A company models its production costs with a linear function. Initially, the base cost function is given by \(f(x) = ax + b\), where \(x\) is the number of units produced, \(a\) represents the variable cost per unit, and \(b\) represents fixed costs. \(a\) and \(b\) are positive constants.
Due to a new tax policy, an additional tax of \(t\) dollars per unit is imposed, transforming the cost function to \(g(x) = (a+t)x + b\). Assume \(t\) is a positive constant.
Answer the following parts:
(a) Describe the meaning of the parameters \(a\) and \(b\) in the base cost function within a financial context.
(b) Explain how the introduction of a per-unit tax \(t\) transforms the cost function. Describe the graphical impact of this change.
(c) Determine the production level \(x\) at which the original cost function \(f(x)\) and the new cost function \(g(x)\) intersect, and explain the implication of a non-zero tax rate on this intersection.
Question
Given the base cost function \(f(x) = ax + b\), describe how the parameters \(a\) and \(b\) influence the cost structure. Explain what each parameter represents in a financial context.
Hint
Hint
Hint
Consider the roles of fixed costs and variable costs.
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Answer
Answer
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Explanation
Explanation
- Formula Booklet
Complete Answer
Parameter \(a\) represents the variable cost per unit, while \(b\) represents the fixed cost.
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Question
Due to a new tax policy, an additional tax of \(t\) dollars per unit is added, transforming the cost function to \(g(x) = (a+t)x + b\). Explain the transformation that has occurred and describe its graphical effect on the cost function.
Hint
Hint
Hint
Focus on the change in the slope of the function and how it affects the graph.
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Answer
Answer
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Explanation
Explanation
- Formula Booklet
Complete Answer
The transformation is an increase in the variable cost per unit by \(t\). Graphically, this results in an increase in the slope of the cost function from \(a\) to \(a+t\), making the graph steeper. The y-intercept remains unchanged.
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Question
Determine the production level \(x\) at which the original cost function \(f(x) = ax + b\) and the new cost function \(g(x) = (a+t)x + b\) intersect. Describe the effect of a non-zero tax \(t \neq 0\) on this intersection point.
Hint
Hint
Hint
Set \(f(x)\) equal to \(g(x)\) and solve for \(x\). Consider what happens when \(t\) is not zero.
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Answer
Answer
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Explanation
Explanation
- Formula Booklet
Complete Answer
The functions intersect at \(x = 0\). A non-zero tax \(t
eq 0\) ensures that the only intersection point is at \(x=0\), meaning that for any positive production level, the new cost function \(g(x)\) will always be higher than the original cost function \(f(x)\).
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