Question

An island economy has a labor endowment of 60 units to produce consumption goods x and y. The production functions are x = 0.2Lx and y = 0.10Ly, where Lx and Ly represent the labor allocation. The utility function is u(x,y) = 2x + 8y.

(a.) What is the equation for the PPF?

(b.) What is the optimal basket (x*,y*) in a closed economy?

(c.) In an open economy, if the price of x is Px = $6 and the price of y is Py = $5, then find the optimal production pair (xp*, yp*) and optimal consumption pair (xc*,yc*).

Answer 1

Lx+Ly= 60--(a)

x = 0.2Lx--(b)

y = 0.10Ly--(c)

a. From b we get, x/0.2= Lx and from c we get, y/0.1=Ly

Put these in (a) we get PPF

x/0.2+ y/0.1= 60

b. In closed economy; equilirbrium will be where slope of utility function, that is MRS, is equal to slope of PPF, which is MRT

Now, MRS= (du/dx)/(du/dy)= 2/8= 1/4

MRT= (1/0.2)/(1/0.1)= 1/2

So, MRS<MRT this means equilibrium will be where x*=0, So, y*= 60*(0.1)= 6

(x*,y*)= (0,6)

c. If the price of x is Px = $6 and the price of y is Py = $5, Then budget line will be: 6x+5y= m (coz income is not given in the equation).

then consumer's optimal point is where Px/Py= MRS

But, 6/5>1/2 so slope of budget line is steeper than slope of IC this means optimal consumption is where y*=0, x*= m/6

slope of PPF= 1/4 is flatter than sope of cost line= 6/5, so optimal production point is where x*=0, y*= 6.