Economics of Consumer Behaviour

Assumption of Rationality
Assumption of rationality is the point of departure in the theory of the consumer’s beahviour.
Rationality in this case means utility optimization. According to this postulate, the consumer
is assumed to choose among the available alternatives in such a manner that the satisfaction
derived from consuming commodities is as large as possible. This implies that he is aware of
the alternatives facing him and is capable of evaluating them. The principle assumption upon
which the theory of consumer behavior and demand is built is that, a consumer attempts to
allocate his/her limited money income among available goods and services so as to maximize
his/her utility (satisfaction).

Utility Analysis
Cardinal Measurement of Utility
Nineteenth century economists, such as W. Stanley Jevons, Leon Walras, Alfred Marshall,
and others assumed that utility is measurable in a cardinal sense. It means that the consumer
is assumed to be capable of assigning to every commodity or combination of commodities a
numerical number representing the amount or degree of utility associated with it. According
to this approach, numbers used to denote utility were supposed not to show only the level of
satisfaction; but, also these can be used to compare utility levels.

Quantity Utility
1 45
2 15

As in the above case, numbers in the table, if viewed from cardinal approach, implies that;
while consumption of first unit of some specific good gives 45 utils, second unit yields 15
utils. Moreover, utility got from consumption of first unit of the good gives three times more
utility than the second unit and vice versa.

Ordinal Measurement of Utility

Economists following the lead of J. R. Hicks, R. G. D. Allen, Slutsky, Vilfredo Pareto, and
others believe that utility is measurable in ordinal sense. In case of ordinal measurement also,
utility is measured in or expressed by numerical numbers. But, contrary to cardinal
measurement of utility, in case of ordinal measurement, these numbers only show the rank of
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satisfaction or preference order. That is, these numbers are in no way comparable. As in case
of previous example, numbers in the table, if viewed from ordinal approach, implies that;
only deduction which can be made is that, consumption of first unit of some specific good
gives 45 utils and the second unit yields 15 utils.

Existence of Utility Function

A consumer’s preferences must satisfy certain conditions in order to be represented by a
utility function. Axioms of consumer preferences theory are created for the purposes of:
iii) Using mathematical representation of utility functions;
ii) Portraying rational behavior (rationality in this case means optimizing);
iii) Deriving well-defined demand curves.
A set of sufficient conditions for the existence of utility function can be expressed by the
following axioms:

Axiom 1: Axiom of Completeness

According to this axiom preferences are complete. That is, for any two combinations, A and
B, a consumer can establish a preference ordering. In other words, for any comparison of
combinations, he/she will choose one and only one of the following: A is preferred to B; or B
is preferred to A; or A is indifferent to B. Without this property, preferences are undefined.

Axiom 2: Axiom of Transitivity

According to this axiom preferences are transitive. That is, for any consumer, if, A is
preferred to B and B is preferred to C, then, it must be that A is preferred to C. That is,
consumers are consistent in their preferences.

Axiom 3: Axiom of Continuity

According to this axiom preferences are continuous. That is, for any consumer, if, A is
preferred to B and C lies within A and B, then, A is preferred to C. To derive well-behaved
demand curves, continuity in consumer preferences is needed.

Axiom 4: Axiom of Non-satiation

According to this axiom the consumer is supposed to never get enough. Let, two
combinations A and B composed of two goods, X and Y are given. Moreover, XA = amount
of X in combination A, XB = amount of X in combination B, YA = amount of Y in
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combination A, YB = amount of Y in combination B. Then, if, XA = XB and YA > YB, then,
combination A will be preferred to combination B (regardless of the levels of XA, XB, YA,
YB). This implies that, the consumer always places positive values on more consumption;
and, the indifference curve map stretches out endless and there is no upper limit of utility.
While this axiom is not necessary to construct a well-defined utility function, it is helpful.

Axiom 5: Axiom of Diminishing Marginal Rate of Substitution (MRS)

This axiom is also not necessary to construct a well-defined utility function, but, it generally
captures a fundamental feature of human preferences.

Utility Function
All information pertaining to the satisfaction that the consumer derives from various
quantities of commodities is contained in the utility function. When the consumer’s purchases
are limited to two commodities, his ordinal utility function is, U = f (q1, q2)
Where, U = Total utility
q1 = Quantity of good Q1 consumed, and, q2 = Quantity of good Q2 consumed.

Indifference Curve and Indifference Map

For a given level of utility U 0, above equation becomes U 0 = f (q1, q2)
The locus of all commodity combinations from which the consumer derives same level of
satisfaction forms an indifference curve (IC). An indifference map is a collection of
indifference curves corresponding to different levels of satisfaction. Hence, following
functions constitute what may be called the indifference map.
U 0 = f (q1, q2); U1 = f (q1, q2); U2 = f (q1, q2); ……………………… Un = f (q1, q2);

Rate of Commodity Substitution (RCS)/Marginal Rate of Substitution (MRS)

Total differential of the above mentioned utility function, U = f (q1, q2) is,
dU = f1dq1 + f2dq2 , where,
dU = Change in total utility;
f1 = U/q1 = MU1 = Marginal utility of Q1; dq1 = Change in quantity of Q1 consumed;
f2 = U/q2 = MU2 = Marginal utility of Q2; and, dq2 = Change in quantity of Q2 consumed.
0
When the consumer moves along the same indifference curve, dU = 0, as U = U and, in
that case, dU = f1dq1 + f2dq2 = 0

