The economy is populated with L farmers, that are consumers and producers at the same time. As consumers they buy x and y from the market for...

The economy is populated with L farmers, that are consumers and producers at the same

time.

As consumers they buy x and y from the market for consumption. Their utility function is

u(x, y) = xy.

As producers all farmers are endowed with one unit of labor that they can use to grow crops

and sell them on the market to earn income. All farmers have access to two production

technologies

Qx(l) = αxl and Qy(l) = αyl

and they can choose between them (l is the amount of labor devoted to a specific technology,

Qx is the amount of x produced and Qy is the amount of y produced).

Let us denote Lx the number of people who chose to produce x and Ly the number of people

who chose to produce y.

(a) (1 point) Write down individual earnings of a person who produces x and a person

who produces y (these should be algebraic expressions involving px, py, αx, αy)

(b) (3 points)In equilibrium both x and y are demanded, so both x and y should be

supplied. Producers can choose what to produce, so we want them to choose both (we

want some of them to choose to produce x and others to choose to produce y).

Explain why it can not happen unless αxpx = αypy. (Hint: what would happen if

αxpx > αypy?, what would happen if αxpx < αypy?)

(c) (1 point) Let I be the equilibrium GDP in this economy. Write down the expressions

for the demand for x and y and the supply of x and y (these should be algebraic

expressions involving I, px, pyαx, αy, Lx, Ly).

(d) (1 point)Write down the expressions for I (this should be an algebraic expression

involving αx, αy, px, py, Lx, Ly)