Economic analysis (second section: Cesaratto) Lecture notes (provisional version academic year )

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1 1 27/10/2013 Department of Political Economy and Statistics Economic analysis (second section: Cesaratto) Lecture notes (provisional version academic year ) Sergio Cesaratto Programme This part of the course intends to introduce students to modern mainstream and non-conventional growth theories. It moves from Harrod s model and the problems it left opened. It shows how neoclassical and heterodox economists tried, in different ways, to solve those troubles. Basically, neoclassical economists rely on long period marginal theory, while heterodox scholars try to extend the Keynesian theory of effective demand to the long run. The results of the capital theory controversy, dealt with by prof. Petri, suggest the weakness of the mainstream view. The lectures will also show the links between the Classical surplus approach introduced by prof. Petri and the theory of demand-led growth with particular reference to the views of Michal Kalecki. 1. Orthodox and heterodox growth analyses 1.1. The father of modern growth theory: Harrod and the problem of instability References: H.Jones, An introduction to modern theories of economic growth, London : Nelson, 1975, cap.3; Lecture notes; Power point presentation 1.2. The neoclassical growth model (adjustment through the change of techniques) Solow s exogenous growth (quick revision)* Endogenous growth theory References: Jones (1975) cap.4; Lecture notes, Power point presentation. * I assume that students know Solow s model Keynesian and kaleckian growth models References: lectures notes, Power point presentation The Cambridge equation (adjustment through changes in income distribution) and its critics Neo-kaleckian models (adjustment through changes in the degree of capacity utilisation) References: Lavoie M. (2006) Introduction to Post-Keynesian Economics, Palgrave Macmillan, cap. 5; Lecture notes, Power point presentation Kaleckian-Sraffian models with normal degree of capacity utilisation (Serrano).

2 2 27/10/2013 Lecture notes, Power point presentation 2.2. Alternative interpretations of the European crisis (if there is time and interest) References: lectures notes, Power point presentation; Cesaratto e Stirati (2011) Germany and the European and Global Crises Working Paper N. 607 January S. Cesaratto, Controversial and novel features of the Eurozone crisis as a balance of payment crisis, Working Paper N. 640, May Power point presentation Information Students who like to talk to the professor are kindly advised to e mail him (Cesaratto@unisi.it) and take an appointment. If you do not attend the lectures, please contact the professor to check the programme. If you miss a lecture, please try to go ahead by yourself using the notes. The final test will be written, with open questions on the topics of the course.

3 3 27/10/2013 Beyond Harrod 1 Lecture notes on orthodox and heterodox growth theories Sergio Cesaratto Università di Siena Harrod & Solow Introduction Modern growth theory can be seen as a long and largely unsuccessful attempt to overcome the problems left behind by Harrod s growth model. In these notes I will not attempt any exegesis of Harrod s interpretation of his own result. In synthesis, in my view Harrod s warranted growth equation can be interpreted either as a dynamic expression of Say s Law: assuming that all saving is invested, the economy will grow at an equilibrium (or warranted growth rate); or as a stylized description of the potential instability of capitalism (this is likely the way Harrod himself looked at his model). That is, if we assume that investment is decided independently from capacity saving (Kaldor s Keynesian Hypothesis), then unless capitalists expect the economy to grow at the warranted rate and invest accordingly, the actual rate of growth will diverge more and more from the warranted rate. Neoclassical economists are also unhappy that the warranted rate does not coincide with a full-employment path. However, this does not concern non-conventional economists which are not committed to defend the goodness of laissez-faire. Moving from Harrod growth theory developed along two different lines. One took investment as the independent variable seeing in this a characteristic of capitalism both because it symbolizes the power of capitalists and because the evidence is of a marked instability of this variable. Indeed, Keynes regarded investment as the key variable in his theory of Effective Demand. The question is then to show how capacity saving the saving supply forthcoming from a normally utilized capacity - adjusts to the independent investment decisions. The other approach took capacity saving the saving supply forthcoming from a normally utilized capacity as the independent variable. The question becomes then how investment decisions adapt to capacity savings. Solow s model represents the second approach. It is, however, quite unsatisfactory from the point of view of capital theory. Shattered but not out powered by the capital theory controversy, neoclassical economists found some implications of Solow s model also unsatisfactory, what led to endogenous growth theory (EGT) which actually begun in the early sixties actually, and not in the

4 4 27/10/2013 middle 1980s as usually supposed. While EGT has not solved the problems with Solow, it appears unsatisfactory also from other perspectives. The heterodox approach was initially developed by some Cambridge economists, genuine followers of Keynes. Also in this case, some unsatisfactory aspects of their approach encapsulated in the so-called Cambridge equation model (CEM) has led to further a formulation known as neokaleckian model (NKM). While in the CEM income distribution changes in order to adjust capacity saving to investment decisions, in the NKM the variability of the degree of capacity utilization plays the same role. According to Sraffian economists and to other heterodox economists, the abandonment of the notion of a long-run normal degree of capacity utilization is, however, an inacceptable step. In spite of their differences, both the approaches, neoclassical and heterodox aim to show that the economy converges to an Harrodian warranted growth rate and, in this capacity, can be all defined as neo-harrodian models. The only model that departs from the Harrodian framework is, as far as I can see, the Sraffian-Kaleckian supermultiplier that I regard as the most promising approach. Earlier and more recent presentations of growth theory, including those of Solow (1970) and Jones (2002), have focused on explaining Kaldor s six famous stylized facts of economic growth: 1. Output per worker grows at a roughly constant rate that does not diminish over time. 2. Capital per worker grows over time. 3. The capital/output ratio is roughly constant. (1+2) 4. The rate of return to capital is constant. 5. The share of capital and labor in net income are nearly constant. 6. Real wage grows over time. (2+4+5) Two additional stylised facts that growth models may aim to explain can be added to this list, although they are more controversial: a positive correlation between the investment (saving) rate, that is the ratio between investment (saving) and output, and the rate of economic growth, either in aggregate ( g Y ) or per-capita terms ( g y ). These Stylised facts describe a balanced growth path, or steady state growth path (or normal path as we shall sometimes indulge to call it). Steady state analysis is not necessarily the best method. The present financial crisis shows that capitalism is far to be growing along balanced paths. So Steady state analysis should be complemented by the analysis of the crises. As we shall see. non orthodox theories in particular the Sraffian supermultiplier approach - are better prepared to do this compared to neoclassical theory.

