3. Labor Mode vs Capital Mode (25 points total) A competitive firm which takes as given a price of 1 for its output can produce in one of two production modes: labor mode" or "capital mode." If the firm chooses to produce using labor mode, then its output y is a function of its exogenous productivity z and its chosen labor n according to the function y(n) given by y(n) = zna The firm takes as given the exogenous wage W which it must pay for each unit of labor n in a competitive labor market. If the firm chooses to produce in capital mode, it must pay an exogenous fixed cost $ > 0 to do so. This fixed cost does not vary with the level of production in capital mode, and you should note from above that there is no such fixed cost for labor mode. In capital mode, the firm's output y is a function of its exogenous productivity z and its chosen capital k according to the function y(k) given by y(k)=zk The firm takes as given the exogenous rental rate of capital R$ > 0 which it must pay for each unit of capital k in a competitive spot market for capital. Note that the exogenous productivity z > 0 and the elasticity of production a (0,1) are the same for both labor mode and capital mode. Assume that Rk 0 to do so. This fixed cost does not vary with the level of production in capital mode, and you should note from above that there is no such fixed cost for labor mode. In capital mode, the firm's output y is a function of its exogenous productivity z and its chosen capital k according to the function y(k) given by y(k)=zk The firm takes as given the exogenous rental rate of capital R$ > 0 which it must pay for each unit of capital k in a competitive spot market for capital. Note that the exogenous productivity z > 0 and the elasticity of production a (0,1) are the same for both labor mode and capital mode. Assume that Rk