Volkswagen owns two factories, one in Germany and one in the US. The long run pr

Question

Volkswagen owns two factories, one in Germany and one in the US. The long run production function in Germany is fGermany(K,L) = 9K^2/3 L^1/2 while that in the US is fUS(K,L)=200 (K^2+L^2)^-1/ 2

In the short run, capital is fixed at K = 10 in both factories.

1) Compute expressions for MPL and APL for the German factory and the US factory.

2) Is it true that MPL is decreasing everywhere for the German factory? the US factory? (Hint: You may use your graph to answer this question.)

3) Which factory has the highest marginal product of labor at L = 100 workers? Justify your answer numerically. Check your answer graphically by plotting the MPL curve for the German factory in the previous graph.

Explanation / Answer

a. s fGermany(K,L) = 9K^2/3 L^1/2

In short run

  s fGermany(K,L) = 9(10)^2/3 L^1/2 = 41.77*L^1/2

MPL = df/dL = 1/2*41.77L^-1/2 = 20.88L^-1/2

APL = s fGermany(K,L) /L = 20.88L^1/2

fUS(K,L)=200 (K^2+L^2)^-1/ 2

In Short run

  fUS(K,L)=200 ((10)^2+L^2)^-1/ 2

fUS(K,L)=200 ( 0.01 +L^2)^-1/ 2

MPL = df/dL = -1/2*200 ( 0.01 +L^2)^-3/ 2*L^-3

APL =  fUS(K,L)/L = 200 ( 0.01 +L^2)^-1/ 2/L

b

dMPL Germany/dL = -10.44(L)^-3/2

So, Its decreasing everywhere

dMPL US/dL = 3/2*100 ( 0.01 +L^2)^-5/ 2*L^-3 + -3*-100 ( 0.01 +L^2)^-3/ 2*L^-4

So, Its n't decreasing everywhere

c. MPL Germany = 20.88L^-1/2 = 20.88(100)-1/2 = 2.88

MPL US = -100 ( 0.01 +L^2)^-3/ 2*L^-3 = -100 ( 0.01 +(100)^2)^-3/ 2*(100)^-3

= -100(0.01 + 0.0001)^-3/2*0.000001

= 985*0.0001

= 0.985

So, MPL of Germany is highest.

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