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Consider an investor who has initial wealth w and has to decide how to invest it. There is a riskless asset with rate of return r. The risky asset has return xi with probability pi , i = 1, . . . , n. Denote by the total amount of wealth that the investor puts into the risky asset. Write the investor's problem. Show that if the investor has decreasing absolute risk aversion, then the amount of of wealth invested in the risky asset increases with w (that is, d/ dw > 0.

Use the blue line (circle symbol) to plot Andrew's budget constraint on the following graph. Next, use the orange point (square symbol) to shade the area that represents combinations of juice and cereal that are affordable for Andrew. Finally, place the black point (plus symbol) on the point on Andrew's budget constraint that corresponds to a scenario in which Andrew spends $16 on each good. Note: Dashed drop lines will automatically extend to both axes. 24 O 22 20 BC, ($32) 18 O 16 14 Affordable Region 12 CEREAL (Boxes) $16 on Each A 4 BC, ($40) 2 2 4 6 8 10 12 14 16 18 20 22 24 JUICE (Gallons)8. An individual's budget Suppose Andrew has a weekly budget of $32 to spend on juice and cereal. Juice is priced at $4 per gallon, and cereal is priced at $2 per box. If Andrew spends his entire $32 on juice, he can buy gallons of juice. If he spends his entire $32 on cereal, he can buy boxes of cereal.What does the slope of Andrew's budget constraint represent? O The opportunity cost of an additional box of cereal in terms of gallons of juice O The cost of an additional gallon of juice in terms of dollars O The opportunity cost of an additional gallon of juice in terms of boxes of cereal O The cost of an additional box of cereal in terms of dollars Suppose Andrew receives $8 from his grandmother and decides to dedicate this money to buying more juice and cereal. Using the green line (triangle symbol), draw Andrew's new budget constraint on the previous graph. True or False: Andrew faces a new tradeoff between juice and cereal. O True O False5. Imperfect Competition: The inverse market demand curve for bean sprouts is given by p(y) = 100 - y, and the total cost function for any firm in the industry is given by TC(y) = 4y. (a) What is the marginal cost for any firm in the industry? What is the change in price for a one unit increase in output? (b) If the industry were perfectly competitive, what would the industry output and price be? (c) Suppose that two Cournot firms operated in the market. What are the firms' reaction functions? What would industry output be? What would each firm's output be? And what would the market price be. (d) For the Cournot case, draw the two reaction curves and indicate the equilibrium point. (e) If the two firms decided to collude, what would be industry output? And what would the market price be? (f) Suppose both the colluding firms are producing equal amounts of output. If one of the colluding firms assumes that the other firm would not react to a change in industry output, what would happen to a firmOs own profits if it increased its output by one unit? (g) Suppose one firm acts as a Stackleberg leader and the other firm behaves as a follower. What is the maximization problem for the leader? Solving this problem leads to what output levels for the leader and the follower? This implies what level of industry output and what industry price? 6. The Competitive Fringe: Consider a market with one large firm and many small firms. The supply curve of the small firms taken together is S(p) = 100 + p. The market demand curve for the product is D(p) = 200 - p. The cost function for the one large firm is c(y) = 25y. (a) Suppose that the large firm is forced to operate at a zero level of output. What will be the equilibrium price? What will be the equilibrium quantity? (b) Suppose now that the large firm tries to exploit its market power by setting a profit maximizing price. In order to model this we assume that customers always go first to the competititve firms and buy as much as they are able to and then go to the large firms. In this situation, determine the large firm's residual demand curve, and use this to find the equilibrium price, the quantity supplied by the large firm, and the equilibrium quantity supplied by the competitive firms. What will be the large firm's profits