Economic Growth: Lecture Notes on the Neoclassical Growth Model

Chapter 3 The Neoclassical Growth Model • In the Solow model, agents in the economy (or the dictator) follow a simplistic linear rule for con­ sumption and investment. In the Ramsey model, agents (or the dictator) choose consumption and investment optimally so as to maximize their individual utility (or social welfare). 3.1 The Social Planner • In this section, we start the analysis of the neoclassical growth model by considering the optimal plan of a benevolent social planner, who chooses the static and intertemporal allocation of resources in the economy so as to maximize social welfare. We will later show that the allocations that prevail in a decentralized competitive market environment coincide with the allocations dictated by the social planner. • Together with consumption and saving, we also endogenize labor supply. 3.1.1 Preferences • Preferences are defined over streams of consumption and leisure, x = {xt }∞ t=0 , where xt = (ct , zt ), and are represented by a utility function U : X∞ → R, where X is the domain of xt , such that U (x) = U (x0 , x1 , ...) • We say that preferences are recursive if there is a function W : X × R → R (often called the utility aggregator) such that, for all {xt }∞ t=0 , U (x0 , x1 , ...) = W [x0 , U (x1 , x2 , ...)] We can then represent preferences as follows: A consumption-leisure stream {xt }∞ t=0 induces a utility stream {Ut }∞ t=0 according to the recursion Ut = W (xt , Ut+1 ). • We say that preferences are additively separable if there are functions υt : X → R such that U (x) = ∞ � υt (xt ). t=0 We then interpret υt (xt ) as the utility enjoyed in period 0 from consumption in period t + 1. • Throughout our analysis, we will assume that preferences are both recursive and additively separable. In other words, we impose that the utility aggregator W is linear in ut+1 : There is a function U : R → R and a scalar β ∈ R such that W (x, u) = U (x) + βu. Hence, Ut = U (xt ) + βUt+1 . or, equivalently, Ut = ∞ � τ =0 • β is called the discount factor, with β ∈ (0, 1). β τ U (xt+τ ) • U is sometimes called the per-period utility or felicity function. We let z > 0 denote the maxi­ mal amount of time per period. We accordingly let X = R+ × [0, z]. We finally impose that U is neoclassical, by which we mean that it satisfies the following properties: 1. U is continuous and (although not always necessary) twice differentiable. 2. U is strictly increasing and strictly concave: Uc (c, z) > 0 > Ucc (c, z) Uz (c, z) > 0 > Uzz (c, z) 2 < Ucc Uzz Ucz 3. U satisfies the Inada conditions lim Uc = ∞ and lim Uz = ∞ and c→0 z →0 lim Uc = 0. c→∞ lim Uz = 0. z →z 3.1.2 Technology and the Resource Constraint • We abstract from population growth and exogenous technological change. • The time constraint is given by zt + lt ≤ z. We usually normalize z = 1 and thus interpret zt and lt as the fraction of time that is devoted to leisure and production, respectively. • The resource constraint is given by ct + it ≤ yt • Let F (K, L) be a neoclassical technology and let f (κ) = F (κ, 1) be the intensive form of F. Output in the economy is given by yt = F (kt , lt ) = lt f (κt ), where κt = kt lt is the capital-labor ratio. • Capital accumulates according to kt+1 = (1 − δ)kt + it . (Alternatively, interpret l as effective labor and δ as the effective depreciation rate.) • Finally, we impose the following natural non-negativitly constraints: ct ≥ 0, zt ≥ 0, lt ≥ 0, kt ≥ 0. • Combining the above, we can rewrite the resource constraint as ct + kt+1 ≤ F (kt , lt ) + (1 − δ)kt , and the time constraint as zt = 1 − lt , with ct ≥ 0, lt ∈ [0, 1], kt ≥ 0. 3.1.3 The Ramsey Problem • The social planner chooses a plan {ct , lt , kt+1 }∞ t=0 so as to maximize utility subject to the resource constraint of the economy, taking initial k0 as given: max U0 = ∞ � β t U (ct , 1 − lt ) t=0 ct + kt+1 ≤ (1 − δ)kt + F (kt , lt ), ∀t ≥ 0, ct ≥ 0, lt ∈ [0, 1], kt+1 ≥ 0., ∀t ≥ 0, k0 > 0 given. 3.1.4 Optimal Control • Let µt denote the Lagrange multiplier for the resource constraint. The Lagrangian of the social planner’s problem is L0 = ∞ � β t U (ct , 1 − lt ) + t=0 ∞ � µt [(1 − δ)kt + F (kt , lt ) − kt+1 − ct ] t=0 • Let λt ≡ β t µt and define the Hamiltonian as Ht = H(kt , kt+1 , ct , lt , λt ) ≡ U (ct , 1 − lt ) + λt [(1 − δ)kt + F (kt , lt ) − kt+1 − ct ] • We can rewrite the Lagrangian as L0 = ∞ � t β {U (ct , 1 − lt ) + λt [(1 − δ)kt + F (kt , lt ) − kt+1 − ct ]} = t=0 or, in recursive form, Lt = Ht + βLt+1 . ∞ � t=0 β t Ht • Given kt , ct and lt enter only the period t utility and resource constraint; (ct , lt ) thus appears only in Ht . Similarly, kt ,enter only the period t and t + 1 utility and resource constraints; they thus appear only in Ht and Ht+1 . ∞ Lemma 9 If {ct , lt , kt+1 }∞ t=0 is the optimum and {λt }t=0 the associated multipliers, then H � ��t � (ct , lt ) = arg max H(kt , kt+1 , c, l, λt ) c,l taking (kt , kt+1 ) as given, and Ht + β Ht+1 kt+1 � �� � � � = arg max H(kt , k , ct , lt , λt ) + βH(k , kt+2 , ct+1 , lt+1 , λt+1 ) � k taking (kt , kt+2 ) as given. • We henceforth assume an interior solution. As long as kt > 0, interior solution is indeed ensured by the Inada conditions on F and U. • The FOC with respect to ct gives ∂L0 ∂Ht ∂Ht = βt =0⇔ = 0 ⇔ Uc (ct , zt ) = λt ∂ct ∂ct ∂ct The FOC with respect to lt gives ∂L0 ∂Ht ∂Ht = βt =0⇔ = 0 ⇔ Uz (ct , zt ) = λt FL (kt , lt ) ∂lt ∂lt ∂lt Finally, the FOC with respect to kt+1 gives � � ∂Ht ∂Ht+1 ∂Ht+1 ∂L0 t =β +β = 0 ⇔ −λt + β =0⇔ ∂kt+1 ∂kt+1 ∂kt+1 ∂kt+1 λt = β [1 − δ + FK (kt+1 , lt+1 )] λt+1 • Combining the above, we get Uz (ct , zt ) = FL (kt , lt ) Uc (ct , zt ) and Uc (ct , zt ) = 1 − δ + FK (kt+1 , lt+1 ). βUc (ct+1 , zt+1 ) • Both conditions impose equality between marginal rates of substitution and marginal rate of transfor­ mation. The first condition means that the marginal rate of substitution between consumption and leisure equals the marginal product of labor. The second condition means that the marginal rate of intertemporal substitution in consumption equals the marginal capital of capital net of depreciation (plus one). This last condition is called the Euler condition. • The envelope condition for the Pareto problem is ∂(max U0 ) ∂L0 = = λ0 = Uc (c0 , z0 ). ∂k0 ∂k0 More generally, λt = Uc (ct , lt ) represents the marginal utility of capital in period t and will equal the slope of the value function at k = kt in the dynamic-programming representation of the problem. • Suppose for a moment that the horizon was finite, T < ∞. Then, the Lagrangian would be L0 = �T t t=0 β Ht and the Kuhn-Tucker condition with respect to kT +1 would give ∂L ∂HT = βT ≥ 0 and kT +1 ≥ 0, with complementary slackness; ∂kT +1 ∂kT +1 equivalently µT = β T λT ≥ 0 and kT +1 ≥ 0, with β T λT kT +1 = 0. The latter means that either kT +1 = 0, or otherwise it better be that the shadow value of kT +1 is zero. When T = ∞, the terminal condition β T λT kT +1 = 0 is replaced by the transversality condition lim β t λt kt+1 = 0, t→∞ which means that the (discounted) shadow value of capital converges to zero. Equivalently, lim β t Uc (ct , zt )kt+1 = 0. t→∞ Proposition 10 The plan {ct , lt , kt }∞ t=0 is a solution to the social planner’s problem if and only if Uz (ct , zt ) = FL (kt , lt ), Uc (ct , zt ) Uc (ct , zt ) = 1 − δ + FK (kt+1 , lt+1 ), βUc (ct+1 , zt+1 ) kt+1 = F (kt , lt ) + (1 − δ)kt − ct , (3.1) (3.2) (3.3) for all t ≥ 0, and k0 > 0 given, and lim β t Uc (ct , zt )kt+1 = 0. t→∞ (3.4) • Remark: We proved necessity of (3.1) and (3.2) essentially by a perturbation argument, and (3.3) is just the constraint. We did not prove necessity of (3.4), neither sufficiency of this set of conditions. See Acemoglu (2007) or Stokey-Lucas for the complete proof. • Note that the (3.1) can be solved for lt = l(ct , kt ), which we can then substitute into (3.2) and (3.3). We are then left with a system of two difference equations in two variables, namely ct and kt . The intitial condition and the transversality condition then give the boundary conditions for this system. 3.1.5 Dynamic Programing • Consider again the social planner’s problem. For any k > 0, define V (k) ≡ max ∞ � β t U (ct , 1 − lt ) t=0 subject to ct + kt+1 ≤ (1 − δ)kt + F (kt , lt ), ∀t ≥ 0, ct , lt , (1 − lt ), kt+1 ≥ 0, ∀t ≥ 0, k0 = k given. V is called the value function. • The Bellman equation for this problem is V (k) = max U (c, 1 − l) + βV (k � ) s.t. c + k � ≤ (1 − δ)k + F (k, l) k � ≥ 0, c ∈ [0, F (k, 1)], l ∈ [0, 1]. • Let [c(k), l(k), G(k)] = arg max{...}. These are the policy rules. The key policy rule is G, which gives the dynamics of capital. The other rules are static. • Define k by the unique solution to k = (1 − δ)k + F (k, 1) and note that k represents an upper bound on the level of capital that can be sustained in any steady state. Without serious loss of generality, we will henceforth restrict kt ∈ [0, k]. • Let B be the set of continuous and bounded functions v : [0, k] → R and consider the mapping T : B → B defined as follows: T v(k) = max U (c, 1 − l) + βv(k � ) s.t. c + k � ≤ (1 − δ)k + F (k, l) k � ∈ [0, k], c ∈ [0, F (k, 1)], l ∈ [0, 1]. The conditions we have imposed on U and F imply that T is a contraction mapping. It follows that T has a unique fixed point V = T V and this fixed point gives the solution. • The Lagrangian for the DP problem is L = U (c, 1 − l) + βV (k � ) + λ[(1 − δ)k + F (k, l) − k � − c] The FOCs with respect to c, l and k � give ∂L = 0 ⇔ Uc (c, z) = λ ∂c ∂L = 0 ⇔ Uz (c, z) = λFL (k, l) ∂l ∂L = 0 ⇔ λ = βVk (k � ) ∂k � The Envelope condition is Vk (k) = ∂L = λ[1 − δ + FK (k, l)] ∂k • Combining, we conclude Uz (ct , lt ) = Fl (kt , lt ) Uc (ct , lt ) and Uc (ct , lt ) = β [1 − δ + FK (kt+1 , lt+1 )] , Uc (ct+1 , lt+1 ) which are the same conditions we had derived with optimal control. Finally, note that we can state the Euler condition alternatively as Vk (kt ) = β[1 − δ + FK (kt+1 , lt+1 )]. Vk (kt+1 ) 3.2 Decentralized Competitive Equilibrium 3.2.1 Households • Households are indexed by j ∈ [0, 1]. For simplicity, we assume no population growth. • The preferences of household j are given by U0j = ∞ � β t U (cjt , ztj ) t=0 In recursive form, Utj = U (cjt , ztj ) + βUtj+1 . • The time constraint for household j can be written as ztj = 1 − ltj . • The budget constraint of household j is given by cjt + ijt + xjt ≤ ytj = rt ktj + Rt bjt + wt ltj + αj Πt , where rt denotes the rental rate of capital, wt denotes the wage rate, Rt denotes the interest rate on risk-free bonds. Household j accumulates capital according to ktj+1 = (1 − δ)ktj + ijt and bonds according to bjt+1 = bjt + xjt In equilibrium, firm profits are zero, because of CRS. It follows that Πt = 0. Combining the above we can rewrite the household budget as cjt + ktj+1 + bjt+1 ≤ (1 − δ + rt )ktj + (1 + Rt )bjt + wt ltj . • The natural non-negativity constraint ktj+1 ≥ 0 is imposed on capital holdings, but no short-sale constraint is imposed on bond holdings. That is, household can either lend or borrow in risk-free bonds. We only impose the following natural borrowing constraint: −(1 + Rt+1 )bjt+1 ≤ (1 − δ + rt+1 )ktj+1 + ∞ � qτ wτ . q t+1 τ =t+1 where qt ≡ 1 = (1 + Rt )qt+1 . (1 + R0 )(1 + R1 )...