Macroeconomic Theory I. Cesar E. Tamayo Department of Economics, Rutgers University

Transcription

1 Macroeconomic Theory I Cesar E. Tamayo Department of Economics, Rutgers University ctamayo@econ.rutgers.edu Class Notes: Fall 200

2 Contents I Deterministic models 4 The Solow Model 5. Dynamics Balanced growth The golden rule Quantitative implications Solow growth accounting Optimal growth 9 2. Optimal growth in discrete time Assumptions The sequential method: Lagrange The recursive method: dynamic programming The Envelope Theorem approach Balanced growth and steady state Linearization Equilibrium growth and welfare theorems Extensions to the optimal growth model Assets in the OGM The role of government in OGM Optimal Growth in continuous time Steady State Tobin s q Overlapping generations (OLG) OLG in economies with production (Diamond s) Log-utility and Cobb-Douglas technology Steady state Golden rule and dynamic ine ciency The role of Government Social security Restoring Ricardian equivalence OLG in pure exchange economies (Samuelson s) Homogeneity within generation The role of money Fiscal policy and the La er curve Monetary equilibria with money growth Within generation heterogeneity The real bills doctrine

3 II Stochastic models 4 4 Stochastic Optimal growth Uncertainty in the neoclassical OGM Non-stochastic steady state Stationary distribution Log-linear approximation Solution method : Blanchard-Khan Impulse response functions (IRF) RBC models The baseline RBC model The general case CRRA utility and Cobb-Douglas production The log-linear system Labor productivity (King, Plosser & Rebelo, 988) Solution method 2: Sim s GENSYS Varieties of RBC models Asset pricing models (Lucas, Shiller) Calibration Estimation methods Generalized method of moments (GMM) III Appendixes 56 A The dynamic programming method 57 A. Guess and verify A.2 Value function iteration A.3 Solving for the policy functions A.4 Properties of the BFE A.5 The Envelope Theorem: an application B The Maximum Principle 66 B. Discrete time B.2 Continuous time B.2. Current value vs. present value Hamiltonian C First-Order Di erence Equations and AR() 70 C. The AR() process C.. Representation and properties C..2 Conditional Distribution C..3 Unconditional Distribution C.2 Linear First-Order Di erence Equations (FODE) C.2. Induction & Geometric Series C.2.2 Homogeneous part and General solution C.2.3 Asymptotic Stability C.3 Systems of linear FODE (or VDE) C.3. Asymptotic stability

4 Summary These notes summarize the material of a rst semester graduate course in Macroeconomic theory. The rst sections focus on deterministic growth models and models of overlapping generations (OLG). Later sections are dedicated to stochastic models, including neoclassical growth and real business cycles models. The appendixes cover some mathematical material required for solving simple macro models. The notes are freely based on: Acemoglu (2008), Romer (200), Stokey and Lucas (989), Ljungqvist and Sargent (2004), Dave and De Jong (20), Dixit (990), Levy (992) and lecture notes from Professor Roberto Chang from Rutgers University. Naturally, all errors and omissions are my own. A word on notation: Throughout these notes, x will be used to denote a speci c solution (optima), x will be used for steady states in di erence (di erential) equations and ^x will be used for ln (x=x). When applicable, upper cases will stand for economy-wide values of variables, while lower cases will stand for per-capita (per-e ective labor) variables. Finally, in stochastic models E 0 or simply E will denote unconditional expectations while E t will stand for expectations conditional on information available at t: 3

5 Part I Deterministic models 4

6 Chapter The Solow Model In the original Solow model, time is continuous and the horizon is in nite. Without loss of generality (WLOG) assume that time is indexed t 2 (0; ). At each point in time, there is only one fnal good Y (t). Assumption The nal good is produced with Harrod-neutral or labour augmenting technology: Y (t) = F (K(t); A(t)L(t)) Assumption 2 F () is twice di erentiable and F (K; AL) = F (K; AL). This implies constant returns to scale or no gains from specialization and that one can write the production function in intensive form: K y = f(k) = F AL ; Assumption 3 f(0) = 0; f 0 (k) > 0; f 00 (k) < 0, lim k! f 0 (k) = 0 and lim k!0 f 0 (k) = Assumption 4 Savings are a constant fraction of income. Assumption 5 Existence of a representative household To ilustrate the assumptions about the production technology: Example The Cobb-Douglas production function satis es assumptions -3. To see this consider: F (K(t); A(t)L(t)) = K (AL) K F AL ; = K (AL) AL K = AL f(k) = k and note that f(k) = k = f(k). Also note that f 0 (k) = k > 0 since > 0 and k > 0: Likewise, f 00 (k) = ( ) k 2 < 0 Furthermore lim k = and lim k!0 k! k = 0 since < 0: Remark Notice that: F K (K; AL) = K (AL) = K AL = k = f k (k) 5

7 . Dynamics Suppose that inputs grow as follows: Labor: _ L(t) = nl(t) so that e ln L(t) = L(t) = L(0)e nt Technology: _A(t) = ga(t) so that e ln A(t) = A(t) = A(0)e gt Capital law of motion: _K(t) = sy (t) K(t) To derive the last expression in intensive form apply the quotient rule and the product rule to the expression: d (K=AL) dt the key equation of the Solow model..2 Balanced growth _K(t) K(t) h = A(t)L(t) [A(t)L(t)] 2 sy (t) K(t) = k(t)n k(t)g A(t)L(t) _k(t) = sf(k(t)) k(t) [n + g + ] A(t) L(t) _ + A(t)L(t) _ i Suppose that the economy nds itself in a path in which K(t) and A(t)L(t) are growing at the same rate. This is a special case of balanced growth which itself induces a so-called steady state for k since _ k(t) = 0: Thus sf( k(t)) = k(t) [n + g + ] (.) and one can see that starting from any level of capital per e ective worker, k! k: Furthermore, at the level k one can see that: _k(t) = 0 ) and given the assumption of homogeneity (CRTS): _ K(t) K(t) = n + g _k(t) = 0 ) _y(t) = 0 ) _ Y (t) Y (t) = n + g K(t) nally, note that _ L(t) = g = Y _ (t) L(t). That is, the economy reaches a Balanced Growth Path (BGP), where each variable fy; K; A; Lg is growing at a constant rate..3 The golden rule Suppose starting from the BGP, there s a shift in s. Then k _ jumps since sf(k(t)) > k(t) [n + g + ] and then falls gradually until k! k new : In turn Y (t) L(t) grows by g and k _ > 0 so that Y (t) L(t) jumps but falls gradually too. Consumption C AL, falls by de nition since s jumps: c (t) = ( s) f(k(t)) The generic notion of balanced growth path is a situation in which all variables growth at a constant rate over time (though this rate need not be the same across variables). A special case of a balanced growth path is a steady state, in which the growth rate of all variables is equal to zero. 6

