15 γ K γ K wbγ K γ K wb n n wb n n wb eb 1 e b 1 ψψ nb ψ ψK Figure 2 The

15 γ k γ k wbγ k γ k wb n n wb n n wb eb 1 e b 1

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- γ * K > γ K , wb γ * K < γ K , wb n * < n wb n * > n wb e b * 1 > 0 e b * 1 < 0 ψ ψ n b ψ ψ K Figure 2 – The interplay between bubble, fertility and capital growth 2. In Corollary 2, we have shown that if γ * > γ wb , we get n * < n wb . Now, without comparing the growth factors, we deduce, using (16) and (35), that n * < n wb if and only if ψ > ψ n , with: ψ n μ ( 1 - α ) 1 + β ( α + β ) (43) 3. Recall now that γ denotes the growth factor of the capital-labor ratio. It is often more usual to define the growth factor of capital per capita (or GDP per capita). This last one is defined by K t / N t = k t L t / N t = k t ( 1 - ψ + 1/ n t ) . On a BGP, since n is constant, the growth factor of capital per capita is also equal to γ and the growth factor of capital is defined by: γ K K t + 1 K t = k t + 1 k t N t + 1 N t = γ n Using (31), γ K = α A γ * K at the asymptotic bubbly BGP and, using (14) and (16), γ K = μα A ψ ( 1 + αβ )+ αμ γ K , wb at the bubbleless BGP. We deduce that γ * K > γ K , wb if and only if ψ > ψ K , with: ψ K μ ( 1 - α ) 1 + αβ (44) Using (41), (43) and (44), we have that ψ n < b ψ < ψ K . 11 This allows us to draw Figure 2, which is useful to understand why the bubble raises or not 11 One can further note that under Proposition 3, max { ψ ; ψ nwb } < ψ n and ψ K < min { ψ b ; 1 } , which means that the critical values we focus on belong to the interval of ψ considered in Proposi- tion 3. 16
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growth of capital per capita. As follows from this figure, there are three main configurations according to the value of ψ : ψ < ψ n , ψ n < ψ < ψ K and ψ > ψ K . Before studying these different configurations, it is useful to note that since γ = γ K / n , the threshold value ψ p above which γ * > γ wb belongs to ( ψ n , ψ K ) . Of course, γ * > γ wb for ψ > ψ K and γ * < γ wb for ψ 6 ψ n . 1. ψ < ψ n : Crowding-out effect Since ψ is small, we have seen that the total time cost of rearing children ψ n is large, whether or not the bubble exists. As a consequence, the labor supply and the income at middle-age are low. When the speculative asset is positively valued, young households buy the bubble to transfer income to the middle-age. Therefore, they invest less in capital when there is a bubble. Growth is lower when the bubble exists. This also reduces the cost of having children ψ w t + 1 in terms of consumption, which explains that population growth is larger at the asymptotic bubbly BGP. 2. ψ n < ψ < ψ K : Indeterminate crowding effect Both ψ and ψ n * take intermediate values. The main mechanism at stake is clearly different than in the previous configuration. To understand what happens, let us assume ψ = b ψ . At the asymptotic bubbly BGP, young households neither buy, nor sell the bubble, i.e. e b * 1 = 0. Since e b * 2 > 0 to finance consumption when old, a middle-age household reduces the expenditures on children, implying a lower population growth at the asymptotic bubbly BGP. Then, productive investment a can be larger or lower at the asymptotic bubbly BGP than at the bubbleless one because of two opposite effects: on the one hand, fewer children expenses have to be covered; on the other hand, the purchase of the bubble to finance consumption when old has to be financed.
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  • Spring '10
  • Economics, Capital accumulation

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