SOLUTION: Question: The demand curve a firm faces is p(q) = 5q^-0.5 where q is the quantity produced. The cost of producing q units of output varies nonlinearly with the quantity and is eq

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Question 394851: Question:
The demand curve a firm faces is p(q) = 5q^-0.5 where q is the quantity produced. The cost of producing q units of output varies nonlinearly with the quantity and is equal to c(q)= q^0.8. Firm also pays a fixed cost of 2 to cover administrative jobs and rents. What is the level of output that maximizes profit for this firm?
Thank-you in advance.
Found 2 solutions by robertb, richard1234:
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
**** To the OTHER tutor: The operative assumption here is everything happens in a PERFECTLY COMPETITIVE MARKET ( which enables us to make mathematical calculations.). So the DEMAND curve tells us that what producers manufacture actually came from the DEMAND of consumers, which THEY WILL BUY eventually. In addition, total profit in a perfectly competitive market reaches its maximum point where MARGINAL revenue equals MARGINAL cost. ***




c%28q%29=+q%5E0.8 and p%28q%29+=+5q%5E%28-0.5%29
Then the profit function is P%28q%29+=+qp%28q%29+-+c%28q%29+-+2=+5q%5E0.5+-+q%5E0.8+-+2.
Get the derivative and set it to zero, and then solve the resulting equation:
dP%2Fdq+=+2.5q%5E%28-0.5%29+-+0.8q%5E%28-0.2%29+=+0
==> 2.5q%5E%28-0.5%29+=+0.8q%5E%28-0.2%29
==> 3.125++=+q%5E0.3
==> q = 44.62, to 2 decimal places.
Use the 2nd derivative test to see if profit P is maximized at q = 44.62.

==> d%5E2P%2Fdq%5E2+=+-1.25q%5E%28-1.5%29+%2B+0.16q%5E%28-1.2%29.
Incidentally at q = 44.62, d%5E2P%2Fdq%5E2+%3C+0, and hence P is maximized.
Therefore q = 44.62 is the level of output that maximizes profit for this firm.

Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
graph%28300%2C+300%2C+0%2C+20%2C+0%2C+20%2C+5x%5E%28-1%2F2%29%2C+x%5E%284%2F5%29%29

This question does not make sense. The green curve shows the cost of production, for producing q units. However, the red curve is a demand curve, and demand curves show how prices change with demand. In other words, it has to show the price per unit with a demand of q. The two curves have different y-axes (price per unit and total cost) and they cannot be easily combined in any way.

In any economic market, I don't think p(q) would represent the profit because there is an asymptote at q = 0 (i.e. maximum profit occurs when there is zero production).

It is fairly likely that p(q) is the price for a unit(however it would not be a demand curve, it would be a supply curve since it models supply, q, over price). If this is so, then the profit is most likely explained as y+=+q%2Ap%28q%29+-+c%28q%29+-+2. You can work from there on, taking the derivative of y in terms of q just like the other tutor did. However this may not be true since q is the amount *supplied*, and not necessarily the amount actually sold.

If the two graphs are legitimate supply and demand curves (that is, the red curve shows how price is affected by demand, and the green curve shows how price is affected by supply), then the optimal profit usually occurs at equilibrium, where the demand and supply are equal. Then you can set p(q) = c(q) and solve.

Or, p(q) shows the demand with a given supply q. However, this makes the question unsolvable, as the y-axes of the green and red curves show the total cost and the amount demanded (also note that it is the amount demanded, not the amount actually sold).