MODELING TIME AND MACROECONOMIC DYNAMICS. Preliminary, November 27, 2004

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1 MODELING TIME AND MACROECONOMIC DYNAMICS Alexis Anagnostopoulos Cryssi Giannitsarou Preliminary, November 27, 2004 Abstract. Wen analyzing dynamic macroeconomic models, a commonly eld view is tat assuming continuous or discrete time is a matter of convenience witout any economic importance. Te aim of tis paper is to callenge tis view by demonstrating tat tere is indeed a significant difference between a discrete and a continuous time version of te same model, in terms of structure, solution and economic interpretation. Specifically, we consider a general framework tat incorporates many well known models, and sow tat te Euler equations of te two setups are different in te following way. Wile investment decisions in te continuous time setup depend on present rates of return, in te discrete time setup tey depend on expected future rates of return. Te reason for tis is tat in continuous time, te ouseold s decisions are allowed to adjust continuously to possible canges in te economic environment, wile in discrete time, ouseolds are committed to teir decision until te beginning of te following period. Furtermore, we illustrate via some examples, tat tis fundamental difference between te two modeling assumptions as important implications for te stability properties of te steady state. In turn, tis difference may imply contradicting macroeconomic policy recommendations under te two setups. JEL classification: E00, C62 Keywords: Discrete and continuous time, modeling, macroeconomic dynamics.. Introduction Wen analyzing dynamic macroeconomic models, te decision of weter to model time as continuous or discrete is often based upon te metodological needs of te researcer. Typically, a continuous time setting yields a set of equilibrium conditions tat are easier to work wit compared to tose deriving from a discrete time setting. On te oter and, wen te aim is to explore quantitative issues and te analysis needs to be done numerically, assuming discrete time is more appropriate since computers cannot do exact representations of continua. Altoug occasionally some economic interpretation is put fort to justify te use of one or te oter, te general consensus is tat tere sould not be any difference between neiter te qualitative nor te quantitative results deriving from te two assumptions. In tis paper, we callenge tis view by demonstrating tat tere is indeed a significant difference between a discrete and a continuous time version of te same model, in terms of structure, solution and economic interpretation. Tis is sown in te framework of a generic London Business Scool, Regent s Park, London NW 4SA, UK. aanagnostopoulos@london.edu. Web: ttp://pd.london.edu/aanagnostopoulos. University of Cambridge, Faculty of Economics, Austin Robinson Building, Sidgwick Avenue, Cambridge, CB3 9DD, United Kingdom. cg349@cam.ac.uk Web: ttp:// We tank Tiago Cavalcanti, Omar Licandro, Andrew Scott, Flavio Toxvaerd and Josep Zeira for elpful discussions and comments.

