Question

Given the total-cost function \(C=Q^{3}-5 Q^{2}+12 Q+75\), write out a variable- cost (VC) function. Find the derivative of the VC function, and interpret the economic meaning of that derivative.

Short answer

The VC function is \(VC = Q^3 - 5Q^2 + 12Q\), and its derivative is \(VC' = 3Q^2 - 10Q + 12\); this derivative is the marginal cost, indicating the cost to produce one more unit.

Step-by-step solution

Identify Total-Cost Function Components

We begin by identifying the components of the total cost (TC) function, which is given as \(C = Q^3 - 5Q^2 + 12Q + 75\). The total cost function typically consists of a variable cost (VC) component and a fixed cost (FC) component. In this function, the fixed cost is the constant term, which is 75.

Formulate the Variable-Cost Function

Knowing that the fixed cost is 75, we can find the variable-cost function by subtracting the fixed cost from the total cost. Thus, the variable cost function (VC) is \(VC = C - 75 = Q^3 - 5Q^2 + 12Q\).

Differentiate the Variable-Cost Function

To find the derivative of the variable-cost function, we take the derivative of \(VC = Q^3 - 5Q^2 + 12Q\) with respect to \(Q\). The derivative is \(VC' = \frac{d}{dQ}(Q^3 - 5Q^2 + 12Q) = 3Q^2 - 10Q + 12\).

Interpretation of the Derivative

The derivative \(VC' = 3Q^2 - 10Q + 12\) represents the marginal cost (MC). It indicates how the variable cost changes for a small change in output \(Q\). Economically, it is the additional cost incurred to produce one more unit of the good. A positive MC means cost increases with additional units, and a negative MC implies cost decreases.

Key concepts

Total-Cost Function

The total-cost function is a fundamental concept in economics used to determine the overall expense associated with producing a particular quantity of goods, denoted by \( Q \). In our given function \( C = Q^3 - 5Q^2 + 12Q + 75 \), **Total Cost (TC)** captures both fixed and variable costs.
- **Fixed Costs (FC)** are constants in the function, like rent or equipment, which do not change with output. In this case, it is 75.
- **Variable Costs (VC)** vary with production levels, which means parts of the function change with different inputs of \( Q \).

Understanding this separation allows us to analyze each section of costs individually, providing insight into how different factors influence overall expenses. Separation is vital for effectively managing resources and predicting future costs.

Variable-Cost Function

Having identified the fixed costs in the total-cost function, we move to express the variable-cost function by excluding these fixed portions.
To derive the **Variable-Cost (VC) function**, we subtract the constant (fixed cost) from the total-cost function. In the provided example, this means our VC function becomes:
\[VC = C - 75 = Q^3 - 5Q^2 + 12Q.\]
This formula highlights the costs that fluctuate with the level of output, encompassing aspects that require change or adjustment as production scales. This is crucial for businesses when planning for different production scenarios, allowing for accurate cost projections when expanding or reducing product output.

Derivative

Finding the derivative of a function is like unwrapping a layer to see how one element changes over another. In economics, especially with cost functions, the derivative shows the rate of change of costs.
For our variable-cost function \(VC = Q^3 - 5Q^2 + 12Q\), differentiating with respect to \(Q\) gives us:
\[VC' = \frac{d}{dQ}(Q^3 - 5Q^2 + 12Q) = 3Q^2 - 10Q + 12.\]
This process, known as differentiation, helps understand how small changes in input \( Q \) affect the total cost. Insights from derivatives are especially valuable in optimizing production processes and minimizing expenses.

Marginal Cost

The concept of marginal cost, derived from the variable-cost function's derivative, represents the added expense of producing an additional unit. Simply put, marginal cost helps decide if producing that next item is financially worthwhile.
In our context, the derivative \( VC' = 3Q^2 - 10Q + 12 \) reveals how costs change as \( Q \) increases by one unit.
- **Positive Marginal Cost**: Indicates rising costs with each additional unit, common in scenarios where resources get scarce.
- **Negative Marginal Cost**: Suggests a decrease in costs, sometimes seen when production efficiencies are present.
This metric is vital for managerial decision-making, enabling firms to adjust production for optimal profits and resource allocation, balancing cost and output seamlessly.