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In [[mathematics]], particularly [[algebraic topology]] and [[homology theory]], the '''Mayer–Vietoris sequence''' is an [[algebra]]ic tool to help compute [[algebraic invariant]]s of [[topological space]]s, known as their [[Homology group|homology]] and [[cohomology group]]s. The result is due to two [[Austria]]n mathematicians, [[Walther Mayer]] and [[Leopold Vietoris]]. The method consists of splitting a space into pieces, called [[Subspace topology|subspaces]], for which the homology or cohomology groups may be easier to compute. The sequence relates the (co)homology groups of the space to the (co)homology groups of the subspaces. It is a [[Natural (category theory)|natural]] [[long exact sequence]], whose entries are the (co)homology groups of the whole space, the [[direct sum of abelian groups|direct sum]] of the (co)homology groups of the subspaces, and the (co)homology groups of the [[intersection (set theory)|intersection]] of the subspaces.
{{Technical|date=August 2011}}
{{Development economics sidebar}}
'''Endogenous growth theory''' holds that [[economic growth]] is primarily the result of [[endogeny|endogenous]] and not external forces.<ref>{{cite journal |journal = [[The Journal of Economic Perspectives]] |volume= 8 |issue= 1 |year= 1994 |url= http://links.jstor.org/sici?sici=0895-3309%28199424%298%3A1%3C3%3ATOOEG%3E2.0.CO%3B2-H |first= P. M. |last= Romer |title= The Origins of Endogenous Growth |authorlink= Paul Romer |page = [http://www.iset.ge/old/upload/Romer%201994.pdf 3]  }}</ref> Endogenous growth theory holds that investment in [[human capital]], [[innovation]], and knowledge are significant contributors to economic growth. The theory also focuses on [[positive externalities]] and [[spillover effects]] of a knowledge-based economy which will lead to economic development. The endogenous growth theory also holds that policy measures can have an impact on the long-run growth rate of an economy. For example, [[subsidies]] for [[research and development]] or [[education]] increase the growth rate in some endogenous growth models by increasing the incentive for innovation.


The Mayer–Vietoris sequence holds for a variety of [[cohomology theory|cohomology]] and [[homology theory|homology theories]], including [[singular homology]] and [[singular cohomology]]. In general, the sequence holds for those theories satisfying the [[Eilenberg–Steenrod axioms]], and it has variations for both [[Reduced homology|reduced]] and [[Relative homology|relative]] (co)homology.  Because the (co)homology of most spaces cannot be computed directly from their definitions, one uses tools such as the Mayer–Vietoris sequence in the hope of obtaining partial information. Many spaces encountered in [[topology]] are constructed by piecing together very simple patches. Carefully choosing the two covering subspaces so that, together with their intersection, they have simpler (co)homology than that of the whole space may allow a complete deduction of the (co)homology of the space. In that respect, the Mayer–Vietoris sequence is analogous to the [[Seifert–van Kampen theorem]] for the [[fundamental group]], and a precise relation exists for homology of dimension one.
==Models in Endogenous Growth==


==Background, motivation, and history==
In the mid-1980s, a group of growth theorists became increasingly dissatisfied with common accounts of [[exogenous]] factors determining long-run growth. They favored a model that replaced the exogenous growth variable (unexplained technical progress) with a model in which the key determinants of growth were explicit in the model. The initial research was based on the work of [[Kenneth Arrow]] (1962), [[Hirofumi Uzawa]] (1965), and [[Miguel Sidrauski]] (1967).<ref>{{cite web |url= http://www.newschool.edu/nssr/het/essays/growth/moneygrowth.htm |title=Monetary Growth Theory  |work=newschool.edu |year=2011 [last update] |accessdate=11 October 2011}}</ref> [[Paul Romer]] (1986), [[Robert Emerson Lucas, Jr. |Lucas]] (1988),<ref>{{cite journal |url= http://www.fordham.edu/economics/mcleod/LucasMechanicsEconomicGrowth.pdf |title= On the mechanics of Economic Development |first= R. E. |last= Lucas |authorlink= Robert Emerson Lucas, Jr. |journal = [[Journal of Monetary Economics]] |year=1988 |volume = 22  }}</ref> and Rebelo (1991)<ref>{{cite journal |url=http://www.nber.org/papers/w3325 |title= Long-Run Policy Analysis and Long-Run Growth |first= Sergio |last= Rebelo  |journal = [[Journal of Political Economy]] |year=1991 |volume = 99 |issue= 3 |page= 500 }}</ref><ref>{{cite web |url= http://www.econ2.jhu.edu/people/ccarroll/public/lecturenotes/Growth/RebeloAK.pdf |title= The Rebelo AK Growth Model  |first= C.|last= Carroll |work=econ2.jhu.edu |year=2011 [last update] |accessdate=11 October 2011 |quote= the steady-state growth rate in a Rebelo economy is directly proportional to the saving rate.}}</ref> omitted technological change. Instead, growth in these models was due to indefinite investment in [[human capital]] which had [[spillover effect]] on economy and reduces the diminishing return to [[capital accumulation]].<ref name= "BX">{{cite book |first1= R. J. |last1= Barro |first2= Xavier |authorlink2= Xavier Sala-i-Martin |last2= Sala-i-Martin |title= Economic Growth |isbn= 978-0-262-02459-4 |date= 1998-11-20 |url= http://mitpress.mit.edu/books/chapters/0262025531chap1.pdf  }}</ref>


[[Image:Vietoris4343.jpg|Right|thumb|Leopold Vietoris on his 110th birthday]]
The [[AK model]], which is the simplest endogenous model, gives a constant-saving-rate of endogenous growth. It assumes a constant, exogenous saving rate and fixed level of the technology. It shows elimination of diminishing returns leading to endogenous growth. However, the endogenous growth theory is further supported with models in which agents optimally determined the consumption and saving, optimizing the resources allocation to research and development leading to technological progress. Romer (1987, 1990) and significant contributions by Aghion and Howitt (1992) and Grossman and Helpman (1991), incorporated [[imperfect market]]s and R&D to the growth model.<ref name= "BX"/>