8 To see what happens when the economy reaches the new BGP: and di erentiate w.r.t. s c = ( s) f( k(t)) = f( k(t)) sf( k(t)) = f( k(t)) k(t) [n + g + ] by de nition (s; n; g; ) = f 0 ( k(s; n; g; ) (n + g + k(s; n; g; since the last term is unambiguously positive, the depends on whether f 0 ( k(s; n; g; )? (n + g + ). In fact, the BGP level of capital (per AL) that brings: f 0 ( k(s; n; g; )) = (n + g + ) = 0 (BGP-consumption is at its maximum) is called the golden rule level of capital, k Gold : Therefore, if k > k Gold one has that f 0 ( k) < f 0 ( k Gold ) = (n + g + ) and therefore the economy can increase c by dis-saving. Exercise 2 Suppose that f(k(t)) = k(t). Show that for some, the Solow model can predict overaccumulation of capital in the sense that k > k Gold : Solution. Simply note that (.2) is (s; n; g; ) = k so one needs to show that k one has i (n + g + k(s; n; g; (.2) (.3) (n + g + ) < 0 (and < 0). Using (.) k = s n + g + and replacing k in the golden rule condition, it can be seen that < s ) k > k Gold and the Solow model predicts overaccumulation of capital..4 Quantitative implications In order to quantify the e ect of savings on long-run growth (i.e., = f 0 ( k(s; n; g; to it su ces to di erentiate implicitly the (BGP) equation of _ k = 0: substituting: ds ds f( k) + s df( k) ds f 0 ( k) + f( k) = d k(s; n; g; ) = = d (n + g + ) k(s; n; g; ) + d k(s; n; g; ) (n + g + ) ds ds (n + g + ) f( k) (n + g + ) sf 0 = f 0 ( k)f( k) (n + g + ) sf 0 ( k) 7

9 simplify multiplying by s y obtain: and using y = f( k) and s = k[n+g+] f( k) from the equation _ k = 0 to = = kf 0 ( k)=f( k) kf 0 ( k)=f( k) k k.5 Solow growth accounting To obtain equation () in growth form di erentiate w.r.t. time (recall dy dt rule and omitting the (t): = _ Y ), using the chain _Y Y _Y = F k _K + F k L _ + FA _A _Y = F k K _ + F k L _ Y Y Y + F _ A A Y _Y = KF k _K Y Y K + LF k _L Y L + AF A _A Y A _Y _K = " k Y K + " _L L L + " _A A {z A } _L L = " _K k K + ( " L L ) _ L + X Note that this equation cannot be estimated with data for K and obtaining the residual as X since the residual is correlated (by construction) with capital per worker. Instead, rearrange:! X = _y _K _L " k y K L and now one has everything measurable in the RHS. This is the key equation of growth accounting used to measure Solow s residual. Usually " k is the share of capital on the economy and in a Cobb-Douglas like the example above, " k = : _ K 8

10 Chapter 2 Optimal growth 2. Optimal growth in discrete time Suppose that, savings are not a xed share of income but rather that households decide how much to consume and how much to save on each period. For now, assume that there is neither technical change nor poppulation growth (g = n = 0) so that aggregate production takes the form F (K t ; L t ): As before, suppose that F () is homogeneous of degree one so that production can be written in per labor units: y = f(k) = F K AL ; : Next, suppose there exists a representative household (RH) or, equivalently, that preferences can be aggregated economy-side. Then the RH solves the (discrete-time) problem: max c t;k t+ X t u(c t ) t=0 c t + k t+ f(k t ) + ( s:t: (2.) k 0 given )k t so that the RH chooses a consumption-saving plan fc t ; k t+ g t=0 resource constraint holds at every t = 0; ; ::: under the condition that the 2.2 Assumptions Assumption 6 Assumptions -3 about f() from section. hold. Assumption 7 The objective function u() is continuous, twice di erentiable and satis es the Inada conditions: u(0) = 0; u 0 (k) > 0; u 00 (k) < 0; lim k! u0 (k) = 0 and lim u 0 (k) = k!0 Assumption 8 The constraint set fk t+ j k t+ f(k t ) + ( )k t c t g is compact, convex. Assumption 9 Preferences are additive. This ensures dynamic consistency. Under these assumptions any organization of markets and production will yield the same competitive equilibrium allocation. Hence the competitive equilibrium is unique and the rst and second welfare theorems hold. That is, the competitive equilibrium will be Pareto e cient and the planner s problem can be descentralized as the outcome of a competitive equilibrium as will be shown below. 9

11 2.3 The sequential method: Lagrange The problem above can be approached by the in nite-horizon Lagrange method: X L = t fu(c t ) + t [f(k t ) + ( )k t c t + k t+ ]g t=0 t = 0 =) t u 0 (c t ) = t+ = 0 =) t = t+ (f 0 (k t ) + t+ = 0 =) c t + k t+ = f(k t ) + ( )k t (2.2c) replacing t and using the fact that t+ = t+ u 0 (c t+ ) one obtains the Euler Equation: u 0 (c t ) = u 0 (c t+ ) (f 0 (k t ) + ) (2.3) Along with the resource constraint (F.O.C. (2.2c)) these two di erence equations fully characterize the solution to the optimal plan. The boundary condition required for the solution to (2.2c)-(2.3) to exist is the transversality condition: lim tk t+ = lim t u 0 (c t )k t+ = 0 t! t! 2.4 The recursive method: dynamic programming This problem can also be solved as a discrete time, deterministic, stationary, dynamic programming one. In fact, when the problem is stationary (i.e. the problem faced at every period is identical), the sequential and recursive methods are equivalent. If one separates the problem into periods, each problem depending on the state k t and consisting on choosing controls c t ; k t+ ; the Bellman functional equation (BFE) for the iterated (recursive) one-period optimization problem is: V (k t ) = max c t [u(c t ) + V (k t+ )] s:t: (2.4) y t = c t + i t = f(k t ) k t+ = ( )k t + i t k 0 given In fact, this problem can be seen to have only one control variable since choosing c t is equivalent to choosing k t+ given the combination of the resource constraint and the capital accumulation equation. To be more precise notice that the problem is (2.4) is equivalent to: V (k t ) = max k t+ [u(f(k t ) k t+ ( )k t ) + V (k t+ )] (2.5) V (k t ) = max c t [u(c t ) + V (( )k t + f(k t ) c t )] (2.6) and associated transversality condition: lim t! t+ V 0 (k t+ )k t+ (2.7) 0

