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Deriving the Profit-Maximizing Price with Constant Unit Cost

Article Quant Q&A · Author: user61871
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Summary

The document addresses a pricing formula from a quantitative finance course. Demand is represented by a power function, and the question asks how calculus yields an optimal price in terms of production cost and the demand exponent. The asker attempts to maximize revenue and equate marginal revenue with marginal cost, but applies a constant marginal cost value directly in the calculation.

The accepted response identifies that assumption as the source of the error: the marginal cost should not have been set to the value used in the attempt. Correcting it and setting the relevant expression to zero gives the stated price of 10/3. The exchange signals the need to distinguish production cost from marginal cost when deriving an optimum, but it does not show the corrected algebra or establish the general formula’s assumptions. Readers should treat it as a brief correction, not a complete derivation for other demand or cost specifications.

Key ideas

  • The question concerns optimal pricing under a power-function demand model.
  • The stated approach seeks a profit maximum by equating marginal revenue and marginal cost.
  • The accepted response attributes the incorrect result to an assumption about marginal cost.
  • The corrected calculation is reported to produce the price 10/3.
  • The exchange omits the detailed algebra and conditions needed to generalize the result.

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# General Equation for price optimisation where cost is constant


# General Equation for price optimisation where cost is constant












I'm currently working on the Quantitative Finance course offered on Coursera by Wharton and in one example it states that "through calculus, one can obtain the optimal value of price when p(opt)=(c*b)/(1+b) where c is the production cost and b is the exponent in the power function."

But it fails to explain how that is so? Could someone here please enlighten me? My logic below didn't work out...

- in my understanding profit max is when MR-MC=0,

- R=QP=(60000P^(-2.5))*P

- Pmax=MR-2=(dR/dQ)-2=(-90000P^(-2.5))-2

- so Pmax = (-1/45000)^(-2/5) but it doesn't :( -- answer given is 3.33

(Apologies for poor formatting, I tried to LaTeX but I don't think it works here? Or maybe I just didn't do it correctly...)

[edit: for the sake of clarity, I thought I'd just include print screens]

## Answer by user61871 (score 0, accepted)

https://quant.stackexchange.com/a/23225

OK solved my own problem! The issue was assuming MC = 2... instead,

and when you set that equal to zero you get... p = 10/3 :)

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