Consider a rm that operates using only a single variable input, labor (denoted f). The production function of this rm is given by: cw) = q = 2d? Where C] denotes output. The rm is a price-taker in both the output market and the input market for labor. Each unit of labor costs the rm w > 0. The market price for the rm's output is given by P > O. (a) Show that the rm's total cost function is TC (q ; w) = ng. 2 (b) Show that the rm's prot function is if" (P, w) = PE. Show that 11\"\" (P, w) is homogeneous of degree one (with respect to output and input prices) and interpret this result. (c) Derive the rm's supply function, (3* (P, w). Show that q* (P, w) is homogeneous of degree zero (with respect to output and input prices) and interpret this result. ((1) Derive the rm's (unconditional) labor demand function, {*(P, w). Show that {*(P, w) is homogenous of degree zero (with respect to output and input prices) and interpret this result. (e) Derive the comparative statics terms 6q*/6P, 6q*/aw, 63* /6P, and 61'?\" /6w and interpret these results