Like the [[fundamental group]] or the higher [[homotopy group]]s of a space, homology groups are important topological invariants. Although some (co)homology theories are computable using tools of [[linear algebra]], many other important (co)homology theories, especially singular (co)homology, are not computable directly from their definition for nontrivial spaces. For singular (co)homology, the singular (co)chains and (co)cycles groups are often too big to handle directly. More subtle and indirect approaches become necessary. The Mayer–Vietoris sequence is such an approach, giving partial information about the (co)homology groups of any space by relating it to the (co)homology groups of two of its subspaces and their intersection.
== The AK Model ==
{{main|AK model}}


The most natural and convenient way to express the relation involves the algebraic concept of [[exact sequence]]s: sequences of [[Object (category theory)|objects]] (in this case [[Group (mathematics)|groups]]) and [[morphism]]s (in this case [[group homomorphism]]s) between them such that the [[Image (mathematics)|image]] of one morphism equals the [[Kernel (algebra)|kernel]] of the next. In general, this does not allow (co)homology groups of a space to be completely computed. However, because many important spaces encountered in topology are [[topological manifold]]s, [[simplicial complex]]es, or [[CW complex]]es, which are constructed by piecing together very simple patches, a theorem such as that of Mayer and Vietoris is potentially of broad and deep applicability.
The model works on the property of absence of diminishing returns to capital. The simplest form of production function with diminishing return is:
[[File:Ak model.png|thumb|figure 1.1]]
:<math>Y = AK\,</math>


Mayer was introduced to topology by his colleague Vietoris when attending his lectures in 1926 and 1927 at a local university in [[Vienna]].<ref>{{harvnb|Hirzebruch|1999}}</ref> He was told about the conjectured result and a way to its solution, and solved the question for the [[Betti number]]s in 1929.<ref>{{harvnb|Mayer|1929}}</ref> He applied his results to the [[torus]] considered as the union of two cylinders.<ref>{{harvnb|Dieudonné|1989|p=39}}</ref><ref>{{harvnb|Mayer|1929|p=41}}</ref> Vietoris later proved the full result for the homology groups in 1930 but did not express it as an exact sequence.<ref>{{harvnb|Vietoris|1930}}</ref> The concept of an exact sequence only appeared in print in the 1952 book ''Foundations of Algebraic Topology'' by [[Samuel Eilenberg]] and [[Norman Steenrod]]<ref>{{harvnb|Corry|2004|p=345}}</ref> where the results of Mayer and Vietoris were expressed in the modern form.<ref>{{harvnb|Eilenberg|Steenrod|1952|loc=Theorem 15.3}}</ref>
where
{{-}}
:<math> A\,</math> , is a positive constant that reflects the level of the technology.  


==Basic versions for singular homology==
:<math> K \,</math> capital (broad sense to include human capital)
Let ''X'' be a [[topological space]] and ''A'', ''B'' be two subspaces whose [[Interior (topology)|interiors]] cover ''X''. (The interiors of ''A'' and ''B'' need not be disjoint.) The Mayer–Vietoris sequence in [[singular homology]] for the triad (''X'', ''A'', ''B'') is a [[long exact sequence]] relating the singular homology groups (with coefficient group the integers '''Z''') of the spaces ''X'', ''A'', ''B'', and the [[intersection (set theory)|intersection]] ''A''∩''B''.<ref>{{harvnb|Eilenberg|Steenrod|1952|loc=§15}}</ref> There is an unreduced and a reduced version.


===Unreduced version===
:<math>y = AK\,</math> , output per capita and the average and marginal product are constant at the level <math>A>0\,</math>
For unreduced homology, the Mayer–Vietoris sequence states that the following sequence is exact:<ref name="Hatcher149">{{harvnb|Hatcher|2002|p=149}}</ref>


<br /><math>\begin{align}
If we substitute <math>\frac{f(k)}{k}=A \,</math> in equation of transitional Dynamics of Solow-Swan model ([[Exogenous growth model]]) which shows how an economy’s per capita incomes converges toward its own steady-state value and to the per capita incomes of other nations.
\cdots\rightarrow H_{n+1}(X)\,&\xrightarrow{\partial_*}\,H_{n}(A\cap B)\,\xrightarrow{(i_*,j_*)}\,H_{n}(A)\oplus H_{n}(B)\,\xrightarrow{k_* - l_*}\,H_{n}(X)\xrightarrow{\partial_*}\\
&\quad\xrightarrow{\partial_*}\,H_{n-1} (A\cap B)\rightarrow \cdots\rightarrow H_0(A)\oplus H_0(B)\,\xrightarrow{k_* - l_*}\,H_0(X)\rightarrow\,0.
\end{align}</math>


Here the maps ''i'' : ''A''∩''B'' ↪ ''A'', ''j'' : ''A''∩''B'' ↪ ''B'', ''k'' : ''A'' ↪ ''X'', and ''l'' : ''B'' ↪ ''X'' are [[inclusion map]]s and <math>\oplus</math> denotes the [[direct sum of abelian groups]].
Transitional Dynamics equation, where Growth rate on <math> k\,</math> is given by,


===Boundary map===
:<math>\gamma_K=\dot{k}/k = s.f(k)/ k - (n+\delta)\ ,</math>
[[Image:Mayer Vietoris sequence boundary map on torus.png|thumb|280px|right|Illustration of the boundary map ∂<sub>*</sub> on the torus where the 1-cycle ''x'' = ''u'' + ''v'' is the sum of two 1-chains whose boundary lies in the intersection of ''A'' and ''B''.]]
 
