$5.$ Two countries, Richland and Poorland, are described by the Solow growth model. They have the same Cobb$$Douglas production function, F$($K$,$ L$)5$ A KaL$12$a$,$ but with different quantities of capital and labor. Richland saves $32$ percent of its income, while Poorland saves $10$ percent. Richland has population growth of $1$ percent per year, while Poorland has population growth of $3$ percent. $($The numbers in this problem are chosen to be approximately realistic descriptions of rich and poor nations.$)$ Both nations have technological progress at a rate of $2$ percent per year and depreciation at a rate of $5$ percent per year. a$.$ What is the per$-$worker production function f$($k$)?$ b$.$ Solve for the ratio of Richland$$s steady$-$state income per worker to Poorland$$s$.($Hint: The parameter a will play a role in your answer.$)$ c$.$ If the Cobb$$Douglas parameter a takes the conventional value of about $1/3,$ how much higher should income per worker be in Richland compared to Poorland? d$.$ Income per worker in Richland is actually $16$ times income per worker in Poorland. Can you explain this fact by changing the value of the parameter a$?$ What must it be$?$ Can you think of any way of justifying such a value for this parameter? How else might you explain the large difference in income