2 2 A. Anagnostopoulos and C. Giannitsarou model tat incorporates a large class of well-known dynamic macroeconomic models. We sow tat tere are two sources of possible differences between te two setups. Te first is te mere fact tat te stability properties of systems described by differential equations (corresponding to te equilibrium conditions in continuous time) are different from tose of systems described by difference equations (corresponding to te equilibrium conditions in discrete time). Te second, and peraps more important, is te fact tat investment decisions in te continuous time setup depend on present rates of return, wereas in te discrete time setup tey depend on expected future rates of return. We begin by sortly describing te difference between systems of differential and difference equations. It is a well known matematical result tat te conditions required for stability (in te matematical sense) of a system of difference equations are stricter tan tose required for stability of te corresponding system of differential equations. In terms of systems of macroeconomic variables, tis result is important for specifying weter a steady state is locally indeterminate or not. In particular, even if tere is no oter difference between te equilibrium conditions of a model in continuous and discrete time, tis matematical result is enoug to render a steady state indeterminate in continuous time, wile it migt be determinate under discrete time. Next, we introduce a generic economy populated by a representative ouseold tat maximizes lifetime utility, over a vector of state variables and a vector of control variables, subject to its budget constraint. Time is modeled wit discrete time intervals of lengt. We ten derive te equilibrium conditions tat come from te ouseold s maximization and take te limits of tese for 0 and, in order to get te corresponding continuous and discrete time systems of equations. Te second difference between te two settings stems from te difference in te two Euler equations tat describe te intertemporal decision process of te ouseolds. Rewriting tese in suc a way tat te time-preference discount rate is equal to te total rate of return to te state variable, we ten explain ow, in discrete time, ouseolds make investment decisions based on expected future rates of return wile in continuous time tey base teir decisions on current rates of return. Tis result furter implies tat, even if te steady state of te economy is determinate under bot modeling assumptions, tere are possibly differences between te equilibrium pats leading to te steady state. Consequently, tere migt be differences in te volatility of te variables. Tere are two (equivalent) ways of interpreting te difference between te discrete and continuous time models. Bot derive from te fact tat essentially, tese are two different models wit different fundamental modeling assumptions, rater tan two versions of te same story. Te first interpretation comes from realizing tat in te discrete time model tere is an inerent time-to-build delay since current stock variables (e.g. capital) become operative in te subsequent period, wile in te continuous time model stock variables become operative instantaneously. Te second interpretation comes from understanding te time line of events in te asset market (e.g. te market for capital goods). In te discrete time model, agents formulate teir market demand by looking aead and anticipating teir optimal decision as of te end of te current period (basing teir decisions on expected future returns on te asset). Alternatively, one could assume tat prices and rates of return adjust so te agents optimize given te asset levels at te beginning of te period; te limit of suc beavior, as te time period srinks, corresponds to te standard continuous time model. In

3 Modeling Time 3 Foley s (975) terminology, wit discrete time we ave an "end-of-period" equilibrium, wile wit continuous time we ave a "beginning-of-period" equilibrium, and tese two ave quite different properties. Finally, to illustrate te importance of te differences between te two modeling assumptions, we present tree examples of well known models tat fit into our framework, namely te real business cycle model of Hansen (985), te model of balanced-budget rules of Scmitt-Groé and Uribe (997) and te model wit increasing returns of Benabib and Farmer (994) and Farmer and Guo (994). Te first example is used to give a more detailed and intuitive explanation for te differences between te two timing formulations. Te last two of examples ave been sown to exibit indeterminacies; exploring tis fact, we ten demonstrate tat for tese models te difference between te two assumptions results in different ranges of indeterminacies, for reasonable parameterizations of te models. Related literature tat explicitly addresses te importance of ow time is modeled in macroeconomics includes te following work. Foley (975) considers a simple model wit two assets (capital and money) and sows tat te "beginning-of-period" and "end-of-period" specifications of a discrete time model are in general inconsistent wit eac oter in te limit (i.e. as time becomes continuous). Carlstrom and Fuerst (2003) point out te possibility of te difference between discrete and continuous time modeling by studying te local determinacy of a monetary model for interest rate rules. Hintermaier (2003) sows tat te existence of sunspot equilibria in discrete time dynamic stocastic general equilibrium models may depend on te lengt of te time period considered. Finally, Bambi and Licandro (2004) consider an extension of te continuous time model of Benabib and Farmer (2004) wit time-to-build delay and find tat even small time-to-build delays rule out local indeterminacy. 2. Stability of Dynamic Systems We begin by exploring te differences in te stability of difference and differential equations. Consider te difference equation w t+ = Dw t () were w is an n vector of variables and D is a known n n matrix. We can rewrite as w t+ w t+ w t =(D I n )w t Cw t = w t+ = Cw t Terefore, te corresponding differential equation will be ẇ t = Cw t (2) Stability and indeterminacy of a system. In pysics, a problem of great importance is tat of determining te beavior of a system in te neigborood of an equilibrium state. For example, suppose tat a pysical system is described by te vector w t. If te system returns to te equilibrium state after a small disturbance, it is called stable, if not it is called unstable. Similarly, in economics an important issue is to determine te beavior of a system in te neigborood of a steady state. Typically, w t corresponds to a vector tat contains te

