Advanced Macroeconomics An Easy Guide Growth theory preliminaries

Growth theory preliminaries Why do we care about growth ?

It is hard to put it better than Nobel laureate Robert Lucas did as he mused on the importance of the study of economic growth for and for anyone interested in economic The diversity across countries in measured per capita income levels is literally too great to be . Rates of growth of real per capita are also diverse , even over sustained periods , For we observe , for example India , per year Egypt , 34 South Korea , the United States , the industrial economies averaged . An Inc ian will , on average , be twice as well off as his grandfather a Korean 32 times . I do not see how one can look at like these without seeing them as representing possibilities . Is there some action a government of India could take that would lead the Indian economy to grow like Indonesia or Egypt ?

If so , what , exactly ?

If not , what is it about the nature of Inc ia that makes it so ?

The human welfare involved in questions like these are simply staggering Once one starts to think about them , it is hard to think about anything Lucas Ir . 1988 ) emphasis added ) While it is common to think about growth today as being somehow natural , even expected in fact , if wor growth falls from to , it is perceived as a big crisis it is worthwhile to edge at this was not always the case . Pretty much until the end of the century growth was quite low , if it happened at all . In fact , it was so low that people could not see it during their times . They lived in the same world as their parents and grandparents . For many years it seemed that growth was actually behind as people contemplated the feats of antiquity without ing how they could have been accomplished . Then , towards the turn of the century , as shown in Figure 21 something happened that created explosive economic growth as the world had never seen before , Understanding this transition will be the purpose of Chapter 10 . Since then , growth has become the norm . This is the reason the first half of this book , in fact up to Chapter 10 , will deal with understanding growth . As we proceed we will ask about the of capital lation ( Chapters through , as well as and ) and discuss the process of technological progress ( Chapter ) Institutional factors will be addressed in Chapter . The growth process raises many questions should we expect this growth to continue ?

Should we expect it eventually to ?

Or , on the contrary , will it accelerate without bound ?

How to cite this book chapter , and , A . 2021 . Advanced An Easy Guide . Growth theory preliminaries , London Press . DOI . License .

GROWTH THEORY PRELIMINARIES Figure The evolution of the world per capita over the years 8000 ) per capita ( 1990 ) 43 250 500 1000 1250 1500 1730 2000 Year Figure Log per capita of selected countries ( 11 Log per capita 1310 1300 1300 1900 1950 1940 1900 1900 2000 Year Brazil India Spain United States China Republic of Korea United Kingdom But the fundamental point of Lucas quote is to realise that the differences in income per capita across countries are to a large extent due to differences in growth rates over time and the power of exponential growth means that even relatively small differences in the latter will build into huge differences in the former . Figures and make this point . The richest countries

GROWTH THEORY PRELIMINARIES Figure Log per capita of selected countries ( 11 to Log per capita oo 19 ao 1990 zobo Year India Singapore have been growing steadily over the last two centuries , and some countries have managed to converge to their income levels , Some of the performances are really stellar . Figure shows how South Korea , with an income level that was 16 of that of the in 1940 , managed to catch up in just a few . Today it income is compared to the . Likewise , Spain income in 1950 was 23 that of the , Today it is 57 , At the same time other countries lagged , Argentina for example dropped from an income level that was 57 of US . income at the turn of the century to today . Figure shows some diversity during recent times , The spectacular performances of , Singapore or , more recently , of China and India , contrast with the stagnation of , Argentina or Venezuela . In 1960 the income of the average ( as someone from is called ) was only as rich as the average Venezuelan , In 2018 he or she was 48 richer ! These are crucial reasons why we will spend about the initial half of this book in understanding growth . But those are not the only reasons ! You may be aware that disagree on a lot of things however , the issue of economic growth is one where there is much more of a consensus . It is thus helpful to start off on this relatively more solid footing , Even more importantly , the study of nomic growth brings to the forefront two key ingredients of essentially all of analysis general equilibrium and dynamics . First , understanding the behaviour of an entire economy requires thinking about how different markets interact and affect one another , which inevitably requires a eral equilibrium approach . Second , to think seriously about how an economy evolves over time we must consider how today choices affect tomorrow in other words , we must think dynamically ! As such , economic growth is the perfect background upon which to develop the main methodological tools in the model of , known as the neoclassical growth model ( for short , also known as the Ramsey model ) and the overlapping generations model ( we call it the model ) A lot of what we will do later , as we explore different policy issues , will involve applications of these dynamic tools that we will learn in the context of studying economic growth . So , without further delay , to this we turn ,

GROWTH THEORY PRELIMINARIES The facts VVhat are the key stylised facts about growth that our models should try to match ?

