Final answer:
For a monopolistic firm selling widgets with a market demand p = 160 - 4Q and constant marginal/average cost of $100, the equilibrium quantity sold is 7.5 widgets and the equilibrium price is $130.
Step-by-step explanation:
The market demand for widgets is given as p = 160 - 4Q, where p is the price and Q is the quantity. Since there is only one firm selling widgets, this scenario can be seen as a monopolist setting the quantity such that its marginal cost (MC) equals its marginal revenue (MR). Given that the marginal cost and average cost (AC) is $100, the firm will seek to produce and sell widgets where MC = MR.
For a monopolist, MR is the change in total revenue for a unit increase in quantity sold, which can be derived from the demand curve. Since the market demand is p = 160 - 4Q, the total revenue (TR) is TR = p × Q = (160 - 4Q) × Q = 160Q - 4Q². To find MR, we differentiate TR with respect to Q, yielding MR = 160 - 8Q. Setting MR equal to MC, we have 160 - 8Q = 100, so the quantity Q equals 7.5.
Substituting Q into the demand equation to find p, we have p = 160 - 4(7.5), which equals $130. Therefore, the equilibrium price is $130, and the quantity sold is 7.5 widgets when there is only one firm acting as a monopolist.