uses labour L to produce two outputs and , according to the production functions

Question

uses labour L to produce two outputs and , according to the production functions 1011-0.5 r82-0752 where L' is labour allocated to sector i = 1,2. She wishes to maximise the value of total output where 4 and 5 are the world prices of ri and r2, respectively. (o) Assuming that there are 12 units of labour available. Find the optimal labour allocation and the shadow price of labour. (15 marks) (b) Assume now that there are 20 units of labour available. What is the new optimal labour allocation and (15 marks) the shadow price of labour? (c) Explain your results in each case (8 marks)

Explanation / Answer

So, here the objective function is “V=4*X1+5*X2”, where “X1=10*L1-0.5*L1^2” and “X2=8*L2-0.75*L2^2”. So, the above function become as follows.

=> V = 4*X1+5*X2 = 4*(10*L1-0.5*L1^2)+5*(8*L2-0.75*L2^2)

=> 40*L1-2*L1^2+40*L2-3.75*L2^2 = V.

SO, here the problem is given by, “V = 40*L1 - 2*L1^2 + 40*L2 - 3.75*L2^2”, subject to “L1+L2=m”, where “m” be the available labor.

So, the lagrange function is given by.

=> L = 40*L1 - 2*L1^2 + 40*L2 - 3.75*L2^2 + c*(m-L1-L2).

So, the FOC for maximization are given by, “dL/dL1=dL/dL2=0”.

=> dL/dL1 = 40 - 2*2*L1+c*(-1) = 0, => 40 - 4*L1=c ……………(1).

=> dL/dL2 = 40 – 7.5*L2+c*(-1) = 0, => 40 – 7.5*L2=c ……………(2).

So, from (1) and (2) we get the following g conditions.

=> 40 - 4*L1 = 40 - 7.5*L2, => 4*L1 = 7.5*L2, => L1 = 1.875*L2.

Now, putting the above condition in to the constraint we get.

=> L1+L2=m, => 1.875*L2+L2=m, => 2.875*L2=m, => L2 = m/2.875=0.35*m.

So, here “L1=1.875*L2=1.875*(m/2.875) = 0.65*m = L1.

So, if “m=12”, => the optimum labor allocation is given by.

=> L1=0.65*m=0.65*12 =7.8=L1 and L2=0.35*m=0.35*12=4.2=L2.

Now, the objective function is given by, “V=40*L1 - 2*L1^2 + 40*L2 - 3.75*L2^2 = 40*7.8 - 2*(7.8)^2 + 40*(4.2) - 3.75*(4.2)^2= 312 – 121.68 + 168 – 66.15= 292.17=V*.

Now, let’s assume that “m” increases to “13”, => “L1=0.65*m=0.65*13 =8.45=L1 and L2=0.35*m=0.35*13=4.55=L2. So, the objective function is given by, “V=40*L1 - 2*L1^2 + 40*L2 - 3.75*L2^2 = 40*8.45 - 2*(8.45)^2 + 40*(4.55) - 3.75*(4.55)^2= 338 – 142.81 + 182 – 77.63= 299.56=V*.

So, the shadow price of labor is given by, (299.56-292.17)/(13-12) = 7.39.

b).

Similarly, if “m=20”, => the optimum labor allocation is given by.

=> L1=0.65*m=0.65*20 =13=L1 and L2=0.35*m=0.35*20=7=L2.

Now, the objective function is given by, “V=40*L1 - 2*L1^2 + 40*L2 - 3.75*L2^2 = 40*13 - 2*(13)^2 + 40*7 - 3.75*(7)^2= 520 – 338 + 280 – 183.75= 278.25=V*.

Now, let’s assume that “m” increases to “21”, => “L1=0.65*m=0.65*21 =13.65=L1 and L2=0.35*m=0.35*21=7.35=L2. So, the objective function is given by, “V=40*L1 - 2*L1^2 + 40*L2 - 3.75*L2^2 = 40*13.65 - 2*(13.65)^2 + 40*(7.35) - 3.75*(7.35)^2= 546 – 372.65 + 294 – 202.58= 264.77=V*.

So, the shadow price of labor is given by, (278.25-264.77)/(21-20) = (-13.48).

c).

So, in the 1st part we see that the total labor is “12” and the optimum labor choice is “L1=7.8” and “L2=4.2”. Now the shadow price is “7.39”, => if the labor supply increases by “1 unit”, => the objective function will increase by “7.39 units”.

Now, in the 2nd part we see that the total labor is “20” and the optimum labor choice is “L1=13” and “L2=7”. Now the shadow price is “-13.48”, => if the labor supply increases by “1 unit”, => the objective function will decrease by “13.48 units”.

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