Question

Suppose that the average cost of mining a ton of coal varies with the total weight of coal mined each day in the way shown in Table 1 . $$ \begin{array}{|l|l|l|l|l|l|l|} \hline \text { Tons mined } & 100 & 101 & 102 & 103 & 104 & 105 \\ \hline \text { Average Cost per Ton } & \$ 300 & \$ 299 & \$ 298 & \$ 297 & \$ 296 & \$ 295 \\ \hline \end{array} $$ Compute the marginal cost at each level of production of mining a ton of coal.

Short answer

The marginal cost at each level of production (from 101 to 105 tons mined) is $299.

Step-by-step solution

Define Marginal Cost

Marginal cost represents the additional cost of producing one more unit (one more ton, in this case). Mathematically, it is the first derivative of the total cost with respect to the quantity: \(MC(Q) = \frac{d(TC)}{dQ}\) Where - \(MC\) is the marginal cost, - \(Q\) is the quantity (tons mined), and - \(TC\) is the total cost. Since we are given the average cost, we can find the total cost by multiplying the average cost by the quantity. Then, we will compute the marginal cost for each level of production.

Compute Total Cost

For each row in the table, multiply the tons mined by the average cost per ton to obtain the total cost. Add this information to the table: $$ \begin{array}{|l|l|l|l|l|l|l|} \hline \text { Tons mined } & 100 & 101 & 102 & 103 & 104 & 105 \\ \hline \text { Average Cost per Ton (\$)} & 300 & 299 & 298 & 297 & 296 & 295 \\ \hline \text { Total Cost (\$)} & 30000 & 30299 & 30598 & 30897 & 31196 & 31495 \\ \hline \end{array} $$

Compute the Marginal Cost

The marginal cost is the change in the total cost as production increases by one unit (one ton). For each interval in the production level, subtract the total cost at the previous level of production from the total cost at the current level of production. Marginal cost for 101 tons: \(\frac{\Delta TC}{\Delta Q}=\frac{30299 - 30000}{1} = 299\) Repeat this process for all levels of production and add the marginal costs to the table: $$ \begin{array}{|l|l|l|l|l|l|l|} \hline \text { Tons mined } & 100 & 101 & 102 & 103 & 104 & 105 \\ \hline \text { Average Cost per Ton (\$)} & 300 & 299 & 298 & 297 & 296 & 295 \\ \hline \text { Total Cost (\$)} & 30000 & 30299 & 30598 & 30897 & 31196 & 31495 \\ \hline \text { Marginal Cost (\$)} & - & 299 & 299 & 299 & 299 & 299 \\ \hline \end{array} $$ The marginal cost at each level of production (from 101 to 105 tons mined) is $299.

Key concepts

Average Cost

Understanding average cost is crucial for businesses to assess the efficiency of their production processes. Average cost, often denoted as AC, is the cost of producing one unit of output and is calculated by dividing the total cost of production by the quantity of units produced.

In the context of our exercise, the average cost per ton of coal mined provides a straightforward way to determine the per-unit expense at various levels of production. For example, if the average cost for mining 100 tons of coal is \(300 per ton, it suggests that the total expense incurred for 100 tons is \)30,000. As production increases, economies of scale might kick in, reducing the average cost, which signals increased efficiency up to a certain point.

It's important for students to not only multiply the average cost by the quantity to find the total cost but also to understand how changes in average cost reflect the overall efficiency of production.

Total Cost

Total cost (TC) is a foundational concept in economics, representing the sum of all costs - both fixed and variable - associated with the production of a certain quantity of goods. It integrates expenses such as raw materials, labor, and overhead.

To calculate total cost in our exercise, we multiply the number of tons mined by the average cost per ton. As production scales, the total cost will generally increase, but the rate at which it increases may provide insights into the efficiency and cost structure of the production process. Recognizing the relationship between total cost and quantity produced is vital for making informed decisions about production and pricing strategies.

Quantity of Production

Quantity of production refers to the total output a company produces over a given period. It is an essential metric since it directly affects both the total and average costs and is linked to revenue.

In our example with coal mining, the quantities range from 100 to 105 tons. As quantity changes, it is crucial to understand how this impacts costs and the optimal level of production. Companies need to balance the quantity produced to meet demand while minimizing costs to maximize profits. Tracking how quantity affects costs aids businesses in determining the most cost-effective production levels.

Economic Production Levels

The concept of economic production levels relates to the quantity of production that maximizes profitability. It's a point where the cost to produce an additional unit is lowest, ideally where marginal cost intersects with average cost.

Companies aim to operate at an output level that ensures economies of scale are exploited without falling into diseconomies of scale, where additional production increases average costs. In the coal mining exercise, if the marginal cost remains constant while the average cost decreases with increased quantity, there may be an optimal economic level of production where profits are maximized.

Understanding the delicate balance of production volume and cost efficiency is vital for any business striving to optimize its operations for long-term sustainability and profitability.