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Linear Functions
Application of Linear Equations
1.
Keynesian Consumption Function
Keynesian Consumption Function is a single mathematical function used to express consumer
spending.
The function is used to calculate the amount of total consumption in an economy.
Keynesian consumption Function is also known as the Absolute Income Hypothesis as its only
bases consumption on current income and ignores potential future income.
The function can be written in variety ways: C  a  b(Y  T )
C c c Yd
0 1
C=total consumption
c =Autonomous consumption ( c  0 ) or the level of consumption that would still exist even
0
0
if income was $0. In estimation, this is always assumed to be positive.
C1 =marginal propensity to consume( MPC)(which is a ratio of consumption changes to income
changes) (0  C1  1)
MPC measures the rate at which consumption is changing when income is changing.
Geometrically, MPC is the slope of the consumption function. MPC is also assumed to be
positive.
Y d =disposable income
(e.g. income after government intervention e.g. benefits, taxes, transfer payments or
Y  (G  T ) )
Example on the study guide:
If given:
C  4141  0.78Y
1
C=consumption,
Y=National Income
To find (MPC) marginal propensity to consume
MPC 
DC
DY
Dc  C  C
1 0
C  originalconsumption
0
C  consumption
1
Marginal Propensity to save:
MPS  1 MPC
If we increase income by N$1 for example then the new consumption will be:


C  4141  0.78 Y  1
1
0
MPC=additional consumption from additional dollar of disposable income
MPS=additional saving from an additional dollar of disposable income
INCOME DETERMINATION MODELS
Income determination models generally express the equilibrium level of income in a four-sector
economy as:
Y  C  I  G  (X  Z)
Y  income
C  consumption
I  investment
G  government expenditure
X  exports
Z  imports
2
Example:
Assume a simple two-sector economy where:
Y  C  I,
C  C  bY and I  I
0
0
Assume further that:
C  85, b  0.9 and I  55.
0
0
The equilibrium level of income can be calculated in terms of:
The general parameter: Y  C  I and substituting C and I in the general parameter,
Y  C0  bY  I0 we get the reduced form, e.g.
Solving for Y:
Y  bY  C  I
0 0
Y 1  b   C  I
0 0
C I
Y 0 0
1 b
The term
1
is called the autonomous expenditure multiplier in economics’
1 b
It measures the multiple effect each dollar of autonomous spending has on the equilibrium
level of income. Since b  MPC in the income determination model, the multiplier=
1
1  MPC
.
IS-LM ANALYSIS:
Seeks to find the level of income and the rate of interest at which both the commodity market
and the money market will be in equilibrium.
NB: We use the technique of solving equations simultaneously.
EXAMPLE 1:
The commodity market for a simple two – sector economy is in equilibrium when
3
Y  C  I.
The money market is in equilibrium when the supply of money  M s  equals the demand for
 d  which in turn is composed of the transaction – precautionary demand for money
money M
 Mt  and the speculative demand for money  M z  .
Assume a two – sector economy where
C  48  08Y ,
I  98  75i, M s  250, Mt  0.3Y and M z  52 150i .
We can calculate the level of income
Y  and the rate of interest  i 
that will concurrently
bring equilibrium to the economy as follows:
Commodity equilibrium  IS  exists when:
Y CI
Y  48  0.8Y  98  75i
Y  0.8Y  146  75i
0.2Y  75i 146  0
Monetary equilibrium ( LM ) exists when M s  Mt  M z
250  0.3Y  52 150i
0.3Y 150i 198  0
Solving simultaneously:
0.4Y  150i  292
0.3Y 150i 198  0
6
Y  700 , i   0.08
75
The commodity and money market will be in simultaneous equilibrium when
Y  700 and i 
6
 0.08
75
By substituting the available values in the given equations, one can obtain the values as follows:
4
C  48  0.8(700)  608
I  98  75(0.08)  92
M t  0.3(700)  210
M z  52 150(0.08)  40
C  I  608  92  700
And
Mt  M z  210  40  250  M s
Example 2:
Find:
(a)
the level of income and the rate of interest that concurrently bring equilibrium to the
economy and
(b)
estimate the level of consumption C, investment I, the transaction – precautionary
demand for money M t , and the speculative demand for money M w ,when the money
supply M s =300 and
C  102  0.7Y , I  150 100i
, Mt  0.25Y , M w  124  200i
Work out:
INCOME DETERMINATION MODELS
EXAMPLE 1:
(a)
Find the reduced –form equation and
(b)
solve for the equilibrium level of income
1. directly and
2. with the reduced form equation, if given:
Y  C  I G
C  C  bY
0
I  I G G
0
0
Where C  135, b  0.8, I  75, and G  30
0
0
0
5
From Y  C  I  G
(a)
Substituting the given values and solving for Y in terms of the parameters and exogenous
variables,
C0 , I 0 and G0
Y  C0  bY  I 0  G0
Y  bY  C0  I 0  G0
Y (1  b)  C0  I 0  G0
Y
C0  I 0  G0
(1  b)
This is the reduced form.
(b) 1.
Y  C  I G
Y  135  0.8Y  75  30
Y  0.8Y  240
0.2Y  240
Ye  1200
b. (2)
C0  I 0  G0
(1  b)
135  75  30
Y
(1  0.8)
240
Y
 1200
0.2
Y
4.1.6 Find the equilibrium level of income Ye , given
Y  C  I  G when C  125  0.8Y , I  45 and G  90
We know that the reduced form equation is:
Y
C0  I 0  G0
(1  b)
6
Given that C  125  0.8Y and we also know that
C  C0  bY
Then


