Question

Consider a public pension program and how it affects two types of people: patient and myopic. Both types face the following maximization problem: max c1 ,c2 ln(c1 )+βln(c2 ) subject to c1 +s=100&c2 =(1+r)s where c1 is consumption in period 1,β is the discount rate, c2 is consumption in period 2,s is savings, r is the interest rate. (a) What is the optimal ratio of consumption in period 2 to consumption in period 1(c2/c1 ) as a function of β and r ? Explain this relationship intiuitively. [2 mark] (b) Suppose that the interest rate r=0.2 for the rest of the problem. What is the optimal amount of savings s for patient people who have β=1 [2 marks] (c) What is the optimal amount of savings s for myopic people who have β=0 [2 marks] (d) Suppose the government imposes a lump sum tax of τ=50 paid in period 1 and pays benefits of b=50(1+r) back in period 2. Note that people can't borrow against these period 2 benefits (i.e., they cannot undo the tax in period 1) i. How much do patient individuals save? Does this plan make them better off? [2 mark] ii. How much do mypotic is the optimal amount of savings s for myopic people who have β=0

Answer 1

(a) The optimal ratio of consumption in period 2 to consumption in period 1 (c2/c1) is β/(1+β) in terms of β and r.

(b) For patient individuals with β=1 and r=0.2, the optimal amount of savings (s) is 50.

(c) For myopic individuals with β=0, the optimal amount of savings (s) is 100.

(d) i. Patient individuals save s=50, and the plan makes them better off as their utility increases.

ii. Myopic individuals do not save (s=0), and the plan does not affect their utility.

Step-by-step explanation:

(a) The optimal consumption ratio is found by solving the maximization problem. The utility function ln(c1) + βln(c2) is subject to the budget constraint c1 + s = 100 and c2 = (1+r)s. The optimal solution yields the ratio c2/c1 = β/(1+β).

(b) For patient individuals (β=1) and a given interest rate (r=0.2), the optimal savings (s) is found by solving the budget constraint equations, resulting in s=50.

(c) For myopic individuals (β=0), the intertemporal budget constraint simplifies to c1 + s = 100, and the optimal savings is the full income in period 1, s=100.

(d) i. Patient individuals, facing a lump sum tax in period 1 and receiving benefits in period 2, save s=50. The plan makes them better off as the benefits offset the tax.

ii. Myopic individuals do not save in this scenario (s=0), and the plan has no impact on their utility, as they do not consider future consequences.