(1 + Rt ) This constraint simply requires that the net debt position of the household does not exceed the present value of the labor income he can attain by working all time. • Simple arbitrage between bonds and capital implies that, in any equilibrium, the interest rate on riskless bonds must equal the rental rate of capital net of depreciation: Rt = rt − δ. If Rt < rt − δ, all individuals would like to short-sell bonds, and there would be excess supply of bonds. If Rt > rt − δ, nobody in the economy would invest in capital. • Households are then indifferent between bonds and capital. Letting ajt = bjt + ktj denote total assets, the budget constraint reduces to cjt + ajt+1 ≤ (1 + Rt )atj + wt ltj , and the natural borrowing constraint becomes ajt+1 ≥ at+1 , where at+1 ∞ 1 � ≡− qτ wτ qt τ =t+1 • We assume that {Rt , wt }∞ t=0 satisfies ∞ 1 � qτ wτ < M < ∞, qt τ =t+1 for all t, so that at is bounded away from −∞. Note in particular that if �∞ τ =t+1 qτ wτ at any t, the agent could attain infinite consumption in every period τ ≥ t + 1. was infinite j j j ∞ • Given a price sequence {Rt , wt }∞ t=0 , household j chooses a plan {ct , lt , kt+1 }t=0 so as to maximize lifetime utility subject to its budget constraints max U0j = ∞ � β t U (cjt , 1 − ltj ) t=0 s.t. cjt + ajt+1 ≤ (1 + Rt )ajt + wt ltj cjt ≥ 0, ltj ∈ [0, 1], ajt+1 ≥ at+1 • Let µjt = β t λjt be the Lagrange multiplier for the budget constraint, we can write the Lagrangian as Lj0 = ∞ � ∞ � � � �� j j j j j j j β U (ct , 1 − lt ) + λt (1 + Rt )at + wt lt − at+1 − ct = β t Htj t t=0 where t=0 � � Htj = U (cjt , 1 − ltj ) + λjt (1 + Rt )ajt + wt ltj − ajt+1 − cjt • The FOC with respect to cjt gives j ∂Lj0 t ∂Ht = β =0 ∂cjt ∂cjt ⇔ Uc (cjt , ztj ) = λjt • The FOC with respect to ltj gives j ∂Lj0 t ∂Ht = β =0 ∂ltj ∂ltj • Combining, we get ⇔ Uz (cjt , ztj ) = λjt wt Uz (cjt , ztj ) = wt . Uc (cjt , ztj ) That is, households equate their marginal rate of substitution between consumption and leisure with the (common) wage rate. • The Kuhn-Tucker condition with respect to ajt+1 gives � � j j ∂H ∂Lj0 ∂H t = βt + β jt+1 ≤ 0 ∂ajt+1 ∂ajt+1 ∂at+1 ⇔ λjt ≥ β [1 + Rt ] λjt+1 , with equality whenever ajt+1 > at+1 . That is, the complementary slackness condition is � λjt − β [1 + Rt ] λjt+1 �� � ajt+1 − at+1 = 0 • Finally, if time was finite, the terminal condition would be � � µjT ≥ 0, ajT +1 ≥ aT +1 , µjT ajT +1 − aT +1 = 0, where µjt ≡ β t λjt . Now that time is infinite, the analogous condition is given by � � lim β t λjt ajt+1 − at+1 = 0 t→0 • Using λjt = Uc (cjt , ztj ), we can restate the Euler condition as Uc (cjt , ztj ) ≥ β[1 + Rt ]Uc (cjt+1 , ztj+1 ), with equality whenever ajt+1 > at+1 . That is, as long as the borrowing constraint does not bind, households equate their marginal rate of intertemporal substitution with the (common) return on capital. On the other hand, if the borrowing constraint is binding, the marginal utility of consumption today may exceed the marginal benefit of savings: the household would like to borrow, but it can’t. • For arbitrary borrowing limit at , there is nothing to ensure that the Euler condition must be satisfied with equality. But if at is the natural borrowing limit, and the utility satisfies the Inada condition Uc → ∞ as c → 0, then a simple argument ensures that the borrowing constraint can never bind. Suppose that at+1 = at+1 . Then cjτ = zτj = 0 for all τ ≥ t, implying Uc (cjt+1 , ztj+1 ) = ∞ and therefore necessarily Uc (cjt , ztj ) < β[1 + Rt ]Uc (cjt , ztj ), unless also cjt = 0 which in turn would be optimal only if at = at . But this contradicts the Euler condition, proving that a0 > a0 suffices for at > at for all dates, and hence for the Euler condition to be satisfied with equality. • Moreover, if the borrowing constraint never binds, iterating λjt = β [1 + Rt ] λjt+1 implies β t λjt = qt λj0 . We can therefore rewrite the terminal condition as lim β t λjt ajt+1 = lim β t λjt at+1 = λj0 lim qt at+1 t→∞ t→∞ t→∞ But note that qt at+1 = ∞ � qτ w τ τ =t and �∞ τ =0 qτ wτ < ∞ implies limt→∞ �∞ τ =t qτ wτ = 0. We thus arrive to the more familiar version of the transversality condition: lim β t λjt ajt+1 = 0, t→∞ or, equivalently, lim β t Uc (cjt , ztj )ajt+1 = 0. t→∞ • It is useful to restate the household’s problem in a an Arrow-Debreu fashion: max ∞ � β t U (cjt , ztj ) t=0 s.t. ∞ � qt cjt t=0 + ∞ � qt wt ztj ≤ x t=0 where x ≡ q0 (1 + R0 )a0 + ∞ � qt wt < ∞. t=0 Note that the intertemporal budget constraint is equivalent to the sequence of per-period budgets together with the natural borrowing limit. The FOCs give β t Uc (cjt , ztj ) = µqt β t Uz (cjt , ztj ) = µqt wt , where µ > 0 is Lagrange multiplier associated to the intertermporal budget. You can check that these conditions coincide with the one derived before. �∞ Proposition 11 Suppose the price sequence {Rt , rt , wt }∞ t=0 satisfies Rt = rt − δ for all t, t=0 qt < ∞, and �∞ j j j ∞ t=0 qt wt < ∞, . The plan {ct , lt , at }t=0 solves the individual household’s problem if and only if Uz (cjt , ztj ) = wt , Uc (cjt , ztj ) Uc (cjt , ztj ) = 1 + Rt , βUc (cjt+1 , ztj+1 ) cjt + ajt+1 = (1 + Rt )ajt + wt ltj , ltj + ztj = 1, for all t ≥ 0, with boundary condition aj0 > 0 given and lim β t Uc (cjt , ztj )ajt+1 = 0. t→∞ j j ∞ j j j j Given {ajt }∞ t=1 , an optimal portfolio is any {kt , bt }t=1 such that kt ≥ 0 and bt = at − kt . • Remark: For a more careful discussion on the necessity and sufficiency of these conditions, check Stokey-Lucas. 