12 Under familiar assumptions (see Appendix A), the solution to the Bellman equation, will yield time-invariant, sate-dependent rules for consumtion and capital accumulation, i.e., policy functions: c t = h(k t ) (2.8) k t+ = g(k t ; c t ) = g(k t ; h(k t )) (2.9) and notice that g() is precisely the state-transition function that results after consolidating the two restrictions in (2.4); that is: k t+t = ( )k t + f(k t ) c t = g(k t ; c t ) However, since (2.4) is a functional equation, one needs to solve for V () in order to obtain (2.8)-(2.9). There are mainly three approaches to solve for V (): Guess and verify the value function Value function iteration or method of successive approximations, and, Policy functions iteration or Howard s improvement algorthm. These three approaches are developed in Appendixes A.-A.3, and some examples are provided. The main conditions for the existence of a unique solution, V (); to the BFE (2.4) are: Condition Assumptions -3 at the begining of this chapter hold. Condition 2 The control space and the state space are convex, compact sets, Condition 3 The operator V 7! T (V ) in (2.4) maps the set of continuous, bounded, real-valued functions M into itself. Condition 4 The operator V 7! T (V ) (2.4) is a contraction mapping de ned on a complete metric space of continuous, bounded, real-valued functions M. Condition 3 requires in turn that u () ; f () are continuous and bounded and that the correspondence (k t ) = fc t jc t ( )k t + f(k t ) k t+ g is UHC, LHC, non-? and compact-valued. Appendix A.4 provides the relevant proofs and also proves some additional properties of the operator T () : Condition (iv) requires a contraction mapping; recall that a contraction is a mapping, '; on a metric space (W; d) that satis es d (' (f) ; ' (g)) d (f; g) for all f; g 2 M and some < : Since M above is a complete metric space, conditions (i)-(iv) would su ce for a unique solution to the BFE since the Banach xed point theorem asserts that every contraction mapping on a complete m.s. has a unique xed point, i.e., 9 V such that V = T (V ). Moreover, the sequence de ned by V i+ = T (V i ) converges to the unique xed point V : Now, the Blackwell conditions for an operator to be a contraction mapping are: Monotonicity: ' > (w.r.t. box metric)) T (') > T ( ) whenever '; 2 M. Discounting: for any constant function A = h(), and ' 2 M one has T ('+A) T (')+A for some < : In this case the assumption that V is bounded is redundand since the compactness of the control space ensures this property.

13 Both conditions are easily seen to hold in this case: Assuming that one obtains V () by any of the abovementioned methods, it is easy to obtain the F.O.C. for the problem (2.6): the associated F.O.C. is: and the resource constraint: V (k t ) = max c t [u(c t ) + V (( )k t + f(k t ) c t )] u 0 (c t ) V 0 (k t+ ) = 0 (2.0) c t + k t+ = f(k t ) + ( )k t (2.) At this point one can use the knowledge of V () in order to obtain V 0 () and solve the system of di erence equations (2.0)-(2.). Example 3 (Guess and verify) Cobb-Douglas technology with full depreciation and logarithmic utility. The problem is: the F.O.C is therefore: If one guesses the form of V () as: V (k t ) = max c t;k t+ [ln c t + V (k t+ )] s:t: (2.2) c t + k t+ = k t k 0 given c t = V 0 (k t+ ) kt k t+ = V 0 (k t+ ) V (k t ) = F + G log k t so that one replaces for V 0 (k t+ ) in the F.O.C.: k t = G k t+ and using the resource constraint arrives to the policy functions (with undetermined coe cient G): now using this in the BFE: V (k t ) = F + G log k t = log k t+ = c t = G + G k t k t+ G + G which solving for the undetermined coe cients F,G yields: G = F = k t G G kt + F + G log + G + G k t log ( ) + ( ) log() 2

14 so that nally one can generate optimal plans fc t ; k t+ g t= from the fully speci ed policy functions: c t = ( ) k t k t+ = () k t Finally, note that the transversality condition is satis ed: lim t! t+ V 0 (k t+ )k t+ = lim t+ t! 2.4. The Envelope Theorem approach A di erent avenue that avoids dealing with the value function explicitely is as follows. If conditions -4 above regarding the objects in the problem are satis ed (i.e., concavity, convexity, continuity, compactness, monotonicity, discounting), then the unique solution to the BFE V (), would be continuous, concave, increasing and importantly, di erentiable. Hence, the Envelope theorem applies and one can use the envelope condition for the value function (see the Appendix A for a more elaborate example). To see how the Envelope theorem works in this particular case, recall the policy functions: then the value function becomes: c t = h (k t ) k t+ = g(k t ; c t ) = ( )k t + f(k t ) h (k t ) = 0 V (k t ) = u(h (k t )) + V (( )k t + f(k t ) h (k t )) di erentiating w.r.t. k t : V 0 (k t ) (k (k t ) + V 0 (k t+ ) ( (k t (k t t (k (k t ) + V 0 (k t+ ) ( t) V 0 (k t+ (k (k t t = V 0 (k t+ ) ( t) (k (kt )) V 0 (k t+ (k t ) but the F.O.C. (2.0) (k t (k t ) = u 0 (c t ) = V 0 (k t+ ) and therefore: V 0 (k t ) = V 0 (k t+ ) ( t = u 0 (c t ) ( t and since this holds for every period: V 0 (k t+ ) = u 0 (c t+ ) [f 0 (k t+ ) + ( )] 3

