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 3; is a function of its exogenous productivity 2 and its chosen labor a according to the function ytn} given by yErr} = 251\". The firm takes as given the exogenous wage W which it must pav for each unit of lab-or n in a competitive labor market. If the firm chooses to produce in capital mode. it must pav an exogenous fixed cost or x t] to do so. This fixed cost does not varsr with the level of production in capital mode. and vou should note from shove that there is no such fixed cost for labor mode. In capital mode. the firm's output 3; is a function of its exogenous productivity 2 and its chosen capital it: according to the function yr] given by yc} = zfc\". The firm takes as given the exogenous rental rate of capital R'\" r: c which it must pav for each unit of capital it in a competitive spot market for capital. Note that the exogenous productivity 2 3: ti and the elasticity of production :1 E (o. 1} are the same for both labor mode and capital mode. Assume that H" a: W, Le. assume that capital is a cheaper input than labor at the margin. {a} {T points} If the firm chooses to produce in labor mode. solve the profit maximization problem for the firm maxyn} W-n. fl. and provide expressions for the optimal level of labor n. the resulting optimal level of output pm}. and the resulting optimal level of profits {n) 2 gm} We as functions of exogenous values and parameters alone. {bi [T points}L If the firm chooses to produce in capital mode. solve the profit maximization problem for the firm my art-'5) Raf: - ti: and provide expressions for the optimal level of capital it, the resulting optimal level of output stir}. and the resulting optimal level of profits [k) = we] R'vi: o as functions of exogenous values and parameters alone