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Comment on 'Asset Bubbles and Overlapping Generations'

Ngoc-Sang Pham, Alexis Akira Toda

Abstract

Tirole (1985) studied an overlapping generations model with capital accumulation and showed that the emergence of asset bubbles solves the capital over-accumulation problem. His Proposition 1(c) claims that if the dividend growth rate is above the bubbleless interest rate (the steady-state interest rate in the economy without the asset) but below the population growth rate, then bubbles are necessary in the sense that there exists no bubbleless equilibrium but there exists a unique bubbly equilibrium. We show that this result (as stated) is incorrect by presenting an example economy that satisfies all assumptions of Proposition 1(c) but its unique equilibrium is bubbleless. We also restore Proposition 1(c) under the additional assumptions that initial capital is sufficiently large and dividends are sufficiently small. We show through examples that these conditions are essential.

Comment on 'Asset Bubbles and Overlapping Generations'

Abstract

Tirole (1985) studied an overlapping generations model with capital accumulation and showed that the emergence of asset bubbles solves the capital over-accumulation problem. His Proposition 1(c) claims that if the dividend growth rate is above the bubbleless interest rate (the steady-state interest rate in the economy without the asset) but below the population growth rate, then bubbles are necessary in the sense that there exists no bubbleless equilibrium but there exists a unique bubbly equilibrium. We show that this result (as stated) is incorrect by presenting an example economy that satisfies all assumptions of Proposition 1(c) but its unique equilibrium is bubbleless. We also restore Proposition 1(c) under the additional assumptions that initial capital is sufficiently large and dividends are sufficiently small. We show through examples that these conditions are essential.

Paper Structure

This paper contains 14 sections, 13 theorems, 49 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Tirole (1985)'s model
  3. Counterexample to Proposition 1(c)
  4. Restoring Proposition 1(c)
  5. Proofs
  6. Proof of Lemma \ref{['lem:system']}
  7. Proof of Lemma \ref{['lem:impossible2']}
  8. Proof of Lemma \ref{['lem:curse']}
  9. Proof of Proposition \ref{['prop:C']}
  10. Proof of Theorem \ref{['thm:restore']}
  11. Proof of Theorem \ref{['thm:restore2']}
  12. Systematic literature search
  13. Data collection
  14. Evaluation

Key Result

Lemma 2.1

If Assumption asmp:U holds, the equation has at most one solution $x=g(k,p)>0$, which satisfies $g_k>0$ and $g_p<0$ on its domain. Letting $(k_t,p_t,d_t)=(K_t,P_t,D_t)/G^t$, given $k_0>0$, an equilibrium has a one-to-one correspondence with the system

Figures (1)

  • Figure 1: The graph of $\phi(k)=k\log(1+1/k)$.

Theorems & Definitions (24)

  • Lemma 2.1: Equilibrium system
  • Example 1: Logarithmic utility
  • Lemma 2.2: Equilibrium monotonicity
  • Lemma 2.3: Unique, bubbleless equilibrium
  • Lemma 3.1: Resource curse
  • Proposition C: Counterexample to Tirole1985, Proposition 1(c)
  • Theorem 1: Bubble necessity with small dividends
  • Example 2: Importance of initial condition
  • Example 3: Large dividends imply resource curse
  • Theorem 2: Bubble necessity with arbitrary dividends
  • ...and 14 more