15 replacing in the F.O.C. one obtains once more the system of di erence equations that (under the assumptions above) will characterize the solution to the RH problem: u 0 (c t ) = u 0 (c t+ ) [f 0 (k t+ ) + ( )] (2.3) c t + k t+ = f(k t ) + ( )k t (2.4) Alternatively, one can use the policy functions (2.8)-(2.9) to express the Euler equation in terms of k t : u 0 (g(k t )) = u 0 (g(h(k t ))) [f 0 (h(k t )) + ( )] Finally, note that V 0 (k t+ ) is something similar to a shadow price associated with the resource constraint. Remark 2 Note that in this problem it is the case that 6= and therefore, f 0 (k t ) + = R t. If one was to consider the case where =, as will be the case in some sections below then one would have: f 0 (k t ) = r t = R t : 2.5 Balanced growth and steady state As mentioned earlier, the generic notion of balanced growth path is a situation in which all variables grow at a constant rate over time (though this rate need not be the same across variables). Perhaps the simplest example of balanced growth that solves the optimal growth problem is: Example 4 Consider the so-called AK model.under CRRA utility and full depreciation. The social planner s problem is: the BFE for this problem is: with F.O.C.: and envelope condition: so the Euler equation becomes: or, using the functional form for u (c t ) : max s:t: y t c t + i t X t c t t=0 y t = f (k t ) = Ak t i t = k t+ V (k t ) = max k t+ fu (Ak t k t+ ) + V (k t+ )g u 0 (c t ) + V 0 (k t+ ) = 0 V 0 (k t+ ) = u 0 (c t+ ) A u 0 (c t ) = u 0 (c t+ ) A c t+ c t = A Since c t = Ak t k t+ : (Ak t+ k t+2 ) = A (Ak t k t+ ) 4

16 A second order di erence equation that, without a (on k) that, without a terminal condition has multiple solutions for any given k 0 : The interest is in that which satis es the TVC (2.7) which in this case becomes (using the EC): lim t! t+ V 0 (k t+t ) k t+ = lim t! t+ u 0 (c t+ ) Ak t+ Now to nd conditions that ensure holding of the TVC note that in a BGP capital and consumption grow at the same constant rate so k t+ = k t and c t+ = c t for some : Thus: (c t ) c t = A ) = (A) = which brings positive growth iif > =A. Next, rewrite u 0 (c t+ ) = c t =A and k t+ = t+ k 0 in the TVC: lim t! t+ u 0 (c t+ ) Ak t+ = lim t+ ct t! A At+ k 0 = lim t+ ct t! t+ k 0 = lim t! () t c t k 0 then the TVC holds iif < )if (A) = < : Hence by assuming that < A + one can ensure that there is balanced growth and the TVC holds. A special case of a balanced growth path is a steady state, in which the growth rate of all variables is equal to zero. From (2.3)-(2.4) one can compute the steady state by noting that in the SS, x t = x t+ = x for x = fc; k; yg. Hence: u 0 (c) = u 0 (c) f 0 ( k) + ( ) = f 0 ( k) + ( ) de ning a modi ed golden rule for capital accumulation, and c k = f( k) = y 2.6 Linearization Next it is possible to study the behavior of the model around the steady state (SS). Recall that in the SS, x t = x t+ = x for x = [c; k] 0 and ^x t = (x t x)=x. So, linearize the system around the SS. Recall that to linearize G(x; y) = 0 around the SS (x; y)x ^x t + (x; y)y ^y t = t Linearizing (2.3): ^c t = ^c t+ + ^k t+ (2.5) where division by u 0 (c) on both sides has been used and the fact that = = f 0 ( k) + ( ) from the SS and ; are the elasticities of u 0 () and f 0 (), respectively. Finally, the linear constraint is linearized as: c^c t + k^k t+ = f( k) + ( ) k^kt!^c t + ^k t+ = [ +! + ] ^k t (2.6) 5

17 where! = c y, = k y and is the elasticity of f():now one can suppose that the linearized policy functions are of the form: so that replacing in (2.5)-(2.6) one obtains: ^c t = ^kt ^k t+ = 2^kt = 2 + 2! + 2 = [ +! + ] and one can solve for the undetermined coe cients ; 2 as functions of parameters of the model (;!; ). Note that the rst of these equations will be a cuadratic one so one has two solutions, from which one selects 2 < since this is the only solution that satis es the transversality condition. With this log-linear policy functions and the "true" coe cients as functions of the parameters of the model, one can generate optimal sequences f^c t ; ^k t+ g t=0, that is sequences of the variables in deviation from SS form. 2.7 Equilibrium growth and welfare theorems The solution to the optimal growth model can in fact be deduced as the outcome of a competitive equilbrium. To see this, state the problem of the RH and the rm separately. Households The representative household maximizes lifetime discounted utility subject to its resource constraint. Households own the factors of production k; l and own the rms. For simplicity, suppose there is full depreciation ( = ). At each period, the RH receives income from renting all of its available capital, working all its endowed labor and earning pro ts from the rms. 2 With this income, the RH and decides how much to consume and how much to invest (save): Firms max c t X t u(c t ) t=0 s:t: c t + k h t+ r t k h t + w t l h t + t = y h t Firms produce a single good by renting production factors from the RH and maximize pro ts subject to their production technology: X X max t = max p t y f t w t l f t r t k f t t=0 s:t: t=0 y f t F (k f t ; l f t ) where F () is continuous, di erentiable, strictly increasing and homogeneous of degree one. Since there is only one good in the economy, there are no relative prices and one can set p t =. Also, 2 The assumption that households work their entire endowment of labor re ects the fact that there is no disutility of labor in this model. The RBC models surveyed in the sections below relax this assumption so that labor becomes in fact a choice variable. 6

