CONSUMPTION 43 Exercise 8 Suppose that labor income y is generated by the following stochastic process: y t = Îy t−1 + x t−1 + ε 1t , x t = ε 2t , where x t (= ε 2t ) does not depend on its own past values ( x t−1 , x t−2 , ) and E (ε 1t · ε 2t )=0.x t−1 is the only additional variable (realized at time t − 1)whichaffects income in period t besides past income y t−1 . Moreover, suppose that the information set used by agents to calculate their permanent income y P t is I t−1 = { y t−1 , x t−1 } , whereas the information set used by the econometrician to estimate the agents’ permanent income is  t−1 = { y t−1 } . Therefore, the additional informat ion in x t−1 is used by agents in forecasting income but is ignored by the econometrician. (a) Using equation (1.7) in the text (lagged one period), find the changes in perma- nent income computed by the agents (y P t ) and by the econometrician ( ˜ y P t ), considering the different infor mation set used (I t−1 or  t−1 ). (b) Compare the variance of y P t e  ˜ y P t , and show that the variability of permanent income according to agents’ forecast is lower than the variability obtained by the econometrician with limited information. What does this imply for the interpreta- tion of the excess smoothness phenomenon? Exercise 9 Consider the consumption choice of an individual who lives for two periods only, with consumption c 1 and c 2 and incomes y 1 and y 2 . Suppose that the utility function in each per iod is u(c)=  ac − (b/2)c 2 for c < a/b; (a 2 /2b) for c ≥ a/b. (Even though the above utility function is quadratic, we rule out the possibility that a higher consumption level reduces utility.) (a) Plot marginal utility as a function of consumption. (b) Suppose that r = Ò =0,y 1 = a/b, and y 2 is uncertain: y 2 =  a/b + Û, with probability 0.5; a/b − Û, with probability 0.5. Write the first-order condition relating c 1 to c 2 (random variable) if the consumer maximizes ex pected utility. Find the optimal consumption when Û =0, and discuss the effect of a higher Û on c 1 .  FURTHER READING The consumption theory based on the intertemporal smoothing of optimal consump- tion paths builds on the work of Friedman (1957) and Modigliani and Brumberg (1954). A critical assessment of the life-cycle theory of consumption (not explicitly 44 CONSUMPTION mentioned in this chapter) is provided by Modigliani (1986). Abel (1990, part 1), Blanchard and Fischer (1989, para. 6.2), Hall (1989), and Romer (2001, ch. 7) present consumption theory at a technical level similar to ours. Thorough overviews of the theoretical and empirical literature on consumption can be found in Deaton (1992) and, more recently, in Browning and Lusardi (1997) and Attanasio (1999), with a particular focus on the evidence from microeconometric studies. When confronting theory and microeconomic data, it is of course very important (and far from straight- forward) to account for heterogeneous objective functions across individuals or house- holds. In particular, empirical work has found that theoretical implications are typi- cally not rejected when the marginal utility function is allowed to depend flexibly on the number of children in the household, on the household head’s age, and on other observable characteristics. Information may also be heterogeneous: the information set of individual agents need not be more refined than the econometrician’s (Pischke, 1995), and survey measures of expectations formed on its basis can be used to test theoretical implications (Jappelli and Pistaferri, 2000). The seminal paper by Hall (1978) provides the formal framework for much later work on consumption, including the present chapter. Flavin (1981) tests the empirical implications of Hall’s model, and finds evidence of excess sensitivity of consumption to expected income. Campbell (1987) and Campbell and Deaton (1989) derive theor- etical implication for saving behavior and address the problem of excess smoothness of consumption to income innovations. Campbell and Deaton (1989) and Flavin (1993) also provide the joint interpretation of “excess sensitivity” and “excess smoothness” outlined in Section 1.2. Empirical tests of the role of liquidity constraints, also with a cross-country perspective, are provided by Jappelli and Pagano (1989, 1994), Campbell