application of differentiation in economics

helloprajna

How to solve the following problem?
A revenue maximizing monopolist requires profit of at least 1500. Its demand and cost functions are p=304-2q and c=500+4q+8q^2. Determine the output level and price.

MarkFL

Profit is revenue minus cost. Revenue is the product of price per unit times units demanded, hence:

$$\displaystyle P(q)=q\cdot p(q)-C(q)$$

$$\displaystyle P(q)=q\(304-2q$$-$$500+4q+8q^2$$\)

$$\displaystyle P(q)=304q-2q^2-500-4q-8q^2=-10q^2+300q-500=-10\(q^2-30q+50$$\)

To find the maximum profit using differentiation, we find the output level for which marginal profit is zero, which may be done as follows:

$$\displaystyle \frac{d}{dq}\(q^2-30q+50$$=0\)

$$\displaystyle 2q-30=0$$

$$\displaystyle q=15$$

This is the output level which maximizes profit. Now to find the price at this output level, we compute:

$$\displaystyle p(15)=304-2(15)=274$$

To find the output level corresponding to a profit of at least 1500, we solve the inequalilty:

$$\displaystyle 1500\le-10\(q^2-30q+50$$\)

$$\displaystyle 150\ge q^2-30q+50$$

$$\displaystyle q^2-30q-200\le0$$

$$\displaystyle (q-10)(q-20)\le0$$

From this, we determine $$\displaystyle 10\le q\le20$$. This give the range of prices as:

$$\displaystyle 304-2(20)\le p\le304-2(10)$$

$$\displaystyle 264\le p\le284$$

helloprajna

But it's a revenue maximising monopolist not a profit maximising one... :roll:

CRGreathouse

Forum Staff
helloprajna said:
But it's a revenue maximising monopolist not a profit maximising one... :roll:
Set profit equal to 1500 and find the two solutions. Take the one giving greater revenue.

MarkFL

helloprajna said:
But it's a revenue maximising monopolist not a profit maximising one... :roll:
Well, excuuuuuse me! :lol:

helloprajna

I want to solve the problem like this:-

The monopolist requires a profit(P) of at least 1500.
So, $$\displaystyle P \ge1500$$
or$$\displaystyle -10q^2+300q-500 \ge1500$$
or $$\displaystyle 10\le q\le20$$
So the monopolist must produce his output in the range of 10 to 20 units.It's given that the monopolist maximizes his revenue. So we have to find at what value of q, revenue is maximum.

At x=10, Revenue(R)= 304q-2q^2 = 2840
At x=20, Revenue(R)=5280
Clearly, as the monopolist goes on increasing his output from 10 units to 20 units, its revenue goes on increasing and becomes maximum at 20 units i.e., at x=20. So we conclude that the output level for the monopolist is x=20.

Here I've combined the suggestions of you two.
Please tell me if I'm right or wrong.

helloprajna

MarkFL said:
helloprajna said:
But it's a revenue maximising monopolist not a profit maximising one... :roll:
Well, excuuuuuse me! :lol:
I accidentally clicked on that "rolling eyes" smiley.