Problem (Three period quasi-hyperbolic discounting model with sophisticated consumers who face
liquidity constraints)
Consider a person who lives for three periods, t = 1, 2, 3. To simplify
exposition, refer to three selves of this individual. These selves are
indexed by their respective periods of control over the individual’s
consumption decision. At t = 1, self one chooses c1 . At t = 2, self two
chooses c2 . At t = 3, self three chooses c3 . At time t = 1 the agent has
the following utility function:
ln c1 + βδ ln c2 + βδ 2 ln c3
These preferences would be standard if we set β = 1; in which case
the discount structure would be exponential. However, we’ll assume
0 < β < 1. At time t = 2 the agent has the following utility function:
ln c2 + βδ ln c3
Finally, at time t = 3 the agent has the following utility function:
ln c3
• The agent begins life (t = 1) with assets x1 = y1 = 1.
• In period t = 2, the agent receives labor income 1 ≤ y2 ≤ 1/β.
• In period t = 3, the agent receives labor income y3 = 1.
• The agent has a standard dynamic budget constraint:
xt+1 = R (xt − ct ) + yt+1
• To simplify the algebra, set δ = R = 1.
a. Assume that the agent faces a liquidity constraint, such that
ct ≤ xt
In other words, the agent can’t consume more than the cash-on-hand
(xt ) that she has available at any given point in time. Prove that in
equilibrium ct = yt for all t. Explain why consumption tracks income
even though income is highly volatile.
Clearly, c∗3 = x3 . From Problem 3, we know that the unconstrained solution
A2
A2
, leaving c3 = β 1+β
, where A2 = x2 + 1 is the
for the period-2 self sets c2 = 1+β
present value of self-2’s assets. Self-2 can achieve this unconstrained maximum
1
2 +1
if c2 = x1+β
≤ x2 . Solving for the critical value of x2 , the liquidity constraint
binds if x2 < β1 . So
(
x2 +1
if x2 ≥ β1
∗
c2 = 1+β
x2
if x2 < β1
Since x2 = x1 −c1 +y2 , self-2 will be liquidity-constrained iff c1 > y2 − β1 +x1 .
Suppose (for contradiction) that the optimal c1 ≤ y2 − β1 + x1 . Self-2 is not
constrained in that case, so the equilibrium will be the unconstrained solution
A1
1 −c1
1 −c1
, c2 = A1+2β
, c3 = A1+2β
, where A1 = y2 + 2 is the
from Problem 3: c1 = 1+2β
y2 +2
> 1 ≥ y2 − β1 + x1
present value of self-1’s assets. But in that case, c1 = 1+2β
a contradiction. Therefore, the optimal c1 must satisfy c1 > y2 − β1 + x1 , and
self-2 will be constrained in equilibrium. So Self-1 solves:
max
1
+x1 <c1 ≤x1
y2 − β
ln (c1 ) + β ln (y2 + 1 − c1 ) + β ln (1)
β
2 +1
which has F.O.C. c11 = y2 +1−c
, which has solution yβ+1
> 1 = x1 . At the
1
equilibrium, therefore, self-1 is constrained to set c1 = x1 .
We have shown that, in equilibrium, ct = yt for t = 1, 2, 3. Self-1 would
like to have a downward-sloping consumption profile, but the income path gives
“too much” of the lifetime wealth to the later selves (from self-1’s and self-2’s
point of view). Selves 1 and 2 would like to consume more of the lifetime wealth
than they can, so they end up consuming as much as they can, namely their
current income. They leave no savings for future selves, so each self ends up
consuming its current income.
b. Suppose that you offer an agent an extra ∆ dollars in period 1 or
an extra ∆0 in period 2. Assuming that ∆ is small (in a neighborhood
of zero) and y2 = 1, solve for ∆0 /∆ such that the agent is indifferent
(from the period 1 perspective) between these two rewards.
Since both self-1 and self-2 are liquidity-constrained in equilibrium, each will
consume 100% of the marginal dollar of current income. For small ∆, we want
to find ∆0 such that
ln (1 + ∆) + β ln (1) + β ln (1) = ln (1) + β ln (1 + ∆0 ) + β ln (1)
i.e.
ln (1 + ∆) = β ln (1 + ∆0 )
∆0
∆
Since ln (1 + z) ≈ z for z close to 0 (by first-order Taylor approximation),
= β1 .
c. Suppose that you offer an agent an extra ∆ dollars in period 2 or
an extra ∆0 in period 3. Assuming that ∆ is small and y2 = 1, solve for
2
∆0 /∆ such that the agent is indifferent (from the period 1 perspective)
between these two rewards. Relate parts b. and c. to the experimental literature on discounting. In this example, the frequency of
income receipt is the same as the frequency of consumption (i.e., income is received one per period and consumption is chosen once per
period). Is that a realistic assumption? Why or why not? How is
this timing assumption driving the analysis?