The boundary maps ∂<sub>*</sub> lowering the dimension may be made explicit as follows.<ref name="Hatcher 2002 150">{{harvnb|Hatcher|2002|p=150}}</ref> An element in ''H''<sub>n</sub>(''X'') is the homology class of an ''n''-cycle ''x'' which, by [[barycentric subdivision]] for example, can be written as the sum of two ''n''-chains ''u'' and ''v'' whose images lie wholly in ''A'' and ''B'', respectively. Thus ∂''x'' = ∂(''u'' + ''v'') = 0 so that ∂''u'' = &minus;∂''v''. This implies that the images of both these boundary (''n'' &minus; 1)-cycles are contained in the intersection ''A''∩''B''. Then ∂<sub>*</sub>([''x'']) is the class of ∂''u'' in ''H''<sub>n&minus;1</sub>(''A''∩''B''). Choosing a different representative ''x′'' does not affect ∂''u'' since ∂''x′'' = ∂''x'' = 0; nor does choosing another decomposition ''x'' = ''u′'' + ''v′'' since then ∂''u'' + ∂''v'' &minus; ∂''u′'' &minus; ∂''v′'' = 0 which implies ∂''u'' = ∂''u′'' and ∂''v'' = ∂''v′''. Notice that the maps in the Mayer–Vietoris sequence depend on choosing an order for ''A'' and ''B''. In particular, the boundary map changes sign if ''A'' and ''B'' are swapped.
on substituting <math> A\,</math>, we get,
:<math>\gamma_K= sA -(n+\delta)\ ,</math>


===Reduced version===
We return here to the case of zero technological progress, <math> x=0\,</math>, because we want to show that per capita growth can now occur in the long-run even without exogenous technological change. The figure 1.1 explains the perpetual growth, with exogenous technical progress.  The vertical distance between the two line, <math> sA\,</math>and n+&delta; gives the<math>\gamma_K\,</math>
For [[reduced homology]] there is also a Mayer–Vietoris sequence, under the assumption that ''A'' and ''B'' have [[non-empty]] intersection.<ref>{{harvnb|Spanier|1966|p=187}}</ref> The sequence is identical for positive dimensions and ends as:


<br /><math>\cdots\rightarrow\tilde{H}_0(A\cap B)\,\xrightarrow{(i_*,j_*)}\,\tilde{H}_0(A)\oplus\tilde{H}_0(B)\,\xrightarrow{k_* - l_*}\,\tilde{H}_0(X)\rightarrow\,0.</math>
As, <math> sA>\, </math>n+&delta;, so that<math>\gamma_K > 0\,</math>. Since the two line are parallel, <math>\gamma_K\,</math>is constant; in particular, it is independent of <math>K\,</math>. In other words,<math>K\,</math> always grows at steady states rate,<math>\gamma_K^*= sA -(n+\delta)\ ,</math>.


===Analogy with the Seifert–van Kampen theorem===
Since
There is an analogy between the Mayer–Vietoris sequence (especially for homology groups of dimension 1) and the [[Seifert–van Kampen theorem]].<ref name="Hatcher 2002 150"/><ref>{{harvnb|Massey|1984|p=240}}</ref> Whenever ''A''∩''B'' is [[path-connected]] the reduced Mayer–Vietoris sequence yields the isomorphism
:<math>y = AK\,</math>,<math>\gamma_K\,</math> equals <math>\gamma_K^*\,</math>  


:<math>H_1(X) \cong (H_1(A)\oplus H_1(B))/\text{Ker} (k_* - l_*)</math>
at every point of time. In addition, since
:<math>c= (1-s) y\,</math>,


where, by exactness,  
the growth rate of
:<math>c\,</math> equals <math>\gamma_K^*\,</math>.


:<math>\text{Ker} (k_* - l_*) \cong \text{Im} (i_*, j_*).</math>
Hence, the entire per capita variable in the model grows at same rate, given by
:<math>\gamma^*= sA -(n+\delta)\ ,</math>


This is precisely the [[Commutator subgroup#Abelianization|abelianized]] statement of the Seifert–van Kampen theorem. Compare with the fact that ''H''<sub>1</sub>(''X'') is the abelianization of the [[fundamental group]] π<sub>1</sub>(''X'') when ''X'' is path-connected.<ref>{{harvnb|Hatcher|2002|loc=Theorem 2A.1, p. 166}}</ref>
However, we can observe that<math>y = AK\,</math> technology displays a positive long-run per capita growth without any exogenous technological development. The per capita growth depends on behavioural factors of the model as the saving rate and population. It is unlike neoclassical model, which is higher saving, s, promotes higher long-run per capita growth <math>\gamma^*\,</math>.<ref>Economic Growth, 2nd Edition Robert J. Barro and Xavier Sala-i-Martin</ref>


==Basic applications==
== Endogenous versus exogenous growth theory ==
In neo-classical growth models, the long-run rate of growth is [[exogeny|exogenous]]ly determined by either the savings rate (the [[Harrod–Domar model]]) or the rate of technical progress ([[Solow model]]). However, the savings rate and rate of technological progress remain unexplained. Endogenous growth theory tries to overcome this shortcoming by building macroeconomic models out of [[Microfoundations|microeconomic foundations]]. Households are assumed to maximize utility subject to budget constraints while firms maximize profits. Crucial importance is usually given to the production of new technologies and [[human capital]]. The engine for growth can be as simple as a constant return to scale production function (the AK model) or more complicated set ups with [[Knowledge spillover|spillover]] effects (spillovers are positive externalities, benefits that are attributed to costs from other firms), increasing numbers of goods, increasing qualities, etc.


===''k''-sphere===
Often endogenous growth theory assumes constant marginal product of capital at the aggregate level, or at least that the limit of the marginal product of capital does not tend towards zero. This does not imply that larger firms will be more productive than small ones, because at the firm level the marginal product of capital is still diminishing. Therefore, it is possible to construct endogenous growth models with [[perfect competition]]. However, in many endogenous growth models the assumption of perfect competition is relaxed, and some degree of [[monopoly]] power is thought to exist. Generally monopoly power in these models comes from the holding of patents. These are models with two sectors, producers of final output and an R&D sector. The R&D sector develops ideas that they are granted a monopoly power. R&D firms are assumed to be able to make monopoly profits selling ideas to production firms, but the [[free entry]] condition means that these profits are dissipated on R&D spending.
[[Image:SphereCoverStriped.png|thumb|250px|right|The decomposition for ''X'' = ''S''<sup>2</sup>]]
To completely compute the homology of the [[n-sphere|''k''-sphere]] ''X'' = ''S''<sup>''k''</sup>, let ''A'' and ''B'' be two hemispheres of ''X'' with intersection [[homotopy equivalent]] to a (''k'' &minus; 1)-dimensional equatorial sphere. Since the ''k''-dimensional hemispheres are [[homeomorphic]] to ''k''-discs, which are [[contractible]], the homology groups for ''A'' and ''B'' are [[Trivial group|trivial]]. The Mayer–Vietoris sequence for [[reduced homology]] groups then yields