1
Ye 
C  I G
0
1 b 0 0
1
Ye 
125  45  90 
1  0.8
Ye  1300
4.17
Find
(a)
the reduced form
(b)
the value of the equilibrium level of income Ye , and
(c)
The effect on the multiplier, given the following model in which
not autonomous but a function of income.
investment is
Y CI
C  C  bY
I  I  aY
0
0
and C  65, I  70, b  0.6, and a  0.2
0
0
4.19
Find
(a)
the reduced form
(b)
the numerical value of Ye and
(c)
the effect on the multiplier if an income tax T with a proportional component t is
incorporated into the model, given
Y CI
C  C  bY
0
d
T  T  tY
0
And
C  85, I  30, T  20, b  0.75 and t  0.2
0
0
0
7
Y  Y T
d
I I
0
ECONOMIC APPLICATION OF GRAPHS AND EQUATIONS:
ISOCOST LINES
Represent the different combinations of two inputs or factors of production that can be
purchased with a given sum of money.
The general formula is:
P K  P L  E,
L
k
K  Capital
L  Labor
P  CapitalPrice
k
P  Labor Price
L
E=amount allotted(agreed to) to expenditures
In ISOCOST analysis:
The individual price and the expenditure are initially held constant. Only the different
combinations of inputs are allowed to change.
P K P LE
L
k
P K  EP L
L
k
EP L
L
K
P
k
E  PL 
K

L
P
P 
K  k
This is the familiar linear function to the form:
y  mx  c where
c
P 
E
E  PL 
E
in K 

L  K   L  L 
P
P
P 
P 
P
K
K  k
K
 k
P
P
k
m   L is the slope.
8
From the graph, the effects of a change in any one of the parameters are easily noticeable.
An increase in the expenditure from E to E will increase the vertical intercept and cause the
ISOCOST line to shift out to the right but parallel to the old line,
Change in P will change the slope but not C. Change in P will change slope and C.
L
k
Example 1:
A person has N$120 to spend on two goods ( X , Y ) whose respective prices are N$3 and N$5.
(a)
Draw a budget line showing all the different combinations of the two goods
can be bought with the given budget (B).
SOLUTION:
The general function for a budget line is:
9
that
Px X  PyY  B
If Px  3 and Py  5 and B  120
then
3 X  5Y  120
Tograph,weput instandardform
3
Y  24  X
5
coordinates :  0;24  and  40;0 
What happens to the original budget line if:
(b)
the budget falls by 25%?
Solution:
10
75% 120  90
3 X  5Y  90
3
y   x  18
5
coordinates :  0;18 (30;0)
Lowering the budget causes the budget line to shift parallel to the left.
(c)
If the price of X doubles?
Px  3 2
6 X  5Y  120
6
Y   X  24
5
coordinates :  0;24   20;0 
11
Vertical intercept remains the same. Slope changes and becomes steeper. With a higher price
of X, less X can be bought within given budget.
(d)
If the price of Y falls to 4?
3 X  4Y  120
3
y  30  X
4
coordinates:  0;30 (40;0)
12
With a change in Py both the vertical intercept and the slope change.
Example 2:
Either coal (C) or gas (G) can be used in the production of steel. The coast of coal is 100, the
cost of gas is 500. Draw an isocost curve showing the different combinations of gas and coal
that can be purchased.
GRAPHS IN THE INCOME DETERMINATION MODEL
MACRO-ECONOMIC MODEL
Generally express the equilibrium level of income in a four-sector economy as:
Y  C  I  G  ( X  Z ), where
Y  income
C  consumption
I  investment
G  Government exp enditures
X  exp orts
M or Z  imports
3.4
NON-LINEAR FUNCTIONS- TYPES OF NON-LINEAR FUNCTIONS:
13
3.5
Application of Quadratic functions to economic theory


