1.
Utility Function* *(MKM
C20/449-50:
420-1 & C21/472: 440)

i - Assumptions

The atom-like unit, the
utile, is a measure of the pleasure or pain enjoyed by a consumer in
consuming a good or service. The utile exhibits distinct characteristics. First, and for
Economics most important, the willingness of a person to pay a money
price for a good or service measures how many utiles of happiness
they expect to receive in exchange. This is *reification* –
making concrete (money) something that is abstract (happiness).
Furthermore, it is assumed that goods are infinitely divisible.

Second,
total utility is the satisfaction of consuming a total quantity of a
good or service.

Third,
marginal utility is the additional utility yielded by consuming one
more unit of that good or service.

Fourth,
diminishing marginal utility means that at some level of consumption an
additional unit yields less satisfaction than the preceding unit, *
i.e.* total utility increases but eventually at a decreasing
rate. Furthermore, diminishing marginal utility eventually turns
negative becoming pain not pleasure, too much of a good thing;

Fifth,
diminishing marginal utility means a person does not consume just one
good. One does not live by bread alone. Assuming rationality, a
person chooses that combination of two or more goods and services
that maximizes total utility. This is calculatory rationality
meaning every choice is a calculation of the number of utiles, *a.k.a.,* happiness or pleasure.

Furthermore, in theory, the
consumer is assumed rational, *i.e.,* one chooses between
alternative commodity combinations to maximize utility assuming:

a)
perfect knowledge, that
is, the consumer is aware of all alternative commodity combinations,
their prices and resulting utility;

b) competence, that is, a
consumer is capable of evaluating the alternatives; and,

c) taste of a consumer is
transitive or consistent, that is, if a consumer likes A as much as
B and B as much as C then one likes C as much as A.

It is also assumed that the
consumer is only able to order commodity combinations by level of
utility, 1st, 2nd, 3rd *etc*. This is called ordinal measurement or
rank ordering. Thus in consumption one does not specify the actual
numeric level or utility known as cardinal measurement.

Putting all these
definitions and assumptions together we generate the consumption
function as:

(1)
U = *f *(x, y) where:

U is the utility derived
from consuming combinations of x and y;

*
f *
is the unique taste function
of a consumer;

U is continuous meaning
there are infinite combinations yielding the same level of utility.
Put another way, U is a dense set;

a number assigned to
commodity combinations such as U_{5} indicates only that it is
preferable to combinations with a lower number, *e.g.,* U_{4} and
inferior to U_{6}. In other words, we can rank order
preferences but a U_{# }has no cardinal meaning; and,

U is defined for a specified
timeframe that is long enough to allow substitution between
commodity combinations but short enough to insure constancy of taste

*
ii -
Indifference Curves *
(MKM C20/465-70:
435-39)

For any level of utility say
U^{1} =* f* (x, y) there is a locus of commodity combinations
which graphically form an indifference curve, first displayed in
Edgeworth's *Mathematical Psychics*. All combinations on
that curve have the same level of utility meaning the consumer is
‘indifferent’ to any point on the curve. The indifference curve is
also called a preference or utility curve.

Usually an indifference
curve is ‘convex’ in shape reflecting that an increase in x can only
be obtained by a reduction in y, and *vice versa*
(B&P
7th Ed. Fig. 9.3;
MKM Fig. 21.2).
The amount of *y* that must be given up to get more *x* while
maintaining the same level of utility is the slope (rise over run)
of the curve at that point. It is called the *marginal rate
of substitution*, *i.e.,*

(2)
MRS = MUx/MUy where:

MRS = marginal rate of
substitution

MUy = marginal utility of y

MUx = marginal utility of x
(P&B
7th Ed Fig. 9.4;
MKM Fig. 21.2).

As noted above ‘*f*’ in
the equation U = *f* (x, y) is the taste function which is
different for each consumer. Thus each consumer will have a
uniquely shaped indifference curve and different MRS. When we plot
all possible levels of U we get a set of curves forming a consumer’s
*indifference map*. The transitivity assumption ensures the
curves do not intersect but rather rise higher and higher.

2.
Budget Constraint
(MKM
464-65: 433-35)

Before considering the
Budget Constraint it is appropriate to consider the nature of goods
& services purchased by a consumer and how they are able to pay their prices.
Commodities are called ‘goods’ because they satisfy human want,
needs and desires.

i -
Goods (MKM
C21/470 & 473)

Goods can be classified in
different ways. __First__, there are complementary and
substitute goods. Complementary goods are ones that are consumed
together, *e.g.,* hamburgers and French fries or IPods and IPod
docking stations (MKM
Fig. 21.5b). Substitute goods are alternatives to one another,
*e.g*., a bicycle is a substitute for a car in transportation.
There are near and distant substitutes for most goods, *e.g.,*
a car or a truck are near substitutes while a car and a bicycle is a
distant substitute (MKM
Fig. 21.5a).

