FINC3024 Personal Finance and Superannuation S1 2017
Assignment 1 part 2
Due: 9:30am, Monday 10th April 2017
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Assignment 1 has three parts consisting of three problem sets worth equal marks, and contributes 20% of your final grade in total. This tutorial problem set is the second part of assignment 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 + dlnC2 + d2lnC3. 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 find expressions for optimal consumption in terms of wealth.
(c) (1 mark) Under what conditions for d will Tom’s level of consumption be increasing, flat or decreasing over time? What does d show about Tom’s impatience?
Josh is endowed with $200 in year one and $200 in year two. He can borrow and lend at an interest rate of 3% p.a. Josh has a utility function given by U(C1,C2) = -e-aC1C2
(a) (1 mark) Write down Josh’s marginal rate of inter-temporal substitution and budget constraint.
(b) (1 mark) Find Josh’s optimal consumption bundle ( and ). Is Josh a borrower or a lender? What is the value of his utility function at the optimal bundle if a = 1/10000?
(c) (1 mark) Suppose that the interest rate Josh faces falls to 1% p.a.? What consumption bundle will he choose under these conditions? What is the value of his utility function at the new optimum?
Josh’s optimal value of C1 has changed from the amount he chose when interest rates were 3% to a different amount now interest rates are 1%. This change can be decomposed into an income effect (as a lender (borrower), Josh has less (more) income overall when interest rates fall) and a substitution effect (lower interest rates make current consumption relatively more attractive compared with future consumption).
(d) (2 marks) Compute the substitution effect of the interest rate fall by calculating the level of C1 that Josh would choose if interest rates were 1% but his total utility stayed at the same level as his optimum when interest rates were 3% (as if he stays on the same indifference curve but at a different interest rate). By implication, what is the income effect?
(e) (1 mark) Explain in words the direction of the income and substitution effects if Josh was a borrower instead of a lender and interest rates fell?