4 4 A. Anagnostopoulos and C. Giannitsarou predetermined (state) and te control (jump) variables. Te steady state of te economic system is ten called indeterminate if, given te predetermined variables, tere exists more tan one trajectory tat leads to te steady state. Te notion of indeterminacy of a steady state for an economic system is equivalent to te notion of stability of a pysical system. In tis section we will use te term stability of a generic system w t in tis sense. Let λ C i and λ D i be te eigenvalues of C and D correspondingly. Ten te conditions for stability of te system defined by () is tat λ D i <, foralli, and te conditions for stability of te system described by (2) is tat Re λ C i < 0, orequivalentlytatre λ D i <, for all i. 2 Tis means tat te conditions for stability of a difference equation are stricter tan te conditions for stability of te corresponding differential equation. Turning into a typical dynamic macroeconomic model, suppose for example tat te economy is described by a univariate state variable x t and a univariate control variable y t and tat after linearizing te system of equilibrium conditions around a steady state, we can reduce te model in a system of two difference equations (if time is discrete) µ µ xt+ xt = D (3) y t+ y t or a system of differential equations (if time is continuous) µ µ ẋt xt = C ẏ t y t (4) In tese two cases, te conditions for indeterminacy are λd i <, fori =, 2 for te discrete time system (3) and Re λ C i < 0, fori =, 2 for te continuous time system (4). Clearly, even if it is te case tat C = D I 2, i.e. even if tere is an exact correspondence between te continuous and te discrete time setting, te conditions for indeterminacy are not te same. Specifically, if te steady state is locally indeterminate for te discrete time setting, it will also be locally indeterminate in te continuous time setting, but te opposite is not true. In oter words, if C = D I 2, indeterminacy would occur less often in te discrete time setting tan in te continuous time setting. However, it is not always true tat C = D I 2 (in a later section we will sow tat for many well known models, tis is indeed not true). Wen C 6= D I 2, tere is an inerent difference in te economic interpretation of te equilibrium conditions tat reduce to te systems (3) and (4). In tis case, te conditions required for indeterminacy in te two settings may, in principle, be completely unrelated. Tis is a somewat unfortunate correspondence of terminologies, because in economics we use te term stability to describe a system tat is saddle-pat stable, i.e. a system for wic, given te predetermined variables, tere is a unique trajectory tat leads to te steady state. 2 Tis is because te eigenvalues of te two matrices C and D are related in te following way. If λ C i are te eigenvalues of C ten 0 = det ³C λ C i I n = det λ D i = +λ C i ³ ³ D +λ C i =det ³D I n λ C i I n I n =

5 Modeling Time 5 3. A General Discrete Time Model Foley (975) asserts tat "No substantive prediction or explanation in a well-defined macroeconomic [discrete time] model sould depend on te real time lengt of te period. [...] If te results of a [discrete time] model do not depend in any important way on te period, te model can be formulated as a continuous model. Te metod used to accomplis tis is to retain te lengt of te period as an explicit variable in te matematical formulation of a [discrete time] model and to make sure tat it is possible to find meaningful limiting forms of te equations as te period goes to zero. [...] tis procedure sould be routinely applied as a test tat any [discrete time] model is consistent and well formed were no particular calendar time is specified as a natural period". Following tis precept, we first consider a general discrete time model, wit an arbitrary period lengt. Te generic model is described by an n vector of control (flow) variables, y t and an m vector of state (stock) variables x t. Te stock variables are measured at te beginning of te period, i.e. at te beginning of te time interval. Te model migt also contain oter (flow) variables, summarized by an l vector z t, tat migt be endogenously determined but are not coice variables for te representative ouseold. We consider te following maximization problem X µ t max u(yt ) (5) {y t,x t+ } +ρ t=0 mx s.t. (x i,t+ x i,t )=Q(y t,x t ; z t ) (6) i x 0 given Tis setup covers a wide range of standard dynamic macroeconomic models, suc as variations of te real business cycle model (possibly wit increasing returns to scale), as well as many standard monetary models. In te above maximization problem, y t is interpreted as te vector containing te rates of flow of te control variables tat are assumed constant over period t,wilex t is interpreted as te vector of te levels of te state variables tat are measured at te beginning of period t (i.e. at te beginning of te interval [t, (t + )]). In (5) we ave multiplied u(y t) wit because te cumulative utility over a period is te product of te instantaneous rate of cange times te lengt of te period. A similar argument applies to multiplying te rigtand-side of (6) wit. We also make te usual assumption for concavity of te utility function, altoug suc an assumption does not ave any direct implication for te issue we study ere. Te Lagrangian of tis problem is L = X µ t " Ã m!# X u(y t ) (x i,t+ x i,t ) Q(y t,x t ; z t ) +ρ t=0 i