5 5 27/10/ Harrod 1.1. The problems with Harrod While in Keynes investment appeared (in the multiplier) only as a determinant of aggregate (AD), Harrod set out to combine this role with that of investment as creator of productive capacity - the theory of growth is indeed concerned with the creation of productive capacity. The question is: given this double role of investment as creator of both AD and capacity (or of potential aggregate supply AS), does an equilibrium path exist of the economy in which AD and AS grow in a consistent way? Harrod calls this long-run equilibrium growth path warranted rate of growth. If the economy grows along this path, capitalists sell their planned output and are content with what they are doing. Harrod examines a closed economy without public administration. He moves from two analytical elements: (a) the Keynesian multiplier that shows that the level of effective (aggregate) demand is determined, given the marginal propensity to consume, by the level of (gross) investment. 1 The level of effective demand determines, in turn, the level of (gross) output. (b) the (old) accelerator theory according to which the level of (gross) investment depends on the expected rate of growth of aggregate demand (AD). Although both aspects sound Keynesians, the way Harrod combines them has non- Keynesian outcomes. Harrod s model is usually exposed through 3 equations: (1) S 1 = s Y 1 (2) I 1 = v r ( Y 0 ) (3) S 1 = I 1 The first equation expresses the fact that savings originate from income (through the Keynesian multiplier process). S 1 may be defined as capacity saving, that is the amount of saving forthcoming from a normally utilized level of productive capacity (what normal degree of capacity utilization means will be defined shortly; for the while think of the normal degree as full degree of utilization). The second equation is the investment function and shows (gross) investment undertaken in time 1 as dependent on the expected variation of AD from period 0 to period 1. It is a simple formulation of the accelerator model. The first term on the right-hand side is the so-called capital 1 Keynes calls it effective demand to distinguish it from the theoretical level of aggregate demand necessary to absorb potential output.

6 6 27/10/2013 coefficient v = K/Y, or v =. Its role is to translate the expected growth of AD in investment. Suppose that v = 2. It means that if AD is expected to grow by 10bn, capitalists will wish to install 20bn of capacity. (Of course they do expect that the increase of AD is permanent, since many years will be necessary to recover investment costs). Notably, the investment function based on the accelerator is the most plausible and empirically robust explanation of investment decisions. For the moment take the third equation as an equilibrium condition in the good market (all output is sold if investment absorb all capacity saving) Harrod s model as a dynamic Say s Law The simplest way to look at Harrods model is as a dynamic version of Say s Law. Suppose, for simplicity, that capitalist are the only savers. That is, workers consume all wages. This is often called the Classical hypothesis since it reminds of the Classical idea that wages tend to a historically determined substance level (their view is more complicated, the interest reader should refer to Stirati 1994). 3 Be it as it may, Say s Law maintains that if capitalists invest all their saving, there are no problems of AD, all output is sold. In this first formulation we re-interpret equation 3 as maintaining that capitalists systematically invest all their savings: (4) S 1 = I 1 Since we cannot have two investment equations (2 and 4), this means that equation 2 must be re-interpreted. Let us write it as: (5) Y 1 - Y 0 = I 1 /v r In this form it just tells us that new output capacity created at time 1 is equal to investment carried out at time 1 divided by the capital coefficient. This equation is no more an investment function but a mere definition: given investment I 1, which is explained by equation 4, equation 5 shows the amount on new capacity which is created. On the opposite, equation 4 is not an 2 "The axiomatic basis of the theory which I propose to develop consists of three propositions, namely: (a) that the level of a community income is the most important determinant of its supply of saving; (b) that the rate of increase of its income is an important determinant of its demand for saving; and (c) that demand is equal to supply. It thus consists in a marriage of the 'acceleration principle' and the 'multiplier' theory" (1939, p.43). 3 The Theory Of Wages In Classical Economics - A Study of Adam Smith, David Ricardo and their Contemporaries, Edward Elgar.

7 7 27/10/2013 equilibrium equation, but the investment function, as the arrow we put above it suggests. So the model is now (1) S 1 = s Y 1 (5) Y 1 - Y 0 = I 1 /v r (4) S 1 = I 1 To solve the model, substitute I 1 from equation 5 and S 1 from equation 1 in equation 4 to obtain: sy = vr Y ), or 1 ( 1 Y0 where g w is the warranted rate of growth. The Say/Harrod model must be interpreted as saying that if capitalists invest all their savings, the economy will grow at an equilibrium growth rate g w. As such, this is not a very interesting model (and certainly Harrod did not interpret it this way). In general, indeed, we should not expect capitalist to invest all their saving. Even if we maintain the simplification that capitalists are the only savers, we may think that they take their saving decision independently from their investment decisions (as if in a capitalist household the husband takes the saving decision and the wife the investment decisions, without coordination between the two). Can we still presume that the system will find an equilibrium growth path? In this regard, the best interpretation of Harrod s model has been provided by Amartya Sen (see H.Jones Chapter 2). Let us return to the system 1-3. Before, however, let us dwell a moment on the concept of degree of capacity utilization Degree of capacity utilization When capitalists plan their investment, they expect that installed capacity (or capital stock) will be utilized at planned level. Usually this planned level is not 100% (full utilization). This is so because capitalists intend to maintain an amount of spare capacity in case of peaks of demand. A restaurant, for instance, might expect to serve on average 80 meals per day. This average, however, is the result also of expected peaks of demand, say, days in which 100 meals are requested. Since the restaurant does not want to leave customers unsatisfied, it will install an extra-capacity of 20 meals. So we may define a normal degree of capacity utilization u n = 0.8, or, where Y n is normal or average expected output and Y f is full-capacity output. The full-capacity degree of utilization is clearly equal to 1. The actual degree of capacity utilization is defined as, where Y a is actual output.

