Answer to Question #194235 in Financial Math for Muhammad Umair

Question

Consider the following small open economy model with production. At dates 1 and 2, the home country receives exogenous fixed endowments y₁ and y₂ respectively. The home country has access to the international capital market at a fixed interest rate r* at which it can save or borrow. Let the net saving of the home country be s1 and its consumption stream in two periods be given by

c₁ and c₂ respectively.the following maximization problem:

Max ln c₁ + ln c₂

s.t. C₁ + S₁ = Y₁

C₂ = C₁(1+r*) + Y₂

Derive optimal consumption and current account functions and carefully interpret it in terms of a two-period Fisherian graph.

Refer to (a). Suppose the home country also has an access to an investment

technology which means that if it invests k units at date 1, it produces y2 = Ak^a units of output

in the next period where A >0 and 0 < a< 1 . Modify budget constraints for (a)Derive the optimal investment and saving rules for this

economy assuming the same logarithmic utility function as in (a). Interpret your results

Answer 1

(i)

consumer will maximize utility when budget constraint is

$C_1+\frac{C_2}{1+r}=Y_1+\frac{Y_2}{1+r}$

solve the budget constraint for $C_2$

$C_2=Y_1(1+r)+Y_2-(1+r)C_1$

substitute the constraint into objective function

$U=u(C_1)+β(Y_1(1+r)+Y_2-(1+r)C_1)$

maximize the objective function wrt $C_1$

$U(C_1)-βu(C_2)(1+r)=0$

$U(C_1)\frac{1}{1+r}=β(C_2)$

$Bu(C_2), u(C_1)=\frac{1}{1+r}$

the expression shows how much $C_1$ is willing to trade for one more unit with $C_2$ given constant utility and also how much $C_1$ have to give up to get one more unit $C_2$ .

current account for any given r

$CA(r;Y_1;Y_2)=Y_1-C_1(r;Y*_1,Y*_2)$

Current account

$Y_1-C=\frac{Y_1-Y_2}{1+r}$

(ii)

optimal investment and saving rules

new budget constraint

$C_1+\frac{C_2}{1+r}=Y_1+\frac{Y_2}{1+r}+\frac{K_1-r(K_2-K_1)}{1+r}$

rule

$\frac{\delta m}{\delta r}=-\frac{Y_2-rK_2}{1-r_e}-\frac{K_2}{1+r}=-\frac{Y_2+K_2}{(1+r)_2}$

the slope is negative meaning investment and saving is decreasing with r as consumption increases.