Discussion Paper No. 274
On the Concavity of the Consumption Function witha Quadratic Utility under Liquidity Constraints
Shin-Ichi Nishiyama
Ryo KatoSeptember 2011

TOHOKU ECONOMICS RESEARCH GROUPGRADUATE SCHOOL OF ECONOMICS AND

MANAGEMENT TOHOKU UNIVERSITY
KAWAUCHI, AOBA-KU, SENDAI,
980-8576 JAPANOn the Concavity of the Consumption Function
with a Quadratic Utility under Liquidity Constraints
Shin-Ichi Nishiyamay

Ryo Katoz

Tohoku University

Bank of Japan

This Version: September, 2011
First Version: September, 2004

Abstract
This paper demonstrates the concavity of the consumption function of in…nitely
living households under liquidity constraints who are not prudent — i.e. with aquadratic utility. The concavity of the consumption function is closely related tothe 3-convexity of the value function.
Key Words: Consumption Function, Liquidity Constraints
JEL Classi…cation: C6, D91, E21
We thank Hiroshi Fujiki, Keiko Murata, Makoto Saito and, especially, Miles Kimball for theirhelpful comments and suggestions. This paper was written while the …rst author was at the Institutefor Monetary and Economic Studies, Bank of Japan.

The views expressed in this paper are those ofthe authors and do not necessarily re‡ect those of the Bank of Japan or the Institute for Monetary andEconomic Studies.
y27-1 Kawauchi, Aoba-ku, Sendai-shi, Miyagi-ken, 980-8576, JAPAN, Tel.: +81-22-795-3471, E-mail:nishiyama@econ.tohoku.ac.jpz2-1-1 Hongokucho, Nihonbashi, Chuo-ku, Tokyo, 103-8660, JAPAN, E-mail: ryou.katou@boj.or.jp

11Introduction
Since the numerical illustration by Deaton (1991), researchers have been aware thatliquidity constraints generate a concavity in the consumption function. However, analytics of the concavity due to liquidity constraints has remained unknown until recently.
Carroll and Kimball (2001) made the …rst important attempt in setting an analyticalfoundation and showed the concavity of the consumption function when the consumer’s
optimization has a …nite horizon.

Technically, they exploit the convexity of marginalvalue function in the terminal period and use the backward induction to show the convexity of marginal value function in the current period.

However, in the context of

in…nite horizon, this particular approach is not applicable since the terminal period’svalue function is not de…ned.
In this paper, we o¤er an analytical foundation of the concavity of the consumptionfunction in the context of in…nite horizon, when consumer’s utility is quadratic. Taking
a di¤erent approach to Carroll and Kimball (2001), we directly prove the 3-convexity
or Levinson’s Inequality (Levinson 1964) of the value function and show that the consumption function is concave. The concept of 3-convexity is extremely convenient whencharacterizing the value function, especially with the in…nite horizon. Thus, we regardthis 3-convexity approach as a complement, rather than a substitute, to Carroll andKimball’s (2001) backward induction approach in the …nite horizon setting.
Finally, it should be emphasized that under the model that we consider – i.e. consumer’s utility is quadratic – , the concavity is not generated by prudence of the consumer, but is solely generated by the presence of liquidity constraints1 . By the virtueof this set-up, we can solely focus on the analytical mechanism how liquidity constraints
generate the concavity in the consumption function. The rest of the paper is organized1

For the concavity of the consumption function generated by prudence, see Carroll and Kimball(1996).

2

as follows. Section 2 discuss the set-up of the model. Section 3 clari…es the concept of
3-convexity, shows the 3-convexity of the value function and proves the concavity of theconsumption function. Section 4 provides some concluding remarks.

2The Model
We assume a very simple in…nite horizon dynamic optimization problem where con-sumer’s utility is quadratic and time-separable. Further, consumer faces no uncertaintyin terms of rate-of-return on the net wealth and in terms of labor income.

The only

source that makes the dynamic optimization problem non-standard is the existence ofliquidity constraints – the net wealth cannot be negative. Thus, consumer’s dynamic
optimization problem can be formulated as follows.
Vt (wt ) = max
ct+i1XU (ct+i )

(1)

i=0

s:t: wt+i+1 = R (wt+iwt+i+1ict+i ) + y0

where ct stands for consumption, which is the control variable of the consumer, andwt stands for the net wealth, which is the state variable of the optimization problem.
Period-by-period utility is de…ned as a quadratic function in consumption, i.e. U (ct ) =act(b=2) c2t , where a and b are positive constant parameters. Discount rate ; interest

rate R, and labor income y, are assumed to be time-invariant2 .
2

Recursive nature ofUncertainty in rate-of-return or labor income can be easily introduced in the model set-up, butwill not alter the main implication.

Indeed, since the very message of this paper is the concavity of

consumption function under deterministic environment, introduction of uncertainty will only obscure
the main point of this paper.

