You own and operate PC’s R Us, a firm that manufactures customized computers to meet the specifications of universities that offer online MBA programs. Nearly 70% of your customers are Webster University MBA students. Your firm is not the only business that makes the custom computers, but competes with other firms that sale via retail channels as well as by mail-order and online. You differentiate your product from that of your competitors by offering 2 years of free onsite repair for any system you sell. Because you wish to know more about the underlying market conditions that impact your business, you decide to trade a Webster Business 6120 student a free computer for a study on your firm’s demand and cost structure. The study results are:

Yearly cost of producing computers is: C(Q) = 20,000 + 2Q2 , where ‘Q’ represents the number of computer systems produced.
Marginal Cost (MC) = 4Q
Yearly demand for computers is: Q = 1,000 – P, where ‘P’ represents the selling price of a computer system.

a. How many PC’s should you produce to maximize profits?

b. If you charge the profit maximizing price, what is your firm’s profit or loss?

c. How much does the last unit you produce cost you to make?

d. How can you keep your profits from eroding over time as the market matures?

Take a shot. What do you think?

Hint: always always always, maximize profits where MC=MR.
Hint 2: Total revenue (TR) is P*Q

Ok well here is what I have so far:

a.
MC (Q) = MR (Q) P = MC
P = 1000 – Q & C (Q) = 20,000 + 2Q2
MR = a + 2bQ
MR = 1000 – 2Q
MC = 4
1000 – 2Q = 4
6Q = 1000
Q = 167 units
= 20,000 + 2(167) ^2
=$75,778
MC = 668
MR = 666
P = $833

b.
Profits are given by the difference between revenues and costs
= P*Q* - C (Q*)
= 833* 167 – 20,000 + 2(167) ^2
= 139,111 – 75,778
= $63,333 Profit

c.
The marginal cost curve intersects the average total cost curve and the average variable cost curve at their minimum points. The reason is that ATC and AVC are averages of the cost of the first unit of output, the second unit, and so on. If the average is falling, the last unit must have a cost below the average, in order to be bringing down the average. If the average is rising, the last unit must have a cost above the average, in order to be bringing up the average

d.
Due to the free entry of monopolistic markets, if we are earning a profit in the short run additional firms will probably enter the market too, in the long run to capture some of those profits. As new firms enter the market they will make different PC’s and offer other service plans, setting themselves apart from us. Some of our consumers will then use the newer firms PCs and substitute for our PC. As the demand curve decreases where it is just tangent to our average cost curve, we’ll be earning probably no profits and other firms won’t enter the market. So, for us to be successful over time we’ll have to charge a price that exceeds the MC of producing our PCs.

How close am I?

To answer these questions, we need to analyze the cost and demand functions provided in the study. Let's go step by step:

a. To maximize profits, we need to determine the quantity of computers we should produce. In order to do that, we need to find the quantity that equates marginal cost (MC) with marginal revenue (MR). In this case, since the demand function is given, we can find MR by differentiating the demand equation with respect to Q:

MR = d(Q*P)/dQ
= P + Q * dP/dQ
= P + Q * (-1) [since dP/dQ = -1 from the demand function]

Now, we can equate MR to MC to find the profit-maximizing quantity:
P + Q * (-1) = 4Q

Solving for Q:
P - Q = 4Q
P = 5Q
Q = P/5

Since Q represents the number of computer systems produced, the profit-maximizing quantity will be one-fifth of the selling price.

b. To find the firm's profit or loss at the profit-maximizing price, we need to calculate the total revenue (TR) and total cost (TC) at that quantity.

TR = P * Q
= P * (P/5)
= P^2/5

TC = C(Q)
= 20,000 + 2Q^2
= 20,000 + 2(P/5)^2
= 20,000 + (2P^2)/25

Profit (π) = TR - TC
= P^2/5 - (20,000 + (2P^2)/25)

To determine if it's a profit or loss, we need to compare this equation with zero.

c. The cost of the last unit produced can be calculated using the marginal cost (MC) function provided:
MC = 4Q
Plug in the profit-maximizing quantity Q = P/5 into the MC function to find the cost of the last unit produced.

d. To prevent profits from eroding over time as the market matures, you can consider the following strategies:

1. Enhance the product: Continuously improve the features and quality of your computers to differentiate them from competitors.

2. Offer additional services: Besides the free onsite repair, consider adding value-added services like extended warranties, technical support, or software packages.

3. Build customer loyalty: Develop strong relationships with your existing customers, providing exceptional customer service and personalized support.

4. Target new markets: Explore opportunities to expand your customer base beyond Webster University MBA students. Consider partnering with other universities or targeting different industries that require customized computers.

5. Pricing strategies: Continuously evaluate your pricing strategy to ensure it remains competitive while still covering your costs and maintaining profitability. Monitor market trends and adjust pricing accordingly.

6. Efficient operations: Streamline production processes, optimize inventory management, and minimize operating costs to improve overall efficiency and profitability.

Remember, these strategies should be tailored to the specific market conditions and needs of your business. Regular market research and analysis will also help you stay informed about any changes in the industry and adapt your strategies accordingly.