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DCDM BUSINESS SCHOOL
FACULTY OF MANAGEMENT
ECONOMIC TECHNIQUES 102
LECTURE 1
THE SRAIGHT LINE AND APPLICATIONS
EQUATION OF THE STRAIGHT LINE
The following preliminaries will be discussed:
•
•
•
•
The standard equation of the line.
The Intercept.
The Gradient.
The graphing of a straight line.
DEMAND AND SUPPLY
Demand and supply decisions by consumers, firms and the government determine the
level of economic activity within the economy.
There are several variables that determine the general demand function. However, the
simplest demand function is Q = f (P). Therefore the demand of a good depends on price
only.
EXAMPLE 1
Consider the demand function Q = 200 – 2P. The inverse demand function is defined by
P = 100 –0.5Q. This definition is useful since in economics, the graphing of the Demand
function necessitates that Price is on the vertical axis. However, from now on we will not
differentiate from the Demand and Inverse Demand functions. The inverse demand
function is graphed below:
The General Linear Demand function
P = a - bQ such as P =100 - 0.5Q
P
120
a = 100
100
Slope = - b = - 0.5
D
80
P = 100 - 0.5Q
60
40
20
Q
200
240
120
160
80
40
0
0
a
= 200
b
Figure 2.19 Demand function, P = 100 - 0.5Q
Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd
15
SUPPLY FUNCTION
The Supply function P = h (Q) can be modeled by the simple linear equation
P = c + dQ where c and d are constants.
EXAMPLE 2
The Supply Function P = 10 + 0.5Q is graphed below:
Supply Function P = 10 + 0.5Q
Calculate and plot the supply schedule P = 10 + 0.5Q
Table 2.4 Supply schedule
P = 10 + 0.5Q
P
70
S
60
50
40
P = 10 + 0.5Q
30
20
Slope = 0.5
10
80
100
60
40
0
20
0
-20
Q: Quantity
P: P = 10 +0.5Q
0
10: P =10 +0.5(0)
20
20: P = 10+0.5(20)
40
30
60
40
80
50
100
60
Intercept (vertical) = 10
c = 10
Slope = 0.5
Figure 2.22
Horizontal intercept = - 20
Q
NOTE: the supply function may be plotted by simply joining the intercepts
16
Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd
MONEY MARKET LINE
Consider an individual confronted by the net cash flows C0 in period-0 and C1 in period1. Assume the individual can borrow and lend at the constant interest rate r . She may
borrow at most
C1
in period-0 and increase current consumption to
(1 + r )
C0 +
C1
.
(1 + r )
(6)
In period-1, the loan must be repaid with accrued interest. The amount is:
C1
)(1 + r ) = C1 ,
1+ r
(
(7)
which is exactly the amount available to her in period-1. Consumption in period-1 is zero.
The present value (PV) of this consumption mix is given by:
PV = C0 +
C1
1+ r
(8)
where PV is the maximum the individual can consume in period-0.
Next, (8) is rewritten as:
C1 = ( PV − C 0 )(1 + r ) ,
(9)
which is a function that indicates for a given PV (of consumption) the set of possible
consumption bundles ( C0 , C1 ).
By borrowing and lending at r , the consumer can move along the line as she wishes,
adjusting the consumption bundle without changing the parameters hence the name the
money-market line.
C1
C1 = ( PV − C 0 )(1 + r )
C0
Note:
C1 = PV (1 + r ) ; when C 0 = 0
C 0 = PV ; when C1 = 0
PRICE ELASTICITY OF DEMAND
Price elasticity of demand measures the responsiveness (sensitivity) of quantity
demanded to changes in the good’s own price. It is also referred to as own-price
elasticity. It is represented by the general elasticity formula:
εd =
Percentage change in quantity demanded
Percentage change in price
εd =
% ΔQ d ΔQ P
=
•
ΔP Q
% ΔP
Given the linear demand function P = a – bQ, the formula for point elasticity of demand
at any point ( P0 , Q0 ) is:
εd = −
1 P0
b Q0
Interpretation of coefficient of price elasticity
Coefficient of price elasticity of demand
− ∞ < ε d < −1
−1 < εd < 0
ε d = −1
Description
Elastic demand: demand is strongly
responsive to changes in price. When
ε d → −∞ , demand is perfectly elastic.
Inelastic demand: demand is weakly
responsive to changes in price. When
ε d = 0 , demand is perfectly inelastic.
Unit elastic demand: % change in demand
is equal to % change in price.
BUDGET AND COST CONSTRAINTS
How to plot any Linear Budget Constraint
xPX + yPY = M
‹
‹
‹
‹
→
y=
M ⎛ PX ⎞
−⎜
⎟x
PY ⎝ PY ⎠
Rearrange the equation in the form y = mx + c (see above)
Plot y on the vertical axis, against x on the horizontal axis
Calculate and plot the vertical and horizontal intercepts
Join the points and label the graph
Quantity of good Y , y
M
PY
40
30
20
M
PX
10
Quantity of good X , x
0
0
30
60
90
2
Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd
EXAMPLE 3
Example: Budget Constraint: x ( PX ) + y ( PY ) = M
‹
‹
‹
‹
‹
‹
‹
‹
‹
Example:
PX =£2: PY = £6: M = 180:
x × PX
+
y ×
PY = M
x × 2
+
y ×
6 = 180
(units of X) (price per unit) + (units of Y )(price per unit) = budget limit
This is the budget equation:
For plotting, rearrange the equation into the form y = mx + c:
Hence, 2x + 6y = 180 is rearranged as: y = 30 - 0.33x
In this form, it is easy to read off intercepts
Vertical intercept = 30 (from the equation above):
Horizontal intercept = 90 since -c/m = -(30)/(-0.33) = 90
Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd
3
It is important to notice changes:
•
•
•
Price of good X.
Price of good Y.
In the budget amount.
EXAMPLE 4
The equation of a demand function is given by Q = 64 – 4P, where Q is the number of
helicopter flights demanded daily; P is the price/flight in 00’s of pounds.
(a)
(b)
(c)
(d)
Plot the demand function with Q on the vertical axis
What is the demand (Q) when price increases by 1 unit?
What is the demand when P = 0?
What is the price when Q = 0?
EXAMPLE 5
Given the supply function, P = 500 + 2Q, where P is the price of a bottle of cognac in
pounds, Q number of litres supplied.
(a) Graph the supply function. What is the meaning of the value at the vertical
intercept?
(b) What is the value of Q when P = 600 pounds?
(c) What is the value of P when Q = 20?
EXAMPLE 6
Given the demand function, Q = 250 – 5P where Q is the number of children’s watches
demanded at P pounds each,
(a) Derive an expression for the point elasticity of demand in terms of P only.
(b) Calculate the point elasticity at each of the following prices, P = 20; 25; 30.
(c) Describe the effect of price changes on demand at each of these prices.
EXAMPLE 7
P = 90 – 0.05Q is the demand function for calculators at a university.
(a) Derive expressions for ε d in terms of (i) P only, (ii) Q only.
(b) Calculate the value of ε d when calculators are priced at P =20; 30; 70.
(c) Determine the number of calculators demanded when ε d =-1; ε d = 0.