Download Econ 712 HW #4: Asset Prices & Future Contracts in Lucas Tree Economy and more Assignments Introduction to Macroeconomics in PDF only on Docsity! Econ 712 - Homework # 4 (Version 3 - Issued 10/3/07) Rody Manuelli Due on October 11, 2007 [give it to Kyoung Jin] Problem 1 (Asset Prices and Martingales) It is sometimes claimed that efficient markets imply that the change in the price of an asset cannot be fore- cast. The argument goes along the following lines: “If a price change can be anticipated, then this provides an arbitrage opportunity and individuals will take advantage of that until price changes are not forecastable.” Let us define a fore- cast as a conditional expectation. Thus the forecast, at time t, of next period’s stock price is p̄st ≡ Et[pst+1], where Et stands for the conditional expectation operator. Thus, one interpreta- tion of the popular claim is that p̄st = p s t , or Et[p s t+1 − pst ] = 0. 1. Consider a Lucas tree economy. Discuss whether the assertion is correct in the case that {pst} is the equilibrium price of an asset which is a claim to a stream of dividends {dt}. 2. Does your answer to the previous point depend on the nature of the {dt} process? In particular, do your findings apply to bonds as well as stocks that display arbitrary patterns of correlation of their dividend payments with equilibrium consumption? 3. Let {Xt} be any sequence of random variables (a stochastic process if you will). {Xt} is a martingale if Xt = Et[Xt+s], for s ≥ t. In terms of this notation, your analysis in 1 determined whether asset prices are martingales. Consider the following random variable: Yt = β tu0(ct)pst + tX j=0 βju0(cj)dj , 1 where pst is the price of a claim to the stream of dividends {dt}. Show that in equilibrium {Yt} is a martingale. Problem 2 (Future Contracts) Consider a Lucas tree economy. Assume that markets are complete (i.e. claims to the fruit that drops from the trees and any other assets (e.g. bonds) are traded). Preferences are given by U = E[ ∞X t=0 βtu(ct)], where 0 < β < 1, u is strictly concave and “nice”. An individual wants to lock, at time t, the interest rate he will have to pay for a one period (risk free) loan at time t+s (i.e. he borrows at t+s and repays the loan at t+ s+ 1). 1. Go as far as you can determining that rate. [Hint: Use arbitrage argu- ments] 2. Go as far as you can describing the impact on this interest rate of news that affect the distribution of consumption at t+ j, with j < s, but do not change the distribution of equilibrium consumption for t+j+k, for k > 0. Problem 3 (Markets and Inequality) Consider a Lucas tree economy in which there are two types of trees. Let X be a random variable whose value will be realized at time 1. At t = 0, its distribution is known, and it is given by Pr[X = 0] = Pr[X = 1] = 1/2. Both types of trees drop one unit of consumption at t = 0. From then on, the dividends (fruit) is given by dAt = ½ 2 if X = 1, for all t ≥ 1 0 if X = 0, for all t ≥ 1 and dBt = ½ 0 if X = 1, for all t ≥ 1 2 if X = 0, for all t ≥ 1 Assume that there are two types of consumers, I and II. Both types have the same preferences given by U = E[ ∞X t=0 βtu(ct)], where 0 < β < 1, u is strictly concave with u(0) = 0, and u0(0) <∞. Consumers differ in terms of their initial endowment of trees. Type I consumers each owns one type A tree, and type II consumers each own a type B tree. 1. Assume that at t = 0 trees (and any other asset) can be freely traded. Go as far as you can describing equilibrium tree prices, interest rates, and consumption allocations for all t. 2
