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Introducing Advanced Macroeconomics: Chapter 3 – second lecture Growth and business cycles CAPITAL ACCUMULATION AND GROWTH: THE BASIC SOLOW MODEL ©The McGraw-Hill Companies, 2005 The basic Solow model (repetition) Yt BK t L1t 1 Kt rt B Lt Kt wt 1 B Lt St sYt Kt 1 Kt St Kt Lt 1 1 n Lt , n 1 • Parameters: B, ,s, and n . B constant: we focus on capital accumulation • Given K0 and L0 the model determines Yt , Kt , Lt , rt , wt and St ©The McGraw-Hill Companies, 2005 Government sector • Kt 1 Kt St Kt follows from the identity/definition Kt 1 Kt I t Kt and the national accounting identity Yt Ct It Yt Ct It St It • Same construction with a public sector: Yt Ctp Ctg I tp I tg Stp Stg Yt Tt Ctp Tt Ctg Itp I tg • Insert into the identity Kt 1 Kt I tp I tg Kt to get St Kt 1 Kt Stp Stg Kt ©The McGraw-Hill Companies, 2005 • We can use the model as it stands. We just have to reinterpret K t as the sum of private and government capital stock and St as the sum of private and public savings. Similarly the equation St sYt should be reinterpreted as: St Stp Stg sYt Ctp Ctg 1 s Yt • The essential assumption underlying the Solow model interpreted to include a government is thus that the sum of private and public consumption as a fraction of GDP is a constant, 1 s . • This seems plausible empirically. And s 0.2 seems plausible for many Western countries: ©The McGraw-Hill Companies, 2005 The consumption share of GDP in several Western countries USA 1 United Kingdom 1 Ct / Yt Ct / Yt 0.8 0.8 p Ct / Yt p 0.6 0.6 0.4 Ct / Yt 0.4 Cgt / g Ct / Yt Yt 0.2 0.2 0 0 1960 1965 1970 1975 1980 1985 1990 1995 2000 Netherlands 1 1965 1970 1975 1980 1985 1990 1995 2000 1985 1990 1995 2000 Belgium 1 Ct / Yt 0.8 1960 Ct / Yt 0.8 p 0.6 p Ct / 0.6 Yt 0.4 Ct / Yt 0.4 Cgt / Yt g Ct / Yt 0.2 0.2 0 0 1960 1965 1970 1975 1980 1985 1990 1995 2000 1960 1965 1970 1975 Finland 1 1980 Denmark 1 0.8 Ct / Yt 0.8 Ct / Yt 0.6 Ct / Yt p 0.6 p Ct / Yt 0.4 0.4 g g 0.2 0 1960 Ct / Yt Ct / Yt 1965 0.2 1970 1975 1980 1985 1990 1995 2000 0 1960 ©The McGraw-Hill Companies, 2005 1965 1970 1975 1980 1985 1990 1995 2000 The basic Solow model, short version Yt BK t L1t Kt 1 Kt sYt Kt Lt 1 1 n Lt • In previous lecture we saw that the Solow model leads to the transition equation: kt 1 • 1 syt 1 kt 1 n and to the Solow equation: 1 sBkt n kt kt 1 kt 1 n ”technical term” appearing because of discrete time savings per capita syt Replacement investment to compensate for depreciation and growth of labour force ©The McGraw-Hill Companies, 2005 The Solow diagram (repetition) 1 kt 1 kt sBkt n kt 1 n (n + )kt sBkt k* kt Why does growth in kt and yt have to stop? Diminishing returns! ©The McGraw-Hill Companies, 2005 Comparative analysis in the Solow diagram 1. The economy is initially in steady state with parameters B, ,s, and n . What happens if the savings rate increases permanently from s to a new and higher level, s' ? (n + )kt s’Bkt sBkt k0 k* kt ©The McGraw-Hill Companies, 2005 2. The economy is initially in steady state. No parameters change, but an exogenous event, e.g., a war or natural disaster, reduces the capital stock to half size ”over night”. How is this analysed in the Solow diagram? (n + )kt sBkt k* / 2 k* kt ©The McGraw-Hill Companies, 2005 Steady state • Last time we saw that the Solow model implies convergence to a unique steady state. From the Solow equation 1 kt 1 kt sBkt n kt 1 n one easily computes k B * and then 1 1 s n 1 1 1 s n 1 1 s r* B k * n y B k * * w* 1 B B 1 1 1 1 s n 1 1 y* ©The McGraw-Hill Companies, 2005 Some main lessons (repeated from previous lecture) 1 lny* B lns ln n . 1 1 1 : 1 2 • The elasticity of y wrt. s is an increase in the savings rate of 10%, e.g. from 20 to 22%, gives a long run increase in income per worker of around 5% according to the basic Solow model! * * • The elasticity of y wrt. B is 1 3 1 ! 1 2 Note that the effect is stronger than one-to-one due to capital accumulation. • Another steady state prediction concerns the real interest rate… ©The McGraw-Hill Companies, 2005 The ”natural” interest rate 1 1 s s * r* n n • The real rate of interest is determined by productivity and thrift in the long run (Knut Wicksell): – Higher capital is more productive demand of capital per worker increases (ceteris paribus) higher equilibrium interest rate (the price of capital). – Higher s / n supply of capital per worker increases the equilibrium interest rate decreases. • Reasonable parameter values on an annual basis, 1 / 3, s 0.22, n 0.005, 0.05, imply r* 8.3% * and * 3.3% . This value for is very close to empirical observations of the real interest rate! ©The McGraw-Hill Companies, 2005 Structural policy 1. Crowding out: • • • • Consider