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Or, f2dq2 = ‒ f1dq1
𝑑𝑞2 𝑓1 𝑑𝑞2 𝑓1
Or, =‒ Or, − =
𝑑𝑞1 𝑓2 𝑑𝑞1 𝑓2

𝑑𝑞2
Here, is the rate at which a consumer would be willing to substitute Q1 for Q2 per unit
𝑑𝑞1

of Q1 in order to maintain a given level of utility. It, therefore, expresses the slope of the
𝑑𝑞2
indifference curve. Negative of the slope − is the rate of commodity substitution (RCS)
𝑑𝑞1

or marginal rate of substitution (MRS). This is again, equal to the ratio of the partial
𝑓1
derivatives of the utility function ( ); that is ratio of marginal utility of two goods.
𝑓2

Budget Equation, Budget line and Slope of Budget Line

A budget equation is represented as, Y = p1q1 + p2q2
Where, p1 = Price of good Q1, p2 = Price of good Q2 and, Y = Consumer’s income. A
budget line is graphical presentation of a budget equation with a fixed level of income.
Accordingly, an equation for a budget line takes the form, Y0 = p1q1 + p2q2 ; (Y0 = fixed level
of consumer income). This implies that, budget line is a straight line.

From above budget equation, Y0 = p1q1 + p2q2, we get,

Y0 𝑝1
q2 = − q1,
𝑝2 𝑝2
𝑑𝑞2 𝑝1 𝑑𝑞2 𝑝1
Therefore, =− Or, − =
𝑑𝑞1 𝑝2 𝑑𝑞1 𝑝2

That is, slope of a budget line can be expressed by the ratio of prices of two goods under
consideration. Moreover, it is implied that budget line is downward sloping.

Consumption Duality/Consumers’ Optimizing Behaviour

Duality means looking at a particular thing from two different alternative ways. Thus, in
consumer theory, duality is the alternative ways of looking at consumer’s utility
maximization decision. In microeconomic analysis, duality refers to the relationship between
quantities and prices that arise as a consequence of hypotheses of optimization and convexity.
In consumer theory, consumer’s optimizing behavior can be examined either in the

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perspective of utility maximization subject to fixed income or in the perspective of
expenditure minimization subject to attainment of a fixed level of utility. In other words,
there are two ways to solve a consumer’s choice problem. That is, we can either fix
a budget and obtain the maximum utility from it (primal demand) or set a level of utility we
want to achieve and minimize expenditure (dual demand).

Thus, the duality concept related to consumer behavior is based on the fact that consumer
preferences can be represented by two forms other than the utility function. These are the
indirect utility function and the expenditure function. While indirect utility function gives
Marshallian demand function, expenditure function gives Hicksian demand function. Concept
of duality shows that there is a clear relation between the primal and the dual problems. In
consumer theory, primal problem is maximization of utility subject to given budget and dual
problem is minimization of expenditure to attain a fixed level of utility.

Utility maximization subject to budget constraint

Let, the consumer’s utility function be, U = f (q1, q2) ………………(1)
And the budget constraint is, Y0 = p1q1 + p2q2 …….…………………(2)
According to Lagrange multiplier method of constrained optimization, corresponding
Lagrange function takes the following form:
V = f (q1, q2) + λ(Y0 ‒ p1q1 ‒ p2q2 ) .………………..(3) (λ = marginal utility of money)
Now, first order condition of utility maximization is obtained by setting the first partial
derivatives of ‘V’ with respect to q1, q2 and λ equal to zero. That is,

𝜕𝑉
= f1 ‒ λp1 = 0 …………………….(4) Or, f1 = λp1 ……………(4/)
𝜕𝑞1
𝜕𝑉
= f2 ‒ λp2 = 0 ………..……………(5) Or, f2 = λp2 ……………(5/)
𝜕𝑞2
𝜕𝑉
𝜕𝜆
= Y0 ‒ p1q1 ‒ p2q2 = 0 ……………...(6)
𝑓1 𝑝1
Equation (4/) divided by (5/) gives, = . That is, ratio of marginal utilities of the two
𝑓2 𝑝2

goods must equal the ratio of their prices for maximization of utility. Again, from (4/), we
𝑓1 𝑓2 𝑓1 𝑓2
get, = λ; and from (5/), we get, = λ , Or, = = λ, implying, marginal utility
𝑝1 𝑝2 𝑝1 𝑝2

divided by price must be same for all goods.

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According to second order condition, the bordered Hessian determinant |𝐇| is to be positive.
𝑓11 𝑓12 − 𝑝1
|H| = |
𝑓21 𝑓 22 − 𝑝2
| >0
−𝑝1 − 𝑝2 0

Expenditure minimization subject to utility constraint

Let, the consumer’s utility function be, U 0 = f (q1, q2) ………………….(1)
And the budget equation, Y = p1q1 + p2q2 ……………………………(2)
Following the Lagrange method of constrained optimization, corresponding Lagrange
function is formed as follows,
1 1
L = p1q1 + p2q2 + μ{U 0 − f (q1, q2)} …………..…(3) (μ = 𝜆 = 𝑀𝑎𝑟𝑔𝑖𝑛𝑎𝑙 𝑢𝑡𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑚𝑜𝑛𝑒𝑦)

First order condition of expenditure minimization is obtained by setting the first partial
derivatives of L with respect to q1, q2 and μ equal to zero. That is,
𝜕𝐿
= p1 − μ f1 = 0 ………………….(4) Or, p1 = μ f1 …………………..(4/)
𝜕𝑞1
𝜕𝐿
= p2 − μ f2 = 0 ………..………...(5) Or, p2 = μ f2 …………………..(5/)
𝜕𝑞2
𝜕𝐿
𝜕𝜇
= U 0 − f (q1, q2) = 0…..………..(6)
𝑓1 𝑝1
Equation (4/) divided by (5/) gives, = , that is, the ratio of marginal utilities must equal
𝑓2 𝑝2

the ratio of prices for maximization of utility. Again, from equation (4/), we have
𝑓1 1 𝑓2 1 𝑓1 𝑓2 1
= = λ, and from equation (5/), we have, = = λ. Or, = = = λ.
𝑝1 𝜇 𝑝2 𝜇 𝑝1 𝑝2 𝜇

That is, marginal utility divided by price must be the same for all commodities.

