# Theory of consumer behavior. Types of utility functions and a critical analyses of the theory of demand

## 54 Pages

Excerpt

The basis of the theory of consumer behavior rests upon the theory of demand. Theory of
demand begins with the `law of demand' which states that the quantity demanded of a
commodity varies inversely with its price,
ceteris paribus (keeping other factors unchanged,
e.g., taste, preference, income etc. of the consumer).
Now in the theory of consumer behavior, we assume the consumer is a rational individual who
wants to maximize her utility from a basket of consumable goods given her income unchanged.
There are two basic approaches in microeconomic theory for measuring the utility levels from
the commodity basket and they are: (1)
The cardinalist approach and (2) The ordinalist
approach.
THE CARDINAL UTILITY THEORY
Assumptions:
1. Rationality: The consumer is a rational individual who wants to maximize her total
utility subject to her budget constraint.
2. Cardinal Utility: The utility levels of commodities are CARDINALLY measurable
(in terms of money).
3. Diminishing marginal utility: The `law of diminishing marginal utility' states that
the total utility of a commodity falls beyond certain `saturation' point of consumption.
4. The total utility derived from a `basket of goods' is directly related to the different
quantities consumed. Formally, U
i
=
f ( x
i
, m
*, p
*
) (i = 1,2,...,n)
where U
i
= Utility derived from i-th commodity,
x
i
= Quantity of i-th commodity,
p*= Prices of other commodities (constant)
with U
i
/ x
i
> 0 and
2
U
i
/ x
i
2
< 0.
2

The
consumer's equilibrium condition is expressed as:
MU
1
/ p
1
= MU
2
/ p
2
= ... = MU
n
/ p
n
in a `n' ­ commodity world,
where MU
i
is the marginal utility of commodity i and p
i
is the price of commodity i.
(i = 1, 2, ..., n)
Consumer Preferences
Consumer preferences are guided by two axioms and defined by the preference relation `
' based on rationality and they are:
(i)
Completeness : For all x and y X, we have x y or y x (or both);
(By the notation x y, we mean that "x is at least good as y" or simply "x is
indifferent to y")
(ii)
Transitivity: For all x, y, and z X, if x y and y z, then x z.
In other words, if x is `weakly preferred' to y and y is `weakly preferred' to z,
then x is `weakly preferred' to z.
The strong axiom of revealed preference theorem states that for all x, y and z X,
if we have x > y and y > z, then x > z.
Continuity
For all x. y X, the sets {x: x y} and {x: x y} are
closed sets and {x: x y} and
{x: x y} are
open sets. In other words, in the superset X, the indifferent choices are
closed sets and the
strongly preferred choices are open sets.
3

Utility Functions
A function U: x is a utility function which represents preference relation
(indifferent choices between goods) if, for all x, y X, U (x) U (y) with x y.
In a similar fashion, f (u(x)) f (u(y))
if and only if U(x) U(y).
The major assumptions related to the preferences are:
Weak monotonicity: If x y then x y.
Strong monotonicity: If x y then x > y given that x y.
Strong monotonicity obviously conforms to strong axiom of revealed preference theorem
where the preference for x will always supersede the preference for z provided that none
of the goods in the subset is an inferior good or a cause of externalities in consumption,
e.g., pollution.
Local nonsatiation
Given any x, y and X and > 0, then the Euclidian distance between x and y,
that is, |x - y| < such that y > x.
In other words, in the Euclidian set X, the preferential difference between commodity
bundles x and y must be less than the utility level of , that is, there is a probability that
may lie in a higher subset in the Euclidian set x.
4

Convexity
Given any x, y and z X such that x z and y z, then it follows that
tx + (1-t) y z for all 0 t 1 (t = scalar quantity).
The concept of convexity may be explained in this manner ­ in a finite Euclidian set X, if
two commodity bundles x and y are at least as good as commodity z, then in the finite
budget space of a consumer, the sum of respective shares of income spent on two bundles
x and y would yield the same utility from commodity bundle z.
Strict Convexity
Given x, y and z x with x z, y z and x y, then tx + (1-t)y > z for all 0 < t < 1 .
In the case of strict convexity , within the finite Euclidian set X , if two commodity
bundles x and y are at least as good as commodity bundle z, then in the finite budget
space of a consumer, the sum of respective shares of income spent on two bundles x
and y would yield more utility from commodity bundle z.
Ordinal utility theory and indifference curve
The concept of utility function and the related assumptions discussed earlier jointly
form the framework of ordinal utility theory and the concept of indifference curve.
An
indifference curve is a locus of points within the finite budget space of a rational
consumer in a two-commodity world which yields constant utility.
The indifference curve is negatively sloped and convex to the origin. Convex
preferences by a consumer may exhibit `flat spots' on the curve but strictly convex
preferences exhibit a rotund shape.
5

To put it in a very simple way, the equation of the indifference curve is given by:
U
¯
= f (x, y) where U = U
¯
= Utility level (fixed).
By taking total differential, we have:
f
x
dx + f
y
dy = dU
¯
= 0
dy / dx|
IC
= - f
x
/ f
y
< 0 as f
x
, f
y
> 0.
In ordinal utility theory, a rational consumer may have a `two-fold' objective:
(i)
Either to maximize her utility function derived from two commodity bundles x
and y ( x, y X) subject to her budget constraint,
Or
(ii)
To minimize her budget (within the finite budget space) subject to her utility
constraint derived from two commodity bundles x and y (x, y X).
For the first case, the objective function of the consumer may be expressed as:
Maximize V = f (x, y) subject to m p
x
x
+ p
y
y
where m= Income of the consumer,
p
x
= Price of commodity bundle x,
p
y
= Price of commodity bundle y.
The required Lagrangian equation would be:
= f(x, y) + [m
¯
- p
x
x ­ p
y
y ]
( 0 ) ( = Lagrangian multiplier)
6

