Macroeconomics: Diamond’s Overlapping Generations Model PDF

Macroeconomics Diamond’s Overlapping Generations Model Alejandro Cuñat  References These notes are based on D. Romer (2018): Advanced Macroeconomics, 5th edition, McGraw Hill....

Macroeconomics Diamond’s Overlapping Generations Model Alejandro Cuñat  References These notes are based on D. Romer (2018): Advanced Macroeconomics, 5th edition, McGraw Hill. 2  Outline Introduction Firms & households Equilibrium Dynamics The possibility of dynamic inefficiency Pension systems Bubbles Appendix 3  Introduction Very tractable framework to model dynamic behaviour and obtain economic intuition. [Not so amenable for quantitative purposes.] Optimising agents ⇒ The model enables us to think about welfare. Applications: – Bubbles – Fiscal policy – Growth (similar in many respects to the Solow model) – Intergenerational issues (transmission of wealth) – Pension systems 4  Firms All markets are perfectly competitive. ⇒ Zero profits Many identical firms: – produce output Yt = F(Kt, At Lt ), which is used for Ct and It; Pt = 1 ∀t; – hire labour at wage wt per worker; – rent capital at rate Rt per unit of capital; we assume δ = 0. ⇒ Capital owners earn rt = Rt. Yt = F(Kt, Lt ): neoclassical production function; A0 = 1, g = 0; x ≡ X /L Under CRS, if firms maximise profits and markets are competitive, inputs are paid their marginal products and factor payments exhaust all output. FK (Kt, Lt ) = f′(kt ) = rt, FL(Kt, Lt ) = f(kt ) − kt f′(kt ) = wt 5    Households Time is discrete: variables are defined for t = 0,1,2,... For simplicity, the model assumes each individual lives for only 2 periods. Key assumption: existence of turnover in the population Lt individuals are born in t (with zero net financial wealth). Lt = (1 + n)Lt−1, n ≥ 0 constant At t there are Lt individuals in the first period and Lt−1 = Lt /(1 + n) individuals in the second period of their lives. Individuals die at the end of the 2nd period of their lives. 6  Demographics of the OLG model t t+1 t+2 t+3 t+4... Old Cohort t − 1 Young Old Cohort t Young Old Cohort t + 1 Young Old Cohort t + 2 Young Old Cohort t + 3 Young Cohort t + 4... 7  Households When young, each individual born in t supplies one unit of labour. The resulting income is divided between consumption and saving. When old, the individual consumes the saving and any interest she earns. C1t + St = wt C2t ⏟ ⇒ IBC: C1t + = wt 1 + rt+1 C2t+1 = (1 + rt+1)St = (1 + rt+1)(wt − C1t ) The young individual’s wealth is simply wage income: Ω1t = wt. 1 Individual’s utility: Ut = u(C1t ) + βu(C2t+1), β≡ , ρ > − 1. 1+ρ If ρ > 0, individuals are “impatient”: they value one unit of utility from C1t more than one unit of utility from C2t. Perfect foresight 8  Households max u(C1t ) + βu[(1 + rt+1)(wt − C1t )] ⇒ u′(C1t ) = (1 + rt+1)βu′(C2t+1) C1t – C1t = μ1t Ω1t = (1 − st )wt, 1 − st = μ1t, st: saving rate – C2t+1 = (1 + rt+1)st wt C 1−θ − 1 1−θ 1 u(C) = , θ>0 ⇒ 1/μ1t = 1 + (1 + rt+1) θ βθ 1−θ 1 1−θ (1 + ρ) θ (1 + rt+1) θ – st = s(rt+1) = 1 − 1 1−θ = 1 1−θ (1 + ρ) + (1 + rt+1) θ θ (1 + ρ) + (1 + rt+1) θ θ – θ > 1 ⇒ s′(r) < 0; θ < 1 ⇒ s′(r) > 0 C 1−θ − 1 1 – lim = ln C ⇒ s= , s′(r) = 0 θ→1 1 − θ 2+ρ 9       Equilibrium Kt+1 = s(rt+1)wt Lt: amount saved by the young in t; depends on wt and thereby on Kt, the amount saved by the young in t − 1. ⇒ Dynamics Kt+1 1 1 = s(rt+1)wt Lt ⇔ kt+1 = s(rt+1)wt Lt+1 (1 + n)Lt 1+n rt+1 = f′(kt+1), wt = f(kt ) − kt f′(kt ) 1 ⇒ kt+1 = h(kt ) = s[ f′(kt+1)][ f(kt ) − kt f′(kt )] 1+n 1 u(C) = ln C ⇒ s = ⏟ 2+ρ 1 1 ⇒ kt+1 = (1 − α)ktα 1+n 2+ρ f(k) = k α ⇒ f(kt ) − kt f′(kt ) = (1 − α)ktα [Romer (2018) discusses results under general functional forms.] 