Second order condition implies that the bordered Hessian determinant |𝐇| is positive.
−μ𝑓11 −μ𝑓12 − 𝑓1
|H| = |
− μ 𝑓21 − μ𝑓22 − 𝑓2
| <0
−𝑓1 − 𝑓2 0

Mathematically, it can be proved that, this second order condition turns out to be:
𝑓11 𝑓12 − 𝑝1
|H| = |
𝑓21 𝑓22 − 𝑝2
| >0
−𝑝1 − 𝑝2 0

This is, same as the condition for constrained utility maximization.

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Derivation of Ordinary/Marshall’s Demand Function from Utility function
Let, the consumer’s utility function be U = q1 q2 ………………..(1)
And the budget constraint, Y0 = p1q1 + p2q2 ……..……………...(2)
Now, following the Lagrange method of constrained optimization, corresponding Lagrange
function is formed as follows,
V = q1 q2 + λ (Y0 ‒ p1q1 ‒ p2q2 )…………………………………..(3)
First order condition of utility maximization is obtained by setting the first partial derivatives
of V with respect to q1, q2 and λ equal to zero. That is,
𝜕𝑉
= q2 ‒ λp1 = 0 …………………….(4)
𝜕𝑞1
𝜕𝑉
= q1 ‒ λp2 = 0 ………..……………(5)
𝜕𝑞2

𝜕𝑉
= Y0 ‒ p1q1 ‒ p2q2 = 0 …………….(6)
𝜕𝜆

Rearranging equations (4) and (5) and then, by dividing,

𝑞2 𝑝1
we get, = Or, p1q1 = p2q2
𝑞1 𝑝2

Now, from equation (6), we can write, Y0 ‒ p1q1 ‒ p1q1 = 0

Y0
Or, 2p1q1 = Y0, or, q1 = , this is the ordinary/Marshall’s demand function of good Q1.
2𝑝1

Similarly, from equation (6), we can write, Y0 ‒ p2q2 ‒ p2q2 = 0

Y0
Or, 2p2q2 = Y0 , hence, q2 = 2𝑝 , this is ordinary/Marshall’s demand function of good Q2.
2

Derivation of Compensated/Hicks’ Demand Function from Utility function

Let, the consumer’s utility function be U0 = q1 q2 ………………….(1)
And the budget constraint, Y = p1q1 + p2q2 …………………..……(2)
Following the Lagrange method of constrained optimization, corresponding Lagrange
function is formed as follows,
L = p1q1 + p2q2 + μ (U0 − q1 q2) ……….. (3)
The first order condition of expenditure minimization is obtained by setting the first partial
derivatives of L with respect to q1, q2 and μ equal to zero. That is,
𝜕𝐿
= p1 − μ q2 = 0………………………(4) Or, p1 = μ q2 ………………..(4/)
𝜕𝑞1
𝜕𝐿
= p2 − μ q1 = 0..………..…………..(5) Or, p2 = μ q1 ………………..(5/)
𝜕𝑞2
𝜕𝐿
𝜕𝜇
= U0 − q1q2 = 0...…………………...(6)

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 𝑞2 𝑝1
Equation (4/) divided by (5/) gives, =
𝑞1 𝑝2
𝑝1 𝑞1
Or, p1q1 = p2q2 Therefore, q2 =
𝑝2

Putting this value of q2 in equation (6), we get,

𝑝1 𝑞1 𝑝1 𝑞12 𝑈 0 𝑝2 𝑈 0 𝑝2
0
U − q1 =0 Or, U −0
=0 Or, 𝑞12 = , that is, q1 =√
𝑝2 𝑝2 𝑝1 𝑝1

This is the compensated/Hicks’ demand function of good Q1.

𝑈 0 𝑝1
Similarly, we can derive, q2 =√
𝑝2
.

This is compensated/Hicks’ demand function of good Q2.

Derivation of Inverse Price-Demand Relationship

Cardinal Analysis
Cardinal utility analysis is the oldest theory of demand which provides an explanation of
consumer’s demand for a product and derives the law of demand which establishes an inverse
relationship between price and quantity demanded of a product. This analysis is based on two
laws: the law of diminishing marginal utility and law of equi-marginal utility.

The law of equi-marginal utility states that the consumer will distribute his money income
between the goods in such a way that the utility derived from the last unit of money spent on
each good is equal. With given income, a unit of money has a certain utility for him; this
utility is the marginal utility of money. The consumer will go on purchasing goods until the
marginal utility of money expenditure on each good becomes equal to marginal utility of
money to him.

