Hi I need a response for the following question:

Suppose you own a circular island with radius R which can be used for
either growing pineapples or renting to tourism industry. The
production technology for pineapples is:

Q = min{alpha1 * L, alpha2 * A}

with Q being the quantity of pineapples grown, L for labor, A for area
of land used for pineapples. The wage for labor is w, while the price
for pineapples is p.

The rental price P for the land to the tourist industry is a
decreasing function of distance of land from edge of island, r: P =
P(r), P'(r) <0 for all r.

How will you use the land? What will be the shape of the plot of land
for pineapple farming and what fraction of the total island will it
occupy?

What is your supply function for pineapples?

Thanks. I need a response as soon as possible. I will TIP an answer
that I believe requires more compensation than that already given.

Dear la2005,

Because renting land to the tourist industry involves essentially no
labor or capital investment, we will want to exploit coastal areas of
the island to this end. But land that is far enough from the coast can be
more profitably used to grow pineapples, so we will also want to devote
some of the land to pineapple farming.

The plot of land used for pineapple farming must be circular, like the
island itself, with its center coinciding with that of the island. This 
is because the nearer a parcel of land is to the center of the island,
the less it is worth to the tourist industry. The border between
the pineapple-farming circle and the tourist-area perimeter strip
demarcates the point where tourism and pineapple cultivation are equally
profitable. Pineapples are more profitable on the inland side of this
line, while tourism is more profitable on the coastal side.

Assuming that the only cost of growing Q pineapples is the labor cost Lw,
and given that y is the radius of the pineapple-farming plot, the profit
function for pineapple farming alone is

    profit_0(y)  =  Qp - Lw 

                 =  min{alpha1 * L, alpha2 * A}p - Lw

                 =  min{alpha1 * L, alpha2 * pi*y^2} - Lw .

The width of the perimeter strip rented to the tourist industry is
R-y. Thus, we gain a revenue of

    profit_1(y)  =  P(R-y)
    
from tourism rentals. The total profit that we earn from both uses of
the land can be expressed as

    profit_T(y)  =  profit_0(y) + profit_1(y)

                 =  Qp - Lw + P(R-ry)

                 =  min{alpha1 * L, alpha2 * pi*y^2} * p - Lw + P(R-y) .
             
The radius r' of the pineapple plot, and therefore the fraction of the
total island 

    pi*y^2 / A  =  pi*y^2 / pi*(R-y)^2
    
devoted to pineapple farming, is determined by the value of y that
maximizes the total profit function profit_T(y).

Using this value y_max, the supply function for pineapples becomes

    Q = min{alpha1 * L, alpha2 * pi * y_max * y_max} .
    
Regards,

leapinglizard

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