Logic programming for knowledge-intensive interactive applications
10 Pages
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Logic programming for knowledge-intensive interactive applications

Downloading requires you to have access to the YouScribe library
Learn all about the services we offer
10 Pages


  • mémoire - matière potentielle : usage
  • exposé
  • dissertation - matière potentielle : series
  • system interface
  • logic programming for knowledge
  • f.a.h. van harmelen prof
  • 8.8 implementation
  • 9.1.5 discussion
  • 8.1 introduction
  • prolog
  • systems
  • programming



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Language English


The Diamond Model
How does this Overlapping Generations Model explain the basic questions about growth? Dennis Paschke Course: Topics in Economic Theory 2 (EC4307) Lecturer/Tutor: Dr Laurence Lasselle
1 Introduction_____________________________________________________________ 3 2 The Diamond Model ______________________________________________________ 3 3 Growth in the Diamond Model ______________________________________________ 4 4 Deficiencies of the Diamond Model __________________________________________ 8 5 Conclusions_____________________________________________________________9 Notes _____________________________________________________________________ 9 References ________________________________________________________________ 10
1 Introduction This essay serves to develop the Diamond Model which was first developed by Peter A. Diamond (1965). However, the emphasis is not placed on the development of the model itself 1 which can be found in many advanced textbooks but on a careful presentation of the model to be able to understand how this model explains the basic questions of economic growth. These are: (1)
Where does economic growth comes from? (2) Why have some countries higher growth rates than others? (3) Why are some countries richer than others? Section 2 presents the diamond model as far as it is useful for later discussions. Section 3 argues how the diamond model answers the basic questions (1)-(3) about economic growth. Section 4 discusses benefits and deficiencies of the model in explaining economic growth. Section 5 is to summarise and to conclude.
2 The Diamond Model The Diamond Model is a so-called Overlapping Generations Model (OLG). In every period t witht=0,1,2,..., i.e. time is
discrete, there are always two types of households, young and old who are continually born or are continually dying, respectively. Moreover, every household is assumed to live for only two periods. The population is assumed to grow at rate n, i.e. inhabitants born in period t and period t-1, respectively, Ltand Lt-byL1n)L and 1, respectively, are determinedt=(+t1
L L n, respectively. Each young inha t1=t/(1+supplies one) bitant unit of labour and distributes the resulting income to first-period consumption and savings, respectively. Each old inhabitant only consumes first-period savings including interest. Every household maximizes utility by means of a constant-relative-risk-aversion utility function
Growth in the Diamond Model
1θ1θ C1C 1t2t+1 u(C,C) . t1t2t+1= +withθ>0;ρ> −1 1θ1+ρ1θ Where C1tand C2t+1is the consumption of a household born in period t when it is young and old, respectively.θis the constant-relative-risk-aversion coefficient which determines the willingness of a household to shift first-period consumption to the second period. The smaller isθthe more willing is the household to shift consumption to gain utility, i.e. the higher isθthe more risk-averse the household is which means he prefers to consume current income in the current period and would ask for a high return on savings before he shifts a small fraction of income. Finally, is the rate of depreciation. If is greater (smaller)
than zero the household places a greater (lower) weight on first-2 period consumption than on second-period consumption. A production functionY=F(K,A Lwhere K represents) , t t t t capital, A the technology or the effectiveness of labour and L labour itself, is assumed. It follows the usual and well-known properties. Finally, markets are assumed to be perfect, i.e. every input earns its marginal product and firms make zero profits.
