3. Consider the problem of a rm that invests in intangible capital (brand capital) in addition to physical capital and labor. The rm's operating prots at time t is arm], 3,, L,) where K}, B, and L, are physical capital, brand capital and labor at time t, respectively. The rm invests in physical capital 1!" and brand capital 1,3 every period. The two capital stocks evolve as follows: Kr+1=(1' aler + If 3,.1 = (l 53)B, +1r. where 5\" and 53 are the exogenous depreciation rates of physical capital and brand capital, respectively. The rm hires H, amount of labor every period. However, a xed rate of labor. denoted by p, leave every period for voluntary reasons. Firm incurs costs when adjusting stocks of physical and brand capital of the form C (K,, 8,, 15.1,\"). This cost function is convex and increasing in investments of both capital units. There is no adv justment cost involved in hiring labor. Labor is paid at the wage rate W,. The prices of both physical and brand capital are constant and normalized to unity. The rm takes all prices includ- ing the wage rate as given. Finally, time is discrete and the rm lives forever. I = 0. 1, 2 00. The rm discounts time using the constant factor [3. 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 rm? [15%] b. State the optimization problem of the rm and solve it to obtain the system of equations that characterize rm's optimal investment policy in all variable inputs. You should provide an interpretation of these equations. [20%] c. If operating prots net of all costs are taxed at the rate r, how does investment in physical and brand capital respond to changes in this corporate tax rate? [15%]