Since self-2 is liquidity-constrained in equilibrium and self-3 lives in the last
period, each will consume 100% of the marginal dollar of current income. For
small ∆, we want to find ∆0 such that
ln (1) + β ln (1 + ∆) + β ln (1) = ln (1) + β ln (1) + β ln (1 + ∆0 )
By inspection, ∆0 = 1.
The experimental literature that measures discounting in humans is typically
conducted by offering choices between monetary rewards at different points in
time. When a quasi-hyperbolic agent is not liquidity-constrained, he or she
will discount money in a time-consistent way at the market rate of interest
(even though he or she discounts utility flows quasi-hyperbolically) – because
the money simply gets added into the present value of assets. However, parts b
and c of this problem show that a liquidity-constrained quasi-hyperbolic agent
will discount money quasi-hyperbolically. The reason is that the money gets
consumed immediately, so it is, in effect, a utility flow. This reasoning suggests
that a crucial reason why typical experiments uncover time-inconsistent discounting of money is that subjects are liquidity-constrained. This makes sense,
given that most experiments are conducted with college students as subjects.
This reasoning also predicts less extreme and less time-inconsistent discounting
over money for subjects who are not liquidity-constrained.
In the reality consumption choices are more frequent than income receipt.
The timing assumption matters because receiving a constant y every period
means that (in equilibrium) the agent will consume c = y every period (if they
are liquidity constrained). Therefore consumption will be constant (until it is
perturbed by the offers made in the experiment). Once y comes infrequently
(say once every 30 periods if each period is a day and the agent is paid once per
month), then consumption will vary over the monthly pay cycle (as in Shapiro
2005) and it will no longer be true that that consumption is constant.
d. Repeat part b, but now extend the problem in the following way.
First, assume that the agent has an infinite horizon (with unit income in every period).1 Second, assume that the agent can buy (but
can’t sell) d durables (with price one per durable, so d durables cost
d dollars). Assume that each durable has a q > 0 chance of permanently breaking down per period. Assume each durable generates
1 If you are worried about convergence, you can assume that we are still in a Önite horizon
problem, but the horizon T is very large. But this is a technical issue that you can ignore in
this problem.
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use value of q per period, so the expected total stream of use value
(undiscounted) of d durables is
h
i
q
2
3
d q + (1 − q) q + (1 − q) q + (1 − q) q + . . . = d
=d
1 − (1 − q)
(Verify the first equality in the previous equation.) The payout
stream starts after the period the durable is purchased (so a durable
purchased at time t starts generating use value at time t + 1). Now
assume that ∆ = 1/β − 1, so ∆ is no longer in a neighborhood of zero.
Assume that q is small. Now solve for ∆0 and show that
ln (1/β) ' β [ln (1/β) + ∆0 − (1/β − 1)]
(1)
[Hint: First remember that ln (1) = 0. Second, note that when self
2 receives ∆0 she has so much liquidity that she chooses to consume
c2 ' 1/β and to pass ∆0 − (1/β − 1) to her future selves by buying
d = ∆0 − (1/β − 1) durables that start to pay out in period 3. Since the
amount that she passes to later selves is spread out over many periods
(q is small) the extra amount that is consumed per period is small
(bounded above by d · q). The expected sum of marginal utility due to
this durable purchase is d, since ∂ ln (c) /∂c = 1/c = 1 when c = 1. Since
self 1 is sophisticated, self 1 anticipates all of this when choosing ∆0 .]
Equation (1) implies that
∆0
' 1 − ln β < 1/β
∆
(Verify the inequality in the previous equation.) Compare these
results to part b. We have found that large stakes rewards yield
lower implied discount rates than low stakes rewards. Why is this?
Explain the economic intuition behind this result. What is the role
of commitment in this example?
Comparing these results to part b. We have found that large stakes
rewards yield lower implied discount rates than low stakes rewards.
Why is this? Explain the economic intuition behind this result. Finally, explain why the agent would not buy the durable in part b (if
it had been on offer).
Consider
self-1’s unconstrained problem when self-1 receives extra income
1
∆ = β − 1 . Self-1’s F.O.C.s for an optimum imply u0 (c1 ) = βu0 (ct ) for all
t = 2, 3, . . ., that is, ct = βc1 . Self-1 can attain this first-best consumption path
by consuming all current wealth x1 = 1+∆ = β1 in period 1 and then consuming
income ct = yt = 1 in subsequent periods. Therefore self-1 will not buy any of
the durable good.
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On the other hand, when self-2 receives extra income ∆0 , self-1 remains
liquidity-constrained and consumes c1 = 1. In order for self-1 to be indifferent
between receiving ∆ in period 1 or ∆0 in period 2, it must be that ∆0 > ∆. As
a result, self-2 will not want to consume all of current wealth, but will prefer to
save some of it. Since self-2 prefers all future selves to consume equally, self-2
would rather save by purchasing the durable good (an illiquid asset) rather than
by passing on cash to self-3. In fact, in the limit as q → 0, the durable good
spreads consumption perfectly smoothly over all future periods, and it has the
same price as nondurable consumption, so it is a dominant way to save. Since
q is small, we can therefore approximate self-2’s problem as
max
2
ln
(c
+
d
q)
+
β
ln
(1
+
(1
−
q)
d
q)
+
ββ
ln
1
+
(1
−
q)
d
q
+ ...