3 Growth in the Diamond Model – a Special Case of Logarithmic Utility and Cobb-Douglas Production Assume as a special case a Cobb-Douglas production function
α in intensive formf(k)=kwheref(k)is the output per unit of 3 effective labour andk is capital per unit of effective labour. Suppose further thatθ=1 in the households utility function 4 which then simplifies to a logarithmic utility function. This means that a household has constant time preferences to consume. Put in other words, savings which determine second-period consumption are independent of the interest rate r, i.e. the saving rate, ceteris paribus, is constant. With these assumptions it can be
Growth in the Diamond Model
shown that capital per unit of effective labour in period t+1, kt+1, 5 is determined as follows:
1 1 α k(1 )kt+1= −αt (1+n)(1+g) 2+ρ
Where g is the growth rate of the technology. Consider a special case where 0 k rises above k k1= =k,t+1 tfor small t+t
* khe optim values of kt, crosses a 45° line whenkt=, t al capital
* per unit of effective labour, and remains below ktfor allk>k. t Figure 3.1 shows the values of kt+1a function of k as tthis for special case.
* k k2
Figure 3.1: Capital per unit of effective labour in the special case of logarithmic utility and Cobb-Douglas production.
Two equilibria exist in this case. One at the origin which is not stable and one at point E which is stable. To see this consider an original capital per unit of effective labour k0 as in Figure 3.1. The corresponding capital endowment in period 1 can then be found on the vertical axis. Take this k1as given the corresponding
Growth in the Diamond Model
endowment in period 2, k2, can again be found on the vertical axis. No matter where the original capital per unit of effective labour endowment is in period 0 the economy will smoothly * converge to E. The same applies for every k0greater than k . This special case provides an answer for the first and the second question about growth. Countries experience economic growth because of the convergence to their optimal capital endowment. For the second question note that the growth rate of capital is the difference between the kt+1function and the 45° line * (see Figure 3.1). With k0smaller (greater) than k the growth rate is positive (negative). It is obvious that the growth rate is zero at the origin. Is then positive and growing if kt diverges from the * origin and converges to k . After a maximum growth rate (in Figure 3.1 around k0) the growth rate remains positive but * decreases as ktgets closer to k . Therefore, as Figure 3.1 shows, a country with an initial capital endowment k0has a higher growth rate than a country with an initial capital endowment k2. This can explain the differences between growth rates in poor countries, transition countries and rich countries. Growth rates are usually low in poor countries, high in transitional countries and again low in rich countries (see Table 3-1 for some empirical examples). However, so far the third question can be answered only partially. During convergence a country with a higher initial capital endowment remains richer than a country with a lower initial capital endowment. Mathematically, the convergence process takes an infinite number of periods, i.e. the equilibrium will never be reached. That means a rich country will always be richer than a poor country. However, after some periods the difference between rich and poor countries should converge and will eventually become small enough that it cannot be noticed anymore.
Growth in the Diamond Model
Table 3-1: Average growth rates in poor countries, transition countries and developed countries between 1990-1999 Country Annual average real growth of GDP in 1990s [%] Algeria 1,6 Angola 0,8 Brazil 2,9 Kenya 2,2 Chad 2,3 Colombia 3,3 South Africa 1,9 Uzbekistan 2,0 Venezuela 1,7 Zimbabwe 2,4 Chile 7,2 China 10,7 India 6,1 Malaysia 6,3 Myanmar (Burma) 6,3 Poland 4,7 Singapore 8,0 South Korea 5,7 Taiwan 6,3 Vietnam 8,1 Australia 3,8 Belgium 1,7 Canada 2,3 France 1,7 Germany 1,4 Italy 1,2 Japan 1,4 Spain 2,2 United Kingdom* 2,2 United States of A. 3,4 * includes North Ireland Source: Fischer Weltalmanach 2002
If an economy has reached a stable equilibrium it is on a so-called balanced growth path. Like in other growth models, 6 namely the Solow Model and Ramsey Model, the saving rate is then constant. A more satisfying answer to the third question can then be developed. Suppose, the discount rate is lower in
economy i than in economy j, i.e. the young in economy i save more than the young in economy j. It is obvious from the kt+1-
i j kk. function that everything else being equal1>k+1for allt>0 t+t i * Figure 3.2 shows that E comes along with a higher k compared j to E . Unless either economy i saves less permanently or economy
j saves more permanently, economy i will be richer than economy
j permanently.
Deficiencies of the Diamond Model
j E
j* k
i E
i* k
i kt+1-function
j kt+1-function
i j <
Figure 3.2: Optimal capital per unit of effective labour in two economies with different saving rates.
