The minimal marginal propensity to consume #
Carroll and Shanker's Theoretical Foundations of Buffer Stock Saving, draft dated 2025-12-05, Lemma 1 (Limiting MPCs), gives consumption-share bounds in their buffer-stock model: https://econ-ark.github.io/BufferStockTheory/BufferStockTheory.pdf
Here the minimal bound is proved for the finite capped-household model, under explicit positivity and cap-slack hypotheses,
κ · m ≤ c(m), κ = 1 - Þ/R, Þ = (βR)^(1/γ),
for CRRA utility at a zero borrowing limit. Þ is Carroll's absolute patience factor, and κ
is the MPC of a perfect-foresight consumer with no human wealth.
Why this one and not the other #
The MAXIMAL MPC is 1 - ℘^(1/γ)Þ/R, where ℘ is the probability of a zero-income event. This
model has minIncome_pos, so ℘ = 0, the maximal MPC is 1, and the bound it gives —
c(m) ≤ m — is the budget constraint and nothing more. That is not a defect of the estimate: a
liquidity-constrained household at the borrowing limit really does consume everything, and our
own corner condition says so. So only the minimal MPC has content here.
The proof is one Euler step, and the bound is self-reproducing #
Suppose the continuation's consumption function already satisfies c ≥ κ m. At an interior
choice A > 0 the Euler inequality u'(c) ≤ βR E[u'(c'(m'))] applies, and m' ≥ R A because
income is positive, so
c^(-γ) ≤ βR (κ R A)^(-γ), hence c ≥ (κR/Þ) A = (κR/Þ)(m - c),
and (κR/Þ)/(1 + κR/Þ) = κ precisely because 1 - κ = Þ/R. At the corner A = 0 the household
consumes everything, so c = m ≥ κ m outright. The bound therefore survives the Bellman
operator, and holds at the zero continuation because the household with no future eats
everything.
No limit of a sequence of MPCs is taken: κ is a fixed point of the recursion
κ⁻¹ ↦ 1 + (Þ/R)κ⁻¹, so the single constant reproduces itself.
The sign of the rate #
Everything below is stated with 0 < κ as the hypothesis, not 0 ≤ r. Return impatience
βR < 1 at a non-negative rate gives 0 < κ (minMPC_pos), but at γ = 1 one has
κ = 1 - β (minMPC_one) at EVERY rate r > -1, so the log bounds — and with them the
capital ceiling — run into negative rates, where Aiyagari's K(r) has to be excluded too.
What it is for #
Two consequences the development already had homes for. policy_lt_self_of_consumption_lower_bound
turns it into the DECLINE condition, which was a raw assumption; and
aggregateCapital_le_of_consumption_bound turns it into a quantitative ceiling on capital
supply, in place of the asset cap. Both conditions ask for (1-κ)R < 1, and (1-κ)R = Þ, so
both are exactly βR < 1.
The minimal marginal propensity to consume, κ = 1 - Þ/R with Þ = (βR)^(1/γ).
Instances For
The absolute patience factor is positive when the household discounts at all.
Impatience is exactly Þ < 1.
(1 - κ)R = Þ: the factor by which assets grow under the bound is the patience factor.
At γ = 1 the patience factor is βR itself, so κ = 1 - β: log and log Stone--Geary
need no root at all.
Sufficient conditions for a positive minimal MPC: impatience and a nonnegative rate.
Here βR < 1 and R ≥ 1 imply the required return-impatience bound; these conditions are
not necessary, as minMPC_one also applies at negative admissible rates.
The bound along the iteration #
One shifted-CRRA Euler step propagates the minimal consumption bound.
One Euler step of the minimal-MPC bound. If a continuation v satisfies
κ m ≤ c_v(m) everywhere, with consumption positive and the cap slack against it, then so does
bellman v. Extracted from the induction so that it can also drive a SIMULTANEOUS induction on
cap slack (crra_minMPC_of_cap), which is what large β needs.
The shifted-CRRA version #
With u'(c) = (η + c)^(-γ) the same argument runs on η + c against η + m:
κ (η + m) ≤ η + c(m),
with the SAME κ. The step produces c ≥ κ m + η(κ - Þ)/R, and what has to be checked is that
this is at least κ m - η(1-κ), which holds because κ ≥ 0. So the subsistence level costs
nothing in the constant; it only shifts what the constant multiplies.
Read as a bound on saving this is policy ≤ (1-κ)(η + m), so the household behaves like a CRRA
household with income raised by η — which is what a subsistence level is.
The minimal-MPC bound for shifted CRRA, at every iterate.
The shifted-CRRA bound at the value function.
The minimal-MPC bound, at every iterate. Consumption is at least κ times cash on hand,
uniformly in the horizon — so it passes to the value function.
The hypotheses beyond CRRA are the two the Euler inequality needs and nothing else: consumption
positive at each iterate, and the saving cap slack there. Both are fields (or immediate
consequences of fields) of Calibrated.
At the fixed point, and what it buys #
The bound is uniform in the horizon, so it survives the limit: tendsto_policyOf says argmaxes
follow uniform limits when the limit has concave slices, which the value function does.
The minimal-MPC bound at the value function.
The decline condition, from primitives. Above an explicit asset level the household runs
its assets down, because it consumes at least the share κ of cash on hand and
(1-κ)R = Þ < 1. This is the hypothesis hdecl of exists_exhaust_of_decline, which the
development has carried as an assumption.
Cap slack from the bound itself #
The induction above ASSUMES the cap is slack at every iterate. At small β that is supplied by
an oscillation bound; at Aiyagari's β = 0.96 no such bound is available. But the minimal-MPC
bound itself implies cap slack — saving is at most (1-κ) m, and (1-κ)(maxIncome + R·cap) < cap once the cap is large enough — so the two can be proved TOGETHER, one induction carrying
both. The cap is then bookkeeping again: any cap > (1-κ)maxIncome / (1 - (1-κ)R) will do, and
1 - (1-κ)R = 1 - Þ > 0.
Saving under the minimal-MPC bound is at most (1-κ) m, hence below the cap once the cap
clears (1-κ)(maxIncome + R·cap).
The minimal-MPC bound and cap slack, by one induction. Nothing is assumed about the cap beyond one inequality among the primitives.
The minimal-MPC bound at the value function, with the cap from primitives.
The cap is slack at the fixed point, from primitives.
The decline condition for shifted CRRA. Saving is at most (1-κ)(η + m), so the
household behaves like a CRRA household whose income is raised by the subsistence level.
The Doeblin exhaustion data for shifted CRRA, with no assumption on the income process.
The exhaustion data, without IidIncome #
exists_exhaust_of_impatient (the capped worst-state decline argument) also supplies
the decline half from
impatience, but it gets there by comparing income states: consumption is lowest where income is
lowest, which needs IidIncome so that the continuation does not depend on today's state. The
minimal-MPC route compares nothing. It is a statement about the budget and the Euler inequality
at ONE state, so it holds whatever the income process does.
What it costs instead is the two iterate-level hypotheses, which the Açıkgöz route asks for only at the fixed point.
The Doeblin exhaustion data from the minimal MPC. Only the corner condition and one threshold inequality are left to the calibration — and no assumption is made about the income process.