Business and Economic Applications

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Business and Economic Applications
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Summary of Business Terms and Formulas

• x is the number of units produced (or sold)
• p is the price per unit
• R is the total revenue from selling x units
• R xp
• C is the total cost of producing x units

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Summary of Business Terms and Formulas

• P is the total profit from selling x units
• P R C
• The break-even point is the number of units for
  which R C.

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Marginals
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Using Marginals as Approximations

• A manufacturer determines that the profit derived
  from selling x units of a certain item is given
  by P 0.0002x3 10x.
• a. Find the marginal profit for a production
  level of 50 units
• b. Compare this with the actual gain in profit
  obtained by increasing the production from 50 to
  51 units.

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Demand Function

• The number of units x that consumers are willing
  to purchase at a given price p is defined as the
  demand function
• p f(x)

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Finding the Demand Function

• A business sells 2000 items per month at a price
  of 10 each. It is predicted that monthly sales
  will increase by 250 items for each 0.25
  reduction in price. Find the demand function
  corresponding to this prediction.

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Steps

• First find the number of units produced

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Steps

• Now solve this equation for p

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Finding Marginal Revenue

• A fast-food restaurant has determine that the
  monthly demand for its hamburgers is
• Find the increase in revenue per hamburger
  (marginal revenue) for monthly sales of 20,000
  hamburgers.

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Finding Marginal Revenue

• Because the total revenue is given by
• R xp, you have
• and the marginal revenue is

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Finding Marginal Revenue

• When x 20,000 the marginal revenue
• Now lets look at the graph. Notice that as the
  price decreases, more hamburgers are sold. (Make
  sense?)

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Finding Marginal Profit

• Suppose that the cost of producing those same x
  hamburgers is
• C 5000 0.56x
• (Fixed costs 5000 variable costs are 56 per
  hamburger)

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Finding Marginal Profit

• Find the total profit and the marginal profit for
  20,000, for 24,400, and for 30,000 units.
• Solution Because P R C, you can use the
  revenue function to obtain

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Finding Marginal Profit
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Finding Marginal Profit

• Now do the derivative to find a marginal

Demand 20,000 24,400 30,000
Profit 23,800 24,768 23,200
Marginal profit 0.44 0.00 - 0.56
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Finding Maximum Profit

• In marketing a certain item, a business has
  discovered that the demand for the item is
• The cost of producing x items is given by C
  0.5x 500. Find the price per unit that yields a
  maximum profit.

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Finding Maximum Profit

• From the given cost function, you obtain
• P R C xp (0.5x 500).
• Substituting for p (from the demand function)
  produces

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Finding Maximum Profit

• To find maximums using calculus, you now do a
  derivative of the profit equation and then set it
  equal to zero.

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Finding Maximum Profit

• This gives us the number of units needed to be
  produced in order to get the maximum profit. How
  do we find the price?

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Minimizing the Average Cost

• A company estimates that the cost (in dollars) of
  producing x units of a certain product is given
  by
• C 800 0.4x 0.0002x2.
• Find the production level that minimizes the
  average cost per unit.

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Minimizing Average Cost

• Substituting from the given equation for C
  produces

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Practice

• Your turn
• Homework (From CD Appendix G)
• p. G5 problems 1 33 odd