Patience has a utility function , where c₁ is her consumption in period 1 and c₂ is her consumption in period 2. Her income in period 1 is twice as large as her income in period 2. At what interest rate will she choose to consume the same amount in period 1 as in period 2? A) 0.40 B) 0.10 C) 0.20 D) 0 E) 0.30

Question

Quizplus · Accepted Answer

Since Patience's income in period 1 is twice as large as her income in period 2, we can set her income in period 2 as the numeraire (i.e. normalize it to 1) and express her consumption in period 1 as a function of her consumption in period 2:

cS1U1B11S1U1B0 = 2cS1U1B12S1U1B0.

Thus, Patience's utility function becomes:

u(cS1U1B11S1U1B0, cS1U1B12S1U1B0) = cS1U1B12S1U1B0^(0.5) + 2*cS1U1B12S1U1B0^(0.5).

To find the interest rate at which Patience will consume the same amount in period 1 as in period 2, we can use the Euler equation:

u1(cS1U1B11S1U1B0, cS1U1B12S1U1B0) = β(1+r)u1(cS1U1B12S1U1B1, cS1U1B13S1U1B1).

Since we've normalized income in period 2 to 1, we can drop the subscripts and write this as:

u1(2c, c) = β(1+r)u1(c, c/(1+r)).

Taking the first derivative of u with respect to c1 and solving for r, we get:

r = 0.2.

Therefore, Patience will choose to consume the same amount in period 1 as in period 2 if the interest rate is 0.2.

Patience Has a Utility Function , Where C1 Is

Patience Has a Utility Function , Where C₁ Is