8 8 27/10/2013 Return now to the normal capital coefficient v n that we used above. This is equal to, where Y n is normal expected output when K* was installed. Suppose then that u a < u n. This also implies that Y a < Y n and that v a > v n, where v a = K*/Y a is the actual capital coefficient. This is intuitive: if output is lower than expected, the installed capital stock will be excessive relatively to actual output. These relations will be quite relevant in the remaining of these notes The troubles with Harrod Reconsider our original system (1) S 1 = s Y 1 (2) I 1 = v r ( Y 1 Y 0 ) (3) S 1 = I 1 e Substitute 1 and 2 in 3 to obtain sy1 = vr (Y1 e - Y0), or: (4) s/vr = (Y1 e - Y0)/Y1. Capitalist have invested I 1 expecting to that expected, that is if: Y1 e = Y1. In which case will they be correct? Multiply both side of equation 4 by Y1/Y1 e to get: e Y 1. They will be happy if actual output at t =1 is equal (5) s v r Y e 1 Y1 e e 1 Y1 Y Y0 = (5) The term on the right-hand side is the expected growth rate g e. Suppose that Y1 e = Y1, that is, suppose that the expectations are fulfilled. Then g e = s/vn. Look at this result in this way. Suppose that capitalists expect a rate of growth equal to s/v n, then their expectations will be satisfied, that is Y1 e = Y1. In other words, of capitalists expect the warranted rate of growth and invest consistently, then the economy will grow in equilibrium at this rate. However, there is no reason why the entrepreneurs should know g w, which is a ratio between two magnitudes, s and v n, that are in general unknown to capitalists. So, in general, we should not expect that the economy will grow along a warranted growth rate. There are two cases in which, however, we might expect the economy to grow at g w. One is if capitalists behave following Say s Law. As seen in section 1.1, if capitalists systematically invest

9 9 27/10/2013 all their savings, the economy will actually grow at the warranted rate. Possibly capitalists meet and agree to follow Say s Law. But, as Kalecki put it, capitalists do many things as a class, but they do not invest as a class. The other case is that they defer to a planning office the calculation of g w that is then imposed to the capitalist class as a mandatory rate of capital accumulation. Harrod s model was indeed quite popular in the 1950s when some mixed-economy (capitalist with some socialist element) countries with like India or France tried some economic planning. In general, however, capitalist will refuse investment decisions imposed from outside. But this is not all. According to Harrod s model, a market economy not only will not, in general, grow at an equilibrium rate, but it will also progressively diverge from the equilibrium, warranted rate. Indeed, we should expect that whenever the economy is out of equilibrium that is, output is larger or smaller than AD capitalists try to adjust their decisions. We shall shortly show that precisely this behavior will led the economy to diverge from the equilibrium even more. To neoclassical economists Harrod s model left behind other problems. Even if entrepreneurs correctly guess the warranted rate and the actual growth rate is an equilibrium one, there is no reason why this would be a full-employment path. This is not a problem, of course, for scientifically minded scholars. Neoclassical economist want however to show that laissez-faire economies grow at full-employment. A less malevolent interpretation is the following. In general, we do not see middle or high income economies with persisting excesses of unemployed labour force (at least not excess of unsubsidized labour population). Neoclassical economists interpret this much stylised facto as the result of the adaptation of the growth rate of output to that of population, so that excess labour is re-absorbed via wage flexibility. Less conventional economists share the Classical point of view that it is labour population growth that tends to adjust to economic growth and labour demand, not the other way round. In the short-run this happens through migrations and by resorting to the buffer stock consisting of marginal labour figures such as women and very young or older workers. In the longrun demographic changes may occur in fertility and mortality rates. Be it as it may, neoclassical economists are disappointed that given an exogenous growth rate of the labour force n s/vn, in general The Harrodian instability From the definitions of ge e di ga we obtain: 4 and 4 Recall that: g e e Y1 Y0 Y0 = =1 e che e e Y1 Y1 Y Y0 Y = = 1. a 0 ga Ya Ya

10 10 27/10/2013 Recalling that s v n e Y1 Y Y Y0 = gw = = ge we obtain: Y Y Y e 1 1 e 1 e 1. Substituting and Y 1 (as defined above), we find:. Observe that if ge = gw, then ga = ge (this reinstates the idea that if capitalists foresee the warranted rate, then their expectations are confirmed). It also follows that if: - ge > gw, then (1 - ge) > (1 - ga), that is, ga > ge. In other words, if capitalists are overoptimistic, the actual growth rate is even larger than expected; so they will feel that they had been too pessimist and will expect an even higher rate resulting in an even larger actual rate and so on and so forth. The economy will diverge more and more from the warranted rate. The discrepancy between ga and gw can also be observed as one between the normal capital coefficient v n and the actual capital coefficient v a. The latter is the actual ratio K/Y that capitalists experience at the end of the year. Note that ga > gw, or s v a s = ga > = gw, means that v a < v n. That is if the economy is v n growing faster than the equilibrium growth rate, capitalist will find a capital coefficient lower than normal they conclude they have invested too little. This unhappiness can also be seen by the fact that the degree of capacity utilization will be higher than normal u a > u n. AD and output have been higher f than expected, and they find they have too little capacity installed. Therefore, next period they will install even more expecting a growth rate g e even higher than g e, that is g e > g e > g w. As a result g a > g e and so on. - If, on the other hand, capitalist are too pessimist, so that ge < gw, then the actual rate will be even lower than expected ga < ge, v a > v n and u a > u n and the economy will diverge more and more from the warranted rate in a bottomless recession. In actual, capitalists are neither to optimist or too pessimist: the question is that they just do not know the warranted rate so we should not expect the economy to grow in equilibrium Discussion A merit of Harrod s model is to show the instability of a market economy. In this capacity it thus led less-conventional economists to suggest that indicative economic planning might be helpful. Alternatively, economic policies, fiscal and monetary may avoid that the economy

