increasing long-run average costs, and a firm experiencing increasing returns will have decreasing long-run average costs. However, this relationship breaks down if the firm does not face perfectly competitive factor markets (i.e., in this context, the price one pays for a good does depend on the amount purchased). For example, if there are increasing returns to scale in some range of output levels, but the firm is so big in one or more input markets that increasing its purchases of an input drives up the input's per-unit cost, then the firm could have diseconomies of scale in that range of output levels. Conversely, if the firm is able to get bulk discounts of an input, then it could have economies of scale in some range of output levels even if it has decreasing returns in production in that output range. Formal definitions Formally, a production function is defined to have: Constant returns to scale if (for any constant a greater than 0) (Function F is homogeneous of degree 1) Decreasing returns to scale if (for any constant a greater than 1) Increasing returns to scale if (for any constant a greater than 1) where K and L are factors of production—capital and labor, respectively. In a more general set-up, for a multi-input-multi-output production processes, one may assume technology can be represented via some technology set, call it , which must satisfy some regularity conditions of production theory. In this case, the property of constant returns to scale is equivalent to saying that technology set is a cone, i.e., satisfies the property . In turn, if there is a production function that will describe the technology set it will have to be homogeneous of degree 1. Formal example The Cobb–Douglas functional form has constant returns to scale when the sum of the exponents is 1. In that case the function is: where and . Thus Here as input usages all scale by the multiplicative factor a, output also scales by a and so there are constant returns to scale. But if the Cobb–Douglas production function has its general form with and then there are increasing returns if b + c > 1 but decreasing returns if b + c < 1, since which for a > 1 is greater than or less than as b+c is greater or less than one. See also Diseconomies of scale and Economies of scale Economies