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1. Suppose a consumer values consumption and leisure according to the utility function, u(c,) = 0% + log(). (a) Derive the marginal utility of consumption. Is it always positive or always negative? Demonstrate mathematically. (b) Derive the marginal utility of leisure. Is it always positive or always negative? Demonstrate mathematically. (c) Is marginal utility of consumption decreasing in 0? Demonstrate mathematically. (d) Is marginal utility of leisure decreasing in L'? Demonstrate mathematically. 2. Suppose that Peter has a utility function u(c, E) = log 0 + log 6 He has one unit of time h. = 1 and two sources of income, wage income mNs [where N 3 is hours worked) and non-wage income 11'. (a) Suppose that Peter's wage rate 10 = 20,000 and his non-wage income is it = 5, 000. Plot his budget constraint. (b) Compute his optimal consumption and leisure. Using this, nd N 3, his optimal supply of labor. (0) Suppose that Peter's non-wage income is the dividend he receives from stocks. This year the stocks he bought did not do very well. His non-wage income was 11' = 2,000. Plot his new budget constraint. Does his feasible set expand or contract? (d) What is Peter's optimal labor supply and consumption in this case? Do they increase or decrease compared with part (a)? (e) Suppose that Peter's friend Linda has a utility function 11.02, E) = log 0 + 210g(') Both Peter and Linda have a wage rate of 20, 000 and non-wage income of 5, 000. Will Linda work more hours than Peter or less? Can you give an explanation to your