Final answer:
To solve for equilibrium output, find the level at which MC equals MR. Then, use this output to calculate the price. The first question results in Q = 65 and p* = $4,226.92, while the second question yields Q = 20 and p* = $2.01.
Step-by-step explanation:
To solve for equilibrium output, we need to find the level of output at which marginal cost (MC) equals marginal revenue (MR). We can then use this equilibrium output to calculate the price (p*).
1. First, find the level of output at which MC equals MR: In this case, the intersection occurs at Q = 65.
2. Next, use the equilibrium output to find the price (p*): We can use the given formula TC = Q3, along with the equilibrium output of Q = 65, to calculate the total cost (TC). By substituting the values into the formula, we find that TC = 65^3 = 274,625.
Finally, divide TC by the equilibrium output to find the price:
p* = TC/Q = 274,625/65 = $4,226.92 (rounded to the nearest cent).
Following the same steps for the second question:
1. At the intersection of MC and MR, we find Q = 20.
2. Using Q = 20, we can calculate the total cost (TC) using the given formula AC = 5/Q + 2Q. By substituting the value of Q into the formula, we get AC = 5/20 + 2*20 = 0.25 + 40 = 40.25.
Finally, divide AC by the equilibrium output (20) to find the price:
p* = AC/Q = 40.25/20 = $2.01 (rounded to the nearest cent).