and Mankiw (1989, 1991) and Attanasio (1995, 1999). Blanchard and Mankiw (1988) stress the importance of the precautionary saving motive, and Caballero (1990) solves analyt- ically the optimization problem with precautionary saving assuming an exponential utility function, as in Section 1.3. Weil (1993) solves the same problem in the case of constant but unrelated intertemporal elasticity of substitution and relative risk aver- sion parameters. A precautionary saving motive arises also in the models of Deaton (1991) and Carroll (1992), where liquidity constraints force consumption to closely track current income and induce agents to accumulate a limited stock of financial assets to support consumption in the event of sharp reductions in income (buffer-stock saving). Carroll (1997, 2001) argues that the empirical evidence on consumers’ behav- ior can be well explained by incorporating in the life-cycle model both a precautionary saving motive and a moderate degree of impatience. Sizeable responses of consump- tion to predictable income changes are also generated by models of dynamic inconsis- tent preferences arising from hyperbolic discounting of future utility; Angeletosetal. (2001) and Frederick, Loewenstein, and O’Donoghue (2002) provide surveys of this strand of literature. The general setup of the CCAPM used in Section 1.4 is analyzed in detail by Campbell, Lo, and MacKinley (1997, ch. 8) and Cochrane (2001). The model’s empir- ical implications with a CRRA utility function and a lognormal distribution of returns and consumption are derived by Hansen and Singleton (1983) and extended by, among others, Campbell (1996). Campbell, Lo, and MacKinley (1997) also provide CONSUMPTION 45 a complete survey of the empirical literature. Campbell (1999) has documented the international relevance of the equity premium and the risk-free rate puzzles,origi- nally formulated by Mehra and Prescott (1985) and Weil (1989). Aiyagari (1993), Kocherlakota (1996), and Cochrane (2001, ch. 21) survey the theoretical and empirical literature on this topic. Costantinides, Donaldson, and Mehra (2002) provide an explanation of those puzzles by combining a life-cycle perspective and borrowing constraints. Campbell and Cochrane (1999) develop the CCAPM with habit for mation behavior outlined in Section 1.4 and test it on US data. An exhaustive survey of the theory and the empirical evidence on consumption, asset returns, and macroeconomic fluctuations is found in Campbell (1999). Dynamic programming methods with applications to economics can be found in Dixit (1990), Sargent (1987, ch. 1) and Stokey, Lucas, and Prescott (1989), at an increasing level of difficulty and analytical rigor.  REFERENCES Abel, A. (1990) “Consumption and Investment,” in B. Friedman and F. Hahn (ed.), Handbook of Monetary Economics, Amsterdam: North-Holland. Aiyagari, S. R. (1993) “Explaining Financial Market Facts: the Importance of Incomplete Markets and Transaction Costs,” Federal Reserve Bank of Minneapolis Quarterly Review, 17, 17–31. (1994) “Uninsured Idiosyncratic Risk and Aggregate Saving,” Quarterly Journal of Eco- nomics, 109, 659–684. Angeletos, G M., D. Laibson, A. Repetto, J. Tobacman, and S. Winberg (2001) “The Hyperbolic Consumption Model: Calibration, Simulation and Empirical Evaluation,” Journal of Economic Perspectives, 15(3), 47–68. Attanasio, O. P. (1995) “The Intertemporal Allocation of Consumption: Theory and Evidence,” Carnegie–Rochester Conference Series on Public Policy, 42, 39–89. (1999) “Consumption,” in J. B. Taylor and M. Woodford (ed.), Handbook of Macroeco- nomics, vol. 1B, Amsterdam: North-Holland, 741–812. Blanchard, O. J. and S. Fischer (1989) Lectures on Macroeconomics, Cambridge, Mass.: MIT Press. and N. G. Mankiw (1988) “Consumption: Beyond Certainty Equivalence,” American Eco- nomic Review (Papers and Proceedings), 78, 173–177. Browning, M. and A. Lusardi (1997) “Household Saving: Micro Theories and Micro Facts,” Journal of Economic Literature, 34, 1797–1855. Caballero, R. J. (1990) “Consumption Puzzles and Precautionary Savings,” Journal of Monetary Economics, 25, 113–136. Campbell, J. Y. (1987) “Does Saving Anticipate Labour Income? An Alternative Test of the Permanent Income Hypothesis,” Econometrica, 55, 1249–1273. (1996) “Understanding Risk and Return,” Journal of Political Economy, 104, 298–345. (1999) “Asset Prices, Consumption and the Business Cycle,” in J. B. Taylor and M. Wood- ford (ed.), Handbook of Macroeconomics, vol. 1C, Amsterdam: North-Holland. and J. H. Cochrane (1999) “By Force of Habit: A Consumption-Based Explanation of Aggregate Stock Market Behavior,” Journal of Political Economy, 2, 205–251. 