3

this in…nite horizon problem allows us to reduce it into the following Bellman equation.
V (w) =maxb 2c + V (w)~2

acc w+y=R

s:t: w~ = R (w(2)c) + yIt should be noted that since the optimization horizon is in…nite, the value functions inthe subsequent period will converge to a certain function V ( ) as can be seen on both sidesof the Bellman eq. (2). Furthermore, this converged value function is a consequence ofrecursive optimization under liquidity constraints from the future period and therefore
should be distinguished from the value function under liquidity unconstrained case3 . In
other words, the value function under liquidity constraints will no longer be a quadratic
function even under quadratic utility, which is in sharp contrast to the case withoutliquidity constraint whose value function is, of course, quadratic.
Taking the …rst-order condition of eq. (2) with respect to consumption will yield thefollowing equation.
c(w) =abb

RV 0 (w)~(3)

The function c(w) on the left-hand side characterizes the optimal consumption as afunction of the current net wealth.

Further, by invoking the envelope theorem (orBenveniste-Scheinkman formula) on eq. (2), we can derive the following relation between
the current shadow price of the net wealth –i.e. the marginal value function evaluatedat the current net wealth –and the future shadow price.
V 0 (w) = RV 0 (w)~3(4)

We assume the regularity condition on discount rate to ensure the Contraction Mapping Theoremto hold.

For more rigorous treatment on this issue, see for Stokey and Lucas (1989).

Also, for theexistence of converged value function under the case where control variables are constrained, see for
instance Chmielewski and Manousiouthakis (1996).

4

Combining eq. (3) and eq. (4), we obtain the following key equation.
c(w) =ab

1 0
V (w)
b(5)

The virtue of eq. (5) is that it relates the current optimal consumption to the current
shadow price of the net wealth rather than the future shadow price of the net wealth asin eq. (3). This key relationship enables us to infer the characteristics of the optimal
consumption function by investigating the nature of the marginal value function. Or
putting it another way, it su¢ ces to characterize the marginal value function in orderto characterize the optimal consumption function4 .
Some remarks are in order. If the value function is three times di¤erentiable, then 3convexity of the value function is equivalent to positiveness of the third derivative of thevalue function –i.e., V 000 ( )0. However, as pointed out by Carroll and Kimball (2001),

the marginal value function in eq. (5) is ‘kinked’under the deterministic environment
with liquidity constraints that V 000 ( ) is not well de…ned. Thus, it is not appropriate torely on third order di¤erentiability of V ( ) in proving the concavity of the consumptionfunction, especially in our case of consideration. Fortunately, the concept of 3-convexity
is more general in its applicability. As long as the value function is once di¤erentiable,
3-convexity implies the convexity of marginal value function even if the marginal valuefunction is kinked and this implication does not require the function to be 2 or 3 times
di¤erentiable. It is this property of 3-convexity that makes it relatively easy to show
the concavity of the consumption function in the presence of liquidity constraints. The
following section clari…es the concept of 3-convexity and then shows that the valuefunction is 3-convex.
4

It should be noted that when the value function is quadratic (as in the case when preference isquadratic and without liquidity constraints), the optimal consumption function will be linear. This canbe easily seen from eq. (5).

53Main Results
We …rst de…ne the notion of 3-convexity.

De…nition 1 (3-convexity) A function f : [a; b] ! < is said to be 3-convex on [a; b]
if for 8x1 ; x2 2 [a; b] such that x1 6= x2
f|x1 + x2
2

1
1f (x1 ) + f (x2 )22{z}Af (x2 )|1f2

1x1 + x2+ f
22{z3x2Bx12:}(6)

The inequality (6) is a special case of Levinson’s inequality (Levinson 1964) which
can be regarded as a higher-order Jensen’s inequality. As Jensen’s inequality is closely
related with the notion of convexity, so is Levinson’s inequality with 3-convexity. The
intuition of the inequality (6) can be vividly captured by Figure 1. The left-hand sideof the inequality (i.e. denoted A) represents the di¤erence between the value of function
evaluated at the mid-point of x1 and x2 to the mid-point of the chord from x1 and x2 .
It is possible to interpret A as a magnitude of concavity of a function in the domain[x1 ; x2 ]. The right-hand side of the inequality (i.e. denoted B) can be interpreted in the
2 3x2 x1
similar fashion with a di¤erence that domain is now [ x1 +x]. Thus, intuitively
2 ;2

speaking, the function will be 3-convex if the magnitude of concavity decreases as x
increases5 .
Next, we state the lemma that links 3-convexity of the function to convexity of themarginal function. The following lemma is a special case of the more general theorem
that links n-convexity to convexity of (nk)th derivative of a function. Rigorous proofof the theorem is well beyond the scope of this paper and will be omitted.
Lemma 1 If a function f : < ! < is 3-convex on [a; b], then the …rst derivative f 0 :< ! < exists and is convex on [a; b].
5Or, in the continuous analouge, f 00 (x) is increasing in x

