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Understanding Monopoly w/ Uniform Pricing

  1. Jun 10, 2012 #1
    1. The problem statement, all variables and given/known data

    This actually returns to an older question I posted here (and got answer, thanks!). I have hard time with the interpretation of the solution and part (2) of the question.

    A monopolist can produce a widget at a cost C. The monopolist then has to find a price that would maximize his expected profit, but he doesn't know how much buyers are willing to pay for the widget. The monopolist does know that buyers' willingness to pay is randomly distributed and that a buyer would be willing to pay a given price only if his valuation is greater than 1/a * p (where 0 < a <1).
    (1) What would be the price that maximizes the monopolist expected payoff?
    (2) Can I show that because of the buyer's margin constraint (only pays when v > 1/a *p) that although production has net expected positive value, the monopolist may not produce?


    2. Relevant equations

    The solution is:
    Maximize profits when 1-F(p/a) = p/a * f(p/a).

    3. The attempt at a solution

    (1) The farthest I went with the interpretation is that at the margin, expected loss (the marginal buyer who drops out) has to be equal to the revenue generated by the remaining buyers. What I can't get is why the expected loss is p/a * f(p/a) and not simply p *f(p/a).

    (2)
    I know that ∫(v-C)dv>0 because expected net value is positive. In (1) I found the maximal price, so that now the challenge would be to show that there is some distribution of net positive values in which (1-F(p/a))p < 0 while also ∫(v-C)dv>0, but I am not sure how to do it.
     
  2. jcsd
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