CHAPTER10:The Theory of the Firm

Section 1 : Introduction

(1) The Break-Even Model 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.

(2 ) The Linear Programming Model (LP) associated with Management Sciences to have a very specific objective - for example ,

to maximise profits decision variables -- how many units of each pr

oduct 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 Assumption

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 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

(b) The Total Revenue Curve TR = price x quantity = pq

(i) TR starts at 0 when q equals 0 ; (ii) TR initially increases as q increases , but then de

creases as q increases (iii) TR equals 0 again , when q equals 250.

TR curve is not a straight line (nolinear equation) TR = p x q = ( 500 - 2q ) q

= 500q - 2q2

QUADRATIC equation

TR information is available from both types of diagrams but it comes in different ways : 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.

(c) The Marginal Revenue Curve MR = TR / q

TR curve is rising ---- MR is positive TR curve is falling ----MR is negative

MR curve 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 : MR = d(TR)/dq

Relationship between MR and TR (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

(d) The Average Revenue Curve AR = TR/q AR = TR/q = pq/q = p

AR is just the same as p price equation /demand curve /AR curve are comple

tely interchangeable terminology

[B] Cost Considerations

(a) The Total Cost Curve 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 isexpanded.

(3)a final stage where total costs rise rapidly as output gets even bigger.

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.

(b) The Average Cost Curve Average costs (AC) have been defined as

follows : AC = TC/q

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 (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.

(c) The Marginal Cost Curve MC= TC/ q

based upon differential calculus as follows :

MC = q2 - 54q + 801 To derive the MC curve we proceed as follows :

(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 (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.

Applying this rule to the individual components of our TC curve we get: (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.

[C] Profit Considerations Profit = TR - TC

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 nooutput 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.

(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 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) The CPW for this problem:

q , p , TR , TC , Profit, AC , MR , MC q --company's decision variable

quantity range -- 0 ~ 250. a worksheet with 252 rows

The symbol ' ^ ' is used in order to raise some quantity to some power

Diagrams can be created by using the standard spreadsheet GRAPHICS facilities. (1)When drawing Diagram 10.8 , we need a Y-

scale 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 Y-scale that starts at 0 and is just big enough to catch the top of the AR curve

DATA/SORT facility in Excel in ascending order in descending order

Section 4 : The Answer

Using DATA/SORT facility, the spreadsheet is re-ordered according to the profit maximising criterion as follows:

Answer to the profit maximising problem (a) the company should produce 43 units of output. (b) the company should charge a selling price of £414 p

er unit. (c) the company earns a total revenue of £17802 - (£41

4)(43). (d) the company is faced with total costs of £12022. (e) the company earns Profits of £5780 = (£17802)-(£12

022). (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.

Profit curve--the highest point on the Profit curve. 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 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.

TR curve 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.

AR curve move vertically upwards until we meet the dem

and curve move horizontally across until we meet the ver

tical 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 ar

ea of the rectangle defined by the point (43,414) and the origin.

AC curve move vertically upwards until we meet the AC

curve move horizontally across until we meet the vert

ical axis. This gives an AC/unit of 280. In terms of TC=ACxq , TC value can be seen a

s the area of the rectangle defined by the point (43,280) and the origin.

Profit = TR – TC The area of the rectangle by ( 43 , AR-AC ).

MR curve and MC curve MR=MC=328 at q=43 At the profit maximising output level, MR = MC. General result:

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.

In standard economics textbooks, the rule MR=MC is seen as a pre-condition for finding the optimal quantity

Two points satisfy MR=MC : 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.