6 6 A. Anagnostopoulos and C. Giannitsarou and te first order conditions are u yj (y t ) = Q yj (y t,x t ; z t ), for all j =,...,n (7) = +ρ ( + Q x i (y t+,x t+ ; z t+ )) +, for all i =,...,m (8) mx (x i,t+ x i,t ) = Q(y t,x t ; z t ) (9) i Note tat substituting (7) into (8) yields te usual Euler equations. Rearranging (8), we obtain 3 + = (ρ Q x i (y t+,x t+ ; z t+ )) +Q xi (y t+,x t+ ; z t+ ) Finally, te equilibrium conditions are summarized by te following set of equations (0) u yj (y t ) = Q yj (y t,x t ; z t ) () mx i + x i,t+ x i,t = (ρ Q x i (y t+,x t+ ; z t+ )) +Q xi (y t+,x t+ ; z t+ ) (2) = Q(y t,x t ; z t ) (3) Te model closes wit l additional equations tat determine te evolution of z t. Note tat to obtain te equilibrium conditions for te usual discrete time setup wit a unit lengt time period, we set =in () - (3) to get were denotes te difference operator. u yj (y t ) = Q yj (y t,x t ; z t ) (4) + = (ρ Q x i (y t+,x t+ ; z t+ )) (5) +Q xi (y t+,x t+ ; z t+ ) mx x i,t+ = Q(y t,x t ; z t ) (6) i 4. Te Standard Continuous Time Model Our next step is work out te continuous time analogue of te discrete time model. To obtain te equilibrium conditions for te continuous time setup we let 0 in () - (3) to get 3 For te derivation, see appendix A. u yj (y t ) = Q yj (y t,x t ; z t ) (7) = (ρ Q xi (y t,x t ; z t )) (8) mx ẋ t = Q(y t,x t ; z t ) (9) i

7 Modeling Time 7 Te difference between te continuous and discrete time boils down to understanding te difference between (5) and (8). To work out te intuition for tis, it is convenient to assume tat tere is only one (asset) stock variable x. We rewrite (2) as follows 4 ρ = Q x (y t+,x t+ ; z t+ ) Ten, for we get and for 0 we get ρ = Q x (y t+,x t+ ; z t+ ) (20) ρ = + Q x (y t,x t ; z t ) (2) Te multiplier λ corresponds to te sadow price of te asset x. In equilibrium, tis asset price adjusts suc tat te asset s total rate of return balances out te time-preference (discount) rate ρ. Tis is exactly te interpretation of equations (20) and (2). Te rigt and side of bot relations represents te total rate of return to olding one unit of te asset x, decomposed into two components. Te first term in bot expressions represents te rate of cange of asset gains. Turning into te second term, in te continuous time setting it represents te rate of cange of te asset. In te discrete time setting, it represents te future rate of cange of te asset, in terms of te current period s price. Te difference between te discrete and continuous time setup is tus te fact tat te rate Q x (tat influences investment decisions) is known to te ouseolds at every instance wen time is continuous, wile wen time is discrete, te ouseolds get a return on teir investment based on next period s rate. In oter words, in continuous time, if te rate Q x canges, ouseolds may adjust teir investment decisions instantaneously, wile in discrete time tey are "committed" to teir decision until te beginning of te next period. To understand better te intuition beind tis difference, it is useful to closely look at te time line of events witin one period (for te discrete time model) and ten compare it wit analogue continuous time setting. We do tis in te next section, in te context of a familiar simple model wic will facilitate te comparison. 5. A First Example: te Real Business Cycle Model We start by briefly describing te model, wic is along te lines of Hansen (985). We set up te model in te general discrete time setting, i.e. wit time periods of lengt. Te representative ouseold maximizes lifetime discounted utility, subject to its budget constraint: X max ( +ρ ) t [log ct An t ] s.t. t=0 c t + k t+ k t = (r t δ)k t + w t n t k 0 given 4 Te discussion follows te arguments of Obstfeld (992).

