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problems dealing with a monopoly marginal cost

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I need help on these two microeconomic theory problems dealing with a monopoly marginal cost. See attached file for full problem description.

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2. Suppose the quantity demanded is given by qd=90-p and a monopolist's marginal cost is given by MC=q (i.e. the marginal cost of the first unit (q=1) is 1; the marginal cost of the second unit (q=2) is 2, etc.).
a. What is the monopolist's marginal revenue curve?
b. What is the monopolist's profit maximizing quantity of production? What is the optimal price for him to charge? Assuming the monopolist has no fixed costs, what is his maximum profit?
c. Assuming the monopolist is maximizing his profits, what is consumer surplus (in dollars)?
d. What is the socially optimal (or efficient) level of production? What would total surplus be at that level (in dollars)?
e. What is the dead weight loss from the existence of a monopoly in this market (in dollars)?

3. In problem 2, indicate whether CS, PS, total surplus and dead weight loss would each increase or decrease if:
a. The government imposed a 50% tax on all economic profits.
b. The government imposed a $6 tax on the production of good y.
c. The government imposed a $6 tax on the consumption of y.
d. The monopolist can perfectly (first degree) price discriminate - i.e. the monopolist can tell how much each consumer is willing to pay and can charge each consumer a different price.
e. The government set a price ceiling at $45. (Hint: How does this change the marginal revenue curve?)

Question 2

Part a
In order to find the marginal revenue curve, we must first find the inverse demand function:

Now we can use the following property. When the inverse demand function is given by
, then marginal revenue is given by . Therefore, in this case, we get that the marginal revenue must be:

Part b
The profit maximizing quantity of production is found by equating marginal revenue (MR) to marginal cost (MC). We're told that . Therefore, the profit maximizing quantity is the solution to:

Thus the monopolist should produce 30 units. In order to find the price charged, we simply plug this quantity into the inverse demand function. We thus get that the price charged is .

Now, since the monopolist has no fixed costs and since we're told that MC = Q, we can find ...

Solution Summary

This post assists with problems dealing with a monopoly marginal cost.