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Non-concavity problems in the dynamic macroeconomic models: A note
Journal of Banking & Finance 33 (2009) 568–572
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Journal of Banking & Finance journal homepage: www.elsevier.com/locate/jbf
Non-concavity problems in the dynamic macroeconomic models: A note Ryoji Hiraguchi *,1 Institute of Economic Research, Yoshida-honmachi, Sakyo-ko, Kyoto University, Kyoto, Japan
a r t i c l e
i n f o
Article history: Received 2 June 2008 Accepted 1 September 2008 Available online 9 September 2008 JEL classification: E21 O40 Keywords: Non-concavity Shopping-time model Human capital
a b s t r a c t This note provides a method to convert the dynamic models in Cysne [Cysne, Rubens P., 2006. A note on the non-convexity problem in some shopping-time and human-capital models. Journal of Banking and Finance 30 (10), 2737–2745] and in Cysne [Cysne, Rubens P., 2008. A note on ‘‘inflation and welfare”. Journal of Banking and Finance 32 (9), 1984–1987] to concave optimization problems. We do this by introducing new control and state variables in the models. Cysne (2006, 2008) restrict attention to continuous time models and derive parametric conditions to use Arrow’s sufficiency theorem. When the sufficient conditions presented in Cysne (2006) are satisfied (but not under the sharper sufficient conditions presented in Cysne (2008)) we can rewrite these models as concave optimization problems even if time is discrete. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction In the dynamic macroeconomic models, the first order conditions (FOCs) play a crucial role in characterizing the equilibrium paths. However, some of the models have the non-concavity (or non-convexity) problems. In these models, the FOCs may not be sufficient for optimality. Cysne (2006) investigated two continuous time non-concave models (the shopping-time models and the human-capital models) and found that under some parametric conditions, Arrow’s theorem is available. Hence, if Cysne’s conditions hold, a path satisfying the FOCs and the transversality conditions is still optimal. Recently Cysne (2008) reinvestigated the shopping-time model in Cysne (2006) and found the sharper parametric conditions to use Arrow’s theorem. In this note, we propose another approach to solve the problems, which is to introduce new control and state variables in the non-concave models for making them concave. We consider the two non-concave models in Cysne (2006) and show that if the parametric conditions in Cysne (2006) hold, we can rewrite these models as concave optimization problems. Ladrón-deGuevara et al. (1999) and Takahashi (2008) also introduced new variables in their human-capital growth models. Here we show that a similar approach is available in the shopping-time model. We also show that the FOCs of the original problem and of the converted concave problems are the same. * Tel.: +81 75 753 7133; fax: +81 75 753 7193. E-mail address: [email protected] 1 We thank an anonymous referee and Editor for the detailed comments. 0378-4266/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jbankfin.2008.09.001
One advantage of our approach is that the conversion method is available even if time is discrete, as long as the model satisfies the parametric conditions in Cysne (2006). Therefore, we can use the Lagrangian to characterize the optimal path. To use Arrow’s theorem, Cysne (2006) had to restrict his attention to the continuous time models, although many of the recent monetary models are discrete (see Lucas (2000)). Another advantage is that the conversion method is sometimes possible in models where Arrow’s theorem is unavailable. As an example, we investigate the growth model in which human-capital accumulation exhibits increasing returns to scale. In this model, we cannot use Arrow’s theorem. However, by introducing new variables, we can still rewrite the model as a concave optimization problem. The conversion method has one limitation which needs to be noted. As we mentioned above, Cysne (2008) reinvestigated the shopping-time model and found a sharper sufficient condition for optimality regarding the coefficient of the relative risk aversion r. However, the conversion method cannot obtain the same conditions. Cysne (2008) showed that if r > 0:0085, the Hamiltonian is concave and Arrow’s theorem is available, while our paper and Cysne (2006) require a more strict condition r P 1=2. Hence, our approach cannot dominate the Arrow-theorem method in the shopping-time model. In the human capital model, on the other hand, the parametric conditions to use Arrow’s theorem and the conditions to use the conversion method coincide. The rest of the paper is organized as follows. Section 2 uses a simple example to explain the concavification. Section 3 describes the shopping-time models. Section 4 describes the human-capital
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growth model. Sections 5 and 6 investigates the human-capital model with externalities. Conclusion is outlined in Section 7.