2
2
2
2
0
c2 +d2 ≤1+∆
Denoting the Lagrange multiplier on the budget constraint by λ and approximating ln (1 + z) ≈ z (since q is small), the F.O.C.’s are
λ=
λ=
1
c2 + d2 q
q
q
2
+ β (1 − q) q + β (1 − q) q + . . . =
+ β (1 − q) q
c2 + d2 q
c2 + d2 q
Substituting for λ and rearranging, 1 = q +β (1 − q) (c2 + d2 q). Substituting
1+∆0 − 1
the budget constraint c2 + d2 = 1 + ∆0 and solving, d2 = 1−q β and c2 =
1
−q (1+∆0 )+ β
are arbitrarily close to optimal for q arbitrarily close to 0. When
1−q
self-2 receives extra income ∆0 , self-1’s utility is (approximately)
ln (1) + β ln (c2 + d2 q) + β (1 − q) d2 = β
−q (1 + ∆0 ) +
1−q
1
β
+q
1 + ∆0 −
1
β
!
1−q
1 + ∆0 − β1
+ β (1 − q)
1−q
1
1
0
+∆ + 1−
= β ln
β
β
To solve for ∆0 , we equate the utilities for self-1 receiving extra income ∆
and self-2 receiving extra income ∆0 :
1
1
1
0
≈ β ln
+∆ + 1−
ln
β
β
β
0
Rearranging and dividing by β1 − 1, ln β1 = 1∆−1 − 1. Substituting ∆ =
β
1
β
− 1.
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∆0
= 1 − ln β
∆
Since ln z < z −1 for all z > 1, it follows that 1−ln β = 1+ln β1 < 1+ β1 −1 =
1
β.
The reason that the agent has a lower effective discount rate over largestakes rewards than over small-stakes rewards is that the larger rewards relax the
agent’s liquidity constraint. As a result, the agent transfers some consumption
to the future rather than consuming it all now. The conflict between self-1 and
self-2 is about consumption in period 1 versus period 2, not about consumption
in future periods. Selves-1 and 2 agree about consumption in future periods.
For this reason, self-1 is willing to have self-2 make consumption decisions when
self-2 is allocating future consumption. When self-2 is liquidity-constrained,
however, self-2’s marginal propensity to consume (MPC) is 1, which maximizes
conflict between the two selves, making self-1 maximally reluctant to pass on
any wealth to self-2 (as is part b). In this problem, ∆0 is large enough that self2 is no longer liquidity-constrained, so self-2’s MPC is less than 1. Although
self-1 still disagrees with how self-2 will split the income between consumption
today and future consumption, they agree about the quantitative utility value of
passing on wealth to the future (and its allocation among future selves). Since
self-2 is not liquidity-constrained, self-2’s decisions are more in line with self1’s interests, so self-1 needs less compensation for the marginal dollar given to
self-2.
This general intuition also lies behind the hyperbolic Euler equation,
∂ct+1
∂ct+1
0
βδ + 1 −
δ u0 (ct+1 )
u (ct ) = R
∂At+1
∂At+1
This equation shows that self-t’s effective discount factor between t and t + 1
∂ct+1
. If self-(t + 1) is liquidity-constrained so
depends on self-(t + 1)’s MPC, ∂x
t+1
∂ct+1
that ∂x
= 1, then self-t is maximally impatient. The reason is that, because
t+1
self-(t + 1) will entirely consume the marginal dollar, self-t is literally trading
off utils today versus utils tomorrow. Self-t values utils tomorrow only βδ as
much as utils today. However, when self-(t + 1) is not liquidity-constrained so
∂ct+1
that ∂x
< 1, then self-t is happier to pass on wealth to self-(t + 1) (and more
t+1
∂ct+1
so the smaller is ∂x
). Self-t values utils in periods after tomorrow by a factor
t+1
of only δ less than self-(t + 1) does, so the conflict is reduced. Self-t’s effective
discount factor between t and t + 1 is a weighted average of βδ and δ, with the
weight depending on self-(t + 1)’s MPC.
Commitment through durable goods is a way to transfer resources to future
period without generating any temptation of over-consumption. Indeed, since
the agent cannot disinvest any unit of d, period 1 agent can partially solve the
conflict between period 1 and 2 by buying more d.
The agent in part b would not have bought the durable good (had it been
on offer) because the agent was liquidity-constrained and preferred to consume
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more in the current period relative to all future periods.
e. Explain why the agent would not have purchased the durable in
part b (if it had been offered to her).
This is because the agent is liquidity-constrained in part b). As a consequence, her marginal propensity to consume is 1 and she would consume 100%
of an extra dollar received. This is equivalent to say that spending on d subtracting money from current consumption would be suboptimal.
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