CHAPTER10:
The Theory of the
Firm
Section 1 : Introduction
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(1) The Break-Even Model
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Associated with Accountancy
To find the level of output where profits are zero
No clear company objective in the break-even
model
The break-even model assumes a constant
variable cost per unit irrespective of the number of
units that are made.
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(2 ) The Linear Programming Model (LP)
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associated with Management Sciences
to have a very specific objective - for example ,
to maximise profits
decision variables -- how many units of each
product should be made in order to achieve the
stated objective.
LP technique then finds the optimal values of
the relevant decision variables.
Section 2 : The Economist's Model
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Assumption
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company is seen as producing one product only
company is to MAXIMISE PROFITS
decision variable-- the level of product output
To find the level of output , q ( the DECISION
VARIABLE ) , that maximises profit ( the
OBJECTIVE FUNCTION ), within the
particular confines of the revenue and cost
conditions which it faces.
[A] Revenue Considerations
 (a) demand curve for its own single
product
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p = 500-2q
How wide should the quantity (horizontal) axis be ?
Should we have a range of q from 0-10 ? or q from
0-50 ? or q from 0 - 100 ? or q from 0 - 700 ? or
WHATEVER ?
0 <— q —> 250
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(b) The Total Revenue Curve
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TR = price x quantity = pq
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(i) TR starts at 0 when q equals 0 ;
(ii) TR initially increases as q increases , but then
decreases as q increases
(iii) TR equals 0 again , when q equals 250.
TR curve is not a straight line (nolinear
equation)
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TR = p x q = ( 500 - 2q ) q
= 500q - 2q2
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QUADRATIC equation
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TR information is available from both types of
diagrams but it comes in different ways :
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for a demand curve diagram , the TR associated
with any particular output level is given by the area
of the appropriate rectangle ;
for a TR type diagram , the TR value for any output
level is given simply by the height of the curve at that
output level.
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(c) The Marginal Revenue Curve
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MR = TR /  q
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TR curve is rising ---- MR is positive
TR curve is falling ----MR is negative
MR curve
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for a straight line demand curve , the associated
MR curve has the same intercept but slopes
down twice as quickly as the linear demand
curve.
MR = 500 - 4q
MR curve is the first order derivative of the TR curve ,
that is :
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MR = d(TR)/dq
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Relationship between MR and TR
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(i)when TR is rising , MR is positive,
(ii) when TR is falling , MR is negative,
(iii) when TR reaches its maximum value MR is zero.
(iv) MR is zero and TR reaches its maximum value
when q = 125
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(d) The Average Revenue Curve
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AR = TR/q
AR = TR/q = pq/q = p
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AR is just the same as p
price equation /demand curve /AR curve are
completely interchangeable terminology
[B] Cost Considerations
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(a) The Total Cost Curve
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The total cost (TC) structure for the company is assumed to
be given by :
 TC =q3 /3 - 27q2 + 801q + 1000 (CUBIC equation )
TC curve has distinct phases :
 (1) an initial stage where total costs rise with output.
 (2) an intermediate stage where total costs are
relatively flat such that total costs do not rise by very
much when output is
expanded.
 (3)a final stage where total costs rise rapidly as output
gets even bigger.
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The first phase can be seen as inefficiency
stemming from the under-utilisation of resources.
The second phase can be seen as the company
using its productive resource in an efficient manner.
The third stage can be interpreted as resources
being over-stretched and being asked to work at
levels which are bigger than their accepted
productive potential.
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(b) The Average Cost Curve
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Average costs (AC) have been defined as
follows :
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AC = TC/q
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AC for any specific output level is given by the slope
of the straight line ray from the origin to the point on
the TC curve that is vertically above the output level
of interest.
Some observations on AC curve
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(i) AC is always POSITIVE.
(ii) As output INCREASE , AC FALLS ;
At some point AC reaches a MINIMUM value
As quantity INCREASES , AC RISES.
AC curve is a U-shaped curve
For any point on the AC curve , TC is given by the
area of the associated rectangle.
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(c) The Marginal Cost Curve
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MC=  TC/  q
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based upon differential calculus as follows :
MC = q2 - 54q + 801
To derive the MC curve we proceed as follows :
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(i)if any component does not depend upon q , then this
component is dropped completely and plays no role in MC.
if a component does depend upon q , then to find the role
that it plays in the MC curve, we perform the following
manipulations
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(a) to find the new coefficient, multiply the old coefficient by
the old power of q
(b) to find the new power of q , subtract one from the old
power of q.
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Applying this rule to the individual components of our
TC curve we get:
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(1) q3/3 , becomes
3 q2/3 = q2
(2) -27q2, becomes -(2)(27)q' =-54q
(3) 801q , becomes (l)(801)q° = 801
MC = q2 - 54q + 801
The typical shape for the MC curve is shown by our
specific example in Diagram 10.6.
