Section 4'4 Applications to Marginality

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Section 4.4Applications to Marginality
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• Many decisions in business are based on
  maximizing profits
• We have used the derivative to find maximums
• Profits are determined by revenue and production
  cost
• The cost function, C(q), gives the total cost of
  producing a quantity q of some good
• The revenue function, R(q), gives the total
  revenue received by a firm from selling a
  quantity q of some good
• The profit function, p(q), gives the total profit
  from producing and selling q items and is given
  by p(q) R(q) - C(q)

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• Lets think about the shape of these functions
• What would a cost function look like?
• What would a revenue function look like?
• What would the resulting profit function look
  like?

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Marginal Analysis

• Often a companies decision to continue to produce
  goods is based on how much additional revenue
  they gain versus the additional cost
• The Marginal Cost is the average rate of change
  of adding one more unit
• Therefore it can be approximating by the
  instantaneous rate of change
• Marginal Cost MC C(q)
• Marginal Revenue MR R(q)
• Where is profit maximized (or minimized)?
• Where C(q) R(q)

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Example

• What is the marginal cost of q if fixed costs are
  3000 and the variable cost is 225 per item?
• What is the marginal revenue if you charge 375
  per item?

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Example

• Find the quantity q which maximizes profit if the
  total revenue, R(q), and total cost, C(q) are
  given in dollars by
• Does the value of q give you a local max or a
  local min? How can you tell?

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Example

• A hotel if they charge 300 per night for a
  hotel room, they can rent out a total of 20
  rooms. They find that for each 25 decrease in
  price, they can rent an additional room. How
  many rooms should they rent out to maximize their
  revenue? What is their maximum revenue?