Necessity of Hyperbolic Absolute Risk Aversion for the Concavity of Consumption Functions
aa r X i v : . [ ec on . T H ] N ov Necessity of Hyperbolic Absolute Risk Aversionfor the Concavity of Consumption Functions
Alexis Akira Toda ∗ November 10, 2020
Abstract
Carroll and Kimball (1996) have shown that, in the class of utilityfunctions that are strictly increasing, strictly concave, and have nonnega-tive third derivatives, hyperbolic absolute risk aversion (HARA) is suffi-cient for the concavity of consumption functions in general consumption-saving problems. This paper shows that HARA is necessary, implying theconcavity of consumption is not a robust prediction outside the HARAclass.
Keywords: concavity, consumption function, hyperbolic absolute riskaversion, robust predictions.
JEL codes:
C65, D11, D14.
The notion that the marginal propensity to consume decreases with wealth,or that the consumption function is concave, dates back at least to Keynes(1936). This observation is important in macroeconomics because the effectof a fiscal transfer of one dollar to a wealthy household is smaller than thatto a poor household, implying that fiscal policies need to account for house-hold heterogeneity. In an important contribution, Carroll and Kimball (1996)have shown that in the class of utility functions that are strictly increasing, arestrictly concave, and have nonnegative third derivatives, hyperbolic absoluterisk aversion (HARA) is sufficient for the concavity of consumption functions infinite-horizon consumption-saving problems without liquidity constraints. Theirresult has been extended in several directions: Carroll and Kimball (2001, Sec-tion 5) obtain concavity under finite-horizon, HARA utility, and liquidity con-straints; Nishiyama and Kato (2012) obtain concavity under infinite-horizon,quadratic utility, and liquidity constraints; Ma et al. (2020, Proposition 2.5 andRemark 2.1) obtain concavity under infinite horizon, constant relative risk aver-sion (CRRA) utility, and liquidity constraints. In all of these theoretical papers, the utility function is restricted to beHARA. Thus a natural question is whether it is possible to obtain the concavity ∗ Department of Economics, University of California San Diego. Email: [email protected]. It is clear from the proof technique of Ma et al. (2020) that concavity obtains in finite- orinfinite-horizon, with or without liquidity constraints, and HARA utility if the domain of theutility function is appropriately modified. See Section 3 for more discussion.
1f consumption functions under weaker assumptions: does concavity hold in alarger class than HARA, or is concavity a non-robust prediction that fails tohold outside the HARA class? This paper provides a definitive answer to thisquestion, which is the latter. More precisely, I show that if the utility functionis strictly increasing, is strictly concave, has a positive third derivative, but isnot HARA, then there exists a finite-horizon (in fact, one period) consumption-saving problem such that the consumption function is not concave. Combinedwith the earlier results on the concavity of consumption functions just cited,my result shows that in the class of natural utility functions, HARA is bothnecessary and sufficient for the concavity of consumption functions in generalconsumption-saving problems.
Following Carroll and Kimball (1996), I consider a general consumption-savingproblem with a finite horizon. The problem can be informally described asfollows. The agent is endowed with initial wealth w > T periods.The period utility function is u : (0 , ∞ ) → R , which is strictly increasing andconcave. The agent receives income Y t ≥ t . Thegross return on wealth between time t − t is R t >
0. The agent discountsutility between time t − t using the discount factor β t >
0, where β ≡ c t > t and w t ≥ t , the agent’s objective is to solvemaximize E T X t =0 t Y s =0 β s ! u ( c t ) (1a)subject to w t +1 = R t +1 ( w t − c t ) + Y t +1 ≥ , (1b)where the initial wealth w = w > β t , return on wealth R t , and income Y t can all be stochastic.If a solution to the consumption-saving problem (1) exists, the time t con-sumption c t can be viewed as a function of the current wealth w t , which we callthe consumption function. The main result of this paper is that if the utilityfunction satisfies some regularity conditions and the consumption function isalways concave regardless of the specification of the process ( β t , R t , Y t ) Tt =1 , then u must exhibit hyperbolic absolute risk aversion (HARA).We now make this statement more precise. If the consumption functionis concave regardless of the specification of the process ( β t , R t , Y t ) Tt =1 , then inparticular the consumption function in a one period problem ( T = 1) mustalways be concave. We can then rewrite (1) as the static maximization problemmax c u ( c ) + E[ βu ( R ( w − c ) + Y )] , (2)where β, R, Y > c satisfies R ( w − c )+ Y ≥ β, R, Y are arbitrary, in particular we may restrict attention toproblems in which β, R, Y take finitely many values. In this case the expectationin (2) is a finite sum and always well-defined.2he following lemma shows that under standard monotonicity, concavity,and Inada conditions, the consumption and saving functions can be unambigu-ously defined, which are differentiable and strictly increasing. Lemma 1.
