A startup firm, Loopy Transit is planning to construct a hyperloop between Pittsburgh and Columbus, Ohio.A hyperloop is a tube through which people or goods can be transported at very rapid speeds.The hyperloop would rely upon three inputs: capital, labor, and fuel.A production function for the hyperloop has been estimated as:

T = aK^bL^cF^d = 0.02K^0.25L^0.2 F^0.55

where T = the number of trips taken through the loop annually, K is capital, L is labor, and F is fuel.

a.If Loopy employs 100 units of capital, 500 units of labor and 20,000 units of fuel, calculate the marginal products if Loopy increases input use by 1 for K, L and F.

b.What is Loopy's marginal rate of technical substitution (MRTS) between K and L?What about MRTS between K and F?Would Loopy try to ensure that all its MRTSs are equal?Why?

c.Does the law of diminishing returns apply to K in the production of trips between Pittsburgh and Columbus?What about to L or F?Why do you say this?Would the law of diminishing returns apply to L if c = -0.2 instead of 0.2?What if c = 1.2?

d.Loopy's production function exhibit constant, increasing or decreasing returns to scale?Why do you say this?Would your answer change if c = -0.2 instead of 0.2?What if c = 1.2?