You work for a company that makes specialized computer hardware. There is a competing firm that makes many of the same products as you, but sells a GPU that you have $-$ so far $-$ not been able to manufacture. One of your engineers approaches you, reporting that they have developed a GPU with equivalent performance. If you take the product to market, you will initiate a price war with your competitor. The low cost producer will drive the market price down to just below the $MC$ of the highest cost producer $-$taking the entire market, and earning a unit profit that represents the difference between the two firm's marginal costs. Your engineers estimate that each GPU has a marginal cost of $$3,000$ to produce. You do not know your competitor's marginal cost. However, your marketing department has estimated the demand

elasticity for the GPU $(=-1\mathrm{.}2),$ and you that know your competitors GPUs sell for $$30,000.$ And then, you remember the inverse elasticity pricing rule you learned in OTM $732$:

$\frac{p^{*}-mc}{mc}=(\frac{-1}{+1}),$ and

$p^{*}=(\frac{mc}{1+\frac{1}{}}).$

A$.(10$ points$)$ Assuming that your competitor is setting optimal prices as a monopolist, use the inverse elasticity pricing rule to estimate their marginal cost. Can you offer a lower price than your competitor in a price war?

B$.(5$ points$)$ Assume that you can charge a price equal to your competitor's marginal cost $($you are now engaging in price competition so the IEPR no longer applies$),$ how much variable profit would you make on each unit?

C$.(10$ points$)$ Recall that the elasticity of demand depends on current prices and quantities. If your price war results in a lower price, the quantity sold may also increase dramatically. You want to generate a back of the envelope estimate of the conditions under which you can cover your fixed costs. If face a fixed cost of $$10m$ to scale up production of the GPU. What is the minimum number of units would you need to sell for entry to be profitable?