Suppose that (i) the utility function u : [0 , ∞ ) → R ∪ {−∞} istwice continuously differentiable on (0 , ∞ ) and satisfies u ′ > , u ′′ < , and lim x ↓ u ′ ( x ) = ∞ , and (ii) the random vector ( β, R, Y ) ≫ has finite support.Then for any initial wealth w > , the consumption-saving problem (2) has aunique solution, and the consumption function c = c ( w ) and saving function s ( w ) = w − c ( w ) satisfy the Euler equation u ′ ( c ( w )) = E[ βRu ′ ( Rs ( w ) + Y )] . (3) Furthermore, c, s are continuously differentiable and c ′ ( w ) , s ′ ( w ) ∈ (0 , . In what follows, we maintain the assumptions of Lemma 1. Since the savingfunction s is continuously differentiable and strictly increasing, its range S := s ((0 , ∞ )) is an open interval of R . Since u ′ is strictly decreasing, by (3) foreach s ∈ S there exists a unique c ∈ (0 , ∞ ) such that u ′ ( c ) = E[ βRu ′ ( Rs + Y )].Therefore we can unambiguously define the function g : S → (0 , ∞ ) by g ( s ) := ( u ′ ) − (E[ βRu ′ ( Rs + Y )]) . (4)The function g in (4) returns the consumption level c > s ∈ S . Since u ′ is strictly decreasing, g is clearly strictly increasing.Ma et al. (2020, Proposition 2.5) show that the concavity of g is sufficient forthe concavity of the consumption function. The following lemma shows thatthe concavity of g is actually necessary. Lemma 2. If c is concave, then so is g in (4) . The following lemma, which is closely related to Hardy et al. (1952, Section3.16), plays a crucial role in the proof of the main result.
Lemma 3.
Let I ⊂ R be an open interval and φ : I → R be a twice differentiablefunction such that φ ( I ) = (0 , ∞ ) , φ ′ < , and either φ ′′ > on I or φ ′′ ≡ on I . For p, x, v ∈ R N with p, v ≫ and x ∈ I N , let g ( s ; p, x, v ) := φ − N X n =1 p n φ ( x n + v n s ) ! , which is well-defined in the neighborhood of s = 0 . Then the following statementsare true:1. If g ( s ; p, x, v ) is concave in the neighborhood of s = 0 for arbitrary p, x, v ,then φ and I take one of the following forms: φ ( x ) = c ( ax + b ) − /a , < a < , I = ( −∞ , − b/a ) , (5a) φ ( x ) = − bc e − x/b , b > , I = R , or (5b) φ ( x ) = c ( ax + b ) − /a , a > , I = ( − b/a, ∞ ) , (5c) where c > is arbitrary. . Conversely, if φ and I take one of the forms in (5) , then g ( s ; p, x, v ) isconcave in s on its domain for arbitrary p, x, v . Note that in either case in Lemma 3, log-differentiating φ , we obtain φ ′ ( x ) φ ( x ) = − ax + b , (6)where a > − I = { x ∈ R : ax + b > } . We can now state the main result. Theorem 4.
