Advanced Macroeconomics An Easy Guide Overlapping generations models

CHAPTERS Overlapping generations models The neoclassical growth model ( with its and identical individuals , is very useful for analysing a large number of topics in , as we have seen , and will continue to see , for the remainder of the book . However , there are some issues that require a departure from those assumptions , An obvious example involves those issues related to the understanding of the interaction of individuals who are at different stages in their life cycles . If lives are and not , as in the , individuals are not the same ( or at a minimum are not at the same moment in their lives ) This diversity opens a whole new set of issues , such as that of optimal consumption and investment over the life cycle , and the role of . It also requires a of . Not only because we need to address the issue of how to evaluate welfare when agents have different utility functions , but also because we will need to check if the properties of the prevail . For example , if there are poor instruments to save , yet people need to save for retirement , can it be possible that people accumulate too much capital ?

This richer framework will provide new perspectives for evaluating policy decisions such as , taxation , and discussing the impact of demographic changes . Of course , the analysis becomes more nuanced , but the added difficulty is not an excuse for not tackling the issue , particularly because in many instances the fact that individuals are different is the key aspect that requires attention , To study these very important issues , in the next three chapters we develop the overlapping ( model , the second workhorse framework of modern . We will see that , when bringing in some of these nuances , the implications of the model turn out to be very different from those of the , This framework will also allow us to address many of the most relevant rent policy debates in , including low interest rates , secular stagnation , and topics in and monetary policy . The model The model by assuming two generations young and old . The young save for retirement , and this is the capital stock next period . The dynamics of capital will be by a savings equation of the form ( This savings equation will allow us to trace the evolution of capital over time . How to cite this book chapter , and , A . 2021 . Advanced An Easy Guide . Overlapping generations models , London Press . DOI License .

I OVERLAPPING GENERATIONS MODELS Here we present a discrete time model initially developed by Diamond ( 1965 ) building on earlier work by ( 1958 ) in which individuals live for two periods ( young and old ) The economy lasts forever as new young people enter in every period . We first characterise the competitive equilibrium of the model . We then ask whether the market solution is the same as the allocation that would be chosen by a central planner , focusing on the of the golden rule , which will allow us to discuss the possibility of dynamic ( excessive capital accumulation ) The decentralized equilibrium The market economy is composed of individuals and Individuals live for two periods . They work for , receiving a wager They also lend their savings to , receiving a rental rate . An individual born at time consumes cu in period and , in period , and derives utility ( Note that the subscript refers to consumption when young , and labels consumption when old . Individuals work only in the period of life , supplying one unit of labour and earning a real wage of . They consume part of their income and save the rest to their retirement consumption . The saving of the young in period the capital stock that is used to produce output in period in combination with the labour supplied by the young generation 14 The time structure of the model appears in Figure The number of individuals born at time and working in period tis Population grows at rate so that , Figure Time structure of overlapping generations model Generation Work Retire Tim Generation Work Retire mim Generation Work Retire

OVERLAPPING GENERATIONS MODELS Firms act competitively and use the constant returns technology ( For simplicity , assume that capital fully after use , which is akin to assuming that ( is a net production function , with depreciation already accounted for . As before , output per worker , is given by the production function ( where is the ratio . This production function is assumed to satisfy the conditions . Each , taking the wage rate , and the rental rate on capital , as given . We now examine the problem of individuals and and derive the market equilibrium , Individuals Consider an individual born at time His problem is max ( cU ( subject to CU , where , is the wage received in period tand is the interest rate paid on savings held from period to period . In the second period the individual consumes all his wealth , both interest and principal . Note that this assumes that there is no altruism across generations , in that people do not care about leaving to the coming generations . This is crucial ) The condition for a maximum is ( 85 ) which can be rewritten as ( Cu I This is the equation for the generation born at time Note that this has the very same intuition , in discrete time , as the equation ( Ramsey rule ) we derived in the context of the , Next , using ( and ( to substitute out for , and , and rearranging we get , We can think of this as a saving function 05 , 05 , or . Saving is an increasing function of wage income since the assumption of separability and concavity of the utility function ensures that both goods ( consumption in both periods ) are normal . The effect of an increase in the interest rate is ambiguous , however , because of the standard income and substitution

