Final answer:
1. Each firm's fixed cost is 50 and its variable cost is 1/2q². The equation for average total cost is (50 + 1/2q²)/q. 2. The minimum point on the average total cost curve is at quantity 10. At this quantity, marginal cost and average total cost are both equal to 10. 3. Each firm's supply curve is identical to its marginal cost curve above the minimum point on the average variable cost curve.
Step-by-step explanation:
1. Each firm's fixed cost can be determined by substituting zero for the quantity produced (q) in the total cost equation (TC = 50 + 1/2q²). Thus, the fixed cost is TC = 50. Variable cost can be found by calculating the difference between total cost and fixed cost, so variable cost = TC - fixed cost = (50 + 1/2q²) - 50 = 1/2q². The equation for average total cost can be obtained by dividing total cost by the quantity produced: ATC = TC/q = (50 + 1/2q²)/q.
2. Graphing the average total cost curve and the marginal cost curve for quantities from 5 to 15, we find that the minimum point on the average total cost curve is at q = 10. At this quantity, marginal cost and average total cost are both equal to 10.
3. Each firm's supply curve is identical to its marginal cost curve above the minimum point on the average variable cost curve. Using the given marginal cost equation (MC = q), the firm's supply curve can be written as S = MC = q.
4. The market supply curve for the short run, with a fixed number of firms, is the horizontal sum of the individual firms' supply curves. Therefore, the market supply curve is the sum of all the individual firms' supply curves, which can be written as Sᴹ = 9*S = 9*q.
5. To find the equilibrium price and quantity in the short run, we set the market supply equal to the market demand (Qᴰ = Qᴹ). Substituting the given equations for demand (Qᴰ = 120 - P) and market supply (Qᴹ = 9*q), we have 120 - P = 9*q. Solving for P, we get the equilibrium price P = 120 - 9*q. To find the equilibrium quantity, we substitute this equilibrium price into the market supply equation and solve for q: 120 - 9*q = 9*q. Solving for q, we get the equilibrium quantity q = 6.
6. In this equilibrium, each firm produces the quantity determined by the market supply curve, which is q = 6. To calculate each firm's profit or loss, we subtract the average total cost from the equilibrium price and multiply the difference by the quantity produced: Profit = (P - ATC)*q = (120 - ATC)*6. Substituting the equation for average total cost (ATC = (50 + 1/2q²)/q), we can calculate each firm's profit or loss.
7. In the long-run equilibrium with free entry and exit, firms will enter or exit the market until economic profits are driven to zero. This occurs when price equals the minimum average total cost. Therefore, in the long-run equilibrium, the equilibrium price is equal to the minimum average total cost. To find the equilibrium quantity, we substitute this equilibrium price into the market demand equation and solve for quantity. The number of firms in the market will adjust until supply equals demand.
8. In the long-run equilibrium, each firm produces the quantity determined by the market supply curve, which is the same as the market demand curve. The number of firms in the market will adjust until supply equals demand.