46 CONSUMPTION and A. Deaton (1989) “Why is Consumption So Smooth?” Review of Economic Studies, 56, 357–374. and N. G. Mankiw (1989) “Consumption, Income and Interest Rates: Reinterpreting the Time-Series Evidence,” NBER Macroeconomics Annual, 4, 185–216. (1991) “The Response of Consumption to Income: a Cross-Country Investigation,” European Economic Review, 35, 715–721. A. W. Lo, and A. C. MacKinley (1997) The Econometrics of Financial Markets,Princeton: Princeton University Press. Carroll, C. D. (1992) “The Buffer-Stock Theory of Saving: Some Macroeconomic Evidence,” Brookings Papers on Economic Activity, 2, 61–156. (1997) “Buffer-Stock Saving and the Life Cycle/Permanent Income Hypothesis,” Quarterly Journal of Economics , 102, 1–55. (2001) “A Theory of the Consumption Function, With and Without Liquidity Constraints,” Journal of Economic Perspectives, 15 (3), 23–45. Cochrane, J. H. (2001) Asset Pricing, Princeton: Princeton University Press. Costantinides G. M., J. B. Donaldson, and R. Mehra (2002) “Junior Can’t Borrow: A New Perspective on the Equity Premium Puzzle,” Quar terly Journal of Economics, 117, 269–298. Deaton, A. (1991) “Saving and Liquidity Constraints,” Econometrica, 59, 1221–1248. (1992) Understanding Consumption, Oxford: Oxford University Press. Dixit, A. K. (1990) Optimization in Economic Theory, 2nd edn, Oxford: Oxford University Press. Flavin, M. (1981) “The Adjustment of Consumption to Changing Expectations about Future Income,” Journal of Political Economy , 89, 974–1009. (1993) “The Excess Smoothness of Consumption: Identification and Interpretation,” Revie w of Economic Studies, 60, 651–666. Frederick S., G. Loewenstein, and T. O’Donoghue (2002) “Time Discounting and Time Prefer- ence: A Critical Review,” Journal of Economic Literature, 40, 351–401. Friedman, M. (1957) A Theory of the Consumption Function,Princeton:PrincetonUniversity Press. Hall, R. E. (1978) “Stochastic Implications of the Permanent Income Hypothesis: Theory and Evidence,” Journal of Political Economy, 96, 971–987. (1989) “Consumption,” in R. Barro (ed.), Handbook of Modern Business Cycle Theory, Oxford: Basil Blackwell. Hansen, L. P. and K. J. Singleton (1983) “Stochastic Consumption, Risk Aversion, and the Temporal Behavior of Asset Returns,” Journal of Political Economy, 91, 249–265. Jappelli, T. and M. Pagano (1989) “Consumption and Capital Market Imperfections: An Inter- national Comparison,” American Economic Review, 79, 1099–1105. (1994) “Saving, Growth and Liquidity Constraints,” Quarterly Journal of Economics, 108, 83–109. and L. Pistaferri (2000), “Using Subjective Income Expectations to Test for Excess Sensitiv- ity of Consumption to Predicted Income Growth,” European Economic Review 44, 337–358. Kocherlakota, N. R. (1996) “The Equity Premium: It’s Still a Puzzle,” Journal of Economic Literature, 34(1), 42–71. CONSUMPTION 47 Mehra, R. and E. C. Prescott (1985) “The Equity Premium: A Puzzle,” Journal of Monetary Economics , 15(2), 145–161. Modigliani, F. (1986) “Life Cycle, Individual Thrift, and the Wealth of Nations,” American Economic Review, 76, 297–313. and R. Brumberg (1954) “Utility Analysis and the Consumption Function: An Inter- pretation of Cross-Section Data,” in K. K. Kurihara (ed.), Post-Keynesian Economics,New Brunswick, NJ: Rutgers University Press. Pischke, J S. (1995) “Individual Income, Incomplete Information, and Aggregate Consump- tion,” Econometrica, 63, 805–840. Romer, D. (2001) Advanced Macroeconomics, 2nd edn, New York: McGraw-Hill. Sargent, T. J. (1987) Dynamic Macroeconomic Theory, Cambridge, Mass.: Harvard University Press. Stokey, N., R. J. Lucas, and E. C. Prescott (1989) Recursive Methods in Economic Dynamics, Cambridge, Mass.: Harvard University Press. Weil, P. (1989) “The Equity Premium Puzzle and the Risk-Free Rate Puzzle,” Journal of Monetary Economics, 24, 401–421. (1993) “Precautionary Savings and the Permanent Income Hypothesis,” Review of Economic Studies, 60, 367–383. 