Pricing Models (Applied Calculus)
Toy demand curves, marginal profit, the best quantity, elasticity, and why models need checking.
Pen and paper is fine · no calculator needed why?
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the idea
A toy model writes price as a function of units sold: p(q) = 100 − q means each extra unit lowers the price by $1. Revenue is q × p(q), and profit is revenue minus total cost.
The derivative π′(q) is how fast profit changes per extra unit: positive means rising, and zero marks the peak, unless capacity stops you first. Elasticity compares the % change in quantity with the % change in price.
A model's answer is only as good as its guesses.
techniques
Price, revenue, cost, profit
- Price: put q into p(q).
- Revenue: q × that price.
- Cost: put q into the cost formula, fixed part included.
- Profit: revenue minus cost.
worked example
In a toy model, unit price is p(q) = 80 − q dollars when q units are sold per month, and total cost is 20q + 300 dollars. What is monthly profit at q = 20?
- Price: 80 − 20 = $60.
- Revenue: 20 × $60 = $1,200.
- Cost: 20 × 20 + 300 = $700.
- Profit: $1,200 − $700 = $500.
Answer: $500
Derivative, then zero
- For π(q) = mq − bq² − F, the derivative is m − 2bq.
- Put in q for the rate at that point. Negative means profit is falling.
- Set m − 2bq to zero and solve for the peak.
- If capacity is below the peak, the best quantity is the capacity.
worked example
Profit is π(q) = 80q − 2q² − 500 dollars, and capacity limits q to at most 15. What whole quantity maximizes profit?
- π′(q) = 80 − 4q, which is zero at q = 20.
- Capacity stops q at 15, where profit is still rising.
Answer: 15
Elasticity: quantity over price
- Write the % change in quantity, with a minus sign for a drop.
- Divide it by the % change in price.
- A size above 1 means buyers react more than in proportion.
worked example
Near the current price, a 4% price increase goes with about a 6% drop in quantity sold. What is the approximate price elasticity of demand? Give a plain number, with a minus sign if it's negative.
- Quantity −6%, price +4%.
- −6 ÷ 4 = −1.5.
Answer: −1.5
watch out for
- Stopping at revenue, or treating the unit price as total revenue.
- Dropping the 2 when differentiating. The derivative of 2q² is 4q, not 2q.
- Taking the peak when capacity is lower. Then capacity is the answer.
- Treating a precise optimum as fact. A guessed demand curve gives a guessed answer.
practice
Price and profit in a toy model
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
- Price = 100 − 2 × 46 = $8; revenue = 46 × $8 = $368.
- Cost = 5 × 46 + 500 = $730; profit = $368 − $730 = −$362.
Marginal profit (the derivative)
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
- π′(q) = 90 − q, so π′(125) = 90 − 125 = −35.
- Profit is falling by about $35 per extra unit at this point.
Profit-maximizing quantity
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
- π′(q) = 130 − 2q = 0 at q = 65. Capacity stops q at 64, where profit is still rising (π′(64) = 2 > 0).
- So the best feasible quantity is 64.
Price elasticity
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
- Elasticity ≈ % change in quantity ÷ % change in price = −10% ÷ 5% = −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?
worked example
A calculation says the best price is $115, but its demand curve was guessed. What should management do before acting on it?
- Use it as is: the math is exact
- Test the demand and cost assumptions and check a range of prices
- Ignore the model: math can't help with this
- Average it with the current value
Answer: Test the demand and cost assumptions and check a range of prices
- A guessed demand curve gives a guessed optimum.
- 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.