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June 30, 2016
Posted by **Long Weekend** on Friday, July 31, 2009 at 6:45pm.

Consumers income per capita is expected to be $20,000 and total fixed cost is $100,000

#1. How many carpets should the firm produce in order to maximize profit?

#2. What is the profit maximizing price of carpets?

#3. What is the maximum amount of profit that the firm can earn selling carpets?

#4. Answer parts a through c if consumers income per capita is expected to be $30,000 instead.

- Economics -
**economyst**, Friday, July 31, 2009 at 10:58pmThis is a straight monopoly problem; set MC=MR and solve for Q, and then determine P. The tricky part is that the total cost function is a cubic.

Since M is given, rewrite the demand equation as Q= 212,000 - 500P. Next rewrite so P is a function of Q. so P=424 - .002Q

Now then TR=P*Q=424Q - .002Q^2

MR is the first derivitive, so MR=424-.004Q

TVC = AVC*Q = 2000Q-.012Q^2 + .000002Q^3

MC is the first derivitive

MC=2000 - .024Q + .000006Q^2

My quadratic equation is 0=1576 - .02*Q + .000006Q^2.

Which, by my calculations, does not have a solution, which means no maxima. Please Please check my math. I am confident in my methodology, less so in my arithmitic. - Economics -
**LongWeekend**, Saturday, August 1, 2009 at 3:53pmEconomyst, could you elaborate on #1 and #4. I'm stuck!

- Economics -
**economyst**, Monday, August 3, 2009 at 9:45amfor #1 Always always always, maximize when MC=MR. (Since you have a quadratic for an MC equation, you may have two points where MC=MR. One will represent where profits are maximized, the other where profits are minimized. In calculas, check second-order conditions. Or, for simplicity, test both answers and pick the one with the highest profit.) So, I set MR=MC as 424-.004Q=2000-.024Q+.000006Q^2. Rearranging terms, I got my quadritic equation; the next step is to solve for Q

for #4. It is the exact same procedures as #1-#3, except with a different cost function. Here Q=262,000 - 500P

I hope this helps. - Managerial Economics -
**Jorge**, Friday, December 11, 2009 at 10:06pmThe reason this cannot be worked out is because there is an extra 0 on the first term in the AVC = 2000 0.012Q + 0.000002Q2. It should really be AVC = 200 0.012Q + 0.000002Q2.

- Economics -
**Jorge**, Friday, December 11, 2009 at 11:25pmThe estimated demand equation is Q = 112,000 500P + 5(20,000) = 212,000 500P.

The inverse demand function is P = 424 0.002Q

The marginal revenue is MR = 424 0.004Q

The average variable cost is AVC = 200 0.012Q + 0.000002Q2

The marginal cost is SMC = 200 0.024Q + 0.000006Q2

a. How many carpets should the firm produce in order to maximize profit?

To maximize profit, marginal revenue is set to equal marginal cost.

MR = SMC = 424 0.004Q = 200 0.024Q + 0.000006Q2 = 224 + 0.020Q 0.000006Q2

Solving for Q, it is equal to 4666.67 and 8000. Since there can be no negative output, the firm should produce 4667 carpets.

b. What is the profit-maximizing price of carpets?

The profit maximizing price P = 424 0.002Q, when Q = 4667, the price is $414.33.

c. What is the maximum amount of profit that the firm can earn selling carpets?

The maximum amount of profit the firm can earn selling carpets is:

(P * Q) [(AVC * Q) + TFC] = (414.33 * 4667) [(187.56 * 4667) + 100,000)] = $958,341.48

d. Answer parts a through c if consumers income per capita is expected to be $30,000 instead.

The estimated demand equation is Q = 112,000 500P + 5(30,000) = 262,000 500P.

The inverse demand function is P = 524 0.002Q

The marginal revenue is MR = 524 0.004Q

The average variable cost is AVC = 200 0.012Q + 0.000002Q2

The marginal cost is SMC = 200 0.024Q + 0.000006Q2

To maximize profit, marginal revenue is set to equal marginal cost.

MR = SMC = 524 0.004Q = 200 0.024Q + 0.000006Q2 = 1476 0.020Q + 0.000006Q2

Solving for Q, it is equal to 5833.33 and 9166.67. Since there can be no negative output, the firm should produce 5834 carpets.

The profit maximizing price P = 524 0.002Q, when Q = 5834, the price is $512.33.

The maximum amount of profit the firm can earn selling carpets is:

(P * Q) [(AVC * Q) + TFC] = (512.33 * 5834) [(198 * 5834) + 100,000)] = $1,733,801.22. - Economics -
**Ramesh**, Tuesday, February 23, 2010 at 11:59pmDemand Equation Q= 612000- 500P