18 since there is no discounting, lifetime pro ts are maximized, pro ts are maximized at every period t: 3 Equilibrium For simplicity suppose that lt h = : Then a competitive equilibrium consists of a set of prices fp t = ; w t ; r t g t=0 and allocations fk t ; lt = ; yt ; c t g t=0 such that 8 t :. The rm maximizes pro ts. To do so, notethat since F () is strictly increasing, the technology constraint will hold with equality y f t = F (k f t ; l f t ). Thus, the F.O.C.s of the rm f t ; l f t f f t ; l f t f t = 0 =) w t = F l (k f t ; l f t ) = 0 =) r t = F k (k f t ; l f t ) 2. The RH maximizes utility. The F.O.C.s for the RH are, as before: 3. Markets clear in all periods (t = ; 2:::): u 0 (c t ) = u 0 (c t+ )r t+ c t + k h t+ = r t k h t + w t l h t + t y h t = y f t = F (k f t ; l f t ) l h t = l f t = k h t = k f t Next, replace the F.O.C.s for the rm in the pro t function at t and recalling that l t = : t = F (k t ; l t ) F l (k t ; l t ) F k (k t ; l t )k t and because F () is homogeneous of degree one, Euler s theorem (x rf(x) = f(x)) implies that t = 0 so that: w t = F (kt ; lt ) F k (kt ; lt )kt and P t=0 t = 0. Replacing in the F.O.C.s for the RH yields: u 0 (c t ) = u 0 (c t+ )F k (k t ; l t ) and c t + kt+ = F k (kt ; lt )kt + F (kt ; lt ) F k (kt ; lt )kt = F (kt ; lt ) which are of course, the same optimality conditions derived under the centralized approach in the section above. Hence one has found a vector of prices that delivers the (planned) Pareto optimal allocation which results as a solution to (2.3)-(2.4). That is, the optimal allocation has been descentralized as a competitive equilibrium of the economy This is an ilustration of the second fundamental theorem of welfare economics It is straightforward to extend this model to the case where rms discount future pro ts. A natural candidate for discounting would be where R is the gross interest rate (in this economy all assets would earn R). 4 R Recall that the rst welfare theorem states that whenever households are non-satiated, a competitive equilibrium allocation is Pareto optimal. 7

19 2.8 Extensions to the optimal growth model 2.8. Assets in the OGM Recall the budget constraint for the RH is, in general: and if one assumes l t = 8 t: k t+ = ( + r t )k t + w t l t + t c t k t+ = ( + r t ) {z } k t + w t c t + t (2.7) R t = gross return Now, allow for assest a t that HH carry from previous periods. Note that in aggregate a t = k t +b t where we may have b t < 0 at some t. So we can re-write the ow budget constraint as: c t + a t+ = R t a t + w t + t c t + a t+ R t R t = a t + w t + t At this point one carries out forward substitution of a t+ from the equation for a t+2, then substitute the latter from the equation for a t+3 and so on. For some nite T ; the intertemporal budget constraint (IBC) is given by: where: TX t=0 R t c t + a T + R T = ( + r 0 )a 0 + R t = ty R j = j= TX t=0 ty ( + r j ) j= R t (w t + t ) (2.8) But since we re assuming an in nite horizon, we must prevent a T + R T! as T! (build debt forever) which would result in P t=0 R t c t! : So we impose the additional no Ponzi games condition: a T + lim = 0 T! R T so that when the horizon is in nite, taking limits on both sides of (2.8) and using the no-ponzi condition, the IBC can be expressed as: P V X c t R t=0 t {z } = ( + r 0 )a 0 {z } + X (w t + t ) R t=0 t {z } consumption = initial asset income + PV non-asset income (2.9) Note that a stream of consumption that satis es the (IBC) also satis es the ow constraint at each t. To see this, consider a consumption plan f^c t g t=0 that satis es the IBC. At some, by (2.8) the household will have accumulated assets: a + R = ( + r 0 )a 0 + X t=0 R t (w t + t ^c t ) X = ( + r 0 )a 0 + (w t + t ^c t ) + (w + ^c ) (2.20) R t R t=0 8

20 obviously since: a X = ( + r 0 )a 0 + (w t + t ^c t ) R R t=0 t ) X t=0 we can replace this in (2.20) to get: R t (w t + t ^c t ) = ( + r 0 )a 0 + a R a + R = R (w + ^c ) + a R simply multiply this last expression by R on both sides to obtain: a + = w + ^c + R a which is of course the sequential BC for period. Using the IBC just derived, the optimal growth problem can be stated more compactly as: " # X X L = t X u(c t ) + ( + r 0 )a 0 + (w t + t ) c t R t R t t=0 t=0 with associated F.O.C. (noting that in equilibrium t = t t ) t u 0 (c t ) = R t+ t+ ) t+ u 0 (c t+ ) = R t+ so equating them yields the same Euler equation derived earlier: u 0 (c t ) = R t+ u 0 (c t+ ) = u 0 (c t+ )( + r 0 ) which naturally relies on the fact that in equilibrium all rates of return are equilized (so all assets receive and are discounted to the same rate) The role of government in OGM To introduce the government, derive restrictions similar to (2.7) and (2.9): t=0 b t+ = (g t t ) + R t b t where b t+ = debt outstanding at t +, (g t t ) = net income or primary de cit at t and R t b t =debt service in t: [ Note that the same rate of return is used for the government, all nancial assets and capital which is the case if there is no uncertainty (no default risk) and perfect complete nancial markets!]. Next, set up the IBC for the government: X t g t ( + r 0 )b 0 lim b t+ t! R t Naturally, if we allow lim t! R t b t+!, the LHS of this inequality approaches and therefore the government could run primary de cits forever. Hence, the following no-ponzi condition for the Gvt.is imposed: lim b t+ = 0 t! R t 9 t=0 R t R t

21 so that the IBC for the government is: ( + r 0 )b 0 Note that now HH would have to pay taxes so: t=0 X t=0 X t t=0 R t c t = ( + r 0 )a 0 + R t g t R t X wt + t t=0 and replacing PV taxes from the Gvt. IBC: X X wt + t c t = ( + r 0 )a 0 + R t = ( + r 0 )k 0 + t=0 R t X wt + t t=0 R t R t X t=0 X t=0 t R t g t R t ( + r 0 )b 0 g t R t (2.2) where the de nition of assets a 0 = b 0 + k 0 has been used. This is a crucial result, for, it shows that only the (PV) level of government spending matters and not the means through which it is nanced (debt or taxes). This is naturally a Ricardian Equivalence statement. The new IBC for the HH (2.2) can be used again in the Lagrangian as above to derive the system of equilibrium conditions for the optimal growth problem. Furthermore, one can see that for a given initial stock a 0, at any t : k t + b t = a t = R t a 0 and since a 0 is constant, " b t )# k t and/or " R t which are both ways of crowding out. Example 5 (OGM with assets and government) The problem can be stated as: max c t X t u(c t ) t=0 s:t: a t+ = ( + r)a t + w t c t so that the state variable is a t and the control variable again c t : Note that this problem is not well de ned even with the transversality condition: since it may happen that a t+! above: lim t! t+ V 0 (a t+ )a t+ = 0. Therefore, one needs the no-ponzi condition introduced a T+ lim = 0 T! R T so that the TVC holds. With the problem well speci ed, the solution can be found by solving: with F.O.C.: V (a t ) = max fu(( + r)a t + w t a t+ ) + V (a t+ )g u 0 (( + r)a t + w t a t+ ) = V 0 (a t+ ) u 0 (c t ) = V 0 (a t+ ) 20