I OVERLAPPING GENERATIONS MODELS effects with which you are familiar from micro theory . An increase in the interest rate decreases the relative price of consumption , leading individuals to shift consumption from the to the second period , that is , to substitute for consumption . But it also increases the feasible consumption set , making it possible to increase consumption in both periods this is the income effect . The net effect of these substitution and income effects is ambiguous . If the elasticity of substitution between consumption in both periods is greater than one , then in this model the substitution effect dominates and an increase in interest rates leads to an increase in Saving . Firms Firms act competitively , renting capital to the point where the marginal product of capital is equal to its rental rate , and hiring labour to the point where the marginal product of labour is equal to the wage ( where , is the firms ratio . Note that ( is the marginal product of labour , because of constant returns to scale . Goods and factor market equilibrium The goods market equilibrium requires that the demand for goods in each period be equal to supply , or equivalently that investment be equal to saving , The side is net investment the change in the capital stock between and . The side is net saving the term is the saving of the young the second is the of the old . Eliminating , from both sides tells us that capital at time is equal to the saving of the young at time Dividing both sides by , gives us the equation of motion of capital in per capita terms ( The services of labour are supplied the supply of services of capital in period is mined by the savings decision of the young made in period I . Equilibrium in the factor markets obtains when the wage and the rental rate on capital are such that wish to use the available amounts of labour and capital services . The factor market equilibrium conditions are therefore given by equations ( and ( The dynamics of the capital stock The capital accumulation equation ( together with the factor market equilibrium conditions ( and ( implies the dynamic behaviour of the capital stock ,

OVERLAPPING GENERATIONS MODELS 01 , 814 ) This last equation implies a relationship between , and , We will describe this as the savings locus . The properties of the savings locus depend on the derivative ( The numerator of this expression is positive , the fact that an increase in the capital stock in period increases the wage , which increases savings The denominator is of ambiguous sign because the effects of increases in the interest rate on savings are ambiguous If , then the denominator in ( is positive , and then so is , The savings locus in Figure both the dynamic and the behaviour of the economy . The line in Figure 842 is the line along which steady states , at which , must lie Any point at which the savings locus crosses that line is a steady state . We have drawn a locus that crosses the line only once , and hence guarantees that the steady state capital stock both exists and is unique But this is not the only possible configurations The model does not , without further restrictions on the utility production functions , guarantee either existence or uniqueness of a equilibrium with positive capital stock If there exists a unique equilibrium with positive capital stock , will it be stable ?

To answer this , evaluate the derivative around the steady state ( Figure The capital stock OVERLAPPING GENERATIONS MODELS be less than one in absolute value ( Local ) stability requires that ( Again , without further restrictions on the model , the stability condition may or may not be . To obtain results on the comparative dynamic and properties of the model , it is necessary either to specify functional forms for the underlying utility and production functions , or to impose conditions for uniqueness of a positive capital example In this , we analyse the properties of the model under a fairly simple set of log utility ( the limit case where ' and production . This is sometimes referred to as the canonical model . This permits a simple of both dynamics and the steady state . With this assumption on preferences , the saving function is , so that savings is proportional to wage income . Notice that the interest rate cancels out in the case of log utility , but not otherwise . This is a case in which the savings rate will be constant over time ( as in the model ) though , once again , here this is the result of an optimal choice ( as in the version of the AK model that we studied in Chapter ) With technology , the firms rules for optimal behaviour ( and ( become , and ( Using ( and ( in ( yields , which is the new law of motion for capital . Define as usual the steady state as the situation in which . Equation ( implies that the steady state is given by ( so that we have a unique and positive capital stock . This stock is decreasing in ( the rate of discount ) and ( the rate of population growth ) Note the similarities with the and the model .

OVERLAPPING GENERATIONS MODELS Similarly , we can write income as ( or . Again , this level is decreasing in and Will the system ever get to the steady state ?