8 8 A. Anagnostopoulos and C. Giannitsarou were ρ is te preference discount rate. Capital depreciates at rate 0 < δ <. Consumption, instantaneous utility, labor income and capital income are flow variables. Terefore, at period t defined by te interval [t, (t+)], tecumulativeflow of any of tese variable is te (fixed) rate of flow of te variable, i.e. c t, log c t An t, w t n t,and(r t δ)k t respectively, times te period s lengt, werer t and w t denote te rates of return on capital and labor. Finally, te total capital over te same interval is k t+ k t. 5 Te firm as a production a Cobb- Douglas production function wit constant returns to scale F (k t,n t )=k s k t n s n t, s k + s n =, and maximizes profits period by period. Since firm profits are a flow variable as well, te firm s problem is max [F (k t,n t ) r t k t w t n t ] wic implies te first order conditions r t = F k (k t,n t ) and w t = F n (k t,n t ). To write tis example in te general form (5) - (6) we rewrite te ouseolds budget constraint as k t+ k t =((r t δ)k t + w t n t c t ) So tat te state variable is x t k t and te control variables are y t (c t,n t ). Also, z t (r t,w t,τ t ). Furtermore, Terefore, te equilibrium conditions are Q(c t,n t,k t ; r t,w t,τ t ) (r t δ)k t + w t n t c t u(c t,n t ) log c t An t c t = A = w t + = ρ (r t+ δ) +(r t+ δ) k t+ k t = (r t δ)k t + w t n t c t r t = F k (k t,n t ) w t = F n (k t,n t ) 5 Note tat wen =tis is te typical disrete time model. Te continuous time version can be obtained by taking limits as 0. lim ( 0 +ρ ) t =exp( ρt) so te objective becomes Z U t = exp( ρt)(u(c t ) υ(n t ))dt t=0 and te budget constraint c t + k t+ k t k t+ k t = (r t δ)k t + w t n t = (r t δ)k t + w t n t c t k t+ k t k t = lim =(r t δ)k t + w tn t c t 0

9 Modeling Time 9 Concentrating on te Euler equation we ave tat for =and 0 ρ = (r t+ δ) ρ = +(r t δ) To understand te key difference between discrete and continuous time let s consider te timeline of events witin a period [t, t + ]. Since all te variables apart from capital are flows, te ouseolds work, receive income, save and consume continuously over te period, at fixed rates. On te oter and, wile capital is being produced and accumulates during te current period, new additions to te capital stock only become operative (i.e. put into production) in te next period. In particular, capital k t is being rented out once at te beginning of te period; watever is saved trougout te period remains inoperative in te possession of consumers. At te end of te period, te rented (depreciated) capital returns to te possession of te ouseolds and is added to te newly accumulated capital. Tis new capital stock k t+ remains in te possession of te ouseolds until te beginning of next period, wen it is rented out again. For tis reason, te ouseolds are interested in te return tey will get for teir capital once all of it becomes operative. Terefore, wen optimizing in te current period, tey coose ow muc to invest so tat teir subjective discount rate ρ is balanced out by te growt rate of capital gains, plus te next period s rental rate (in terms of tis period s capital sadow price). Tis argument implies tat in te discrete time model, tere is a inerent time-to-build delay of one period, since it takes one period for capital to become operative and agents make teir decisions based on tis fact. Turning to te limiting case of 0 (continuous time), capital is made operative at te very instance tat it is produced (i.e. tere is no time-to-build delay), so tat at eac instance, ouseolds decide ow muc of tis capital to rent out according on te current rental rate tey get. To summarize, te two modeling assumptions (discrete or continuous time) essentially imply two different models wit different dynamics and possibly different predictions. Te next section provides two more examples were indeed te two models provide different conclusions regarding te local determinacy of te steady state. 6. More Examples: Local Indeterminacy In tis section, we present two examples of dynamic macroeconomic models, namely te model of balanced-budget fiscal policy of Scmitt-Groé and Uribe (997) and te model wit increasing returns to scale as in Benabib and Farmer (994) and Farmer and Guo (994). Bot tese models are well known for exibiting local indeterminacies. By studying te dynamics of te continuous and discrete time versions we illustrate ow tere are certain (empirically plausible) parameter regions for wic te predictions of te two models are contradicting. 6.. A model wit balanced-budget fiscal policy. We consider te model of Scmitt- Groe and Uribe (997), were time is measured in increments of lengt. Tis model is an extension of te model of Hansen (985) wit a government wic maintains a balanced budget and finances its expenditures by taxing labour income.

