As the components of Marshallian model have
been laid out let us look at the model building
exercise, and we will derive the demand function
from this mathematical model .
So, first we will start with the net utility
function. Let us define net utility function
as v, and that is basically the benefit or
the utility which one derives out of the consumption
minus, there is a loss because as this person
is consuming q units of the quantity, p q
is the total expense that he incurreds he
or she incurreds on that good .
So, this amount of money is forgone to consume
the good q. Now this amount of money could
have been used somewhere else to buy some
other commodity maybe, then there is some
marginal utility of money that we need to
apply here in front of this expenses on that
good, they need to be multiplied. And together
this lambda p q you will talk about the loss
or cost of consumer . Now there is a benefit
portion here in front . So, this is the gain
or benefit and we needs to be maximized with
the decision variable .
So, the net utility function v has to be maximized
with respect to the decision variable q. Now
let us have a quick graph to see the situation
. So, here we have units of commodity consumed
q, and we are plotting the u q and lambda
p q both . So, of course, ah concave to origin
and cost curve we will take the shape of a
straight line , like this because lambda is
constant, p is constant.
So, it will be a straight line with slope
lambda p q . So, of course, these gap between
these 2 curves aiming to maximize the net
utility. And that could be done when the gap
between these 2 curves is maximum . So, basically
we are going to have a parallel straight line
with the same slope at this point, where it
makes the tangency to the total utility function
or u q we find the consumers equilibrium .
Because it is add that point say denoted by
point A , we find a q star such that the utility
sorry the net utility v is maximize. Now we
can represent the consumers equilibrium A
in a different manner as well . So, these
alternative graphical representation we will
be in terms of the marginal utility function.
So, in the upper panel we have already plotted
total utility function, and we have seen the
consumers equilibrium denoted by the consumption
level q star. Now let us going to have an
alternative graphical representation of consumer
equilibrium under Marshallian assumptions
through marginal utility curve . So, for that
let us now look at panel b diagram. So, let
us call this panel b in the above one is the
previous 1 is panel A .
So now in the case of panel b diagram we have
plotted a downward slopping marginal utility
curve. This time we have drawn it as convex
it does not matter, it may have been a straight
line also. Now these horizontal line horizontal
to q axis is basically the constant that we
have used in the cost line. So, that is the
slope of that coastline lambda p here we are
going to plot that. So, this is a fixed number
of course, because you know we are talking
about a particular ah straight line from origins.
So, of course, in only one slop possible.
So, where this lambda p ah horizontal straight
line intersects the marginal utility curve,
that is where we get our optimal consumption
q star for the consumer. Why? Because we have
already seen that at the consumer equilibrium
q start as we observed in panel A. At that
point the slope of the total utility function
which is the marginal utility function equals
the slope of the coastline which is lambda
p; so, this is an alternative representation
.
Now with this let us also look at how one
can derive 
demand function in Marshalls model . Now 
we will start with the mathematical first
order condition from this optimization exercise.
So, we have already spoken about the ah equality
of the slope of the total utility function
and the loss function. So, that means, we
are talking about this first order condition
using calculus , right .
So, the negative slope of demand function
is derived from the second order condition.
So, for the second order condition what do?
We need to do we need to differentiate again.
So, here we differentiate then we can rewrite
or we can also inform from this this particular
expression dp dq equals q double prime U divided
by lambda. And note that that this is all
at the optimal level of consumption q star.
So, you have to put a q star here, and a q
star there, and a q star here . So now, note
this assumption law diminishing marginal utility
gives the sign of this particular second order
derivative, and the sign would be negative
.
Because the margin utility is a downward sloping
function . Whereas, the lambda is a non-negative
number. So, together with we can say that
the slope dp dq has to be negative. Now what
is this entity? This is the slope of the demand
function , right? So now, we have established
that demand function has to be a downward
sloping curve. Now we are not commenting on
the curvature whether it straight line or
it is convex or concave it has to be a downward
slopping curve .
Now, from the ah Marshallian analysis can
we derive the demand function. Now note that
from the first order condition that we have
earlier we can write ; so, from the first
order condition we can write p equal to marginal
utility divided by lambda. Now of course,
marginal utility is a downward slopping curve
, it can be even a straight line lambda is
a constant number non negative. So, of course,
we can draw something like this a curve marginal
utility divided by lambda. And hence in this
diagram now we can change the value of p.
Suppose we start with the high value of p
and the corresponding level of demand is this
much 
q 1.
So, if we now lower the price of the commodity
to p 2 , say p 2 here, we can get another
point along this curve which will give me
the consumers equilibrium. At a lower level
of price, we see we obtain another point b
as the consumers equilibrium on the m u over
lambda curve. Let me denote the first position
of equilibrium as point B. So, as per point
B we can see that as price as fallen the consumer
has demanded for a higher level of q q 2 start
so that he or she maximizes is are utility
or net utility .
So, Marshall assumes lambda equal to 1. So,
if that is the case, then the marginal utility
curve becomes demand function . 
Now note that this is a very interesting result.
So, what we see? So, in a single good Marshallian
model the consumer as to spend the entire
money on that particular commodity only. So,
the p times q has to always equal to m the
money income. So, what we can see here is
that this pq combinations along that the convicts
origin marginal utility curve will always
give a fixed level of ex penditure or money
expenditure. That is why in a Marshallian
word we can say that the demand function is
a rectangular hyperbolum. So, with this we
have concluded our discussion on one good
Marshallian ah theory of consumer behavior.
Now we are going to discuss other things in
the next lecture .