That there is growth in output and capital per worker with relatively stable income shares . The modern study of economic growth starts in the period and was mostly motivated by the experience of the developed world , In his classical article ( 1957 ) Nicolas stated some basic facts that he observed economic growth seemed to satisfy , at least in those countries . These came to be known as the facts , and the main challenge of growth theory as initially constituted was to account simultaneously for all these facts . But , what were these facts ?

Here they are ?

Output per worker shows continuous growth , with no tendency to fall . The ratio is nearly constant . But what is capital ?

Capital per worker shows continuous growth ( follows from the other two ) The rate of return on capital is nearly constant ( real interest rates are ) Labour and capital receive constant shares of total income . The growth rate of output per worker substantially across countries ( and over time , we can add , miracles and disasters ) Most of these facts have aged well . But not all of them . For example , we now know the constancy of the interest rate is not so when seen from a big historical sweep , In fact , interest rates have been on a secular downward trend that can be dated back to the ( 2019 ) Of course rates are way down now , so the question is how much lower can they go ?

We will show you the data in a few pages . In addition , in recent years , particularly since the early , the labour share has fallen in most countries and industries . There is much argument in the literature as to the reasons why ( see and ( 2014 ) for a discussion on this ) and the whole debate about income distribution trends recently spearheaded by ( 2014 ) has to do with this issue . We will come back to it shortly . As it turns out Robert established a simple model ( 1956 ) that became the first ing model of economic contribution became the foundation of the , and the backbone of modern growth theory , as it seemed to fit the facts . Any study of growth must start with this model , reviewing what it explains and , just as crucially , what it fails to The VVe outline and solve the basic model , introducing the key concepts of the neoclassical production function , the balanced growth path , transitional dynamics , dynamic inefficiency , and convergence . Consider an economy with only two inputs physical capital , and labour , The production function is (

GROWTH THEORY PRELIMINARIES I where Yis the of output produced , Assume output is a homogeneous good that can be consumed , or invested , I , to create new units of physical capital . Let be the fraction of output that is saved that is , the saving rate so that is the fraction of output that is consumed . Note that . Assume that capital at the constant rate . The net increase in the stock of physical capital at a point in time equals gross investment less depreciation ( where a dot over a variable , such as , denotes differentiation with respect to time . Equation ( determines the dynamics of for a given technology and labour force , Assume the population equals the labour force , It grows at a constant , exogenous rate , Ifwe the number ofpeople at time to , then , where , is labour at time If is given from ( and technological progress is absent , then ( determines the time paths of capital , and output , Such behaviour depends crucially on the properties of the production function , Apparently minor differences in assumptions about ( can generate radically different theories of economic growth . The ( neoclassical ) production function For now , neglect technological progress . That is , assume that ( is independent of This assumption will be relaxed later . Then , the production function ( takes the form ( Assume also the following three properties are satisfied . First , for all and , exhibits positive and diminishing marginal products with respect to each input ap i ex am , Second , exhibits constant returns to scale ( for all . Third , the marginal product of capital ( or labour ) approaches as capital ( or labour ) goes to and approaches as capital ( or labour ) goes to oo , These last properties are called conditions . We will refer to production functions satisfying those three sets of conditions as neoclassical .

GROWTH THEORY PRELIMINARIES The condition of constant returns to scale has the convenient property that output can be written as ( where is the ratio , and the ( is to equal ( The duction function can be written as ( where is per capita output . One simple production function that all of the above and is often thought to provide a reasonable description of actual economies is the function , AK , where A is the level of the technology , and It is a constant with . The function can be written as Ak . Note ( oof ( and , 00 . In short , the the properties of a neoclassical production function . The law of motion of capital The change in the capital stock over time is given by ( If we divide both sides of this equation by , then we get ( The side contains per capita variables only , but the side does not . We can write as a function by using the fact that 11 ( alt , where . If we substitute ( into the expression for then we can rearrange terms to get ( The term on the side of ( can be thought of as the effective depreciation rate for the ratio , If the saving rate , were , then would decline partly due to depreciation of at the rate and partly due to the growth of at the rate Figure shows the workings of ( The upper curve is the production function , The term ( looks like the production function except for the multiplication by the positive fraction . The ( curve starts from the origin ( has a positive slope ( and gets as rises ( because ( The conditions imply that the ( curve is vertical at and becomes perfectly as approaches The other term in ( appears in Figure as a straight line from the origin with the positive slope .

