The quantity, q, of a product manufactured depends on the number of workers. IV,

Question

The quantity, q, of a product manufactured depends on the number of workers. IV, and the amount of capital invested. K, and is given by: q= 6W^3/4K1/4. Labor costs arc dollar 10 per worker and capital costs are dollar 20 per unit, and the budget is dollar 3000. What are the optimum number of workers and the optimum number of units of capital? Recompute the optimum values of W and K when the budget is increased by one dollar. Check that increasing the budget by dollar 1 allows the production of X extra units of the good, where X is the Lagrange multiplier.

Explanation / Answer

3000=(q)(10+20)

q = 3000/30

   q = 100

dq = 6*3/4*w -0.25 K0.25 + 6w0.75 * 0.25K -0.75 = 0

18/4*K= 6/4 w

3K = w...................(1)

but from the original equation

100 = 6 w 0.75 * (w/3)0.25

100 = 6 w (1/3)0.25

100 = 4.55901 w

Hence , w = 100 / 4.55901

w = 21.9346

Now from (1), putting w into (1), we get :

k = w/3

    = ( 21.9346 ) / 3

Hence, k = 7.3115

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