Symbolically, therefore, condition of consumer’s equilibrium can be expressed as:

MU1 MU2 MUn
= =..…. = MUM = λ , MUi = Marginal utility of ith good (i = 1,2………….n);
P1 P2 Pn

Pi = Price of ith good (i = 1,2………….n); and, MUM = λ = Marginal utility of money. Now,
MU1
to derive demand curve of good X, expression = λ is used as shown below.
P1

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 Derivation of Demand Curve

𝐌𝐔𝐗 𝐌𝐔𝐗 𝐌𝐔𝐗 𝐌𝐔𝐗

𝐏𝐗 𝐏𝐗𝟏 𝐏𝐗𝟐 𝐏𝐗𝟑

MUM = λ
E1 E2 E3

O Q1 Q2 Q3 Quantity of X

Price
D
𝐏𝐗 𝟏 a
𝐏𝐗 𝟐 b
𝐏𝐗 𝟑 c
D
O Q1 Q2 Q3 Quantity of X

Here, the consumer buys OQ1 of good X when price is PX1 ; when price of good X falls to PX2 ;
the consumer increases quantity demanded to OQ2 and so on. Lower panel of the same figure
shows how demand curve for X is derived.

Ordinal Analysis
Given the consumer’s money income and his indifference map, it is possible to draw his
demand curve for any commodity from the PCC. This method has an edge over the former
cardinal approach. It arrives at the same result without making the unrealistic assumption of
measurability of utility and constant marginal utility of money.

This analysis assumes that, money to be spent by consumer is given and constant; only price
of good X falls and prices of other related goods do not change; and, consumer’s tastes and
preferences remain constant.

As in the figure below, it is considered that, money income and price of good Y remaining
fixed; starting from a level of price, P1, there is gradual fall in price of good X to P2 and P3.
Consequent budget lines are AB, AC and AD; which are tangent respectively to indifference
curves IC1, IC2 and IC3. Hence, consumer’s equilibrium points are E1, E2 and E3.
Accordingly, the consumer respectively buys OX1, OX2 and OX3 units of X.

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 Derivation of demand curve
Y IC1 IC2 IC3
A

E1 E2 E3

O X1 X2 (B) X3 (C) D X
Price
D
P1 a
P2 b
P3 c
D
O X1 X2 X3 X

Price-demand schedule of the consumer for good X got from upper panel is as follows: when
price of good X is P1, the consumer buys OX1 amount of it (combination ‘a’); when price is
P2, he buys OX2 amount (combination ‘b’); and when price is P3, he buys OX3 amount
(combination ‘c’). Now, joining these three points, we get downward sloping demand curve
DD, implying inverse relation between demand and price.

Monotonic and Non-monotonic Utility

In calculus, a function defined on a subset of the real numbers with real values is
called monotonic if and only if, it is either entirely non-increasing, or entirely non-
decreasing. That is, a function that increases monotonically does not exclusively have to
increase, it simply must not decrease (1st figure). Likewise, a function that decreases
monotonically does not exclusively have to decrease, it simply must not increase (2nd figure).
Conversely, a non-monotonic function (3rd figure) first falls, then rises, then falls again.

Y Y Y

O X O X O X
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Monotonic Utility
Monotonicity of preference simply means that an individual prefers more quantity of a good
as compared to less. Monotonic utility function has no decreasing part which means it has no
peak. Traditional monotonic utility function gives downward sloping and convex to the origin
indifference curves showing combinations of goods for which the consumer has the same
preference. Accordingly, we get indifference map consisting of indifference curves stretching
from left to right infinitely and indicating higher and higher level of satisfaction. As a result,
with higher and higher income, consumers are able to attain higher and higher level of utility.

Consumers’ Equilibrium
A rational consumer desires to purchase a combination of goods X and Y from which he
derives the highest level of satisfaction. However, his income being limited, he is not able to
purchase unlimited amounts of the commodities.

In case of indifference curve analysis, according to the first order/necessary condition of

utility maximization; at the equilibrium point, budget line is to be tangent to an indifference
curve. That is, at this point, slope of the budget line = slope of the indifference curve.
𝑑𝑌 𝑓1 𝑝1
Symbolically, first order condition of consumers’ equilibrium stands as: − = = .
𝑑𝑋 𝑓2 𝑝2

According to the second order/sufficient condition of utility maximization, at the equilibrium

point, corresponding indifference curve must be convex to the origin. This implies that, at the
point of equilibrium, the rate of change of marginal rate of substitution must be negative.
𝑑 dY 𝑑2𝑌 𝑑2𝑌
That is, at equilibrium point, (‒ dX ) = − 𝑑𝑋 2 < 0 or,
𝑑𝑋 2
> 0.
𝑑𝑋

As is evident from the figure below, starting from a lower level of income designated by
budget line AB, the consumer is able to be on indifference curve IC1 (equilibrium point E1)
and able to consume respectively X1 and Y1 amounts of goods X and Y. When consumer’s
income increases (designated by budget line CD), the consumer is able to be on indifference
curve IC2 (equilibrium point E2) and able to consume respectively X2 and Y2 amounts of
goods X and Y. Similarly, when consumer’s income increases (designated by budget line
GH), the consumer is able to be on indifference curve IC3 (equilibrium point E3) and able to
consume respectively X3 and Y3 amounts of goods X and Y.

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 Y
G
IC1
C
Y3 A E3
Y2 E2 IC3
Y1 E1 IC2
O X1 X2B X3 D H X

We have seen that, monotonic utility function has no decreasing part which means it has no
peak. Accordingly, we get indifference map consisting of indifference curves stretching from
left to right infinitely and indicating higher and higher level of satisfaction. As a result, with
higher and higher income, consumers are able to attain higher and higher level of utility. This
assertion of endlessly increasing utility got from increased level of consumption violates/
ignores law of diminishing marginal utility, a vital foundation of microeconomic analysis.