GROWTH THEORY PRELIMINARIES Figure Dynamics in the model ( Finding a balanced growth path A balanced growth path ( is a situation in which the various quantities grow at constant rates , In the model , the corresponds to in ( We it at the intersection of the ( curve with the ( line in Figure . The corresponding value of is denoted . Algebraically , the condition ( Since is constant in the , and are also constant at the values ( and ( respectively . Hence , in the model , the per capita quantities , and do not grow in the it is a growth model without ( growth ! Now , thats not quite right the constancy of the per capita means that the levels of variables , and grow in the at the rate of population growth , 11 . In addition , changes in the level of technology , represented by shifts of the production function , in the saving rate , in the rate of population growth , 71 and in the depreciation rate , all have effects on the per capita levels of the various quantities in the . We can illustrate the results for the case of a production function . The labour ratio on the is determined from ( as ( Note that , as we saw graphically for a more general production function ( rises with the saving rate , and the level of technology , A , and falls with the rate of population growth , and the rate , Output per capita on the is given by ( Thus , is a positive function of and A and a negative function of and .

GROWTH THEORY PRELIMINARIES Transitional dynamics Moreover , the model does generate growth in the transition to the . To see the implications in this regard , note that dividing both sides of ( by implies that the growth rate of is given by ( Equation ( 215 ) says that equals the difference between two terms , and ( which we plot against in Figure . The first term is a curve , which to at and approaches as tends to . The second term is a horizontal line crossing the vertical axis at . The vertical distance between the curve and the line equals the growth rate of capital per person , and the crossing point corresponds to the . Since and ( falls monotonically from to , the curve and the line intersect once and only once . Hence ( except for the trivial solution , where capital stays at zero forever ) the ratio exists and is unique . Note also that output moves according to ( A formal treatment of dynamics follows . From ( one can calculate ( We want to study dynamics in the neighbourhood of the , so we evaluate this at ' Figure Dynamics in the model again Growth rate (

GROWTH THEORY PRELIMINARIES The capital stock will converge to its when and when . Hence , this . requires that In the case the condition is simple . Note that ) so that requires . That is , reaching the requires diminishing returns , With diminishing returns , when is relatively low , the marginal product of capital , is high . By assumption , households save and invest a constant fraction , of this product . Hence , when is relatively low , the marginal return to investment , is relatively high . Capital per worker , effectively at the constant rate . Consequently , the growth of capital , is also relatively high . In fact , for it is positive , Conversely , for it is negative . Suppose that the economy is initially on a with capital per person . Imagine that the ment then introduces some policy that raises the saving rate permanently from 51 to a higher value 52 . Figure shows that the ( schedule shifts to the right , Hence , the intersection with the line also shifts to the right , and the new capital stock , exceeds . An increase in the saving rate generates temporarily positive per capita growth rates . In the long run , the levels of and are permanently higher , but the per capita growth rates return to , A permanent improvement in the level of the technology has similar , temporary effects on the per capita growth rates . If the production function , shifts upward in a proportional manner , then the Figure The of an increase in the savings rate SA , I I ( I I (

GROWTH THEORY PRELIMINARIES ( curve shifts upward , just as in Figure 26 . Hence , again becomes positive temporarily In the long run , the permanent improvement in technology generates higher levels of and , but no changes in the per capita growth rates . Dynamic inefficiency For a given production function and given values of and , there is a unique value for each value of the saving rate , 54 Denote this relation by ( with ( The level of per capita consumption on the is ( We know from ( that ( hence we can write an expression for as ( Figure shows the relation between and that is implied by ( 220 ) The quantity is increasing in for low levels of and decreasing in for high values of . The quantity attains its maximum when the derivative vanishes , that is , when ( Since , the term in brackets must equal If we denote the value of by that corresponds to the maximum of , then the condition that determines is ( The corresponding savings rate can be denoted as , and the associated level of per capita consumption on the is given by ( and is is called the golden rule consumption rate . If the savings rate is greater than that , then it is possible to increase consumption on the , and also over the transition path . We refer to such a situation , where everyone could be made better off by an alternative allocation , as one of dynamic . In this case , this dynamic is brought about by everyone could be made better off by choosing to save less and consume more . But this naturally begs the question why would anyone pass up this opportunity ?

