Consider the firm Chipmunch, a potato chips producer with production function F(K,L) = K1/ 4L1/ 4 (as usual, K represents the production factor capital and L labour). Let the price of capital that Chipmunch faces be r =1 and the price of labour w =1. [Notice that you can do questions (c), (d) and (e) below, even if you do not know how the method of Lagrange works.]
(a) (1) Give a mathematical representation of the cost minimization problem of Chipmunch (assume Chipmunch feels like producing q units of output). (2) Write down the so-called Lagrangian of Chipmunch' cost minimization problem.
(b) (1) Derive the necessary conditions for an optimum using the Lagrangian. (2) Prove that MP K/ r= MP L /w in the optimum, and explain intuitively why this equation holds. (c) Derive the cost function of Chipmunch.