Rational bubbles in a monetary economy

Ge Zhou

doi:10.1515/bejm-2016-0242

Article

Rational bubbles in a monetary economy

Ge Zhou

Published/Copyright: September 4, 2017

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From the journal The B.E. Journal of Macroeconomics Volume 18 Issue 1

Abstract

This study explores rational bubbles in a monetary economy using an endogenous growth model with status seeking. Rational bubbles may arise when the money growth rate is higher than some threshold level. In a bubbly economy, a higher money growth rate leads to a larger bubble size, while the monetary policy is super-neutral. However, in a bubbleless economy, the monetary policy is non-neutral. Based on a comparative analysis of the calibrated model, I argue that an optimal monetary policy that maximizes the social welfare of a bubbleless economy may trigger bubbles and hurt the economy.

Keywords: monetary policy; rational bubbles; status seeking

JEL Classification: E52; G12; O11

Acknowledgement

I would like to thank the Fundamental Research Funds for the Central Universities in China for their financial support.

Appendix

A Proof of Proposition 1

In the bubbleless BGP, the value of q is equal to 0, and I suppose that c = θk, m = ϕk, and π is equal to some constant. This means that

c˙c=k˙k=m˙m≡gn.

From equation (7), we have that

(9)gn=ηθ1+ϕ+αA−δ−ρ.

From equation (6), we get that

(10)γθϕ=αA−δ+π.

From equation (4) and equation (5), we know that

(11)gn=gM−π,

(12)gn=A−δ−θ.

From equation (10), equation (11), and equation (12), we have that

γθgM−(1−α)A+θ=ϕ.

Using this result to replace ϕ in equation (10), we can find that

(1−α)A−θ=ηθ[gM+θ−(1−α)A]gM+θ−(1−α)A+γθ−ρ.

Simplifying the above equation, we have that

0=[η+(1+γ)]θ2+[(η+1)κ−(1+γ)Θ]θ−κΘ,

where κ ≡ g_M − (1 − α)A, Θ ≡ ρ + (1 − α)A > 0. The solutions for this equation are given by

θ1=(1+γ)Θ−(η+1)κ+[(1+γ)Θ−(η+1)κ]2+4(η+1+γ)κΘ2(η+1+γ),

θ2=(1+γ)Θ−(η+1)κ−[(1+γ)Θ−(η+1)κ]2+4(η+1+γ)κΘ2(η+1+γ).

When g_M ≥ (1 − α)A, then κ ≥ 0. Thus, to guarantee that the value of θ is nonnegative, θ should be equal to (1+γ)Θ−(η+1)κ+[(1+γ)Θ−(η+1)κ]2+4(η+1+γ)κΘ2(η+1+γ).

When g_M < (1 − α)A, then κ < 0. Suppose that

θ2=(1+γ)Θ−(η+1)κ−[(1+γ)Θ−(η+1)κ]2+4(η+1+γ)κΘ2(η+1+γ)>0.

It is also easy to find that

θ2+κ<(1+γ)Θ−(η+1)κ−[(1+γ)Θ−(η+1)κ]2+4(η+1+γ)κΘ2(η+1+γ)+κ=(1+γ)Θ−(η+1)κ−[(1+γ)Θ+(η+1)κ]2−4ηγΘκ2(η+1+γ)+κ<(1+γ)Θ−(η+1)κ−[(1+γ)Θ+(η+1)κ]22(η+1+γ)+κ.

If (1 + γ)Θ + (η + 1)κ > 0, then θ₂ + κ < γκ/(1 + γ + η) < 0. If (1 + γ)Θ + (η + 1)κ ≤ 0, then θ2+κ<(1+γ)Θ+(η+1)κ+γκη+1+γ<0. Thus, the value of m is negative, which has no economic meaning.

Therefore, we should rule out the solution of θ₂. ■

B Proof of Proposition 2

At the bubbly BGP, I guess that c = θk, m = ϕk, q = vk, and π is equal to some constant. Thus:

c˙c=k˙k=m˙m=q˙q≡g=αA−δ.

From equation (4), we obtain that

π=gM−(αA−δ).

While from equation (5), we can solve

θ=(1−α)A.

Together with equation (6), we have that

ϕ=γ(1−α)AgM.

Substituting the above results into equation (7), we have that

v=(ηρ−γgM)(1−α)A−1.

Given the parameter restriction

η(1−α)A>ρ,

to guarantee that bubbles exist (v > 0), the restriction on the growth rate of money is given by

gM<0,

or,

gM>γρ(1−α)Aη(1−α)A−ρ≡g¯M.

■

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Published Online: 2017-9-4

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https://doi.org/10.1515/bejm-2016-0242

Keywords for this article

monetary policy; rational bubbles; status seeking