Suppose a firm at the first stage has employed 4 labors and 16 capitals. At the second stage, the

manager cannot hire or fire any of the labors and cannot buy or sell any amount of capital. He just needs

to decide on how many plants to build and then allocate the labors and capital among different plants of

his firm equally. If you need, assume that the price of the firm's product is 1, the labor wage is w and the

rental price of capital is r. Suppose the production function in each plant is given by Y = K²L², where Y,

K, and L denote production, capital, and labor, respectively.

(i) Denote the number of plants by N, construct the profit maximization problem of the firm and discuss

the optimal number of plants needed to build. Assume building any extra plant has no cost for the firm.