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May 29, 2013

Piggy Banks

Donald P. Morgan and Katherine A. Samolyk

What do banks do? Ask an economist and you’ll get a variety of answers. Banks play
a vital role in allocating capital by linking savers and borrowers; they
produce information by screening and monitoring borrowers; they create liquidity;
they share and distribute risk; they engage in maturity transformation by
borrowing short and lending long. What you won’t usually hear is that banks may
help people stick to an optimal savings plan that they might not be able to stick
to if they invested their money themselves. In other words, banks may serve as piggy banks by preventing people from
consuming assets when the return to investing is high, even when the temptation
to consume is strong.

The Set-Up
In a New York Fed staff
, now forthcoming in the Journal of Financial Services Research, we develop this notion of
banks as a commitment device, or piggy bank. Our study uses a modified version
of a classic
by Douglas Diamond and Phillip Dybvig. Their
model features savers who have one unit of wealth to invest to support their
consumption over time. Savers have two assets to invest in: a liquid,
short-term asset that pays off relatively soon and a long-term, illiquid asset
that pays off with a delay. A key feature of the Diamond-Dybvig model is that
savers have random demand for liquidity, that is, there’s some chance that they’ll 
have to consume early, before the long-term
asset pays off. People like that are called, aptly, early consumers, but it
should be clear that “early” could be tomorrow, next year, or before
retirement. The other people are called late consumers. The random demand for
liquidity forces savers to trade off liquidity and returns because the
long-term asset pays more, but the short-term asset pays sooner.

    We deliberately alter the Diamond-Dybvig
set-up to make our point about piggy banks. In our version, both assets are
freely traded, so the delayed payoff on the long-term asset isn’t really a
problem; once a saver realizes she needs to consume early, she could just trade
any long-term assets she held for short-term assets. A second, key difference
in our set-up is that the return on the short-term asset is random-it can be
low or high, higher even than the return on the long-term asset. That
assumption drives our “piggy bank” result, but it doesn’t seem like a stretch
to say that short-term assets have random returns and that they can, in some
events, be higher than the return on long-term assets.

The First-Best Solution
A solution to the model is an allocation of savers’ wealth across the two
assets, the resulting consumption stream for early and late consumers and a
reinvestment strategy for rolling over any short-term assets that don’t get
consumed by early consumers. We first solve for the first-best solution, that
is, the allocation that an omnipotent, benevolent planner would choose to
maximize the expected well-being of the typical person in the economy. The
planner solves for all that ex ante, that
is, before consumption preferences and asset returns in the next period are
realized. Not surprisingly, the planner reinvests some of the short-term asset
when its return over the next period turns out to be high, but she doesn’t roll
over any of the asset when its return turns out to be low. Accordingly, late consumers get more of the returns from
short-term investments when returns turn out to be high. That’s too bad for the
early consumers, but they get more consumption when returns to reinvesting turn
out to be low. On average, this
solution yields the best expected outcome for all consumers ex ante. In
particular, when the return over the next period turns out to be high, the loss
to early consumers from reinvesting some of their short-term asset will be outweighed
by the resulting gain to late consumers.

Savers’ Commitment Problem
How do savers allocate their wealth and consume if they invest directly in
the two assets and then trade them once they learn their consumption needs? Our
key result is that when savers invest directly (instead of through a bank),
early consumers consume too much (relative to the first-best) when the
short-term interest rate turns out to be high. That’s the essence of the
commitment problem; if choosing on their own, early consumers, who only care
about early consumption, will consume all their short-term assets, even when
the next period’s return on the short-term asset is high. That’s their optimal strategy
ex post, that is, after consumption
needs and returns are realized, but it’s suboptimal compared with the first-best
solution, which calls for some of the short-term asset to be reinvested.

Banks and Other Financial Intermediaries Can Help with Commitment Problems
We go on to show that savers can solve, or at least ameliorate, their
commitment problem by locking their savings in a bank. The bank offers a
two-period time deposit in which the penalty for early withdrawal is contingent
upon the short-term interest rate prevailing at that time; if the short-term
rate rises, the penalty is larger, so the bank can roll over more of the
short-term asset in that event. While the model delivers a very specific
solution, we think that other intermediaries, such as pension funds and even
social security, by locking funds out of the reach of savers, may constitute
real-world examples of this commitment role. We conclude that, apart from their
many other functions, banks and other financial intermediaries can serve a
piggy bank role as well.

The views expressed in this post are those of the authors and do not necessarily reflect the position of the Federal Reserve Bank of New York or the Federal Reserve System. Any errors or omissions are the responsibility of the authors.


Donald P. Morgan
is an assistant vice president in the Federal Reserve Bank of New York’s Research and Statistics Group.


Katherine A. Samolyk is a senior economist at the Consumer Financial Protection Bureau.

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