Question 1
Tom must decide how much to consume over a three year period. His year 1 wealth is A1 and he does not earn any income in year 2 or year 3. Tom’s three-year utility function is U = lnC1 + δlnC2 + δ2lnC3. His savings earn an annual interest rate of r.
(a) (1 mark) Write down Tom’s budget constraint in terms of A1, C1, C2 and C3.
(b) (2 marks) Tom wants to maximise the utility of C1, C2 and C3 subject to his budget constraint. Assume that the interest rate r = 0. Calculate what proportion of Tom’s wealth A1 should be spent in each period so as to maximize his utility.
Hint: Write down the utility function and the budget constraint in terms of consumption in each period and use either substitution or the method of Lagrange multipliers to ï¬nd expressions for optimal consumption in terms of wealth.
(c) (1 mark) Under what conditions for δ will Tom’s level of consumption be increasing, flat or decreasing over time? What does δ show about Tom’s impatience?
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