Use of factorization, graphical method, quadratic formula, completion of squares:
Cobb-Douglas production function
COBB-DOUGLAS PRODUCTION FUNCTION
Optimization of Cobb-Douglas production function:

q  AK  L
( A  0; 0   ,   1)
q  quantity of outputinphysicalunits
K  quantity of capital
L  quantity of labor
 (the output elasticity of capital)
Elasticity is……….
 measures the % change in q for a 1% change in k while L is held constant.
 (the output elasticity of labor) is exactly parallel
A= efficiency parameter reflecting the level of technology
A strict Cobb – Douglas function, in which   
 1 , exhibit constant returns to scale.
A generalized Cobb – Douglass function in which (    1) exhibits increasing returns
to scale if     1 and decreasing returns to scale if     1.
14
QUESTIONS FROM PAST EXAMS PAPERS
1.
The market for a 6-pack of Aquafina bottled water (manufactured by Pepsi) is
composed of the following demand and supply equations respectively:
QD  120  40 P
QS  60  50 P
The calculated equilibrium price and quantity will be:
2.
3.
A.
P  N $2; Q  40units
C.
P  N $1.50; Q  60units
B.
P  N $10; Q  18units
D.
P  N $1.00; Q  50units
Consider the relationship between total profit (π) and output (Q) is given by the
function   f (Q) . In this function, Q is described as the
A. Dependent variable
B. Independent variable
C. Slope
D. Intercept
Suppose the production of flat screen televisions is represented by the
following
production function Q 
of
production output is
0.4
AL K
Q1 units.
0.4
. The manufacturer’s new level
If all the inputs are doubled, the new level of
output
will be equal to:
A.
4.
Q1  20.4 Q
B.
Q1  20.8 Q
C.
Q1  0.8Q
D.
Q1  1.6Q
The following is a national income function for a four-sector economy at the
equilibrium level of income:
Y C  I G X Z
It has the following equations to describe the above four-sector economy
Y  income, C  consumption, I  investment ,
Where:
G  government exp enditure,  X  Z   net exports
15
For the IS sector
C  900  0.80YD ; YD  Y  T ; T  250  0.25Y
I  220  40i; G  1100 and ;  X  Z   800
where T  taxes and YD  disposable income
For the LM sector
M d  600  0.85i and M s  596  4i
where M d =money demanded and M s =money supplied
4.1
Calculate the level of income
Y  and the rate of interest  i 
that will
concurrently bring equilibrium to the economy
at equilibrium
Y  C  I  G   X  Z  and M s  M d
Y  900  0.8 Y   250  0.25Y    220  40i  1100  800
Y  900  0.8Y  200  0.2Y  220  40i  1100  800
0.4Y  40i  1220    eqn1
Ms  Md
600  0.85i  596  4i
i  0.824742268
substitute i into eqn1
0.4Y  40(0.824742268)  1220
0.4Y  1187.010309
Y  2967.525775
i  0.824  82.4% and Y  N $2967.53
4.2
At the equilibrium level of income and interest rate computed in 4.1,
determine the new level of consumption,
16
Ce
Ce  900  0.8 Y   250  0.25Y  
Ce  700  0.6Y
Ce  700  0.6  2967.53
Ce  N $2480.52
4.3
Evaluate the MPC at this new level of income
Ce  900  0.8 Y   250  0.25Y  
Ce  700  0.6Y
MPC 
5.
dC
 0.6
dY
A Cobb-Douglas production output function for Tangeni Holdings is given as
1
3
2
3
Q1  AK L
5.1
Determine the cost of production output
Q1 if the cost of technology is N$55,
cost of capital is N$105 and that of labour is 30% the cost of capital.
1
3
2
3
Q1  AK L
1
3
Q1  55(105) (31.5)
2
3
 2588.0111241
cost of production  N $2588.01
5.2
Show that the new level of production will be
Q2  3.18Q1
If the cost of capital is doubled and the cost of labour is quadrupled
17
1
3
2
3
Q1  AK L
now
1
3
Q2  A(2 K ) (4 L)
2
3
1 2


