Advanced Macroeconomics An Easy Guide The neoclassical growth model

CHAPTERS The neoclassical growth model The VVe will solve the optimal savings problem underpinning the Neoclassical Growth Model , and in the process introduce the tools of dynamic we will use throughout the book . VVe will also encounter , for the first time , the most important equation in the equation . We have seen the lessons and shortcomings of the basic model . One of its main assumptions , as you recall , was that the savings rate was constant . In fact , there was no involved in that model , and welfare statements are hard to make in that context . This is , however , a very rudimentary assumption for an able policy maker who is in possession of the tools of dynamic . Thus we tackle here the challenge of setting up an optimal program where savings is chosen to maximise welfare . As it turns out , British philosopher and mathematician Frank Ramsey , in one of the two seminal contributions he provided to economics before dying at the age of 26 , solved this problem in 1928 ( Ramsey ( 1928 ) The trouble is , he was so ahead of his time that economists would only catch up in the 19605 , when David Cass and independently revived Ramsey contribution ?

That is why this model is often referred to either as the Ramsey model or the model . It has since become ubiquitous and , under the grand moniker of Neoclassical Growth Model ( it is the foremost example of the type of dynamic general equilibrium model upon which the entire of modern is built . To make the problem manageable , we will assume that there is one representative household , all of whose members are both consumer and producer , living in a closed economy ( we will lift this tion in the next chapter ) There is one good and no government . Each consumer in the representative household lives forever , and population growth is as before . All quantities in letters are per capita . Finally , we will look at the problem as solved by a benevolent central planner who the welfare of that representative household , and evaluates the utility of future consumption at a discounted rate . At this point , it is worth stopping and thinking about the models assumptions . By now you are already used to outrageously unrealistic assumptions , but this may be a little too much . People How to cite this book chapter , and , A . 2021 . Advanced An Easy Guide . The neoclassical growth model , London Press . DOI License .

24 THE NEOCLASSICAL GROWTH MODEL obviously do not live forever , they are not identical , and what this business of a benevolent central planner ?

Who are they ?

Why would they discount future consumption ?

Let us see why we use these shortcuts . We will look at the central planner problem , as opposed to the equilibrium , because it is easier and gets us directly to an allocation . We will show that , under certain conditions , it provides the same results as the equilibrium . This is due to the welfare theorems , which you have seen when studying , but which we should perhaps restate here a . A competitive equilibrium is Optimal . All Optimal allocations can be as a competitive equilibrium under some convexity assumptions . Convexity of production sets means that we can not have increasing returns to scale . If we do , we need to depart from competitive markets . There only one household ?

Certainly this is not very realistic , but it is okay if we think that typically people react similarly ( not necessarily identically ) to the parameters of the model . do people respond similarly to an increase in the interest rate ?

If you think they do , then the assumption is okay . Do all the people have the same utility function ?

Are they equal in all senses ?

Again , as above , not really . But , we believe they roughly respond similarly to basic . In addition , as shown by and Ventura ( 2000 ) one can incorporate a lot of sources of heterogeneity ( namely , individuals can have different tastes , skills , initial wealth ) and still end up with a household representation , as long as that heterogeneity has a certain structure . The assumption also means that we are , for the most part , ignoring distributional concerns , but that paper also shows that a wide range of distributional are compatible with that representation . We will also raise some points about inequality as we go along ) Do they live ?

Certainly not , but it does look like we have some links . 1974 ) suggests an individual who cares about the utility of their child ( 43 , If that is the case , substituting gives an utility of the sort we have posited . And people do think about the future . Why do we discount future utility ?

To some extent it is a revealed preference argument interest rates are positive and this only makes sense if people value more today consumption than tomorrow , which is what we refer to when we speak of discounting the future . On this you may also Want to check Caplin and ( 2004 ) who argue that a utility such as that in ( imposes a sort of tyranny of the present past utility carries no weight , whereas future utility is discounted . But does this make sense from a planner point of view ?