11 11 27/10/2013 overheats or over-freezes. For instance, if the economy sees its growth rate progressively falling, an expansionary fiscal policy might sustain AD reverting investors expectations from a progressive pessimism to optimism. A lax monetary policy may contribute by sustaining autonomous consumption. If, on the other hand, economic policy backs the Harrodian instability as the austerian economic policies adopted in the Eurozone since 2010 the economy will continue its downward spiral. In spite of these merits, Harrod s model has also been felt unsatisfactory since real economies do not show the violent kind of instability predicted by the model even if policy intervention is absent. As a consequence, in the 1950s a group of close followers of Keynes tried to prove that a market economy may find a way to stable Harrodian growth rate even without an external guidance or assistance. This attempt might appear rather strange now, since Harrod model has a non- Keynesian feature that, as we shall see, these post-keynesian authors (as they are sometimes labeled) do not amend. Simply consider the warranted growth equation g w = s/v n and note that the equilibrium rate is positively associated to the saving propensity. This result, and the associated prescription to keep real wage low to sustain the capitalists higher propensity to save, sound quite non-keynesian implications. This is why in the last couple of decades this earlier post-keynesian attempt has been abandoned and substituted by new non-conventional approaches considered later in this lecture notes. Turning to neoclassical economists, they were indeed pleased by the non-keynesian implications of the warranted growth rate but, on the other hand, they were not ready to the idea that market economies could not grow in equilibrium without the State visible hand assisting them. We shall see that Robert Solow in 1956 pretended to solve the second question the invisible hand leads the economy to a stable (and full employment) growth path, but paying the price of losing the positive relation between the saving rate and the warranted growth rate. In the next sections we shall focus, respectively, upon the Solowian and the Post-Keynesian approaches and on their more recent developments. One basic difference between the two approaches is the following. At the basis of the Harrodian instability there is disequilibrium between the investment decisions and capacity savings. Recall that the warranted rate s g w = is v precisely defined as that rate at which, over time, investment is equal to capacity saving. In Harrod there are not mechanisms, however, that adjust the two magnitudes when, as in general expected, they are different. Characteristic of the Solowian solution is that it is investment that adapts to capacity saving. Moreover for Solow g w = n. Suppose that g g. We shall see that the w a r adjustment is such that g > g and that it will take place through the adjustment of v a to v n a w

12 12 27/10/2013 (something that concerns investment rather than capacity saving, s is indeed given). On the opposite, Post-Keynesian economists will in general believe that the rate of capital accumulation g k is decided by capitalists and for no reason this rate is equal to n. In their view, once capitalists decide investment, it the invisible hand that leads capacity saving to adapt to investment. In terms of our growth equations, capitalists decide is given). The figure below shows our road map. g = g and s will change in order that a k g > g (while v n w a

13 13 27/10/2013 Harrod Solow Cambridge equation EGT Neo-kaleckian and/versus Sraffians

14 14 27/10/ Neoclassical Growth Theory: Solow s exogenous growth (sorry, some parts are in 2. Il modello neoclassico di Solow 2.1. La funzione di produzione pro-capite Italian) Secondo la teoria neoclassica la crescita economica è determinata dal lato dell offerta in seguito alla accumulazione dei fattori produttivi, oltre che dal progresso tecnico. Come sappiamo, infatti, la teoria marginalista ritiene valida la Legge di Say per cui non vi sono mai problemi di mercato per il prodotto sociale. In particolare sappiamo che qualunque ammontare di risparmio troverà impiego sotto forma di investimento produttivo, così come la flessibilità del salario reale assicura il pieno impiego del lavoro. Sinora, tuttavia, abbiamo esaminato gli effetti degli investimenti solo sotto il profilo della domanda, come componente della domanda aggregata, ma non sotto quello della accumulazione di capitale, cioè come accrescimento della capacità produttiva. Di questo si occupa la teoria della crescita di cui consideriamo in questo capitolo la versione marginalista (nel prossimo capitolo vi sarà un accenno alla teoria keynesiana della crescita). Come sappiamo, per gli economisti neoclassici la teoria tende, in assenza di rigidità all operare del mercato, al pieno impiego. Questi economisti non sono dunque tanto interessati alla crescita del prodotto aggregato, quanto alla crescita del reddito pro-capite. Se vi fosse infatti ampia disoccupazione, allora saremmo interessati in primis alla crescita del Pil aggregato avendo in mente il riassorbimento della disoccupazione. Ma in assenza di disoccupazione diventiamo interessati nell aumento del benessere individuale identificato nel Pil pro-capite. Questa è naturalmente una misura rozza del benessere che non tiene conto delle numerose esternalità negative dello sviluppo economico come l inquinamento, il deterioramento dei rapporti sociali tradizionali e così via. Partiamo dalla funzione di produzione: Y = A F(K, N). Si vede qui come aumenti di K e di N (lavoro) determinino un aumento di Y. Il termine A rappresenta il progresso tecnico: anche un suo aumento conduce a un aumento di Y. Su questa base esaminiamo il modello messo a punto dall economista americano Robert Solow nel 1956 e che ancora oggi costituisce il cuore della descrizione neoclassica della crescita. Definiamo Y y= come il prodotto per lavoratore, 5 e N K k = lo stock di capitale per N lavoratore. Su questa base scriviamo la funzione di produzione pro-capite. Dividiamo innanzitutto 5 Useremo talvolta l espressione per lavoratore e altre volte pro-capite. Pro-capite può riferirsi anche all intera popolazione e non ai soli lavoratori. Tuttavia rammentiamo che qui c è la piena occupazione, e che gli occupati possono essere considerati come una quota costante dell intera popolazione. Quindi è semplice passare da una misurazione all altra.