According to this law, in a short period of time if someone goes with increase in consumption
of a good or a combination of good then though total utility increases but it increases at a
decreasing rate; thus, making marginal utility positive but declining. So, it is clear that, while
analyzing consumer behaviour if assumption of monotonic utility is made, then it simply
implies ignoring of this vital law of microeconomics. Therefore, traditional utility function
which gives downward sloping and convex to the origin indifference curves showing
combinations of goods for which the consumer has tshe same preference cannot yield the
optimal consumption. It is, therefore, not fit for microeconomic analysis. Instead, assumption
of non-monotonic utility function is more suitable and realistic.

Non-monotonic Utility and Consumers’ Equilibrium

Non-monotonic utility function also gives indifference curves; but they are different from
those under monotonic utility function. Graphically, indifference curves associated with non-
monotonic utility function can be visualized as located along a mountain seen from an
airplane just above the peak of that mountain. The larger oval is the lower altitude of the
mountain and also indicates lower level of pleasure. The smaller oval is the higher altitude of
the mountain which yields greater level of pleasure. The peak point designates highest
altitude of the mountain which yields maximum level of pleasure (U*).
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 Y Y
G
I
C

A Y4 E3
E2 U* Y3 E4
Y2
Y1 E1

O X1 X2 B D X O X3X4 J H X
Higher Income yields greater pleasure Higher Income may yield same pleasure/
Optimal income yields highest pleasure

Let, in the above graphs, outermost/largest oval shows level of utility U1, next inner
one/second largest oval shows level of utility U2, next inner one/third largest oval shows level
of utility U3, and so on. The central point being representing peak of the mountain, shows
maximum utility (U*). With AB budget line, the consumer’s equilibrium is at point E1,
consumption of goods X and Y are respectively X1 and Y1 and utility level is U2. When
consumer’s income rises (budget line CD), he is at equilibrium at point E2, consuming
respectively X2 and Y2 amounts of goods X and Y, and having a higher utility, U4.

Let, consumer’s income rises still further (shown by budget line GH in right hand figure), he
is at equilibrium at point E3, consuming respectively X3 and Y3 amounts of goods X and Y,
but having the same utility, U4. That is, even after consuming greater amount of X and Y (as
X3 > X2 and Y3 > Y2); the consumer is getting same level of utility (U4). Now, if considered a
case of fall in budget (shown by the line IJ), then the situation is that, the consumer is in
equilibrium at point E4; furthermore, even though he is consuming lesser amounts than before
(X4 < X3 and Y4 < Y3); he is enjoying not only relatively higher utility than before, but is
having highest level of utility possible (U*).

In sum, contrary to the concept of monotonic utility, where utility rise endlessly with rise in
income; under non-monotonic utility, it may rise with rise in income (move from E1 to E2)
remain constant even after rise in income (move from E2 to E3) or may even increase after
fall in income (move from E3 to E4).
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Derivation of Demand Function in ‘n’ Commodity Case
Let, the consumer’s utility function be U = q1q2q3q4……………… qn
And the budget constraint is, Y0 = p1q1 + p2q2 + p3q3 +.…………………+ pnqn

According to Lagrange multiplier method of constrained optimization, corresponding

Lagrange function takes the following form:

V = q1q2q3q4……………… qn + λ(Y0 ‒ p1q1 ‒ p2q2 ‒ p3q3 ‒ .………….. ‒ pnqn)

Now, first order condition of utility maximization is obtained by setting the first partial
derivatives of ‘V’ with respect to q1, q2 ,……….qn and λ equal to zero. That is,

𝜕𝑉
= q2q3q4……………… qn ‒ λp1 = 0 ..……(1) Or, q2q3q4……… qn = λp1……….(1/)
𝜕𝑞1
𝜕𝑉
= q1q3q4……………… qn ‒ λp2 = 0 ………(2) Or, q1q3q4……….qn = λp2……….(2/)
𝜕𝑞2
𝜕𝑉
= q1q2q4……………… qn ‒ λp3 = 0……….(3) Or, q1q2q4……….qn = λp3……….(3/)
𝜕𝑞3

……………………………………………….
𝜕𝑉
= q1q2q3q4…………..qn‒1 ‒ λpn = 0 ……….(n)
𝜕𝑞𝑛

𝜕𝑉
𝜕𝜆
= Y0 ‒ p1q1 ‒ p2q2 ‒ .…. ‒ pnqn = 0 …….(n + 1)

𝑞2 𝑝1
Dividing equation (1/) by (2/), we get, = Or, p1 q1 = p2 q2
𝑞1 𝑝2
𝑞3 𝑝2
Similarly, dividing equation (2/) by (3/), we get, = Or, p2 q2 = p3 q3
𝑞2 𝑝3

Proceeding in same fashion, we finally get, p1q1 = p2q2 = p3q3 = ………………..= pnqn
Hence (n + 1)th equation becomes:
Y0 ‒ np1q1 = Y0 ‒ np2q2 = Y0 ‒ np3q3 = ….= Y0 ‒ npnqn = 0

Y0 Y0 Y0
Here from we get, q1 = , q2 = ,………………………….………………. qn = ,
𝑛𝑝1 𝑛𝑝2 𝑛𝑝𝑛

14
Indirect utility function
In an ‘n’ commodity case utility function can be expressed as: U = f (q1, q2, q3,.………… qn)
And the budget equation as, Y = p1q1 + p2q2 + p3q3 +.…………………+ pnqn

𝑝1 𝑝2 𝑝3 𝑝𝑛 𝑝
Let, V1 = , V2 = , V3 = ,………………………………Vn = . That is, Vi = 𝑖 .
𝑌 𝑌 𝑌 𝑌 𝑌

Dividing above budget equation through by ‘Y’, we get,

𝑌 𝑝1 𝑞1 𝑝2 𝑞2 𝑝3 𝑞3 𝑝𝑛 𝑞𝑛
= + + +………………………
𝑌 𝑌 𝑌 𝑌 𝑌

Or, 1 = V1q1 + V2q2 + V3q3 +.…………………+ Vnqn = ∑𝑛

1 𝑉𝑖 𝑞𝑖

Or, 1 ‒ ∑𝑛
1 𝑉𝑖 𝑞𝑖 = 0

Since optimal solutions are homogeneous of degree zero in income and prices, nothing
essential is lost by this transformation to normalized prices.

