Question

Given: \(\mathrm{TC}=\mathrm{FC}+\mathrm{VC}\) prove that: \(\quad \mathrm{ATC}=\mathrm{AFC}+\mathrm{AVC}\)

Short answer

To prove that \(\mathrm{ATC} = \mathrm{AFC} + \mathrm{AVC}\), start with the given equation: \(\mathrm{TC} = \mathrm{FC} + \mathrm{VC}\). Divide both sides by output (Q) to get: \(\frac{\mathrm{TC}}{Q} = \frac{\mathrm{FC}}{Q} + \frac{\mathrm{VC}}{Q}\). Now, substitute the formulas for ATC, AFC, and AVC into the equation: \(\mathrm{ATC} = \frac{\mathrm{TC}}{Q}\), \(\mathrm{AFC} = \frac{\mathrm{FC}}{Q}\), and \(\mathrm{AVC} = \frac{\mathrm{VC}}{Q}\). This gives us the desired result: \(\mathrm{ATC} = \mathrm{AFC} + \mathrm{AVC}\).

Step-by-step solution

Defining Costs

Let's begin with our given equation: \(\mathrm{TC} = \mathrm{FC} + \mathrm{VC}\) Here, Total Cost (TC) is comprised of Fixed Costs (FC) and Variable Costs (VC). Now we need to find ATC, AFC, and AVC, the average versions of these costs with respect to output.

Finding Average Costs

First, let's find the Average Total Cost (ATC). We calculate ATC by dividing the Total Cost (TC) by the output, which we represent as Q: \(\mathrm{ATC} = \frac{\mathrm{TC}}{Q}\) Next, we find the Average Fixed Cost (AFC) by dividing the Fixed Costs (FC) by the output (Q): \(\mathrm{AFC} = \frac{\mathrm{FC}}{Q}\) Finally, we find the Average Variable Cost (AVC) by dividing the Variable Costs (VC) by the output (Q): \(\mathrm{AVC} = \frac{\mathrm{VC}}{Q}\) Now that we have the average cost formulas, let's move on to the proof.

Proving the Equation

We know that: \(\mathrm{TC}=\mathrm{FC}+\mathrm{VC}\) Now, divide both sides of the equation by Q: \(\frac{\mathrm{TC}}{Q} = \frac{\mathrm{FC} + \mathrm{VC}}{Q}\) Now, we can separate the terms on the right side: \(\frac{\mathrm{TC}}{Q} = \frac{\mathrm{FC}}{Q} + \frac{\mathrm{VC}}{Q}\) Now, recall the formulas we derived for ATC, AFC, and AVC: \(\mathrm{ATC} = \frac{\mathrm{TC}}{Q}\) \(\mathrm{AFC} = \frac{\mathrm{FC}}{Q}\) \(\mathrm{AVC} = \frac{\mathrm{VC}}{Q}\) Replace the terms in the equation with these cost formulas: \(\mathrm{ATC} = \mathrm{AFC} + \mathrm{AVC}\) Using this, we have now successfully proven that the average costs add up as expected: \(\mathrm{ATC} = \mathrm{AFC} + \mathrm{AVC}\)

Key concepts

total cost

When discussing total costs in economics, it's essential to understand they are the sum of all the expenses a firm incurs to produce a certain level of output. Think of it like mixing two ingredients: fixed costs and variable costs. Total costs give you the complete picture of how much it costs to produce goods or services.
Remember this handy formula: \( \text{TC} = \text{FC} + \text{VC} \). Here, \( \text{TC} \) stands for Total Cost, \( \text{FC} \) for Fixed Costs, and \( \text{VC} \) for Variable Costs.

• Fixed Costs (FC): These are costs that remain constant, regardless of the level of output. For example, rent or salaries of permanent staff. They do not change even if you produce more or less.


• Variable Costs (VC): These costs change with the level of output. Think of raw materials or utility bills. They increase as you produce more and decrease when you produce less.

By combining these two types of costs, you get the total cost of production, which helps in pricing decisions and profit calculation.

fixed costs

Fixed costs represent the expenses that do not fluctuate with the level of production. Imagine you're running a bakery. Whether you bake 10 or 1000 bread loaves a day, some costs like rent or the salary of your full-time employees do not change. They are predictable, making budget planning easier.
Consider these points when you think of fixed costs:

• They provide stability in cost management since they remain unchanged in the short term.


• Though they don't change with production levels, as the scale of operations increases, they can effectively spread their burden across more units, lowering average fixed costs.

The formula used to find Average Fixed Costs is \( \text{AFC} = \frac{\text{FC}}{Q} \), where \( Q \) is the output quantity. As production rises, the Average Fixed Cost decreases because the same cost gets distributed over more products.

variable costs

Variable costs are the dynamic component of a firm's cost structure. These expenses vary directly with the level of production. Every time you decide to make more goods, your variable costs will rise. Think of the bakery example—buying more flour or using more electricity means an increase in variable costs.
These points are key to understanding variable costs:

• Variable costs are often linked to the marginal cost of producing an additional unit. Thus, they directly affect decisions on how much to produce.


• They fluctuate with the volume of production, meaning they provide flexibility in scaling operations up or down.

To calculate the Average Variable Cost, use the formula \( \text{AVC} = \frac{\text{VC}}{Q} \). This shows how variable costs spread over units produced, offering insight into efficiency and cost control as production scales.