<Table> <Tr> <Td> </Td> <Td> This article needs additional citations for verification . Please help improve this article by adding citations to reliable sources . Unsourced material may be challenged and removed . (July 2016) (Learn how and when to remove this template message) </Td> </Tr> </Table> <Tr> <Td> </Td> <Td> This article needs additional citations for verification . Please help improve this article by adding citations to reliable sources . Unsourced material may be challenged and removed . (July 2016) (Learn how and when to remove this template message) </Td> </Tr> <P> In economics, returns to scale and economies of scale are related but different terms that describe what happens as the scale of production increases in the long run, when all input levels including physical capital usage are variable (chosen by the firm). The term returns to scale arises in the context of a firm's production function . It explains the behavior of the rate of increase in output (production) relative to the associated increase in the inputs (the factors of production) in the long run . In the long run all factors of production are variable and subject to change due to a given increase in size (scale). While economies of scale show the effect of an increased output level on unit costs, returns to scale focus only on the relation between input and output quantities . </P> <P> The laws of returns to scale are a set of three interrelated and sequential laws: Law of Increasing Returns to Scale, Law of Constant Returns to Scale, and Law of Diminishing returns to Scale . If output increases by that same proportional change as all inputs change then there are constant returns to scale (CRS). If output increases by less than that proportional change in inputs, there are decreasing returns to scale (DRS). If output increases by more than the proportional change in inputs, there are increasing returns to scale (IRS). A firm's production function could exhibit different types of returns to scale in different ranges of output . Typically, there could be increasing returns at relatively low output levels, decreasing returns at relatively high output levels, and constant returns at one output level between those ranges . </P>

Prove that production function is decreasing returns to scale in terms of input factor k