Consider the following two$-$period overlapping generations model. The size

$of$ each cohort $is$ normalised $to1$ and there $isno$ population growth. Young

agents supply one unit $of$ labour inelastically and save, while old agents

retire and consume. Agents maximise expected consumption when old

$u_t=E_t{c}_{t+1}.$

Final output $is$ produced $by$ a large number $of$ firms according $to$

$As{L}_t=1it$ follows that $y_t=\frac{Y_t}{L_t}=Y_t=k_t^{}.$ Moreover, capital $is$ depreciated

entirely after one period.

Assume that there are productive and unproductive agents, denoted $by$

$P$ and $U,$ respectively. Both types $of$ agents differ with respect $to$ their

productivity $to$ build $up$ physical capital $in$ the sense that $U\text{'}s$ marginal

product $of$ capital $is$ just a share $ofP\text{'}s$ marginal product. The population

share $ofU-$agents $is1-and$ the one $ofP$ agents $is.$

Young agents have the additional option $of$ purchasing bubbles $(pyramid$

schemes$),$ where $b_t$ denotes the stock $of$ existing bubbles until $t$ and $b_t^{NP}+$

$b_t^{\frac{?}{NU}}$ represents the stock $of$ new bubbles created $int.(iii)$ Derive the $E_t{x}_{t+1}-$equation for $UU-$agents invest $in$

$bubbles{b}_t^{NP},b_t^{\frac{?}{NU}}>0.$

$(iv)$ Derive from your above equation $[(iii)]$ a constraint for $which$

excludes unfeasible bubbles. Explain all relevant steps. Assume

that $b_t^{NP},b_t^{\frac{?}{NU}}=0$