Prove that a profit-maximizing firm will always minimize costs.
A profit-maximizing firm will always minimize costs because the goal of a profit-maximizing firm is to maximize profits. This means that the firm will seek to minimize costs in order to maximize profits.
The profit of a firm is equal to total revenue minus total costs. Therefore, if a firm wants to maximize its profits, it must minimize its costs. This is because if the firm can reduce its costs, it can increase its profits.
For example, if a firm can reduce its costs by 10%, it can increase its profits by 10%. This is because the total revenue will remain the same, but the total costs will be reduced by 10%. Therefore, the firm will be able to increase its profits by 10%.
Therefore, it is clear that a profit-maximizing firm will always seek to minimize costs in order to maximize profits.