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The Solow model Stylised facts of growth The Solow model Steady state and convergence

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The Solow model Until now, when output was changing, it was due to economic fluctuations in the IS-LM or AS-AD models. Long run growth, however, determines the capacity of the economy to produce goods and services, and ultimately welfare: 1913 : Argentina’s GDP is 70% larger than Spain’s. 2000 : Spain’s GDP is 50% larger than Argentina’s. 1945 : Ghana’s GDP is 60% larger than Korea ’s. 2000 : South Korea’s GDP is 100% larger than Ghana’s. 1970 : Italy’s GDP is 50% larger than Ireland’s. 2000 : Ireland’s GDP passes Italy’s GDP. What are the causes of economic growth? How can one maintain growth?

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The Solow model 5 Stylised facts The Solow model Convergence to the steady state Growth and convergence

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Stylised fact 1 : Sudden acceleration of output US Industrial production index (Source: NBER) 0 500 1000 1500 2000 2500 17901795180018051810181518201825183018351840184518501855186018651870187518801885189018951900190519101915

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Stylised fact 2 : Medium run fluctuations in growth Real GDP per capita (1950 =100) Source: Penn Tables 6.1

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Stylised fact 3 : Persistent lags and catch-up ARG AUS AUT BEL BOL BRA CAN CHE COL CRI DNK EGY ESP ETH FIN FRA GBR GTM HND IND IRL ISL ISR ITA JPN KEN LKA LUX MAR MEX MUS NGA NIC NLD NOR NZL PAK PAN PER PHL PRT SLV THA TTO TUR UGA URY USA VEN ZAF 0 10000 20000 30000 40000 GDP per capita 2000 0200040006000800010000 GDP per capita 1950

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Stylised fact 3 : Persistent lags (USA=100) 0 10 20 30 40 50 60 70 80 90 100 196019621964196619681970197219741976197819801982198419861988199019921994199619982000 CameroonIvory CoastGabonRwandaSenegal Real GDP per capita Source: Penn Tables 6.1

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Stylised fact 3 : Catch-up (USA=100) Real GDP per capita (1950 =100) Source: Penn Tables 6.1

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Stylised fact 4 : Increased inequality between countries Source: Bourguignon & Morrison (2003) 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 18201850187018901910192919501960197019801992 Inequality between countriesInequality within countries

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Stylised fact 5 : Biased technical change The technological evolutions linked to growth seem to favour skilled labour, leading to a loss of jobs in traditional sectors This is called “skill-biased technical change”. This increases income inequality because it changes the structure of the demand for labour. Keeping labour supply unchanged this leads to either An increase in unemployment A fall in relative wage between skilled/unskilled labour This phenomenon is neither universal or permanent The post-war boom did not affect unskilled labour negatively

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5 Stylised facts 1.World output has seen an abrupt acceleration over the long run. 2.GDP per capita and productivity can fluctuate significantly in the medium run. These fluctuations are not necessarily synchronised across countries. 3.Some countries have been able to catch up with the living standards of the richest countries, while other countries have stagnated relative to rich countries. 4.Inequalities have increased and shifted from inequalities within countries to inequalities between countries. This has slowed down since the 90’s, mainly because of the take-off of the Chinese and Indian economies. 5.Technical progress is biased as in increases income inequalities, either by reducing the wages of the unskilled labourers, either by increasing unemployment (i.e. By reducing their employability).

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The Solow model 5 Stylised facts The Solow model Convergence to the steady state Growth and convergence

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The Solow model The Solow model is based on several simplifying assumptions Joan Robinson ironically referred to the lack of realism of these assumptions by referring to the “Kingdom of Solovia” A1Factors of production are substitutes and not complements. A2Savings generate investments, which is consistent with the neoclassical interpretation of the savings/investment balance. A3The interest rate is perfectly flexible and instantaneously adjusts investment and savings. A4Wages adjust such that the supply of labour (set exogenously by the growth rate of the population) and the demand for labour adjust perfectly

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The Solow model The macroeconomic production function Production is a function of capital K and L (with exogenous growth rate n ) It exhibits constant returns to scale Simplification : By dividing by the amount of labour L, one can express the variables “per capita”:

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The Solow model Capital per worker Output per worker Decreasing marginal returns: each extra unit of capital per worker reduces the marginal productivity of capital 1 Output y = f(k)

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The Solow model Capital per worker Output per worker Output y = f(k) Investment i= s × f(k) y Output per worker c Consumption per worker i Investment per worker Income is either spent or saved : Additionally, savings are equal to investment : Therefore:

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The Solow model This tells us that given a production technology and a level of population, the level of output will depend only on the available stock of capital. This stock is determined by two flows: Investment : the capital stock increases when firms purchase new equipment. We have just seen how this is determined. Capital consumptions, which reduce the stock of capital available to workers. This is what we look at next. Capital stock per worker Investment Capital consumptions

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The Solow model Capital consumptions 1: Discounting Capital stock is reduced by depreciation. As the capital stock grows older, its value is discounted The amount of discounting is given by the discount rate δ. For example, if the expected life of a piece of equipment is 20 years, the discount rate is around 5%. This gives δ≃0,05. With a capital stock k, the size of the discount is equal to δk

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The Solow model Capital consumptions 2: Population growth In the long run, populations are not constant. This creates a second capital consumption, as one needs to provide capital to the new workers: Lets assume a fixed capital stock K : If the population grows at a rate n, the expenditure required to keep the the capital stock per worker equal to k is equal to nk

