As head of the planning commission of Eastern Motors, your job is to determine where to locate a new plant.The only inputs used in your cars are steel and labor and the production function is Cobb-Douglas where f (S, L)= S.5L.5, where S is tons of steel and L is units of labor.You can locate your plant either in country A or country B.In country A, steel costs $7 a ton and labor costs $7 per unit.In country B, steel costs $8 per ton and labor costs $6 per unit.In which country should the company locate its new plant so as to minimize costs per unit of output?
A)Country A.
B)Country B.
C)It doesn't matter, because the two locations are equally costly.
D)Country A if output is greater than 14, and country B otherwise.
E)There is not enough information to enable us to tell.

Question

Quizplus · Accepted Answer

To minimize costs per unit of output, we need to compare the cost of producing one unit of output in each country. The cost function for one unit of output is:
C(1)A=7*Sa+7*La
C(1)B=8*Sb+6*Lb
where Sa and La are the amounts of steel and labor required in country A and Sb and Lb are the amounts required in country B.

Now, we can use the production function to express La and Lb in terms of Sa and Sb:
La=Sa0.5Lb=Sb0.5
Substituting these expressions into the cost functions above, we get:
C(1)A=7*Sa+7*Sa0.5
C(1)B=8*Sb+6*Sb0.5
To compare the two costs, we can divide them:
C(1)A/C(1)B=[7*Sa+7*Sa0.5]/[8*Sb+6*Sb0.5]
To find the critical point where the two costs are equal, we can set this expression equal to 1 and solve for Sa/Sb:
[7*Sa+7*Sa0.5]/[8*Sb+6*Sb0.5]=1
7*Sa+7*Sa0.5=8*Sb+6*Sb0.5
Solving for Sa/Sb, we get:
Sa/Sb=(6/7)2
Sa/Sb=1.2245
This means that if Sa/Sb is greater than 1.2245, it is cheaper to produce in country A, and if Sa/Sb is less than 1.2245, it is cheaper to produce in country B.

Now, to find the actual ratio of Sa/Sb for the company's production function, we can use the fact that the production function is Cobb-Douglas and homogeneous of degree one. This means that the ratio of the inputs will be the same as the ratio of the output:
Sa/Sb=f(Sa, La)/f(Sb, Lb)=(Sa/Sb)0.5(La/Lb)0.5
Since the ratio of output is not given, we cannot determine the exact value of Sa/Sb. However, we can use the production function to determine the optimal location depending on the level of output. The total cost function for a given level of output Q is:
C(Q)A=7(Sa/Q)0.5(Q/La)0.5+7(Q/La)C(Q)B=8(Sb/Q)0.5(Q/Lb)0.5+6(Q/Lb)
To minimize this function, we can take the derivative with respect to Q and set it equal to zero:
dC/dQ=((-3.5*7*Sa/Q1.5)/(2*La/Q0.5))+7/(La/Q)=((-4*8*Sb/Q1.5)/(2*Lb/Q0.5))+6/(Lb/Q)
Simplifying, we get:
(Q/La)/(Sa/Q1.5)=1.578(Q/Lb)/(Sb/Q1.5)
(Q/La)2=2.489(Q/Lb)2
Q/La=1.579(Q/Lb)
This means that if Q/La is greater than 1.579, it is cheaper to produce in country A, and if Q/La is less than 1.579, it is cheaper to produce in country B.

Therefore, the company should locate its new plant in country B if the level of output is low enough for Q/La to be less than 1.579, and in country A if the level of output is high enough for Q/La to be greater than 1.579.

As Head of the Planning Commission of Eastern Motors, Your