004: Macroeconomic Theory

Transcription

1 004: Macroeconomic Theory Lecture 13 Mausumi Das Lecture Notes, DSE October 17, 2014 Das (Lecture Notes, DSE) Macro October 17, / 18

2 Micro Foundation of the Consumption Function: Limitation of the Static Approach In the last class we have seen that to derive the aggregate consumption function of the form C = C + cy (where the marginal propensity to consume out of current income is a constant) from optimizing behaviour of the households requires several restrictive assumptions: It requires specific functional forms of the utility function; It requires the future variables (e.g., P, r) to treated as be exogenous in the current period. This second assumption is particularly problematic because in a proper dynamic (intertemporal) framework current decisions will have implications for future and the future in turn will impact upon the current decision variables. Treating all future variables as exogenous shuts down this two-way feedback mechanism from present to future. Das (Lecture Notes, DSE) Macro October 17, / 18

3 Consumption Function: A Dynamic Approach The above mentioned limitation arises here because we have posited the household s intertemporal decision- making problem in a static set up. In order to allow for various intertemporal feedback mechanisms to operate we must pose the problem in a dynamic framework. A proper micro-founded dynamic model of consumption choice will take into account the feedback from current to future and vice versa. Here we briefly sketch the outline of such a dynamic model. We ignore the labour-leisure choice here and focus only on consumption choices in various time periods. We also ignore the price dynamics and assume that the price level remains constant at unity. Thus all variables that we consider below are expressed in terms of their real values. Das (Lecture Notes, DSE) Macro October 17, / 18

4 Consumption Function: A Dynamic Approach (Contd.) Consider a one member household where the agent lives for a finite time period T ( 2). The agent has a given stream of income over his life-time, denoted by: y 0, y 1,..., y T. Given the stream of income the agent, sitting at time 0, wants to optimally determine his consumption path over this entire time period (denoted by {c t } T t=0 ) so as to maximise his life-time utility function represented by: U 0 = u(c 0 ) + βu(c 1 ) + β 2 u(c 2 ) β T u(c T ) = where β (0, 1) represents the discount factor. T β t u(c t ) t=0 Das (Lecture Notes, DSE) Macro October 17, / 18

5 Consumption Function: A Dynamic Approach (Contd.) In each period, the agent can use his current income to consume or to save. (Savings in any period can be negative too, which means the household is borrowing). Let r be the given time-invariant rate of interest in the economy. Savings in time period t generates an additional interest earnings of (1 + r)s t in time period t + 1 over and above the current income in that period (y t+1 ). Altenatively, borrowing at time period t (b t = s t ) implies that (1 + r)b t amount will be deducted as interest payments in time period t + 1 from his current income y t+1. Either way, the flow budget constraint of the agent at any time period t + 1 is can be written as: c t+1 + s t+1 = y t+1 + (1 + r)s t. Das (Lecture Notes, DSE) Macro October 17, / 18

6 Consumption Function: A Dynamic Approach (Contd.) Notice that even though there are two choice variables in any time period (c t and s t ), they are not really independent. Given the RHS (which is determined by history), once we choose c t,the corresponding s t is automatically determined form the corresponding flow budget constraint. We shall call c t the control variable, and s t the state variable. Note that there are altogether T + 1 such flow budget constraint faced by the the household, which are given by c 0 + s 0 = y 0 ; c 1 + s 1 = y 1 + (1 + r)s 0 ;... c T 1 + s T 1 = y T 1 + (1 + r)s T 2 ; c T = y T + (1 + r)s T 1, Das (Lecture Notes, DSE) Macro October 17, / 18

7 Consumption Function: A Dynamic Approach (Contd.) In writing these flow budget constraints, we have assumed that the household does not start with any past stock of savings or past debt, i.e., s t 1 = 0 at t = 0. This is an initial conition imposed on the state variable s t. Since the agent is going to die at the and of period T, saving at this terminal period does not make sense. Hence we have also imposed a terminal condition (transversality condition) on the state variable: s T = 0. Notice however that this last logic (that s t should be zero at the terminal time period) is not valid if s t represents the amount of borrowing instead of savings. In fact in that case the household would want to borrow an infinite amount in the terminal periond which he never has to pay back (because he is not there the next period!), i.e., s T =. Das (Lecture Notes, DSE) Macro October 17, / 18

8 Consumption Function: A Dynamic Approach (Contd.) If we allow this possibility then our optimization problem becomes meaningless! Anybody can borrow a huge amount any point of time to maintain an arbitarily high level of consumption level all through and thus pile up an infinite amount of debt which he can push to the terminal period. And then he can escape paying altogether! We want to impose additional conditions to rule out this possibility. Setting the terminal condition s T = 0 indeed rules this out. Terminal condition s T = 0 means either that households are not allowed to borrow ( s t 0 for all t, including the terminal time T ) or that households are allowed to borrow at various points of time but at the terminal time they must clear all their debt ( s t 0 for any t < T, but at the terminal time s T = 0). For simplicity we shall assume here that borrowing is not allowed. We shall change this assumption to allow borrowing (subject to an additional condition) later in the course. Das (Lecture Notes, DSE) Macro October 17, / 18

9 Consumption Function: A Dynamic Approach (Contd.) In order to determine its optimal consumption path, the household has to maximise T t=0 constraints specified above. β t u(c t ) subject to all the (T + 1) flow budget Das (Lecture Notes, DSE) Macro October 17, / 18