Suppose that (i) the utility function u : [0 , ∞ ) → R ∪ {−∞} is three times differentiable on (0 , ∞ ) and satisfies u ′ > , u ′′ < , u ′′′ > , lim x ↓ u ′ ( x ) = ∞ , and lim x →∞ u ′ ( x ) = 0 , and (ii) the random vector ( β, R, Y ) ≫ has finite support. If the consumption function in Lemma 1 is concave forarbitrary distribution of ( β, R, Y ) , then u exhibits constant relative risk aver-sion (CRRA). Conversely, if u is CRRA, then the consumption function of theconsumption-saving problem (1) is concave.Proof. Let c, s be the consumption and saving functions. By Lemma 1, c, s arecontinuously differentiable, strictly increasing, and S = s ((0 , ∞ )) is an openinterval of R . Let us show 0 ∈ S . If 0 / ∈ S , then either S ⊂ (0 , ∞ ) or S ⊂ ( −∞ , S ⊂ (0 , ∞ ), then s ( w ) > w . Since c ( w ) + s ( w ) = w , we have c ( w ) < w . Using the Euler equation (3) and u ′′ <
0, we obtain u ′ ( w ) < u ′ ( c ( w )) = E[ βRu ′ ( Rs ( w ) + Y )] ≤ E[ βRu ′ ( Y )] < ∞ . Letting w ↓
0, we obtain a contradiction because u ′ (0) = ∞ .If S ⊂ ( −∞ , s ( w ) < w . Since c ( w ) + s ( w ) = w , we have c ( w ) > w . Again by (3) and u ′′ <
0, we obtain u ′ ( w ) > u ′ ( c ( w )) = E[ βRu ′ ( Rs ( w ) + Y )] ≥ E[ βRu ′ ( Y )] > . Letting w ↑ ∞ , we obtain a contradiction because u ′ ( ∞ ) = 0.The above argument shows that 0 ∈ S regardless of the specification of( β, R, Y ). Since ( β, R, Y ) has finite support, we can write g ( s ) in (4) as g ( s ) = ( u ′ ) − N X n =1 π n β n R n u ′ ( R n s + Y n ) ! , (7)where π n > n . Since ( β n , R n , Y n ) ≫ φ = u ′ , I = (0 , ∞ ), p n = π n β n R n , x n = Y n , and v n = R n , yielding u ′ ( x ) = φ ( x ) = c ( ax + b ) − /a for some a, c > b = 0.Since the multiplicative constant c does not affect the ordering of utility, wemay assume u ′ ( x ) = x − γ with γ = 1 /a >
0, so u is CRRA.The sufficiency of CRRA for the concavity of consumption can be shown bythe same argument as in Ma et al. (2020, Proposition 2.5 and Remark 2.1).One limitation of Theorem 4 is that to derive the strong conclusion that theutility function is CRRA, we require the strong assumption that the distributionof ( β, R, Y ) is arbitrary. The following corollary shows that we obtain the sameconclusion even if the discount factor β is exogenously fixed at a constant value.4 orollary 5. Suppose that the assumptions of Theorem 4 hold and the discountfactor β > is exogenously given. If the consumption function in Lemma 1 isconcave for arbitrary distribution of ( R, Y ) , then u is CRRA.Proof. If the consumption function c is concave, by Lemma 2 and the proof ofTheorem 4, g in (7) is concave in the neighborhood of 0 ∈ S . Therefore setting β n = β and s = ks for any constant β, k >
0, the function h ( s ) := g ( ks ) = ( u ′ ) − N X n =1 π n βR n u ′ ( R n ks + Y n ) ! = ( u ′ ) − N X n =1 p n u ′ ( x n + v n s ) ! (8)is concave in the neighborhood of 0 ∈ S , where p n = π n βR n , x n = Y n , and v n = R n k. (9)Since p n v n = βπ n k and P Nn =1 π n = 1, it must be k = β P Nn =1 p n /v n . (10)Therefore given any p, x, v ≫
0, we can choose
R, Y ≫ π suchthat (9) holds by choosing k as in (10) and setting R n = v n /k , Y n = x n , and π n = kp n βv n . This argument shows that if the consumption function is concavefor arbitrary distribution of ( R, Y ), then the function h in (8) is concave in theneighborhood of s = 0 for arbitrary p, x, v ≫
