# Find optimal price from demand function?

I've been struggling with this for hours, trying to figure out how to solve this.

"A perfectly competitive market has the marginal cost function, c (cost is C(y)=cy) and is facing the demand function: D(p)=A-a*p"

I have found the price elasticity: (-ap)/D(p) = (-ap)/(-a*p+A)

Can someone explain how I can find the optimal price?

So I have tried to find the revenue, by multiplying D(p) and p (price). R(p) = pD(p) --> pD(p) = p*(A-ap) --> p*(A-ap) = -ap^2+Ap Then I have taken the derivative of R(p), which is the marginal revenue R'(p) = A-2*ap Then I assume the optimal price would be to isolate p?!

• Is this a monopoly or perfectly competitive market? – Herr K. Oct 8 '18 at 0:43
• Sorry forgot to mention. It is perfectly competitive market – Taquito Oct 8 '18 at 1:57
• In a perfectly competitive market, price is equal to marginal cost, since everyone is a price-taker. I'm not sure what you mean by "optimal" then; optimal in what sense? – Herr K. Oct 8 '18 at 3:29

This logic is something that I've seen many beginners struggle with, so I will try to give a thorough answer.

The optimal price is not derived by maximizing revenue, as you suggest, but rather by maximizing profits. So you want to maximize revenue minus costs.

If you were to include costs in your approach you would come to the correct conclusion that the solution is "Marginal Revenue = Marginal costs". Mathematically, maximizing Profit = Revenue - Costs means taking the derivative and setting it to zero. The derivative of revenue and costs are marginal revenue and marginal cost respectively.

This rule is always true. Nevertheless, you must take the type of competition into account. Perfect competition has a different revenue expression than a monopoly.

Suppose we have perfect competition. This means the firm is a price taker. In turn, the firm cannot take the demand function into account when making its decision (by optimizing). So the firm maximizes:

$$Profit = p*y - c*y$$.

The solution is $$p=c$$.

Taking the first order condition will hence give you the famous "price = marginal cost" rule. In this case, the marginal revenue is just $$p$$. Here, the revenue for each extra unit sold is just whatever that unit costs. There could be price effects from selling an additional unit, since a larger supply means a lower price. Nevertheless, the optimizing firm in perfect competition cannot take such price effects into account. It must take the price as given.

The monopolist, however, does not take the price as given, but can choose it. This is done by taking the (inverse) demand function into account. I find it easier to use inverse demand, which is $$P(y)$$. You can find this by rearranging your demand function, which is $$D(p)=y(p)$$. We have to maximize:

$$Profit = P(y)*y - c*y$$.

The solution here is:

$$P(y) + P'(y)* y = c$$

Therefore, we have marginal revenue equals marginal cost. This is what I believe you were attempting to do and it only works for monopolies.