3.3 Marginal functions

(Section 4.3 of Jacques)

· Calculate marginal revenue and marginal
cost

· Calculate marginal product of labour

· Study the law of diminishing returns

· Partial derivatives

3.3.1 Revenue and cost

Recall:

Total
revenue is

*TR = P**´ Q*

where *P* is the price of a good and *Q* is quantity demanded

Given a
demand function, e.g.

*P = 100 - 2Q*

we can write
*TR* in terms of *Q*:

*TR = PQ
= 100Q - 2Q ^{2}.*

* *

In order to
study how *TR *reacts to a small change in *Q*, we look at the derivative of *TR* with respect to *Q*.

DEFINITION:

Marginal
revenue *MR *is the derivative of total revenue
with respect to demand. That is,

_{}.

Example:

In the
example above, we had

*TR = 100Q - 2Q ^{2}.*

Differentiating
gives

*MR = 100 - 4Q.*

This formula
allows us to calculate *MR *at any value of *Q*.

NON-CALCULUS
DEFINITION

In some
books, *MR *is defined as

* *

the
change in the value of *TR* brought about by a one
unit increase in the value of demand *Q*.

This
definition approximates the slope of the graph of *TR
*(i.e. the tangent slope) by the slope of a nearby line which cuts the curve at two
points. However this approximation is not always accurate.

The
definition of differentiation tells us that

_{}.

So if D*Q* is small, then

_{}. (*)

In
particular, if D*Q*
= 1, this gives the non-calculus version of *MR:*

_{}.

If we read
equation (*) from right to left, we see that we can use *MR *to *predict*
the change in *TR *brought about by a small change
in demand *Q*:

_{}

or

change in total revenue ~ marginal revenue X change in demand

Example

Given the
demand function *P = 60 - Q, *calculate *TR* and then

(1)
find *MR *in terms of *Q*, and evaluate *MR *at *Q = 50.*

(2)
Calculate *TR* at *Q =
50 *and at *Q = 51* and hence confirm that the
one-unit-increase approach gives a reasonable approximation to the exact value of *MR.*

Solution: We
have

*TR = PQ = (60 - Q)Q = 60Q - Q ^{2}.*

(1)
Differentiating
this expression gives

_{}.

When *Q = 50, *this gives

*MR = 60 - 2(50) = -40.*

* *

(2)
When *Q* = 50, *TR
= 60(50) - (50) ^{2} = 500.*

When *Q = 51, TR = 60(51) - (51) ^{2} = 459.*

Then using
the approximation formula

_{},

we get

*MR **~ 459 - 500 = -41,*

which is a
good approximation to the actual value of -40.* *

Example

The total
revenue of a good is

*TR = 100Q - Q ^{2}.*

Write down
an expression for marginal revenue.

If the
current demand is *Q=60, *estimate the change in *TR* brought about by a 2 unit increase in *Q*.

Solution

Differentiating
gives

*MR = 100 - 2Q.*

At *Q = 60, *this gives

*MR = -20*

Now we use
the formula

_{}

This
predicts that the change in *TR *will be

*D*(*TR*)
~ -20 ´ 2 = -40.

A two-unit
increase in demand leads to a 40 unit decrease in total revenue.

Exercise

The total
revenue function for a good is given by

*TR = 1000Q - 4Q ^{2}.*

Write down
an expression for the marginal revenue.

If demand is
currently 30, estimate the change in *TR* due to
a

(1)
three-unit increase
in *Q*

(2)
two-unit decrease
in *Q*

What we have
done here for total revenue can be done for any economic function. For example, given the
total cost function *TC*, we define the **marginal cost ***MC by*

_{}

marginal cost is the derivative of
total cost

with respect to output *Q*

Just as we
did for revenue, we can derive the approximation

change in total cost ~ marginal cost X change in output

Exercise

Find the
marginal cost given the average cost function

_{}.

If the
current output is 15, estimate the effect on *TC *of
a three-unit decrease in *Q*.

(Remember
that _{}.)

3.3.2 Production

We studied
production functions of the form

*Q = f(K,L),*

i.e. the
production function *f* determines output *Q* in terms of labour *L* and capital *K*.

Here, we
will consider the case where *K* does not change,
so that *Q* depends only on *L:*

*Q = f(L).*

* *

Then the
derivative of *Q* with respect to *L* tells us how *Q* reacts to changes in *L*. This derivative is called **marginal product of labour:**

_{}.

marginal product of labour is the
derivative of output

with respect to labour.

As usual,

change in output ~ marginal product of labour X
change in labour

Example

If the
production function is

*Q = 300**ÖL - 4L,*

calculate
the value of *MP _{L} *when

(a)
*L = 1*

(b)
*L = 9*

(c)
*L = 100*

(d)
*L = 2500.*

Comment on
the results.

Solution

We have

*Q = 300 L ^{1/2} - 4L*

so

_{}

(a)
When *L = 1*,

_{}

(b)
When *L = 9*,

_{}

(c)
When *L = 100*,

_{}

(d)
When *L = 2500*,

_{}

Thus we see
that *MP _{L} *is decreasing, and
eventually becomes negative.