10 0 A. Anagnostopoulos and C. Giannitsarou Te economy is populated by a continuum of ouseolds, a firm and a government. Te representative ouseold maximizes lifetime discounted utility, subject to its budget constraint: max X ( +ρ ) t [log ct An t ] (22) t=0 s.t. c t + k t+ k t = (r t δ)k t + ( τ t )w t n t k 0 given Te notation and description of te model is identical to our previous example, wit te added constraint for te government: in period t defined by te interval [t, (t + )] te government as a constant rate of flow of expenditures G (wic is time invariant). Te cumulative flow of government expenditures G is financed by taxing labor income, i.e. G = τ t w t n t,wereτ t is te labor tax rate. To write tis example in te general form (5) - (6) we rewrite te ouseolds budget constraint, using te first order conditions from te firm s maximization, as k t+ k t =((r t δ)k t +( τ t )w t n t c t ) So tat te state variable is x t k t and te control variables are y t (c t,n t ). Also, z t (r t,w t,τ t ). Furtermore, Q(c t,n t,k t ; r t,w t,τ t ) (r t δ)k t +( τ t )w t n t c t u(c t,n t ) log c t An t Terefore, te equilibrium conditions for te discrete time setting are c t = A = ( τ t )w t + = ρ (r t+ δ) +(r t+ δ) k t+ = (r t δ)k t +( τ t )w t n t c t r t = F k (k t,n t ) w t = F n (k t,n t ) G = τ t w t n t wile in continuous time (as in Scmitt-Groé and Uribe (997)), te equilibrium conditions

11 Modeling Time are c t = A = ( τ t )w t = (ρ (r t δ)) k t = (r t δ)k t +( τ t )w t n t c t r t = F k (k t,n t ) w t = F n (k t,n t ) G = τ t w t n t Let lower case letters denote te steady state values of variables and denote wit s i = δk/f(k, n), s c = c/f (k, n). Log-linearizing te two systems around te steady state and eliminating all variables apart from te state variable and te Lagrange multiplier, we obtain for te continuous time setting µ µ kt kt = C (23) were µ c c C 2 c 2 c 22 = Ã (ρ + δ) τ s k τ δ (ρ + δ) s nτ s k τ i δ sn ( τ) s i s k τ + s c (ρ + δ) s n( τ) s k τ! (24) and for te discrete time setting µ kt+ + µ kt = B (25) were 6 µ b b B 2 b 2 b 22 µ = c c 2 c 2 +c 2 c +ρ c 22 c 22 +c 2 c 2 +ρ c 22 (26) It is apparent from te previous expression tat B 6= C, i.e. te discrete time system (25) is not te direct analogue of te continuous time system (23). In oter words te stability dynamics of te two models will be different. Te conditions for indeterminacy of te continuous time system are tat Re λ C i < 0, wile te conditions for indeterminacy of te discrete time system are tat 2 <λ B i < 0. Fixing all te parameters of te model apart from te steady state level of labor tax rate τ, as in Scmitt-Groé and Uribe (997), i.e. setting s k =0.3, δ =0., ρ =0.04, wefind (numerically) tat te discrete time model is indeterminate for te range τ (0.38, 0.75), wile te continuous time model is indeterminate for τ (0.3, 0.75). In oter words, tere is no common prediction of te two models for te range (0.3, 0.38). 6 For te derivation see appendix B.

