To derive the total cost (TC) and total profit (TP) functions, we begin by defining some basic economic concepts.
Total Cost Function (TC)
The total cost function can be expressed as:
\[ TC = FC + VC \]
Where:
- \( TC \) = Total Cost
- \( FC \) = Fixed Costs (costs that do not change with the level of output)
- \( VC \) = Variable Costs (costs that vary with the level of output)
If we assume that variable costs per unit are constant, we can express variable costs more explicitly:
\[ VC = v \cdot Q \]
Where:
- \( v \) = Variable cost per unit of output
- \( Q \) = Quantity of output produced
Thus, the total cost function can be expressed as:
\[ TC = FC + v \cdot Q \]
Total Revenue Function (TR)
Total revenue (TR) can be determined as follows:
\[ TR = P \cdot Q \]
Where:
- \( TR \) = Total Revenue
- \( P \) = Price per unit of the product
- \( Q \) = Quantity sold
Total Profit Function (TP)
Total profit is defined as the difference between total revenue and total cost:
\[ TP = TR - TC \]
Substituting the previously defined equations for TR and TC, we have:
\[ TP = (P \cdot Q) - (FC + v \cdot Q) \]
This can be simplified to:
\[ TP = P \cdot Q - FC - v \cdot Q \]
\[ TP = (P - v) \cdot Q - FC \]
Summary
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Total Cost Function: \[ TC = FC + v \cdot Q \]
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Total Revenue Function: \[ TR = P \cdot Q \]
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Total Profit Function: \[ TP = (P - v) \cdot Q - FC \]
These equations are fundamental in microeconomics and can be used to analyze the profitability of a business operation.
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