To solve the optimization problem, we need to use the Euler equation and the budget constraint.
1. Given Y1, Y2, and R, we can set up the following optimization problem:
maximize U(C1, C2) = 2ln(C1) + 2ln(C2) subject to the budget constraint:
Y1 = C1 + R*C2
To solve this problem, we can use the Lagrangian method. The Lagrangian function is:
L = 2ln(C1) + 2ln(C2) + λ(Y1 - C1 - R*C2)
Taking the partial derivatives with respect to C1, C2, and λ and setting them equal to zero:
dL/dC1 = 2/C1 - λ = 0
dL/dC2 = 2/C2 - λR = 0
dL/dλ = Y1 - C1 - R*C2 = 0
From the first equation, we have 2/C1 = λ.
From the second equation, we have 2/C2 = λR.
Dividing these two equations, we get C2/C1 = R.
Substituting this into the budget constraint, we have:
Y1 = C1 + R*C2
Y1 = C1 + C1
C1 = Y1/2
Similarly, C2 = Y1/2 * R = Y1/2 * (3/5) = 3Y1/10.
Therefore, the optimal consumptions are C1 = Y1/2 and C2 = 3Y1/10.
2. To find the maximized utility given the real interest rate r, we need to consider the new budget constraint:
Y1 = C1 + (1+r)C2
Using the same optimization process as in part 1, we find that the optimal consumptions are:
C1 = Y1/2
C2 = Y1/2 * R = Y1/2 * (3/5) = 3Y1/10.
Substituting these into the utility function U(C1, C2), we get:
U = 2ln(C1) + 2ln(C2)
= 2ln(Y1/2) + 2ln(3Y1/10)
= 2ln(Y1) - ln(2) + 2ln(3) + ln(10)
= 2ln(Y1) + ln(90) - ln(2).
3. When the real interest rate changes to rₙ, the new budget constraint becomes:
Y1 = C1 + (1+rₙ)C2
Using the same optimization process as in parts 1 and 2, we find that the new optimal consumptions are:
C1ₙ = Y1/2
C2ₙ = Y1/2 * Rₙ
To find Rₙ, we can use the formula:
Rₙ = (1 + rₙ)/(1 + r)
Plugging in the given values, we have:
Rₙ = (1 + 39/121)/(1 + 3/5)
= (160/121)/(8/5)
= (160/121) * (5/8)
= 800/968
= 100/121.
Therefore, C2ₙ = Y1/2 * Rₙ = Y1/2 * (100/121) = 100Y1/242.
4. The substitution effect measures the change in consumption due to a change in relative prices while keeping utility constant. In this case, the change in C1 and C2 purely due to the substitution effect is zero because the relative price, R, has not changed.
5. The income effect measures the change in consumption due to a change in real income while keeping prices constant. In this case, the change in C1 and C2 purely due to the income effect can be obtained by comparing the initial consumption levels (C1 and C2) with the new consumption levels (C1ₙ and C2ₙ):
Change in C1 = C1ₙ - C1 = Y1/2 - Y1/2 = 0
Change in C2 = C2ₙ - C2 = 100Y1/242 - 3Y1/10.
6. To find the maximized utility given the new real interest rate rₙ, we again substitute C1ₙ and C2ₙ into the utility function U(C1, C2):
Uₙ = 2ln(C1ₙ) + 2ln(C2ₙ)
= 2ln(Y1/2) + 2ln(100Y1/242)
= 2ln(Y1) - ln(2) + 2ln(100Y1/242)
= 2ln(Y1) - ln(2) + ln(100) + ln(242).
Comparing this with the previous maximized utility U, we can see that Uₙ is different from U because the new real interest rate rₙ affects the optimal consumption levels C1ₙ and C2ₙ. Whether the new maximized utility is lower or higher depends on the specific values of Y1, Y2, and rₙ.