Question

Drive the tr.tc and total profit function

Answer 1

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

1. **Total Cost Function:**
\[
TC = FC + v \cdot Q
\]

2. **Total Revenue Function:**
\[
TR = P \cdot Q
\]

3. **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.