# MGMT 1P93 Lecture Notes - Numerical Analysis, Cubic Function, Fixed Cost

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4 Feb 2013

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PROFIT MAXIMIZATION

Profit = Total Revenue (P*Q) – Total Costs

Accountants calculate Profit as operating Profit

= Total Revenue – Operating Costs

= Total Revenue – Variable Costs (in economic terminology)

(Operating Profit = Producer Surplus in economic terminology)

→ Accountants compare this operating profit with Capital to find the rate of return on capital

Economics calculate Economic Profit from all costs, including the cost of capital calculated

as the opportunity cost of Capital.

Definition: Economic Profit = Total Revenue – (Variable Costs + Fixed Costs)

Definition: Normal Profit = Opportunity Cost of Capital

→ 0 Economic Profit = Normal Profit since it includes the Opportunity Cost of Capital

→ Economic Loss => Economic Profit < 0 (less than the Opportunity Cost of Capital)

→ Economic Profit => Economic Profit > 0 (greater than the Opportunity Cost of Capital)

Short-run Profit Maximization

=> Find the output that maximizes Profit (TR – TC)

=> ∆(TR – TC)/∆q = 0 [d(TR – TC)/dq = 0]

=> ∆TR/∆q – ∆TC/∆q = 0 [dTR/dq - dTC/dq = 0]

=> MR = MC

[More formally:

Profit = TR – TC = PQ – (wL + rK) (w = wage rate and r = opportunity cost of Capital)

Profit Maximization => ∆(PQ – wL – rK)/ ∆Q = 0

→ P + Q∆P/∆Q – w∆L/∆Q = 0

- 1 -

→ P + Q∆P/∆Q = MC since w∆L/∆Q = MC

→ MR = MC for all firms since P + Q∆P/∆Q = MR]

Marginal Revenue equal to Marginal Cost gives the profit maximizing output but this

expression also includes the loss minimizing output if the price is below Average Cost. (There is

one exception to this rule as we will see).

E.g. Suppose that the following functions describe the Variable Cost and Fixed Cost of a

competitive firm VC = q3 – 59q2 + 1315q FC = 2000

This cubic function gives the correct shape for the Marginal Cost function but we will

eventually work only with more quadratic Variable Cost functions. Note that a change in

quantity changes Variable Cost but has no effect on Fixed Cost. Total Cost is the sum of both

functions.

Suppose further that the firm’s Total Revenue function is TR = 1000q – 2q2.

TC = q

3

- 59q

2

+1315q +2000; TR = 1000q - 2q

2

0 10 20 30 40 50

Quantity

0

10

20

30

40

50

60 $000

TC

TR

Max (TR - TC) => Max Profit

Slope TR = Slope TC => MR = MC

q*

Max Profit

Total Economic Profit is the difference between the Total Revenue and Total Cost functions

as shown in the diagram below. Total Economic Profit is greatest at the output where Total

Revenue – Total Cost is greatest. This occurs where the slopes of the two functions are equal, or

- 2 -

when Marginal Revenue equals Marginal Cost since these are the slopes of Total Revenue and

Total Cost respectively.

The diagram below shows Economic Profit as a function of q for this firm.

Total Profit = 1000q - 2q

2

- (q

3

- 59q

2

+1315q +2000)

0 10 20 30 40 50

Quantity

0

5

10

15

-5

$000

Total Profit (TR - TC)

q*

Max Profit

Comparison Marginal Revenue and Marginal Cost shows the maximum profit output more

clearly than comparison of Total Revenue and Total Cost.

Marginal Revenue = 1000 – 4q Marginal Cost = 3q2 - 118q + 1315

- 3 -