ðP 1 Þ :
max
fdt ;st ;mtþ1 g1 t¼0
s:t: 2. Conversion method In this section, we use an example to explain the conversion method. The objective of the method is to introduce new variables in the non-concave models for expressing them only with concave functions. Let us consider the following two dimensional optimization problem:
ðPNC Þ : maxðx1=2 þ y3=2 Þ s:t: x þ y3 6 2: x;y
Clearly the problem ðPNC Þ is non-concave. Now let z ¼ y3 . Using x and z, we can express the optimization problem ðPNC Þ as
pffiffiffi pffiffiffi ðPC Þ : maxð x þ zÞ s:t: x þ z 6 2: x;z
The new problem ðP C Þ is concave and ðx; zÞ ¼ ð1; 1Þ is the unique solution. Hence, ðx; yÞ ¼ ð1; 1Þ is the unique solution to the original non-concave problem ðPNC Þ. Below we call the problem ðP C Þ as the converted concave problem. The two problems ðP NC Þ and ðP C Þ are equivalent simply because there exists a one-to-one correspondence between y and z. By investigating the strictly concave problem ðPC Þ, we easily see that the solution to the non-concave problem ðPNC Þ exists, is interior and is unique. Although this example is simple, such a conversion method is sometimes available in the complicated dynamic models with many variables.
3. Shopping-time model In this section, we show that we can rewrite the shopping-time model as a concave optimization problem under several parametric restrictions. 3.1. Model Here we consider the following model of Lucas (2000):
ðP1 Þ :
1 X
max
fct ;st ;mtþ1 g1 t¼0
s:t:
t¼0
bt
r1 c1 t ; 1r
1 ðmt ct st mt Þ; 1 þ ptþ1 ct 6 mt f ðst Þ ¼ jmt st ;
mtþ1 6
ð1Þ ð2Þ ð3Þ
where b is a discount factor, ct is consumption, r is the coefficient of the relative risk aversion, mt is real money balance, pt is money growth rate, st is time devoted to transaction, mt is the lump-sum tax, f ðsÞ ¼ js is the linear transacting technology and j > 0 is constant. The initial amount of money m0 is assumed to be given. In Eq. (3), the function mf ðsÞ ¼ jms is not concave. Therefore, the Lagrangian
L¼ 1 X t¼0
" t
b
ð1rÞ
ct
1
1r
# mt ct st mt þ kt mtþ1 þ /t ðjmt st ct Þ ; 1 þ ptþ1
is also non-concave. Here kt P 0 and /t P 0 are the multipliers on the budget constraint and on the transacting technology constraint, respectively. pffiffiffiffi Now if we define a new control variable dt ¼ ct , the problem ðP1 Þ is written as
1 X t¼0
bt
ðdt Þ2ð1rÞ 1 ; 1r
1 ½mt ðdt Þ2 st mt ; mtþ1 6 1 þ ptþ1 pffiffiffiffiffiffiffiffiffiffiffiffiffi dt 6 jmt st :
ð4Þ ð5Þ
We have the following lemma. Lemma 1. There exists a one-to-one correspondence between a path 1 fct ; st ; mtþ1 g1 t¼0 in the model ðP 1 Þ and a path fdt ; st ; mtþ1 gt¼0 in the model ðP1 Þ. Hence, the model ðP1 Þ is equivalent to the model ðP1 Þ. pffiffiffiffi 1 Proof. First, for any path fct ; st ; mtþ1 g1 t¼0 , a path f c t ; st ; mtþ1 gt¼0 is 1 uniquely determined. Next, for any path fdt ; st ; mtþ1 gt¼0 , a path 2 fdt ; st ; mtþ1 g1 t¼0 is also uniquely determined. Therefore, there exists and a one-to-one correspondence between fct ; st ; mtþ1 g1 t¼0 h fdt ; st ; mtþ1 g1 t¼0 . Note that Lemma 1 holds even if the path is boundary solution such that ct ¼ 0 for some t. Now let us derive a condition to ensure that the problem ðP 1 Þ is 2 concave. First, in Eq. (4), the function mt dt st is concave in dt , st pffiffiffiffiffiffiffiffiffiffi and mt . Next, the function jms in Eq. (5) is also concave. Finally, if
r P 1=2;
ð6Þ 2ð1rÞ
then 2ð1 rÞ 6 1 and the utility function ðd 1Þ=ð1 rÞ is concave. Therefore, if the condition (6) holds, then ðP1 Þ is concave and the following Lagrangian
" 2ð1rÞ ! 2 dt 1 mt dt st mt þ kt mtþ1 1r 1 þ ptþ1 t¼0 # pffiffiffiffiffiffiffiffiffiffiffiffiffi þ/t jmt st dt ;
L ¼
1 X
bt
ð7Þ