Shouldn we Figure Feasible consumption GROWTH THEORY PRELIMINARIES think of a better model of how people make their savings decisions ?

We will see about that in the next chapter . Absolute and conditional convergence Equation ( implies that the derivative of with respect to is negative i ( Other things equal , smaller values of are associated with larger values of , Does this result mean that economies with lower capital per person tend to grow faster in per capita terms ?

Is there convergence across economies ?

We have seen above that economies that are structurally similar in the sense that they have the same values of the parameters , and and also have the same production function , have the same values and . Imagine that the only difference among the economies is the initial quantity of capital per person , The model then implies that the economies with lower values of ( and ( have higher growth rates of This hypothesis is known as conditional convergence within a group of structurally similar economies ( with similar values for , and and production function , poorer economies will grow faster and catch up with the richer one . This hypothesis does seem to match the data think about how poorer European countries have grown faster , or how the South has caught up with the North , over the second half of the century . An alternative , stronger hypothesis would posit simply that poorer countries would grow faster without conditioning on any other characteristics of the economies . This is referred to as absolute convergence , and does not seem to fit the data Then again , the model does not predict absolute convergence ! Can the model account for income differentials ?

VVe have seen that the model does not have growth in per capita income in the long run . But can it help us understand income differentials ?

We will tackle the empirical evidence on economic growth at a much greater level of detail later on . However , right now we can ask whether the simple model can account for the differences in income levels that are observed in the world . According to the World Bank calculations , the range of 2020 income levels vary from per capita in or in Norway , all the way down to 700 in . Can the basic model explain this difference in income per capita of a factor of more than 100 times or even close to 200 times ?

In order to tackle this question we start by remembering what output is supposed to be on the ( Assuming A and this to ( GROWTH THEORY PRELIMINARIES The ability of the model to explain these large differences in income ( in the ) as can be seen from the expressions above , will depend critically on the value of oz . If The standard ( rough ) estimate for the capital share is and Prescott ( 2002 ) however , claim that the capital share in is much larger than usually accounted for because there are large capital assets In fact , they argue that the share of investment in is closer to rather than the more traditional . The reasons for the unaccounted investment are ( their estimates of the relevance of each in parenthesis ) Repair and maintenance ( of ) of ) multiplied by three ( of ) to take into account perfecting the manufacturing process and launching new products ( the times three is not well substantiated ) Investment in software ( of ) Firms investment in organisation capital . They think 12 is a good number . Learning on the job and training ( 10 of ) Schooling ( of ) They claim all this capital has a return and that it accounts for about 56 of total ! At any rate , using the equation above an ( 22 50 125 41 100 300 73 200 800 ( But even the 800 we get using the estimate seems to be way too low relative to what we see in the data . Alternatively , the differences in income may come from differences in total factor productivity ( as captured by A . The question is how large do these differences need to be to explain the output differentials ?

Recall from ( that ( GROWTH THEORY PRELIMINARIES So if or , as suggested by and Prescott ( 2002 ) then AW . Now , let forget about , for example , by assuming they are the same for all countries ) and just focus on differences in A . Notice that if is , of the level in the other country , this indicates that the income level is then , and Prescott ( 2002 ) use this to estimate , for a group of countries , how much productivity would have to differ ( relative to the United States ) for us to replicate observed relative incomes over the period Country Relative Income Relative UK 60 86 Colombia 22 64 Paraguay 16 59 Pakistan 10 51 These numbers appear quite plausible , so the message is that the model requires substantial differences in productivity to approximate existing differences in income . This begs the question of what makes productivity so different across countries , but we will come back to this later , The model with exogenous technological change VVe have seen that the model does not have growth in per capita income in the long run . But that changes if we allow for technological change . Allow now the productivity of factors to change over time . In the case , this means that A increases over time , For simplicity , suppose that , Out of the , output then evolves according to ) A ic . A ! ark ( On the , where is constant , a . This is a strong prediction of the model in the long run , technological change is the only source of growth in per capita income . Let now embed this improvement in technology or in workers . We can define labour input as broader than just bodies , we could call it now human capital by , where is the amount of labor in units . The production function is ( To put it in per capita terms , we .

