2 .A perfectly competitive firm that uses quantities Z₁ and Z₂ of inputs 1 and 2, at prices W₁ and W₂, and sells the resulting output in a perfectly competitive market at price p. The production function is f(z₁ z₂) = z₁^1/4 z₂^1/4.

2.1 Formulate the profit maximization problem for this firm and find the first-order conditions. Find the input demand functions for z1 and z2 and the pro t-maximizing quantity of output.

2.2 Formulate the cost-minimization problem for the firm, find the conditional demand functions, and find the cost function for the firm. Using this cost function, the firm solves the problem

max_q pq C(q)

Show that the solution to the above problem gives the firm the same output as in 2.1. [These results illustrate that, when working with a profit maximizing firm, it does not matter whether one takes an output-oriented as in 2.2 or an input-oriented approach as in 2.1.]