a permanent fall in s caused by a permanent increase in government consumption as a percentage of GDP. What happens on impact? Yt is unaffected and still grows at the rate of n, and yt is unaffected. But savings decrease and consumption increases. There is full crowding out. What happens in the longer run? During a transitional period Yt grows more slowly than at the rate of n and yt falls down to a new lower steady state level. There is more than full crowding out. And the real interest rate increases. The government cannot increase GDP by raising government expenditure in the long run. How about the short run? (Book Two) ©The McGraw-Hill Companies, 2005 2. Motives for tax-financed public services from a long run perspective • Public investments (that would not be made by private agents) For government consumption: • Public (non-rival and possibly non-excludable) goods • Public consumption, e.g., on education and health care, replacing private consumption, which means that s is not affected • Distributive reasons • Externalities (education) • General productivity effects of public consumption, e.g., judicial system, health care etc. ©The McGraw-Hill Companies, 2005 3. Incentive policies: – Policies that do not affect model parameters directly through government expenditure/revenue, but indirectly through the way they affect private behaviour. We cannot analyze incentive policies explicitly because private behaviour has not been derived from optimization. – Golden rule: 1 s 1 * 1 y B n c* B 1 1 s 1 s n 1 The s that maximizes c* , which is s** , is called the golden rule savings rate. – The model suggests structural policies that • promote technology • encourage savings (assuming that s is considerably below s** ): institutions and incentive • reduce n ©The McGraw-Hill Companies, 2005 Growth in the basic Solow model • The long run prediction of the Solow model is its steady state. What is the growth rate of GDP per capita in steady state? • Zero! Not in accordance with stylized facts. • What is the growth rate of Yt in steady state? Since Yt / Lt yt and yt is constant, Yt and Lt must grow at the same rate, n . GDP grows, but only at the same rate as the labour force. Why is that? • Assume that the economy initially is below steady * state implying that kt k . Then sBkt n kt , implying that capital per worker increases. But, once again, because of diminishing returns the growth in kt and yt will ultimately cease. • However, there is transitory growth. How long-lasting ©The McGraw-Hill Companies, 2005 is that? Simulation • Initially we are in steady state with the following parameter values: B 1, 1 / 3, 0.05, n 0.03, s 0.08 representing a developing country. This implies that k* y* 1. • With effect first time in period 1, the savings rate increases permanently to s' 0.22 corresponding to the savings rate of a typical Western economy, implying k * ' 5.20 and y* ' 1.73. • Starting with k0 1 in period zero and one we now simulate 1 s' Bkt 1 kt 1 n over t 2,3,... . We also calculate yt Bkt , ct 1 s' yt and gty lnyt lnyt 1and drawt the 1, 2evolutions ,3,... etc. for of these kt 1 variables in the following figures. ©The McGraw-Hill Companies, 2005 The evolution of yt , ct and gty after the increase in s . ©The McGraw-Hill Companies, 2005 The evolution of yt , ct and gty after the increase in s . (continued) The figures show that transitory growth is relatively long-lasting. ©The McGraw-Hill Companies, 2005 • In the Solow model, the transition towards steady state is at least as important as the steady state itself. And during this transition there is growth in kt and yt . Hence, the basic Solow model is a growth model! • It is easy to find the growth rate of kt : 1 kt 1 kt sBkt n kt 1 n kt 1 kt 1 sBkt 1 n kt 1 n This is called the modified Solow equation. y k g g y • The growth rate of t follows from t t . ©The McGraw-Hill Companies, 2005 The modified Solow diagram kt 1 kt 1 1 sBk n t kt 1 n ©The McGraw-Hill Companies, 2005 • Growth in GDP per worker is higher the further below steady state the economy is. This is in accordance with conditional convergence. • A permanent increase in s gives a jump upwards in the growth rate of GDP per worker. ©The McGraw-Hill Companies, 2005 Conclusions based on the basic Solow model • What can a (poor) country do to create a transitory growth in GDP per worker resulting in a permanently higher level of income and consumption per worker? The basic Solow model provides the following answers: – – – – Increase the savings rate Reduce the growth rate of the labour force Reduce the rate of depreciation, i.e. invest better Improve the level of technology • How useful are these recommendations? ©The McGraw-Hill Companies, 2005