2 Dynamic Models of Investment Macroeconomic IS–LM models assign a crucial role to business investment flows in linking the goods market and the money market. As in the case of con- sumption, however, elementary textbooks do not explicitly study investment behavior in terms of a formal dynamic optimization problem. Rather, they offer qualitatively sensible interpretations of investment behavior at a point in time. In this chapter we analyze investment decisions from an explicitly dynamic perspective. We simply aim at introducing dynamic continuous-time optimization techniques, which will also be used in the following chapters, and at offering a formal, hence more precise, interpretation of qualitative approaches to the behavior of private investment in macroeconomic models encountered in introductory textbooks. Other aspects of the subject matter are too broad and complex for exhaustive treatment here: empirical applications of the theories we analyze and the role of financial imperfections are men- tioned briefly at the end of the chapter, referring readers to existing surveys of the subject. As in Chapter 1’s study of consumption, in applying dynamic optimiza- tion methods to macroeconomic investment phenomena, one can view the dynamics of aggregate variables as the solution of a “representative agent” problem. In this chapter we study the dynamic optimization problem of a firm that aims at maximizing present discounted cash flows. We focus on technical insights rather than on empirical implications, and the problem’s setup may at first appear quite abstract. When characterizing its solution, however, we will emphasize analogies between the optimality conditions of the formal problem and simple qualitative approaches familiar from undergraduate textbooks. This will make it possible to apply economic intuition to mathematical for- mulas that would otherwise appear abstruse, and to verify the robustness of qualitative insights by deriving them from precise formal assumptions. Section 2.1 introduces the notion of “convex” adjustment costs, i.e. techno- logical features that penalize fast investment. The next few sections illustrate the character of investment decisions from a partial equilibrium perspective: we take as given the firm’s demand and production functions, the dynamics of the price of capital and of other factors, and the discount rate applied to future cash flows. Optimal investment decisions by firms are forward looking, and should be based on expectations of future events. Relevant techniques and mathematical results introduced in this context are explained in detail in the INVESTMENT 49 Appendix to this chapter. The technical treatment of firm-level investment decisions sets the stage for a discussion of an explicitly dynamic version of the familiar IS–LM model. The final portion of the chapter returns to the firm-level perspective and studies specifications where adjustment costs do not discourage fast investment, but do impose irreversibility constraints, and Section 2.8 briefly introduces technical tools for the analysis of this type of problem in the presence of uncertainty. 2.1. Convex Adjustment Costs In what follows, F (t) denotes the difference between a firm’s cash receipts and outlays during period t. We suppose that such cash flows depend on the capital stock K (t) available at the beginning of the period, on the flow I (t)of investment during the period, and on the amount N(t)employedduringthe period of another factor of production, dubbed “labor”: F (t)=R(t, K (t), N(t)) − P k (t)G(I (t), K (t)) − w(t)N(t). (2.1) The R(·) function represents the flow of revenues obtained from sales of the firm’s production flow. This depends on the amounts employed of the two factors of production, K and N, and also on the technological efficiency of the production function and/or the strength of demand for the firm’s product. In (2.1), possible variations over time of such exogenous features of the firm’s technological and market environment are taken into account by including the time index t alongside K and N as arguments of the revenue function. We assume that revenue flows are increasing in both factors, i.e. ∂ R(·) ∂ K > 0, ∂ R(·) ∂ N > 0, (2.2) as is natural if the marginal productivity of