Local stability requires that be less than one in , absolute value , which in this case implies a ( which is always . Hence , if the initial capital stock is larger than zero it will gradually converge to . Convergence is depicted in Figure . The economy starts out at ko and gradually moves toward the capital stock . The effects of a shock Suppose next that the economy is at the steady state and at some time the discount rate falls from to , where . This shock is unexpected , and will last forever . From ( 821 ) we see that the new per capita capital stock will clearly rise , with In Figure 84 we show the dynamic adjustment toward the new stationary position . The economy starts out at and gradually moves toward Income per capita rises in the transition and in the new steady state . Figure Convergence to the steady state

OVERLAPPING GENERATIONS MODELS Figure Fall in the discount rate new old new The distinctive characteristic ofthe model is that the interest rate may be smaller than the growth rate . In this case , there is a potential gain of reducing the stock of capital . The model can lead to dynamic ' We now ask how the market allocation compares to that which would be chosen by a central planner who an social welfare function . This raises a basic question , that of the social welfare function . When individuals have horizons and are all alike , it is logical to take the social welfare function to be their own utility function . But here the generations that are alive change as time passes , so it is not obvious what the central planner should maximise . The marginal product of capital In any event , as in the model , there is something we can say about here . Notice that , at the steady state , the marginal product of capital is a ( Notice that this interest rate depends on more parameters than in the . The relationship between the discount factor and the interest rate is still there . A higher discount factor implies less savings today and a higher interest rate in equilibrium . But notice that now that the population growth affects the interest rate . Why is this the case ?

The intuition is simple . A higher growth rate of population OVERLAPPING GENERATIONS MODELS decreases the stock of capital thus increasing the marginal product of capital . How does this compare with the golden rule off ( From the above it is clear that if , which in turn implies ( That is , if I is low ( or , alternatively , if is high ) the capital stock in the equilibrium can exceed that of the golden rule . Dynamic Suppose a benevolent planner found that the economy was at the steady state with and , Suppose further that . Is there anything the planner could do to redistribute consumption across that would make at least one generation better off without making any generation worse off ?

Put differently , is this steady state ?

Let resources available for consumption ( of the young and old ) in any period I , be given by , Note next that in any steady state , Note that , by construction , is the that , since , The initial situation is one , so that . Suppose next that , at some point , the planner decides to allocate more to consumption and less to savings in that period , so that next period the capital stock is . Then , in period , resources available for consumption will be ( In every subsequent period , resources available for consumption will be , Clearly , in available resources for consumption will be higher than in the status quo , since . Note next that , this should be obvious , since at time those alive can consume the difference between and ) Therefore , in resources available will also be higher than in the status quo . We conclude that the change increases available resources at all times , The planner can then split them between the two generations alive at any point in time , ensuring that everyone is at least as well off as in the original status quo , with at least one generation being better off . Put differently , the conclusion is that the solution leading to a steady state with a capital stock of is not . Generally , an economy with ( alternatively , one with ) is known as a dynamically economy Why is there dynamic ?

If there is perfect competition with no or other market failures , why is the competitive solution ?

Shouldn the First Welfare Theorem apply here as well ?

The reason why this isn the case is the of agents involved , while the welfare theorems assume a number of agents ?

OVERLAPPING GENERATIONS MODELS An alternative way to build this intuition is that when the interest rate is below the growth rate of the economy , budget constraints are and not , making our economic restrictions meaningless , This gives the planner a way of redistributing income and consumption across generations that is not available to the market . In a market economy , individuals wanting to consume in old age must hold capital , even if the marginal return on capital is low . The planner , by contrast , can allocate resources between the current old and young in any manner they desire . They can take part of the fruit of the labour income of the young , for instance , and transfer it to the old without forcing them to carry so much capital . They can make sure that no generation is worse off by requiring each succeeding generation to do the same ( and remember , there are many of them ) And , if the marginal product of capital is sufficiently low ( lower than , so the capital stock is above the golden rule ) this way of transferring resources between young and old is more than saving , so the planner can do better than the decentralized allocation . Are actual economies dynamically ?

Recall that in the equilibrium we had ( 830 ) so the rental rate is equal to the marginal product of capital . Notice also that the rate of growth of the economy is ( income is constant , and the number of people is growing at the rate ) Therefore , the condition for dynamic is simply that be lower than the rate of growth of the economy , or , taking depreciation into account ( which we have ignored here ) that the rate of interest minus depreciation be lower than the rate of growth of the economy Abel et al . 1989 ) extend the model to a context with uncertainty ( meaning that there is more than one observed interest rate , since you have to adjust for risk ) and show that in this case a condition for dynamic is that net capital income exceeds investment . To understand why , notice that the condition for dynamic is that the marginal product of capital ( exceeds the growth rate of population ( which happens to be the growth rate of the economy So , is the total return to capital and is total investment , so the condition can be tested by comparing the return on capital new investment the net out of . Their evidence from seven industrialised countries suggests that this condition seems to be comfortably in practice . However , a more recent appraisal , by ( 2013 ) suggests that this picture may have actually changed or never been quite as sanguine . He updates the Abel et al . data , and provides a different treatment to mixed income and land rents , With these adjustments , he that , in general , tries are in dynamically positions , though some countries such as and South Korea are in a dynamically state ! And Australia joins the pack more recently ) In other words , it seems at the very least that we can not so promptly dismiss dynamic as a curiosity . Why is this important ?

