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Contents The Overlapping Generations Model and the Pension System Contents 1. Introduction 3 2. The overlapping generations model 4 3. The steady state 7 4. Is the steady state Pareto-optimal? 8 5. Fully funded versus pay-as-you-go pension systems 10 6. Shifting from a pay as-you-go to a fully 13 funded system 7. 16 Conclusion 17 References Please click the advert The next step for top-performing graduates Masters in Management Designed for high-achieving graduates across all disciplines, London Business School’s Masters in Management provides specific and tangible foundations for a successful career in business. This 12-month, full-time programme is a business qualification with impact. In 2010, our MiM employment rate was 95% within 3 months of graduation*; the majority of graduates choosing to work in consulting or financial services. As well as a renowned qualification from a world-class business school, you also gain access to the School’s network of more than 34,000 global alumni – a community that offers support and opportunities throughout your career. For more information visit www.london.edu/mm, email mim@london.edu or give us a call on +44 (0)20 7000 7573. * Figures taken from London Business School’s Masters in Management 2010 employment report Download free ebooks at bookboon.com 2 Introduction The Overlapping Generations Model and the Pension System 1. Introduction This note presents the simplest overlapping generations model. The model is due to Diamond (1965), who built on earlier work by Samuelson (1958). Overlapping generations models capture the fact that individuals do not live forever, but die at some point and thus have finite life-cycles. Overlapping generations models are especially useful for analysing the macro-economic effects of different pension systems. The next section sets up the model. Section 3 solves for the steady state. Section 4 explains why the steady state is not necessarily Pareto-efficient. The model is then used in section 5 to analyse fully funded and pay-as-you-go pension systems. Section 6 shows why a shift from a pay-as-you-go to a fully funded system is never a Pareto-improvement. Section 7 concludes. Download free ebooks at bookboon.com 3 The Overlapping Generations Model and the Pension System The overlapping generations model 2. The overlapping generations model The households Individuals live for two periods. In the beginning of every period, a new generation is born, and at the end of every period, the oldest generation dies. The number of individuals born in period t is Lt . Population grows at rate n such that Lt+1 = Lt (1 + n). The utility of an individual born in period t is: Ut = ln c1,t + 1 ln c2,t+1 1+ρ with ρ > 0 (1) Please click the advert Teach with the Best. Learn with the Best. Agilent offers a wide variety of affordable, industry-leading electronic test equipment as well as knowledge-rich, on-line resources —for professors and students. We have 100’s of comprehensive web-based teaching tools, lab experiments, application notes, brochures, DVDs/ CDs, posters, and more. See what Agilent can do for you. www.agilent.com/find/EDUstudents www.agilent.com/find/EDUeducators © Agilent Technologies, Inc. 2012 u.s. 1-800-829-4444 canada: 1-877-894-4414 Download free ebooks at bookboon.com 4 The Overlapping Generations Model and the Pension System The overlapping generations model c1,t and c2,t+1 are respectively her consumption in period t (when she is in the first period of life, and thus young) and her consumption in period t + 1 (when she is in the second period of life, and thus old). ρ is the subjective discount rate. In the first period of life, each individual supplies one unit of labor, earns labor income, consumes part of it, and saves the rest to finance her second-period retirement consumption. In the second period of life, the individual is retired, does not earn any labor income anymore, and consumes her savings. Her intertemporal budget constraint is therefore given by: c1,t + 1 c2,t+1 = wt 1 + rt+1 (2) where wt is the real wage in period t and rt+1 is the real rate of return on savings in period t + 1. The individual chooses c1,t and c2,t+1 such that her utility (1) is maximized subject to her budget constraint (2). This leads to the following Euler equation: c2,t+1 = 1 + rt+1 c1,t 1+ρ (3) Substituting in the budget constraint (2) leads then to her consumption levels in the two periods of her life: c1,t = c2,t+1 = 1+ρ wt 2+ρ 1 + rt+1 wt 2+ρ (4) (5) Now that we have found how much a young person consumes in period t, we can also compute her saving rate s when she is young:1 s = = wt − c1,t wt 1 2+ρ (6) The firms Firms use a Cobb-Douglas production technology: Yt = Ktα (At Lt )1−α with 0 < α < 1 (7) where Y is aggregate output, K is the aggregate capital stock and L is employment (which is equal to the number of young individuals). A is the technology Download free ebooks at bookboon.com 5 The Overlapping Generations Model and the Pension System The overlapping generations model parameter and grows at the rate of technological progress g. Labor becomes therefore ever more effective. For simplicity, we assume that there is no depreciation of the capital stock. Firms take factor prices as given, and hire labor and capital to maximize their net present value. This leads to the following first-order-conditions: Yt Lt Yt α Kt (1 − α) = wt (8) = rt (9) ... such that their value in the beginning of period t is given by: Vt = Kt (1 + rt ) (10) Every period, the goods market clears, which means that aggregate investment must be equal to aggregate saving. Given that the capital stock does not depreciate, aggregate investment is simply equal to the change in the capital stock. Aggregate saving is the amount saved by the young minus the amount dissaved by the old. Saving by the young in period t is equal to swt Lt . Dissaving by the old in period t is their consumption minus their income. Their consumption is equal to their financial wealth, which is equal to the value of the firms. Their income is the capital