20 GROWTH THEORY PRELIMINARIES So . 232 ) so . Notice that in this equilibrium income per person grows even on the , and this accounts for all six facts . What have we learned ?

The model shows that capital accumulation by itself can not sustain growth in per capita income in the long run . This is because accumulation runs into diminishing marginal returns . At some point the capital stock becomes large enough and its marginal product correspondingly small enough that a given savings rate can only provide just enough new capital to replenish ongoing ation and increases in labour force . Alternatively , if we introduce exogenous technological change that increases productivity , we can generate growth in income per capita , but we do not really explain In fact , any differences in growth rates come from exogenous differences in the rate of technological change we are not explaining those differences , we are just assuming them ! As a result , nothing within the model tells you what policy can do about growth in the long runs That said , we do learn a lot about growth in the transition to the long run , about differences in income levels , and how policy can affect those things There are clear lessons about ( i ) convergence the model predicts conditional convergence ( ii ) dynamic it is possible to save too much in this model and ( iii ) differences in income they seem to have a lot to do with differences in productivity . Very importantly , the model also points at the directions we can take to try and understand term growth . We can have a better model of savings behaviour how do we know that individuals will save what the model says they will save ?

And , how does that relate to the issue of dynamic ?

GROWTH THEORY PRELIMINARIES We can look at different assumptions about technology maybe we can escape the shackles of returns to accumulation ?

Or can we think more carefully about how technological progress comes about ?

These are the issues that we will address over the next few chapters , Notes Lucas words hold up very well more than three decades later , in spite of some evidently dated examples , Once we are done with our study of economic growth , you can check the new facts proposed by ones and ( 2010 ) which update the basic empirical based on the progress over the subsequent or so . For those of you who are into the history of economic thought , at the time the framework to study growth was the model , due to the independent contributions of ( you ably guessed it ) 1939 ) and ( 1946 ) It assumed a production function with between labour and capital ( as it is known to economists ) and predicted that an economy would generate increasing unemployment of either labour or capital , depending on whether it saved a little or a lot . As it turns out , that was not a good description of the real world in the period . eventually got a Nobel prize for his trouble , in 1987 also for his other contributions to the study of economic growth , to which we will return . An Australian economist , Trevor Swan , also published an independently developed paper with very similar ideas at about the same time , which is why sometimes the model is referred to as the model . He did not get a Nobel prize . We will population growth in Chapter 10 , when discussing growth theory . The is often referred to as a steady state , borrowing terminology from classical physics . We have noticed that talk of steady state tends to lead students to think of a situation where variables are not growing at all . The actual refers to constant growth rates , and it is only in certain cases and for certain variables , as we will see , that this constant rate happens to be zero . You should try to show mathematically from ( that , with a neoclassical production function , the only way we can have a constant growth rate is to have , Or does it ?

More recently , et al . 2021 ) have argued that there has been a move towards absolute convergence in the data in the century Stay tuned ! References , 1946 ) Capital expansion , rate of growth , and employment . 1939 ) An essay in dynamic theory . The Economic , 49 ( 193 ) I . 2010 ) The new facts Ideas , institutions , population , and human capital . American Economic , 1957 ) A model of economic growth , Economic , 67 ( 268 ) 2014 ) The global decline of the labor share . The Quarterly Journal , 129 ( Willis , You , 2021 ) Converging to convergence . Macro Annual

22 GROWTH THEORY PRELIMINARIES Lucas , 1988 ) On the mechanics development . Journal Economics , 22 , Prescott , 2002 ) Barriers to riches . MIT Press . 2014 ) Capital in the century . Harvard University Press . 2019 ) Eight centuries of global real rates , and the decline , Available at or . 1956 ) A contribution to the theory of economic growth . The Quarterly , 70 (