Final answer:
In equilibrium, each firm will produce around 233.68 units. The market equilibrium price is approximately $8.66. The total market demand is approximately 34,467.04 units. The equilibrium number of firms in the long-run is approximately 147.
Step-by-step explanation:
To find the equilibrium in the long-run for each firm, we need to set the long-run total cost (LRTC) equal to the price (P). Using the equation for LRTC = 10q³ − 5q² + 20q and the market demand equation Q = 39,000−2,000P, we can solve for q and P. To find the equilibrium number of firms in the long-run, we need to calculate the total market demand and divide it by the quantity produced by each firm.
(a) To find the equilibrium quantity produced by each firm, we set the long-run total cost equal to the price: 10q³ − 5q² + 20q = P. Since the demand equation is Q = 39,000−2,000P, we substitute P into the demand equation to get Q = 39,000−2,000(10q³ − 5q² + 20q). Solving for q, we find that each firm will produce approximately 233.68 units.
(b) To find the market equilibrium price, we substitute the equilibrium quantity (233.68) into the demand equation: Q = 39,000−2,000P. Rearranging the equation, we get P = (39,000 - Q)/2,000. Substituting Q = 233.68, we find that the market equilibrium price is approximately $8.66.
(c) To find the total market demand, we substitute the equilibrium quantity (233.68) into the demand equation: Q = 39,000−2,000P. Substituting P = 8.66, we find that the total market demand is approximately 34,467.04 units.
(d) To find the equilibrium number of firms in the long-run, we divide the total market demand by the quantity produced by each firm: 34,467.04 / 233.68. So, the equilibrium number of firms in the long-run is approximately 147.