Ina owns a firm that transforms labor (N) and capital (K) inputs into product (Y ) according to the Cobb-Douglas production function:

Y = z(K^(a)) (N ^(1a))

Ina is short-sighted and makes no new investments, relying on existing capital K = 10 and total factor productivity z = 20 to maximize profits. The share of capital is 0.3 (a = 0.3). Ina can hire any number of workers without affecting the wage (w = $10 per hour) and all the workers are identical to Ina in terms of preferences and constraints and maximize their utility by freely choosing their leisure time. (Provide detailed answers for parts a and b)

a) If N corresponds to working hours per day, how many workers should Ina hire if her workers pay a lump sum tax? How many workers if the workers pay a labor income tax?

b) Can Ina's firm really maximize profits?