:<br /><math> \cdots\rightarrow 0 \rightarrow \tilde{H}_{n}\left(S^k\right) \xrightarrow{\partial_*}\, \tilde{H}_{n-1}\left(S^{k-1}\right) \rightarrow 0 \rightarrow \cdots \! </math>
==Implications==
An Endogenous growth theory implication is that policies which embrace openness, competition, change and innovation will promote growth.<ref>{{cite journal|last=Fadare|first=Samuel O.|title=Recent Banking Sector Reforms and Economic Growth in Nigeria|journal=Middle Eastern Finance and Economics|issue=Issue 8 (2010)|url=http://www.eurojournals.com/MEFE_8_12.pdf}}</ref>  Conversely, policies which have the effect of restricting or slowing change by protecting or favouring particular existing industries or firms are likely over time to slow growth to the disadvantage of the community. [[Peter Howitt (economist)|Peter Howitt]] has written:
<blockquote>
Sustained economic growth is everywhere and always a process of continual transformation. The sort of economic progress that has been enjoyed by the richest nations since the Industrial Revolution would not have been possible if people had not undergone wrenching changes. Economies that cease to transform themselves are destined to fall off the path of economic growth. The countries that most deserve the title of “developing” are not the poorest countries of the world, but the richest. [They] need to engage in the never-ending process of economic development if they are to enjoy continued prosperity. (Conclusion, "Growth and development: a Schumpeterian perspective", 2006 [http://www.cdhowe.org/pdf/commentary_246.pdf]).
</blockquote>


Exactness immediately implies that the map ∂<sub>*</sub> is an isomorphism. Using the [[reduced homology]] of the [[0-sphere]] (two points) as a [[Mathematical induction|base case]], it follows<ref>{{harvnb|Hatcher|2002|loc=Example 2.46,  p. 150}}</ref>
== Criticisms ==<!-- This section is linked from [[Economics]] -->


:<br /><math>\tilde{H}_n\left(S^k\right)\cong\delta_{kn}\,\mathbb{Z}=\left\{\begin{matrix}
One of the main failings of endogenous growth theories is the collective failure to explain conditional convergence reported in the empirical literature.<ref> See {{cite journal |last=Sachs |first=Jeffrey D. |first2=Andrew M. |last2=Warner |year=1997 |title=Fundamental Sources of Long-Run Growth |journal=[[American Economic Review]] |volume=87 |issue=2 |pages=184–188 |doi= |jstor=2950910 }}</ref> Another frequent critique concerns the cornerstone assumption of diminishing returns to capital. Some contend that ''new growth theory'' has proven no more successful than [[exogenous growth model|exogenous growth theory]] in explaining the income divergence between the [[developing nation|developing]] and [[developed nation|developed]] worlds (despite usually being more complex).<ref>See for instance, Professor Stephen Parente's 2001 review, ''The Failure of Endogenous Growth'' ([https://netfiles.uiuc.edu/parente/The%20Failure%20of%20Endogenous%20Growth.pdf Online] at the [[University of Illinois at Urbana-Champaign]]). (Published in [http://www.metapress.com/(gqdg4dzadovmv5fnfj5jki2x)/home/main.mpx Knowledge Technology & Policy] Volume XIII, Number 4.)</ref>
\mathbb{Z} & \mbox{if } n=k  \\
0 & \mbox{if } n \ne k  \end{matrix}\right.</math>


where δ is the [[Kronecker delta]]. Such a complete understanding of the homology groups for spheres is in stark contrast with current knowledge of [[homotopy groups of spheres]], especially for the case ''n'' > ''k'' about which little is known.<ref>{{harvnb|Hatcher|2002|p=384}}</ref>
== See also ==
{{-}}
* [[Economic growth]]
 
* [[Human capital]]
===Klein bottle===
* [[Paul Romer]]
[[Image:KleinBottle2D covered by Möbius strips.svg|thumb|200px|right|The Klein bottle ([[fundamental polygon]] with appropriate edge identifications) decomposed as two Möbius strips ''A'' (in blue) and ''B'' (in red).]]
* [[Neoclassical growth model|Exogenous growth model]]
A slightly more difficult application of the Mayer–Vietoris sequence is the calculation of the homology groups of the [[Klein bottle]] ''X''. One uses the decomposition of  ''X'' as the union of two [[Möbius strip]]s ''A'' and ''B'' [[Quotient space|glued]] along their boundary circle (see illustration on the right). Then ''A'', ''B'' and their intersection ''A''∩''B'' are [[Homotopy#Homotopy equivalence and null-homotopy|homotopy equivalent]] to circles, so the nontrivial part of the sequence yields<ref>{{harvnb|Hatcher|2002|p=151}}</ref>
* [[Mahalanobis model]]
 
* [[Ramsey–Cass–Koopmans model]]
:<br /><math> 0 \rightarrow H_{2}(X) \rightarrow\, \mathbb{Z}\ \xrightarrow{\alpha} \ \mathbb{Z} \oplus \mathbb{Z} \rightarrow \, H_1(X) \rightarrow 0 \! </math>
 
and the trivial part implies vanishing homology for dimensions greater than 2. The central map α sends 1 to (2, &minus;2) since the boundary circle of a Möbius band wraps twice around the core circle. In particular α is [[Injective function|injective]] so homology of dimension 2 also vanishes. Finally, choosing (1, 0) and (1, &minus;1) as a basis for '''Z'''<sup>2</sup>, it follows
 
:<br /><math>\tilde{H}_n\left(X\right)\cong\delta_{1n}\,(\mathbb{Z}\oplus\mathbb{Z}_2)=\left\{\begin{matrix}
\mathbb{Z}\oplus\mathbb{Z}_2 & \mbox{if } n=1\\
0 & \mbox{if } n\ne1    \end{matrix}\right.
</math>
{{-}}
 