0. The conclusion then followsfrom Lemma 3 and the proof of Theorem 4.
Theorem 4 essentially shows that for the consumption functions in generalconsumption-saving problems to be concave, constant relative risk aversion(CRRA) is necessary and sufficient. This statement may appear to be in conflictwith Carroll and Kimball (1996), who have shown that hyperbolic absolute riskaversion (HARA) is sufficient for concavity. However, there is no contradictionbetween the two results. This is because in Theorem 4, to avoid unnecessarycomplications, I have restricted the utility function to have domain (0 , ∞ ) andsatisfy appropriate Inada conditions. If the domain is changed to ( − b/a, ∞ ),then it is straightforward to adopt the proof of Theorem 4 to show that u isHARA, as we can see from (6) with φ = u ′ . This argument shows that for theconcavity of consumption functions, HARA is necessary and sufficient (amongthe class of utility functions with u ′ > u ′′ <
0, and u ′′′ ≥ necessity of HARA for the concavity of consumption functions;the possibility of borrowing is not required for the sufficiency of HARA for con-cavity, and in fact Ma et al. (2020, Remark 2.1) show that HARA is sufficienteven in the presence of borrowing constraints. Theorem 4 and the subsequent discussion suggest that we cannot expect theconcavity of consumption functions unless the stochastic process ( β t , R t , Y t ) Tt =1 is somehow restricted. Indeed, Gong et al. (2012) show that in a finite-horizon,deterministic consumption-saving problem in which the discount factor β t andthe gross risk-free rate R t satisfy β t R t ≥
1, the concavity of the absolute risktolerance − u ′ ( x ) /u ′′ ( x ) is sufficient for the concavity of consumption functions.This condition is much weaker than HARA, as the absolute risk tolerance isaffine when u is HARA. However, note that the condition βR ≥ A Proof of lemmas
Proof of Lemma 1.
Fix w >
0. Suppose ( β, R, Y ) takes the value ( β n , R n , Y n )with probability π n >
0, where n = 1 , . . . , N . Define¯ c = sup { c > ∀ n ) R n ( w − c ) + Y n ≥ } = w + min n Y n R n ≥ w > , which is well-defined because R n , Y n >
0. Define f : (0 , ¯ c ) → R as the objectivefunction in (2). Since f ′ ( c ) = u ′ ( c ) − E[ βRu ′ ( R ( w − c ) + Y )] ,f ′′ ( c ) = u ′′ ( c ) + E[ βR u ′′ ( R ( w − c ) + Y )] < u ′′ < f is strictly concave. By the Inada condition, we have lim c ↓ f ′ ( c ) = ∞ and lim c ↑ ¯ c f ′ ( c ) = −∞ . Since f ′ is continuous and strictly decreasing, by theintermediate value theorem there exists a unique c ∗ ∈ (0 , ¯ c ) satisfying f ′ ( c ∗ ) = 0.By the strict concavity of f , this c ∗ =: c ( w ) uniquely solves the optimizationproblem (2). Letting s ( w ) = w − c ( w ), the first-order condition f ′ ( c ( w )) = 0implies the Euler equation (3). Strictly speaking, Ma et al. (2020, Remark 2.1) concerns CRRA utility, but this is becausethe domain of the utility function is restricted to (0 , ∞ ). It is straightforward to handle HARAutility by shifting the domain.
6o show the continuous differentiability of c , let F ( w, c ) = u ′ ( c ) − E[ βRu ′ ( R ( w − c ) + Y )] . Since u ′′ <
0, we obtain ∂F∂w = − E[ βR u ′′ ( R ( w − c ) + Y )] > ,∂F∂c = u ′′ ( c ) + E[ βR u ′′ ( R ( w − c ) + Y )] < . By the implicit function theorem, c is continuously differentiable and c ′ ( w ) = − ∂F/∂w∂F/∂c = − E[ βR u ′′ ( R ( w − c ) + Y )] − u ′′ ( c ) − E[ βR u ′′ ( R ( w − c ) + Y )] ∈ (0 , . Then s ′ ( w ) = 1 − c ′ ( w ) ∈ (0 , Proof of Lemma 2.
By the discussion preceding Lemma 2, we have g ( s ( w )) = c ( w ) (11)for all w >
0. If g is not concave, we can take s , s ∈ S and α ∈ [0 ,
1] such that g ((1 − α ) s + αs ) < (1 − α ) g ( s ) + αg ( s ) . (12)Let w j = s − ( s j ) for j = 1 ,
2. Then s j = s ( w j ) and g ( s j ) = c ( w j ) by (11).Define ¯ c := (1 − α ) c ( w )+ αc ( w ) and ¯ w := (1 − α ) w + αw . Since by assumption c is concave, we have c ( ¯ w ) = c ((1 − α ) w + αw ) ≥ (1 − α ) c ( w ) + αc ( w ) = ¯ c. (13)Therefore c ( ¯ w ) ≥ ¯ c = (1 − α ) c ( w ) + αc ( w ) ( ∵ (13))= (1 − α ) g ( s ) + αg ( s ) > g ((1 − α ) s + αs ) ( ∵ (11) , (12))= g ((1 − α )( w − c ( w )) + α ( w − c ( w )))= g ( ¯ w − ¯ c ) ≥ g ( ¯ w − c ( ¯ w )) ( ∵ (13) , g increasing)= g ( s ( ¯ w )) = c ( ¯ w ) , ( ∵ (11))which is a contradiction. Proof of Lemma 3.
