Q$2$Growth Theories$$

Consider the classical Solow model, which is based on the following concepts and assumption

$($a$)$ Macroeconomic balance: S$($t$)1($t$),$ where S$($t$)$ represents savings at date tand $1($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$).$

$(1)$ First, let's consider the Solow model without technical progress by adding the following assumption. There are two inputs in production: capital K$(1)$ and population P$(1).$ The production function is $70)-$KUP The population grows at the rate n$.$

i$)$ Please explicitly solve for the steady$-$state value of the per capita output. How does the value change in response to a rise in the rate of saving s and the population growth rate?

ii$)15$ points$)$ What is the long$-$run $($steady state$)$ growth rate of per capita output?

$(2)$ Next, let's consider the Solow model with technical progress using the following production function inatead of the one mentioned in $(1)$: Y$($t$)=$ K$($t$)"($E$($t$)$P$($t$))",$ where E$($t$)$ represents the efficiency of an individual at time e$,$ and this grows at the rate $.$ The level of efficiency at time $0$ is denoted as $8(0).$ The population still grows at the rate

iii$)$ Please explicitly solve for the steady$-$state value of the per capita output at date t$.$ How does the value change in response to a rise in the rate of saving s and the population growth rate #$?$

iv$)$ What is the long$-$run $($steady state$)$ growth rate of per capita output?