Having trouble figuring out this problem: Optimal price. Any help is appreciated.?

Question

Q=50P^-1.5 I ^.5

Q=quantity demanded P=product price
I=per capita disposable income Marginal cost is $10/product

Product price is $20, is this the optimal price, why or why not? If not what is??

Answer 1

The optimal price for the manufacturer is the price that returns the most profit. So, for each price P you need to compute total cost and total income.

Total income is just total number sold times the price, but this is a function of I, complicating things a bit.

Total cost is the integral of the marginal costs over the quantity manufactured. In this case, the marginal cost is a very unrealistic constant, making your life simpler (mostly to make the presence of I in the income function work).

So, you can compute profit as a function of P and I. For each I, the optimal P is one that makes the derivative of the profit function equal 0. (Minima and maxima of continuous functions always occur at points where the derivative is 0)

Solve the equation for optimal price in terms of I. If I turns out not to be a factor, then you win and the question has a real answer (i.e. a number instead of a function of I). Otherwise, the question would seem to be ill-formed (at least until you come up with a value for I).

Answer 2

what is the product? in the real world, price fluctuates depending on demand. its this kind of stupidity that creates engineers. no wonder we are in the position we are in now.