12 2 A. Anagnostopoulos and C. Giannitsarou 6.2. A model wit increasing returns. In tis section, we describe te economy of Benabib and Farmer (994) and Farmer and Guo (994). It is very similar in structure to te standard real business cycle model. Te difference lies in te aggregate production function tat exibits increasing returns to scale F (k t,n t )=k α t n β t were α + β>. Tis can be interpreted as a setup were competitive firms face constant returns tecnologies but te economy wide tecnology as increasing returns due to production externalities. A second interpretation assumes monopolistic competition wit increasing returns to scale tecnologies in te intermediate goods sector. But tese goods are combined to produce a final good in a perfectly competitive sector 7. Wen calibrating te model s parameters, we follow Farmer and Guo (994) in adopting te second interpretation. Te consumer problem and te resulting equilibrium conditions are invariant to te coice of production structure. Te representative consumer faces a standard maximization problem as follows " # X max ( +ρ ) t log c t A n γ t γ t=0 s.t. c t + k t+ k t = (r t δ)k t + w t n t + π t k 0 given were te only differences compared to te previous example are tat utility is now nonlinear in labour and tat firm profits π t are also present in te consumer s budget constraint. Te correspondence to te general form (5) - (6) is straigtforward. Te state variable is x t k t, te control variables are y t (c t,n t ) and z t (r t,w t,π t ) are additional endogenously determined variables. Te rate of cange in te state variable Q is given by and te utility function Q(c t,n t,k t ; π t,r t,w t ) (r t δ)k t + w t n t c t u(c t,n t ) log c t A n γ t γ depends only on te control variables. Equilibrium conditions in discrete time are given by c t = An γ t = w t + = ρ (r t+ δ) +(r t+ δ) k t+ = (r t δ)k t + w t n t + π t c t 7 For a detailed discussion of te model wit increasing returns and te alternative production structures, see Benabib and Farmer (994).

13 Modeling Time 3 and in continuous c t = An γ t = w t = (ρ (r t δ)) k t = (r t δ)k t + w t n t + π t c t Additional restrictions are provided by te firm s maximization problem (tese are te same for te two cases) and specify tat factors are paid teir sares in national income F (k t,n t ) r t = s k k t F (k t,n t ) w t = s n n t and profits are π t = F (k t,n t ) w t n t r t k t Let s i = δk/f(k, n) and s c = c/f (k, n). Log-linearizing te two systems around te steady state and eliminating all variables apart from te state variable and te Lagrange multiplier, we obtain for te continuous time setting µ µ kt kt = C were µ Ã c c C 2 = c 2 c 22 and for te discrete time setting µ kt+ + were i ρ+δ s k (α + αβ β γ ) δ δ β s i β γ + s c (ρ + δ)(α + αβ β γ ) (ρ + δ) β β γ µ µ b b B 2 = b 2 b 22 µ kt = B c c 2 c 2 +c 2 c +ρ c 22 c 22 +c 2 c 2 +ρ c 22 Note tat tis relationsip between B and C, te matrices describing equilibrium dynamics in discrete and in continuous time respectively, is common in bot te examples considered. We examine te presence of indeterminacies in te solution of te model under te two timing conventions. We fix all parameters to te values used by Farmer and Guo (994) except for te parameter λ, wic measures te degree of monopoly power in te markets for intermediate goods. Tus we set δ =0.025, ρ =0.0, γ =0. For factor sares we adopt te Baxter and King (994) coices of s k =0.3 and s n =0.63. Finally, we let λ vary in te range (0, ) and set α = s k λ and β = sn λ. Steady states are found to be determinate under bot timing conventions for λ<s k or λ>s n.wens k <λ<s n, te continuous time steady state is always indeterminate wereas te discrete time one can be eiter determinate or indeterminate. Obviously, watever te coice of λ, te solutions are different, as can be seen by inspection of te matrices C and D.!