__Second__,
there are normal and inferior goods. A normal good is one the
consumption of which increases as income increases. An inferior
good is one the consumption of which decreases as income increases.
An example of an inferior good is cheap wine. As one’s income
increases consumption of cheap wine tends to decline.

__Third__,
there are conspicuous consumption or Veblen goods (named after
economist
Thorstein Veblen) and normal consumption goods. Veblen
goods are rare and their consumption goes up as price goes up in
distinction from normal goods whose consumption goes down as price
goes up. Conspicuous consumption goods are bought to demonstrate to
others that one can afford such luxuries.

ii -
Prices

To buy a good a consumer must pay its price. Further to Bentham’s
assumption, the money price one is willing to pay for a good or
service equals the satisfaction or utility one believes can be
extracted from that good.

iii -
Income/Work

To pay a price, however, one must have income. Income is earned
through work which in the standard model is disutility*, i.e*.,
pain. One does not work for enjoyment (if one does one earns
‘psychic’ income) but rather for the monetary income used to buy
goods and thereby derived satisfaction.

iv -
Constrained Maximization
(MKM C21/470 & 472:
439-41)

While a consumer wants to
rise as high up the indifference map as possible, one is constrained
by income (I) and the price of x and y. Thus for a given level of
income and prices a budget line or constraint can be drawn. This
constraint shows all combinations that can be purchased that exhaust
income*, i.e., *

(3)
I = PxX + PyY where:

I = income

P = prices

X & Y =
quantity of goods
(P&B
7th Ed Fig. 9.1;
MKM Fig. 21.6).

One cannot consume above the
constraint and, in this model, it is irrational to consume below
(keeping cash on hand) because utility is derived *only* from
consuming goods & services.

The maximum amount of
*x* or *y*
one can afford (with a given income and prices) is shown by the
intercepts of the budget line and respective axes,* i.e.*, I/Px and I/Py

(4)
Price Ratio = Px/Py

The slope of
the Budget Constraint is
∆Y/∆X.
To get the price ratio, however, an example will serve. Let us
assume I = $10, Px = $2 and Py =$5. The maximum X a consumer
can buy is 5 units; the maximum Y is 2 units. The slope of the
Budget Constraint is ∆Y = (2-0) and ∆X = (5-0) or 2/5. Notice
that this is equivalent to Px/Py = 2/5. Given the slope of the
budget constraint is downward it is negative slope and the price
ratio is
expressed as (Px/Py).

If income varies while
prices remain fixed then a new higher budget line becomes available
to the consumer parallel to the original. In other words higher
income relaxes the constraint on one’s happiness. If, on the other
hand, the price of X (or Y) decreases the slope of the budget line
and therefore the price ratio changes. For example if Px goes
down then the intercept which measures
the maximum amount of X a consumer can afford increases even if income
remains constant.

3. Equilibrium
(MKM C21/470-2:
439-41)

The combination of
*x* and *y*
that maximizes a consumer’s utility is the one on the budget line
tangent or just touching the highest attainable indifference curve
.
This is the ‘best affordable point’ (P&B
7th Ed Fig. 9.6;
MKM Fig. 21.6)
and satisfies the following conditions:

(5)
MRS = MUx/MUy = (Px/Py)

that is the slope of the
indifference curve or Marginal Rate of Substitution equals the slope
of the Budget Line or the inverse of the price ratio (Px/Py) and at
this point the ‘rationale’ consumer equates the MU per dollar of
each commodity consumed or

(6)
MUx/Px = MUy/Py where

dollar-for dollar the
additional utility from an additional unit of *x* is equal
dollar-for-dollar to the additional satisfaction from one more unit
of *y*.

Consumers will remain at
this point, *i.e.,* be in equilibrium, as long as taste, income
and prices remain fixed. This is the initial equilibrium. It is a
condition which once achieved continues indefinitely unless one of
the variables is altered.
For our purposes there are two types:

a) stable equilibrium: which
refers to a condition which once achieved continues indefinitely
unless there is a change in some underlying conditions. Changes in
economic conditions will be followed by reestablishment of the
original equilibrium. Example: a ball resting at the bottom of a
cup; shake it and the ball moves; stop shaking and it returns to the
bottom of the cup; and,

b) unstable equilibrium:
which refers to a condition which once achieved will continue
indefinitely unless one of the variables changes but the system will
not return to the original equilibrium. Example: a ball resting on
the top of an overturned cup - shake it and the ball falls off never
to return to the same place.

**
***
Summary of Demand*

(1)
U = *f *(x, y)
Utility Function

(2)
MRS = MUx/MUy
Marginal Rate of Substitution

(3)
I = PxX + PyY
Budget Constraint

(4)
Px/Py
Price Ratio

(5)
MRS = MUx/MUy
= Px/Py Consumer Equilibrium

(6)
MUx/Px = MUy/Py
Equilibrium Condition

We will now change
assumptions one by one and see what happens to equilibrium.