Would this make sense from the perspective of tomorrow ?

In fact , Ramsey argued that it was unethical for a central planner to discount future Having said that , let go solve the problem . The consumer problem The utility function is I ( where denotes consumption per capita and ( is the rate of time Assume ( and conditions are .

THE NEOCLASSICAL GROWTH MODEL 25 The resource constraint The resource constraint of the economy is ' with all variables as in the previous chapter . Notice that for simplicity we assume there is no depreciation . In particular , is a neoclassical production function hence neoclassical growth model . You can think of household production household members own the capital and they work for themselves in producing output . Each member of the household supplies one unit per unit oftime . This resource constraint is what makes the problem truly dynamic . The capital stock in the future depends on the choices that are made in the present . As such , the capital stock constitutes what we call the state variable in our problem it describes the state of our dynamic system at any given point in time . The resource constraint is what we call the equation of motion it the evolution of the state variable over time . The other key variable , consumption , is what we call the control variable it is the one variable that we can directly choose . Note that the control variable is jumpy we can choose whatever ( feasible ) value for it at any given moment , so it can vary discontinuously . However , the state variable is sticky we can not change it discontinuously , but only in ways that are consistent with the equation of motion . Given the assumption of constant returns to scale , we can express this constraint in per capita terms , which is more convenient . Dividing ( through by we get ( I where ( has the usual properties . Recall . Combining the last two equations yields , which we can think of as the relevant budget constraint . This is the shape of the equation of motion of our dynamic problem , describing how the variable responsible for the dynamic nature of the problem in this case the per capita capital stock , evolves over time . Solution to consumer problem The household problem is to maximise ( subject to ( for given ko . Ifyou look at our appendix , you will learn how to solve this , but it is instructive to walk through the steps here , as they have intuitive interpretations . You will need to set up the ( current value ) for the problem , as follows ( Recall that is the control variable ( jumpy ) and is the state variable ( sticky ) but the brings to the forefront another variable , the variable . It is the multiplier associated with the budget constraint , analogously to the of static .

26 THE NEOCLASSICAL GROWTH MODEL Just like its cousin , the variable has an intuitive economic interpretation it is the marginal value as of time ( ie . the current value ) of an additional unit of the state variable ( capital , in this case ) It is a ( shadow ) price , which is also jumpy conditions ( are ) What do these conditions mean ?

First , should be familiar from static differentiate with respect to the control variable , and set that equal to zero . It makes sure that , at any given point in time , the consumer is making the optimal decision otherwise , she could obviously do better . The other two are the ones that bring the dynamic aspects of the problem to the forefront . Equation ( is known as the transversality condition ( It means , intuitively , that the consumer wants to set the optimal path for consumption such that , in the end of times ( at infinity , in this case ) they are left with no capital . As long as capital has a positive value as given by , otherwise they don really care ) If that weren the case , I would be dying with valuable capital , which I could have used to consume a little more over my lifetime . Equation ( is the with respect to the state variable , which essentially makes sure that at any given point in time the consumer is leaving the optimal amount of capital for the future , But how so ?

As it stands , it has been obtained mechanically However , it is much nicer when we derive it purely from economic intuition . Note that we can rewrite it as follows ' I ( This is nothing but an arbitrage equation for a typical asset price , where in this case the asset is the capital stock of the economy Such arbitrage equations state that the opportunity cost of holding the asset ( in this case ) equals its rate of return , which comprises the dividend yield ( plus whatever capital gain you may get from holding the asset ( If the opportunity cost were higher ( resp . lower ) you would not be in an optimal position you should hold less ( resp , more ) of the asset . We will come back to this intuition over and over again . The balanced growth path and the equation We are ultimately interested in the dynamic behaviour of our control and state variables , and , How can we turn our into a description of that behaviour ( preferably one that we can represent graphically ) We start by taking ( and differentiating both sides with respect to time ( Divide this by ( and rearrange ( it .