At this point you may be scratching your head asking why we seem to be spending so much time with the question of dynamic . The reason is that it is actually very relevant for a number of issues . For example , a dynamically economy is one in which policy has more leeway Any debt level will eventually be wiped out by growth , 2019 ) 1197 ) takes this point seriously and argues

OVERLAPPING GENERATIONS MODELS 125 the current , situation , in which safe interest rates are expected to remain below growth rates for a long time , is more the historical norm than the exception . If the future is like the past , this implies that debt , that is the issuance of debt without a later increase in taxes , may well be feasible . Put bluntly , public debt may have no cost . Not surprisingly , during , in response to the pandemic , many countries behaved as if they could tap unlimited resources through debt issuing . Dynamic , if present and expected to remain in the future , would say that was feasible . If , on the contrary , economies are dynamically , the increases in debt will required more taxes down the road . The second issue has to do with the possibility of bubbles , that is , assets with no intrinsic value . By arbitrage , the asset price of a bubble will need to follow a typical pricing equation ( assuming for a constant interest rate . The solution to this equation is , simply replace to check it is a solution ) The price of the asset needs to grow at the rate of interest rate ( you may hold a asset , but you need to get your return ! In an where , this asset can not exist , because it will eventually grow to become larger than the economy . But if this is not the case , and the bubble can exist , We will come back to this later . What are examples of such assets ?

Well , you may have heard about Bitcoins and cryptocurrency . In fact , money itself is really a bubble ! Finally , notice that the model can deliver very low interest rates , So , it is an appropriate setup to explain the current world of low interest rates . We will come back to this in our chapters on and monetary policy . Before this , however , we need to provide a version of the model , to provide continuity with the framework we have been using so far , and because it will be useful later on . Overlapping generations in continuous time The model can be modelled in continuous time through an ingenious mechanism a constant probability of death and the possibility of your assets upon death in exchange for a payment while you . This provides cohorts and behaviour that make the model tractable . Eyen so , the details get a bit eerie . This section is only for the . The trick to model the model in a framework is to include an probability of dying , By the law of large numbers this will also be the death rate in the population . Assume a birth rate 71 Together these two assumptions imply that population grows at the rate 71 This assumption is tractable but captures the spirit of the model not everybody is the same at the same time . As in ( 1985 ) we assume there exist companies that allow agents to insure against the risk of death ( and , therefore , of leaving behind unwanted ) This means that at the time of death all of an individual assets are turned over to the insurance company , which in turn pays a return of on savings to all agents who remain alive . If , is the interest rate , then from the point of view of an individual agent , the return on savings is ,

OVERLAPPING GENERATIONS MODELS We will also assume logarithmic utility which will make the algebra easier , As of time I the agent of the generation born at time I log cMe ( subject to the budget constraint an ( Cus ( 834 ) where an is the stock of assets held by the individual and is labour income . The other constraint is the game condition requiring that if the agent is still alive at time , then lim aMe ' co If we integrate the first constraint forward ( look at our Mathematical ! and use the second constraint , we obtain ( 00 cUe ' I where I , can be thought of as human capital . So the present value of consumption can not exceed available assets , a constraint that will always hold with equality . With log utility the individual equation is our familiar em ( which can be integrated forward to yield , Using this in the budget constraint gives us the individual consumption function If chief ?

if ( 11 11 ) 11 Cu ( al , so that the individual consumes a share of available assets , as is standard under log utility That completes the description of the behaviour of the representative agent in each generation . The next task is to aggregate across generations or cohorts , Let be the size at time of the cohort born at . Denoting the total size of the population alive at time as , we can write the initial size of the cohort born at ( that is , the newcomers to the world at ) as , In addition , the probability that someone born at is still alive at is ) It follows that ,

