Question $1$

Consider $an$ economy where aggregate output $is$ produced with physical

capital $K_t,$ human capital $H_t,$ and a fixed amount $of$ labor $N$ according $to$

the production function:

$Y_t=K_t^{}{H}_t^{}(A_{t}N{)}^{1--}.$

Both physical and human capital can $be$ accumulated $by$ investing final

output $in$ new capital $(or$ equivalently, each type $of$ capital $is$ produced

$in$ the same way $as$ final output$).$ The laws $of$ motion for the two capital

stocks are given $by$:

$K_{t+1}=(1-{}_K)K_t+s_K{Y}_t,$

$H_{t+1}=(1-{}_H)H_t+s_H{Y}_t,$

where $g>0A_{t+1}=(1+g)A_t{A}_{t}N0.$ Labor

productivity grows $at$ the exogenous rate $g>0$ :

$A_{t+1}=(1+g)A_t$

$(a)$ This model exhibits a balanced growth path $in$ which both physical

and human capital grow $at$ the same rate $as$ effective labor units $A_{t}N.$

Write down laws $of$ motion for physical capital per effective labor

unit and for human capital per effective labor unit.

$(b)$ Solve for the balanced$-$growth levels $of$ the two types $of$ capital per

effective labor unit. What $is$ the ratio $of$ human $to$ physical capital $in$

the balanced growth path? What $is$ the growth rate $of$ output? and $0.$ Labor

productivity grows $at$ the exogenous rate $g>0$ :

$A_{t+1}=(1+g)A_t$

$(a)$ This model exhibits a balanced growth path $in$ which both physical

and human capital grow $at$ the same rate $as$ effective labor units $A_{t}N.$

Write down laws $of$ motion for physical capital per effective labor

unit and for human capital per effective labor unit.

$(b)$ Solve for the balanced$-$growth levels $of$ the two types $of$ capital per

effective labor unit. What $is$ the ratio $of$ human $to$ physical capital $in$

the balanced growth path? What $is$ the growth rate $of$ output?