Consider a University that operates with the following production function: , where stands for the size of campus, denotes hours of teaching and stands for the number of university graduates (to simplify calculations, assume that is not necessarily

Consider a University that operates with the following production function: $y=K^{0,25} L^{0,5}$, where $K$ stands for the size of campus, $L$ denotes hours of teaching and $y$ stands for the number of university graduates (to simplify calculations, assume that $y$ is not necessarily integer).\na) What is the type of returns to scale for this technology? Can we observe diminishing marginal returns to capital services? Labor services?\nb) Let the prices of capital and labor be $r$ and $w$, respectively. Suppose that in the short run the size of University campus is fixed at $\mathrm{K}=10000$. Derive the University's short-run cost function.\nc) Assuming that the University gets a constant $\$ p$ for every graduate that it produces, derive the University's short-run direct supply function and illustrate its short-run supply curve.\nd) Derive the University's conditional demand functions for labor and capital services and write down its long-run cost function. Derive the University's long-run direct supply function and illustrate its long-run supply curve.