all factors and the market price of the product are positive. To prevent the optimal size of the firm from diverging to infinity, it is necessary to assume that the revenue function R(·) is concave in K and N. If the price of its production is taken as given by the firm, this is ensured by non-increasing returns to scale in production. If instead physical returns to scale are increasing, the revenue function R(·) can still be concave if the firm has market power and its demand function’s slope is sufficiently negative. The two negative terms in the cash-flow expression (2.1) represent costs pertaining to investment, I , and employment of N.Astothelatter,inthis chapter we suppose that its level is directly controlled by the firm at each point in time and that utilization of a stock of labor N entails a flow cost w per unit time, just as in the static models studied in introductory microeconomic courses. As to investment costs, a formal treatment of the problem needs to 50 INVESTMENT be precise as to the moment when the capital stock used in production during each period is measured. If we adopt the convention that the relevant stock is measured at the beginning of the period, it is simply impossible for the firm to vary K (t)attimet. When the production flow is realized, the firm cannot control the capital stock, but can only control the amount of positive or negative investment: any resulting increase or decrease of installed capital begins to affect production and revenues only in the following period. On this basis, the dynamic accumulation constraint reads K (t + t)=K (t)+I (t)t − ‰K (t)t, (2.3) where ‰ denotes the depreciation rate of capital, and t is the length of the time period over which we measure cash flows and the investment rate per unit time I (t). By assumption, the firm cannot affect current cash flows by varying the available capital stock. The amount of gross investment I(t) during period t does, however, affect the cash flow: in (2.1) investment costs are represented by a price P k (t) times a function G(·) which, as in Figure 2.1, we shall assume increasing and convex in I (t): ∂G(·) ∂ I > 0, ∂ 2 G(·) ∂ I 2 > 0. (2.4) The function G(·) is multiplied by a price in the definition (2.1) of cash flows. Hence it is defined in physical units, just like its arguments I and K . For example, it might measure the physical length of a production line, or the number of personal computers available in an office. The investment Figure 2.1. Unit investment costs INVESTMENT 51 rate I (t) is linearly related to the change in capital stock in equation (2.3) but, since G(·) is not linear, the cost of each unit of capital installed is not constant. For instance, we might imagine that a greenhouse needs to purchase G(I, K ) flower pots in order to increase the available stock by I units, and that the quantities purchased and effectively available for future production are different because a certain fraction (variable as a function of I and K )ofpots purchased break and become useless. In the context of this example it is also easy to imagine that a fraction of pots in use also break during each period, and that the parameter ‰ represents this phenomenon formally in (2.3). While such examples can help reduce the rather abstract character of the formal model we are considering, its assumptions may be more easily justified in terms of their implications than in those of their literal realism. For pur- poses of modeling investment dynamics, the crucial feature of the G(I, K ) function is the strict convexity assumed in (2.4). This implies that the average unit cost (measured, after normalization by P k ,bytheslopeoflinessuchas OA and OB in Figure 2.1) of investment flows is increasing in the total flow invested during a period. Thus, a given total amount of investment is less costly when spread out over multiple periods than when it is concentrated in a single period. For this reason, the optimal investment policy implied by convex adjustment costs is to some extent gradual. The functional form of investment costs plays an important role not only when the firm intends to increase its capital stock, but also when it wishes to keep it constant, or decrease it. It is quite natural to assume that the firm should not bear costs when gross investment