10       Dynamics kt+1 k* k2 1 1 kt+1 = h(kt ) = (1 − α)ktα 1+n 2+ρ k1 45∘ O k0 k1 k2 k* kt 11  Dynamics On the 45∘-line, kt+1 = kt: – kt+1 = kt = 0: (boring) steady state; – kt+1 = h(kt ) implies kt+1 > kt for 0 < kt < k*; – steady state at the intersection of h(kt ) and 45∘-line: kt+1 = kt = k*; – kt+1 = h(kt ) implies kt+1 < kt for kt > k*. k* is globally stable: provided the economy departs from k0 > 0, it converges to k*. For g > 0, the balanced growth path is like that of the Solow model: s and K /Y are constant; Y/L etc. grow at rate g; Y/(AL) etc. grow at rate 0. The concavity of f(kt ) implies concavity of h(kt ). 12 Convergence  Dynamics The saving of the young is a constant fraction of their income, and their income is a constant fraction of total income. However, the dissaving of the old as a fraction of total income, kt /f(kt ), is not constant over the transition to the steady state: – Due to diminishing returns to capital, kt /f(kt ) is increasing in k. – Since this term enters negatively into saving, total saving as a fraction of output is a decreasing function of k. – Thus, total saving as a fraction of output is above its steady-state value when k < k*, and is below when k > k*. 13  A decrease in ρ kt+1 kt+1 = h(kt ) kt+1 = h(kt ) ρH > ρL 1 1 kt+1 = h(kt ) = (1 − α)ktα 1+n 2+ρ 45∘ O k* k* kt 0 1 14 An increase in productivity  A decrease in ρ Consider a fall in discount rate ρ when the economy is initially on its balanced growth path: – The fall in the discount rate causes the (now more patient) young to save a greater fraction of their labour income. ⇒ h(kt ) shifts up. – The upward shift of h(kt ) raises k*. – k rises monotonically from k* to k*. 0 1 – The change shifts the paths over time of y and k permanently up, but it leads only to temporary increases in the growth rates of these variables. 15  Possibility of dynamic inefficiency 1 1 Under u(C) = ln C and f(k) = k α, k* = (1 − α)(k*)α yields 1+n 2+ρ 1 [1 + n 2 + ρ ] 1 1 1−α α−1 α k* = (1 − α) , f′(k*) = α(k*) = (1 + n)(2 + ρ) 1−α On a balanced-growth path with g = 0, consumption per worker, f(k) − nk, is maximised at kGR, such that f′(kGR) = n. Important result: here there is no reason to rule out the possibility that α n k* > kGR ⇔ f′(k*) < n ⇔ (2 + ρ) <. 1−α 1+n A high n (and a high growth rate of productivity, absent here), a low α and a low ρ make it more likely that the economy be dynamically inefficient. 16     Possibility of dynamic inefficiency Suppose the economy is on its balanced-growth path with k* > kGR ⇔ f(k*) − nk* < f(kGR) − nkGR. Suppose that at t0 more resources are allocated to consumption and fewer to saving so that kt0+1 = kt0+2 = kt0+3 =... = kGR. Under this plan: – At t0, f(k*) + (k* − kGR) − nkGR > f(kGR) − nkGR. – For t > t0, f(kGR) − nkGR > f(k*) − nk*. Consumption can be reallocated between the young and the old each period so as to make every generation better off. ⇒ k* > kGR is Pareto inefficient. Given that markets are competitive and there are no externalities, how can the usual result that competitive equilibria are Pareto efficient fail? 17  Possibility of dynamic inefficiency The standard efficiency result assumes not only perfect competition and no externalities, but also a finite number of agents. Assume the economy is run by a “benevolent social planner” who: – avoids economic agents’ market interactions; – dictates the allocation of consumption over time to each cohort; – seeks to do so in the interest of every cohort. The infinity of generations enables the planner to provide for the C2 of the old in a way: – unavailable to the market; – that can improve on the decentralised allocation. 