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).

12

Excerpt out of 54 pages

- 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

Publish now - it's free

Comments