Thus the **size of the increase** in output brought about by a
one-unit increase in labour *L *is **decreasing**.

Worse still,
at the stage when *MP _{L }< 0*, an
increase in labour

This is what
we would expect in the real world; problems such as overcrowding on the workshop floor and
administrative overheads decrease the efficacy of increasing the size of the workforce.

This example
illustrates the **law of diminishing marginal
productivity** (also known as the **law of
diminishing returns).**

** **

**"The increase in output due to a
one-unit increase in labour will eventually decline"**

Translating
this into maths, we obtain

the marginal product of labour *MP _{L }*will eventually be a decreasing
function.

That is, for
sufficiently large values of *L*, we should have

_{}

Since we
know that *MP _{L }*is the derivative of
output

_{}

Summary:

To show that
a production function satisfies the law of diminishing returns, we must show that

_{}

holds for
all sufficiently large values of *L.*

Example

Show that
the Cobb-Douglas production function

*Q = 5L ^{1/2}K^{1/2},*

with *K = 100*, satisfies the law of diminishing returns.

3.3.3
Partial Derivatives

(cf.
Jacques, Section 5.2)

*Q = 5L ^{1/2}K^{1/2}.*

This
depends on both *L *and *K*.

If we take
*K* to have a
fixed value and allow *L *to vary, we
can analyse changes in *Q *by looking
at the derivative *dQ/dL*.

Similarly,
If we take *L* to have a
fixed value and allow *K *to vary, we
can analyse changes in *Q *by looking
at the derivative *dQ/dK*.

To
emphasise the fact that *Q* depends
on both variables, we call these derivatives **partial derivatives **and use a
different notation:

_{}=
derivative of *Q* with
respect to *L, *holding *K *constant.

_{}=
derivative of *Q* with
respect to *K, *holding *L *constant.

Example:

(i)
Calculate _{} and _{} for the Cobb-Douglas function

*Q = 5L ^{2}K^{3/2}.*

(ii) If
the current values are *L=10, K = 12*, which
would bring about a bigger increase in *Q*:
increasing *L *by one
unit, or increasing *K *by one
unit?

3.3.4Consumption
and Savings

Y =
national income, C = consumption, S = savings connected
by Y= C + S

To
analyse changes in C and S due to changes in Y use **marginal
propensity to consume **MPC and **marginal
propensity to save** MPS.

_{}

These
agree with the special case of linear consumption and savings curves given earlier.

We
can find MPC given MPS (or *vice versa*): we know
that

*Y =
C + S*

Ddifferentiate
both sides with respect to *Y:*

*1 =
MPC + MPS*

Example: Given

*C =
0.01Y ^{2}+0.2Y + 50*

Find
MPC and MPS when Y=30.

_{}

When
Y=30,

*MPC
= 0.02(30) + 0.2 = 0.8*

To
find *MPS: MPC + MPS = 1 =>*

*MPS
+ 0.8 = 1 => MPS = 0.2*

3.4
Elasticity

**Elasticity**
deals with the sensitivity of demand (supply) to changes in price. Remember that *TR = P**´Q *and
in our models, P decreases => Q increases.

Change
in TR depends on **percentage** changes in P and
Q.

Demand
is said to be **elastic**, if the percentage change in Q is higher
than the percentage change in P.

Then
we can increase revenue by **lowering** price.

Demand
is said to be **inelastic**, if the percentage
change in Q is lower than the percentage change in P.

Then
we can increase revenue by **raising** price.

E =
- __ (% change in demand)
__

(%
change in price)

Minus
sign is a convention to ensure E > 0.

Demand
is

· **inelastic
**if
E < 1

· **unit
elastic **if
E = 1

· **elastic
**if
E > 1**
**

Change
in P = DP,
change in Q = DQ

%
change in P = 100 (DP/P)

%
change in Q = 100 (DQ/Q)

__ __

_{}

Elasticity
at P:

Let
the arc PQ shrink down to P by moving Q closer to P

Thus
DP
and DQ
shrink to zero and

_{}

Price
elasticity of demand at a point is given by

_{}

Demand
function is usually P = *f* (Q);

_{}

To
find _{}, we use

_{}.

Example:
Given the demand function

*P =
- Q ^{2} - 4Q + 96,*

(i)
Find price elasticity of demand when
P = 51.

(ii)
If this price rises by 2 %, find the corresponding percentage
rise in demand.

First,
find Q corresponding to P = 51

- Q^{2}
- 4Q + 96 = 51

roots:
Q = - 9 (reject), Q = 5.

Next,
we calculate _{}.

_{} = * -2Q - 4* => _{}

Q =
5 gives _{} = -1/14

Elasticity:
_{}

(ii)
recall that:

_{}

We
found E = 0.73, we are told (% change in P) =
2. Thus

_{}

=>
% change in Q = - 1.46 %

Q
decreases by 1.46 %

** **

_{}

**or**

_{}