===Wedge sums===
[[Image:WedgeSumSpheres.png|right|300px|thumb|This decomposition of the wedge sum ''X'' of two 2-spheres ''K'' and ''L'' yields all the homology groups of ''X''.]]
Let ''X'' be the [[wedge sum]] of two spaces ''K'' and ''L'', and suppose furthermore that the identified [[basepoint]] is a [[deformation retract]] of [[Neighbourhood (mathematics)|open neighborhoods]] ''U'' ⊂ ''K'' and ''V'' ⊂ ''L''. Letting ''A'' = ''K''∪''V'' and ''B'' = ''U''∪''L'' it follows that ''A''∪''B'' = ''X'' and ''A''∩''B'' = ''U''∪''V'', which is [[contractible]] by construction. The reduced version of the sequence then yields (by exactness)<ref>{{harvnb|Hatcher|2002|loc=Exercise 31}}</ref>
:<math>\tilde{H}_n(K\vee L)\cong \tilde{H}_n(K)\oplus\tilde{H}_n(L)</math>
for all dimensions ''n''. The  illustration on the right shows ''X'' as the sum of two 2-spheres ''K'' and ''L''. For this specific case, using the result [[Mayer–Vietoris sequence#k-sphere|from above]] for 2-spheres, one has
:<math>\tilde{H}_n\left(S^2\vee S^2\right)\cong\delta_{2n}\,(\mathbb{Z}\oplus\mathbb{Z})=\left\{\begin{matrix}
\mathbb{Z}\oplus\mathbb{Z} & \mbox{if } n=2  \\
0 & \mbox{if } n \ne 2  \end{matrix}\right.</math>
{{-}}
 
===Suspensions===
[[Image:0-Sphere Suspension - Mayer-Vietoris Cover.svg|right|500px|thumb|This decomposition of the suspension ''X'' of the 0-sphere ''Y'' yields all the homology groups of ''X''.]]
If ''X'' is the [[Suspension (topology)|suspension]] ''SY'' of a space ''Y'', let ''A'' and ''B'' be the [[Complement (set theory)|complements]] in ''X'' of the top and bottom 'vertices' of the double cone, respectively. Then ''X'' is the union ''A''∪''B'', with ''A'' and ''B'' contractible. Also, the intersection ''A''∩''B'' is homotopy equivalent to ''Y''. Hence the Mayer–Vietoris sequence yields, for all ''n'',<ref>{{harvnb|Hatcher|2002|loc=Exercise 32}}</ref>
:<math>\tilde{H}_n(SY)\cong \tilde{H}_{n-1}(Y)</math>
 
The illustration on the right shows the 1-sphere ''X'' as the suspension of the 0-sphere ''Y''. Noting in general that the ''k''-sphere is the suspension of the (''k'' &minus; 1)-sphere, it is easy to derive the homology groups of the ''k''-sphere by induction, [[Mayer–Vietoris sequence#k-sphere|as above]].
{{-}}
 
==Further discussion==
 
===Relative form===
A [[relative homology|relative]] form of the Mayer–Vietoris sequence also exists. If ''Y'' ⊂ ''X'' and is the union of ''C'' ⊂ ''A'' and ''D'' ⊂ ''B'', then the exact sequence is:<ref>{{harvnb|Hatcher|2002|p=152}}</ref>
 
<br /><math>\cdots\rightarrow H_{n}(A\cap B,C\cap D)\,\xrightarrow{(i_*,j_*)}\,H_{n}(A,C)\oplus H_{n}(B,D)\,\xrightarrow{k_* - l_*}\,H_{n}(X,Y)\,\xrightarrow{\partial_*}\,H_{n-1}(A\cap B,C\cap D)\rightarrow\cdots</math>
 
===Naturality===
The homology groups are [[Natural (category theory)|natural]] in the sense that if ''ƒ'' is a [[Continuous function (topology)|continuous]] map from ''X''<sub>1</sub> to ''X''<sub>2</sub>, then there is a canonical [[pushforward (homology)|pushforward]] map ''ƒ''<sub>∗</sub> of homology groups ''ƒ''<sub>∗</sub>&nbsp;:&nbsp;''H''<sub>''k''</sub>(''X''<sub>1</sub>)&nbsp;→&nbsp;''H''<sub>''k''</sub>(''X''<sub>2</sub>), such that the composition of pushforwards is the pushforward of a composition: that is, <math>(g\circ h)_* = g_*\circ h_*</math>. The Mayer–Vietoris sequence is also natural in the sense that if ''X''<sub>1</sub> = ''A''<sub>1</sub>∪''B''<sub>1</sub> to ''X''<sub>2</sub> = ''A''<sub>2</sub>∪''B''<sub>2</sub> and the mapping ''ƒ'' satisfies ''ƒ''(''A''<sub>1</sub>) ⊂ ''A''<sub>2</sub> and ''ƒ''(''B''<sub>1</sub>) ⊂ ''B''<sub>2</sub>, then the connecting morphism ∂<sub>∗</sub> of the Mayer–Vietoris sequence commutes with ''ƒ''<sub>∗</sub>.<ref>{{harvnb|Massey|1984|p=208}}</ref>  That is,<ref>{{harvnb|Eilenberg|Steenrod|1952|loc=Theorem 15.4}}</ref> the following diagram [[Commutative diagram|commutes]] (the horizontal maps are the usual ones):
[[Image:Mayer-Vietoris naturality.png|center|740px]]
 
===Cohomological versions===
 
The Mayer–Vietoris long exact sequence for [[singular cohomology]] groups with coefficient [[group (mathematics)|group]] ''G'' is [[Duality (mathematics)|dual]] to the homological version. It is the following:<ref>{{harvnb|Hatcher|2002|p=203}}</ref>
 
<br /><math>\cdots\rightarrow H^{n}(X;G)\rightarrow H^{n}(A;G)\oplus H^{n}(B;G)\rightarrow H^{n}(A\cap B;G)\rightarrow H^{n+1}(X;G)\rightarrow\cdots</math>
 
where the dimension preserving maps are restriction maps induced from inclusions, and the (co-)boundary maps are defined in a similar fashion to the homological version. There is also a relative formulation.
 