THE NEOCLASSICAL GROWTH MODEL 27 Next , as the elasticity of substitution in consumption . Then , becomes a , A ) Finally , using ( in ( we obtain This dynamic condition is known as the Ramsey rule ( or rule ) and in a more general context it is referred to as the equation . It may well be the most important equation in all of it encapsulates the essence of the solution to any problem that trades off today versus But what does it mean intuitively ?

Think about it in these terms if the consumer postpones the enjoyment of one unit of consumption to the next instant , it will be incorporated into the capital stock , and thus yield an extra ( However , this will be worth less , by a factor of They will only consume more in the next instant ( if the former compensates for the latter , as mediated by their proclivity to switch consumption over time , which is captured by the elasticity of substitution , Any dynamic problem we will see from now on involves some variation upon this general theme the optimal growth rate trades off the rate of return of postponing consumption ( investment ) against the discount rate , Mathematically speaking , equations ( and ( constitute a system of two differential equations in two unknowns , These plus the initial condition for capital and the fully characterise the dynamics of the economy once we have ) and , we can easily solve for any remaining variables of interest . To make further progress , let us characterise the of this economy Setting ( equal to zero yields ( which obviously is a function in , space . The dynamics of capital can be understood with reference to this function ( Figure ) for any given level of capital , if consumption is higher ( resp . lower ) than the level , this means that the capital stock will decrease ( resp . increase ) By contrast , setting ( equal to zero yields , This equation pins down the level of the capital stock on the , and the dynamics of consumption can be seen in Figure for any given level of consumption , if the capital stock is below ( resp . above ) its level , then consumption is increasing ( resp , decreasing ) This is because the marginal product of capital will be relatively high ( resp , low ) Expressions ( and ( together yield the values of consumption and the capital stock ( both ) in the , as shown in Figure . This already lets us say something important about the behaviour of this economy Let recall the concept of the golden rule , from our discussion of the

28 THE NEOCLASSICAL GROWTH MODEL Figure Dynamics of capital Figure Dynamics of consumption model the of consumption on the . From ( we see that this is tantamount to setting ( Recall here we have assumed the depreciation rate is zero ( If we compare this to ( we see that the the optimal level of capital per capita is lower than in the golden rule from the model ( Recall the properties of the neoclassical production function , and that we assume ) Because of this comparison , is sometimes known as the golden rule Why does require that consumption be lower on the than what would be prescribed by the

THE NEOCLASSICAL GROWTH MODEL 29 Figure Steady state golden rule ?

Because future consumption is discounted , it is not optimal to save so much that consumption is it is best to consume more along the transition to the . Keep in mind that it is ( not ( that describes the optimal allocation . The kind of that is possible in the model disappears once we consider optimal savings decisions . Now , you may ask is it the case then that this type of is not an issue in practice ( or even just in theory ) Well , we will return to this issue in Chapter . For now , we can see how the question of dynamic efficiency relates to issues of inequality . A digression on inequality Is right ?

It turns out that we can say something about inequality in the context of the , even though the representative agent framework does not address it directly Let start by noticing that , as in the model , on the output grows at the rate of population growth ( since capital and output per worker are constant ) In addition , once we solve for the equilibrium , which we sketch in Section below , we will see that in that equilibrium we have ( where is the interest rate , or equivalently , the rate of return on capital . This means that the condition for dynamic efficiency , which holds in the , can be interpreted as the condition made famous by ( 2014 ) in his Capital in the Century . The condition is what calls the Fundamental Force for Divergence an interest rate that exceeds the growth rate of the economy In short , he argues that , if holds , then there will be a tendency for inequality to explode as the returns to capital accumulate faster than overall income grows . In words This fundamental inequality ( will play a crucial role in this book , In a sense , it sums up the overall logic of my conclusions . When the rate of return on capital exceeds the growth rate of the economy ( then it logically follows that inherited wealth grows faster than output and (

30 THE NEOCLASSICAL GROWTH MODEL Does that mean that , were we to explicitly consider inequality in a context akin to the we would predict it to explode along the ?

