Returns to scale is a concept that examines how output responds when a firm proportionately increases all of its inputs in the long run. This concept is crucial for understanding production efficiency and economies of scale. A proportionate increase in inputs means that all the inputs are increased by the same percentage or factor in the production process. For example, if a firm decides to double its inputs, it would increase its labor force and capital investment by 100%, maintaining the same ratio among these inputs as before the increase.

Increasing returns to scale occurs when a proportionate increase in all inputs leads to a larger proportional increase in output. Increasing returns to scale occurs when a proportional increase in all inputs leads to a more than proportional increase in output. Mathematically, if all inputs are multiplied by a factor λ > 1, output increases by more than λ.

Two primary reasons for increasing returns to scale are:

1. Specialization: As production scales up, workers can focus on specific tasks, enhancing overall efficiency.

2. Indivisibility of large inputs: Certain inputs, like factories or advanced machinery, are only efficiently utilized at larger production scales. These large-scale inputs can only be effectively utilized once production reaches a certain level.

It is important to note that increasing returns to scale doesn't continue indefinitely. As firms grow larger, they often encounter managerial diseconomies or other factors that lead to constant or even decreasing returns to scale.