Posted by Long Weekend on Friday, July 31, 2009 at 6:45pm.
This 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.
Economyst, could you elaborate on #1 and #4. I'm stuck!
for #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.
The 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.
The 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 consumer’s 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.
Demand Equation Q= 612000- 500P