is zero (and capital may evolve over time only as a consequence of exogenous depreciation at rate ‰). Hence, as in Figure 2.1, G(0, ·)=0, and the positive first derivative assumed in (2.4) implies that G (I, ·) < 0 for I < 0: the cost function is negative (and makes positive contributions to the firm’s cash flow) when gross investment is negative, and the firm is selling used equipment or structures. In the figure, the G(·) function lies above a 45 ◦ line through the origin, and it is tangent to it at zero, where its slope is unitary: ∂G(0, ·)/∂ I =1. This property makes it possible to interpret P k as “the” unit price of capital goods, a price that would apply to all units installed if the convexity of G(I, ·) did not deter larger than infinitesimal investments of either sign. When negative investment rates are considered, convexity of adjustment costs similarly implies that the unit amount recouped from each unit scrapped (as measured by the slope of lines such as OB) is smaller when I is more negative, and this makes speedy reduction of the capital stock unattractive. 52 INVESTMENT Comparing the slope of lines such as OA and OB, it is immediately apparent that alternating positive and negative investments is costly: even though there are no net effects on the final capital stock, the firm cannot fully recoup the original cost of positive investment from subsequent negative invest- ment. First increasing, then decreasing the capital stock (or vice versa) entails adjustment costs. In summary, the form of the function displayed in Figure 2.1 implies that investment decisions should be based not only on the contribution of capital to profits at a given moment in time, but also on their future outlook. If the relevant exogenous conditions indexed by t in R(·) and the dynamics of the other, equally exogenous, variables P k (t), w(t), r (t) suggest that the firm should vary its capital stock, the adjustment should be gradual, as will be set out below. Moreover, if large positive and negative fluctuations of exogenous variables are expected, the firm should not vary its investment rate sharply, because the cost and revenues generated by upward and downward capital stock fluctuations do not offset each other exactly. Convexity of the adjustment cost function implies that the total cost of any given capital stock variation is smaller when that variation is diluted through time, hence the firm should behave in a forward looking fashion when choosing the dynamics of its investment rate and should try to keep the latter stable by anticipating the dynamics of exogenous variables. 2.2. Continuous-Time Optimization Neither the realism nor the implications of convex adjustment costs depend on the length t of the period over which revenue, cost, and investment flows are measured. The discussion above, however, was based on the idea that current investment cannot increase the capital stock available for use within each such period, implying that K (t) could be taken as given when evaluating opportunities for further investment. This accounting convention, of course, is more accurate when the length of the period is shorter. Accordingly, we consider the limit case where t → 0, and suppose that the firm makes optimizing choices at every instant in continuous time. Optimiza- tion in continuous time yields analytically cleaner and often more intuitive results than qualitatively similar results from discrete time specifications, such as those encountered in this book when discussing consumption (in Chap- ter 1) and labor demand under costly adjustment (in Chapter 3). We also assume, for now, that the dynamics of exogenous variables is deterministic. (Only at the end of the chapter do we introduce uncertainty in a continuous- time investment problem.) This also makes the problem different from that discussed in Chapter 1: the characterization offered by continuous-time [...]... optimality conditions in (2.6), the time-varying elements of the system formed by (2.12) and (2. 