Consider the problem of a firm that invests in intangible capital (brand capital) in addition to physical capital and labor. The firms operating profits at time is ( , , ) where , and are physical capital, brand capital and labor at time , respectively. The firm invests in physical capital and brand capital every period. The two capital stocks evolve as follows: =(1)+ +1 =(1)+ +1 where and are the exogenous depreciation rates of physical capital and brand capital, respectively. The firm hires amount of labor every period. However, a fixed rate of labor, denoted by , leave every period for voluntary reasons. Firm incurs costs when adjusting stocks of physical and brand capital of the form ( , , , ). This cost function is convex and increasing in investments of both capital units. There is no ad- justment cost involved in hiring labor. Labor is paid at the wage rate . The prices of both physical and brand capital are constant and normalized to unity. The firm takes all prices includ- ing the wage rate as given. Finally, time is discrete and the firm lives forever, = 0, 1, 2 ... . The firm discounts time using the constant factor . Based on this information, answer the following questions: a. Write down the law of motion for labor and state the expression for dividends. What is the value of the firm? [15%] b. State the optimization problem of the firm and solve it to obtain the system of equations that characterize firms optimal investment policy in all variable inputs. You should provide an interpretation of these equations. [20%] c. If operating profits net of all costs are taxed at the rate , how does investment in physical and brand capital respond to changes in this corporate tax rate?