a) Use the method of Lagrange multipliers to find the cost function c(r,w,y). Find the average and marginal cost.

To minimize output given the budget is:
Min C = wL + rK
s.t. Y = K^a L^b
Lagrange function is
F = wL + rK - D( K^a L^b - Y)
Where D is the Lagrange multiplier.

First order condition:
dF/dK= r - D aK^(a-1) L^b = 0 (1)
dF/dL = w - D bK^a L^(b-1) = 0 (2)
dF/dD = - K^a L^b + Y = 0 (3)

from (1): r = D aK^(a-1) L^b (4)
from (2): w = D bK^a L^(b-1) (5)

(4)/(5):
r /w = a/b * L/K
K = a/b * w/r * L
substitute into (3):
-K^a L^b + Y = 0
K^a L^b = Y
(a/b * w/r * L)^a L^b = Y
(aw/br)^a L^(a+b) = Y ...

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