of expected income Y^e, (ii) is the equilibrium
condition that consumption plans are realized, and (iii) is the
equilibrium condition that income expectations are fulfilled:
The consumption function (i) postulates that consumers plan to
consume some amount "a" plus some portion "b" of their expected
disposable income [1-t]Y^e.
Similarly, equation (3) can be interpreted as the
combination of one behavioral relation and two equilibrium
relations:

(i)*          I^D = e + uY^e  ;

(ii)*         I^D = I;

(iii)*        Y^e = Y,

where (i)* is an investment function giving investment demand I^D
as a function of expected income Y^e; (ii)* is the equilibrium
condition that investment plans are realized: and (iii)* is the
equilibrium condition that income expectations are fulfilled The
investment function (i)* postulates that firms plan to invest
some amount "e" plus some amount proportional to expected real
GDP Y^e, where Y^e could be acting as a proxy for "future
expected profits".

NOTE: In actual macroeconometric models, investment demand
is often taken to depend positively on "expected future profits,"
most simply approximated by expected income or expected after-tax
income, as well as being a decreasing function of the expected
real interest rate on financial assets.  In Chapter 7, Hall and
Taylor make the simplifying assumption that investment demand
depends only on the expected real interest rate on "bonds".

Part B: [6 Points]  Explain carefully how equations (1) through
(3) #together# guarantee that the economy exhibits "spending
balance" (or "product market clearing")?

Answer Ouline for Part B: [See HT6.]

"Spending balance" (or "product market equilibrium") is said
to hold when aggregate demand equals aggregate supply in the
product market and all expectations are fulfilled.
Aggregate demand Y^D is defined to be the sum of consumption
demand, investment demand, planned government expenditures, and
net export demand.  For the economy at hand, government
expenditures and net exports are assumed to be exogenous.  Thus,
aggregate demand is given by Y^D = C^D + I^D + G + NE.  As
explained in Part B, model (1)-(3) can be interpreted as a
reduced form model incorporating the restrictions C^D=C and I^D=I
as well as the fulfilled expectations assumption Y^e = Y.  Given
equation (1) plus Y^D = C^D + I^D + G + NE, the relations C^D=C
and I^D=I imply that Y^D = Y.  Thus, equations (1)-(3) guarantee
that the economy exhibits spending balance.

Part C:  [15 Points]  Use the model equations (1) through (3)
to carefully derive an #algebraic# expression for the solution
Y^o for real GDP Y as a function only of the exogenous variables
of the model. Show your work.

Equations (1) through (3) represent three equations in the
three unknowns Y, C, and I.  Use equations (2) and (3) to
substitute out for C and I in equation (1), leaving one equation
in the one unknown Y:

(1)*  Y  =  (a  +  b[1-t]Y)  +  (e + uY)  +  G  +  NE

Collect terms in Y:

(*)      Y x (1 - b[1-t] - u)  =  a   +   e   +   G   +  NE.

Finally, divide through both sides of equation (*) by the term in
parentheses on the left hand side in order to solve for Y, yielding

a  +  e  +  G  +  NE
(Y Solution)    Y^o  =  -----------------------     .
1 - b[1-t] - u

Note that the right hand side expression in this Y solution
involves only the exogenous variables of the model, as required.

Part D: [20 Points]  Show carefully how to derive an
#algebraic# expression for the #government spending multiplier#
for the model described by equations (1)-(3).  Also, determine
whether this multiplier would be smaller, larger, or unchanged if
investment did not depend on Y, that is, if u were equal to 0 in
equation (3).  Provide an #economic# interpretation for your
findings.

By definition, the government spending multiplier is given
by the derivative of Y^o with respect to G.  Using the Y solution
determined in Part C, this derivative takes the form

dY^o                1
(**)     ------    =   -----------------
dG            1 - b[1-t] - u

By assumption, b[1-t] is positive, and u is positive and strictly
less than 1-b[1-t].  It follows that the denominator in (**) is
positive but less than one, implying that the government spending
multiplier (**) is a positive number bigger than 1.

If investment did not depend on income, that is, if u were
equal to zero in equation (3), then the denominator in (**) would
be larger (more positive).  Thus, the government spending
multiplier (**) would be #smaller# if u were zero.  However,
since b and t are both restricted to be positive and less than 1,
1-b[1-t] is positive and less than 1.  Hence, even if u were
zero, the government spending multiplier (**) would still be
greater than 1.

An intuitive economic interpretation can be given for these
findings as follows.

From equation (1), an increase in G, say an increase of
\$100, results in a "first round" direct increase in Y by the same
amount, \$100.  However, equations (2) and (3) then show that this
\$100 increase in Y then leads to an increase in C by an amount
b[1-t]\$100 and an increase in I by an amount u\$100.  These
increases in C and I result in a further increase in Y by
equation (1), which in turn results in further increases in C and
I by equations (2) and (3), and so on for infinitely many rounds.

The government spending multiplier dY^o/dG is the sum of all
of the increases in Y from all of these rounds.  Since the increase
in Y in the first round is the same size as the increase in G,
namely \$100, it is clear that the sum of all of the increases in
Y results in an overall increase in Y that is larger than \$100,
the increase in G.  This is why the government spending
multiplier (**) is larger than 1.

When u is set to zero, one source of increases in Y---
namely, the increases in Y resulting from increases in I---is
eliminated.  Consequently, the government spending multiplier is
smaller when u is zero.  Nevertheless, the first round increase
in Y resulting from the \$100 increase in G is still \$100, and
this increase still leads to an increase in C which in turn leads
to an increase in Y in the second round, which in turn leads to
an increase in C and hence an increase in Y in the third round,
and so forth and so on in all successive rounds.  It follows that
the overall increase in Y resulting from a \$100 increase in G is
still larger than \$100.  This is why the government spending
multiplier (**) is still larger than 1 when u is set to zero.

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