The first order conditions require:
/ y = f
x
­ p
x
= 0 ... (1)
/ y = f
y
­ p
y
= 0 ... (2)
/ = m
¯
- p
x
x ­ p
y
y = 0 ... (3)
From equations: (1) and (2), we have:
f
x
/ f
y
= p
x
/ p
y
MRS
x,y
= f
x
/ f
y
= p
x
/ p
y
... (4)
where MRS
x,y
= Marginal rate of substitution between bundles x and y.
Equation: (4) gives us the
consumer's equilibrium condition.
The second order condition for utility maximization requires the
bordered Hessian
determinant () to be positive, that is,
= f
xx
f
xy
-p
x
f
yx
f
yy
-p
y
> 0 ... (5)
- p
x
- p
y
0
By expanding equation: (5) we have:
2 f
xy
f
x
f
y
- f
xx
p
y
2
- f
yy
p
x
2
> 0 ... (6)
7

Finally by substituting the values of f
x
and f
y
from equation: (4) into equation: (3), we
get the
Ordinary or Marshallian demand functions as x (p
x
, m) and y (p
y
, m).
In the second case, the objective function of the consumer may be expressed as:
Minimize = p
x
x
+ p
y
y subject to U = f (x, y) = U
(fixed).
The required Lagrangian equation would be:
= p
x
x
+ p
y
y + [U
- f (x, y)]
The required first order conditions require:
/ x = p
x
­ f
x
= 0 ... (7)
/ y = p
y
­ f
y
= 0 ... (8)
/ = U
- f (x, y) = 0 ... (9)
Equations: (7) and (8) give us the similar equilibrium condition as shown in equation:
(4) as shown in equation: (4), i.e.,
MRS
x,y
= f
x
/ f
y
= p
x
/ p
y
.
However the demand functions derived for x and y would be somehow different from
Marshallian demand functions and they are known as
Compensated or Hicksian
demand functions.
8

Illustration:
Let us take a Cobb - Douglas utility function as:
U = x
y
1 -
( < 1)
The consumer's equilibrium condition requires:
MRS
x,y
= p
x
/ p
y
Thus MRS
x,y
= f
x
/ f
y
= x
­ 1
y
1 ­
/ (1 ­ ) x
y
­
= ( / 1 ­ ) * (y / x)
Hence ( / 1 ­ ) * (y / x) = p
x
/ p
y
gives us:
x = ( / 1 ­ ) (p
y
y / p
x
) and
y = ( / 1 ­ ) (p
x
x / p
y
).
By substituting the values of x and y in the budget equation, we get:
m
= p
x
x /
x
*
= Optimal
Ordinary or Marshallian demand function for x = m
/ p
x
,
and y
*
= Optimal
Ordinary or Marshallian demand function for y = (1 ­ ) m
/ p
y
.
Thus x
*
= x (p
x
, m) and y
*
= y (p
y
, m).
Similarly, if we try to derive the
Compensated or Hicksian demand function, we
get:
p
x
/ p
y
= ( / 1 ­ ) * (y / x)
p
x
x = ( / 1 ­ ) p
y
y and p
y
y = (1 ­ / ) p
x
x
9

Hence y = ( / 1 ­ ) * (p
x
x / p
y
)
By substituting the value of y in the utility constraint equation, we get:
U
= x
[( / 1 ­ ) * (p
x
x / p
y
)]
1 ­
U
= [( / 1 ­ ) * (p
x
/ p
y
)]
1 ­
x
x
*
= [(1 ­ ) ]
1 ­
(p
y
/ p
x
)
1 ­
U
= k (p
y
/ p
x
)
1 ­
U
where k = [(1 ­ ) ]
1 ­
= constant, and
x
*
= Optimal
Compensated or Hicksian demand function for x = x (p
x
, p
y
, U
).
Similarly, we derive:
y
*
= (1 ­ / )
(p
x
/ p
y
)
U
= k
(p
x
/ p
y
)
U
= k
(p
x
/ p
y
)
U
where k
= ( 1 ­ )
= constant, and
y
*
= Optimal
Compensated or Hicksian demand function for y = y (p
x
, p
y
, U
).
10

Good 2
Maximizes utility
Minimizes expenditure
Good 1
Figure 1
Corner solutions for indifference curve approach
Good 2
IC
Good 1
Figure 2 (a): When the consumer consumes good 1 only.
11

Good 2
IC
Good 1
Figure 2 (b): When the consumer consumes good 2 only.
Indirect utility
Properties of the indirect utility function:
(1)
If v (p, m) is the indirect utility function, then it is
nonincreasing in p, that is,
if p
> p, then v (p
, m) v (p, m). Similarly, v (p, m) is
nondecreasing in m;
(2)
v (p, m) is
homogeneous of degree zero in (p, m) space;
(3)
v (p, m) is
quasiconvex in p, that is, {p: v (p, m) k} is a convex set for all
values of k;
(4)
v (p, m) is
continuous at all p >> 0, m > 0 (monotonic increase in p).
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Details

Title
Theory of consumer behavior. Types of utility functions and a critical analyses of the theory of demand
Author
Year
2017
Pages
54
Catalog Number
V379198
ISBN (eBook)
9783668578647
ISBN (Book)
9783668578654
File size
707 KB
Language
English
Notes
Resubmitted the old manuscript after a small rectification.
Tags
theory, types
Quote paper
Debasish Roy (Author), 2017, Theory of consumer behavior. Types of utility functions and a critical analyses of the theory of demand, Munich, GRIN Verlag, https://www.grin.com/document/379198 