2. (20 pts.) You will now solve a two-period problem analytically. Suppose there are two countries, France and Germany. Suppose they both have log-utility: V(C) = ln(C). In Germany, lifetime utility is given by In(C) + B In(C). France also has the same utility function, just use C1, C, and 8* instead, where the superscript * refers to France. (a) (3 pts.) Write down Germany's optimization problem by stating (i) the objective function to be maximized, (ii) the relevant constraints in each period and (iii) explicitly stating what variables the representative household must choose. 2 (b) (3 pts.) Continuing with Germany, combine the budget constraints in each period to derive the lifetime budget constraint which states that the net-present value of lifetime consumption must equal the net-present value of lifetime income. Note that this eliminates one choice variable, B. (c) (5 pts.) Still in the case of Germany, derive the key Euler equation that determines the optimal trade-off between consumption in each period as it relates to the interest rate. Hint: You may use either the substitution method or the Lagrangian method. For the substitution method you must do the following. (i) Rewrite the lifetime budget constraint in terms of either C or in terms of C. (ii) Then substitute your rearranged budget constraint into the objective (lifetime utility) function in place of either C or C2, depending on what you did in step (*). (0) Now that the substitution restates the objective function in terms of one choice variable, either C2 or C1, you can take the derivative of the whole function with respect to either C2 or C. (iv) Set the derivative to zero and rearrange that ention to met the ner mution 2. (20 pts.) You will now solve a two-period problem analytically. Suppose there are two countries, France and Germany. Suppose they both have log-utility: V(C) = ln(C). In Germany, lifetime utility is given by In(C) + B In(C). France also has the same utility function, just use C1, C, and 8* instead, where the superscript * refers to France. (a) (3 pts.) Write down Germany's optimization problem by stating (i) the objective function to be maximized, (ii) the relevant constraints in each period and (iii) explicitly stating what variables the representative household must choose. 2 (b) (3 pts.) Continuing with Germany, combine the budget constraints in each period to derive the lifetime budget constraint which states that the net-present value of lifetime consumption must equal the net-present value of lifetime income. Note that this eliminates one choice variable, B. (c) (5 pts.) Still in the case of Germany, derive the key Euler equation that determines the optimal trade-off between consumption in each period as it relates to the interest rate. Hint: You may use either the substitution method or the Lagrangian method. For the substitution method you must do the following. (i) Rewrite the lifetime budget constraint in terms of either C or in terms of C. (ii) Then substitute your rearranged budget constraint into the objective (lifetime utility) function in place of either C or C2, depending on what you did in step (*). (0) Now that the substitution restates the objective function in terms of one choice variable, either C2 or C1, you can take the derivative of the whole function with respect to either C2 or C. (iv) Set the derivative to zero and rearrange that ention to met the ner mution