A consumer has an utility function function u(x, y) = x⁴ y . There is an initial endowment of 81 units of good X. Good Y is a produced using X as an input. The production function is q=6z ^0.5 .

Determine the competitive (Walrasian) equilibrium: the prices; the input z used and output q produced by the firm; firm's revenue, cost, and profit; and the consumption bundle (x, y)