Final answer:
To determine the profit function with given demand and average cost functions, we must first solve for price in terms of quantity, calculate total revenue, and then subtract total costs. The profit function is π = (25 - 2Q)Q - (5Q + 32), and to find the break-even points, we set the profit function to zero and solve for Q.
Step-by-step explanation:
To find an expression for the profit function given the demand function 2Q + P = 25 and the average cost function ATC = 5 + 32/Q, we must first solve the demand function for P to find the price as a function of quantity, Q. Rearranging the demand function, we get P = 25 - 2Q.
Once we have the price, we can calculate total revenue (TR) by multiplying the price (P) by the quantity (Q), TR = P × Q = (25 - 2Q)Q.
To find profit (π), we subtract total cost (TC) from total revenue. Total cost is the average total cost (ATC) times quantity, so TC = ATC × Q = (5 + 32/Q) × Q. Therefore, the profit function is given by:
π = TR - TC = (25 - 2Q)Q - (5Q + 32)
To find the values of Q for which the firm breaks even, we set the profit function to zero (0 = TR - TC) and solve for Q. When price equals average total cost, the firm is breaking even, which means it earns zero economic profit.
If TR = TC, then (25 - 2Q)Q = 5Q + 32. Simplifying this, we find the break-even points for the quantity Q.