Final answer:
To derive a cost function from a production function, inputs needed for any quantity of output are inverted to show the number of workers required. Then, to maximize profit, differentiate the profit equation with respect to the output, solve for the optimal production level, and compare it to the cost function.
Step-by-step explanation:
To derive the cost function from a production function, we need to relate output quantity to the inputs used to produce it. The production function in the question seems to be represented by c(w, y) = x1w1 + x2w2, where c is the cost, w represents the wage rate, and x1 and x2 represent the quantities of labor used. From the given production table, we can invert the function to show the number of workers (L) needed for a quantity of widgets (Q), represented as L = f(Q). For instance, producing 0.2 widgets requires 1 worker, and producing 1 widget requires 3.25 workers. We can use this inverted function to establish the relationship between production quantity and cost.
To solve the profit maximization problem max py - c(w, y), where p represents the price of the product and y represents the quantity, one would differentiate the profit equation with respect to y, set the derivative equal to zero, and solve for y. The result is the optimal level of production y(p, w) that maximizes profit, which is represented by π(p, w) = py(p, w) - c(w, y).