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Which of the following mathematical expressions best describes a mixed cost?


A$Y = bX$
B$Y = a$
C$Y = a + bX$
D$Y = a_{i}$

Verified Answer :

  • A mixed cost is defined as a cost containing both fixed and variable elements.
  • The linear cost equation \(Y = a + bX\) identifies \(a\) as the fixed cost and \(bX\) as the total variable cost.
  • Option C (\(Y = a + bX\)) is the only expression that correctly includes both components.
  • The correct choice is \(\boxed{C}\).

Explanation

  1. Defining Mixed Cost

A mixed cost, also known as a semi-variable or semi-fixed cost, is a type of cost that contains both a fixed component and a variable component. The fixed component remains constant regardless of the level of activity, while the variable component changes in direct proportion to the level of activity. In cost accounting, this is represented by a linear cost function:
\[Y = a + bX\]
where:

  • \(Y\) is the total cost.
  • \(a\) is the fixed cost component (the intercept on the y-axis, representing the cost when the activity level is zero).
  • \(b\) is the variable cost per unit of activity (the slope of the line).
  • \(X\) is the activity level or volume (the independent variable).
  1. Analyzing the Options

We evaluate the provided options against our definition of a mixed cost:

  • Option A (\(Y = bX\)): This expression represents a purely variable cost, as it only contains a variable component proportional to the activity level \(X\), with no fixed cost component (\(a = 0\)).
  • Option B (\(Y = a\)): This expression represents a purely fixed cost, as it does not change with the activity level \(X\) (the variable component is zero).
  • Option C (\(Y = a + bX\)): This expression includes both a constant term \(a\) (fixed cost) and a term \(bX\) (variable cost), which is the standard definition of a mixed cost.
  • Option D (\(Y = a_i\)): This is not a standard representation of a cost function in this context.

Therefore, the expression that best describes a mixed cost is \(Y = a + bX\).

  1. Final Conclusion

Based on the analysis, we identify option C as the correct representation of a mixed cost. The total cost consists of a fixed portion \(a\) and a variable portion \(bX\), which aligns perfectly with the definition of a semi-variable or mixed cost. Thus, the correct choice is C.
\[\boxed{C}\]

Examples

Consider a delivery service company. They have a fixed monthly lease payment of \(\$1,000\) for their trucks (the fixed cost, \(a\)) and they pay \(\$0.50\) in fuel and maintenance for every mile driven (the variable cost per unit, \(b\)). If the company drives \(X\) miles in a month, the total cost for the month \(Y\) would be calculated as:
\[Y = 1000 + 0.5X\]
This formula represents a mixed cost, where the total expenditure combines constant monthly fees with costs that fluctuate based on actual operational mileage.

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