As an important special case when ''G'' is the group of [[real number]]s '''R''' and the underlying topological space has the additional structure of a [[smooth manifold]], the Mayer–Vietoris sequence for [[de Rham cohomology]] is
 
<br /><math>\cdots\rightarrow H^{n}(X)\,\xrightarrow{\rho}\,H^{n}(U)\oplus H^{n}(V)\,\xrightarrow{\Delta}\,H^{n}(U\cap V)\,\xrightarrow{d^*}\,H^{n+1}(X)\rightarrow\cdots</math>
 
where {''U'', ''V''} is an [[open cover]] of ''X'', ''ρ'' denotes the restriction map, and Δ is the difference. The map ''d*'' is defined similarly as the map ''∂''<sub>*</sub> from above. It can be briefly described as follows. For a cohomology class [''ω''] represented by [[closed and exact differential forms|closed form]] ''ω'' in ''U''∩''V'', express ''ω'' as a difference of forms ''ω<sub>U</sub>'' - ''ω<sub>V</sub>'' via a [[partition of unity]] subordinate to the open cover {''U'', ''V''}, for example. The exterior derivative ''dω<sub>U</sub>'' and ''dω<sub>V</sub>'' agree on ''U''∩''V'' and therefore together define an ''n'' + 1 form ''σ'' on ''X''. One then has  ''d*''([''ω'']) = [''σ''].
 
===Derivation===
Consider the [[Homological algebra#Functoriality|long exact sequence associated to]] the [[short exact sequence]]s of [[chain group]]s (constituent groups of [[chain complex]]es)
 
:<math>0 \rightarrow C_n(A\cap B)\,\xrightarrow{\alpha}\,C_n(A) \oplus C_n(B)\,\xrightarrow{\beta}\,C_n(A+B) \rightarrow 0 </math>
 
where α(''x'') = (''x'', &minus;''x''), β(''x'', ''y'') = ''x'' + ''y'', and ''C''<sub>''n''</sub>(''A'' + ''B'') is the chain group consisting of sums of chains in ''A'' and chains in ''B''.<ref name="Hatcher149"/> It is a fact that the singular ''n''-simplices of ''X'' whose images are contained in either ''A'' or ''B'' generate all of the homology group ''H''<sub>''n''</sub>(''X'').<ref>{{harvnb|Hatcher|2002|loc=Proposition 2.21,  p. 119}}</ref> In other words, ''H''<sub>''n''</sub>(''A'' + ''B'') is isomorphic to ''H''<sub>''n''</sub>(''X''). This gives the Mayer–Vietoris sequence for singular homology.
 
The same computation applied to the short exact sequences of vector spaces of [[differential form]]s
 
:<math>
0\rightarrow\Omega^{n}(X)\rightarrow\Omega^{n}(U)\oplus\Omega^{n}(V)\rightarrow\Omega^{n}(U\cap V)\rightarrow0
</math>
 
yields the Mayer–Vietoris sequence for de Rham cohomology.<ref>{{harvnb|Bott|Tu|1982|loc=§I.2}}</ref>
 
From a formal point of view, the Mayer–Vietoris sequence can be derived from the [[Eilenberg–Steenrod axioms]] for [[homology theory|homology theories]] using the [[long exact sequence in homology]].<ref>{{harvnb|Hatcher|2002|p=162}}</ref>
 
===Other homology theories===
The derivation of the Mayer–Vietoris sequence from the Eilenberg–Steenrod axioms does not require the [[dimension axiom]],<ref>{{harvnb|Kōno|Tamaki|2006|pp=25–26}}</ref> so in addition to existing in [[List of cohomology theories#Ordinary homology theories|ordinary cohomology theories]], it holds in [[extraordinary cohomology theories]] (such as [[topological K-theory]] and [[cobordism]]).
 
===Sheaf cohomology===
From the point of view of [[sheaf cohomology]], the Mayer–Vietoris sequence is related to [[Čech cohomology]]. Specifically, it arises from the [[Spectral sequence|degeneration]] of the [[spectral sequence]] that relates Čech cohomology to sheaf cohomology (sometimes called the [[Mayer–Vietoris spectral sequence]]) in the case where the open cover used to compute the Čech cohomology consists of two open sets.<ref>{{harvnb|Dimca|2004|pp=35–36}}</ref> This spectral sequence exists in arbitrary [[Topos|topoi]].<ref>{{harvnb|Verdier|1972}} (SGA 4.V.3)</ref>
 
==See also==
*[[Excision theorem]]
*[[Zig-zag lemma]]


==Notes==
==Notes==
{{reflist|colwidth=30em}}
{{reflist}}
 
==References==
 
*{{citation
|last1=Bott
|first1=Raoul
|author1-link=Raoul Bott
|last2=Tu
|first2=Loring W.
|title=Differential Forms in Algebraic Topology
|publisher=[[Springer Science+Business Media|Springer-Verlag]]
|location=Berlin, New York
|isbn=978-0-387-90613-3
|year=1982}}.
 
*{{citation
|first= Leo
|last= Corry
|authorlink= Leo Corry
|title= Modern Algebra and the Rise of Mathematical Structures
|publisher= Birkhäuser
|year= 2004
|page= 345
|isbn= 3-7643-7002-5
}}.
 
*{{citation
|first= Jean
|last= Dieudonné
|authorlink= Jean Dieudonné
|title= A History of Algebraic and Differential Topology 1900–1960
|publisher= Birkhäuser
|year= 1989
|page= 39
|isbn= 0-8176-3388-X
}}.
 
*{{citation
| last=Dimca
| first=Alexandru
| title=Sheaves in topology
| publisher=[[Springer-Verlag]]
| year=2004
| location=Berlin
| series=Universitext
| isbn=978-3-540-20665-1
| mr=2050072
}}
 
* {{citation
|last1=Eilenberg
|first1=Samuel
|authorlink1=Samuel Eilenberg
|last2=Steenrod
|first2=Norman
|authorlink2=Norman Steenrod
|title=Foundations of Algebraic Topology
|year=1952
|isbn=978-0-691-07965-3
|publisher= [[Princeton University Press]]
}}.
 