4 Deficiencies of the Diamond Model Many questions about economic growth remain unanswered. Four main problems should be mentioned. First, introducing a government sector into the model delivers the following results: a temporary increase in government spending leads to a temporary
change in the rate of growth of capital. Whereas Kocherlakota and Yi (1996) found that a temporary increase of at least certain types of government spendings, e.g. non-military equipment, lead to persistently higher growth rates. Second, the properties of the production function require that kt+1smaller than k is tk if t is sufficiently large. Therefore, unbounded capital growth is
impossible. That implies that total economic growth per worker in the long run cannot be explained by the growth rate of capital because this is eventually zero, i.e. total economic growth per worker can only be explained by increasing effectiveness of labour which is exogenous. Third, it can be shown that the equilibrium is not pareto efficient. In fact, it can be shown that there is an equilibrium where every household were better off. Fourth, after a complete convergence towards the equilibrium the
economy remains on its balanced growth path where the growth
rate of capital per unit of effective labour is zero and the capital/output ratio as well as the output per worker is growing at a constant rate. But even in developed countries, who are supposed to be close to an equilibrium, growth is not constant but varies over time. Such variations cannot be explained with the Diamond Model. Historical investigations support the view that a single balanced growth path does not exist but that growth on such a path varies itself (Nick Crafts 1995). Crafts suggests that macroeconomic inventions, which occur intermittent, give rise to endogenous technological improvements. This cannot be
explained with a model in which the parameter for technological
improvements is exogenous.
5 Conclusions The Diamond model answers basic questions about economic growth in parts. Even a special case can explain short run growth, growth differences, and the differences between rich and poor countries. A more general case can give even better answers to questions about multiple equilibria. But other explanations are missing. First and foremost depends long run economic growth on an exogenous variable and thus cannot be explained at all within the model. Moreover, some empirical findings seem to disprove the result of the model that temporary changes in variables which drive the growth process have only temporary effects on growth. Overall, the Diamond model gives answers to basic questions about economic growth but they are not sufficient.
Notes 1 See David Romer (2001, p 75-90) for a general development of the whole Diamond Model. 2 The assumption that>−1ensures that households consume at least something in the second period, i.e. the weight on second-period consumption cannot be zero or even negative. A negative weight would imply that a household would use a credit in the first-period to increase its overall utility.
However, a household is not allowed to use a credit in this model because by assumption a household has no income in the second period to repay any credit. 3 For a derivation of the intensive form of a Cobb-Douglas production function see Romer (2001) p 11-12. 4 See Romer (2001) p 48-49. 5 See Romer (2001) p 76-80. 6 The Solow Model was developed by Robert Solow (1956) and T.W. Swan (1956). The Ramsey Model goes back to F.P. Ramsey (1928).
References Crafts, Nick F.R. (1995),Exogenous or Endogenous Growth? The Industrial Revolution Reconsidered, inThe Journal of Economic History, Volume 55, Issue 4, 745-772. Diamond, Peter A. (1965),National Debt in a Neoclassical Growth Model, in The American Economic Review, Volume 55, Issue 5, 1126-1150. Kocherlakota, Narayana R.; Yi, Kei-Mu (1996),A Simple Time Series Test of Endogenous vs. Exogenous Growth Models: An Application to the United States, inThe Review of Economics and Statistics, Volume 78, Issue 1, 126-134. Ramsey, F.P. (1928),A Mathematical Theory of Saving, inThe Economic Journal, Volume 38, Issue 152, 543-559. Romer, David (2001),Advanced Macroeconomics, McGraw Hill, New York. Solow, Robert M. (1956),A Contribution to the Theory of Economic Growth, inThe Quarterly Journal of Economics, Volume 70, Issue 1, 65-94. Swan, T.W. (1956),Economic Growth and Capital Accumulation, inEconomic Record, Volume 32, Issue 4, 334-361. Miscellaneous Publications: Der Fischerweltalmanach 2002, Frankfurt a.M.