Consider the Malthusian model where $Y_t=S^{}(A_t{L}_t{)}^{1-},A_{t+1}=(1+g)A_t,$ and $L_{t+1}=(1+{}_t)L_t.$ In this case $S$ is the $($constant$)$ quantity of land, $L$ is labor, and $A$ is technological progress. Assume that $(S,L_0,A_0)$ are all equal to $1,g>0,$ and $=\frac{1}{3}.$ In addition, assume that ${}_t=a+b{y}_t,$ where is $b$ is greater than zero.

A$.$ Suppose that the parameters of the model are set as in question $2$ of Assignment $4.$ Suppose that the economy is in the Malthusian steady state $($per capita income and population growth are both constant$).$ Using Excel, simulate the response of the economy to a sudden decrease in land's share in period $5$ to $=0\mathrm{.}1.$ Produce graphs showing the evolution of per capita income and population growth for $100$ periods following this change. Describe in words how this economy responds to this change.

B$.$ Compute the steady state per capita output and population growth rate for the economy with $=0\mathrm{.}1.$

C$.$ Consider the same exercise as in part A $,$ except that there is an upper bound on population growth. That is$,$ suppose that ${}_t=$min$\{a+b{y}_t,{}_{\text{M}\text{a}\text{x}}\}.$ Assume that ${}_{\text{M}\text{a}\text{x}}=0\mathrm{.}3.$ Repeat part A under this assumption.

D$.$ For a given value of $,$ how small must the upper bound on population growth be that would allow for sustained growth in living standards? Explain.

This is question $2$ of Assignment $4$

Consider the Malthusian model where $Y_t=S^{}(A_t{L}_t{)}^{1-},A_{t+1}=(1+g)A_t,$ and $L_{t+1}=(1+{}_t)L_t.$ In this case $S$ is the $($constant$)$ quantity of land, $L$ is labor, and $A$ is technological progress. Assume that $(S,L_0,A_0)$ are all equal to $1,g>0,$ and $=\frac{1}{3}.$ In addition, assume that ${}_t=a+b{y}_t,$ where is $b$ is greater than zero.

a$.$ Suppose that population in the pre$-$industrial revolution doubled every $230$ years. What is the annual growth rate of population in this case? What would the period population growth rate be if we interpret a period to be $25$ years? $($Be exact; don't use an approximation here.$)$

b$.$ Suppose that a period is $25$ years. Find the value of $g$ that is consistent with no growth in living standards if the population doubles every $230$ years. Use this value of $g$ in the remaining parts of this assignment.

c$.$ Find values for the parameters a and $b$ in the equation determining the population growth rate consistent with the following statements: $($i$)$ a period is $25$ years; $($ii$)$ after the industrial revolution, population doubles every $35$ years at the point where per capita income is equal to twice its pre$-$industrial revolution level.