22 and envelope condition: V 0 (a t+ ) = u 0 (( + r)a t+ + w t a t+2 )( + r) = u 0 (c t+ )( + r) and the usual Euler equation: u 0 (c t ) = u 0 (c t+ )( + r) 2.9 Optimal Growth in continuous time State the problem (2.) in continuous time: max Z c(t);i(t) 0 u(c(t))e t dt s:t: (2.22) c(t) + i (t) f(k(t)) _k(t) = i (t) k(t) (2.23) k(0) given The continuous time problem is actually easier to solve using the tools of optimal control; a review of The Maximum Principle and the Hamiltonian approach is presented in Appendix B. Set up the (present-value) Hamiltonian: H = u(c(t))e t + (t) [f(k(t)) k(t) c(t)] (2.24) where (t) is the co-state variable. The condition for c(t) to maximize the Hamiltonian = 0 =) u0 (c(t))e t = (t) (2.25) solving this yelds c (t) as a function of the co-state and parameters. Next, we can replace c (t) in (2.24) to obtain the maximum value function of the Hamiltonian, H, di erentiate w.r.t. state and co-state variables. Alternatively, simply state the Pontryagin conditions similar to those derived in the Appendix B (B.8)-(B.9): ii) = (t) [f 0 (k(t)) ] (2.26) iii) k(t) = = f(k(t)) k(t) c(t) Conditions (2.25)-(2.26)-(2.27) along with the TCV: lim (t)k(t) = 0 t! are the three di erential equations (and terminal condition) that characterize the solution to the HH problem. Example 6 Suppose that u (c (t)) = ln (c (t)). Then from (2.25): c(t) = e t (t) 2

23 so that replacing in (2.27) we get the two di erential equations on k and that characterize the solution: _k(t) = f(k(t)) k(t) _(t) = (t) [f 0 (k(t)) ] e t (t) Alternatively, one can solve the system in terms of di erential equations in c and k, which is more consistent with the idea of policy rules described in previous sections. To do so, di erentiate (2.25) w.r.t. time: u 00 (c(t))e t _c(t) e t u 0 (c(t)) = _(t) (2.28) and replace in (2.26) to obtain: which upon rearranging: u 00 (c(t))e t _c(t) e t u 0 (c(t)) = (t) [f 0 (k(t)) ] u 00 (c(t))e t _c(t) e t u 0 (c(t)) = e t u 0 (c(t)) [f 0 (k(t)) ] _c(t) c(t) = f 0 (k(t)) [c(t)u 00 (c(t))=u 0 (c(t))] (2.29) where the denominator is naturally, the Arrow-Pratt CRRA coe cient: This condition tells us that whenever f 0 (k(t)) is "large", which happens when k(t) is "low", we have _c=c > 0 so that the HH consumes less today and more in the future. Likewise, _c=c < 0 if is large enough which suggests that the HH is more "patient". Finally, the TVC is: 2.9. Steady State Recall, in SS _c(t) = 0 and _ k(t) = 0. Therefore: lim t! = 0 lim u 0 (c(t))k(t) t! = 0 _c(t) c(t) = 0 =) f 0 ( k) = (2.30) which is, again, a modi ed Golden Rule for capital accumulation and, once more, independent from the shape of u(): Next: _k(t) = 0 =) f( k) c = k so that SS-investment just breaks-even (making _ k(t) = 0). Remark 3 Note that (t) is the shadow price of capital, i.e., it measures the impact of a small increase in k t on the optimal value of the program. By comparison, V (k t ) is the value of of the optimal program from t given the level of capital k t : Therefore, we have that: (t) = V 0 (k t ) and note that _(t) represents the appreciation of capital since its the change in the value of a unit of the state variable. Exercise 7 In order to make meaningful comparisons with the Solow model, reintroduce technical change and poppulation growth. (i) Write the equilibrium conditions of the OGM considerng these features and (ii) Show that along the balanced growth path, k < k Gold where k Gold is the Golden Rule level of (per-e ective-labor) capital in the Solow model. 22

24 Solution. (i) First, derive the analogue to (2.30) after reintroducing poppulation growth and technological progress. To do so, write the constraints in absolute levels: derive the capital accumulation equation: and thus: C (t) + I (t) = F (K (t) ; A (t) L (t)) _K (t) = I (t) K(t) _k(t) = d K (t) = [A (t) L (t)] d dt K (t) K (t) d dt [A (t) L (t)] dt A (t) L (t) [A (t) L (t)] 2 _K (t) _k(t) = k (t) (g + n) A (t) L (t) I (t) K(t) = K (t) (g + n) A (t) L (t) = i (t) ( + g + n) k (t) replace with i (t) = f(k(t)) c (t) so that _ k(t) = f(k(t)) c (t) ( + g + n) k (t) : Therefore conditions (2.26) and (2.29) become: _(t) = _c(t) c(t) (ii) Now, along the BGP _c(t) = 0 so which in turn means that k < k Gold : Tobin = (t) [f 0 (k(t)) g n] = f 0 (k(t)) g n f 0 ( k) = + + g + n > + g + n = k Gold Consider the following model of investment under adjustment costs. Assumptions 2 and 3 in section. are satis ed. There s a single nal good and therefore one can normalize its price to. The problem is that of a RH which must decide how much to consume and how much to invest in the representative rm at each given t. However, installing capital is costly. When there are quadratic investment adjustment costs the problem can be stated as (for simplicity ignore depreciation): max fc(t);i(t)g t=0 s:t. Z 0 e t u(c(t))dt (2.3) _ k(t) = I(t) c(t) + I(t) = f(k(t)) where I(t) is investment and > 0 is a constant: 2 I 2 (t) k(t) 23

