Models for dynamic macroeconomics phần 3 ppsx

The next few sections illustratethe character of investment decisions from a partial equilibrium perspective: optimiza-we take as given the firm’s demand and production functions, the dyn

Exercise 8 Suppose that labor income y is generated by the following stochastic process:

yt = Îy t−1+ x t−1+ε 1t ,

xt =ε 2t , where x t (= ε 2t ) does not depend on its own past values ( x t−1, xt−2, ) and E (ε 1t·

ε 2t ) = 0 x t−1is the only additional variable (realized at time t − 1) which affects income

in period t besides past income yt−1 Moreover, suppose that the information set used

by agents to calculate their permanent income y P

t is It−1={y t−1, xt−1}, whereas the information set used by the econometrician to estimate the agents’ permanent income

is t−1={y t−1} Therefore, the additional information 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 nent income computed by the agents ( y P

perma-t ) and by the econometrician ( ˜y P

t ), considering the di fferent information set used (It−1or 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 c1and c2and incomes y1and y2 Suppose that the utility function

(a) Plot marginal utility as a function of consumption.

(b) Suppose that r = Ò = 0, y1= a /b, and y2is uncertain:

FURTHER READING

The consumption theory based on the intertemporal smoothing of optimal 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

consump-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) presentconsumption theory at a technical level similar to ours Thorough overviews of thetheoretical and empirical literature on consumption can be found in Deaton (1992)and, more recently, in Browning and Lusardi (1997) and Attanasio (1999), with aparticular focus on the evidence from microeconometric studies When confrontingtheory 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 onthe number of children in the household, on the household head’s age, and on otherobservable characteristics Information may also be heterogeneous: the informationset 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 testtheoretical implications (Jappelli and Pistaferri, 2000).

The seminal paper by Hall (1978) provides the formal framework for much laterwork 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-countryperspective, are provided by Jappelli and Pagano (1989, 1994), Campbell and Mankiw(1989, 1991) and Attanasio (1995, 1999) Blanchard and Mankiw (1988) stress theimportance of the precautionary saving motive, and Caballero (1990) solves analyt-ically the optimization problem with precautionary saving assuming an exponentialutility function, as in Section 1.3 Weil (1993) solves the same problem in the case ofconstant 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 closelytrack current income and induce agents to accumulate a limited stock of financial

assets to support consumption in the event of sharp reductions in income (bu ffer-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 precautionarysaving 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; Angeletos et al.

(2001) and Frederick, Loewenstein, and O’Donoghue (2002) provide surveys of thisstrand of literature

The general setup of the CCAPM used in Section 1.4 is analyzed in detail byCampbell, 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 returnsand consumption are derived by Hansen and Singleton (1983) and extended by,among others, Campbell (1996) Campbell, Lo, and MacKinley (1997) also provide

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 empiricalliterature on this topic Costantinides, Donaldson, and Mehra (2002) provide anexplanation of those puzzles by combining a life-cycle perspective and borrowing

constraints Campbell and Cochrane (1999) develop the CCAPM with habit formation

behavior outlined in Section 1.4 and test it on US data An exhaustive survey of thetheory and the empirical evidence on consumption, asset returns, and macroeconomicfluctuations is found in Campbell (1999)

Dynamic programming methods with applications to economics can be found inDixit (1990), Sargent (1987, ch 1) and Stokey, Lucas, and Prescott (1989), at anincreasing level of difficulty and analytical rigor

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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-

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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.

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 Bu ffer-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,” Quarterly 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,”

Review 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: Princeton University

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.

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 investmentflows in linking the goods market and the money market As in the case of con-sumption, however, elementary textbooks do not explicitly study investmentbehavior in terms of a formal dynamic optimization problem Rather, theyoffer qualitatively sensible interpretations of investment behavior at a point

in time In this chapter we analyze investment decisions from an explicitlydynamic perspective We simply aim at introducing dynamic continuous-timeoptimization techniques, which will also be used in the following chapters,and at offering a formal, hence more precise, interpretation of qualitativeapproaches to the behavior of private investment in macroeconomic modelsencountered in introductory textbooks Other aspects of the subject matter aretoo broad and complex for exhaustive treatment here: empirical applications

of the theories we analyze and the role of financial imperfections are tioned briefly at the end of the chapter, referring readers to existing surveys ofthe subject

men-As in Chapter 1’s study of consumption, in applying dynamic tion methods to macroeconomic investment phenomena, one can view thedynamics of aggregate variables as the solution of a “representative agent”problem In this chapter we study the dynamic optimization problem of a firmthat aims at maximizing present discounted cash flows We focus on technicalinsights rather than on empirical implications, and the problem’s setup may atfirst appear quite abstract When characterizing its solution, however, we willemphasize analogies between the optimality conditions of the formal problemand 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 ofqualitative 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 illustratethe character of investment decisions from a partial equilibrium perspective:

optimiza-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 andmathematical results introduced in this context are explained in detail in the

