Chapter II: Choosing with the Worst in Mind: A Reference-Dependent Model 35
2.4 Applications
2.4.1 Intertemporal Choice with Non-Binding Borrowing Constraint 59
Let us consider a two-period intertemporal choice model. A consumer lives two periods and her utility function is
UB(c1,c2) = (c1−c1B)µ+ β(c2−c2B)µ,
where β is a discount factor, ci is consumption in periodi ∈ {1,2}, and ciB is the possible minimum consumption level in periodifor the given budget setB. Suppose µ < 1, so we will have diminishing sensitivity.
The consumer earns income yi in period i and she can borrow at most ¯b in the first period at the interest rate z2. We assume the consumer exhausts all her discounted total income M ≡ y1+ 1+y2r. Then the budget set is
B = {(c1,c2) ∈R2+|c1+ c2
1+r = M andc1 ≤ y1+b¯}.
Now we calculate the optimal consumption profile for two cases, without and with a borrowing constraint, and compare them.
Without a Borrowing Constraint: Suppose there is no borrowing constraint; that is, ¯b ≥ 1+y2r. Then the consumer’s maximization problem is:
(c1max,c2)∈BUB(c1,c2)= cµ
1 + βcµ
2, whereB= {(c1,c2) ∈R2+|c1+ c2
1+r = M}and (0,0)is the reference point. By a direct calculation, the optimal consumption levels are
c∗
1 = M
1+ β1−µ1 (1+r)
µ
1−µ andc∗
2= M· β(1+r) 1
1−µ
1+ β1−µ1 (1+r)
µ 1−µ
.
Today
Tomorrow y1+ 1+y2r = M
y1
y2 (0,0)
y1+b¯
Figure 2.11: The Effect of a Non-Binding Borrowing Constraint
Note that, similar to the standard lifetime consumption model, the optimal consumption levels are proportional to M.
With a Borrowing Constraint: Now suppose there is a borrowing constraint; that is, ¯b< 1+y2r. Then the consumer’s maximization problem is:
(c1max,c2)∈BUB(c1,c2)= cµ
1 + β c2−[y2−(1+r)b¯]µ
where
B={(c1,c2) ∈R2+|c1+ c2
1+r =M andc1≤y1+b¯}and(0,y2−(1+r)b)¯ is the reference point. By a direct calculation, the new optimal consumption levels are
c1∗∗= M −(1+y2r −b)¯ 1+ β1−µ1 (1+r)
µ
1−µ = y1+b¯
1+ β1−µ1 (1+r)
µ
1−µ andc2∗∗=M · β(1+r) 1
1−µ +(y2−b(¯ 1+r)) 1+ β1−µ1 (1+r)
µ 1−µ
.
In the standard model, the optimal consumption profiles are the same when the constraint is not binding; i.e.,y1+b¯ > c∗
1. However, in our model, the two cases are different even if the constraint is not binding (see Figure 2.11). The first implication of the model and diminishing sensitivity is the effect of non-binding constraint on the optimal consumption profile.
Observation 2: The consumer decreases her first period consumption; that is, c∗
1 > c∗∗
1 , even if the borrowing constraint is not binding (c∗
1 < y1+b¯). Moreover,
as the constraint gets tighter (as ¯bdecreases), she consumes less in the first period.
In other words,c∗∗
1 is increasing in ¯b.
The intuition of Observation 2 is as follows. The borrowing constraint decreases the consumer’s the maximum consumption level for today and increases minimum consumption level for tomorrow. Then by diminishing sensitivity (µ < 1), the relative value of tomorrow’s consumption with respect to today’s consumption will increase. Therefore, the consumer will decrease her consumption today and increase consumption tomorrow.
It is common to explain deviations from the standard model of consumer choice such asexcess sensitivity of consumption to incomeanda hump-shaped consumption profile with liquidity or borrowing constraints (see Attanasio 1999 and Attanasio and Weber 2010). However, there is no strong empirical evidence that constraints were actually binding at the time of the decision. For example, Jappelli (1990) directly asked consumers whether they applied for and were denied credit and only 12.5 percent of 1982 respondents answered that they were denied credit. Moreover, Deaton (1991) shows that liquidity constraints are rarely binding by simulation. Our model gives a justification for those explanations since liquidity constraints can have an effect on choices even if they are not binding.14
Observation 3: ci∗∗ is positively correlated with yi, fixing M constant. Moreover, ci∗∗ =αiyi+ βi for someαiand βi.
Contrast to permanent income hypothesis, it is well known that consumption depends on current income. For example, Figure 2.12, borrowed from Attanasio and Weber (2010), reports life-cycle profiles for two education groups in the UK. The left hand side of Figure 2.12 shows income and consumption paths for the group with compulsory education; the dotted curve is disposable income and the solid curve is nondurable consumption. Note that the consumption path closely follows the income path. Moreover, Campbell and Mankiw (1991) showed that in many different countries, a large number of consumers who follow a “rule of thumb" set
14A similar behavior (a non-binding constraint matters) is obtained by Hayashi (2008) in the context of choice under ambiguity (i.e., choice over Anscombe-Aumann acts). He also considers a menu-dependent choice, and in his model, an agent responds to non-binding constraints because of anticipated ex post regrets. It is not obvious how to relate our results with Hayashi (2008) since regret aversion is defined in the context of choice under uncertainty and naturally has a dynamic interpretation, while diminishing sensitivity is defined in a deterministic environment.
