Understanding the Cost Relationship: MC and AC

In microeconomics, understanding the relationship between a firm's costs is crucial for analyzing its production decisions. Specifically, the connection between Marginal Cost (MC) and Average Cost (AC) in the short run is fundamental. Marginal Cost is the additional cost incurred by producing one more unit of output, while Average Cost is the total cost divided by the total quantity produced.

Defining Key Cost Concepts

• Total Cost (TC or C): The total expenditure incurred by a firm in producing a certain level of output. It is a function of the quantity produced, denoted as $C = f(X)$.
• Average Cost (AC): The cost per unit of output. It is calculated as $AC = \frac{TC}{X}$.
• Marginal Cost (MC): The change in total cost resulting from producing one additional unit of output. Mathematically, it is the derivative of the Total Cost function with respect to quantity: $MC = \frac{dTC}{dX}$.

Deriving the Relationship Between MC and AC

We can derive the relationship between MC and AC starting from the definitions. We know that Total Cost (TC) can be expressed as the product of Average Cost (AC) and the quantity (X):

Now, let's find the Marginal Cost (MC) by differentiating the Total Cost (TC) with respect to the quantity (X). We need to use the product rule for differentiation, which states that if $y = u \times v$, then $\frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx}$. In our case, $TC$ is a function of $X$, $AC$ is also a function of $X$, and we are differentiating with respect to $X$.

Applying the product rule:

Since $\frac{dX}{dX} = 1$, the equation simplifies to:

Here, $\frac{dAC}{dX}$ represents the rate of change of Average Cost with respect to quantity, which is the slope of the Average Cost curve.

Analyzing the Options

Let's examine the given options in light of our derived relationship, $MC = AC + X \frac{dAC}{dX}$, where $\frac{dAC}{dX}$ is the slope of the AC curve.

• Option 1 & 2: $slope of AC = $\frac{1}{X} [AC-MC]$. Rearranging our derived formula gives $MC - AC = X \frac{dAC}{dX}$, which means $\frac{dAC}{dX} = \frac{MC - AC}{X}$. Options 1 and 2 present $\frac{dAC}{dX} = \frac{AC - MC}{X}$, which is the negative of the correct relationship.
• Option 3: $MC = AC + {X}. {slope of AC}$. This directly matches our derived formula $MC = AC + X \frac{dAC}{dX}$. The notation `{X}. {slope of AC}` implies $X$ multiplied by the slope of AC.
• Option 4: $X^2$. {slope of AC}= $X^2$.MC-C$. This does not align with the standard derivation of the MC-AC relationship.

Conclusion on the Correct Relationship

The correct mathematical expression that represents the relationship between Marginal Cost (MC), Average Cost (AC), and the slope of the Average Cost curve is:

Therefore, Option 3 accurately captures this relationship.