Appendix to this chapter The technical treatment of firm-level investmentdecisions sets the stage for a discussion of an explicitly dynamic version ofthe familiar IS–LM model The final portion of the chapter returns to thefirm-level perspective and studies specifications where adjustment costs donot discourage fast investment, but do impose irreversibility constraints, andSection 2.8 briefly introduces technical tools for the analysis of this type ofproblem 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) employed during the

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’stechnological 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

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 sufficientlynegative

The two negative terms in the cash-flow expression (2.1) represent costs

pertaining to investment, I , and employment of N As to the latter, in this

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 microeconomiccourses As to investment costs, a formal treatment of the problem needs to

be precise as to the moment when the capital stock used in production duringeach 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) at time t 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 capitalbegins to affect production and revenues only in the following period On thisbasis, 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, andt 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):

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

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 ) of pots

purchased break and become useless In the context of this example it is alsoeasy 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 theformal 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, by the slope of lines such as

OA and OB in Figure 2.1) of investment flows is increasing in the total flowinvested during a period Thus, a given total amount of investment is lesscostly 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 onlywhen 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 firmshould not bear costs when gross investment is zero (and capital may evolveover 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 usedequipment 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 adjustmentcosts 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

Comparing the slope of lines such as OA and OB, it is immediately apparentthat alternating positive and negative investments is costly: even though thereare no net effects on the final capital stock, the firm cannot fully recoupthe 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 thatinvestment 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 ofexogenous variables are expected, the firm should not vary its investment ratesharply, because the cost and revenues generated by upward and downwardcapital stock fluctuations do not offset each other exactly Convexity of theadjustment cost function implies that the total cost of any given capital stockvariation 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 thedynamics of exogenous variables

2.2 Continuous-Time Optimization

Neither the realism nor the implications of convex adjustment costs depend

on the lengtht of the period over which revenue, cost, and investment flows

are measured The discussion above, however, was based on the idea thatcurrent 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 wheret → 0, and suppose that the

firm makes optimizing choices at every instant in continuous time tion in continuous time yields analytically cleaner and often more intuitiveresults than qualitatively similar results from discrete time specifications, such

Optimiza-as those encountered in this book when discussing consumption (in ter 1) and labor demand under costly adjustment (in Chapter 3) We alsoassume, 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 thatdiscussed in Chapter 1: the characterization offered by continuous-time

Chap-models without uncertainty is less easily applicable to empirical time observations, but is also quite insightful, and each of the modelingapproaches we outline could fruitfully be applied to the various substantiveproblems considered The economic intuition afforded by the next chapter’smodels of labor demand under uncertainty would be equally valid if applied

discrete-to investment in plant and equipment investment rather than in workers,and we shall encounter consumption and investment problems in continuoustime (and in the absence of uncertainty) when discussing growth models inChapter 4

In continuous time, the maximum present value (discounted at rate r ) of

cash flows generated by a production and investment program can be written

as an integral:

V (0)≡ max

 ∞0

F (t)e−0t r (s )ds dt,

subject to ˙K (t) = I (t) − ‰K (t), for all t. (2.5)The Appendix to this chapter defines the integral and offers an introduction toHamiltonian dynamic optimization This method suggests a simple recipe forsolution of this type of problem (which will also be encountered in Chapter 4).The Hamiltonian of optimization problem (2.5) is

H(t) = e−

t

0r (s )ds (F (t) + Î(t) (I (t) − ‰K (t))) , where Î(t) denotes the shadow price of capital at time t in current value terms (that is, in terms of resources payable at the same time t).

The first-order conditions of the dynamic optimization problem we arestudying are

to those of more familiar static constrained optimization problems Here, wediscuss their economic interpretation The condition

∂ R(·)

simply requires that, in flow terms, the marginal revenue yielded by

employ-ment of the flexible factor N be equal to its cost w, at every instant t This is

quite intuitive, since the level of N may be freely determined by the firm The

condition

P k ∂G(·)

calls for equality, along an optimal investment path, of the marginal value of

capital Î(t) and the marginal cost of the investment flows that determine an

increase (or decrease) of the capital stock at every instant That marginal cost,

in turn, is−P k ∂G(·)/∂ I in the problem we are considering Such

considera-tions, holding at every given time t, do not suffice to represent the dynamic

aspects of the firm’s problem These aspects are in fact crucial in the thirdcondition listed in (2.6), which may be rewritten in the form

r Î = ∂ F (·)

∂ K − ‰Î + ˙Î

and interpreted in terms of financial asset valuation For simplicity, let ‰ = 0

From the viewpoint of time t, the marginal unit of capital adds ∂ F /∂ K to

current cash flows, and this is a “dividend” paid by that unit to its owner

at that time (the firm) The marginal unit of capital, however, also offerscapital gains, in the amount ˙Î If the firm attaches a (shadow) value Î to theunit of capital, then it must be the case that its total return in terms of bothdividends and capital gains is financially fair Hence it should coincide with

the return r Î that the firm could obtain from Î units of purchasing power

in a financial market where, as in (2.5), cash flows are discounted at rate r

If ‰> 0, similar considerations hold true but should take into account that

a fraction of the marginal unit of capital is lost during every instant of time.Hence its value, amounting to ‰Î per unit time, needs to be subtracted fromcurrent “dividends.”