Let us now proceed with the utility function: U = f (q1, q2, q3,.………… qn)………….(1)
And the new budget equation: 1 ‒ ∑𝑛
1 𝑉𝑖 𝑞𝑖 = 0………………………………………...(2)
Next, for constrained utility maximization we form the corresponding Lagrange function as:

L = f (q1, q2, q3,.………… qn) + λ(1 ‒ ∑𝑛

1 𝑉𝑖 𝑞𝑖 )…………………………………………………….(3)
Thus, first order condition of constrained optimization stands as:

𝜕𝐿
= f1 ‒ λV1 = 0 Or, f1 = λV1
𝜕𝑞1
𝜕𝐿
= f2 ‒ λV2 = 0 Or, f2 = λV2
𝜕𝑞2

…………………. …………….. …………………………………………..(4)

…………………. ……………..
𝜕𝐿
= fn ‒ λVn = 0 Or, fn = λVn
𝜕𝑞𝑛
𝜕𝐿
= (1 ‒ ∑𝑛1 𝑉𝑖 𝑞𝑖 ) = 0……………………………………………………………………………………….(5)
𝜕λ

Solution of equation set (4) and equation (5) yields following demand functions:

15
q1 = D1(V1, V2, V3,……………………Vn)
q2 = D2(V1, V2, V3,……………………Vn)
…………………………………………… Or, qi = Di(V1, V2, V3,…………Vn)…….(6)
…………………………………………….
qn = Dn(V1, V2, V3,……………………Vn)

Indirect utility function can now be derived as:

U = f [q1, q2, q3, ………………………qn]
= f [D1(V1, V2,………Vn), D2(V1, V2,………Vn),……….. Dn(V1, V2, ………………Vn)
= g (V1, V2,………Vn)………………………………………………………………….(7)

Hence, while the direct utility function describes preferences independent of market
phenomena; the indirect utility function reflects influence of market prices. More specifically,
here utility is a function of normalized prices.

Choice of Utility Index

The numbers which the utility function assigns to the alternative commodity combinations
need not have cardinal significance; they need only serve as an index of consumer’s
satisfaction. If a particular set of numbers associated with various combinations of Q1 and Q2
is a utility index, any positive monotonic transformation of it is also a utility index. A
function F(U) is a positive monotonic transformation of U, if F(U1) ˃ F(U0) whenever, U1 ˃
U0.

Let, the consumer’s utility function is U = f (q1, q2), and the budget constraint, Y0 = p1q1 +

p2q2 . Now, a monotonic transformation of the above utility function gives the following
utility function: W = F(U) = F[f (q1, q2)]. It is now to prove that, it means and implies the
same thing if either ‘U’ or ‘W’ is maximized with respect to given budget constraint. That is,
monotonic transformation of a utility function doesn’t lead to any change in the first and
second order conditions of constrained optimization. In other words, first and second order
conditions of constrained optimization are equally applicable to the initial utility function and
all its monotonic transformations.

New utility function is, W = F (U) = F [f (q1, q2)] ……………………..(1)

Budget constraint is, y0 = p1q1 + p2q2 ………………………. (2)
16
Following the Lagrangian method of constrained optimization, corresponding Lagrangian
function takes the form: V = F[f (q1, q2)] + λ (y0 − p1q1 − p2q2 ) …….(3)

The first order condition of utility maximization is obtained by setting the first partial
derivatives of V with respect to q1, q2 and λ equal to zero. That is,

∂V ∂F ∂U
= − λp1 = F/f1 − λ p1 = 0 ………….(4)
∂q1 ∂U ∂q1
∂V ∂F ∂U
= − λp2 = F/f2 − λ p2 = 0 …………..(5)
∂q2 ∂U ∂q2
∂V
= y0 − p1q1 − p2q2 = 0………………. ……….. (6)
∂λ

f1 𝑝1
Equation (4) divided by (3) gives, =
f2 𝑝2

Hence, it is proved that first order condition of optimization doesn’t depend on the utility
index. According to second order condition, relevant bordered Hessian determinant |𝐇|
should take positive value. In this case,

∂ ∂V ∂ ∂V ∂ ∂V
( ) ( ) ( )
∂q1 ∂q1 ∂q2 ∂q1 ∂λ ∂q1
| ∂ ∂V ∂ ∂V ∂ ∂V |
|H| = ( ) ( ) ( ) >0
∂q1 ∂q2 ∂q2 ∂q2 ∂λ ∂q2
| ∂ ∂V ∂ ∂V ∂ ∂V |
( ) ( ) ( )
∂q1 ∂λ ∂q2 ∂λ ∂λ ∂λ

