Can anyone tell me the difference between economies of scale and increasing returns to scale. It seems to me that observationally they would be equivalent.
Increasing returns to scale are normally understood to refer to a situation in which, as all inputs are increased k-fold, output increases more than k-fold (see here and here). This is equivalent to saying that a k-fold increase in output can be achieved by increasing all inputs less than k-fold. The difference from economies of scale is that returns to scale are about the relation between inputs and output and nothing is implied about costs.
If a firm producing a good X is a price-taker in the markets for all the inputs it uses to produce X, and if those markets do not offer any form of discount for bulk purchases, then a k-fold increase in input quantities will imply a k-fold increase in cost, and vice versa. In those circumstances, the firm will benefit from economies of scale if and only if it enjoys increasing returns to scale.