Suppose that a firm has a marginal cost function given by \(M C(q)=q+1\) What is this firm's total cost function? Explain why total costs are known only up to a constant of integration, which represents fixed costs. b. As you may know from an earlier economics course, if a firm takes price ( \(p\) ) as given in its decisions then it will produce that output for which \(p=M C(q)\). If the firm follows this profit-maximizing rule, how much will it produce when \(p=15 ?\) Assuming that the firm is just breaking even at this price, what are fixed costs? c. How much will profits for this firm increase if price increases to \(20 ?\) d. Show that, if we continue to assume profit maximization, then this firm's profits can be expressed solely as a function of the price it receives for its output. e. Show that the increase in profits from \(p=15\) to \(p=20\) can be calculated in two ways: (i) directly from the equation derived in part (d); and (ii) by integrating the inverse marginal cost function \(\left[M C^{-1}(p)=p-1\right]\) from \(p=15\) to \(p=20\) Explain this result intuitively using the envelope theorem.

Understanding the total cost function is crucial for businesses trying to comprehend how their expenses behave at different production levels. In economics, the total cost function represents the combined cost of production, including both fixed and variable costs. Fixed costs are expenses that do not change with the quantity of output produced, such as rent or salaries, while variable costs change with production volume.

Given the marginal cost function, which is the cost of producing an additional unit of output, the total cost function can be derived by integration. For example, for a marginal cost function given by \( MC(q) = q + 1 \), the total cost function \( C(q) \) can be found by integrating the marginal cost function with respect to quantity \( q \). The integration gives \( C(q) = \frac{1}{2}q^2 + q + F \), where \( F \) is the integration constant representing the fixed costs.

It is important to underscore why total costs are known only up to a constant of integration. This simply means that we can determine the behavior of the variable costs, but to obtain a complete total cost function, we must also know the fixed costs, which remain constant irrespective of how much is produced. Since fixed costs do not change, they are represented by a constant in the cost function equation.

Profit maximization is a key goal for any business. It represents the process by which a firm determines the price and output level that returns the greatest profit. To achieve profit maximization, firms set production such that the marginal cost of producing an extra unit is equal to the price of that unit, which is where marginal revenue equals marginal cost.

For the firm described in our exercise, the rule for profit maximization is given by the price \( p \) equaling the marginal cost \( MC(q) \). Therefore, when the price, \( p \), is set at 15, the firm will produce the quantity where \( p = MC(q) \), which happens to be 14 units. This quantity maximizes the firm’s profit given the price.

The concept also extends to determining fixed costs when we know that the firm is just breaking even at a certain price, implying total revenue equals total costs. In this scenario, fixed costs can be solved by substituting the profit-maximizing output level and the price into the total cost function and setting the total cost equal to total revenue. This enables us to determine the constant \( F \), which stands for fixed costs in our total cost function.

The envelope theorem is an elegant component of microeconomic theory that helps understand how a company's profits change when the conditions of its business environment alter, such as the selling price of its products. This theorem tells us that the change in profit resulting from a small change in a parameter, like the selling price, is equivalent to the partial derivative of the profit function with respect to that parameter.

Why does this matter? Imagine the firm raising its price from \( p=15 \) to \( p=20 \) and you want to find out how much profit will increase. Using the envelope theorem, we understand that this profit change can be accurately captured either by direct calculation using the profit function, or by integrating the inverse marginal cost function over the range of the price change.

The true beauty of the theorem lies in its ability to simplify complex economic situations. It allows us to bypass the need for detailed information about all variables and focus only on how a change in price impacts profit. When we integrate the inverse marginal cost function from the old price to the new price, we're applying a practical use of the envelope theorem to predict changes in the firm's profits.