3.2.2 Firms • There is an arbitrary number Mt of firms in period t, indexed by m ∈ [0, Mt ]. Firms employ labor and rent capital in competitive labor and capital markets, have access to the same neoclassical technology, and produce a homogeneous good that they sell competitively to the households in the economy. • Let Ktm and Lm t denote the amount of capital and labor that firm m employs in period t. Then, the profits of that firm in period t are given by m m m m Πm t = F (Kt , Lt ) − rt Kt − wt Lt . • The firms seeks to maximize profits. The FOCs for an interior solution require FK (Ktm , Lm t ) = rt . FL (Ktm , Lm t ) = wt . • As we showed before in the Solow model, under CRS, an interior solution to the firms’ problem to exist if and only if rt and wt imply the same Ktm /Lm t . This is the case if and only if there is some Xt ∈ (0, ∞) such that rt = f � (Xt ) wt = f (Xt ) − f � (Xt )Xt where f (k) ≡ F (k, 1). Provided so, firm profits are zero, Πm t = 0, and the FOCs reduce to Ktm = Xt Ltm . That is, the FOCs pin down the capital labor ratio for each firm (Ktm /Lm t ), but not the size of the firm (Lm t ). Moreover, because all firms have access to the same technology, they use exactly the same capital-labor ratio. (See our earlier analysis in the Solow model for more details.) 3.2.3 Market Clearing • There is no supply of bonds outside the economy. The bond market thus clears if and only if � Lt 0= 0 bjt dj. • The capital market clears if and only if � Mt 0 Equivalently, � Mt 0 � Ktm dm Ktm dm = kt , where kt = Kt ≡ 0 �1 0 1 = ktj dj ktj dj is the per-capita capital. • The labor market, on the other hand, clears if and only if � Mt 0 Equivalently, � Mt 0 � Lm t dm Lm t dm = lt where lt = Lt ≡ � Lt 0 = 0 Lt ltj dj ltj dj is the per-head labor supply. 3.2.4 General Equilibrium ∞ Definition 12 An equilibrium of the economy is an allocation {(cjt , ltj , ktj+1 , bjt+1 )j∈[0,Lt ] , (Ktm , Lm t )m∈[0,Mt ] }t=0 and a price path {Rt , rt , wt }∞ t=0 such that j j j j (i) Given {Rt , rt , wt }∞ t=0 , the path {ct , lt , kt+1 , bt+1 } maximizes the utility of of household j, for every j. (ii) Given (rt , wt ), the pair (Ktm , Lm t ) maximizes firm profits, for every m and t. (iii) The bond, capital and labor markets clear in every period • Remark: In the above definition we surpassed the distribution of firm profits (or the stock market). • In the Solow model, we had showed that the decentralized market economy and the centralized dictatorial economy were equivalent. A similar result holds in the Ramsey model. The following proposition combines the first and second fundamental welfare theorems, as applied to the Ramsey model. Proposition 13 The set of competitive equilibrium allocations for the market economy coincide with the set of Pareto allocations for the social planner. • Proof. I will sketch the proof assuming that (a) in the market economy, k0j + bj0 is equal across all j; and (b) the social planner is utilitarian. For the more general case, we need to allow for an unequal initial distribution of wealth across agents. The set of competitive equilibrium allocations coincides with the set of Pareto optimal allocations, each different competitive equilibrium allocation corresponding to a different point in the Pareto frontier (equivalently, a different vector of Pareto weights in the objective of the social planner). For a more careful analysis, see Stokey-Lucas or Acemoglu. a. We first consider how the solution to the social planner’s problem can be implemented as a competitive equilibrium. The social planner’s optimal plan is given by {ct , lt , kt }∞ t=0 such that Uz (ct , 1 − lt ) = FL (kt , lt ), ∀t ≥ 0, Uc (ct , 1 − lt ) Uc (ct , 1 − lt ) = β[1 − δ + FK (kt+1 , lt+1 )], ∀t ≥ 0, Uc (ct+1 , 1 − lt+1 ) ct + kt+1 = (1 − δ)kt + F (kt , lt ), ∀t ≥ 0, k0 > 0 given, and lim β t Uc (ct , 1 − lt )kt+1 = 0. t→∞ Let κt ≡ kt /lt and choose the price path {Rt , rt , wt }∞ t=0 given by Rt = rt − δ, rt = FK (kt , lt ) = f � (κt ), wt = FL (kt , lt ) = f (κt ) − f � (κt )κt , Trivially, these prices ensure that the FOCs are satisfied for every household and every firm if we set cjt = ct , ltj = lt and Ktm /Lm t = kt for all j and m. Next, we need to verify that the proposed allocation satisfies the budget constraint of the households. From the resource constraint, ct + kt+1 = F (kt , lt ) + (1 − δ)kt . From CRS and the FOCs for the firms, F (kt , lt ) = rt kt + wt lt . Combining, we get ct + kt+1 = (1 − δ + rt )kt + wt lt . The budget constraint of household j is given by cjt + ktj+1 + bjt+1 = (1 − δ + rt )ktj + (1 + Rt )bjt + wt ltj , For this to be satisfied at the proposed prices with cjt = ct and ltj = lt , it is necessary and sufficient that ktj + bjt = kt for all j, t. Finally, it is trivial to check the bond, capital, and labor markets clear. b. We next consider the converse, how a competitive equilibrium coincides with the Pareto solution. Because agents have the same preferences, face the same prices, and are endowed with identical level of initial wealth, and because the solution to the individual’s problem is essentially unique (where essentially means unique with respect to cjt , ltj , and ajt = ktj + bjt but indeterminate with respect to the portfolio choice between ktj and bjt ), we have that cjt = ct , ltj = lt and ajt = at for all j, t. By the FOCs to the individual’s problem, it follows that {ct , lt , at }∞ t=0 satisfies Uz (ct , 1 − lt ) = wt , ∀t ≥ 0, Uc (ct , 1 − lt ) Uc (ct , 1 − lt ) = β[1 − δ + rt ], ∀t ≥ 0, Uc (ct+1 , 1 − lt+1 ) ct + at+1 = (1 − δ + rt )at + wt lt , ∀t ≥ 0, a0 > 0 given, and lim β t Uc (ct , 1 − lt )at+1 = 0. t→∞ From the market clearing conditions for the capital and bond markets, the