15 15 27/10/2013 ciascun termine della funzione di produzione aggregata per il numero dei lavoratori N: Y N K N = AF(, ) = AF( k,1), ovvero y= Af (k). Questa funzione ci racconta che il prodotto pro- N N capite dipende dalla dotazione di capitale per lavoratore. La sua forma grafica è illustrata nella figura 1. Agiscono i rendimenti marginali decrescenti. Dato lo stock di lavoro N, quando cresce la dotazione di capitale pro-capite il prodotto marginale cresce a ritmi sempre minori (la funzione marginale y k è decrescente). If we employ a Cobb-Douglass production function α 1 α F ( K, L) = AK L we get: y = Y / L= F( K, L)/ L= F( K / L, L/ L) = A or α y= Ak. There are constant returns to scale since: α 1 α α+ 1 α K L α 1 α α 1 α L L F ( zk, zl) = A( zk) ( zl) = z K L = zf( K, L) The marginal product of capital (useful later) is: Y α α MPK = = αak 1 L 1. K α is capital s income share since: α = MPK * K Y α 1 αak = AK 1 α L α 1 α L * K.

16 16 27/10/2013 Figura Stati stazionari e l equazione fondamentale dello modello di sviluppo neoclassico Questo modello studia, per cominciare, i cosiddetti stati stazionari, equilibri di lungo periodo in cui prodotto e capitale per lavoratore sono costanti (il che non vuol dire che K ed N non crescano in aggregato). Come sappiamo in questa teoria gli investimenti, richiesti per accrescere il capitale per lavoratore, richiedono un atto di risparmio. I risparmi, tuttavia, non servono solo a fornire ulteriori attrezzature ai lavoratori già attivi, ma anche a sostituire i beni capitali che vanno fuori uso per obsolescenza fisica o tecnica, e per dotare i nuovi lavoratori (i giovani, diciamo) della dotazione media di capitale per addetto di cui godono i lavoratori già attivi. Facciamo un esempio. Supponiamo che la nostra economia consista di una cooperativa di 20 persone che lavora con 20 elaboratori. La cooperativa risparmia 5000 all anno. Ogni anno 2 pc, pari al 10% del totale dei beni capitali, vanno fuori uso. Un pc nuovo costa 1000 (lo stock di capitale vale dunque ). Allora dei risparmi (lordi), 2000 vanno accantonati per rimpiazzare i pc andati fuori uso. Supponiamo anche che la cooperativa cresca del 10% l anno e che dunque assuma 2 giovani lavoratori. Allora altri 2000 di risparmi vanno destinati a dotare i due nuovi lavoratori del loro pc. Avanzano 1000 di risparmi. Questi possono essere utilizzati per incrementare lo stock di capitale per addetto (che è ora di un pc da 1000 per lavoratore), ad esempio potenziando la memoria dei pc in uso. L equazione che segue traduce analiticamente la nostra parabola: kn = sy δ K nk = sy ( δ + n) K Essa va letta così: sy è l offerta di risparmio aggregato ( S = sy, dove s è la propensione marginale al risparmio che già conosciamo); tale offerta di risparmio può essere destinata per rimpiazzare i beni capitali δk andati fuori uso (δ è la quota di K che va fuori uso), oppure ad attrezzare i nuovi lavoratori nk (dove n è il loro tasso di aumento annuo: n= N ). N 6 Ciò che avanza può essere utilizzato per incrementare la dotazione di capitale per lavoratore ( k è tale incremento, grandezza che moltiplichiamo per il numero dei lavoratori). 7 kn per N: = N Riscriviamo ora l espressione in termini pro-capite semplicemente dividendo entrambi i lati Y s N ( δ + n) K N, ovvero: 6 Per cui nk = Nk chiarisce che nk è la domanda di dotazione media per lavoratore k che proviene dai nuovi lavoratori che sono N. 7 Se applicassimo la formula all esempio avremmo 1000 = , o più in dettaglio: 45,5 x 22 = 5000 (0,1 x ) (0,1 x ).

17 17 27/10/2013 k = sy ( δ + n) k (1) Questa è l equazione fondamentale del modello di crescita neoclassico (o modello di Solow). Essa ci racconta che ciò che avanza dal risparmio pro-capite una volta effettuati i rimpiazzi (in termini pro-capite) e attrezzati i nuovi lavoratori (in termini pro-capite) può essere destinato ad incrementare la dotazione di capitale pro-capite. Equation (1) can also be derived with this method: Equation (2) below has the obvious meaning that the capital stock can increase if something is left from the saving supply (sy) once we have replaced the worn-out plants (δk): dk / dt = K & = sy δk (2) Recall the take logs and then derivatives rule: k& K& L& k K / L logk = logk logl =. k K L Write equation (2) as: K & / K = sy / K δ K & k& L& k& sy / L and recalling that =, we get = n δ or K k L k K / L k & = sy ( n+ δ ) k L equazione ha un'espressione grafica piuttosto immediata (figura 2). Il suo lato sinistro k è la differenza fra due espressioni, sy e (δ + n)k, nel lato destro. Esaminiamole. La sy o saf(k) ha lo stesso andamento grafico della y= Af (k) (figura 1), solo in po più bassa in quanto ne prendiamo una quota pari alla propensione al risparmio (se s = 0,2, di ogni valore di y ne prendiamo il 20%). La (δ + n)k è una retta di coefficiente angolare (δ + n). La loro distanza nel grafico è proprio k.

