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The Tobin-Hayashi -theory of investment, imperfect competition, and the IS-LM model. Consider a single firm supplying a differentiated good in the amount per time unit at time . The production function is = 1 0 1 (*) where and are capital and labor input at time respectively. The nominal wage and the nominal general price level in the economy faced by the firm are constant over time and exogenous to the firm. So the real wage is an exogenous positive constant, . The demand, , for the firm's output is perceived by the firm as given by = 1where is the price set in advance by the firm (as a markup on expected marginal cost), relative to the general price level in the economy, is the given large number of monopolistically competitive firms in the economy, is the overall level of demand, and is the (absolute) price elasticity of demand. The interpretation is that the firm faces a downward sloping demand curve the position of which is given by the overall level of demand, which is exogenous to the firm. We assume that within the time horizon relevant for the analysis, is constant and the firm keeps fixed, possibly due to menu costs. Moreover, the analysis will ignore uncertainty. The increase per time unit in the firm's capital stock is given by = 0 0 0 given, where is gross investment per time unit at time and is the capital depreciation rate. We assume that is high enough to always be above actual marginal cost so that it always pays the firm to satisfy demand. Assuming the price of the investment goods is 1 for all the cash flow at time is = () where () represents a capital installation cost and the function satisfies (0) = 0 (0) = 0 00() 0 (An example is ()=(2)2 0) In this exercise we assume that is sales while the installation costs are payment for external installation services. a) To obtain = a certain employment level is needed. Find this employment level as a function of and Let your result be denoted ( ) The real interest rate faced by the firm is denoted and is, until further notice, a positive constant. As seen from time 0, the firm solves the following decision problem: max ()=0 0 = Z 0 ( ) () s.t. free (i.e., no restriction on ) = 0 0 given, 0 for all 49 b) Briefly interpret this decision problem, including the constaints. c) Denoting the adjoint variable derive the first-order conditions and state the necessary transversality condition (TVC) for a solution. Hint: the TVC has the standard form for an infinite horizon optimal control problem with discounting. d) The optimal investment level, can be written as an implicit function of Show this. e) Construct a phase diagram for the ( ) dynamics, assuming that a steady state with 0 exists. Let the steady state value of be denoted . For an arbitrary 0 0 indicate in the diagram the movement of the pair ( ) along the optimal path. Assume that until time 1 the economy has been in steady state. Then, unexpectedly, the aggregate demand level, and thereby shifts to a new constant level 0 and is rightly expected to remain at that level for a long time. f) Illustrate by the same or a new phase diagram what happens on impact and gradually over time. Comment on the implied effect on investment on impact and in the long run. Assume instead that it is the interest rate which at time 1 shifts to a new constant level 0 and is rightly expected to remain at that level for a long time. g) Illustrate by the same or a new phase diagram what happens on impact and gradually over time. Comment on the implied effect on investment on impact and in the long run. h) As a modified scenario, imagine that the fall in demand at time 1 considered under f) was in fact due to a rise in the interest rate at time 1 Compare the implied combined effect on investment on impact and in the long run with the isolated effects under f) and g), respectively. i) Relate the results in f) and g) to the signs of the partial derivatives of the investment function in a standard IS-LM model. Comment. VI.5 Consider a single firm with production function = () 1 0 1 where and are output, capital input, and labor input per time unit at time , respectively. Time is continuous and is the technology level, growing over time at a constant rate 0. The increase per time unit in the firm's capital stock is given by = 0 0 0 where is gross investment per time unit at time and is the capital depreciation rate. Cash flow (in real terms) at time is = () 1 where is the real wage and represents capital installation costs given by = 2 2 There is perfect competition in all markets and no uncertainty. The real interest rate faced by the firm is a constant 0. a) Set up the firm's intertemporal production and investment problem as a standard optimal control problem, given that the firm wants to maximize its market value. Let the adjoint variable be denoted b) Derive the first-order conditions and state the necessary transversality condition (TVC). Hint: along the optimal plan the partial derivative of the current-value Hamiltonian w.r.t. the state variable equals the difference between the discount rate multiplied by the adjoint variable and the time derivative of the adjoint variable; the TVC has the standard form for an infinite horizon optimal control problem with discounting. c) What is the economic interpretation of ? On the basis of one of the first-order conditions, express the optimal investment level at time as a function of and From now on, assume that the firm is a representative firm in a small open economy. d) Suppose the government wants to stimulate firms' investment and from time 0 on implements a subsidy 0 1 so that to attain an investment level purchasing the investment goods involves a cost of (1 ) Assuming the subsidy is financed by some tax which neither directly nor indirectly affects firms' behavior, will the government attain its goal? Make sure that you substantiate your answer by a formal proof.