6

Proof. See Peµcarić et al. (1992, pp.16).
We are now in the position to state the key theorem of this paper.
Theorem 1 Let V (w) be the value function stated in (2). Then for any w 2 [0; 1),V (w) is 3-convex.
Proof. Let w1 and w2 be some arbitrary number in [0; 1) such that w1 < w2 . Thenit su¢ ces to show the following inequality.
V

w1 + w2
2

1
1V (w1 ) + V (w2 )22V (w2 )1V2

w1 + w2
21+ V
23w2

w1
2

which is equivalent in showing thatw1 + w2
2

3V

3w2+V

w1
2V (w1 )3V (w2 )0:(7)1Let sequence fc1;t+j g1
j=0 and fc2;t+j gj=0 be the optimal consumption path given state

w1 and w2 , respectively.
12 c1;t+j :Now, de…ne ct+j = 21 c1;t+j + 12 c2;t+j and c^t+j = 23 c2;t+j

Further, de…ne w = 21 w1 + 12 w2 and w^ = 23 w2

12 w1 .

Then from Chmielewski

and Manousiouthakis (1996), the sequence fct+j g1
ct+j g1
j=0 (or f^j=0 ) is feasible, but notnecessarily equal to the optimal consumption path given the state w (or w).
^ Therefore,
P1 j
P1 j
U (^ct+j ). Then from the inequality (7),U (ct+j ) and V (w)^V (w)
j=0
j=0it follows that

3V (w) + V (w)^31X
j=0jU (ct+j ) +V (w1 )1X

j3V (w2 )U (^ct+j )j=0(8)
1X
j=07

jU (c1;t+j )31
X
j=0jU (c2;t+j ):Rearranging the right-hand side of the inequality (8) and from the de…nition of theutility function, it follows that
RHS =

1X

j

j=01
X

=

j=01
X
j
j=01
X

3(act+jjb 2c ) + (a^ct+j2 t+j(ac1;t+jb 2c) + 3(ac2;t+j
2 1;t+ja(c1;t+j + 3c2;t+j )jb 2c^ )2 t+ja(c1;t+j + 3c2;t+j )j=0b 2c)
2 2;t+jb 2(c+ 3c22;t+j )2 1;t+jb 2(c+ 3c22;t+j )2 1;t+j= 0:
Thus, 3V (w) + V (w)^V (w1 )3V (w2 )0. This proves the theorem.

The concavity of the optimal consumption function follows naturally from Lemma 1
and Theorem 1.
Theorem 2 Let c(w) be the optimal consumption function of the dynamic optimization
problem (2). Then for any w in [0; 1), c(w) is concave.
Proof. Let w1 and w2 be some arbitrary number in [0; 1) such that w1 < w2 . Thenit su¢ ces to show,c( w1 + (1where 0 <)w2 )c(w1 ) + (1)c (w2 )< 1. From eq. (5), this is equivalent in showing thatV 0 ( w1 + (1)w2 )V 0 (w1 ) + (1)V 0 (w2 ) .

Now from Theorem 1, V (w) is 3-convex, which in turn implies that V 0 (w) is convex
from Lemma 1. This proves the theorem.

84Concluding Remark
This paper showed, in the context of in…nite horizon, how the presence of liquidity

constraints generate a concavity in the consumption function, even when consumer is
not prudent –i.e. preference is quadratic. In showing the concavity of the consumptionfunction, we directly proved the 3-convexity (also known as Levinson’s inequality (1964))of the value function. This direct approach utilizing 3-convexity of the value function isconvenient in characterizing the consumption function, especially in the in…nite horizon
context and can thought to be a complement, rather than a substitute, to Carroll andKimball’s (2001) backward induction approach for the …nite horizon.

9References[1] Carroll, C.D. and M.S. Kimball, 1996, On the Concavity of the Consumption Function, Econometrica 64, 981-992.
[2] Carroll, C.D. and M.S. Kimball, 2001, Liquidity Constraints and Precautionary Saving, NBER Working Paper 8496.
[3] Chmielewski, D. and V. Manousiouthakis, 1996, On Constrained In…nite-time Linear
Quadratic Optimal Control, Systems and Control Letters 29, 121-129.
[4] Deaton A., 1991, Saving and Liquidity Constraints, Econometrica 59, 1221-1248.
[5] Kato, R., 2001 On the Constrained Dynamic Optimization with a Quadratic Reward
Function: An Application to Monetary Policy with a Zero Bound on Nominal InterestRates, mimeo, Department of Economics, The Ohio State University.
[6] Levinson, N., 1964, Generalization of an Inequality of Ky Fan, Journal of Mathematical Analysis and Applications 8, 133-134.
[7] Peµcarić, J.E., F. Proschan, and Y.L. Tong, 1992, Convex Functions, Partial Orderings, and Statistical Applications (Academic Press, San Diego).
[8] Stokey, N.L. and R.E. Lucas, Jr. with E.C. Prescott, 1989, Recursive Methods in
Economic Dynamics (Harvard University Press, Cambridge).

10f (x)B

Ax1x1 + x 22x2

3 x 2 − x12Figure 1: Illustration of 3-convexityx