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Unformatted text preview: Homework 1 (Due September, 22nd) Question 1 (50 points): Suppose a firm’s business activities generate a random income of \$x. This firm finances its activities by selling two types of claims (or financial securities). An investor who buys the first type of security earns: min (x,F) An investor who buys the second gets a payoff of: max(x-F,0) Here x is random. By this I mean that sometimes is large and sometimes small. F, on the other hand, is fixed and equal to \$500. a) Draw a graph of the payoff that an investor gets for each claim. Put the random income x on the horizontal axis and the payoff on the vertical axis. b) In light of the discussion of bonds versus stocks, how would you label the first security? How about the second security? Interpret F and x. c) Suppose x can be either \$7000 with probability 0.5 or \$600 with probability 0.5. Calculate the expected payoffs of both securities (Note: don’t worry about computing present values, just calculated the expected payoff in “future” prices). Question 2 (50 points): Let’s begin with a refresher about indifference curves and utility functions. This should be material learned in previous classes and there should be no new material in these first paragraphs. Suppose a consumer values two goods, A and B, and ranks combinations of goods A and B according to a utility function U(X A ,X B ). The graphical representation of these preferences is through indifference curves. Each of these indifference curves represents combinations of A and B that yield a constant utility level. An example could be something like this, If an individual has some resources monetary resources M and prices for goods A and B are given by P A and PB , then an individual’s budget constraint is given by: P A X A + P B X B = M As usual, we do not need to keep track of both prices P_A and P_B separately. We can normalize the price of A, say and let the price of B be the relative price P_B/P_A (good A becomes the numeraire ): X A + P B /P A X B = M A Hopefully, nothing so far is new to you. M A now measures resources in units of good A and all we’ve done is just renormalize everything by the price of good A without really altering any of the substance. Graphically that budget constraint is a downward sloping line, the slope is equal to –(P B /P A ), and points on or below that line are all feasible for our X A X B consumer. The optimal allocation is a point on the budget constraint in which the ratio of marginal utilities equals (P B /P A ). In class we have talked about how financial markets (and intermediaries) exist because they allow for “inter-temporal” trade: economic agents want to consume at different points in time; if they have an excess of funds they lend and if they have a deficit of funds, they borrow....
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