10 Consumption Function: A Dynamic Approach (Contd.) Given the dimension of the problem (T + 1 control variables, T state variables, T + 1 constraint functions), it is not easy to employ the Lagrange method of constrained optimization here. Typically, one uses a different technique called dynamic programming. However, given the simple nature of the flow budget constraints here (all terms are linear and additive), we can eliminate s 0, s 1,..., s T 1 by successive iteration, and use the given initial and terminal condition on s t to arrive at the following life-time budget constraint: c 0 + c r c T (1 + r) T = y 0 + y r y T (1 + r) T. This reduces the problem to T + 1 control variables and a single constraint function. Das (Lecture Notes, DSE) Macro October 17, / 18

11 Consumption Function: A Dynamic Approach (Contd.) One can now apply the Lagrange method to get the following (T + 1) FONCs: u (c 0 ) = λ; βu (c 1 ) = λ 1 + r ;... β t u (c t ) = λ (1 + r) t ; β t+1 u (c t+1 ) = β T u (c T ) = λ (1 + r) t+1 ;... λ (1 + r) T. Ideally, we should be able to solve these T + 1 FONCS along with the life-time budget constraint to get the optimal consumption path {c t } T t=0 as well as the Langrangean multiplier λ. Das (Lecture Notes, DSE) Macro October 17, / 18

12 Consumption Function: A Dynamic Approach (Contd.) We shall characterize the precise consumption path for similar dynamic optimization problems later (using dynamic programming technique.) At this point, it suffi ces for you to note that optimal consumption at any two successive points of time t and t + 1 are related through the following equation: u (c t+1 ) = 1 (1 + r) β u (c t ). This is essentially a difference equation (in implicit form) which tells us that today s consumption and tomorrow s consumption are interrelated through a well-defined dynamic equation. Thus one cannot talk about current consumption in isolation (as we did in the previous static analysis) without analysing its implication for future consumption and vice versa. Das (Lecture Notes, DSE) Macro October 17, / 18

13 A Micro-founded Dynamic Theory of Investment? Just as we have posited the household s decision making problem in a dynamic framework, we can do so for the optimization problem of the firm as well. In fact a dynamic set up seems the natural choice if one is to analyse the optimal investment decision of a firm. Why do firms invest? Because investment augments their productive capacity and allows them to sell more and thus earn more profit in future. Thus the current investment decision must be guided by possibilities of future profit earnings of the firm. Here we again sketch a brief outline of such a dynamic model of investment. Das (Lecture Notes, DSE) Macro October 17, / 18

14 A Dynamic Theory of Investment Once again we ignore the price dynamics and assume that the price level remains constant at unity. Thus all variables that we consider below are expressed in terms of their real values. Consider a firm that which produced a final commodity in every period using a production function Y t = F (N t, K t ). The usual diminishing marginal product proprties, CRS property and the Inada conditions are assumed to hold. At time t, the capital stock available to the firm is given. However it can augment its capital stock over time by investing an amount I t (out of its current profit) which augments the capital stock in the next period: K t+1 = I t + (1 δ)k t for all t = 0, 1, 2,.... (1) Das (Lecture Notes, DSE) Macro October 17, / 18

15 A Dynamic Theory of Investment (Contd.) Assuming a competitive market structure (such that the firm is a price taker), its instantaneous net profit at any time period t is given by: Π t = F (N t, K t ) wn t I t. Let r be the given time-invariant rate of interest in the economy. The firm is believed to be infinitely-lived. So the present discounted value of its sum of net profit from time 0 to is given by: V 0 = t=0 Π t (1 + r) t = t=0 F (N t, K t ) wn t I t (1 + r) t. (2) The firm maximizes (2) subject to its period by period capacity constraint (given by (1)), to determine its optimal level of employment in every period (N t ) and its optimal level of investment in every period (I t ). Das (Lecture Notes, DSE) Macro October 17, / 18

16 A Dynamic Theory of Investment (Contd.) However, since choosing current level of investment is equivalent to choosing next period s capital stock, we can use the capacity constraint equation (given by (1)) to eliminate I t and write down the optimization problem of the firm as an unconstraint problem: Max {N t } 0,{K t+1} 0 V 0 = F (N t, K t ) wn t K t+1 + (1 δ)k t t=0 (1 + r) t ; K 0 given. This yields the following set of first order conditions (for t = 0, 1, 2,... ): (i) ( ) V 0 1 t = 0 [F N (N t, K t ) w] = 0 N t 1 + r ( ) V 0 1 t [ FK (N t+1, K t+1 ) + (1 δ) (ii) = 0 K t r 1 + r ] 1 = 0 Das (Lecture Notes, DSE) Macro October 17, / 18

17 A Dynamic Theory of Investment (Contd.) This tells us that the dynamically optimizing firm would choose an employment level and an investment level in every period such that F N (N t, K t ) = w; F K (N t+1, K t+1 ) δ = r. These look similar to the decisions that will undertaken by a firm in a static optimization framework as well (provided we allow it choose its capital stock for the next period). Thus adding a dynamic framework does not add much to the producer s side of the story - unless we add some additional adjustment costs of investment C (I t ). We shall come back to the adjustment cost issue later (if time permits). Das (Lecture Notes, DSE) Macro October 17, / 18

18 Micro- Foundation: References Ben Heijdra: Foundations of Modern Macroeconomics, Chapter 1, Pages 1-8; Chapter 2, Pages William Branson: Macroeconomic Theory and Policy, Chapter 12, Pages (Scanned pages available in CDE Server: Q:/Course/004) Das (Lecture Notes, DSE) Macro October 17, / 18