In addition the equilibrium converges to this BGP Proof See Appendix A This

In addition the equilibrium converges to this bgp

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In addition, the equilibrium converges to this BGP. Proof. See Appendix A. This proposition establishes the existence of a unique asymptotic BGP with a positive bubble and sustained growth. Of course, there is sustained growth under a sufficiently high productivity A , but it also requires a high enough ψ (see equation (33)). Indeed, a high time cost per child ψ reduces the incentive to have children. This implies that only a small amount of adult time is devoted to the total time cost of rearing children ψ n * (see equation (35)) and a large part of household resources can be used to invest in the productive asset, which promotes growth. As we have seen previously, the shares of the bubble hold by both young and middle-age households, and the number of children decrease with the growth factor. This implies that a positive bubble requires a not too large growth. As a result, the cost ψ should not be too high. We now investigate more deeply the properties of this asymptotic bubbly BGP. We start by focusing on whether young and middle-age households are buyer ( e b * i > 0) or rather short-sellers ( e b * i < 0) of the speculative asset. By direct inspection of equation (37), there is no doubt that e b * 2 > 0 at the bubbly BGP. Adult households use the bubble to transfer purchasing power to their last period of life. In contrast, whether young agents are buyers or short-sellers of the bubble needs a deeper analysis (see equation (36)). Corollary 1. Let b ψ ( 1 - α ) μ 1 + α ( β + β 2 ) (41) Under A > A 1 , the asymptotic BGP with sustained growth and positive bubble is characterized by the following: 11
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1. If β + β 2 1 + 2 β + 2 β 2 6 α < β + 2 β 2 1 + 2 β + 3 β 2 , young agents are short-sellers ( e b * 1 < 0 ) for all ψ < ψ < min { ψ b ; 1 } ; 2. If α < β + β 2 1 + 2 β + 2 β 2 , young agents are short-sellers ( e b * 1 < 0 ) for b ψ < ψ < min { ψ b ; 1 } , neither buy nor sell the bubble ( e b * 1 = 0 ) for ψ = b ψ , and buy the bubble ( e b * 1 > 0 ) for ψ < ψ < b ψ . Proof. See Appendix B. A direct implication of this result is that the existence of a bubbly BGP does not always require e b * 1 < 0. An asymptotic bubbly BGP may exist if the young households buy the bubble and are not short-sellers of this asset to finance productive investment ( e b * 1 > 0). Corollary 1 shows that young households are short-sellers of the bubble if either the return of the productive investment α A or the time cost per child ψ are sufficiently large. In the first case, the high return of capital creates an incentive to finance productive investment by selling short the speculative as- set. In the second case, the quite large time cost per child incites households to have only a few number of children. Therefore, with the log-linear utility, the total rearing cost, ψ n * , is relatively low (see equation (35)). This low total time cost of rearing children incites the households to borrow when young to foster productive investment and redistributes income from the middle to the young age. On the contrary, in the second configuration of the corollary, if ψ is low, the total time cost ψ n * is relatively high. Then, young households buy the bubble to finance this rearing cost of having children when adult.
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  • Spring '10
  • Economics, Capital accumulation

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