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# Profit maximization algebra with a monopolist.

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Part 1: What Price Should Apple Charge for the IPod?

The IPod is a computer console designed and manufactured by Apple. This console currently dominates the computer market, selling more than its competitors.

Assume that the following facts are true:
? The demand for IPod consoles can be approximated by the linear equation:
QW= 300 - .5 PriceW
? The IPod price doesn't affect Apple's profits from any other of its products.
? It costs Apple \$160 on average to manufacture a IPod console.
? Consumers purchase the IPod directly from Apple's website.

We will begin by modeling only profits from the IPod, ignoring profits from any other products sold by Apple.

1) Write down the basic equations of Apple's optimization problem in the form:
Choose _put choice variable(s) here______
To maximize (or minimize) _equation for objective _

Note: This is called formulating the optimization model. The only variables appearing in this formulation should be the choice variables.

2) Using Marginal Analysis, find the price Apple should charge for each copy of the IPod.

3) Explain why we do not need to know how much it cost Apple to design and develop the IPod to estimate the profit-maximizing price.

4) Set up the Apple optimization problem in an Excel spreadsheet.

5) Solve this optimization problem using Excel Solver. Is your result the same or different from the one you obtained in question 2?

6) The price Apple presently charges is approximately \$300. According to your model, is this price too low or too high? If the demand curve is accurate, what might your model be missing?

7) Players buy games to use on the console that are also produced by Apple, and Apple earns a higher profit margin on games than on consoles. Assume that Apple earns on average \$170 in profits from games for each console sold. How can we incorporate this assumption in the model you estimated above? What is the new profit-maximizing price given by the model? Explain why the price changed the way it did.

8) Given the recent appreciation of the Yen, the cost of producing the IPod has increased in U.S. dollars. Assuming this translates into a 20% increase in the cost of production, should Apple change the price it charges in the U.S. and if so, what price should it charge? (Assume that Apple is presently charging the optimal price that you calculated in part 5 above.) Use either marginal analysis or your Excel spreadsheet model to answer this question, but be sure to show all your work (i.e. submit the optimized spreadsheet if you use Excel).

https://brainmass.com/math/linear-algebra/227110

#### Solution Preview

Please see attached files for fully formatted explanation.

Part 1: What Price Should Apple Charge for the IPod?

The IPod is a computer console designed and manufactured by Apple. This console currently dominates the computer market, selling more than its competitors.

Assume that the following facts are true:
? The demand for IPod consoles can be approximated by the linear equation:
QW= 300 - .5 PriceW
? The IPod price doesn't affect Apple's profits from any other of its products.
? It costs Apple \$160 on average to manufacture a IPod console.
? Consumers purchase the IPod directly from Apple's website.

We will begin by modeling only profits from the IPod, ignoring profits from any other products sold by Apple.
Note: For this problem I will write QW as Q.

1) Write down the basic equations of Apple's optimization problem in the form:
The choice variable in the short run is: Q, since the short run decision is a production (output) decision.
To maximize (or minimize) _equation for objective _
Apple wants to choose Q, such that Profit is maximized. Profit is defined as follows:
Profit = TR - TC
Profit = P*Q - TC
We can also rewrite this as TC=(P-ATC)*Q but ATC=MC since MC is constant. So,
TC=(P-MC)*Q, this is just "profit per unit" times the ...

#### Solution Summary

The following begins with the process for finding the profit maximizing level of output for a monopolist and the price the monopolist should charge. Optimizations are provided in Excel using solver. A situation is examined where the monopolist produces an addition complementary product and the effect that has on pricing decisions. The problem of a changing exchange rate on costs is examined as well.

\$2.19