The proof is similar to Hardy et al. (1952, Section 3.16),who assume φ ′ > convexity of g . Sincethese authors do not cover all the fine details, I reproduce their argument.To simplify the notation, write g = g ( s ; p, x, v ) and P = P Nn =1 . Since x n ∈ I for all I , I is open, φ ( I ) = (0 , ∞ ), and φ is monotonic, g is well-definedin a neighborhood of 0. Differentiating both sides of φ ( g ( s )) = P φ ( x n + v n s )with respect to s twice, we obtain φ ′ ( g ) g ′ = X p n v n φ ′ ( x n + v n s ) ,φ ′′ ( g )( g ′ ) + φ ′ ( g ) g ′′ = X p n v n φ ′′ ( x n + v n s ) . g ′ , we obtain φ ′ ( g ) g ′′ = φ ′ ( g ) X p n v n φ ′′ ( x n + v n s ) − φ ′′ ( g ) (cid:16)X p n v n φ ′ ( x n + v n s ) (cid:17) . Since by assumption φ ′ <
0, we have g ′′ ( s ) ≤ φ ′ ( g ) X p n v n φ ′′ ( x n + v n s ) − φ ′′ ( g ) (cid:16)X p n v n φ ′ ( x n + v n s ) (cid:17) ≥ . (14)If φ ′′ ≡ I , (14) is trivial. In this case φ is quadratic and (because φ > φ ′ <
0, and φ ( I ) = (0 , ∞ )) we must have φ ( x ) = c ( ax + b ) − /a for a = − / c > φ ′′ > I . If (14) holds for all s in a neighborhood of 0 for arbitrary x ∈ I N and p, v ≫
0, in particular letting s = 0, we obtain φ ′ ( g ) X p n v n φ ′′ ( x n ) − φ ′′ ( g ) (cid:16)X p n v n φ ′ ( x n ) (cid:17) ≥ ⇐⇒ φ ′ ( g ) φ ′′ ( g ) ≥ ( P p n v n φ ′ n ) P p n v n φ ′′ n , (15)where φ n = φ ( x n ) and φ ′ n , φ ′′ n are defined analogously. Applying the Cauchy-Schwarz inequality, we obtain (cid:16)X p n v n φ ′ n (cid:17) = X p p n φ ′′ n v n s p n φ ′ n φ ′′ n ≤ (cid:16)X p n v n φ ′′ n (cid:17) X p n φ ′ n φ ′′ n ! , with equality achieved when v n = kφ ′ n /φ ′′ n for some k < v n > x ∈ I N and p, v ≫ φ ′ ( g ) φ ′′ ( g ) ≥ X p n φ ′ n φ ′′ n for all x ∈ I N and p ≫ . (16)Now define y n = φ ( x n ) and Φ( y ) := [ φ ′ ( φ − ( y ))] φ ′′ ( φ − ( y )) . (17)Noting that φ ( I ) = (0 , ∞ ), (16) is equivalent toΦ (cid:16)X p n y n (cid:17) ≥ X p n Φ( y n ) for all p, y ∈ R N ++ . (18)If we take N = 2, y = x , y = y , p = y/ x , and p = 1 / y ) ≥ y x Φ( x ) + 12 Φ( y ) ⇐⇒ Φ( y ) y ≥ Φ( x ) x . Interchanging the role of x, y , it follows that Φ( y ) /y is constant, and henceΦ( y ) = ky for some constant k > φ ′′ > x = φ − ( y ), we obtain φ ′ ( x ) φ ′′ ( x ) = kφ ( x ) ⇐⇒ φ ( x ) φ ′′ ( x ) − φ ′ ( x ) φ ′ ( x ) = a for x ∈ I, a = 1 /k − > −
1. Integrating both sides, we obtain − φ ( x ) φ ′ ( x ) = ax + b ⇐⇒ φ ′ ( x ) φ ( x ) = − ax + b , where a > − b is such that ax + b > φ > φ ′ < φ and I in (5) follow by integrating both sides with respect to x andconsidering each case − < a < a = 0, and a > φ and I take one of the forms in (5), then we have Φ( y ) = ya +1 in (17), and the inequality (18) is trivial. Then we can go back the argumentto show that g is concave on its domain. References
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