Not so fast . First of all , when taking the model to the data , we could ask what is . In particular , can have a lot of human capital . be the return to labour mostly , and this may help undo the result , In fact , it could even turn it upside down if human capital is most of the capital and is evenly distributed in the population . You may also want to see and Robinson ( 2015 ) who have a thorough discussion of this prediction . In particular , they argue that , in a model with workers and capitalists , modest amounts of social mobility understood as a probability that some capitalists may become workers , and will counteract that force for divergence . Yet the issue has been such a hot topic in the policy debate that two more comments on this issue are due . First , let understand better the of labour and income shares . Consider a typical production function AL ' With competitive factor markets , the for profit would give aAL . 320 ) Computing the labour share using the equilibrium wage gives AL ab a , AL which implies that for a , labour and capital are constant . More generally , if the production function is ( a ( AL ) with , oo ) then is the ( constant ) elasticity of substitution between physical capital and labour . Note that when co , the production function is linear ( and are perfect substitutes ) and one can show that when the production function approaches the technology of proportions , in which one factor can not be substituted by the other at all . From the of we obtain ( a ( AL ) aA ( AL ) the labour share is now ?

AL ) A ' AL . WY ( 1115 MALE ) Notice that as , several things can happen to the labour , and what happens depends on A and a ( If increases , 326 ) THE NEOCLASSICAL GROWTH MODEL I These two equations show that the elasticity of substitution is related to the concept of how essential a factor of production is . If the elasticity of substitution is less than one , the factor becomes more and more important with economic growth . If this factor is labour this may undo the result . This may be ( and this is our last comment on the issue ! the reason why over the last centuries , while interest rates have been way above growth rates , inequality does not seem to have worsened . If anything , it seems to have moved in the opposite direction . In Figure , 2019 ) looks at interest rates since the 13005 and shows that , while declining , they have consistently been above the growth rates of the economy at least until very recently . If those rates would have led to plutocracy , as fears , we would have seen it a long while ago . Yet the world seems to have moved in the opposite direction towards more democratic regimes ?

Figure Real rates , from ( 2019 ) 35 30 25 20 15 10 Nominal loan rates , and resulting real rate trend in Simon van to Edward III , 35 loan to King , 18 2018 actual real Charles II sale and cash discount , 16 loan to Beni savings dell ( Papal bonds , series Treasury ) EE 1310 1369 1428 1487 rate trend Pe ( Real Fate and Austrian loans Spanish crown Papal States venal offices United States personal loans ) 546 1605 1664 1723 1782 Flanders , States General , and . French crown Other and state unions Milan English crown Emperor , and German princes Papal States short term and King of Denmark rate personal loans Transitional dynamics How do we study the dynamics of this system ?

We will do so below graphically . But there are some shortcuts that allow you to understand the nature of the dynamic system , and particularly the relevant question of whether there is one , none , or multiple equilibria .

32 THE NEOCLASSICAL GROWTH MODEL A dynamic system is a bunch of differential equations ( difference equations if using discrete time ) In the mathematical appendix , that you may want to refer to now , we argue that one way to approach this issue is to the system around the steady state . For example , in our example here , Equations ( and ( are a system of two differential equations in two unknowns and To the system around the or steady state we compute the derivatives relative to each variable as shown below ' I where 55 ( 55 and ak , 11 ! 329 ) ak ( ak ( These computations allow us to a matrix with the of the response of each variable to those in the system , at the steady state . In this case , this matrix is , 01 . In the mathematical appendix we provide some tools to understand the importance of this matrix of coefficients . In particular , this matrix has two associated , call them 11 and 12 ( not to be confused with the marginal utility of consumption ) The important thing to remember from the appendix is that the dynamic equations for the variables will be of the form Ae . Thus , the nature of these turns out to be critical for understanding the dynamic properties of the system . If they are negative their effect dilutes over time ( this is called a sink , as converge to their steady state ) If positive , the variable blows up ( we call these systems a source , where variables drift away from the steady state ) If one is positive and the other is negative the system typically blows up , except if the of the positive eigenvalue is zero ( we call these systems ) You may think that what you want is a sink , a system that converges to an equilibrium . While this may be the natural approach in sciences such as physics , this reasoning would not be correct in the realm . Imagine you have one state variable ( and a control variable ( jumpy ) as in this system . In the system we are analysing here is a state variable that moves slowly over time and