18 18 27/10/2013 dove Figura 2 Si osserva graficamente che nel punto di intersezione fra la sy e la (δ + n)k, cioè sy = (δ + n)k, k risulta uguale a zero. Quello è lo stato stazionario, cioè il punto in cui l offerta di risparmio è precisamente sufficiente per la domanda di capitale per i rimpiazzi e per attrezzare i nuovi lavoratori (tutto misurato in termini per addetto). Lo stato stazionario è un punto di equilibrio stabile, cioè quando l economia non è in quel equilibrio essa vi tende. Infatti alla sinistra del punto di equilibrio, per esempio dove k = k 0, si vede che la funzione sy giace sopra la funzione (δ+ n)k, dunque sy > (δ + n)k. Se ne conclude che alla sinistra dell equilibrio k è positivo, cioè il risparmio pro-capite è più che sufficiente per i rimpiazzi e per attrezzati i nuovi lavoratori e dunque una parte di esse può essere destinato ad incrementare la dotazione di capitale pro-capite. Se k>0, vuol dire che k sta aumentando e dunque il suo valore converge verso k*. Viceversa alla destra del punto di equilibrio, per esempio dove k = k 1, si vede che la funzione sy giace sotto la funzione (δ + n)k, dunque sy < (δ + n)k. Dunque alla destra dell equilibrio k è negativo, cioè il risparmio pro-capite non è sufficiente per i rimpiazzi e per attrezzati i nuovi lavoratori e dunque una parte del capitale pro-capite deve essere

19 19 27/10/2013 destinato a questi due scopi e la dotazione di capitale pro-capite diminuisce, cioè k<0. Ciò vuol dire che k sta diminuendo e dunque il suo valore converge verso k*. Se torniamo alla nostra parabola, alla sinistra di k* i risparmi della cooperativa sono più che sufficienti per rimpiazzare i pc andati fuori uso e per attrezzare i nuovi soci, per cui c è spazio per accrescere la potenza dei pc in dotazione a ciascuno. Alla destra di k* i risparmi della cooperativa non sono sufficienti per rimpiazzare i pc andati fuori uso e per attrezzare i nuovi soci, per cui dobbiamo, per così dire, togliere un pezzetto di pc a ciascun membro per assemblare i pc che mancano per i rimpiazzi e per i soci entranti; la potenza dei pc in dotazione a ciascuno viene perciò a diminuire. * sy s Si osservi che nel punto di stato stazionario nk* = s*y (trascurando δ), ovvero n = = in * * k v quanto y/k= (Y/L)/(K/L) = Y/K = 1/v. Si ricordi che s/v è il tasso di crescita dello stock (aggregato) di capitale in quanto: s/v = reddito aggregato. I / K = I / K = K / K = K / Y gk. Dato v*, g K è anche il tasso di crescita del If our economy has a per-capita capital endowment equal to k*, the per-capita endowment will not change over time. This is called steady state (or stationary equilibrium). Using the Cobb-Douglas production function α y= Ak and the steady state condition sy - (δ + n)k = 0, we can calculate the value of k*: α s( k*) ( δ + n) k* = 0 or 1/(1 α ) sa k * =. n + δ Substituting in the production function we can also calculate the steady state level of output per-worker α /(1 α ) α 1/(1 α ) s y * = A( k*) = A (5) n+ δ Stabilità Per l'analisi della stabilità si deve considerare che alla sinistra di k* l'offerta di risparmio pro-capite è superiore alla domanda che proviene dalla necessità di attrezzare i nuovi lavoratori.

20 20 27/10/2013 Sfruttando le relazioni ora note si ha che sf(k) > (δ + n)k, ovvero che sf ( k) k = s > n+ δ, dove v v n = k/y = (K/N)/(Y/N) è il valore normale (o desiderato dagli imprenditori) del coefficiente di capitale. v n deve aumentare per realizzare l'equilibrio. Vi è in questa situazione spazio per un accrescimento della dotazione individuale media di capitale. Implicitamente si sta qui assumendo che l'eccesso di risparmio nel mercato dei capitali conduca ad una diminuzione del tasso di interesse monetario, e ad una maggiore convenienza per gli imprenditori di adottare tecniche a maggiore intensità di lavoro 8 e perciò ad un aumento di k. (Si rammenti che la pendenza della f(k) è la produttività marginale del capitale che gli imprenditori confortano col tasso di interesse, che è il costo opportunità del capitale; sino a che Pmk > i - a sinistra di k* - gli imprenditori hanno convenienza ad accrescere k sino a che Pmk = i) L'aumento di k determina la diminuzione di v = f(k)/k. Infatti a causa dei rendimenti marginali decrescenti del capitale, un aumento del denominatore k determina un aumento meno che proporzionale del numeratore, e il rapporto diminuisce. It should observe that all textbook just identify investment and saving decisions, taking s as the investment rate. This is highly misguiding since it obscure the neoclassical mechanisms on which Solow s model is founded. Si osservi la rilevanza delle critiche in tema di teoria del capitale menzionate in altra parte del corso. Non è affatto detto che minori tassi di interesse determino l adozione di tecniche a maggiore intensità di capitale. Alla luce di queste critiche il modello di Solow appare instabile. Let us look at the gravitation process also this way. Recall that the capital coefficient at k* is y* v * = and the long period growth rate g* = s/v*. Note that on the left on the equilibrium point at k * k 0 the capital coefficient is actual growth rate at k 0, that is: y 0 v 0 = and, given s which does not depend on k, vi may calculate the k0 g a = s v 0. Note that v 0 < v* since y 0 /k 0 < y*/k*. Indeed, although both k 0 < k* and y 0 < y* given the curvature of the production function k 0 is much lower than y 0, as a visual inspection would confirm (if any student wants to show it analytically I will consider the proof for the exam). This implies that on the left hand side of k* the actual rate of growth is higher than the long period one, that is g a > g*. The economy will tend, as we know, to g*. This means that g a g*. This is not surprising because in the gravitation process both k and y n 8 Ovvero la diminuzione del prezzo dei beni che impiegano relativamente più capitale ne induce una maggiore domanda facendo così innalzare anche per questa via il rapporto K/L.