aggregate supply of bonds is zero and thus at = kt . Next, by the FOCs for the firms, rt = FK (kt , lt ) wt = FL (kt , lt ) and by CRS rt kt + wt lt = F (kt , lt ) Combining the above with the FOCs and the budget constraints gives ct + kt+1 = F (kt , lt ) + (1 − δ)kt , ∀t ≥ 0, which is simply the resource constraint of the economy. Finally, and limt→∞ β t Uc (ct , 1 − lt )at+1 = 0 with at+1 = kt+1 implies the social planner’s transversality condition, while a0 = k0 gives the initial condition. This concludes the proof that the competitive equilibrium coincides with the social planner’s optimal plan. � • The equivalence to the planner’s problem then gives the following. Corollary 14 (i) An equilibrium exists for any initial distribution of wealth. The allocation of production across firms is indeterminate, and the portfolio choice of each household is also indeterminate, but the equilibrium is unique as regards prices, consumption, labor, and capital. (ii) If initial wealth k0j + bj0 is equal across all agent j, then cjt = ct , ltj = lt and ktj + bjt = kt for all j. The equilibrium is then given by an allocation {ct , lt , kt }∞ t=0 such that, for all t ≥ 0, Uz (ct , 1 − lt ) = FL (kt , lt ), Uc (ct , 1 − lt ) Uc (ct , 1 − lt ) = β[1 − δ + FK (kt+1 , lt+1 )], Uc (ct+1 , 1 − lt+1 ) kt+1 = F (kt , lt ) + (1 − δ)kt − ct , with k0 > 0 given and limt→∞ β t Uc (ct , 1 − lt )kt+1 = 0. Finally, equilibrium prices are given by Rt = R(kt ) ≡ f � (kt ) − δ, rt = r(kt ) ≡ f � (kt ), wt = w(kt ) ≡ f (kt ) − f � (kt )kt . 3.3 Steady State Proposition 15 There exists a unique (positive) steady state (c∗ , l∗ , k ∗ ) > 0. The steady-state values of the capital-labor ratio, the productivity of labor, the output-capital ratio, the consumption-capital ratio, the wage rate, the rental rate of capital, and the interest rate are all independent of the utility function U and are pinned down uniquely by the technology F , the depreciation rate δ, and the discount rate ρ. In particular, the capital-labor ratio κ∗ ≡ k ∗ /l∗ equates the net-of-depreciation MPK with the discount rate, f � (κ∗ ) − δ = ρ, and is a decreasing function of ρ + δ, where ρ ≡ 1/β − 1. Similarly, R∗ = ρ, r∗ = ρ + δ, y∗ = f (κ∗ ), l∗ where f (κ) ≡ F (κ, 1) and φ(κ) ≡ f (κ)/κ. Uz (c∗ , 1 − l∗ ) , Uc (c∗ , 1 − l∗ ) c∗ y∗ = − δ, k∗ k∗ w∗ = FL (κ∗ , 1) = y∗ = φ(κ∗ ), k∗ • Proof. (c∗ , l∗ , k ∗ ) must solve Uz (c∗ , 1 − l∗ ) = FL (k ∗ , l∗ ), Uc (c∗ , 1 − l∗ ) 1 = β[1 − δ + FK (k ∗ , l∗ )], c∗ = F (k ∗ , l∗ ) − δk ∗ . Let κ ≡ k/l denote the capital-labor ratio at the stead state. By CRS, F (k, l) = f (κ)l FK (k, l) = f � (κ) FL (k, l) = f (κ) − f � (κ)κ where f (κ) ≡ F (κ, 1). The Euler condition then reduces to 1 = β[1 − δ + f � (κ∗ )] or equivalently f � (κ∗ ) − δ = ρ where ρ ≡ 1/β − 1. That is, the capital-labor ratio is pinned down uniquely by the equation of the MPK, net of depreciation, with the discount rate. It follows that the gross rental rate of capital and the net interest rate are r∗ = ρ + δ and R∗ = ρ, while the wage rate is w∗ = FL: (κ∗ , 1). Labor productivity (output per work hour) and the output-capital ratio are given by y∗ = f (κ∗ ) and l∗ y∗ = φ(κ∗ ), k∗ where φ(κ) ≡ f (κ)/κ. Finally, by the resource constraint, the consumption-capital ratio is given by c∗ y∗ ∗ = φ(κ ) − δ = ∗ − δ. k∗ k � • The comparative statics are trivial. For example, an increase in β leads to an increase in κ, Y /L, and s = δK/Y . We could thus reinterpret the exogenous differences in saving rates assumed in the Solow model as endogenous differences in saving rates originating in exogenous differences in preferences. • Homework: consider the comparative statics with respect to exogenous productivity or a tax on capital income. 3.4 Transitional Dynamics • Consider the condition that determined labor supply: Uz (ct , 1 − lt ) = FL (kt , lt ). Uc (ct , 1 − lt ) We can solve this for lt as a function of contemporaneous consumption and capital: lt = l(ct , kt ). Substituting then into the Euler condition and the resource constraint, we conclude: Uc (ct , 1 − l(ct , kt )) = β[1 − δ + FK (kt+1 , l(ct+1 , kt+1 ))] Uc (ct , 1 − l(ct , kt )) kt+1 = F (kt , l(ct , kt )) + (1 − δ)kt − ct This is a system of two first-order difference equation in ct and kt . Together with the initial condition (k0 given) and the transversality condition, this system pins down the path of {ct , kt }∞ t=0 . 3.5 Exogenous labor and CEI • Suppose that leisure is not valued, or that the labor supply is exogenously fixed. Either way, let lt = 1 for all t. Suppose further that preferences exhibit constant elasticity of intertemporal substitution: U (c) = c1−1/θ − 1 , 1 − 1/θ where θ > 0 is the elasticity of intertemporal substitution. • The Euler condition then reduces to ct+1 = [β(1 + Rt+1 )]θ , ct or equivalently ln(ct+1 /ct ) ≈ θ(Rt+1 − ρ). Thus, θ controls the sensitivity of consumption growth to the rate of return to savings Proposition 16 The equilibrium path {ct , kt }∞ t=0 is given by the unique solution to ct+1 = {β[1 + f � (kt+ ) − δ]}θ , ct kt+1 = f (kt ) + (1 − δ)kt − ct , for all t, with initial condition k0 > 0 given and terminal condition lim kt = k ∗ , t→∞ where k ∗ is the steady state value of capital, that is, f � (k ∗ ) = ρ + δ. • Remark. That the transversality condition reduces to the requirement that capital converges to the steady state will be argued later, with the help of the phase diagram. It also follows from the following result, which uses information on the policy function. Proposition 17 For any initial k0 < k ∗ (k0 > k ∗ ), the capital stock kt is increasing (respectively, de­ creasing) over