Part 1A monopoly produces a good with a network externality at a constant marginal and average cost of c = $22. In the first period, its inverse demand curve is p equals 12 minus 1 Upper Qp=12−1Q. In the second period, its inverse demand curve is p equals 12 minus 1 Upper Qp=12−1Q unless it sells at least Q = 99 units in the first period. If it meets or exceeds this target, then the demand curve rotates out by alphaα (it sells alphaα times as many units for any given price), so that its inverse demand curve is p equals 12 minus StartFraction 1 Over alpha EndFraction Upper Qp=12−1αQ. The monopoly knows that it can sell no output after the second period. The monopoly's objective is to maximize the sum of its profits over the two periods. Part 2For what values of alphaα would the monopoly earn a higher two-period profit by setting a lower price in the first period?If alphaα is [input] greater than Your answer is correct. You answered greater than. [input]1.51.5 . (round your answer to two decimal places)

Question

Part 1A monopoly produces a good with a network externality at a constant marginal and average cost of c​ = ​$22. In the first​ period, its inverse demand curve is p equals 12 minus 1 Upper Qp=12−1Q. In the second​ period, its inverse demand curve is p equals 12 minus 1 Upper Qp=12−1Q unless it sells at least Q​ = 99 units in the first period. If it meets or exceeds this​ target, then the demand curve rotates out by alphaα ​(it sells alphaα times as many units for any given ​ price), so that its inverse demand curve is p equals 12 minus StartFraction 1 Over alpha EndFraction Upper Qp=12−1αQ. The monopoly knows that it can sell no output after the second period. The​ monopoly's objective is to maximize the sum of its profits over the two periods. Part 2For what values of alphaα would the monopoly earn a higher​ two-period profit by setting a lower price in the first​ period?If alphaα is [input] greater than Your answer is correct. You answered greater than. [input]1.51.5 .  ​(round your answer to two decimal​ places)

BlackTom AI · Accepted Answer

We are given a two-period monopoly problem with a network externality that rotates the second-period demand if a first-period target of Q = 99 is met or exceeded. The first-period inverse demand is p = 12 − Q, and the second-period inverse demand is p = 12 − Q if the first-period target is not met, but if the target is met, the post-rotation inverse demand becomes p = 12 − (1/α)Q, which in terms of Q is Q = α(12 − p). The cost in both periods is constant at c = 22, so profits in a period are π = (p − c)Q. The monopoly cannot sell after the second period, so it must consider the trade-off across periods.

Option analysis:
- The statement captured by the first blank is just the relational form involving α, i.e., whether α must be greater than some threshold. On its own, the word “greater than” is incomplete because it lacks the numeric cutoff that determines when lowering the first-period price increases total two-period profit. This is incomplete as a full criterion for the decision rule, since the profitability of a low first-period price depends on how large α is and on the relative magnitudes of profits in the two periods.
- The statement captured by the second blank presents a numeric threshold, “1.50,” which is the value that the problem’s solution indicates as the cutoff in order for a lower first-period price to raise the two-period profit when α is above this value. The key issue is not simply whether α is above 1.50 in isolation, but whether α exceeds this specific threshold that equalizes the marginal benefit and cost across periods given the rotated demand curve.

To understand why the threshold matters, consider the following conceptual points:
- If α is large, then after meeting the first-period target, the second-period demand becomes flatter with respect to p (since Q = α(12 − p), a given price change translates into a larger change in Q when α is larger). This can amplify the payoff from gaining more sales in the second period through a lower first-period price, because the rotated demand yields higher Q for the same price, boosting second-period profit more than the first-period loss from a lower price.
- If α is small, the rotated second-period demand is relatively steep and the gains in the second period from a lower first-period price are smaller, so the first-period price reduction is less attractive overall.
- The critical comparison is between the marginal effect of lowering the first-period price on total profit now and the induced shift to the rotated second-period demand that can raise profits later. The numeric threshold around 1.50 emerges from balancing these marginal effects under the given cost and demand parameters.

In summary, the decision rule hinges on α relative to a cutoff; when α exceeds that cutoff, lowering the first-period price improves two-period profits due to the pronounced advantage from the rotated second-period demand. The given numeric value 1.50 represents that cutoff point. Thus, the condition is that α must be greater than 1.50 for the two-period profit to be higher with a lower first-period price.

Note on the structure of the answer inputs: the prompt presents two blanks—the first expecting a relational word like “greater than,” and the second expecting the numeric threshold. The intended interpretation is the combined statement “α is greater than 1.50.” The reasoning here explains why that combination, particularly the numeric threshold, determines the profitability of a lower first-period price.

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