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Tom has preferences over consumption and leisure of the following form: U = ln(c1)+ 2 ln(l)+βln(c2),...

Tom has preferences over consumption and leisure of the following form: U = ln(c1)+ 2 ln(l)+βln(c2), where ct denotes the stream of consumption in period t and l, hours of leisure. He can choose to work only when he is young. If he works an hour, he can earn 10 dollars (he can work up to 100 hours). He can also use savings to smooth consumption over time, and if he saves, he will earn an interest rate of 10% per period. Suppose also that he values the future and the present equally, i.e. β = 1. (Remember the marginal utility of ln(c) is 1/c and marginal utility of leisure is given by 2/3l).
(a) Draw a standard indifference curve on the consumption today and leisure set. What is
the slope of the indifference curve?
(b) Draw a standard indifference curve on the consumption today and consumption tomor- row set. What is the slope of the indifference curve?
(c) Write down the budget constraint in the first period.
(d) Write down the budget constraint in the second period.
(e) Derive the lifetime budget constraint.
(f) Draw the lifetime budget constraint on the consumption today and leisure set. What is the slope of the budget line?
(g) Draw the lifetime budget constraint on the consumption today and consumption to- morrow set. What is the slope of the budget line?
(h) Find Tom’s optimal consumption in each period (c∗1, l∗ and c∗2) and optimal saving rate?
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