Why does the minimum point of the average-totalcost curve show the quantity at which the firm is most efficiently supplying output in the short run?

We begin by diving into the concept of Average Total Cost (ATC). This metric gives businesses insight into the cost of producing each unit of their product. It's calculated by dividing the total cost of production by the number of units produced.

Understanding ATC is crucial as it highlights the efficiency of production at varying scales. An important feature of the ATC curve is its U-shape in the short run. Initially, as production increases, average costs decrease due to the spreading of fixed costs over more units (economies of scale). However, after a certain point, costs start to increase as the disadvantages of scale kick in (diseconomies of scale). The lowest point on this U-shaped curve indicates the quantity where the business achieves the most cost-effective production in the short run.

Marginal Cost (MC) is another pivotal concept in understanding a firm's production costs. It represents the cost of producing one additional unit of output. Mathematically, it's the derivative of the total cost function with respect to quantity.

In practice, as a firm ramps up production, the MC initially falls due to increasing efficiencies, but beyond a certain production level, it starts to rise due to factors such as limited capacity and increasing resource prices. The intersection of the MC curve with the ATC curve at the ATC's lowest point has a significant implication: it indicates that any further increase in production will result in a higher average total cost, thus highlighting the most cost-efficient scale of production for the firm in the short run.

Economic Efficiency is the goal for any firm trying to maximize profits while minimizing costs. It's about producing the right quantity of goods or services using the least amount of resources—or, in other words, getting the most bang for your buck.

In the context of the ATC curve, this efficiency is achieved at the minimum point. Producing less than this quantity means the firm has excess capacity, while producing more means incurring unnecessary additional costs. The firm is economically efficient when it operates at the point where the marginal cost of producing an additional unit equals the minimum average total cost.

Short-run Production Costs are those incurred by a firm during a period where at least one input, such as equipment or buildings, is fixed. During this period, firms can only adjust variable inputs like labor and materials to change their level of output.

The concept of short-run costs establishes a temporal framework within which firms can strive for optimal efficiency - often visualized through cost curves. In the short run, a firm has limited ways to adjust to the ideal production level due to fixed costs being locked in. Therefore, understanding the relationship between the ATC and MC is essential to navigate short-run constraints and still operate efficiently, aiming for the point where the two curves intersect.