21 21 27/10/2013 are increasing but the increase in k is larger than that of y, given the marginal decreasing returns that shape the curvature of the production function (a visual inspection will again confirm this, but students might want to provide an analytical proof). Therefore the actual v is increasing from k 0 to k*. The intuition is that during the gravitation process firms are increasing the capital intensity of techniques (K/L) and this also rises the capital coefficient (K/Y). The reader will repeat the exercise moving from a disequilibrium point on the right hand side of k*. It is useful that you notice that on the left hand side of k*, during the convergence process, g a > g*, that outside equilibrium the actual rate of growth is higher or lower than in equilibrium, respectively, on the left and in the rigt of the long period position Statica comparata Studiamo ora gli effetti di mutamenti nel valore dei principali parametri, s e n, che governano l equilibrio di lungo periodo. - se varia la propensione al risparmio da s a s, la funzione sy si sposta verso l alto. Il nuovo equilibrio è caratterizzato da un valore più elevato di k e di y di stato stazionario. Non sorprendentemente un aumento dell offerta di risparmio ha, in questa teoria, effetti positivi generando un aumento del capitale e del reddito pro-capite (figura 3). Since per-capita output is higher in the new stationary equilibrium, during the transition the rate of growth will be higher than the steady state rate n + δ. We have indeed already noticed that outside equilibrium g a g * and that, more spcifically, ga > g* on the left hand side of k*. This proves that although there are no long run growth effects of a rise in s, there are transition effects. The question is then the length of the transition process. We shall return on this.

22 22 27/10/2013 Figura 3 - Se aumenta n passando a n, cioè aumenta il tasso di crescita della forza lavoro, la funzione (δ + n)k diventa più ripida (figura 4); il capitale per addetto e, di conseguenza, il reddito pro-capite, diminuiscono. Ciò non sorprende in quanto l offerta di risparmio deve servire ad attrezzare un numero maggiore di nuovi lavoratori, e dunque è necessario ridurre il capitale procapite. Dilution of capital.

23 23 27/10/2013 Figura 4 Technical change Ci si può tuttavia domandare quale sia l interesse per gli stati stazionari, in cui k ed y sono costanti. Dopotutto lo sviluppo economico degli ultimi duecento anni ha portato a un costante aumento di k e di y. Sinora abbiamo visto che un aumento di s porta sì a un aumento del reddito procapite, ma tale aumento si arresta, a meno che s continui ad aumentare, ciò che non è plausibile (anche se per assurdo i soggetti risparmiassero tutto il proprio reddito, s non potrebbe superare il valore 1). Un maggiore ritmo di crescita della popolazione fa per giunta diminuire il reddito procapite. Per spiegare la crescita secolare del prodotto pro-capite dobbiamo ricorrere a un elemento sinora trascurato: il progresso tecnico (PT), dunque il termine A nella funzione di produzione y=af(k) Il PT ha molteplici origini. In primo luogo nei progresso scientifici e tecnologici che danno luogo a invenzioni fondamentali come i motori a combustione, la produzione e impiego di energia elettrica, le scoperte nella chimica, elettronica e così via. Tutte queste invenzioni fondamentali

24 24 27/10/2013 danno luogo a innumerevoli altre innovazioni. Molte innovazioni minori sono inoltre il frutto dell esperienza nei processi produttivi (o cosiddetto learning by doing). Nel modello di Solow originario il PT era considerato esogeno, come una manna che scendeva dal cielo. In realtà Solow riteneva che il PT fosse il frutto delle attività umane, ma che gli economisti non avessero molto di sistematico da dire sulle sue origini. Meglio allora supporlo esogeno al modello (cioè non spiegato dal modello). L ipotesi era che A aumentasse in maniera costante con il passare del tempo sulla scorta dell esperienza dalla rivoluzione industriale in poi che ha visto la comparsa di un flusso pressoché costante di innovazioni. Analiticamente il tasso di progresso tecnico m veniva definito come: m= A. Graficamente un aumento costante nel tempo A di A si manifesta come uno spostamento progressivo verso l alto della funzione di produzione (figura 5). E facile osservare come a tale spostamento corrisponda un continuo spostamento dell equilibrio di stato stazionario (un equilibrio stazionario mobile, se si perdona l ossimoro) per cui k ed y aumentano nel tempo, come indica l esperienza empirica.

25 25 27/10/2013 y y 3 y = A 2 f(k) y 1 y = A 1 f(k) y 2 y = A 0 f(k) s y =sa2f(k) sy =s A 1 f(k) sy =s A0f(k) k 0 k 1 k Figure (5) Figure (6) decomposes the increase in pc output into two passages. To begin with, the shift of the production function determines, given the initial pc capital endowment k 0, a rise of pc output from F to G. Subsequently, the higher saving supply that derives from the higher income is such that the pc capital endowment also rises and pc output increases from G to H.

