$[14]$ Consider a two$-$period model $of$ intertemporal choice. $In$ each period $i=1,2,$ the

consumer has income $m_i>0$ and makes a consumption choice $x_{i}0.$ The price $of$ the

consumption good $is1in$ both periods, and the consumer can save and borrow $at$ interest

rate $r>0.$ The consumer's utility from consumption path $(x_1,x_2)isln{x}_1+ln{x}_2,$ where

$01is$ the consumer's discount factor.

$(a)[2]$ Write down the consumer's intertemporal utility maximisation problem.

$(b)[2]$ Argue that the consumer's intertemporal budget constraint holds $asan$ equality $at$

any optimal consumption path $(x_1^{*},x_2^{*}).$

$(c)[2]$ Write down the Lagrangean $of$ the consumer's utility maximisation problem and

derive its first$-$order necessary conditions for a solution with $x_1^{*},x_2^{*}0.$

$(d)[2]Isit$ possible for $x_1^{*}=0tobe$ optimal for the consumer? $Isit$ possible for $x_2^{*}=0$

$tobe$ optimal for the consumer?

$(e)[2]$ Derive the consumer's demand functions.

$(f)[2]$ Suppose that a government wants $to$ promote the consumer's period $2$ expenditures

$(say,$ because the government wants the consumer $tobe$ better off $in$ her old age$).$ One

policy available $to$ the government $isto$ send a cheque for $M$$ $to$ the consumer $in$

period $1.$ What $is$ the effectiveness $of$ this policy? That $is,$ calculate the increase $in$

the consumer's period $2$ expenditures due $to$ the policy.