 2 4  AK 3 L3 


1 2


 3.17480  AK 3 L3 


Q2  3.175Q1
1
3
5.3
2
3
Determine the partial derivative of
1
3
Q1 keeping K constant, that is, determine
Q1
L
2
3
Q1  AK L
1
Q1
2  13
3
 A K  L
L
3
1
1

Q1 2
3
 AK L 3
L 3
6.
The demand for Levi’s blue jeans depends on the own price of the blue jeans, on
consumer income, I and on the level of advertising, A, done by Levi Strauss & Co.
The demand function is given as
A
where p is the price of the Levi’s blue jeans, I is consumer
2
income and A is the advertising budget of the company. The supply of the blue jeans
by the company is given by the function
Qd  100  p  3I 
Qs  40  2 p
Find the equilibrium price and quantity for the blue jeans sold on the market if
I  $20;
A  $60 .
18
7.
An economy has the following prevailing conditions:
C  89  0.6Y ; I  120  150i ; M t  0.1Y ; M w  240  250i ;
M s  M t  M w ; where M s  275
Note that C is the level of consumption, I the investment, Mt the transaction
precautionary demand for money and Mw the speculative demand for money.
Determine:
7.1
the level of income and the rate of interest that concurrently bring equilibrium to the
economy
Y CI
M D  MS  Mt  Mw
and
Y  89  0.6Y  120  150i
275  0.1Y  240  250i
0.4Y  150i  209....1
0.1Y  250i  35.........2
solve simult
Y  N $500
i  0.06  6%
7.2
the levels of
C , I , M t , and M w when M s  275 , at the equilibrium level of
income and interest rate computed in 7.1
8.
The monthly revenue R obtained by selling a particular blend of tea is a function of
the demand x in the market. It is observed that, as a function of price p per packet,
the monthly revenue and monthly demand are given respectively as
R  300 p  2 p 2
8.1
;
x  300  2 p
Show that
R  150 x 
(Hint: Express
p
x2
.
2
in terms of
x and then substitute in the correct equation)
19
8.2
Copy and complete the following table for R  150 x 
x
100
120
140
150
160
x2
.
2
180
200
R
x
8.3
At what value of
is the revenue the highest?
8.4
At what price will the revenue stand at the highest level?
9.
Voltron manufactures flat screen televisions and the weekly production cost of the
televisions is given by:
C  22 x  750
When x units are produced per week. The advertising costs,
C A necessary to sell
these x units per week is given by
CA  18 x  0.1x3  750 ,
And the weekly revenue R resulting from the sale of these x units is given by
R  340x  2.25x2
9.1
Determine the optimum production level, i.e. production level at which maximum
profit occurs and hence determine the maximum weekly profit
9.2
Determine the level of output at which the advertising cost per unit is minimum, and
thereafter calculate this minimum average cost.
10.
Sarah makes customized dresses for a select clientele. In the relevant range of her
production, she can produce dresses according to the production function
Q  0.2M 0.67 L0.33
20
Where
and
Q her output per week is, L is the number of hours each week that she
works
M is the amount of material she uses per week.
Sarah values her labour at N$3 per hour and pays N$6 per unit of materials. She is
currently working 20 hours per week and using 50 units of material per week.
10.1
Determine how many dresses a week Sarah is producing.
Q  0.2(50)0.67 (20)0.33
Q  7.39
7dresses / week
10.2
Calculate how much it costs Sarah to make one dress.
TotalCost  3  20   6  50
 N $360 / dress
10.3
Compute the MRTS for Sarah’s dressmaking business.
MRTS 
MPL
MPK
0.2  0.33  L0.67  M 0.67
0.2  0.67  L0.33  M 0.33
K
 0.493
L
MRTS 
11.
12.
Consider the relationship between total profit (π) and output (Q) is given by the
function   f ( ) . In this function,  is described as the
A. Dependent variable
B. Independent variable
C. Slope
D. Intercept
The market for a 6-pack of Aquafina bottled water (manufactured by Pepsi) is
composed of the following demand and supply equations respectively:
QD  120  40 P
QS  60  50 P
21
The calculated equilibrium price and quantity will be:
13.
A.
P  N $2; Q  40units
C.
P  N $1.50; Q  60units
P  N $10; Q  18units
D.
P  N $1.00; Q  50units
The IS curve shows all combinations of income and
A.
B.
C.
D.
14.
B.
interest rate for which the goods market is in equilibrium
interest rate for which the money market is in equilibrium
price level for which the goods market is in equilibrium
price level for which the money market is in equilibrium
The following is a national income function for a four-sector economy at the
equilibrium level of income:
Y C  I G X Z
It has the following equations to describe the above four-sector economy
Y  income, C  consumption, I  investment ,
Where:
G  government exp enditure,  X  Z   net exports
For the IS sector
C  950  0.85YD ; YD  Y  T ; T  200  0.20Y
I  220  40i; G  1100 and ;  X  Z   800
where T  taxes and YD  disposable income
For the LM sector
M d  600 - 0.80i and M s  590  5i
where Md = money demanded and Ms = money supplied
14.1
Calculate the level of income
Y  and the rate of interest  i 
concurrently bring equilibrium to the economy
22
that will