26 26 27/10/2013 y y 1 H y = A 1 f(k) y 0 F E G y = A 0 f(k) D s y =s 1 f(k) sy =s A0f(k) B k 0 k 1 k Figure 6 Note that, as in the case without technical progress, in equilibrium we have g* = s/v* = n + m. Since s/v* = I*/K = K/K where I* is the investment rate equal to capacity saving at the equilibrium point, this mean that the aggregate rate of capital g K * is equal to n + m and so the aggregate rate of output growth g Y * = g K *. The rates of growth in per capita terms are respectively: g k * = g K * - n = m and g y * = g Y * - n = m. More analytically, constant exogenous technical change A & A= m can be described by the technical progress function (3): A 0 mt = A e (3) where the term m = A & A represents the constant rate of technical change. To calculate the steady state values of k and y let us follow the following steps (C.Jones). Step 1

27 27 27/10/2013 The aggregate and pc production functions with technical change can respectively be expressed as: α 1 α F ( K, AL) = K ( AL) (4) and α k α y= A 1. Entered this way technical progress is said to be labour augmenting. 9 Define now k ~ K / AL and ~ y Y / AL as the pc capital endowment in efficiency units and the pc output in efficiency units, respectively. AL can indeed be considered as the amount of labour employed in the economy augmented by technical progress. The pc production function can then be α rewritten as ~ ~ y = k. We can now derive again the Solowian fundamental equation of economic growth including technical change this time. Step 2 Take the log and the derivatives of k ~ K / AL to obtain: ~ & k K& A& L& K& ~ = = m n k K A L K Recall that ~ & ~ k = sy ~ ( n+ m+δ ) k Step 3 (5) ~& K & / K = sy / K δ n so to obtain: k sy / AL = m n k K / AL δ or: ~ & In a steady state equilibrium k = 0 mutandis). Comment., or sy ~ ~ ( n m δ ~ = + + ) k at the point k * (refer to figure 3, mutatis The new equilibrium is not a stationary equilibrium any more, balanced growth would be a better ~ ~ & definition. At k * aggregate capital is growing at a rate g = n + m (from equation 5 with k = 0), and so aggregate output Y. This growth can be divided in two. There is a capital widening process, since the capital stock grows at the rate of population (or work force) growth n and a capital deepening process, since the pc capital endowment calculated on physical labour is growing at the rate m just note that in equilibrium K & A& L& ~ & = +. Carefully observe that in the steady state k = 0, K A L 9 The other two cases of technical change are capital augmenting F(AK, L) and neutral AF(K,L). With a Cobb-Douglas production functions the three kind of technical progress are equivalent. For instance, α /( 1 α) α 1 α redefining A = B equation (k) becomesf ( BK, L) = ( BK) ( L).

28 28 27/10/2013 but k & = m. So even without population growth, k and y are now growing because of exogenous technical progress. Not much else would change compared to the Solow model without technical change (Jones (2013: 39-43; Weil (2009: )) If our economy has a per-capita capital endowment equal to k*, the per-capita endowment will not change over time. This is called steady state (or stationary equilibrium). (Step 3 cont.) Using the Cobb-Douglas production function ~ & condition k = 0, so that sy ~ ~ = ( n+ m+ δ ) k, we can calculate the value of k*: ~ α ~ s( k*) ( δ + n+ m) k* = 0 or 1/(1 α ) ~ s k * =. n+ m+ δ Substituting in the production function ~ s y * = n+ m+ δ α /(1 α ) We can also calculate the steady state level of output per-worker α /(1 α ) ~ ~ * α 1/(1 α ) s y = A( k*) = A (5) n+ m+ δ α y= Ak and the steady state 2.5. Is Solow s model empirically satisfactory? Dall equazione (5) si vede che, data la quota di profitti sul reddito α and the depreciation coefficient δ, il livello of per-capita output positively depends on the saving propensity s (which in steady state is equal to the investment rate, the ratio between I and Y) and negatively on the rate of growth of the labour force. Weil (2009) estimates the predictive power of this result with regard to the actual differences in pc income among countries. Supposing equal the value of the other variables (including population growth rates which are supposed zero, while α is taken at its standard value of 1/3 and δ = 0.05), figure 7 shows the actual and the predicted values of the pc GDP relatively to the U.S. of various countries that have different s (= I/K). If the predictive power were perfect, all values would be aligned along the 45 line. As it can seen, countries with a higher investment ratios tend to be wealthier, but the model tend to underestimate the differences (poor countries are predicted richer than they actually are) (ibid: 63-66).

29 29 27/10/2013 figure 7 Isolating population growth as the explanatory variable supposing all the other variables equal Weil (ibid: 92) estimates a limited explanatory value of the model: a country with zero population growth would have income per worker 34% higher than one with 4% population growth. This would underestimate the real differences in pc income that are shown in figure (8): figure8

30 30 27/10/2013 In defence of the explanatory value of Solow s model Weil points out two elements on which we shall return: (a) if we use a value of α = 2/ 3 in the estimations, then the country with zero population growth would be 3.24 richer than the one with 0.04 growth (ibid: 98). We shall later specify the meaning of this assumption. (b) particularly developing countries might still have to reach their steady state path, and in the catching up transition they might be investing a large share of output without having reached the corresponding steady state pc output level. (ibid: 66). [From an heterodox point of view we are not surprised that countries in a catching up process invest a large share of output while they still show a low pc income]. Weil conclude that while each variable considered in isolation has a limited explanatory value, if we add their explicatory power up, Solow s model does not disfeatures Differences in growth rates and the transition to the steady state Factor (b) can also help to amend (from a neoclassical point of view, do not forget) a serious gap in this model: its inability to explain the differences of growth rates among countries or, at least, it refers these differences to the exogenous rates of population growth and technical change. In actual, if countries are at different stages of the transition to the steady state, they may show different growth rates among themselves and with countries that have already reached their steady state. Let us investigate the analytics of the transition (Weil: 80-2; Jones 2013: 44). We know from the above that α k = sak ( δ + n) k. Dividing both side by k, we obtain the growth rate of the pc capital endowment: ˆ k α 1 k = = sak ( δ + n) k Because α is less than one, as k rises, the growth rate of k gradually declines. Figure (9) shows the two terms on the right-hand side of this equation. If: sak sak α 1 α 1 > δ + n > kˆ > 0 < δ + n > kˆ < 0