Consider a representative firm who uses labor (Lt) and capital (K) to produce a final good (Y). The firm's output technology is described by the Cobb-Douglas production function F(K, Lt) = A.KL. The firm hires labor in perfectly competitive markets at the rates W and invests (It) in new capital via the competitive resource markets. The firm pays a dead-weight cost of for investment values other than zero. In addition, the firm's investment process has a productivity shock which makes each unit of investment more or less valuable. The firm's profit maximization problem appears as follows: subject to: max II = II + Lt, It Lt+1,It+1 1 (1+r) -Ilt+1 - II = AKL - WLt - It- XI 2Kt It+1 = At+1K+1L++ - W+1Lt+1- It+1 Kt+1=MtIt + (1 - 8) Kt Kt+2 = Mt+1It+1+(1-6) Kt+1 XI+1 2K+1 1.) Re-write the optimization problem as a constrained optimization problem using qand 9t+1 as the Lagrange multipliers. 2.) Take the derivative of the profit maximization problem for the present value of invest- ment and re-arrange the condition to solve for a value of I, in terms of other variables and parameters. 3.) Explain (a verbal argument is sufficient) what the variables qt measures in the context of this problem. 4.) Explain what impact the deadweight cost has on the firm's investment decision. 5.) How would an increase in productivity (the variable +) impact the firm's investment decision?