I. 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:

$Y_t=A_t{x}^{**}{N}_t^{1-}$

where ain$(0,1),A_t$ denotes the productivity level at time $t,x>0$ stands for a fixed aggregate

stock of land used in agriculture and $N_t$ represents the aggregate population.

Let us assume that productivity grows at a constant rate ain$(0,1).$

$\frac{A_{t+1}-A_t}{A_t}=a$

Let $x_t$ stand for the land per capita at time $t$ defined as:

$x_t-=\frac{x}{N_t}$

Let us consider that the birth rate is fued at bin$(0,1)$ and the death rate at time $t$ can be written

as:

$d_t=1-y_t$

where $>0$ is a parameter and $y_t$ denotes the output per capita at time $t$ defined as:

$y_t-=\frac{Y_t}{N_t}$

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$)$

b$.$ Write down the equation of the production function in per capita units. Is this relationship

consistent with Malthus' assumption $1$ given $A_t?(12$ 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 $a$

affect the output per capita growth rate from time $t$ to $t+1.(8$ points$)$

d$.$ Using your answers to questions a and c$,$ derive a relationship showing how the output per

capita at time $t$ affects its rate of change from time $t$ to $t+1$ given the productivity growth

rate $a(4$ points$)$

e$.$ Using your answer to question d$,$ derive the steady$-$state level of output per capita: $y_x$

compatible with no growth in the living standard. $(8$ points$)$

f$.$ Derive the population growth rate at the steady$-$state solution found in question e$.(8$ points$)$

In the absence of productivity growth rate: $a=0$ where $A_{t+1}=A_t=Z,$ derive the new

output per capital steady$-$state solution as well as the associated population growth rate and

population size. $(12$ points$)$