Short answers, pulled from the story.
The three types are constant returns to scale (CRS), decreasing returns to scale (DRS), and increasing returns to scale (IRS). CRS means output increases by the same proportion as inputs; DRS means output increases by less than the proportional change in inputs; IRS means output increases by more than the proportional change in inputs.
Decreasing returns to scale occur primarily because of increased management difficulties associated with a larger scale of production and a lack of coordination across all stages of production. These factors reduce production efficiency as the firm expands.
Increasing returns to scale occur because expanding the scale of production allows firms to adopt more advanced and specialized technologies, resulting in more streamlined and efficient production processes that are not available at smaller scales.
When factor costs are constant and the production function is homothetic, constant returns to scale correspond to constant long-run average costs, decreasing returns correspond to increasing long-run average costs, and increasing returns correspond to decreasing long-run average costs. This relationship breaks down if the firm faces imperfectly competitive input markets.
In the Cobb-Douglas production function, returns to scale depend on the sum of the output elasticities b and c. If b plus c is greater than 1, the firm has increasing returns to scale; if b plus c equals 1, returns are constant; if b plus c is less than 1, returns are decreasing.
Returns to scale is a long-run concept because it requires all factors of production to be variable simultaneously. In the short run, some inputs such as buildings or machinery are fixed. Only in the long run can a firm change all inputs by building new facilities, investing in new machinery, or improving technology.