25 Note that in this problem, the control variables are c(t) and I(t) (or k t+ in the OGM), while the state variable is k(t). On what follows, the present-value optimization problem is solved and then it is expressed in current value terms since the latter form lends itself to intuitive interpretation. First, we get rid of c(t) by using the second constraint. Notice that we can do this only because the constraint is speci ed with equality. Notice also that after eliminating this constraint, the Lagrangian and the Hamiltonian are obviously the same so that in the language of Appendix B, G () dissapears and L I = H I : Thus, set up the present-value Hamiltonian: H pv = e t u f(k(t)) I(t) The F.O.C. w.r.t. the control is simply obtained: (t) = e t u 0 (c (t)) 2 I 2 (t) k(t) I(t) + k(t) + (t)i(t) (2.32) Next, using the Pontryagin conditions corresponding to the present-value problem (B.8)- (B.9): " _(t) = e t u 0 (c (t)) f 0 (k(t)) + # 2 I(t) (2.33) 2 k(t) _k(t) = I(t) (2.34) Or, using the F.O.C. to solve for I (t) we can rewrite (2.34) as: _k(t) = k (t) e t (t) u 0 (c (t)) And just as in the discrete time problem, the TVC is given by: lim t! (t)k(t): Naturally, with an explicit functional form for u, we could solve for c (t) ; I (t) using the F.O.C. (2.32) and the constraint, replace this in (2.33)-(2.34) to obtain a pair of di erential equations on (t) and k(t) that would characterize the solution to the problem. An interesting avenue to take in this problem is to express the equilibrium conditions in current-value terms. To do so, multiply (2.33) by e t on both sides (the other conditions do not involve (t)) and de ne q(t) = e t (t). Then, since _q(t) = _(t)e t + (t)e t, one has that _(t)e t = _q(t) q(t) and therefore the Pontryagin conditions can be written: " _q(t) q(t) = u 0 (c (t)) f 0 (k(t)) + # 2 I(t) 2 k(t) _k(t) = k (t) q(t) u 0 (c (t)) Finally, using (2.35) we can arrive at: " _q(t) q(t) = u 0 (c (t)) f 0 (k(t)) + # 2 I(t) 2 k(t) 2 3 Z q(t) = e (s t) u 0 (c(s)) 4f 0 (k(t)) + 2 I(t) 5 ds t {z } 2 k(t) {z } (2.35) (2.36) disc. marg. util of output marg. prod. of k - marg. adj cost (2.37) that is, Tobin s q summarizes the informarion of the discounted social bene t of installing an additional unit of capital. 24

26 Chapter 3 Overlapping generations (OLG) 3. OLG in economies with production (Diamond s) Assume that a generation is born in every period of time. Time is discrete indexed t = ; 2::: Each generation lives two periods. Therefore, at each t, there are two generations alive; "young households" and "old households", call them HH types and 2. For completeness, suppose that at period t = a generation is already alive (agent type 0). Therefore generation t is that born in period t: Each new generation is larger than the previous one by a factor of (+n), n 2 (0; ). Therefore: L t = ( + n)l t In its rst year of life, each HH works, saves and consumes. In its second year of life each HH only consumes. Therefore: c ;t! period t consumption of a typical HH from generation t ("youngs") c 2;t+! period t + consumption of a typical HH from generation t ("olds") Production is carried out by the use of capital, technology and "young" agents as the labor force with Harrod-neutral production function: F (K t ; A t L t ) Moreover, production technology satis es assumptions )-3) in section. and utility satis es assumptions 2)-4) in section 2.2. Technology follows an exogenous growth process: A t = ( + g)a t Hence, there are three markets; nal goods, capital goods and labor. Markets are competitive and for simplicity, there is full depreciation ( = ). Acordingly, t = 0, F K = + r t = R t and F L = w t : In period t, each HH from generation t works, receives (technology-enhanced) labor income, consumes and saves : A t w t = c ;t + s t Generation t s savings are rented in the form of capital for production at t + so that capital available is: K t+ = L t s t = S t (3.) Since only "youngs" save on each period, there is no s 2;t ("olds" don t save), to simplify notation s t = s ;t =total savings of the economy at t: 25

27 and are returned with the corresponding rental income. Generation t (i.e. agent type 2 in period t + ) then consumes the proceedings: c 2;t+ = R t+ s t = ( + r t+ )s t = ( + r t+ ) (A t w t c ;t ) Generation t then maximizes its (discounted) lifetime utility from consumption: max U(c) c ;t;c 2;t+ = u(c ;t ) + u(c 2;t+ ) (3.2) s:t: A t w t = c ;t + c 2;t+ R t+ (3.3) alternatively, one the RH solves the unconstrained optimization problem: max c ;t U(c) = u(c ;t ) + u(r t+ (A t w t c ;t )) both yielding the same F.O.C. and Euler equation: u 0 (c ;t ) = u 0 (c 2;t+ )R t+ (3.4) which is analogous to the one found in the OGM. Equations (3.3)-(3.4) are the pair of di erence equations describing the solution to the typical generation t HH problem. Next, note that the Euler equation can be expressed as: u 0 (c ;t ) = (; u 0 (c 2;t+ ); R t+ ) and that u 00 < 0 ) u 0 (c ;t ) = c ;t. Therefore: hence, if one replaces in: u 0 (c ;t ) = u 0 (c 2;t+ ) (; u 0 (c 2;t+ ); R t+ ) c ;t = u 0 (c 2;t+ ) (; u 0 (c 2;t+ ); R t+ ) A t w t = c ;t + s t s t = A t w t u 0 (c 2;t+ ) (; u 0 (c 2;t+ ); R t+ ) s t = (A t w t ; R t+ ) (+) (?) where is called the savings function. Consider a rise in labor income A t w t, then, everything else constant, the properties of u () imply that consumption in both periods would rise, which implies that s t increases; thus the sign t =@A t w t is unambiguous. However, consider a rise in R t+. There are two e ects to consider. First, since the opportunity cost of consumption in t rises, the HH may want to substitute current for future consumption (substitution e ect). Second, a rise in R t+ has an income e ect since each unit saved yields higher return so the HH will want to consume more on both periods. The total e ect is ambiguous as is the sign t =@R t+ : Next, to derive the law of motion of capital come back to (3.): K t+ = L t s t = S t K t+ = L t (A t w t ; R t+ ) = L t (A t w t ; F K ) 26

