Question

4.5 Susan is certain to live just two periods and receives an income of 10,000 in the first period, and 15,000 in the second. She has no other assets. The real interest rate is 8%.

(a) As she begins the first period, what is the present value of her lifetime resources?

(b) IF she choose to consume the exact same amount in both periods (c1 = c2), what would be her consumption in the first (and second) period? SHOW YOUR WORK

.(c) Bill has the exact same income (and life expectancy) of Susan, but chooses to consume 15,000 in the first period. How much will he consume in the second.

If, in the problem above, the interest rate rises from 8% to 10%, will Bill be happy (i.e., be made better off) or sad (i.e., be made worse off) by the change? Concisely explain your reasoning. (20 words MAX)

Answer 1

Susan earns 10,000 in the first period, and 15,000 in the second. Real interest rate is = r = 8%

Present value of her lifetime resources = 10,000+15,000/ (1+r) = 10,000 + 15,000/(1.08) = 10,000 + 13,888.89 = $ 23,888.88

Therefore, Present value of her lifetime resources = $23,888.88

Given interest rate r, the consumption in period 2 is equal to the income in period 2 (Y₂) plus the savings from period 1 (Y₁-C₁) and the interest earned on the savings [r(Y₁-C₁)]

C₂ = Y₂ + (Y₁-C₁) + r(Y₁-C₁)

C₂ = Y₂ + (1+r) (Y₁-C₁)

C₁ + C₂/(1+r) = Y₁ + Y₂/(1+r) ..... (1)

The above equation gives the intertemporal budget constraint.

Now, she choose to consume the exact same amount in both periods (c1 = c2). Put the given values in the budget constraint.

C₁ + C₂/(1+0.08) = 10,000 +15,000/(1+0.08)

C₁ + C₁/(1+0.08) = 23,888.88 (since C₁ = C₂)

1.08 C₁ + C₁ = 23,888.88 * 1.08

2.08 C₁ =25,800

C₁ =12,403.85

When she choose to consume the exact same amount in both periods then she can consume $12,403.85 in both periods.

c) Now, Bill consumer 15,000 in the first period. C₁ = 15,000

Put the values in Bill’s budget constraint (as given in equation 1)

15,000 + C₂/(1+0.08) = 10,000 +15,000/(1+0.08)

1.08*15,000 + C₂ = 25,800

C₂ =25,800-16,200

C₂ = $9,600

interest rate rises from 8% to 10%. Then the budget equation becomes:

15,000 + C₂/(1+0.10) = 10,000 +15,000/(1+0.10)

1.10*15,000 + C₂ =23,636.36

C₂ = 23,636.36 -16,500

C₂ =$7,136.36

Since, he can consumes less in the second period, while consuming the same in the first period, this means he is worse off after the interest rates have increased.

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