Question

Consider a two-period small open endowment economy populated by a large number of households with preferences described by the lifetime utility function lnC 1 ​ +lnC 2 ​ where C 1 ​ and C 2 ​ denote, respectively, consumption in periods 1 and 2. Suppose that households receive exogenous endowments of goods given by Q 1 ​ =Q 2 ​ =50 in period 1 and 2 , respectively. Every household enters period 1 with some debt, denoted B 0 ∗ ​ , inherited from the past. Let B 0 ∗ ​ be equal to −10. The interest rate on these liabilities, denoted r 0 ​ , is 10 percent. Finally, suppose that the country enjoys free capital mobility and that the world interest rate on assets held between periods 1 and 2 , denoted r ∗ , is 12 percent.

Answer 1

This two-period small open economy model demonstrates how households in an open economy with free capital mobility optimize their consumption based on their endowments, debt, and interest rates.

Two-Period Small Open Economy

Model Setup:

Consumption: C1, C2

Preferences: ln(C1) + ln(C2)

Endowments: Q1 = Q2 = 50

Period 1 Debt: B0* = -10

Period 1 Interest Rate: r0 = 10%

World Interest Rate: r* = 12%

Household Optimization:

Budget Constraint:

Period 1: C1 + B1 = B0*(1+r0) + Q1

Period 2: C2 + B2 = B1(1+r*) + Q2

Utility Maximization:

Maximize ln(C1) + ln(C2)

Subject to budget constraints

Solution:

Substitute B0* and Q1 into the period 1 budget constraint:

C1 + B1 = -10*(1+0.1) + 50

C1 + B1 = 40

Substitute C1 with the expression from the utility maximization for C1:

ln(C1) + ln(C2) = ln(40-B1) + ln(C2)

Solve for B1:

B1 = 40 - 70C2 / (1 + C2)

Substitute B1 into the period 2 budget constraint:

C2 + B2 = (40 - 70C2 / (1 + C2))(1+0.12) + 50

Solve for C2:

C2 ≈ 38.36

Results:

Optimal Consumption:

C1 ≈ 1.64

C2 ≈ 38.36

Period 1 Debt: B1 ≈ 36.72

Conclusion:

This two-period small open economy model demonstrates how households in an open economy with free capital mobility optimize their consumption based on their endowments, debt, and interest rates. The solution provides insights into the trade-off between consumption in different periods and the equilibrium allocation of resources.