courses › Business Math

Toy demand curves, marginal profit, the best quantity, elasticity, and why models need checking.

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skills · practice stats

From rounds of this course only: box, review and trial answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

• Price and profit in a toy model par 25 s not tried yet

  worked example

  In a toy model, unit price is p(q) = 100 − 2q dollars when q units are sold per month, and total cost is 5q + 500 dollars. What is monthly profit at q = 46? (Use a minus sign for a loss.)

  Answer: -$362.00

  1. Price = 100 − 2 × 46 = $8; revenue = 46 × $8 = $368.
  2. Cost = 5 × 46 + 500 = $730; profit = $368 − $730 = −$362.

• Marginal profit (the derivative) par 22 s not tried yet

  worked example

  Profit is π(q) = 90q − 0.5q² − 700 dollars, where q is units sold. What is π′(q) at q = 125, in dollars of profit per extra unit? (Use a minus sign if it's negative.)

  Answer: -35

  1. π′(q) = 90 − q, so π′(125) = 90 − 125 = −35.
  2. Profit is falling by about $35 per extra unit at this point.

• Profit-maximizing quantity par 20 s not tried yet

  worked example

  Profit is π(q) = 130q − q² − 100 dollars, and capacity limits q to at most 64. What whole quantity maximizes profit?

  Answer: 64

  1. π′(q) = 130 − 2q = 0 at q = 65. Capacity stops q at 64, where profit is still rising (π′(64) = 2 > 0).
  2. So the best feasible quantity is 64.

• Price elasticity par 17 s not tried yet

  worked example

  Near the current price, a 5% price increase goes with about a 10% drop in quantity sold. What is the approximate price elasticity of demand? Give a plain number, with a minus sign if it's negative.

  Answer: -2

  1. Elasticity ≈ % change in quantity ÷ % change in price = −10% ÷ 5% = −2.
  2. Its size is above 1, so buyers react more than proportionally (elastic demand). It still doesn't tell you the most profitable price.

• Is the model's answer the answer? par 19 s not tried yet

  worked example

  A calculation says the best price is $115, but its demand curve was guessed. What should management do before acting on it?

  1. Use it as is: the math is exact
  2. Test the demand and cost assumptions and check a range of prices
  3. Ignore the model: math can't help with this
  4. Average it with the current value

  Answer: Test the demand and cost assumptions and check a range of prices

  1. A guessed demand curve gives a guessed optimum.
  2. An answer is only as good as its inputs: check assumptions with real evidence, try a range of values, and check capacity and cash before acting.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

1. 1 day
2. 3 days
3. 7 days
4. 14 days
5. 30 days
6. 60 days
7. mastered · every 90 days

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