∂ ∂V ∂ ∂ ∂
Now, ( )= (F/f1 − λ p1) = f1 (F/) + F/ (f1)
∂q1 ∂q1 ∂q1 ∂q1 ∂q1

= f1F// f1+ F/ f11 = F// f12 + F/ f11

∂ ∂V ∂ ∂ ∂
Similarly, ( )= (F/f1 − λ p1) = f1 (F/) + F/ (f1)
∂q2 ∂q1 ∂q2 ∂q2 ∂q2

= f1F// f2+ F/ f12 = F// f1f2 + F/ f12

∂ ∂V ∂
( )= (F/f1 − λ p1) = − p1
∂λ ∂q1 ∂λ
∂ ∂V ∂ ∂ ∂
( )= (F/f2 − λ p2) = f2 (F/) + F/ (f2)
∂q1 ∂q2 ∂q1 ∂q1 ∂q1

= f2F// f1+ F/ f21 = F// f1f2 + F/ f21

∂ ∂V ∂ ∂ ∂
( )= (F/f2 − λ p2) = f2 (F/) + F/ (f2)
∂q2 ∂q2 ∂q2 ∂q2 ∂q2

= f2F// f2+ F/ f22 = F// f22 + F/ f22

17
∂ ∂V ∂
( )= (F/f2 − λ p2) = − p2
∂λ ∂q2 ∂λ

∂ ∂V ∂
( )= (y0 − p1q1 − p2q2) = − p1
∂q1 ∂λ ∂q1
∂ ∂V ∂
( )= (y0 − p1q1 − p2q2) = − p2
∂q2 ∂λ ∂q2
∂ ∂V ∂
( )= (y0 − p1q1 − p2q2) = 0
∂λ ∂λ ∂λ

Hence, |H| = F// f12 + F/ f11 F// f1f2 + F/ f12 − p1

F// f1f2 + F/ f21 F// f22 + F/ f22 − p2 > 0
− p1 − p2 0

F/ 𝑓1 F/ 𝑓2
From equations (4) and (5), we get, p1 = and, p2 =
λ λ
Accordingly,
F/ 𝑓1
F//f12 + F/f11 F//f1f2 + F/f12 − λ
F/ 𝑓2
|H| = F//f1f2 + F/f21 F//f22 + F/f22 − λ
>0

F/ 𝑓1 F/ 𝑓2
− λ
− λ
0

F/
Taking out from third row and third column, we can rewrite the above as,
λ
F// f12 + F/ f11 F// f1f2 + F/ f12 − f1
F/
|H| = ( )2 F// f1f2 + F/ f21 F// f22 + F/ f22 − f2 >0
λ

−f1 − f2 0

Again multiplying third row through by F//f1 and adding it to first row, and, multiplying third
row by F//f2 and adding it to second row, we get,
F/f11 F/ f12 − f1
F/
|H| = ( )2 F/f21 F/f22 − f2 >0
λ

− f1 − f2 0

18
 λp1 λp2
Next, inserting − f1 = − and − f2 = −
F/ F/
λp1
F/ f11 F/ f12 −
F/
F/ λp2
|H| = ( )2 F/ f21 F/ f22 − >0
λ F/
λp1 λp2
− − 0
F/ F/
λ
Now, taking out from third row and third column, we can rewrite the above as,
F/

F/ f11 F/ f12 − p1
/
|H| = (Fλ )2( Fλ/)2 F/ f21 F/ f22 − p2 >0

− p1 − p2 0

Next dividing first and second rows by F/ , we get

𝑝1
f11 f12 −
F/
𝑝2
|H| = (F/)2 f21 f22 − >0
F/

− p1 − p2 0

Finally, multiplying third column by F/, we get,

f11 f12 − p1
𝟐
𝐅/
|H| = f21 f22 − p2 > 0
𝐅/

− p1 − p2 0

As, F/ > 0, hence, the condition turns to that the determinant must be positive. This proves
that, conditions of constrained utility maximization are free from the influence of choice of
utility index.

In other words, maximization of utility function, U = f (q1, q2), subject to budget constraint,
y0 = p1q1 + p2q2 , and maximization of W = F(U) subject to same budget constraint lead to
same result. That is, monotonic transformation of a utility function leaves the conditions of
constrained optimization unaltered.

19
Consumer Choice under Condition of Risk and Uncertainty
The traditional theory of consumer behaviour does not include an analysis of uncertain
situations. In this sense, the analysis is unrealistic as it assumes that particular actions on the
part of the consumer are followed by particular, determinate consequences which are
knowable in advance. As for example, all automobiles of the same model and produced in the
same factory do not always have the same performance characteristics. As a result of random
accidents, in the production process some substandard automobiles are occasionally produced
and sold. The consumer has no way of knowing ahead of time whether the particular
automobile which he or she purchases is of standard quality or not. Let ‘A’ represent the
situation in which the consumer possesses a satisfactory automobile, ‘B’ a situation in which
the consumer possesses no automobile, and ‘C’ one in which he or she possesses a
substandard automobile.

It is assumed that the consumer prefers ‘A’ to ‘B’ and ‘B’ to ‘C’. Now the consumer is
presented with a choice between two alternatives: (1) He or she can maintain the status quo
and have no car at all. This is a choice with certain outcome; that is, the probability of the
outcome equals unity. (2) He or she can obtain a lottery ticket with a chance of winning either
a satisfactory automobile (alternative A) or an unsatisfactory one (alternative C).

The consumer may prefer to retain his income with certainty or he may prefer the lottery
ticket with dubious outcome, or he may be indifferent between them. Consumers’ decision
will depend upon the chances of winning or losing in this particular lottery. If the probability
of ‘C’ is very high, the consumer might prefer to retain his income with certainty; and, if, the
probability of ‘A’ is very high; he might prefer the lottery ticket. The triplet of numbers (P,
A, B) is used to denote a lottery offering outcome ‘A’ with probability P (0 < P < 1) and
outcome ‘B’ with probability (1 – P).