*{{citation
|first= Allen
|last= Hatcher
|author-link= Allen Hatcher
|title= Algebraic Topology
|url= http://www.math.cornell.edu/%7Ehatcher/AT/ATpage.html
|year= 2002
|publisher= [[Cambridge University Press]]
|isbn= 978-0-521-79540-1
|mr= 1867354
}}.
 
*{{citation
|title= The Heritage of Emmy Noether
|last=Hirzebruch
|first=Friedrich
|authorlink=Friedrich Hirzebruch
|contribution=Emmy Noether and Topology
|pages=61–63
|editor= Teicher, M.
|series= Israel Mathematical Conference Proceedings
|publisher= [[Bar-Ilan University]]/[[American Mathematical Society]]/[[Oxford University Press]]
|year= 1999
|isbn= 978-0-19-851045-1
|oclc= 223099225
}}.
 
*{{citation
|last=Kōno
|first=Akira
|last2=Tamaki
|first2=Dai
|title=Generalized cohomology
|publisher=[[American Mathematical Society]]
|location=Providence, RI
|series=Iwanami Series in Modern Mathematics, Translations of Mathematical Monographs
|volume=230
|year=2006
|origyear=2002
|edition=Translated from the 2002 Japanese edition by Tamaki
|isbn=978-0-8218-3514-2
|mr=2225848
}}
 
*{{citation
|first= William
|last= Massey
|author-link= William S. Massey
|title= Algebraic Topology: An Introduction
|year= 1984
|publisher= [[Springer Science+Business Media|Springer-Verlag]]
|isbn= 978-0-387-90271-5
}}.
 
*{{citation
|first= Walther
|last= Mayer
|author-link= Walther Mayer
|title= Über abstrakte Topologie
|year= 1929
|journal= [[Monatshefte für Mathematik]]
|url= http://www.springerlink.com/content/x33611021p942518/
|doi= 10.1007/BF02307601
|issn= 0026-9255
|volume= 36
|issue= 1
|pages= 1–42
}}. {{de icon}}
 
*{{citation
|first= Edwin
|last= Spanier
|author-link= Edwin Spanier
|title= Algebraic Topology
|year= 1966
|publisher= [[Springer Science+Business Media|Springer-Verlag]]
|isbn= 0-387-94426-5
}}.
 
*{{citation
|first=Jean-Louis
|last=Verdier
|author-link=Jean-Louis Verdier
|contribution=Cohomologie dans les topos
|editor1-first=Michael
|editor1-last=Artin
|editor1-link=Michael Artin
|editor2-first=Alexander
|editor2-last=Grothendieck
|editor2-link=Alexander Grothendieck
|editor3-first=Jean-Louis
|editor3-last=Verdier
|editor3-link=Jean-Louis Verdier
|title=Séminaire de Géométrie Algébrique du Bois Marie – 1963–64 – Théorie des topos et cohomologie étale des schémas – (SGA 4) – Tome 2
|year=1972
|publisher = [[Springer Science+Business Media|Springer-Verlag]]
|location = Berlin; Heidelberg
|language = French
|series=[[Lecture Notes in Mathematics]]
|volume=270
|isbn=978-3-540-06012-3
|doi=10.1007/BFb0061320
|pages=1
}}


*{{citation
== External links ==
|first= Leopold
* [http://www.stanford.edu/~promer/EconomicGrowth.pdf Economic Growth] by [[Paul Romer]].
|last= Vietoris
* [http://www.eda.gov/ImageCache/EDAPublic/documents/pdfdocs/1g3lr_5f7_5fcortright_2epdf/v1/1g3lr_5f7_5fcortright.pdf New Growth Theory, Technology and Learning: A Practitioner's Guide], [[Economic Development Administration|U.S. Economic Development Administration]].
|author-link= Leopold Vietoris
* [http://tcdc.undp.org/CoopSouth/1998_2/cop9829.pdf Technological Implications of New Growth Theory for the South], [[United Nations Development Programme]].
|title= Über die Homologiegruppen der Vereinigung zweier Komplexe
*[http://mitpress.mit.edu/books/chapters/0262025531chap1.pdf The AK Model] by Economic Growth, 2nd Edition Robert J. Barro and Xavier Sala-i-Martin
|year= 1930
*The Origins of Endogenous Growth, Romer.M Paul,The Journal of Economic Perspectives, Vol. 8, No. 1. (Winter, 1994), pp. 3-22.
|journal= [[Monatshefte für Mathematik]]
|volume= 37
|pages= 159–62
|doi=10.1007/BF01696765
}}. {{de icon}}


==Further reading==
* {{citation
|last1=Reitberger
|first1=Heinrich
|title=Leopold Vietoris (1891–2002)
|url=http://www.ams.org/notices/200210/fea-vietoris.pdf
|format=PDF|year=2002
|journal=[[Notices of the American Mathematical Society]]
|issn=0002-9920
|volume=49
|issue=20
}}.


{{good article}}
[[Category:Economic theories]]
[[Category:Economic growth]]


{{DEFAULTSORT:Mayer-Vietoris Sequence}}
[[ca:Desenvolupament endogen]]
[[Category:Homology theory]]
[[de:Endogene Wachstumstheorie]]
[[fr:Théorie de la croissance endogène]]
[[it:Teoria della crescita endogena]]
[[lo:Endogenous growth model]]
[[nl:Endogene groeitheorie]]
[[pl:Endogeniczny model wzrostu gospodarczego]]
[[fi:Endogeenisen kasvun teoria]]

Revision as of 17:17, 11 August 2014

My name is Winnie and I am studying Anthropology and Sociology and Modern Languages and Classics at Rillieux-La-Pape / France.

Also visit my web site ... hostgator1centcoupon.info Template:Development economics sidebar Endogenous growth theory holds that economic growth is primarily the result of endogenous and not external forces.[1] Endogenous growth theory holds that investment in human capital, innovation, and knowledge are significant contributors to economic growth. The theory also focuses on positive externalities and spillover effects of a knowledge-based economy which will lead to economic development. The endogenous growth theory also holds that policy measures can have an impact on the long-run growth rate of an economy. For example, subsidies for research and development or education increase the growth rate in some endogenous growth models by increasing the incentive for innovation.

