Consider the intertemporal choice between current and future consumption. The typical perfectly

foresighted household lives for two periods and $it$ has $no$ inherited wealth. $It$ receives income $of$

$y_{1}in$ period $1$ and $y_{2}in$ period $2$ which $it$ can consume $or$ save. $In$ period $1it$ consumes $c_1$ out $of$

its income and $it$ saves the balance $of$ its first period income, equal $to{s}_1.It$ lends the savings $on$

the credit market for interest $at$ the interest rate $ofr.It$ gets back its savings with interest $in$ period

$2$ and can consume $itin$ period $2.It$ does not save anything $in$ the second period because $it$ knows

that its life ends with the second period. Thus, the budget constraints for the two periods are:

$c_1+s_1=y_1$

$c_2=y_2+s_1(1+r)$

The household's preferences over consumption $in$ the two periods are represented $by$ the utility

function $u=log{c}_1+\frac{log{c}_2}{1+},$ where $is$ the subjective discount factor $by$ which the household

discounts second period utility. $>0$ indicates preference for current $(period1)$ consumption

over future consumption $(period2).$

$a.$ Derive the two$-$period $or$ lifetime budget constraint $on$ current and future consumption for

this typical household and depict $it$ graphically. What $is$ the slope $of$ this budget constraint?

$b.$ Set $up$ the Lagrangean and derive the first order conditions for choice $of{c}_1$ and $c_2.$

$c.$ Solve for the quantities $of$ current and future consumption.

$d.$ What $do$ you infer regarding impact $of$ increase $in$ interest rate $on$ the choices $of$ current and

future consumption?

$e.$ What happens $to$ the choices $if$ because $of$ a permanent improvement $in$ technology, there $is$

a doubling $of$ income $in$ both periods? What $is$ the improvement $is$ temporary resulting $in$ only

the second period income going $up?$