Solve by using Lagrangian method.

Consider the following two-period model of consumption. Mamoon wants to maximize their ifetime utility that is given as: ln(C1)+ln(C2), where =0.5. Mamoon's first-period income (Y1) is 100 , their second-period income (Y2) is 220 , and the interest rate (r) is 10%. (a) (10 points) Solve for the optimal consumption for Mamoon in the first period and in the second period. Please show the steps of your calculation. (b) (7 points) Suppose that the interest rate increases from 10%. Explain in words how the income effect will affect Mamoon's optimal consumption in the first period? How does the substitution effect affect the optimal consumption in the first period? (c) (3 points) Assume that the interest rate stays at 10\%, but suppose now that Mamoon faces a borrowing constraint, i.e. their not allowed to borrow money in the first period. Does this affect their consumption decisions? What will be the optimal consumption in the first period under the borrowing constraint? Consider the following two-period model of consumption. Mamoon wants to maximize their ifetime utility that is given as: ln(C1)+ln(C2), where =0.5. Mamoon's first-period income (Y1) is 100 , their second-period income (Y2) is 220 , and the interest rate (r) is 10%. (a) (10 points) Solve for the optimal consumption for Mamoon in the first period and in the second period. Please show the steps of your calculation. (b) (7 points) Suppose that the interest rate increases from 10%. Explain in words how the income effect will affect Mamoon's optimal consumption in the first period? How does the substitution effect affect the optimal consumption in the first period? (c) (3 points) Assume that the interest rate stays at 10\%, but suppose now that Mamoon faces a borrowing constraint, i.e. their not allowed to borrow money in the first period. Does this affect their consumption decisions? What will be the optimal consumption in the first period under the borrowing constraint