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The Solow model Capital consumptions 3: Technical progress If new technologies are introduced, workers become more productive. Less labour is required to produce the same amount of output ⇒ Some workers become available for other uses Technical progress is therefore equivalent to an increase in the number of workers, in other words to population growth (we shall call this growth g). The net variation of the capital stock per worker is therefore given by the following equation : Δk = i – (δ+n+g)k

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The Solow model Capital per worker Capital consumption Capital consumption (δ+n+g)k Expenditure required to maintain this level of capital per worker

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The Solow model 5 Stylised facts The Solow model Convergence to the steady state Growth and convergence

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Convergence to the steady state Capital per worker ( k ) Investment & consumption flows Investment i = s×f(k) Capital consumptions ( δ+n+g)×k k2k2 i2i2 (δ+n+g) ×k 2 k1k1 i1i1 (δ+n+g) × k 1 The capital stock increases as investment is higher than capital consumptions The capital stock falls as consumptions are higher than investment (δ+n+g)×k*=i* k* Steady-state level of capital per worker

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Convergence to the steady state Capital per worker ( k ) s 1 ×f(k) …increases the steady-state capital stock k2* k1* New steady- state s 2 ×f(k) Capital consumptions (δ+n+g)k Initial steady- state An increase in the savings ratio… Investment & consumption flows

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Convergence to the steady state

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k s × f(k) (δ+n 1 +g) ×k Capital per worker k1*k1* 1. A higher growth rate of the population… (δ+n 2 +g) ×k 2... Reduces the capital stock per worker… k2*k2* 3. …And therefore reduces the steady-state capital stock. The Solow model predicts that countries with high demographic growth rates should have a lower level of per-capita income, ceteris paribus. Investment & consumption flows

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Convergence to the steady state

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The concept of steady-state has three central implications : An economy at steady state no longer changes. An economy that isn’t at the steady-state will tend to move towards it. It therefore defines the long run equilibrium of the economy. However: the steady state depends on the savings ratio, therefore there is space for an economic growth policy.

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Convergence to the steady state Output, investment, and consumption flows c2c2 i2i2 Capital consumption (δ+n+g)k c1c1 i1i1 Investment i 2 = s 2 × f(k) Investment i 1 = s 1 × f(k) Which of the 2 steady states is socially preferable ? Production y = f(k) The savings ratio and the golden rule Capital stock per worker

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Convergence to the steady state Capital stock per worker Investment i 2 = s 2 × f(k) c2c2 i2i2 Capital consumption (δ+n+g)k Investment i 1 = s 1 × f(k) c1c1 i1i1 Which of the 2 steady states is socially preferable ? Production y = f(k) The savings ratio and the golden rule Output, investment, and consumption flows

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Convergence to the steady state Production y = f(k) Investment i*= s* × f(k*) c* i* Capital consumption (δ+n+g)k The optimal steady- state maximises consumption This occurs when the slope of the production function is equal to the slope of the capital consumption function The savings ratio and the golden rule Capital stock per worker Output, investment, and consumption flows

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Convergence to the steady state t Fall in the savings ratio Investment ( i ) Consumption ( c ) Production ( y ) t0t0 Transition to the golden rule steady-state Starting off with too much Capital

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Convergence to the steady state t Increase in the savings ratio Investment ( i ) Consumption ( c ) Production ( y ) t0t0 Transition crisis, which requires political intervention and arbitrage Transition to the golden rule steady-state Starting off with too little Capital

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The Solow model 5 Stylised facts The Solow model Convergence to the steady state Growth and convergence

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Empirical analysis of growth (%Δ real GDP) Country1948-19721972-19951995-2000 Germany5.72.01.7* Canada2.91.82.7 United States2.21.52.9 France4.31.62.2 Italy4.92.31.4 Japan8.22.61.1 United Kingdom2.41.82.5

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Growth and convergence ARG AUS AUT BDI BEL BEN BFA BGD BOL BRA BRB CAN CHE CHL CHN CIV CMR COG COL COM CPV CRI DNK DOM DZA ECU EGY ESP ETH FIN FRA GAB GBR GHA GIN GMB GNB GNQ GRC GTM HKG HND IDN IND IRL IRN ISL ISR ITA JAM JOR JPN KEN KOR LKA LSO LUX MAR MDG MEX MLI MOZ MUS MWI MYS NER NGA NIC NLD NOR NPL NZL PAK PAN PER PHL PRT PRY ROM RWA SEN SGP SLV SWE SYC SYR TCD TGO THA TTO TUR TZA UGA URY USA VEN ZAF ZMB 2 4 6 8 10 Average annual growth rate 01000200030004000 GDP per capita (1960) Source: Penn Tables 6.1 Convergence (All countries)

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Growth and convergence ARG AUS AUT BEL CAN CHE DNK ESP FIN FRA GBR GRC IRL ISL ISR ITA JPN LUX NLD NOR NZL PRT SWE USA 5 6 7 8 Average annual growth rate 100015002000250030003500 GDP per capita (1960) Source: Penn Tables 6.1 Convergence (OECD Countries)

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Growth and convergence Source: Penn Tables 6.1 Convergence (Non OECD countries)

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Growth and convergence Convergence, as predicted by the Solow model, is not a universal phenomenon. Not all countries seem to be converging… Disparities between groups of countries can be explained by differences in the determinants of the steady state. Rate of investment Growth rate of the population Level of technology Convergence only occurs between countries that have the same steady state!

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