QUESTIONS
NUMBER ONE
Given the following demand function
QX = 100 – 2P2
(a) Calculate the price elasticity of demand when price is Ksh. 2 and when price is
Ksh. 6
(8 marks)
(b) Calculate the price elasticity of demand in the price range Ksh. 3 and Ksh. 5
(5marks)
(c) If the current prevailing price is Ksh. 5 what advice would you give to the
producer in order to increase his revenue, and why?
(7 marks)
NUMBER TWO
(a) Define marginal utility and clearly explain the oxiom of diminishing marginal
utility.
(6 marks)
(b) Illustrate and explain the following:
(i) Consumer equilibrium under the cardinalist approach
(7 marks)
(ii) Consumer equilibrium under the ordinalist approach
(7 marks)
(Total: 20 marks)
ANSWERS
NUMBER ONE
(a) Calculation of price elasticity of demand when price is 2 and when price is 6:
Demand function: QX = 100 – 2P2
Point elasticity of demand as follows:
Pεd = ∆QX • PX
∆PX QX
∆QX = -4P
∆PX
when PX = 2
∆QX = -4(2) = -8
∆PX
5.14
QX = 100 – 2(2)2
(highly) elastic
100 – 2(4)
(100 – 8) = 92 Units
∴ Pεd = (-8 x 2/92) = -16/92 = -4/23 = -0.17
Pεd = 0.17 < 1 : Inelastic
When PX = 6
∆QX = -4(6) = -24
∆PX
QX = 100 – 2(6)2
100 – 2(6)2
(100 – 72) = 28 units
∴ Ped = (-24 x 6/28) = Ped
= 5.14 > 1:
(b) Calculation of the price elasticity of demand in the price range 3 and 5:
- Arc elasticity of demand as follows:
Arc εd = ∆QX • P1 + P2
∆PX Q1 + Q2
When P = 3
QX = 100 – 2(3) 2
100 – 2(9)
(100 – 18) = 82 units
when P = 5
QX = 100 – 2(5) 2
100 – 2(25)
(100 – 50) = 50 units
Demand schedule
PX
QX
5
50
3
82
∆QX = (82 – 50) = (32)
∆PX
3–5
-2
∴ Arc εd = 32
-2
5+3
50 + 82
-16 ( 8) = -128 = -32
132 132 33
= -0.969
Arc εd = 0.97<1: inelastic
(c) Pεd =
∆QX • PX
∆PX
QX
QX = 100 - 2 P2
Where PX = 5, QX = 50
∆QX = -4P = -20 at P = 5
∆PX
∴ Pεd = (-20 x 5/50) = -2
Pεd
Advice:
= 2 > 1 : price elastic demand
 The demand for commodity X is (price) elastic implying that any change in
price causes a more than proportionate change in quantity demanded (and
revenue from sales which is the product of the price and quantity of X
purchased).
 An increase in price of commodity X, in this case, will more than
proportionately reduce the quantity demanded and revenue; a fall in price
would more than proportionately increase the quantity demanded and
revenue from sales.
 Accordingly therefore, since the producer of commodity X seeks to
maximize sales (and profits) it would be very much advisable to either
reduce the price or maintain it stable at 5 but NOT increasing it above 5.
 A diagram can also be used to clearly demonstrate to the producer the
impact of a price change on sales revenue where demand is price elastic.
PX
Where P1 = 5
D
P1
P2
D
0
Q1
Q2
QdX
Fig 13.1: elastic demand curve for commodity X
NUMBER TWO
(a) Marginal utility is the additional satisfaction derived from the consumption
of an extra unit of a commodity. It is measured by the derivative of the total
utility function, that is, change in total utility per unit change in the quantity
(of a commodity) consumed:
MU = dTU/dQ
where MU: Marginal utility
TU: Total utility
Q: Quantity consumed.
This additional satisfaction (marginal utility) decreases as successive units of a
commodity are consumed – thus diminishing marginal utility.
Marginal utility falls under the cardinalist approach of consumer behaviour which
assumes that consumer satisfaction (utility) is measurable in terms of money the
consumer is willing and able to pay for a commodity.
Marginal utility varies from one individual to another e.g. a person in North
Eastern province of Kenya will find a glass of cold juice very satisfying relative to
a person in a cold area like Limuru or Kericho.
Diminishing marginal utility is based on the following assumptions:

Utility is measurable

Constant marginal utility of money



Normality of goods and rationality of the consumer
Successive units are homogenous
Continuity in consumption of the successive units.
When marginal utility is greater than zero, total utility is rising; total utility is
maximum when marginal utility is zero; when marginal utility is less than zero (ve) the total utility falls. Therefore, total utility (TU) increases at a decreasing rate
since marginal utility (mu) decreases at all levels of subsequent consumption of
successive units of a commodity.
Assuming consumption of one commodity, the consumer would be in equilibrium
when the marginal utility of the commodity is equal to the price of the commodity
i.e. MuX = PX where X is the commodity consumed.
Where more than one commodity is consumed (purchased) then the consumer
would be in equilibrium at the point where the marginal utility per shilling spent on
each product is the same (i.e. the point of equi-marginal utility):
MUx/Px = MUy/Py =MUn/Pn Where X: Commodity X
Y: Commodity Y
n: Commodity n
Marginal Utility
(MUX )
0
X1
Units of X(QX )
MU
Fig 14.1: Diminishing Marginal Utility
Units of commodity X
(QX)
0
1
2
3
4
5
6
7
Total
Utility
(TUX )
0
10
18
24
28
29
29
27
Marginal
Utility
(MUX )
10
8
6
4
1
0
-2
(b)(i) Consumer equilibrium under the cardinalist approach:
The cardinalist approach of consumer theory assumes measurable utility in
monetary terms such that the consumer is in equilibrium when marginal
utility derived from the consumption of a commodity is equal to the unit
price of the commodity, that is, MUX = Px.
Where there are more than one commodities, the condition for the
equilibrium of the consumer is the equality of the ratios of the marginal
utilities of the respective commodities to their prices i.e.
MUX = MUY = MUn
PX
PY
Pn
The marginal utility per shilling spent on all commodities is the same.
Assuming one commodity (X), a fall in price distorts the equilibrium of the
consumer which becomes Mux>Px; to go back to equilibrium the consumer
should reduce the marginal utility of X by consuming more of X pursuant to
the oxiom of diminishing marginal utility.
Assuming commodities X and Y, consumer equilibrium is attained where
MUX = MUY: where MUx & MUy: marginal utilities of commodities X and
Y
PX
PY
respectively.
Px & Py : Prices of commodities X and Y respectively.
If for instance, the price of X falls, Mux/Px>Muy/Py and to go back to
equilibrium, Muy should be increased by consuming less of commodity Y or
increasing the consumption of X in order to reduce Mux again pursuant to
the law of diminishing marginal utility.
Therefore, as the price of a commodity (x) increases, the consumer’s
marginal utility falls such that the consumer is now willing and able to
purchase relatively less units of X (in order to increase utility) thereby
reducing the quantity demanded of commodity X.
If however, the price of X falls, Mux increases and therefore the consumer
would be willing and able to buy more of X hence increasing the quantity
demanded of X. Thus a normal demand curve is based on the law of
diminishing marginal utility.
(ii)
Consumer equilibrium under the ordinalist approach:
Consumer equilibrium refers to a specific point in consumption of (two)
goods from which the consumer derives maximum satisfaction subject to a
given budget constraint (determined by the consumer’s income and
commodity prices).
This equilibrium point is achieved at the point of tangency of a budget line
to the highest possible indifference curve; at this point, the slope of the
indifference curve (i.e. marginal rate of substitution – MRS) is equal to the
slope of the budget line (i.e. relative commodity prices) Thus, at equilibrium
MRSxy = Px/Py.
Indifference curve – defined as the locus of possible combinations of two
commodities their consumption from which the consumer derives the same
level of satisfaction. Such curves are negatively sloped, do not intersect and
convex to the origin.
Budget line – refers to the locus of combinations of two goods whose
purchase exhausts the consumer’s budget constraint (money outlay).
Units of
Commodity Y
Units of Y
I/PY
A
I
I/PX
0
Units of Commodity X
Fig 14.2: Indifference Curve
0
B Units of X
Fig: 14.3: Budget line
At the point of tangency, the consumer is said to be in equilibrium as shown
below:
Units of
Commodity Y
A
Ye
●e
I3
I2
I1
B
0
Xe
Units of Commodity X
Fig : 14.4: Consumer equilibrium
Point (e) is consumer equilibrium point where the slope of the budget line (AB)
(Px/Py) is equal to the slope of the indifference curve (I2) (MRSXY) with Xe of X
and Ye of Y. The indifference curve (I2) has its maximum convexity at point (e)
denoting diminishing marginal rate of substitution.
The indifference curve (I1) is attainable but inefficient since it does not maximize
satisfaction, that is, consumer’s income is not fully utilized. Similarly, indifference
curve (I3) is NOT attainable with the present level of income and commodity
prices.
It is therefore at the point of tangency (e) that the consumer maximizes satisfaction
by fully spending the disposable income on Xe of X and Ye of commodity Y,
given the prices of X and Y.