Robinson Crusoe benefited from consumption and the utility function displayed positive but diminishing marginal utility. suppose. Also suppose that Crusoe doesn't like to work. The utility function is as follows:

(, ) = (1 ) L

and 1 - indicate consumption and free time. Suppose Crusoe uses only labor to produce . Production function: = () = . (a) (2 points) Find the marginal rate of substitution. Explain what it means. (b) (2 points) Find the marginal productivity of labor. Is the declining marginal productivity assumption maintained? Explain. (c) Write down the (4 points) Lagrange function. Find first order conditions. (d) (3 points) Find the optimal condition. Explain. (e) (2 points) Find the optimal level of labor ( ) and consumption ( ). (f) (4 points) Suppose there is a positive technology shock. Technological development, labor and consumption (at the optimal level) Analyze the effect on it. Interpret your results by discussing the effects of wealth and substitution.