18  Possibility of dynamic inefficiency Individuals in the market economy must save to consume in old age, even if r is low. The planner needs not have C2 of the old determined by their own saving: – She can take 1 unit of labour income from each young person (born at t) and transfer it to the old (born at t − 1). – Since Lt /Lt−1 = 1 + n, this raises C2 of each old person by 1 + n units. The planner can prevent this from making anyone worse off by having the next generation of young do the same in the following period, and so on. If f′(k*) < n ⇔ k* > kGR, this way of transferring resources between youth and old age is more efficient than saving. 19   Pension systems: pay-as-you-go social security Suppose the government taxes each young individual an amount T and uses the proceeds to pay each old person (1 + n)T. This affects the individual’s budget constraints and thereby her decisions: – C1t + St = wt − T ⏟ C2t n − rt+1 ⇒ C1t + = wt + T – C2t+1 = (1 + rt+1)St + (1 + n)T 1 + rt+1 1 + rt+1 Solving the model with the new budget constraints yields (after a number of steps with the algebra) the following law of motion for kt+1: 1+n [2+ρ ] 1 1 1 + rt+1 + (1 + ρ)(1 + n) kt+1 = (1 − α)ktα − ZtT , Zt = >0 (2 + ρ)(1 + rt+1) h(kt ) shifts down relative to the benchmark case: today’s young save less, since tomorrow’s young finance their retirement. ⇒ k* is reduced. 20  Pension systems: pay-as-you-go social security If the economy was initially dynamically efficient (k* < kGR), a marginal increase in T would: – result in a gain to the old generation that would receive the extra benefits. – reduce k* further below kGR and thus leave future generations worse off, with lower consumption possibilities. If the economy was initially dynamically inefficient (k* > kGR), – the old generation would again gain due to the extra benefits; – the reduction in k* would allow for higher consumption for future generations and would be welfare-improving; – the introduction of the tax would reduce or possibly eliminate the dynamic inefficiency caused by the over-accumulation of capital. 21  Pension systems: fully funded social security Suppose now the government taxes each young person an amount T and uses the proceeds to purchase capital. Individuals born at time t receive (1 + rt+1)T when old. [Assume that T is not greater than the amount of saving each individual would have done in the absence of the tax.] Individual’s budget constraints: – C1t + St = wt − T ⏟ ⇒ C1t + C2t = wt – C2t+1 = (1 + rt+1)(St + T ) 1 + rt+1 The individual’s IBC remains unaffected. ⇒ Her behaviour is not affected by the introduction of social security. 22  Pension systems: fully funded social security Since social security pays the same rate of return as private saving, the government here is simply doing some of the saving for the young. Individuals, indifferent as to who does the saving, offset one-for-one any saving that the government does for them. Hence, there is no effect on the law of motion for kt+1. The balanced-growth-path value of k is the same as it was before the introduction of the fully-funded social security system. 