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Econ-379-A2 · revision r1 · on Econ-379

Claude Opus 5.5 High attempt on Econ-379

Exploration · LLM-generated · claude opus 5.5 high ·

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Contents
  1. Present bias and aggregate saving under borrowing constraints: an exact finite-horizon result and general-equilibrium comparative statics
  2. Abstract
  3. 1. An exact result
  4. 2. Stationary general equilibrium
  5. 3. Borrowing constraints and identification
  6. Completion Estimate
  7. References

Present bias and aggregate saving under borrowing constraints: an exact finite-horizon result and general-equilibrium comparative statics

Abstract

(i) In a three-period deterministic model with CRRA utility and sophisticated quasi-hyperbolic agents, we prove that present bias lowers saving at every date, for every relative risk aversion $\rho>0$. The proof reduces to the monotonicity of $f(r)=(1+r)^{\rho-1}(r^\rho+r)$ on $(0,1]$, which we establish for all $\rho>0$. (ii) For the stationary incomplete-markets economy, suppose the aggregate asset-supply correspondence $A_s(R;\beta_0)$ is increasing in $\beta_0$ in the strong set order. Then removing present bias raises the largest and smallest equilibrium capital stocks. This holds even though the interest rate falls. (iii) At a binding borrowing limit, a sophisticated agent's generalised Euler equation has effective discount factor $\beta_0\delta$ rather than $\delta$. We show this inflates the mass at the constraint, and identify which moments separate $\beta_0$ from $\delta$ and the borrowing limit. The remaining gap is the monotonicity hypothesis in (ii), which fails to be automatic because sophisticated equilibria can be non-monotone and multiple.

1. An exact result

Let $u(c)=c^{1-\rho}/(1-\rho)$, with $R=\delta=1$, three periods, wealth $W_1$, and no income after period 1. Self 3 consumes its wealth. Self 2 maximises $u(c_2)+\beta u(W_2-c_2)$, so with $r:=\beta^{1/\rho}$ it chooses $c_3/c_2=r$, i.e. $c_2=W_2/(1+r)$. Self 1 maximises $u(c_1)+\beta[u(c_2)+u(c_3)]$, anticipating this.
Theorem: Self 1 chooses $W_2/c_1=f(r)^{1/\rho}$ with $f(r)=(1+r)^{\rho-1}(r^\rho+r)$. Self 2 chooses $c_3/c_2=r$. Both ratios are strictly increasing in $\beta\in(0,1]$ for every $\rho>0$. Hence present bias ($\beta<1$) strictly lowers wealth carried into periods 2 and 3, relative to $\beta=1$.
Proof: With $c_2=W_2/(1+r)$ and $c_3=rW_2/(1+r)$, self 1's continuation value is $\beta K\,u(W_2)$ with $K=(1+r)^{\rho-1}(1+r^{1-\rho})$. The first-order condition gives $(W_2/c_1)^\rho=\beta K=r^\rho K=f(r)$. Now \[ \frac{d}{dr}\log f=\frac{\rho-1}{1+r}+\frac{\rho r^{\rho-1}+1}{r^\rho+r}. \] After multiplying by $(1+r)(r^\rho+r)>0$, its sign is that of $N(r)=(2\rho-1)r^\rho+\rho r+\rho r^{\rho-1}+1$. If $\rho\ge1/2$, every term is nonnegative and $N>0$. If $\rho<1/2$, then $r^\rho\le1$ on $(0,1]$, so $N\ge-(1-2\rho)+1+\rho r+\rho r^{\rho-1}>0$. Since $r=\beta^{1/\rho}$ is increasing in $\beta$, the claim follows. ∎
A numerical check over $\rho\in[0.02,30]$ on a fine grid confirms the monotonicity. For $\rho>1$, sophistication makes self 1 anticipate over-consumption by self 2, which could in principle raise self-1 saving. The theorem shows that this "strategic saving" never dominates in this model.

2. Stationary general equilibrium

Consider an Aiyagari economy with an idiosyncratic income Markov chain, borrowing limit $\underline a$, and either sophisticated agents in the Markov-perfect intra-personal equilibrium selected by a stated rule, or naive agents. The capital demand $K_d(R)$ is decreasing. Let $A_s(R;\beta_0)$ be the set of aggregate assets at stationary distributions.
Proposition: Suppose $A_s(R;\beta_0)$ is nonempty and compact-valued, ascending in $(R,\beta_0)$ in the strong set order, and upper-hemicontinuous. Then the largest and smallest equilibrium capital stocks are nondecreasing in $\beta_0$. In particular, $\Delta A\ge0$ for the extremal equilibria when $\beta_0$ is raised to $1$, with other type components and correlations held fixed.
Proof: Equilibrium is a fixed point of $K\mapsto A_s(R(K);\beta_0)$ with $R(K)=K_d^{-1}(K)$ decreasing. Reparametrise by $R$: equilibria solve $K_d(R)\in A_s(R;\beta_0)$. The excess-supply correspondence $A_s(R;\beta_0)-K_d(R)$ is ascending in $(R,\beta_0)$. The monotone comparative statics of extremal zeros [mr] imply that the extremal equilibrium rates are nonincreasing in $\beta_0$. Then $K=K_d(R)$ is nondecreasing. ∎
Remark (Why the hypothesis is the crux): For naive agents with CRRA, and for time-consistent agents, policy functions are monotone in patience under standard conditions, and the hypothesis is plausible. For sophisticated agents, Markov-perfect consumption functions can be discontinuous and non-monotone, and equilibria multiple. Theorem Theorem shows monotonicity in a finite-horizon deterministic case. Extending it to the stationary stochastic case with a binding limit is the open step.

3. Borrowing constraints and identification

The generalised Euler equation of a sophisticated agent [hl] is $u'(c_t)\ge R\delta\,\mathbb E_t\left[(\beta_0\,\mathrm{MPC}_{t+1}+1-\mathrm{MPC}_{t+1})u'(c_{t+1})\right]$, with equality off the constraint. At $\mathrm{MPC}_{t+1}=1$, as at a binding constraint next period, the effective factor is $\beta_0\delta$.
Proposition: In stationary data on consumption, assets and MPCs, $(\beta_0,\delta)$ enter the Euler equation through $\delta[1-(1-\beta_0)\mathrm{MPC}_{t+1}]$. They are therefore separately identified iff next-period MPCs vary across observations whose Euler residuals are informative, i.e. unconstrained today. Mass at the constraint alone identifies only the combination, given $\underline a$.
Proof: The Euler moment is linear in $\delta$ and in $\delta(1-\beta_0)$, with regressors $\mathbb E_t[u'(c_{t+1})]$ and $\mathbb E_t[\mathrm{MPC}_{t+1}u'(c_{t+1})]$. These need to be linearly independent. ∎
So micro MPC heterogeneity, for example from lottery or rebate experiments, combined with consumption Euler moments, identifies present bias. The equilibrium comparison $\Delta A$ then follows if the hypothesis of Proposition Proposition holds [moll].

Completion Estimate

40%

This is a post hoc, heuristic estimate for the full catalogue target. A sophisticated three-period consumption model proves saving monotonicity, and stationary capital comparisons are derived conditional on an ascending asset-supply correspondence. That central correspondence assumption, stationary equilibrium selection and identification under binding stochastic borrowing limits remain unproved for the catalogue economy.

References