OVERLAPPING GENERATIONS MODELS Now taking into account deaths and births , we can write the size of the total population alive at time as a function of the size of the population that was alive at some time in the past , It follows that , 842 ) We conclude that the relative size at time of the cohort born at is simply ne ' For any variable the per capita ( or average ) as , Tne . Applying this to individual consumption from ( we have , a . so that per capita consumption is proportional to per capita assets , where , a , Tne ) and , are and human wealth , respectively . Focus on each , beginning with human wealth , which using the expression for in ( can be written as I 00 , ne ( I Now , if labour income is the same for all agents who are alive at some time , we have I no ' ne ( where the expression in curly brackets is the same for all agents . It follows that , Finally , differentiating with respect to time ( with the help of rule ) we arrive at , which is the equation of motion for human capital . It can also we written as , This has our familiar , intuitive asset pricing interpretation . If we think of human capital as an asset , then the is the return on this asset , including the capital gain hi and the dividend , both

OVERLAPPING GENERATIONS MODELS expressed in proportion to the value , of the asset . That has to be equal to the individual discount rate , which appears on the . Turn next to the evolution of wealth . Differentiating al , from ( with respect to ( again using rule ! we have . my nay , ne ( naw nay Lao ( since am is wealth at birth , which is zero for all cohorts , we have ill , 11 150 ( ne ' 11 ( ne ' too ' Notice that while the individual the rate of return is , for the economy as a whole the rate of return is only , since the is a transfer from people who die to those who remain alive , and washes out once we aggregate . Recall , however , that at is assets per capita , so naturally ( the rate of growth of population , must be subtracted from . The consumption function ( and the laws of motion for per capita human and wealth , and ( completely characterise the dynamic evolution of this economy It can be expressed as a system in the following way Differentiate the consumption function with respect to time in order to obtain , hi ) 854 ) Next use the laws of motion for both forms of wealth to obtain by ( 855 ) Write the consumption function in the following way , and use it to substitute out from the by equation ( 71 ) I 51 by ( 857 ) This is a kind of equation . The first term is standard , of course , but the second term is not . That second term comes from the fact that , at any instant , there are newcomers for each person alive , and they dilute assets per capita by nay since at birth they have no assets . This slows down the average rate of consumption growth .

OVERLAPPING GENERATIONS MODELS This equation plus the law of motion for wealth ( are a dimensional system of differential equations in and at . That system , plus an initial condition and a transversality condition for ) fully describes the behaviour of the economy . The closed economy We have not taken a stance on what kind of asset al is . We now do so . In the closed economy we assume that a , and ) is productive capital that yields output according to the function ) where at . In this context dictates that ) so that our two differential equations become ( a ) In steady state we have ( a , Combining the two yields ( a ) ak which pins down the capital stock . For given , the first equation yields consumption . Rewrite the last equation as ak ) So the level of the ( per capita ) capital stock is smaller than the golden rule level that solves ' implying of This is in contrast to the , in which the golden rule applies , and the model with lives , in which may occur . Before examining that issue , consider dynamics , described in Figure . Along the and move together . If the initial condition is at , then tion will start above its level and both , and will gradually fall until reaching the level . If , by contrast , the initial condition is at , then consumption will start below its state level and both ) and ) will rise gradually until reaching the steady state . extension But how come we have no dynamic inefficiency in this model ?

Just switching to continuous time does away with this crucial result ?

Not really . The actual reason is that the model so far is not quite like what we had before , in another aspect there is no retirement ! In contrast to the standard model , individuals have a smooth stream of labour income throughout their lives , and hence do not need to save a great deal in order to provide for consumption later in life .

OVERLAPPING GENERATIONS MODELS Figure Capital accumulation in the continuous time model ' Introducing retirement ( ie a stretch of time with no income , late in life ) is analytically some , but as ( 1985 ) demonstrates , there is an alternative that is easily modelled , has the same , and delivers the same effects assuming labour income declines gradually as long as an individual is alive Let take a look ( 1985 ) assumes that each individual starts out with one unit of tive labour and thereafter his available labour declines at the rate . At time , the labour ings of a person in the cohort born at is given by ) where , is the market wage per unit of labour at time It follows that individual human wealth for a member of the generation is ' Using the same derivation as in the baseline model , we arrive at a equation , ak , which now includes the parameter . The steady state capital stock is now again pinned down by the expression ( am ( which can be rewritten as ak ( So if is sufficiently large , then the per capita capital stock can be larger than the golden rule level , which is the one that solves the equation ak ' This would imply of capital . The intuition is that the declining path of labour income forces people to save more , too

