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CHAPTER10: The Theory of the Firm

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

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CHAPTER10: The Theory of the Firm

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  1. CHAPTER10:The Theory of the Firm

  2. 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.

  3. (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 product should be made in order to achieve the stated objective. • LP technique then finds the optimal values of the relevant decision variables.

  4. 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.

  5. [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

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

  7. (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) • TR = p x q = ( 500 - 2q ) q = 500q - 2q2 • QUADRATIC equation

  8. 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.

  9. (c) The Marginal Revenue Curve • MR = TR /  q

  10. 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

  11. 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

  12. (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 completely interchangeable terminology

  13. [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.

  14. 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.

  15. (b) The Average Cost Curve • Average costs (AC) have been defined as follows : • AC = TC/q

  16. 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.

  17. (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.

  18. 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.

  19. [C] Profit Considerations • Profit = TR - TC

  20. 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.

  21. (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

  22. AR/MR/AC/MC on the same graph.

  23. 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

  24. 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

  25. Section 4 : The Answer • Using DATA/SORT facility, the spreadsheet is re-ordered according to the profit maximising criterion as follows:

  26. 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 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.

  27. 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.

  28. 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.

  29. AR curve • 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.

  30. AC curve • 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.

  31. 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:

  32. 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.

  33. 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.

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