THE NEOCLASSICAL GROWTH MODEL 33 is the control variable that can jump . So , if you have a sink , you would that any would take you to the equilibrium . So rather than having a unique stable equilibrium you would have alternative equilibria ! Only if the two variables are state variables do you want a sink . In this case the equilibrium is unique because the state variables are uniquely determined at the start of the program . In our case , to pin down a unique equilibria we would need a . Why ?

Because for this there is only one value of the control variable that makes the coefficient of the explosive eigenvalue equal to zero . This feature is what allows to pin the unique converging equilibria . In the below this will become very clear . What happens if all variables are control variables ?

Then you need the system to be a source , so that the control variables have only one possible value that attains sustainability We will many systems like this throughout the book . In short , there is a rule that you may want to keep in mind . You need as many positive as jumpy or variables you have in your system . If these two numbers match you have uniqueness ! Before proceeding , one last rule you may want to remember . The determinant of the matrix is the product of the , and the trace is equal to the sum . This is useful , because , for example , in our model , if the determinant is negative , this means that the have different sign , indicating a saddle path . In fact , in our case , Det ( If Det ( is the product of the of the matrix and their product is negative , then we know that the must have the opposite sign . Hence , we conclude one eigenvalue is positive , while the other is negative . Recall that is a , or sticky , variable , while can jump . Hence , since we have the same number of negative as of sticky variables , we conclude the system is stable , and the convergence to the equilibrium unique . You can see this in a much less abstract way in the the phase diagram in Figure . Figure The phase diagram

34 THE NEOCLASSICAL GROWTH MODEL Notice that since can jump , from any initial condition for ( the system moves vertically ( moves up or down ) to hit the saddle path and converge to the along the saddle path . Any other trajectory is divergent . Alternative trajectories appear in Figure . Figure Divergent trajectories The problem is that these alternative trajectories either eventually imply a jump in the price of capital , which is inconsistent with rationality , or imply levels of the capital stock . In either case this violates the transversality condition . In short , the two dynamic equations provide the dynamics at any point in the ( space , but only the allows us to choose a single path that we will use to describe our equilibrium The effects of shocks Consider the effects of the following shock . At time and unexpectedly , the discount rate forever ( people become less impatient ) From the relevant and it schedules , we see that the former does not move ( does not enter ) but the latter does . Hence , the new will have a higher capital stock . It will also have higher consumption , since capital and output are higher . Figure shows the old , the new , and the path to get from one to the other . On impact , consumption falls ( from point to point A ) Thereafter , both and rise together until reaching point . Similar exercises can be carried out for other permanent and unanticipated shocks . Consider , for example , an increase in the discount rate ( Figure ) The increase is transitory , and that is anticipated by the planner . The point we want to make is that there can be no anticipated jump in the control variables throughout the optimal path as this would allow for capital gains . This is why the trajectory has to put you on the new saddle path when the discount rate goes back to normal .

