Question

Priyanka's company has the production function Q=100K^0.5L^0.5, where Q measures output, K measures machine hours, and L measures labor hours. Suppose that the rental rate of capital is R=$30, the wage rate is W=$15, and the firm wants to produce 5,000units of output. Use the Lagrange method to find the optimal input mix. What the optimal level of K & L?

Answer 1

Lagrangean problem can be set up as min 15L + 30K subject to the constraint 100√(KL) = 5000 , where 15L + 30K is the cost function .

Now, pitting this in the third FOC, 5000 = 100√(KL)

- IL + 8ok + A(coo༠ - lo/vt） ཨིན. }5- A (o༠) （༤) >༠？ FRཅན་ a_ kL dL QSKL ORDER dd = 30 - $(100) (l) = 0 COMO MONS 23KL dk ༤： D༠༠ - 1༠༠[RE ༤༠ ， = 15 = booster and ༡༠：༠ ལ་ པས་མཁ Jc ཆAL ་ཡོད ： ， ） at -L

==> 5000 = 100 √(K(2K)) ==> √(2K2) = 50 ==> 2K2 = 2500

==> K​​​​​2  = 1250 ==> K = 35.36

==> l = 2(35.36) = 70.72