23  Bubbles Bubble: excess of an asset price over what is justified by the asset’s expected future dividends. Under dynamic inefficiency, there are steady states with “bubbles”, which share some similarities with Ponzi schemes or chain letters: – Assets are bought (by today’s young) at a high price in the belief they can be resold at an even higher price (to tomorrow’s young),… – …who will buy them for the same reason, and so on. – This is similar to the scheme whereby the social planner can achieve an allocation superior to that of the decentralised equilibrium. Under dynamic efficiency, a “bubble” cannot exist here in equilibrium: – The young’s income would end up being too small to buy the bubble. – ⇒ The belief assets are always resold at a higher price is not granted. 24  Appendix: Taylor approximations When solving economic models, one often obtains expressions with non- linear functional forms f(xt, yt ) that are hard to manipulate or interpret. In these cases, it is often convenient to linearise f(xt, yt ) by using a Taylor approximation around a point (x, y) for which the value of f(x, y) is known. First-order Taylor approximation of f(xt, yt ) around (x, y): f (xt, yt ) ≈ f (x, y) + fx(x, y)(xt − x) + fy(x, y)(yt − y) Second-order Taylor approximation of f(xt, yt ) around (x, y): f (xt, yt ) ≈ f (x, y) + fx(x, y)(xt − x) + fy(x, y)(yt − y)+ 1 + [fxx(x, y)(xt − x)2 + 2fxy(x, y)(xt − x)(yt − y) + fyy(x, y)(yt − y)2] 2 The closer (xt, yt ) to (x, y), and the higher the Taylor approximation’s order, the more accurate the approximation. 25  Appendix: Convergence 1 1 Under u(C) = ln C and f(k) = k α, kt+1 = h(kt ) = (1 − α)ktα. 1+n 2+ρ 1 [1 + n 2 + ρ ] 1 1 1 1 1−α k* = (1 − α)(k*)α ⇒ k* = (1 − α) 1+n 2+ρ 1 1 Linearising kt+1 = (1 − α)ktα with a 1st-order Taylor expansion 1+n 2+ρ around k*, kt+1 = h(kt ) ≈ h(k*) + h′(k*)(kt − k*): 1 1 α 1 1 kt+1 ≈ (1 − α)(k*) + α (1 − α)(k*)α−1(kt − k*) 1+n 2+ρ 1+n 2+ρ = k* = (k*)1−α Hence, kt+1 ≈ k* + α(kt − k*). Since α ∈ (0,1), k converges smoothly to k*. 26 Back   Appendix: An increase in A kt+1 kt+1 = h(kt ) kt+1 = h(kt ) 1 = AL < AH 1 1 kt+1 = h(kt ) = (1 − α)Aktα 1+n 2+ρ 45∘ O k* k* kt 0 1 27 Back  Appendix: An increase in A Consider a once-and-for-all increase in productivity level A (growth rate g remains at zero) when the economy is initially on its balanced growth path: – f(k) = Ak α ⇒ f(kt ) − kt f′(kt ) = (1 − α)Aktα 1 1 – kt+1 = (1 − α)Aktα 1+n 2+ρ – The increase in A shifts h(kt ) up. – The upward shift of h(kt ) raises k*. – k rises monotonically from k* to k*. 0 1 – The change shifts the paths over time of y and k permanently up, but it leads only to temporary increases in the growth rates of these variables. 28 Back   Food for thought (beyond this course) A.B. Abel, N.G. Mankiw, L. Summers & R. Zeckhauser (1989): “Assessing Dynamic Efficiency: Theory and Evidence,” Review of Economic Studies, 56(1), pp. 1-19 O.J. Blanchard (1985): “Debt, Deficits, and Finite Horizons,” Journal of Political Economy, 93(2), pp. 223-247 A. Cuñat & R. Zymek (2024): “Bilateral Trade Imbalances,” Review of Economic Studies, 91(3), pp. 1537-1583 O. Galor & J. Zeira (1993): “Income Distribution and Macroeconomics,” Review of Economic Studies, 60, pp. 35-43 F. Geerolf (2018): “Reassessing Dynamic Inefficiency,” manuscript A. Martín & J. Ventura (2018): “The Macroeconomics of Rational Bubbles: A User’s Guide,” Annual Review of Economics, 10, pp. 505-539 J. Tirole (1985): “Asset Bubbles and Overlapping Generations,” Econometrica, 53(6), pp. 1071-1100 J. Ventura (2005): “A Global View of Economic Growth,” in P. Aghion & S.N. Durlauf (eds.), Handbook of Economic Growth, vol. 1B 29