Q$2$: Consider the classical Solow model with technical progress, which is based on the following concepts and assumptions:

$($a$)$ Macroeconomic balance: S$($t$)1($t$),$ in which S$($t$)$ represents savings at date t and $/($t$)$ represents investment at date t$.$

$($b$)$ Physical capital accumulation: K$($t$+1)=(1-8)$K$($t$)+1($t$),$ which implies that investment augments the national capital stock $($K$)$ and replaces a fraction & of the capital stock that is wearing out $($depreciating$).$

$($c$)$ The savings rate is total national savings divided by total national income, denoted by s $=$ S$($t$)/$Y$($t$).$

$($d$)$ There are two inputs in production: capital K$($t$)$ and population P$(1).$ The production function is Y$(1)=$K$(0)($E$(1)$P$(1)),$ E$($t$)$ represents the efficiency of an individual at time $1,$ and this grows at the rate. The level of efficiency at time $0$ is denoted as E$(0).$ The population grows at the rate. Assume that at time $1,$ the conomy is at a steady state.

i$)$ We define the effective labor force $($or effective population$)$ as L$($t$)=$ E$(1)$P$(1).$ In Country A$,$ the labor force $($population$)$ is growing at $2\%$ per year $(0\mathrm{.}02).$ The technical progress rate is $3\%$ per year $($r$=0\mathrm{.}03).$ How fast is the effective labor force growing?

ii$)]$ Please solve the value of output per efficiency unit of labor $(1)$Y$(1)/$L$(1))$ at the steady state $().$

iii$)$ At time, the economy is at a steady state. Please solve the value of the per capita output $($y$,$Y$($r$)/$P$($t$)).$

iv$)$ In Country B$,$ the labor force $($population$)$ is growing at $1\%$ per year and the technical progress rate is $2\%$ per year. Countries A and B have the same depreciation rate $(6)$ and the same savings rate $($s$).$ Which country has a higher output per efficiency unit of labor at a steady state? Why?