THE NEOCLASSICAL GROWTH MODEL 35 Figure A permanent increase in the discount rate Figure A transitory increase in the discount rate The equivalence with the equilibrium VVe will show that the solution to the central planner problem is exactly the same as the solution to a equilibrium . Now we will sketch the solution to the problem of the equilibrium in an economy that is identical to the one we have been studying , but without a central planner . We now have households

36 THE NEOCLASSICAL GROWTH MODEL and ( owned by households ) who independently make their decisions in a perfectly competitive environment , We will only sketch this solution . The utility function to be by each household is I ( where , is consumption and ( is the rate of time preference . The consumers budget constraint can be written as , A , rA , where , is population , A , is the stock of assets , A is the increase in assets , is the wage per unit of labour ( in this case per worker ) and is the return on assets . What are these assets ?

The households own the capital stock that they then rent out to in exchange for a payment of they can also borrow and lend money to each other , and we denote their total debt by , In other words , we can A , You should be able to go from ( to the budget constraint in per worker terms , na , Households supply factors of production , and maximise . Thus , at each moment , you should be able to show that equilibrium in factor markets involves , In this model , we must impose what we call a ( What does that mean ?

That means that households can not pursue the easy path of getting arbitrarily rich by borrowing money and borrowing even more to pay for the interest owed on previously contracted debt . If possible that would be the optimal solution , and a rather trivial one at that . The idea is that the market will not allow these Ponzi schemes , so we impose this as a constraint on household behaviour . lim a , oo You will have noticed that this looks a bit similar to the we have seen in the context of the planner problem , so it is easy to mix them up . Yet , they are different ! The is a constraint on it wasn needed in the planners problem because there was no one in that closed economy from whom to borrow . In contrast , the is an condition that is to say , something that is chosen in order to achieve . They are related , in that both pertain to what happens in the limit , as oo . We will see how they help connect the equilibrium with the planners problem .

THE NEOCLASSICAL GROWTH MODEL 37 Integrating the budget constraint The budget constraint in ( holds at every instant It is interesting to out what it implies for the entire path to be chosen by households . To do this , we need to integrate that budget constraint . In future chapters we will assume that you know how to do this integration , and you can consult the mathematical appendix for that . But the first time we will go over all the steps . So lets start again with the budget constraint for an individual family a , This is a differential equation which ( as you can see in the Mathematical Appendix ) can be solved using integrating factors . To see how that works , multiply both sides of this equation by ( a , The side is clearly the derivative of a ( sides between and with respect to time , so we can integrate both ) Taking the co ( and using the condition ) yields 90 I ( which can be written as a standard budget constraint no no ' This is quite natural and intuitive all of what is consumed must be out of initial assets or wages ( since we assume that Ponzi schemes are not possible ) Back to our problem Now we can go back to solve the consumers problem ( at 141 , The now looks like this ( From this you can obtain the and , following the same procedure from the previous case , you should be able to get to un (

38 THE NEOCLASSICAL GROWTH MODEL How does that compare to ( the equation , which is one of our dynamic equations in the central planner solution ?

We leave that to you . You will also notice that , from the equivalent ( and ( we have , or ( Using this in the equivalent of ( yields ( This means that the becomes ) You can show that this is exactly the same as the for the central planner problem . Think about it since all individuals are identical , what is the equilibrium level of by ?

If an individual wants to borrow , would anyone else want to lend ?

Finally , with the same reasoning on the equilibrium level of lay , you can show that the resource constraint also matches the dynamic equation for capital , 35 ) which was the relevant resource straint for the central planner problem . Do we have growth after all ?

Not really . Having seen the workings of the Ramsey model , we can see that on the , just as in the model , there is no growth in per capita variables is constant at such that ( and is constant at ( It is easy to show that once again we can obtain growth if we introduce exogenous technological progress . What have we learned ?

