Economics
posted by Long Weekend .
The Ali Baba Co is the only supplier of a particular type of Oriental carpet. The estimated demand for its carpets is Q= 112,000 – 500P + 5M, where Q= number of carpets, P= price of carpets (dollar per unit), and M= consumers income per capita. The estimated average variable cost function for Ali Baba’s carpets is AVC= 2000 – 0.012Q + 0.000002Q2
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 consumer’s income per capita is expected to be $30,000 instead.

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 secondorder 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 profitmaximizing 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
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