I need help with these economic assessments. Please help with it as ASAP, will rate

$($modified from Brueckner Exercise $3\mathrm{.}2)$

In this problem, you will solve for the equilibrium population of an idealized city experiencing

rural$-$urban migration, following the augmented Harris$-$Todaro model from Chapter $3.$ The in$-$

comes earned in urban employment and in the rural area are y and y$\_($A$),$ respectively, and t is

commuting cost per mile. J is the number of available urban jobs.

$($a$)$ Suppose the city is on an island and is a rectangle $10$ blocks wide with the employment

center at one end. The city spreads out along the length of the island to accommodate its

population, with its edge located bar$($x$)$ blocks from the employment center. It is NOT a circle,

but instead rectangle$-$shaped. Here's a picture which sort of illustrates what the city looks like:

Note, the length of the city above is not meant to be anything specific. The actual length will

depend on the population of the city.

Compute the city's land area in square blocks as a function of bar$($x$).$ Assuming that each urban

resident consumes $0\mathrm{.}005$ square blocks of land, compute the amount of land needed to house the

city's population L$.$ Set the resulting expression equal to the city's land area, and solve for bar$($x$)$ in

terms of L$.$

$($b$)$ With J jobs in the city, the chance of a resident getting one of the jobs is J$/$L$,$ which makes

the expected income of a city resident equal to y$($J$//$L$).$ The expected disposable income net of a

commuting cost for a resident living at the city's boundary is then y$($J$//$L$)-$t bar$($x$).$ The rural$-$urban

migration equilibrium is achieved when this disposable income equals the rural income, as was

explained in chapter $3.$ Write down this equation, and substitute your solution for bar$($x$)$ in terms of

L from $($a$).$ Then multiply through by L to get a quadratic equation that determines L $.$

$($c$)$ Suppose that y$=12,500,$y$\_($A$)=5,000,$t$=100,$ and J$=80,000.$ Substitute these values into

your equation from $($b$),$ and use the quadratic formula to solve for L $($it$\text{'}$s the positive root$).$

The answer gives the city's equilibrium population. Note, remember to multiple by L to get a

quadratic equation for if you did not already do so in part b$.$

$($d$)$ In equilibrium, what is the chance of getting an urban job? What is the implied unemploy$-$

ment rate in the city?

$($e$)$ What is the distance to the edge of the city? How much does a resident living at the edge

spend on commuting?

$($f$)$ Suppose that y rises to $16,000,$ but no increase in the number of jobs $($in the short run$).$

Repeat $($c$),($d$),$ and $($e$).$ Give an intuitive explanation of the changes in your answers to $($c$)$ and

$($d$).$