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MBFM III - Week 9 Tutorial Questions

The objective of this problem is to practice thinking about the equalisation of asset returns.

1    Problem #1

Consider the problem of an individual who lives for two periods. The individual has preference represented by a period utility function u(c) that is defined on consumption and exhibits, diminishing marginal utility of consumption, more consumption is better and the Inada condition that marginal utility approaches infinity as consumption con- verges to zero. The individual discounts second period utility by a discount factor β so that the lifetime objective is to maximize, u(c1 )+ βu(c2 ).

In the first period of life the individual has income of y1 . The individual can use period 1 income to consume, c1 , or save. Period 2 consumption, c2  is finance entirely out of the returns from saving. Savings can come through two types of assets, money and bonds. The rate of exchange of money for goods is Pt  in periods t = 1, 2. Money pays no interest. Nominal bonds on the other hand pays a gross return of $R per bond. Let B1  denote the number of bonds purchased by the individual in period 1 and carried forward to period 2 and let M1  denote nominal money holdings. Each bond costs $1 to purchase. However, there is also a portfolio cost of φ(B) which is expressed in dollar units increasing with an increasing marginal cost per bond, φ\ (B) > 0 and φ\\ (B) > 0. This portfolio cost is paid in period 2 when the bonds mature.

1. Write down the individual’s period 1 and period 2 budget constraints.

2. Derive the individual’s intertemporal consumption trade-o↵ conditions. Interpret each trade-o↵ condition in words.

3. Explain how the rate of return on money savings a↵ects the return to bond holdings.