income on the shares of the firms. From equation (10) follows then that dissaving by the old is equal to Kt (1 + rt ) − Kt rt = Kt . Equilibrium in the goods markets implies then that Kt+1 − Kt = swt Lt − Kt (11) Taking into account equation (8) leads then to: Kt+1 = s(1 − α)Yt (12) It is now useful to divide both sides of equations (7) and (12) by At Lt , and to rewrite them in terms of effective labor units: yt = ktα kt+1 (1 + g)(1 + n) = s(1 − α)yt (13) (14) where yt = Yt /(At Lt ) and kt = Kt /(At Lt ). Combining both equations leads then to the law of motion of k:2 kt+1 = s(1 − α)ktα (1 + g)(1 + n) (15) Download free ebooks at bookboon.com 6 The Overlapping Generations Model and the Pension System The steady state 3. The steady state Steady state occurs when k remains constant over time. Or, given the law of motion (15), when k∗ = s(1 − α)k∗α (1 + g)(1 + n) (16) where the superscript ∗ denotes that the variable is evaluated in the steady state. We therefore find that the steady state value of k is given by: k ∗  =  = s(1 − α) (1 + g)(1 + n)  1 1−α 1−α (2 + ρ)(1 + g)(1 + n)  1 1−α (17) It is then straightforward to derive the steady state values of the other endogenous variables in the model. Download free ebooks at bookboon.com 7 The Overlapping Generations Model and the Pension System Is the steady state Pareto-optimal? 4. Is the steady state Pareto-optimal? It turns out that the steady state in an overlapping generations model is not necessarily Pareto-optimal: for certain parameter values, it is possible to make all generations better off by altering the consumption and saving decisions which the individuals make. To show this, we first derive the golden rule. The golden rule is defined as the steady state where aggregate consumption is maximized. Because of equilibrium in the goods market, aggregate consumption C must be equal to aggregate production minus aggregate investment: ∗ − Kt∗ ] Ct∗ = Yt∗ − [Kt+1 (18) Or in terms of effective labor units: c∗ = y ∗ − [k∗ (1 + g)(1 + n) − k∗ ] (19) The level of k∗ which maximizes c∗ is therefore such that Please click the advert ∂c∗ ∂k∗   = GR ∂y ∗ ∂k∗  GR − [(1 + g)(1 + n) − 1] = 0 (20) You’re full of energy and ideas. And that’s just what we are looking for. © UBS 2010. All rights reserved.  Looking for a career where your ideas could really make a difference? UBS’s Graduate Programme and internships are a chance for you to experience for yourself what it’s like to be part of a global team that rewards your input and believes in succeeding together. Wherever you are in your academic career, make your future a part of ours by visiting www.ubs.com/graduates. www.ubs.com/graduates Download free ebooks at bookboon.com 8 The Overlapping Generations Model and the Pension System Is the steady state Pareto-optimal? where the subscript GR refers to the golden rule.3 For certain parameter values, it turns out that the economy converges to a steady state where the capital stock is larger than in the golden rule. This occurs when the marginal product of capital is lower than in the golden rule, i.e. when ∂y ∗ /∂k∗ < (∂y ∗ /∂k∗ )GR . From equations (13), (17) and (20), it follows that this is the case when α (1 + g)(1 + n)(2 + ρ) < (1 + g)(1 + n) − 1 1−α (21) which is satisfied when α is small enough. If the aggregate capital stock in steady state is larger than in the golden rule, aggregate consumption could be increased in every period by lowering the capital stock. The extra consumption could then in principle be divided over the young and the old in such a way that in every period all generations are made better off. Such economies are referred to as being dynamically inefficient. It may seem puzzling that an economy where all individuals are left free to make their consumption and saving decisions may turn out to be Pareto-inefficient. The intuition for this is as follows. Consider an economy where the interest rate is extremely low. In such a situation, young people have to be very frugal in order to make sure that they have sufficient retirement income when they are old. But when they are old, the young people of the next generation will face the same problem: as the interest rate is so low, they will have to be very careful not to consume too much in order to have a decent retirement income later on. In such an economy, where an extremely low rate of return on savings makes it very difficult to amass sufficient retirement income, everybody could be made better off by arranging that the young care for the old, and transfer part of their labor income to the retired generation. The transfers which the young have to pay are then more than offset by the fact that they don’t have to save for their own retirement, as they realize that they in turn will be supported during their retirement by the next generation. Download free ebooks at bookboon.com 9 The Overlapping Generations Model and the Pension System Fully funded versus pay-as-you-go pension systems 5. Fully funded versus pay-as-you-go pension systems We now examine how pension systems affect the economy. Let us denote the contribution of a young person in period t by dt , and the benefit received by an old person in period t by bt . The intertemporal budget constraint of an individual of generation t then becomes: c1,t + 1 1 c2,t+1 = wt − dt + bt+1 1 + rt+1 1 + rt+1 (22) A fully funded system In a fully funded pension system, the contributions of the young are invested and returned with interest when they are old: bt+1 = dt (1 + rt+1 ) (23) Substituting in (22) gives then: c1,t + 1 c2,t+1 = wt 1 + rt+1 (24) which is exactly the same intertemporal budget constraint as in the set-up in section 2 without a pension system. Utility maximization yields then the same consumption decisions as before. Note that the amount which a young person saves in period t is now wt − dt − c1,t . This means that the pension contribution dt is exactly offset by lower private saving. Or in other words: young individuals offset through private savings whatever savings the pension system does on their behalf. Download free ebooks at bookboon.com 10