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Imagine if we are doing the following experiment, you will earn the following payoff depending on your subsequence/ranking (r) of entering an industry. Profit=$50*[c+1-r)/(1+2+…+c)], where r = 1,2,3 … c and c is the market capacity However, if you are ranked after the market capacity (c), you will need to pay $10. In other words, your profit will be -$10. Assuming that the market capacity is 4, and there are 12 players/entrants. Part (a): What is the total industry profit or loss if there are 2 actual entrants? (Note: Please enter your answer rounded to the nearest whole dollar. For instance, if your answer is $45.78, input 46 into your answer box.) (2 marks) Part (b): What is the total industry profit or loss if there are 5 actual entrants? (Note: Please enter your answer rounded to the nearest whole dollar. For instance, if your answer is $45.18, input 45 into your answer box.) (2 marks) Part (c): At what rank will the total industry profit or loss be zero? (Note: Please enter your answer as a whole number or integer. For example, if your answer is 9, input 9 into your answer box.) (1 mark)

Question

Imagine if we are doing the following experiment, you will earn the following payoff depending on your subsequence/ranking (r) of entering an industry.

Profit=$50*[c+1-r)/(1+2+…+c)], where r = 1,2,3 … c

and c is the market capacity

However, if you are ranked after the market capacity (c), you will need to pay 10.Inotherwords,yourprofitwillbe−10. In other words, your profit will be -10.

Assuming that the market capacity is 4, and there are 12 players/entrants.

Part (a): What is the total industry profit or loss if there are 2 actual entrants? (Note: Please enter your answer rounded to the nearest whole dollar. For instance, if your answer is $45.78, input 46 into your answer box.) (2 marks)

Part (b): What is the total industry profit or loss if there are 5 actual entrants? (Note: Please enter your answer rounded to the nearest whole dollar. For instance, if your answer is $45.18, input 45 into your answer box.) (2 marks)

Part (c): At what rank will the total industry profit or loss be zero? (Note: Please enter your answer as a whole number or integer. For example, if your answer is 9, input 9 into your answer box.) (1 mark)

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Solution

Part (a):

First, we need to calculate the profit for each entrant. Given that the market capacity is 4 and there are 2 actual entrants, we can use the provided formula to calculate the profit for each entrant.

For the first entrant (r=1), the profit is:

Profit = 50∗[(4+1−1)/(1+2+3+4)]=50 * [(4+1-1) / (1+2+3+4)] = 50 * [4 / 10] = $20

For the second entrant (r=2), the profit is:

Profit = 50∗[(4+1−2)/(1+2+3+4)]=50 * [(4+1-2) / (1+2+3+4)] = 50 * [3 / 10] = $15

So, the total industry profit for 2 actual entrants is 20+20 + 15 = $35.

Part (b):

For the first 4 entrants, the profit is calculated as in part (a). For the fifth entrant (r=5), since it is ranked after the market capacity (c=4), it will need to pay 10.So,theprofitforthefifthentrantis−10. So, the profit for the fifth entrant is -10.

So, the total industry profit for 5 actual entrants is 20+20 + 15 + 10(forthethirdentrant)+10 (for the third entrant) + 5 (for the fourth entrant) - 10(forthefifthentrant)=10 (for the fifth entrant) = 40.

Part (c):

The total industry profit or loss will be zero when the sum of the profits of the entrants equals the sum of the losses of the entrants.

From the formula, we can see that the profit decreases as the rank (r) increases. So, the total industry profit or loss will be zero when the rank (r) is such that the profit for that rank is equal to the loss for the next rank (which is $10).

Solving the equation 50∗[(4+1−r)/(1+2+3+4)]=50 * [(4+1-r) / (1+2+3+4)] = 10, we get r = 3.5. Since the rank must be a whole number, we round up to get r = 4.

So, the total industry profit or loss will be zero when the rank is 4.

This problem has been solved

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