28 or in per-e ective worker terms (i.e., dividing by A t+ L t+ since we re deriving t + capital): K t+ = L t (A t w t ; F K(t+) ) A t+ L t+ A t+ L t+ (( + g) w t ; f k(t+) ) k t+ = ( + n) k t+ = (( + g) (f f k k t ); f k(t+) ) ( + n) where, in the above derivation one uses the following: ) R t+ t+ = F K(t+) t+ = f k(t+), 2) L : =L t+ = =(+n), 3) A t =A t+ = =(+g) and 4) w t = f l = F F K = f f k by homogeneity of degree one (recall Taylor s theorem):therefore, the key equation for the law of motion of capital in Diamond s model is: ( + n)k t+ = (( + g) (f f k k t ); f k(t+) ) (3.5) at this point one needs to specify some functional form for utility and production since k t+ shows up in both sides of the above equation and cannot be solved for explicitely. 3.. Log-utility and Cobb-Douglas technology Under u(c it ) = log c it and F (K t ; A t L t ) = Kt (A t L t ) ) f(k) = k one can re-write the (generation t HH) problem as: so the Euler eq. can be re-written as: now using the resource constraint: max c ;t;c 2;t+ U(c) = log c ;t + log c 2;t+ s:t: A t w t = c ;t + c 2;t+ R t+ (3.6) = R t+ c ;t c 2;t+ c 2;t+ = c ;t R t+ A t w t = c ;t + c ;tr t+ R t+ c ;t = A tw t + = + k k a policy function for consumption; replacing in the savings function: A t w t = c ;t + s t A t w t s t = A t w t + s t = A t w t + = + k k 27

29 that is, as in the growth models of the sections above, savings are a constant fraction of HH income. Next, using the equation for capital: ( + n)k t+ = = = = k t+ = s t A t+ A t w t A t+ + w t ( + ) ( + g) (kt kt ) ( + ) ( + g) ( ) kt (3.7) ( + ) ( + g)( + n) a policy function for capital accumulation. Equation (3.7) is the key equation of Diamond s model unde log-utility and Cobb-Douglas production Steady state The steady state value for capital can be computed as: k = ( ) ( + ) ( + g)( + n) k = ( ) ( + ) ( + g)( + n) k (3.8) steady state output, is: y = k = ( ) ( + ) ( + g)( + n) To obtain SS consumption rst note that c it is the amount consumed by one typical household of generation i in its rst year of life. Since now the poppulation size is not normalized to as in the growth models of previous sections, in order to nd economy-wide consumption at t one must compute: C t = L t c t + L t c 2t so that total consumption per (e ective) worker in period t: now, using: c t = c t + C t = Y t S t c 2t ( + n) c t = y t s t c = k ( + n) k so that: c = ( ) ( + ) ( + g)( + n) ( + n) ( ) ( + ) ( + g)( + n) 28

30 3..3 Golden rule and dynamic ine ciency As before, the golden rule for capital accumulation will follow: k Gold = arg max k k ( + n) k {z } with F.O.C. as: =c k = + n + n k Gold = (3.9) or, in the general case: f 0 ( k Gold ) = + n R = + n r = n that is, only when the interest rate equals the rate of poppulation growth, capital is at its golden rule level. 2 The possibility of dynamic ine ciency arises by comparing (3.8) with (3.9): ( ) ( + ) ( + g)( + n) k? = k Gold?= + n therefore, if k 6= k Gold the current allocation of resources would be Pareto ine cient and the planner could make every one else by rising/lowering investment (savings) 3..4 The role of Government. Suppose that for some reason the economy nds itself under dynamic ine ciency, i.e., k > k Gold. Then the Government could reallocate resources as follows: spend G t and fund it entirely with (lump-sum) taxes in that same period. Then key equation (3.7) becomes: k t+ = ( ) ( + ) ( + g)( + n) [k t G t ] which unambiguously leads to a lower k: The intuition is as follows: since the taxes are only on t, then generation t is taxed only on its rst year of life. The HH would have to reduce consumption in its rst period but since it wants to smooth consumption (under CRRA) it would have to reduce its savings too, so that consumption in period t falls less than the full amount of taxes. Consequently, savings fall and the economy moves towards k Gold (which maximizes the SS level of consumption). The Government in turn can reallocate the taxes as G t to increase consumption of those hurt and therefore increase economy-wide consumption. Claim 8 In the OLG presented above, the Ricardian equivalence result does not hold in general. 2 Note that if 6= then this condition becomes ( + r ) = + n or r = n: 29

31 To se why, suppose that in period t; Government purchases G t are introduced and funded by issuing one-period bonds b t = G t. In turn, in period t+ the government levies a tax in order to repay for the bonds, so t+ = ( + r t+ )b t : In the OGM the RH would then save whatever the government spends G t, so that it yields reutrn ( + r t+ )G t = t+. Therefore, the consumption plan operates as if the government was funding its purchases via taxes in period t: Hence, the irrelevance of distinguishing between taxes and bonds. On the other hand, in the OLG model, at period t only the young agents care about future taxes; therefore, introducing G t has real e ects on the consumption plan of generation t : 3..5 Social security Consider the problem of social security in the OLG model. "Old" HH receive bene ts b t that are funded by some sort of contriubution d: There are two systems; fully funded social security and pay-as-you-go (PAYGO). Fully funded social security. A HH of generation t contributes d in its rst period of life, this contribution is invested at market rates and then in its second period of life the HH receives ( + R t )d: The HH therefore faces the problem 3 : max u(c t ) + u(c 2t+ ) (3.0) c t;c 2t+ s:t: c ;t + s t + d A t w t (3.) c 2;t+ R t+ (d + s t ) (3.2) Claim 9 Under the above assumptions and de nitions, the HH problem with fully funded social security is equivalent to the simple HH problem (3.2)-(3.3) of section 3. Proof. To see why, rst solve for s t in the constraint (3.2): and next replace in constraint (3.): c 2;t+ = R t+ (d t + s t ) s t = c 2;t+ R t+ c ;t + s t + d = A t w t c2;t+ c ;t + d + d = A t w t R t+ c ;t + c 2;t+ R t+ = A t w t which is exactly the resource constraint (3.3). Hence, the problem is identical to that presented in section 3. The intuition for this result is simple: if contributions are invested at market rates, then they are simply a di erent way of saving. Therefore, the HH only needs to choose how much it wants to save at t, then pay d as contributions and invest the remaining as s t : Note that, depending on the size of d; it may be optimal for the HH to set s t = 0: Also notice that now the amount invested (which matches exactly the amounf of capital available at t + since = ) is s t + d = ( + n)k t+ 3 In this set up it is assumed that each HH of all generations make the same contribution, i.e., the case where d t 6= d t+ is ruled out. This possibility is explored in Acemoglu (2008) and in fact the case where d t is a choice variable is discussed. d 30