Models in Endogenous Growth

In the mid-1980s, a group of growth theorists became increasingly dissatisfied with common accounts of exogenous factors determining long-run growth. They favored a model that replaced the exogenous growth variable (unexplained technical progress) with a model in which the key determinants of growth were explicit in the model. The initial research was based on the work of Kenneth Arrow (1962), Hirofumi Uzawa (1965), and Miguel Sidrauski (1967).[2] Paul Romer (1986), Lucas (1988),[3] and Rebelo (1991)[4][5] omitted technological change. Instead, growth in these models was due to indefinite investment in human capital which had spillover effect on economy and reduces the diminishing return to capital accumulation.[6]

The AK model, which is the simplest endogenous model, gives a constant-saving-rate of endogenous growth. It assumes a constant, exogenous saving rate and fixed level of the technology. It shows elimination of diminishing returns leading to endogenous growth. However, the endogenous growth theory is further supported with models in which agents optimally determined the consumption and saving, optimizing the resources allocation to research and development leading to technological progress. Romer (1987, 1990) and significant contributions by Aghion and Howitt (1992) and Grossman and Helpman (1991), incorporated imperfect markets and R&D to the growth model.[6]

The AK Model

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The model works on the property of absence of diminishing returns to capital. The simplest form of production function with diminishing return is:

File:Ak model.png
figure 1.1

where

, is a positive constant that reflects the level of the technology.
capital (broad sense to include human capital)
, output per capita and the average and marginal product are constant at the level

If we substitute in equation of transitional Dynamics of Solow-Swan model (Exogenous growth model) which shows how an economy’s per capita incomes converges toward its own steady-state value and to the per capita incomes of other nations.

Transitional Dynamics equation, where Growth rate on is given by,

on substituting , we get,

We return here to the case of zero technological progress, , because we want to show that per capita growth can now occur in the long-run even without exogenous technological change. The figure 1.1 explains the perpetual growth, with exogenous technical progress. The vertical distance between the two line, and n+δ gives the

As, n+δ, so that. Since the two line are parallel, is constant; in particular, it is independent of . In other words, always grows at steady states rate,.

Since

, equals

at every point of time. In addition, since

,

the growth rate of

equals .

Hence, the entire per capita variable in the model grows at same rate, given by

However, we can observe that technology displays a positive long-run per capita growth without any exogenous technological development. The per capita growth depends on behavioural factors of the model as the saving rate and population. It is unlike neoclassical model, which is higher saving, s, promotes higher long-run per capita growth .[7]

Endogenous versus exogenous growth theory

In neo-classical growth models, the long-run rate of growth is exogenously determined by either the savings rate (the Harrod–Domar model) or the rate of technical progress (Solow model). However, the savings rate and rate of technological progress remain unexplained. Endogenous growth theory tries to overcome this shortcoming by building macroeconomic models out of microeconomic foundations. Households are assumed to maximize utility subject to budget constraints while firms maximize profits. Crucial importance is usually given to the production of new technologies and human capital. The engine for growth can be as simple as a constant return to scale production function (the AK model) or more complicated set ups with spillover effects (spillovers are positive externalities, benefits that are attributed to costs from other firms), increasing numbers of goods, increasing qualities, etc.

Often endogenous growth theory assumes constant marginal product of capital at the aggregate level, or at least that the limit of the marginal product of capital does not tend towards zero. This does not imply that larger firms will be more productive than small ones, because at the firm level the marginal product of capital is still diminishing. Therefore, it is possible to construct endogenous growth models with perfect competition. However, in many endogenous growth models the assumption of perfect competition is relaxed, and some degree of monopoly power is thought to exist. Generally monopoly power in these models comes from the holding of patents. These are models with two sectors, producers of final output and an R&D sector. The R&D sector develops ideas that they are granted a monopoly power. R&D firms are assumed to be able to make monopoly profits selling ideas to production firms, but the free entry condition means that these profits are dissipated on R&D spending.

Implications

An Endogenous growth theory implication is that policies which embrace openness, competition, change and innovation will promote growth.[8] Conversely, policies which have the effect of restricting or slowing change by protecting or favouring particular existing industries or firms are likely over time to slow growth to the disadvantage of the community. Peter Howitt has written:

Sustained economic growth is everywhere and always a process of continual transformation. The sort of economic progress that has been enjoyed by the richest nations since the Industrial Revolution would not have been possible if people had not undergone wrenching changes. Economies that cease to transform themselves are destined to fall off the path of economic growth. The countries that most deserve the title of “developing” are not the poorest countries of the world, but the richest. [They] need to engage in the never-ending process of economic development if they are to enjoy continued prosperity. (Conclusion, "Growth and development: a Schumpeterian perspective", 2006 [1]).

Criticisms

One of the main failings of endogenous growth theories is the collective failure to explain conditional convergence reported in the empirical literature.[9] Another frequent critique concerns the cornerstone assumption of diminishing returns to capital. Some contend that new growth theory has proven no more successful than exogenous growth theory in explaining the income divergence between the developing and developed worlds (despite usually being more complex).[10]

See also

Notes

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External links

ca:Desenvolupament endogen de:Endogene Wachstumstheorie fr:Théorie de la croissance endogène it:Teoria della crescita endogena lo:Endogenous growth model nl:Endogene groeitheorie pl:Endogeniczny model wzrostu gospodarczego fi:Endogeenisen kasvun teoria

  1. ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  2. ↑ Template:Cite web
  3. ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  4. ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  5. ↑ Template:Cite web
  6. ↑ 6.0 6.1 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  7. ↑ Economic Growth, 2nd Edition Robert J. Barro and Xavier Sala-i-Martin
  8. ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  9. ↑ See One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  10. ↑ See for instance, Professor Stephen Parente's 2001 review, The Failure of Endogenous Growth (Online at the University of Illinois at Urbana-Champaign). (Published in Knowledge Technology & Policy Volume XIII, Number 4.)
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