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[C] Profit Considerations
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Profit = TR - TC
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observations can be made from the diagram:
 (1)at q = 0 , Profits are a minus figure in that they are
negative FC. In our example , Profits are -1000 when
no
output is being produced. This result is identical to
what we saw in the break-even model. Thus the
Profit curve must go through the point ( 0 , -1000 ).
 (2) At q = ql , we can see that TR = TC. Thus Profits
are 0 at this quantity which is then a BREAK-EVEN
level of output. It follows that the Profit curve must
pass through the point ( ql , 0 ).
 (3) At q = q2 , we can see that TR = TC. So again
this means that q2 must be a break-even output
level. Hence the Profit curve must go through the
point (q2 ,0 ).
 Thus we see that the economist's model of company
behaviour has two Break-Even points.
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(4) Now consider output levels that lie between q1
and q2 ,that is, q1 <— q —> q2. We notice that in
this range TR > TC, hence Profits must be
POSITIVE.
(5)At output levels beyond q2 we note that TR < TC
and hence Profits are negative. Moreover as q
increases the company's losses become larger and
larger.
(6) At output levels between 0 and q1 , TR < TC and
hence Profits are negative in this range.
Profit curve
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Profit
=TR-TC
= (500q - 2q2) - (q3/3 - 27q2 + 801q + 1000)
= -q3 /3+ 25q2 - 301q - 1000
AR/MR/AC/MC on the same graph.
Section 3 : The Conceptual Paper
Worksheet (CPW)
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The CPW for this problem:
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q --company's decision variable
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q , p , TR , TC , Profit, AC , MR , MC
quantity range -- 0 ~ 250.
a worksheet with 252 rows
The symbol ' ^ ' is used in order to raise some
quantity to some power
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Diagrams can be created by using the
standard spreadsheet GRAPHICS facilities.
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(1)When drawing Diagram 10.8 , we need a Yscale that is just big enough to catch the top of
the TR curve , and small enough to catch the
local minimum in the Profit curve.
(2)When drawing Diagram 10.9 , we need a Yscale that starts at 0 and is just big enough to
catch the top of the AR curve
DATA/SORT facility in Excel
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in ascending order
in descending order
Section 4 : The Answer
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Using DATA/SORT facility, the spreadsheet is
re-ordered according to the profit maximising
criterion as follows:
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Answer to the profit maximising problem
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(a) the company should produce 43 units of output.
(b) the company should charge a selling price of £414
per unit.
(c) the company earns a total revenue of £17802 (£414)(43).
(d) the company is faced with total costs of £12022.
(e) the company earns Profits of £5780 = (£17802)(£12022).
(f) average cost per unit at 43 units of output is £280.
(g) marginal revenue at 43 units is £328.
(h) marginal costs at 43 units is £328.
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Profit curve--the highest point on the Profit
curve.
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move vertically downwards until we meet the
quantity axis. At this point q = 43 , the optimal
production level.
move horizontally across from the top of the Profit
curve until we hit the vertical axis, Profit = £5780 at
q=43
TC curve
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move vertically upwards from q=43 until we meet the
TC curve
move horizontally across to the vertical axis
we find that TC=£12022 , at q=43.
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TR curve
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move vertically upward from q=43 until we
meet the TR curve
move horizontally across to the vertical axis
we find that TR=£17802 , at q=43
The difference between TR and TC at q=43 is
£5780.
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AR curve
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move vertically upwards until we meet the
demand curve
move horizontally across until we meet the
vertical axis.
This gives the answer of a selling price per
unit , or AR , of £414 at q=43.
In terms of TR=pq , TR can be seen as the
area of the rectangle defined by the point
(43,414) and the origin.
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AC curve
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move vertically upwards until we meet the AC
curve
move horizontally across until we meet the
vertical axis.
This gives an AC/unit of 280.
In terms of TC=ACxq , TC value can be seen
as the area of the rectangle defined by the
point (43,280) and the origin.
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Profit = TR – TC
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The area of the rectangle by ( 43 , AR-AC ).
MR curve and MC curve
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MR=MC=328 at q=43
At the profit maximising output level, MR = MC.
General result:
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if a company is considering making an extra unit of output
then , in terms of profitability , it must compare the extra
revenue (MR) from making the unit with the extra cost
(MC) of making the unit.
If MR > MC then the additional unit is profitable and it
makes sense to produce it.
If MR < MC then the additional unit costs more to make
than it contributes to revenue - that is , this unit makes a
loss - and it is sensible not to make it.
In terms of finding the optimal production level the
company should continue to make extra units up to the
point where MR = MC.
If the company stops short of this level of output it is
foregoing profitable production units ;
if the company goes beyond this production level it is
making unprofitable units.
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In standard economics textbooks, the rule
MR=MC is seen as a pre-condition for finding
the optimal quantity
Two points satisfy MR=MC :
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at q=7, MR = MC = 472, Profit= £-1996 , Local
minimum point
at q=43, MR = MC = 472, Profit = £5780 , Maximum
point
MR=MC is just a necessary condition and not
sufficient condition for finding the maximum
profit.