3. (40 points) Consider a firm that owns the capital stock it uses to produce a final good. Every period the firm has to decide how much capital to purchase, i.e. how much to invest in order to maximize its profits. Note that investment can also be negative; this simply means that the firm sells some of its capital stock in order to reduce the amount of capital it owns to adjust its production. The timing of events each period is as follows. The amount of capital the firm owns at the beginning of period t is denoted by ke. The firm produces a final good using the production technology y = kt, where y denotes the amount of good it produces. The price of the good produced by the firm is given by p and it is taken as given by the firm. After production is finished the firm decides how much capital to purchase. The purchase price of capital is constant and equal to 1. For simplicity we assume that there is no depreciation, so the capital stock available for production at the beginning of period t +1 will be given by kt+1 = 14 + kit (1) where I, is investment in period t. We assume that it is costly for the firm to increase or decrease its capital stock. The cost of adjusting the capital stock is given by C(11) = {1}. You can think of these adjustment costs as installation costs. If the firm needs to extend its produc- where I, is investment in period t. We assume that it is costly for the firm to increase or decrease its capital stock. The cost of adjusting the capital stock is given by C(1) = {1}. You can think of these adjustment costs as installation costs. If the firm needs to extend its produc- tion capacity, then it needs to hire some engineers to have additional equipment installed. The amount it needs to pay depends on the amount of equipment installed. This cost is represented by C(1.) and it is not included in the pur- chase price of capital. Also, if the firm would like to have some of its equipment removed, then it also needs to hire some engineers which is costly. Hence, the firm's profits in period t are given by pke - 11 - C(I). The firm maximizes the present value of its profits, - ) + rilpkis 1, -C(1)) subject to (1), where r is the interest rate and is assumed to be constant. The initial amount of capital, ko is taken as given by the firm. (a) (10 points) Set up the Lagrangian and derive the first-order conditions of the problem. (b) (10 points) Show that by combining the first-order conditions you get the equilibrium condition 1 +1441 = (1 + r)(1 + I.) -P (c) (10 points) Show that equation (5) implies that investment in period t can be expressed as 1 = (1 + x) 10 + 1 + h): where A=r-p (d) (10 points) Interpret the Lagrange multiplier. 3. (40 points) Consider a firm that owns the capital stock it uses to produce a final good. Every period the firm has to decide how much capital to purchase, i.e. how much to invest in order to maximize its profits. Note that investment can also be negative; this simply means that the firm sells some of its capital stock in order to reduce the amount of capital it owns to adjust its production. The timing of events each period is as follows. The amount of capital the firm owns at the beginning of period t is denoted by ke. The firm produces a final good using the production technology y = kt, where y denotes the amount of good it produces. The price of the good produced by the firm is given by p and it is taken as given by the firm. After production is finished the firm decides how much capital to purchase. The purchase price of capital is constant and equal to 1. For simplicity we assume that there is no depreciation, so the capital stock available for production at the beginning of period t +1 will be given by kt+1 = 14 + kit (1) where I, is investment in period t. We assume that it is costly for the firm to increase or decrease its capital stock. The cost of adjusting the capital stock is given by C(11) = {1}. You can think of these adjustment costs as installation costs. If the firm needs to extend its produc- where I, is investment in period t. We assume that it is costly for the firm to increase or decrease its capital stock. The cost of adjusting the capital stock is given by C(1) = {1}. You can think of these adjustment costs as installation costs. If the firm needs to extend its produc- tion capacity, then it needs to hire some engineers to have additional equipment installed. The amount it needs to pay depends on the amount of equipment installed. This cost is represented by C(1.) and it is not included in the pur- chase price of capital. Also, if the firm would like to have some of its equipment removed, then it also needs to hire some engineers which is costly. Hence, the firm's profits in period t are given by pke - 11 - C(I). The firm maximizes the present value of its profits, - ) + rilpkis 1, -C(1)) subject to (1), where r is the interest rate and is assumed to be constant. The initial amount of capital, ko is taken as given by the firm. (a) (10 points) Set up the Lagrangian and derive the first-order conditions of the problem. (b) (10 points) Show that by combining the first-order conditions you get the equilibrium condition 1 +1441 = (1 + r)(1 + I.) -P (c) (10 points) Show that equation (5) implies that investment in period t can be expressed as 1 = (1 + x) 10 + 1 + h): where A=r-p (d) (10 points) Interpret the Lagrange multiplier