1. Consider the Cobb-Douglas production function =(,)=0.50.5 where is output, is a parameter, is capital, and is labour. a) Explain the role of the parameter in the production function. b) Derive and interpret the marginal products of labour and capital. Does the marginal product of capital depend on the amount of labour used? Explain why that might be the case with a real-world example. c) Solve for the marginal rate of technical substitution (MRTS). In- terpret the meaning of the MRTS for a producer using four units of capital and ten units of labour. d) With capital fixed at 100 units, i.e., = 100, graph the produc- tion function with on the horizontal axis and on the vertical axis. Draw the graph for values of from 0 to 200. Indicate the marginal product of labour on the graph. Does this production function have the property of a diminishing marginal product of labour? Explain. e) Re-draw the graph from the previous part, and then add a new line that corresponds to the short-run production function when = 200. At a fixed value of labour, say 50 units, how does the marginal product of labour differ between the two lines? Explain what your graph demonstrates.