14 4 A. Anagnostopoulos and C. Giannitsarou 7. Closing Comments In tis paper, we ave explored te differences arising from modeling time as discrete or continuous in a wide class of dynamic macroeconomic models. We ave sown tat tere two ways in wic te analysis under te two setups migt differ. Te first is due to te differences arising from studying a set of variables described by a system of differential or difference equations. Te second is due to te fact tat in continuous time, investment decisions are made based on present rates of return and are allowed to adjust continuously, wile in discrete time, investment decisions are made based on future rates of return, so tat te ouseolds are committed to teir decision until te next period wen te new rate of return is realized. Te differences between te two setups ave important implications for te stability properties of steady states, as well as for te equilibrium pats leading to tese steady states. In furter work on tis issue, we are exploring weter it is possible to ave a general framework wic will nest a discrete and continuous time model tat give te same qualitative and quantitative economic predictions. A possible direction for tis is to construct a discrete time model wit a generic period lengt, wic will also incorporate a time-to-build delay parameter, so tat wen taking appropriate limits, we can obtain a continuous time model wic is equivalent to te discrete time model.

15 Modeling Time 5 Appendix A. Derivations for te General Case A.. Expression (0). = + = + = + + +ρ ( + Q x i (y t+,x t+ )) + = +ρ +Q xi (y t+,x t+ ) = +ρ +Q xi (y t+,x t+ ) = = ( + ρ) ( + Q xi (y t+,x t+ )) +Q xi (y t+,x t+ ) = Q xi (y t+,x t+ )+ρ +Q xi (y t+,x t+ ) = Q xi (y t+,x t+ )+ρ +Q xi (y t+,x t+ ) = Q xi (y t+,x t+ )+ρ +Q xi (y t+,x t+ ) = (ρ Q x i (y t+,x t+ )) +Q xi (y t+,x t+ ) = = B. Derivations for Examples B.. Example 2. Te first reduced equation of te log-linearized equation for te discrete time setting is k t+ = c k t + c 2 Te second reduced log-linearized equation for te discrete time setting is + = c 2 +ρ k t+ + c 22 +ρ + wic simplifies to c 2 +ρ k t+ +( c 22 +ρ ) + = c 2 +ρ k t c 22 +ρ

16 6 A. Anagnostopoulos and C. Giannitsarou Tus. µ kt+ + = = = = µ c 2 +ρ ( c 22 +ρ ) 0 µ c 2 +ρ c 22 +ρ µ kt c c 2 µ c 2 +ρ ( c 22 +ρ ) 0 µ kt+ + = µ c2 +ρ c 22 +ρ c c 2! µ c2 +ρ Ã 0 ( c 22 ( c +ρ ) 22 +ρ ) c 2 +ρ c c 2 Ã! µ c c 2 kt c 2 (+c ) c 2 c 2 +c 22 +ρ c 22 +ρ c 22 µ kt µ b b 2 b 2 b 22 µ kt c 22 +ρ µ kt References [] Bambi M. and O. Licandro, "(In)determinacy and Time-to-Build". Mimeograp. [2] Baxter, M. and R. G. King, 99. "Productive Externalities and Business Cycles". Discussion Paper 53, Institute for Empirical Macroeconomics, Federal Reserve Bank of Minneapolis. [3] Benabib, J. and R. E. A. Farmer, 994. "Indeterminacy and Increasing Returns". Journal of Economic Teory, 63, 9-4. [4] Carlstrom, C. T. and T. S. Fuerst, "Continuous versus Discrete-time Modeling: Does it Make a Difference?", mimeograp. [5] Farmer, R. and J.-T. Guo, 994. "Real Business Cycles and te Animal Spirits Hypotesis". Journal of Economic Teory, 63, [6] Foley, D. K., 975. "On Two Specifications of Asset Equilibrium in Macroeconomic Model", Journal of Political Economy, 83, [7] Hansen, G "Indivisible Labor and te Business Cycle", Journal of Monetary Economics, 6, [8] Hintermaier, T., "A Sunspot Paradox", mimeograp. [9] Obstfeld, M., 992. "Dynamic Optimization in Continuous-Time Economic Models (A Guide for te Perplexed)", mimeograp. [0] Scmitt-Groé, S. and M. Uribe, 997. "Balanced-Budget Rules, Distortionary Taxes, and Aggregate Instability", Journal of Political Economy, 05,