Assignment 2 Consicler a speciçic ractors model The Production functions of good

Question

Assignment 2 Consicler a speciçic ractors model The Production functions of good and Y are faictor endotoments are given a (l)suppose ex-2, and R-2-Gompocte the nominal toage rate anel Show haw labor Is Sharedl in Procluction of the tuoo geods CLx, anel Ly Suppase the pnice of X increases such thal uoe have Rx-3 and R :) . Show the effect onthe real voage. Cin terms of Y,and id Suppose the phces are-2,andl R2 as h ce and the endooment of R increases from too to llo, S remains ct loo Compute th neo CL, and Ly Shouo the efrect on the real veturn to labor

Explanation / Answer

a).

Consider the given problem here there are two goods “X” and “Y” and their production function are also given in the question.

=> X = R^0.5*Lx^0.5, => MPXL = 0.5*Lx^(-0.5)*R^0.5 = 0.5*(R/Lx)^0.5, => MPXL= 0.5*(R/Lx)^0.5.

Now, Y = S^0.5*Ly^0.5 = 0.5*Ly^(-0.5)*S^0.5 = 0.5*(S/Ly)^0.5, => MPYL = 0.5*(S/Ly)^0.5.

So, at the equilibrium “VMPXL = Px*MPXL” must be equal to “VMPYL = Py*MPYL”.

=> VMPXL = VMPYL, => Px*MPXL = Py*MPYL, => MPXL = MPYL, since “Px=Py=2”.

=> 0.5*(R/Lx)^0.5 = 0.5*(S/Ly)^0.5, => (100/Lx)^0.5 = (100/Ly)^0.5.  

=> 100/Lx = 100/Ly, => Lx = Ly. Now, Lx + Ly = 200, => Lx = Ly = 100.

So, there are total “200” labors and out of which “100” are employed at “X” and “100” are employed at “Y” industry.

Now, here the nominal wage is given by “W = VMPXL = VMPYL”. So, at the equilibrium.

=> VMPXL = 0.5*(R/Lx)^0.5 = 0.5 = VMPYL, since “R = Lx = 100”. So, here the nominal wage is “W=0.5”.

b).

Now, let’s assume that “Px” increases to “3” and “Py” remain same at “2”. So, under this change at the equilibrium “VMPXL = VMPYL”.

=> Px*MPXL = Py*MPYL, => 3*0.5*(R/Lx)^0.5 = 2*0.5*(S/Ly)^0.5, => 9*(R/Lx) = 4*(S/Ly).

=> 9*(100/Lx) = 4*(100/Ly), => 9*Ly = 4*Lx, => 9*Ly = 4*(200 – Ly), => 9*Ly = 800 – 4*Ly.

=> 13*Ly = 800, => Ly = 800/13 = 61.54, => Lx=138.46 and Ly=61.54.

Now, the nominal wage is given by, “VMPXL = Px*0.5*(R/Lx)^0.5 = 2*0.5*(100/138.46)^0.5.

=> VMPXL = (100/138.46)^0.5 = 0.85, => the nominal wage is “W=0.85”.

So, the real wage in terms of “X” is “W/Px = 0.85/3 = 0.28”. Now, the real wage in terms of “Y” is given by “W/Py = 0.85/2 = 0.425”.

c).

Now, let’s assume that “Px=Py=2” as in “i" and “R” increases to “110”, => at the equilibrium “VMPXL = Px*MPXL” must be equal to “VMPYL = Py*MPYL”.

=> VMPXL = VMPYL, => Px*MPXL = Py*MPYL, => MPXL = MPYL, since “Px=Py=2”.

=> 0.5*(R/Lx)^0.5 = 0.5*(S/Ly)^0.5, => (110/Lx)^0.5 = (100/Ly)^0.5.  

=> 110/Lx = 100/Ly, => 11*Ly = 10*Lx. Now, Lx + Ly = 200, => 11*(200-Lx) = 10*Lx.

=> 2200-11*Lx = 10*Lx, => 2200 = 21*Lx, => Lx = 104.76 and Ly=95.24.

Now, as we know that nominal wage will be determined by the condition “VMPXL=VMPYL” and in both the sector “W” is same. Now, the price of “X” and “Y” both are same, => the real return in both the sector will not change.

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