13) are just q (t) and K (t)—that is, precisely those for whose dynamics we have derived explicit expressions Thus, the dynamics of the two variables may be studied in the phase diagram of Figure 2.2 On the axes of the diagram we measure the dynamic variables of interest On the horizontal axis of this and subsequent... pen on the diagram at point C, and following the dynamic instructions given by (2.12), the pen should slide towards even higher values of q The same reasoning holds for ˙ all points above the q = 0 locus, for example point D, whence an upwardsloping arrow also starts The speed of the dynamic movement represented is larger for larger values of r + ‰, and for greater distances from the stationary locus:... w(0) for ever However, since between t = 0 and t = T the wage is still w(0) and there is no reason to increase N for given K , it cannot be optimal for the firm to behave as in the solution of the exercise above, where the wage decreased permanently to w(t) = w(T ) for all t In order to characterize the optimal investment policy, recall that to avoid divergent dynamics the firm should select a dynamic. .. follows the dynamics implied by the initial parameters until time T , when the dynamic path meets the new saddlepath Intuitively, the firm finds it convenient to dilute over time the adjustment it foresees For larger values of T the height of the initial jump would be smaller, and the apparently divergent dynamics induced by the expectation of future events would follow slower, more prolonged, dynamics... same reasoning to equation (2. 13) enables us to draw ˙ Figure 2 .3 To determine the slope of the locus along which K = 0, note that the right-hand side of (2. 13) is certainly increasing in q since a higher q is ˙ associated with a larger investment flow The effect on K of a higher K is ambiguous: as long as ‰ > 0 it is certainly negative through the second term, Figure 2 .3 Dynamics of K (supposing that... condition (2.7) for plausible forms of the F (·) function Also, starting from points in the lower quadrant of the diagram, the dynamics of the system, driven by arrows pointing left and downwards, can only lead to economically nonsensical values of q and/or K The system’s configuration is much more sensible at the point where the ˙ ˙ K = 0 and q = 0 loci cross, the unique steady state of the dynamic system... dynamic system we are considering Thus, we can focus attention on dynamic paths starting from the left and right regions of Figure 2.4, where arrows pointing towards the steady state allow the dynamic system to evolve in its general direction 60 INVESTMENT Figure 2.5 Saddlepath dynamics As shown in Figure 2.5, however, it is quite possible for trajectories start˙ ˙ ing in those regions to cross the K... negative In the figure, however, a pair of dynamic paths is drawn that start from points to the left and right of the steady state and continue towards it (at decreasing speed) without ever meeting the system’s stationarity loci All points along such paths are compatible with convergence towards the steady state, and together form the saddlepath of the dynamic system For any given K , such as that labeled... depreciation is higher than Pk , because of adjustment costs Phase diagrams are useful not only for characterizing adjustment paths starting from a given initial situation, but also for studying the investment 62 INVESTMENT effects of permanent changes in parameters To this end, one may specify a functional form for cash flows F (·) in (2.1), as is done in the exercises at the end of the chapter, and study... variables INVESTMENT 57 Figure 2.2 Dynamics of q (supposing that ∂ F (·)/∂ K is decreasing in K ) uniquely determined by them, such as the investment rate I = È(q , K )—are time-varying, then each point in (K , q )-space is uniquely associated with their dynamic changes Picking any point in the diagram, and knowing the functional form of the expressions in (2.12) and (2. 13) , one could in prin˙ ˙ ciple . the Permanent Income Hypothesis,” Review of Economic Studies, 60, 36 7 38 3. 2 Dynamic Models of Investment Macroeconomic IS–LM models assign a crucial role to business investment flows in linking. 108, 83 109. and L. Pistaferri (2000), “Using Subjective Income Expectations to Test for Excess Sensitiv- ity of Consumption to Predicted Income Growth,” European Economic Review 44, 33 7 35 8. Kocherlakota,. elements of the system formed by (2.12) and (2. 13) are just q(t) and K (t)—that is, precisely those for whose dynamics we have derived explicit expressions. Thus, the dynamics of the two variables