Understanding Monopoly w/ Uniform Pricing

[ruzbayhhi]

Homework Statement

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?

Homework Equations

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

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.

 

[mighty2000]

it is important to approach this problem with a clear understanding of the concepts involved. In this case, we are dealing with a monopolist, which means that they have control over the production and pricing of a certain product. The cost of production is denoted as C, and the monopolist's goal is to maximize their expected profit. However, the challenge lies in the fact that the monopolist does not know how much buyers are willing to pay for the product. This is where the concept of willingness to pay comes into play.

Willingness to pay is a measure of how much a buyer is willing to pay for a product or service. In this case, the monopolist knows that buyers' willingness to pay is randomly distributed and that a buyer will only be willing to pay a given price if their valuation is greater than 1/a * p (where 0 < a <1). This means that if the price set by the monopolist is too high, buyers will not be willing to pay for the product.

To answer the first question, we need to find the price that maximizes the monopolist's expected payoff. The solution provided in the forum post is to maximize profits when 1-F(p/a) = p/a * f(p/a). This equation is derived from the concept of expected value, which takes into account the probability of different outcomes. In this case, we are looking at the probability that a buyer's valuation is greater than 1/a * p, denoted by F(p/a), and the probability that a buyer will be willing to pay the set price, denoted by f(p/a). By setting these two probabilities equal to each other, we can find the price that maximizes the expected payoff for the monopolist.

To answer the second question, we need to understand the concept of net expected positive value. This is the expected value of a product or service minus the cost of production. In this case, we know that ∫(v-C)dv>0, which means that the expected net value is positive. However, the question is whether the monopolist will actually produce the product even with a positive expected net value, due to the buyer's margin constraint.

The buyer's margin constraint means that the buyer will only pay for the product if their valuation is greater than 1/a * p. If the price set by the monopolist is too high, then the probability that a buyer's valuation is greater than