# ECON 2508 The Marginal Rate of Substitution of Individuals Optimization Essay

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FE II – Week 11 Tutorial Questions
Consider the following problem of an individual faced with two possible future statesof-nature, j 2 {1, 2}. In state-of-nature j = 1 the individual receives income y1 whereas
in state-of-nature j = 2, individual i receives income y2 , y1 6= y2 . Let cj denote the
amount of the consumption good enjoyed if the state-of-nature is j. State-of-nature j
occurs with probability ?j and ?1 + ?2 = 1.
Prior to learning the state-of-nature, individuals have the ability to purchase (or
sell) assets. There are two assets and the returns to both assets are paid in units of a
consumption good. One asset is risky and pays a rate of interest rjb in state-of-nature j
with r1b 6= r2b . Denote the amount of the risky asset purchased by the individual by b. The
other asset is a risk-free asset and pays an interest rate of rf in both states-of-nature.
The amount of the risk-free asset purchased by the individual is given by a. Thus the
gross return of the asset portfolio (a, b) is (1 + rjb )b + (1 + rf )a in state-of-nature j.
The timing of events is as follows: first, the individual enters an asset market and
purchases a portfolio of assets. The individual is endowed with an initial portfolio (a, b)
which can be sold. The relative price of the risky asset is q and the price of the risk-free
asset is normalized to one. Hence it costs q units of the risk-free asset to purchase one
unit of the risky asset. The individual uses the market value of the portfolio endowment
to finance his/her desired optimal portfolio. After purchasing the desired asset portfolio,
nature reveals the state-of-nature. Asset positions are settled and consumption occurs.
The individual chooses the asset his/her asset portfolio to maximize expected utility.
Utility is derived from consuming the consumption good and is represented by a utility
function with the properties that more is better, u0 (c) > 0, there is diminishing marginal
utility to consumption, u00 (c) < 0 and limc!0 u0 (c) = 1 so that having some consumption in each state-of-nature is strictly preferred to having no consumption in any state-ofnature. The individuals objective function is max {?1 u(c1 ) + ?2 u(c2 )} . c1 ,c2 ,a,b 1. Write down the individuals optimization problem. 2. Write down the Lagrangean for the individuals problem. 3. Solve for the individuals optimal trade-o? condition between the risk-free and the risky asset. 4. Now suppose that there many individuals and they are all identical. As they all face identical income risk and have the same endowments, they all choose the same portfolio and enjoy the same consumption in each state-of-nature. With this assumption derive the Consumption CAPM equation that relates the expected return of the risky asset to the expected return of the risk-free asset. Here you can define the expected return of the risk-free asset by Rf = 1 + rf while defining the ? (1+r1b )+?2 (1+r2b )) expected return of the risky asset by Rb = 1 . q 1 5. Suppose that ?1 , ?2 , rf , q, r1b and r2b as such that mean for c1 and c2 ? u0 (c2 ) u0 (c1 ) = 1. What does this 6. Using the individuals optimal trade-o? condition, show what this implies for the risk-free rate and the expected return of the risky asset. Explain this result in words with reference to the CCAPM equation previously derived. 2 Purchase answer to see full attachment Tags: Financial Economics Lagrange Multipliers optimization problem User generated content is uploaded by users for the purposes of learning and should be used following FENTYESSAYS.COM ESSAY's honor code & terms of service.