Malthusian Models $(60$ points$)$Let us consider a Malthusian economy in which the aggregate level of output at time t can be$$described by the following aggregate Cobb$-$Douglas production function:$!=!!$#$$$where a$(0,1),!$ denotes the productivity level at time t $,>0$ stands for a fixed aggregate$$stock of land used in agriculture and $!$ represents the aggregate population.$$Let us assume that productivity grows at a constant rate $(0,1)$ :$!\%$# $!!=$Let $!$ stand for the land per capita at time t defined as:$!!$Let us consider that the birth rate is fixed at $(0,1)$ and the death rate at time t can be written$$as:$!=1!$where $>0$ is a parameter and $!$ denotes the output per capita at time t defined as:$!!!$a$.$ Show that the rate of population growth from time t to t$+1$ can be written as a function of$$the income per capita and identify the level of income per capita compatible with no$$population growth. $(8$ points$)2$b$.$ Write down the equation of the production function in per capita units. Is this relationship$$consistent with Malthus$$ assumption $1$ given $!?(12$ points$)$c$.$ Using your answer to question b$.,$ derive the exact growth relationship showing how the$$aggregate population growth rate from time t to t$+1$ and the productivity growth rate of $$affect the output per capita growth rate from time t to t$+1.(8$ points$)$