Y C I G X Z

0.32Y  40i  1300
Ms  Md
600  0.8i  5905i
i  1.724
Y  N $3847.50
14.2
At the equilibrium level of income and interest rate computed in 14.1, determine the
new level of consumption,
Ce
14.3
Evaluate the MPC at this new level of income
15.
An income tax T with a proportional component t is incorporated into the model,
given
Y CI
C  C  bY
0
d
T  T  tY
0
Y  Y T
d
I I
0
And
C  85, I  30, T  20, b  0.75 and t  0.2
0
0
0
Find
16.
(a)
the reduced form
(b)
the numerical value of Y and
(c)
the effect on the multiplier
An economy has the following prevailing conditions:
C  89  0.6Y ; I  120  150i ; M t  0.1Y ; M w  240  250i ;
M s  M t  M w ; where M s  275
23
Note that C is the level of consumption, I the investment, Mt the transaction
precautionary demand for money and Mw the speculative demand for money.
16.1
Determine:
16.1.1 The level of income and the rate of interest that concurrently bring equilibrium to the
economy
16.1.2 The levels of
C , I , M t , and M w when M s  275 , at the equilibrium level of
income and interest rate computed in 2.1.1
FROM NOV 2014
1.
Sarah makes customized dresses for a select client. In the relevant range of her
production, she can produce dresses according to the production function
Q  0.2M 0.67 L0.33
where
and
Q her output per week is, L is the number of hours each week that she
works
M is the amount of material she uses per week.
Sarah values her labour at N$3 per hour and pays N$6 per unit of materials. She is currently
working 20 hours per week and using 50 units of material per week.
1.1
Determine how many dresses a week Sarah is producing.
1.2
Calculate how much it costs Sarah to make one dress.
1.3
Compute the MRTS for Sarah’s dressmaking business.
2.
A fishing company has a total revenue function given as
R( x)  280 x  2000
And their total cost function is given as
C ( x)  60 x  5600
24
Where x represents the number of units of fish sold
2.1
If there is no production of fish what amount of costs will the company incur?
2.2
If 1500 units were produced, what would be the company’s total revenue?
2.3
Calculate the total profit of this company from the production of 1500 units of fish.
3.
The national income function for a four-sector economy at equilibrium level of
income has the following equations to describe the above four-sector economy
Y  income, C  consumption, I  investment ,
Where:
G  government exp enditure,  X  Z   net exports
For the IS sector
Where Y
C  I G X Z
C  900  0.80YD ; YD  Y  250
I  220  40i; G  1100 and ;  X  Z   800
where i  interest rate; YD  disposable income
For the LM sector
Where
Ms  Md
M d  600  0.85i and M s  596  5.85i
3.1
Write down the national income function for this four-sector economy
3.2
Calculate the level of income
Y  and the rate of interest  i 
concurrently bring equilibrium to the economy
3.3
What is the level of consumption (𝐶) at equilibrium?
25
that will