Such considerations also offer an intuitive economic interpretation of thetransversality condition (2.7), which would be violated if the “financial” value

Î(t) grew at a rate greater than or equal to the equilibrium rate of return r (s )

while the capital stock, and the marginal dividend afforded by the investment

policy, tend to a finite limit In such a case, Î(t) would be influenced by a

speculative “bubble”: the only reason to hold the asset corresponding to themarginal value of capital is the expectation of everlasting further capital gains,not linked to profits actually earned from its use in production Imposingcondition (2.7), we acknowledge that such expectations have no economicbasis, and we deny that purely speculative behavior may be optimal for thefirm

2.2.1 CHARACTERIZING OPTIMAL INVESTMENT

Consider the variable

q (t)≡ Î(t)

P k (t) ,

the ratio of the marginal capital unit’s shadow value to parameter P k, whichrepresents the market price of capital (that is, the unit of cost of investment inthe neighborhood of the zero gross investment point, where adjustment costsare negligible)

This variable, known as marginal q , has a crucial role in the determination

of optimal investment flows In fact, the first condition in (2.6) implies that

∂G(I (t), K (t))

∂ I (t) = q (t) , (2.10)

and if (2.4) holds then∂G(·)/∂ I is a strictly increasing function of I Such a

function has an inverse: let È(·) denote the inverse of ∂G(·)/∂ I as a function

of I Both ∂G(·)/∂ I and its inverse may depend on the capital stock K The

È(q , K ) function implicitly defined by

Since, by assumption, the investment cost function G (I , ·) has unitary slope

at I = 0, zero gross investment is optimal when q = 1; positive investment

is optimal when q > 1; and negative investment is optimal when q < 1.

Intuitively, when q > 1 (hence Î > P k) capital is worth more inside the firmthan in the economy at large; hence it is a good idea to increase the capital

stock installed in the firm Symmetrically, q < 1 suggests that the capital stock

should be reduced In both cases, the speed at which capital is transferredtowards the firm or away from it depends not only on the difference between

q and unity, but also on the degree of convexity of the G (·) function, that is,

on the relevance of capital adjustment costs If the slope of the function in

Figure 2.1 increases quickly with I , even q values very different from unity are

associated with modest investment flows

Exercise 10 Show that, if capital has positive value, then investment would

always be positive if the total investment cost were quadratic, for example if

G (K , I ) = x · I2where P k = 1 and x ≥ 0 may depend on K Discuss the

real-ism of more general specifications where G (K , I ) = x · I‚

for ‚ > 0.

Determining the optimal investment rate as a function of q does not yield a

complete solution to the dynamic optimization problem In fact, in order to

compute q one needs to know the shadow value Î(t) of capital, which—unlike the market price of capital, P k (t)—is part of the problem’s solution, rather

than part of its exogenous parameterization However, it is possible to acterize graphically and qualitatively the complete solution of the problem onthe basis of the Hamiltonian conditions

char-Since we expressed the shadow value of capital in current terms, calendar

time t appears in the optimality conditions only as an argument of the

func-tions, such as Î(·) and K (·), which determine optimal choices of I and N.Noting that

let us define ˙P k (t) /P k (t)≡ k(the rate of inflation in terms of capital), and

recall that ˙Î = (r + ‰)Î − ∂ F (·)/∂ K by the last optimality condition in (2.6) Thus, we may write the rate of change of q as a function of q itself, of K , and

In this expression the calendar time t is omitted for simplicity, but all

variables—particularly those, not explicitly listed, that determine the size of

cash flows F (·) and their derivative with respect to K —are measured at a

given moment in time

Combining the constraint ˙K (t) = I (t) − ‰K (t) with condition (2.11), we obtain a relationship between the rate of change of K , K itself, and the level

of q :

˙

K = È(q , K ) − ‰K. (2.13)Now, if we suppose that all exogenous variables are constant (including the

price of capital P k, to imply that k= 0), and recall that the investment rate

and N depend on q and K through the 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 diagrams, one reads

the level of K ; on the vertical axis, a level of q If only K and q —and variables