Von Neumann and Morgenstern showed that under some circumstances, it is possible to
construct a set of numbers for a particular consumer that can be used to predict his choices in
uncertain situations. Construction of this utility index needs that the consumer abides by
some axioms. Axioms of Von Neumann and Morgenstern utility index are discussed below:

a) Complete ordering axiom

For the two alternatives ‘A’ and ‘B’ one of the following must be true: the consumer prefers
20
The consumer prefers ‘A’ to ‘B’; he prefers ‘B’ to ‘A’; or he is indifferent between them.
Moreover, the consumer’s evaluation of alternatives is transitive: if he prefers ‘A’ to ‘B’, and
‘B’ to ‘C’, he prefers ‘A’ to ‘C’.

b) Continuity axiom
If, ‘A’ is preferred to ‘B’ and ‘B’ to ‘C’; then, according to this axiom, there exists some
probability P (0 < P < 1), such that the consumer is indifferent between outcome ‘B’ with
certainty and a lottery ticket (P, A, C).

c) Independence axiom
Let, the consumer is indifferent between ‘A’ and ‘B’ and that ‘C’ is any outcome whatever. If
one lottery ticket ‘L1’ offers outcomes ‘A’ and ‘C’ with probabilities P and (1− P)
respectively and another lottery ticket ‘L2’ offers outcomes ‘B’ and ‘C’ with the same
probabilities P and (1− P), the consumer is indifferent between the two lottery tickets.
Similarly, if he prefers ‘A’ to ‘B’, he will prefer ‘L1’ to ‘L2’.

d) Unequal-probability axiom
Let, the consumer prefers ‘A’ to ‘B’; and let L1 = (P1, A, B) and L2 = (P2, A, B). The
consumer will prefer L2 to L1 if and only if P2 ˃ P1.

e) Compound-lottery axiom
Let L1 = (P1, A, B) and L2 = (P2, L3, L4), where L3 = (P3, A, B), and L4 = (P4, A, B), be a
compound lottery in which the prizes are lottery tickets. L2 is equivalent to L1 if,
P1 = P2 P3 + (1 – P2) P4

Concept of expected utility

Expected utility for the two-outcome lottery, L = (P, A, B) is:
E[U(L)] = PU(A) + (1 − P)U(B)

Considering the lotteries L1 = (P1, A1, A2) and L2 = (P2, A3, A4), ‘Expected Utility Theorem’
states that if L1 is preferred to L2, then E[U(L1)] ˃ E[U(L2)]. The significance of this theorem
is that uncertain situations can be analyzed in terms of the maximization of expected utility.
As for example, if there are two lotteries L1= (0.5, A1, A2) and L2= (0.4, A3, A4), and given,
U(A1) = 25; U(A2) = 64; U(A3) = 36; and U(A4) = 49. Then,
21
E[U(L1)] = 0.5 x U(A1) + 0.5 x U(A2) = 0.5 x 25 + 0.5 x 64 = 12.5 + 32 = 44.5
E[U(L2)] = 0.4 x U(A3) + 0.6 x U(A4) = 0.4 x 36 + 0.6 x 49 = 14.4 + 29.4 = 43.8
In this case, E[U(L1)] = 44.5 ˃ E[U(L2)] = 43.8; hence, lottery L1 is preferred to L2.

Use of the concept of expected utility to construct utility index

Expected utility formula may be used to construct utility numbers for a person who conforms
to the Von Neumann Morgenstern axioms. Let utility numbers are arbitrarily assigned to two
certain outcomes A1 and A2.

For example if A2 is preferred to A1, let U(A1) = 20 and U(A2) = 1000. Now, consider the
outcome ‘A3’ which lies between ‘A1’ and ‘A2’ in the preference ranking. The consumer is
then asked for a value of ‘P’ such that he is indifferent between ‘A3’ and (P, A1, A2). If now,
P = 0.8, then, U(A3) = 0.8 x U(A1) + 0.2 x U(A2) = 0.8 x 20 + 0.2 x 1000 = 16 + 200 = 216

If A4 is preferred to all above three alternatives, its utility can be obtained by asking the
consumer for a value of ‘P’ such that he is indifferent between A2 and (P, A1, A4).
If, P = 0.6, then, 1000 = 0.6 x 20 + 0.4 x U(A4) Or, U(A4) =2470.
The process can be continued indefinitely, and will not lead to contradictory results as long as
the five axioms are abided by.

Concept of expected value of a lottery

Expected value of a lottery (P, W1, W2), where the ‘Wi’ are different wealth levels, is the sum
of the outcomes, each multiplied by its probability of occurrence,
That is E[W] = PW1 + (1 – P) W2

* A person is risk neutral relative to a lottery if the utility of the expected value of the lottery
equals the expected utility of the lottery, i.e., if,
U[PW1 + (1 – P) W2] = PU(W1) + (1 − P)U(W2)
* A person is risk averter relative to a lottery if the utility of the expected value of the lottery
is greater than the expected utility of the lottery, i.e., if,
U[PW1 + (1 – P) W2] ˃ PU(W1) + (1 − P)U(W2)
* A person is risk lover relative to a lottery if the utility of the expected value of the lottery is
smaller than the expected utility of the lottery, i.e., if,
U[PW1 + (1 – P) W2] < PU(W1) + (1 − P)U(W2)
22