3 ψ ψ K Crowding in effect As we have seen since ψ is large enough ψ nis

3 ψ ψ k crowding in effect as we have seen since ψ

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3. ψ > ψ K : Crowding-in effect As we have seen, since ψ is large enough, ψ n is relatively low, whether or not the bubble exists. Therefore, the main mechanism is exactly the opposite one than in the first configuration. In this case, the labor income when adult is large. When the bubble exists, there is a transfer of re- sources from the adult to the young age. The young household becomes a short-seller of the bubble and increases productive investment. There- fore, growth is larger at the asymptotic bubbly BGP. This larger growth also implies a larger cost of having children ψ w t + 1 in terms of consump- tion. Therefore, population growth is lower when there is a bubble. 8 Concluding remarks We develop a model where a speculative asset (bubble) is used to finance two types of expenditures, a productive investment and costs of rearing children. If the time cost per child ψ is low, the household has a large number of chil- dren, meaning that the total time cost ψ n is high. In this case, the bubble is 17
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mainly used to finance rearing children expenses instead of productive capital. Growth is lower when there is a bubble. If ψ is high, we have the opposite. The total time cost of having children ψ n is low in this case and the bubble is used to finance productive investment, which enhances growth. The comparison between equilibria with and without bubble can also be interpreted in terms of financial development. Such an interpretation of our results may contribute to the literature studying the link between financial de- velopment and growth. 12 In our framework, the bubbleless equilibrium de- scribes an economy where the only asset is capital, whereas the bubbly one describes an economy where financial markets are more developed since there is an adding traded asset valued on such a market. Accordingly, we can say that the bubbleless economy is characterized by a less significant financial de- velopment than the bubbly economy. Under this interpretation, our model suggests that financial development is beneficial for growth only under some circumstances, i.e. if ψ is high enough. Appendix A Proof of Proposition 2 We have n * < 1/ ψ if and only if ψ > ψ , where this inequality ensures that ψ > μ / ( 1 + β + β 2 + μ ) . Then, using (38), we immediately see that B * > 0 if and only if ψ < ψ b . We also note that ψ < 1 because μ < 1 and ψ < ψ b is equivalent to α < β + 2 β 2 1 + 2 β + 3 β 2 . Of course, there is sustained growth because A > A 1 . Let us focus now on the stability properties. Even if ( γ * , B * , 0 ) is an asymp- totic BGP, it is important to note that from the mathematical point of view, ( γ t , B t , λ t ) = ( γ * , B * , 0 ) is a steady state of the dynamic system (28)-(30). There- fore, to analyze the stability properties of the equilibrium ( γ * , B * , 0 ) , we use standard mathematical tools. Differentiating the dynamic system (28)-(30) in the neighborhood of the BGP ( γ * , B * , 0 ) , we obtain a linear system of the following form: dB t + 1 = a 11 dB t + a 12 d γ t + a 13 d λ t d γ t + 1 = a 21 dB t + a 22 d γ t + a 23 d λ t d λ t + 1 = a 33 d λ t Therefore, the characteristic polynomial can be written P ( η ) ( a 33 - η )( η 2 - T η + D ) = 0, with T = a 11 + a 22 and D = a 11 a 22 - a 12 a 21 . We immediately
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  • Spring '10
  • JAMES
  • Economics, Capital accumulation

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