Figure 2.12: Average Income and Consumption by Education, Attanasio and Weber (2010)
their consumption proportional to their income. Observation 3 shows that our model is consistent with the above empirical regularities on consumption paths.
We could apply Observation 3 in a different context. For example, in a context of financial decision, it is known that younger people invest in risky asset less than older people even after controlling the income effect via risk preference. Observation 3 suggests an additional income effect to younger people’s financial decisions. There- fore, the income effect via reference-dependent preference might be an additional cause for younger people to invest less.
We can also make another interesting observation: when ¯bis small,c∗∗
1 is more dependent on y1. In fact, Zeldes (1989) observed that consumers with a low level of assets (low ¯b) more tightly link their consumption to their income.
2.4.2 Observing Different Degrees of Risk Aversion
Now we turn to risky choice. In particular, we focus on binary lotteriesxwhere xdenotes a binary lottery that givesx1dollars with probabilityx2and nothing with probability 1− x2. Now recall that the utility ofxfor a given menu A:
UAx= f u1(x1)−u1(m1A) + f u2(x2)−u2(m2A).
The objective of this subsection is to obtain contradicting risk behavior: an agent makes a risky choice in case 1 and makes a safe choice in case 2 and the estimated degrees of risk aversion (in the sense of EUT) from two cases contradict each other.
Note that by diminishing sensitivity (the strict concavity of f), f introduces an additional concavity to the total value of alternatives since utilities are distorted by f. Moreover, the additional concavity is menu-dependent. Therefore, an agent will act as if her degree of risk aversion (in the standard sense) is high in some menus and as if it is low in other menus. Now let us show the above intuition more formally.
Suppose an agent has the following additive reference-dependent preference:
For any menu Aof binary lotteries and binary lotteryx ∈ A,
U x|A = αlog(x1)−αlog(m1A)µ+ log(x2)−log(m2A)µ.
Hereα is the degree of risk aversion (in the standard sense) because when µ= 1, the model reduces to the expected utility theory: the agent maximizes expected utility x2· xα
1. Moreover, µis independent from the agent’s risk preference since the concavity of f is motivated by riskless choices.
Take four binary lotteriesx,y,x’, andy’withx1> y1andx2 < y2andx0
1 > y0
1
and x0
2 < y0
2. Suppose the agent prefersxovery in the menu{x,y} (choosing the riskier option), but prefersy’overx’in some menuA(choosing the safer option). In the standard model, the two choices contradict each other when log(y
0 2/x0
2) log(x0
1/y0
1) < log(y2/x2)
log(x1/y1)
because choosingxoveryimplies thatα > log(y2/x2)
log(x1/y1) (the agent is risk loving), but choosingy’overx’implies thatα < log(y02/x02)
log(x0
1/y0
1) (the agent is risk averse).
However, in our model the agent could choosey’overx’fromAbecause f(x1) = xµ introduces an additional menu-dependent concavity to utilities of alternatives.
To illustrate, note that choosingx overyimplies α > log(y2/x2)
log(x1/y1), as in the standard model. However, choosingy’overx’from Aimplies
α <(log(y0
2/mA
2))µ− (log(x0
2/mA
2))µ (log(x0
1/mA
1))µ−(log(y0
1/mA
1))µ 1µ
≡ α(µ).
If the reference point of A satisfies log(x
0 1/mA
1) log(y0
1/mA
1) < log(y20/m2A)
log(x0
2/mA
2), then α(µ) converges to infinity as µ → 0. Therefore, as long as log(
x0
1/mA
1) log(y0
1/mA
1) < log(y20/m2A)
log(x0
2/mA
2) and µis small enough,15we can have loglog(y(x2/x2)
1/y1) < α < α(µ). Therefore, whenµis small, the agent could act as if she is very risk-averse even ifαis large.
15For example, let A={x’,y’,z’} where x0
1 >y0
1>z0
1 andx0
2<y0
2 <z0
2. Since (z0
1,x0
2) is the reference point ofA; i.e.,(mA
1,mA
2)=(z0
1,x0
2), the inequality log(x
0 1/mA
1) log(y0
1/mA
1) = loglog(x(y100/z01)
1/z0
1) < log(y20/m2A)
log(x0 2/mA
2) =
log(y0 2/x0
2) log(x0
2/x0
2) =∞always holds.
Last, we briefly discuss the estimation of risk preference when the agent has an additive reference-dependent preference. Because of diminishing sensitivity or the strict concavity of f, measuring risk preferences from observed choices could be misleading. However, if we estimate risk preferences only using binary comparisons, then it provides an unambiguous measure for the degree of risk because the effect of f is cancel out (recall Remark 1 onin Section 2.2).