Define () and assume that labor supply grows at the constant rate 0 e) Show that = ( ) where and are constants f) On the basis of another of the first-order conditions from question b), derive an equation for in terms of and g) Suppose + Draw a phase diagram in the ( ) plane and illustrate the evolution of the economy for 0, assuming that 0 0 where is the steady-state value of Hint: it can be shown that in a neighborhood of the steady state, the = 0 locus is negatively sloped.

Consider a small open economy facing a constant real interest rate, given from the world market. Markets are competitive. Labor supply is inelastic and constant over time and there is no technical progress. The government contemplates introduction of an 'investment subsidy', such that to buy machines, each with a price equal to one unit of account, firms have to pay (1 ) units of account, where is a constant, 0 1. The private sector is in a steady state and is not aware of these governmental considerations. "In this setting, Tobin's -theory of investment predicts that by introducing and maintaining the investment subsidy , the government will be able to stimulate aggregate net investment temporarily, but not permanently." True or false? Why? VI.7 Short question. In a standard Ramsey model and a model based on the -theory of investment the circumstances under which firms optimize are different. Give a brief characterization of this difference and its implications. VI.8 Short question. In many simple macroeconomic models a firm's acquisition of its capital input is described as if the firm solves a sequence of static profit maximization problems. One can imagine circumstances where this description of firms' behavior is not adequate, however. Give a brief account of what such circumstances might be and what alternative approach might be relevant. VI.9 (Re-exam, Febr. 2016.) Consider a single firm that is a price taker in all markets. The firm chooses a plan ( ) =0 to maximize 0 = Z 0 (( ) ( ) ) s.t. 0 free (i.e., no restriction on ) (2) = 0 0 given, (3) 0 for all 0 (4) Here is a neoclassical production function with constant returns to scale and satisfying the Inada conditions. The stock of capital and the input of labor are denoted and respectively; is a function representing the capital installation costs, is gross investment, is the real wage, 0 is a constant real price of investment goods, 0 a constant real interest rate, and 0 a constant capital depreciation rate. The installation cost function satisfies (0 )=0 (0 )=0 () 0 and () 0 for all () a) Set up the current-value Hamiltonian. Let the adjoint variable be denoted Derive the first-order conditions and state the necessary transversality condition (TVC). Hint: in this problem the necessary TVC has the standard form for an infinite horizon optimal control problem with discounting. b) Interpret Show that the optimal level of investment at time is a function of and From now on we consider the special case () = 2 2 c) Express the optimal as a function of and Assume that the firm is a representative firm in a small open economy (SOE) with perfect competition in all markets and free mobility of financial capital, but no mobility of labor across borders. The real interest rate and the price on investment goods that the firm faces are given from the world market and identical to and respectively. The labor force in the SOE is a constant, . d) Determine the equilibrium real wage in the SOE at time e) Show that the firm's first-order conditions result in two coupled differential equations in and f) Construct the corresponding phase diagram. Comment! Hint: it can be shown that the curve representing = 0 has negative slope (at least in a neighborhood of the steady state). 53 g) Determine the steady-state value of h) Illustrate in the phase diagram the path followed by ( ) for an arbitrary 0 0 Comment! i) Assume that until time 0 0 the system has been in steady state with ( )=( ). Then an unexpected shift in to the level 0 occurs. Assume that the new price is rightly expected to stay at the new level forever. Illustrate in the phase diagram (possibly a new one) the path followed by ( ) for 0 Hint: at least for a "small" rise in it can be shown that the new saddle path necessarily will be positioned above the old (a fact which to some people is surprising). j) Illustrate in a new diagram the time profile for and for 0. Give an intuitive explanation of the long-run effect on and respectively, of the rise in.