I. IUUIELII. . 1.!\" Fullll'f Consider the infinite horizon model of a firm facing adjustment costs to investment. The firm pays a corporate income tax, T}, and it also benefits from an investment tax credit, at the rate 5, on its investment expenditures. The investment tax credits a ppl},? to investment expenditures alone and not to the capital installation costs. Moreover, the installation cost is valued at the output price. The firm managers maximizes its present value of the firm, V\" m l W1=E t=n [1 + Tl! {[1T,}F(K,}Fh(1s,]i_+CU,K,}} {1} where PUG}, is the net output from using the capital stock Kt, {F' ,2\": CI and F" a: D}, T, is the corporate tax rate, I, is investment, C i It} is the capital installation cost, Pk, is the unit price of investment, 5, is the investment credit tax rate, r is the interest rate, which is constant. The capital stock evolves according to the following equation: K,_l = K, {1 a} + I, {2} The adjustment cost function has the following expression: I: 2 Ciao) = g (E. 4') {3} where 1r is a parameter, 1* 13- I], and I] if. :5 s: 1 is the depreciation rate of capital. 1. Denote by q, the shadow value of the capital stock and write the firm optimization problem using the the Lagrangian approach {make sure to indicate all decision variables} {3 points}