time and converges to asymptotically to k ∗ . Similarly, the rate of per-capita consumption growth ct+1 /ct is positive and decreasing (respectively, negative and increasing) over time and converges monotonically to 0. • Proof. The dynamics are described by kt+1 = G(kt ), where G is the policy rule characterizing the planner’s problem. The policy rule is increasing and satisfies k = G(k) if and only if k = 0 or k = k ∗ , k < G(k) < k ∗ for all k ∈ (0, k ∗ ), and k > G(k) > k ∗ for all k > k ∗ . (See Stokey-Lucas for the proof of these properties.) The same argument as in the Solow model then implies that {kt }∞ t=0 is monotonic and converges to k ∗ . The monotonicity and convergence of {ct+1 /ct }∞ t=0 then follows � immediately from the monotonicity and convergence of {kt }∞ t=0 together with the fact that f (k) is decreasing. • We will show this result also graphically in the phase diagram, below. 3.6 Continuous Time and Phase Diagram • Taking logs of the Euler condition and approximating ln β = − ln(1+ ρ) ≈ −ρ and ln[1 −δ +f � (kt )] ≈ f � (kt ) − δ, we can write the Euler condition as ln ct+1 − ln ct ≈ θ[f � (kt+1 ) − δ − ρ]. This approximation is exact when time is continuous. Proposition 18 Consider the continuous-time version of the model. The equilibrium path {ct , kt }t∈[0,∞) is the unique solution to ċt = θ[f � (kt ) − δ − ρ] = θ[Rt − ρ], ct k̇t = f (kt ) − δkt − ct , for all t, with k0 > 0 given and limt→∞ kt = k ∗ , where k ∗ is the steady-state capital. • We can now use the phase diagram to describe the dynamics of the economy. See Figure 3.1. (Figure not shown due to unavailable original.) • The k̇ = 0 locus is given by (c, k) such that k̇ = f (k) − δk − c = 0 ⇔ c = f (k) − δk On the other hand, the ċ = 0 locus is given by (c, k) such that ċ = cθ[f � (k) − δ − ρ] = 0 ⇔ k = k ∗ or c = 0 • The steady state is simply the intersection of the two loci: ċ = k̇ = 0 ⇔ {(c, k) = (c∗ , k ∗ ) or (c, k) = (0, 0)} where k ∗ ≡ (f � )−1 (ρ + δ) and c∗ ≡ f (k ∗ ) − δk ∗ . • We henceforth ignore the (c, k) = (0, 0) steady state and the c = 0 part of the ċ = 0 locus. • The two loci partition the (c, k) space in four regions. We now examine what is the direction of change in c and k in each of these four regions. • Consider first the direction of ċ. If 0 < k < k ∗ [resp., k > k ∗ ], then and only then ċ > 0 [resp., ċ < 0]. That is, c increases [resp., decreases] with time whenever (c, k) lies the left [resp., right] of the ċ = 0 locus. The direction of ċ is represented by the vertical arrows in Figure 3.1. (Figure not shown due to unavailable original.) ˙ If c < f (k) − δk [resp., c > f (k) − δk], then and only then k̇ > 0 • Consider next the direction of k. [resp., k̇ < 0]. That is, k increases [resp., decreases] with time whenever (c, k) lies below [resp., above] the k̇ = 0 locus. The direction of k̇ is represented by the horizontal arrows in Figure 3.1. (Figure not shown due to unavailable original.) • We can now draw the time path of {kt , ct } starting from any arbitrary (k0 , c0 ), as in Figure 3.1. (Figure not shown due to unavailable original.) Note that there are only two such paths that go through the steady state. The one with positive slope represents the stable manifold or saddle path. The other corresponds to the unstable manifold. • The equilibrium path of the economy for any initial k0 is given by the stable manifold. That is, for any given k0 , the equilibrium c0 is the one that puts the economy on the saddle path. • To understand why the saddle path is the optimal path when the horizon is infinite, note the following: – Any c0 that puts the economy above the saddle path leads to zero capital and zero consumption in finite time, thus violating the Euler condition at that time. Of course, if the horizon was finite, such a path would have been the equilibrium path. But with infinite horizon it is better to consume less and invest more in period 0, so as to never be forced to consume zero at finite time. – On the other hand, any c0 that puts the economy below the saddle path leads to so much capital accumulation in the limit that the transversality condition is violated. Actually, in finite time the economy has crossed the golden-rule and will henceforth become dynamically inefficient. Once the economy reaches kgold , where f � (kgold ) − δ = 0, continuing on the path is dominated by an alternative feasible path, namely that of stopping investing in new capital and instead consuming c = f (kgold ) − δkgold thereafter. In other words, the economy is wasting too much resources in investment and it would better increase consumption. • Let the function c(k) represent the saddle path. In terms of dynamic programming, c(k) is simply the optimal policy rule for consumption given capital k. Equivalently, the optimal policy rule for capital accumulation is given by k̇ = f (k) − δk − c(k), with the discrete-time analogue being kt+1 = G(kt ) ≡ f (kt ) + (1 − δ)kt − c(kt ). • Finally, note that, no matter the form of U (c), you can write the dynamics in terms of k and λ: λ̇t = f � (kt ) − δ − ρ λt k̇t = f (kt ) − δkt − c(λt ), where c(λ) solves Uc (c) = λ, that is, c(λ) ≡ Uc−1 (λ). Note that Ucc < 0 implies c� (λ) < 0. As an exercise, draw the phase diagram and analyze the dynamics in terms of k and λ. 3.7 Comparative Statics and Impulse Responses 3.7.1 Additive Endowment • Suppose that each household receives an endowment e > 0 from God, so that its budget becomes j = wt + rt ktj + (1 − δ)ktj + e cjt + kt+1 Adding up the budget across households gives the new resource constraint of the economy kt+1 − kt = f (kt ) − δkt − ct + e On the other hand, optimal consumption growth is given again by ct+1 θ = {β[1 + f � (kt+1 ) − δ]} ct • Turning to continuous time, we conclude that the phase diagram becomes ċt = θ[f � (kt ) − δ − ρ], ct k̇t = f (kt ) − δkt − ct + e. • In the steady state, k ∗ is independent of e and c∗ moves one to one with e. • Consider a permanent increase in e by Δe. This leads to a parallel shift in the k̇ = 0 locus, but no change in the ċ = 0 locus. If the economy was initially at the steady state, then k stays constant and c simply jumps by exactly e. On the other hand, if the economy was below the steady state, c will initially increase but by less that e, so that both the level and the rate of consumption growth will increase along the transition. See Figure 3.2. c ehigh }∆e ∆e elow { Figure 3.2 Figure by MIT OCW. k 3.7.2 Taxation and Redistribution • Suppose that the government taxes labor and capital income at a flat tax rate τ ∈ (0, 1). The government then redistributes the proceeds from this tax uniformly across households. Let Tt be the transfer made in period t. • The household budget is cjt + ktj+1 = (1 − τ )(wt + rt ktj ) + (1 − δ)ktj + Tt , implying Uc (cjt ) = β[1 + (1 − τ )rt+1 − δ]. Uc (cjt+1 ) That is, the tax rate decreases the private return to investment. Combining with rt = f � (kt ) we infer ct+1 θ = {β[1 + (1 − τ )f � (kt+1 ) − δ]} . ct • Adding up the budgets across household gives ct + kt+1 = (1 − τ )f (kt+1 ) + (1 − δ)kt + Tt The government budget on the other hand is � Tt = τ j (wt + rt ktj ) = τ f (kt ) Combining we get the resource constraint of the economy: kt+1 − kt = f (kt ) − δkt − ct Observe that, of course, the tax scheme does not appear in the resource constraint of the economy, for it is only redistributive and does not absorb resources. • We conclude that the phase diagram becomes ċt = θ[(1 − τ )f � (kt ) − δ − ρ], ct k̇t = f (kt ) − δkt − ct . • In the steady state, k and c are decreasing functions of τ. A. Unanticipated Permanent Tax Cut ∗ ∗ • Consider an unanticipated permanent tax cut that is enacted immediately. The k̇ = 0 locus does not change, but the ċ = 0 locus shifts right. The saddle path thus shifts right. See Figure 3.3. • A permanent tax cut leads to an immediate negative jump in consumption and an immediate positive jump in investment. Capital slowly increases and converges to a higher k ∗ . Consumption initially is lower, but increases over time, so soon it recovers and eventually converges to a higher c∗ . c Thigh Tlow Figure by MIT OCW. Figure 3.3 k B. Anticipated Permanent Tax Cut • Consider a permanent tax cut that is (credibly) announced at date 0 to be enacted at some date � t > 0. The difference from the previous exercise is that ċ = 0 locus now does not change immediately. t and shifts right only for t > � t. Therefore, the dynamics of c and k will It remains the same for t < � be dictated by the “old” phase diagram (the one corresponding to high τ ) for t < � t and by the “new” t, phase diagram (the one corresponding to low τ ) for t > � • At t = � t and on, the economy must follow the saddle path corresponding to the new low τ, which t, the economy must follow a path will eventually take the economy to the new steady state. For t < � dictated by the old dynamics, but at t = � t the economy must exactly reach the new saddle path. If t, which would violate the that were not the case, the consumption path would have to jump at date � Euler condition (and thus be suboptimal). Therefore, the equilibrium c0 is such that, if the economy follows a path dictated by the old dynamics, it will reach the new saddle path exactly at t = � t. See Figure 3.4. Tlow Thigh Figure 3.4 Figure by MIT OCW. • Following the announcement, consumption jumps down and continues to fall as long as the tax cut is not initiated. The economy is building up capital in anticipation of the tax cut. As soon as the tax cut is enacted, capital continues to increase, but consumption also starts to increase. The economy then slowly converges to the new higher steady state. 3.7.3 Productivity Shocks: A prelude to RBC • We now consider the effect of a shock in total factor productivity (TFP). The reaction of the economy in our deterministic framework is similar to the impulse responses we get in a stochastic Real Business Cycle (RBC) model. Note, however, that here we consider the case that labor supply is exogenously fixed. The reaction of the economy will be somewhat different with endogenous labor supply, whether we are in the deterministic or the stochastic case. • Let output be given by yt = At f (kt ) where At denotes TFP. Note that rt = At f � (kt ) wt = At [f (kt ) − f � (kt )kt ] so that both the return to capital and the wage rate are proportional to TFP. • We can then write the dynamics as ċt = θ[At f � (kt ) − δ − ρ], ct k̇t = At f (kt ) − δkt − ct . Note that TFP At affects both the production possibilities frontier of the economy (the resource constrain) and the incentives to accumulate capital (the Euler condition). • In the steady state, both k ∗ and c∗ are increasing in A. A. Unanticipated Permanent Productivity Shock • The k̇ = 0 locus shifts up and the ċ = 0 locus shifts right, permanently. • c0 may either increase or fall, depending on whether wealth or substitution effect dominates. Along the transition, both c and k are increasing towards the new higher steady state. See Figure 3.5 for the dynamics. c low EIS high A high EIS low A k Figure 3.5 (A) Figure by MIT OCW. c t k,y y k t L = k + Sk i t Figure 3.5 (B) Figure by MIT OCW. B. Unanticipated Transitory Productivity Shock • The k̇ = 0 locus shifts up and the ċ = 0 locus shifts right, but only for t ∈ [0, � t] for some finite � t. • Again, c0 may either increase or fall, depending on whether wealth or substitution effects dominates. I consider the case that c0 increases. A typical transition is depicted in Figure 3.6. c high A low A low A high A Figure 3.6 (A) Figure by MIT OCW. k c c t k,y y k t c i Figure 3.6 (B) t Figure by MIT OCW. 3.7.4 Government Spending • We now introduce a government that collects taxes in order to finance some exogenous level of government spending. A. Lump Sum Taxation • Suppose the government finances its expenditure with lump-sum taxes. The household budget is cjt + ktj+1 = wt + rt ktj + (1 − δ)ktj − Tt , implying that the Euler condition remains Uc (cjt ) = β[1 + rt+1 − δ] = β[1 + f � (kt+1 ) − δ] Uc (cjt+1 ) That is, taxes do not affect the savings choice. • The government budget is Tt = gt , where gt denotes government spending. • The resource constraint of the economy becomes ct + gt + kt+1 = f (kt ) + (1 − δ)kt Note that gt absorbs resources from the economy. • We conclude ċt = θ[f � (kt ) − δ − ρ], ct k̇t = f (kt ) − δkt − ct − gt • In the steady state, k ∗ is independent of g and c∗ moves one-to-one with −g. Along the transition, a permanent increase in g both decreases c and slows down capital accumulation. See Figure 3.7. (Figure not shown due to unavailable original.) • Clearly, the effect of government spending financed with lump-sum taxes is isomorphic to a negative endowment shock. B. Distortionary Taxation • Suppose the government finances its expenditure with distortionary income taxation. The household budget is cjt + ktj+1 = (1 − τ )(wt + rt ktj ) + (1 − δ)ktj , implying Uc (cjt ) = β[1 + (1 − τ )rt+1 − δ] = β[1 + (1 − τ )f � (kt+1 ) − δ]. Uc (cjt+1 ) That is, taxes now distort the savings choice. • The government budget is gt = τ f (kt ) and the resource constraint of the economy is ct + gt + kt+1 = f (kt ) + (1 − δ)kt . • We conclude ċt = θ[(1 − τ )f � (kt ) − δ − ρ], ct k̇t = (1 − τ )f (kt ) − δkt − ct . • In the steady state, k ∗ is a decreasing function of g (equivalently, τ ) and c∗ decreases more than one-to-one with g. Along the transition, a permanent increase in g (and τ ) drastically slows down capital accumulation. See Figure 3.7. (Figure not shown due to unavailable original.) • Clearly, the effect of government spending financed with distortionary taxes is isomorphic to a neg­ ative TFP shock. 3.8 Beyond Growth 3.8.1 The Phase Diagram with Endogenous Labor Supply • Suppose separable utility, U (c, z) = u(c) + v(z), with CEIS for u, and let l(k, c) be the solution to v � (1 − l) = FL (k, l) u� (c) Note that l increases with k, but less than one-to-one (or otherwise FL would fall). This reflects the substitution effect. On the other hand, l falls with c, reflecting the wealth effect. • Substitute back into the dynamic system for k and c, assuming CEIS preferences: ċt = θ[f � (kt /l(kt , ct )) − δ − ρ], ct k̇t = f (kt , l(kt , ct )) − δkt − ct , which gives a system in kt and ct alone. • Draw suggestive phase diagram. See Figure 3.8. (Figure not shown due to unavailable original.) • Note that the ċ is now negatively sloped, not vertical as in the model with exogenously fixed labor. This reflects the wealth effect on labor supply. Lower c corresponds to lower effective wealth, which results to higher labor supply for any given k (that is, for any given wage). 3.8.2 Impulse Responses Revisited • Note that the endogeneity of labor supply makes the Euler condition (the ċ locus) sensitive to wealth effects, but also mitigates the impact of wealth effects on the resource constraint (the k̇ locus). • Reconsider the impulse responces of the economy to shocks in productivity or governement spending. • Government spending.... If financed with lump sum taxes, an increase in g has a negative wealth effect, which increases labor supply. This in turn leads an increase in the MPK and stimulates more investment. At the new steady state the capital-labor ratio remains the same, as it is simply the one that equates the MPK with the discount rate, but both employment and the stock of capital go up... • Note that the above is the supply-side effect of government spending. Contrast this with the demandside effect in Keynesian models (e.g., IS-LM). • Productivity shocks.... 3.8.3 The RBC Propagation Mechanism, and Beyond • Just as we can use the model to “explain” the variation of income and productivity levels in the cross-section of countries (i.e., do the Mankiw-Romer-Weil exercise), we can also use the model to “explain” the variation of income, productivity, investment and employment in the time-series of any given country. Hence, the RBC paradigm. • The heart of the RBC propagation mechanism is the interaction of consumption smoothing and deminishing returns to capital accumulation. Explain.... • This mechanism generates endogenous persistence and amplification. Explain... • Enogenous persistence is indeed the other face of conditional convergence. But just as the model fails to generate a substantially low rate of conditional convergence, it also fails to generate either substantial persistence or substantial amplification. For the model to match the data, we then need to assume that exogenous productivity (the Solow residual) is itself very volatile and persistent. But then we partly answer and partly peg the question. • Hence the search for other endogenous propagation mechanisms. • Discuss Keynesian models and monopolistic competition... Discuss the potential role financial mar­ kets...