OVERLAPPING GENERATIONS MODELS Figure Capital accumulation with retirement ! much in fact . Again , transfers would have been a more way to pay for ment , but they can not happen in the decentralized equilibrium , in the absence of altruism . In this case , dynamics are given by Figure , with the steady state to the right of the level of capital Revisiting the current account in the open economy We can also revisit the small open economy as a special case of interest . For that , lets go back to the case in which , and consider what happens when the economy is open , and instead of being capital , the asset is a foreign bond , that pays the world interest rate In turn , labour income is now , for simplicity , an exogenous endowment for all moments and for all cohorts . The two key differential equations now become , with values ( which together pin down the levels of consumption and foreign assets . The first equation reveals that in steady state the current account must be balanced , with consumption equal to endowment income plus interest earnings from foreign assets . As the second equation reveals , the stock of foreign assets can be positive or negative , depending on whether is larger or smaller than .

OVERLAPPING GENERATIONS MODELS If , individual consumption is always increasing , agents are accumulating over their lifetimes , and the level of foreign assets is positive . If , individual consumption is and they neither save nor foreign assets are zero , Finally , if , individual consumption is always falling , agents are over their lifetimes , and in the steady state the economy is a net debtor . Equilibrium dynamics are given by Figure 87 , drawn for the case It is easy to show that the system is stable if So the diagram below corresponds to the case Along the , the variables , and , move together until reaching the steady state . In this model the economy does not jump to the steady state ( as the model in Chapter did ) The difference is that new generations are constantly being born without any foreign assets and they need to accumulate . The steady state is reached when the accumulation of the young offsets the of the older generation . Figure The current account in the continuous time model A ( I ) A A ( What have we learned ?

In this chapter we developed the second workhorse model of modern the model , This framework allows us to look at questions in which assuming a single representative agent is not a useful shortcut . We will see how this will enable us to tackle some key policy issues , starting in the next chapter . Moreover , we have already shown how this model yields new insights about capital accumulation , relative to the . For instance , the possibility of dynamic inefficiency that is to say , of accumulation of capital emerges . This is a result of the absence of links , which entail that individuals may need to save too much , as it is the only way to meet their consumption needs as their labor income declines over their life cycle . Notes If the production function makes the function hit the line with a negative slope the model can give origin to cyclical behaviour around the , This cycle can be stable or unstable depending on the slope of the curve .

OVERLAPPING GENERATIONS MODELS The First Welfare Theorem can be extended to deal with an number of agents , but this requires a condition that the total value of resources available to all agents taken together be ( at equilibrium prices ) This is not in the economy , which lasts forever . For those of you who are mathematically inclined , the argument is similar to Grand Hotel paradox . If the argument sounds and esoteric , its because it is so much so that some people apparently think the paradox can be used to prove the existence of God ! see ) Mixed income is that which is registered as accruing to capital , because it comes from the residual income of businesses , but that argues should be better understood , at least partly as , returns to entrepreneurial labour , Land rents , which Abel et al . only had for the , should not be stood as capital in their sense , as land can not be accumulated . Suppose , in addition , that the economy starts with a population . rule ?

Why , of course , you recall it from calculus thats how you differentiate an integral . If you need a refresher , here it is take a function ( to is ( a ( 115 . Intuitively , there are three components of the marginal impact of changing on those of increasing the upper and lower limits of the integral ( which are given by evaluated at those limits ) and that of changing the function at every point between those limits ( which is given by jig ) All the other stuff is what you get from your chain rule , Because individuals discount the future ( this is not the same as the golden rule in the model , which consumption on the steady state . In the golden rule , the capital stock is smaller than that which consumption , precisely because earlier consumption is preferred to later consumption . the derivative of with respect References Abel , A , Summers , I , 1989 ) Assessing dynamic efficiency Theory and evidence , The Review Studies , 56 ( 2019 ) Public debt and low interest rates . American Economic Review , 109 ( I , 1985 ) Debt , and horizons , Journal of Political Economy , 93 ( Diamond , A . 1965 ) National debt in a neoclassical growth model , The American Economic Review , 55 ( 2013 ) Reassessing dynamic . Manuscript , Toulouse School of Economics . A , 1958 ) An exact model of interest with or without the social contrivance ofmoney . Journal Economy , 66 (