We are still left with a growth model without growth it was not the of the savings rate that generated the unsatisfactory features of the model when it comes to explaining run growth . We will have to keep looking by moving away from diminishing returns or by modelling technological progress . On the other hand , our exploration of the Ramsey model has left us with a work that is the foundation of a lot of modern . This is true not only of our further explorations that will lead us into endogenous growth , but eventually also when we move to the realm of short term . At any rate , the is a dynamic general equilibrium framework that we will use over and over again . Even in this basic application some key results have emerged . First , we have the equation that encapsulates how consumers make optimal choices between today and tomorrow . If the marginal of reducing consumption namely , the rate of return on the extra capital you accumulate

THE NEOCLASSICAL GROWTH MODEL 39 is greater than the consumers impatience the discount rate then it makes sense to postpone sumption . This crucial piece of intuition will appear again and again as we go along in the book , and is perhaps the key result in modern . Second , in this context there is no dynamic , as consumers would never choose to oversave in an way Most importantly , now we are in possession of a powerful toolkit for dynamic analysis , and we will make sure to put it to use from now on . Notes The other one was to the theory of optimal taxation ( Ramsey 1927 ) See Cass ( 1965 ) and et al . 1963 ) Another interesting set of questions refer to population policies say you impose a policy to reduce population growth . How does that play into the utility function ?

How do you count people that have not been and will not be born ?

Should a central planner count those people ?

We are departing from standard mathematical convention , by using instead of ses to denote time , even though we are modelling time as continuous and not discrete . We think it pays off to be unconventional , in terms of making notation look less cluttered , but we apologise to the purists in the audience nonetheless ! Note that we must assume that , or the problem will not be . Why ?

Because if , the representative household gets more total utility out of a given level of consumption per capita in the future as there will be more then . If the discount factor does not compensate for that , it would make sense to always postpone consumption ! And why do we have in the utility function in the place ?

Because we are incorporating the utility of all the individuals who are alive at time the more , the merrier ! Recall that the elasticity of a variable with respect to another variable is as As such , is the elasticity of the marginal utility of consumption with respect to consumption it measures how sensitive the marginal utility is to increases in consumption . Now , think about it the more sensitive it is , the more I will want to smooth consumption over time , and this means I will be less likely to substitute consumption over time . That is why the inverse of that captures the poral elasticity of substitution the greater is , the more I am willing to substitute consumption over time . This is the analogue of the standard condition that you may have encountered in the marginal rate of substitution ( between consumption at two adjacent points in time ) must be equal to the marginal rate of transformation . At any rate , it may also be argued that maybe we haven seen because was right . After all , the French and US . revolutions may be explained by previous increases in inequality It works the same for a system of difference equation in discrete time , except that the cutoff is with being larger or smaller than one . 10 To rule out the path that leads to the capital stock of when the locus crosses the horizontal axis to the right of the golden rule , notice that A from ( grows at the rate ( so that ) grows at rate ( but to the right ofthe golden rule ( so that the term increases . Given that the capital stock is eventually fixed we conclude that the transversality condition can not hold . The paths that lead to high consumption and a zero capital stock imply a collapse of consumption to zero when the path reaches the vertical axis . This trajectory is not feasible because at some point

40 THE NEOCLASSICAL GROWTH MODEL it can not continue . When that happens the price of capital increases , and consumers would have that jump away , so that that path would have not occurred in the first place . Or should it now be the condition ?

References , Robinson , I . A . 2015 ) The rise and decline of general laws of capitalism . Journal of Economic Perspectives , 29 ( I . 1974 ) Are government bonds net wealth ?

Economy , 82 ( Caplin , A . I . 2004 ) The social discount rate . Economy , 112 ( 1268 . Ventura , I . 2000 ) A representative consumer theory of distribution . American Economic Review , 90 ( Cass , 1965 ) Optimum growth in an aggregative model of capital accumulation . The Review of Economic Studies , 32 ( et al . 1963 ) On the concept economic growth ( tech . rep ) ation for Research in Economics , Yale University . 2014 ) Capital in the century . Harvard University Press . Ramsey , 1927 ) A contribution to the theory of taxation , The Economic , 37 ( 145 ) Ramsey , 1928 ) A mathematical theory of saving . The Economic Journal , 38 ( 152 ) 2019 ) Eight centuries of global real rates , and the decline , Available at or .