is also concave. Here kt and /t are the multipliers on the budget constraint and on the transacting-technology constraint, respectively. In this set-up, the FOCs and the transversality condition are sufficient for optimality. Note that (6) coincides with the condition to ensure Arrow’s theorem in Cysne (2006). 2 Here the Lagrangian (7) includes the quadratic term dt . We can find a similar functional form in the optimal investment models with quadratic adjustment costs. As we pointed out in Introduction, this method has one limitation. Cysne (2008) recently obtained a sharper sufficient condition for optimality regarding the risk aversion parameter r by using Arrow’s theorem. He found that if r > 0:0085, then the Hamiltonian is concave, while our paper and Cysne (2006) require r P 1=2. Therefore, our approach cannot dominate the Arrow-theorem method in Cysne (2008). Here the main advantage of the conversion method is that it is applicable to both the discrete time models and the continuous time models. 3.2. The first order conditions In this subsection, we show that the FOCs of the original problem and of the converted concave problem are the same. Differentiating the Lagrangian L with respect to c, m and s yields the following the FOCs: [FOCs of the original problem ðP 1 Þ]
kt þ /t ; 1 þ ptþ1 ktþ1 þ b/tþ1 jstþ1 ; mtþ1 : kt ¼ b 1 þ ptþ2 kt ¼ /t jmt : st : 1 þ ptþ1 ct : ct r ¼
ð8Þ ð9Þ ð10Þ
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R. Hiraguchi / Journal of Banking & Finance 33 (2009) 568–572
r ¼ Substituting (10) into Eqs. (8) and (9) yields ð1 þ ptþ1 Þc t kt ð1 þ 1=ðjmt ÞÞ and ð1 þ ptþ2 Þkt ¼ bktþ1 ð1 þ stþ1 =mtþ1 Þ. We then get the consumption Euler equation:
ð1 þ ptþ1 Þ
r ctþ1 ct r stþ1 ¼b 1þ : 1 þ 1=ðjmt Þ 1 þ 1=ðjmtþ1 Þ mtþ1
kt dt þ /t ; 1 þ ptþ1 rffiffiffiffiffiffiffiffiffiffiffi ktþ1 b jstþ1 mtþ1 : kt ¼ b þ /tþ1 ; 1 þ ptþ2 2 mtþ1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi kt 1 jmtþ1 st : ¼ / : 1 þ ptþ1 2 t stþ1 2ð1rÞ1
¼
L¼
ð12Þ ð13Þ ð14Þ 2r
Substituting (14) into (12) and (13) yields ð1 þ ptþ1 Þdt ¼ kt ð1 þ 1=ðjmt ÞÞ and ð1 þ ptþ2 Þkt ¼ bktþ1 ð1 þ stþ1 =mtþ1 Þ. Therefore, we obtain:
ð1 þ ptþ1 Þ
2r 2r dtþ1 dt stþ1 : ¼b 1þ 1 þ 1=ðjmt Þ 1 þ 1=ðjmtþ1 Þ mtþ1
t ~ t¼0 b uðc t Þ
ð11Þ
In the Euler Eq. (11), the marginal utility of consumption is discounted by the term on the transaction cost, 1 þ 1=ðjmt Þ. Next we derive the FOCs of the converted concave problem. [FOCs of the converted problem ðP1 Þ]
dt : 2dt
P t > 1 t¼0 b uðct Þ. Therefore, without loss of generality, we can restrict our attention on the case with equality constraints. As Benhabib and Perli (1994) and Cysne (2006) pointed out, this problem is non-concave and the Lagrangian
P1
2
Proposition 1. In the shopping-time model, the FOCs of the original problem and the ones of the converted problem are exactly the same. This is true whether the optimal path is interior or not. pffiffiffiffi In fact, if we let ðkt ; /t Þ ¼ ðkt ; /t = ct Þ, the FOCs of ðP1 Þ, (8)–(10) coincide with the ones of ðP1 Þ, (12)–(14). This is not a coincidence. Here the one-to-one correspondence between fct ; st ; mtþ1 g1 t¼0 and fdt ; st ; mtþ1 g1 t¼0 exists. Hence, the utility maximization by choosing 1 fct ; st ; mtþ1 g1 t¼0 and the one by choosing fdt ; st ; mtþ1 gt¼0 are equivalent and the FOCs must coincide with each other.
a
bt ½uðct Þ þ kt fAkt ðut ht Þ1a þ kt ktþ1 ct g þ lt fdht ð1 ut Þ
t¼0
þ ht htþ1 g þ /t ut þ gt ð1 ut Þ; is also non-concave. However, we can construct a concave problem. Following Ladrón-de-Guevara et al. (1999) and Takahashi (2008), define a new control variable nt ¼ ut ht . Note that ht > 0 for all t, since htþ1 P ht and h0 > 0. We have the following lemma. Lemma 2. There exists a one-to-one correspondence between a path 1 fct ; kt ; ht ; ut g1 t¼0 and a path fc t ; kt ; ht ; nt gt¼0 , whether these paths are interior or not. 1 Proof. First, for any path fct ; kt ; ht ; ut g1 t¼0 , a path fc t ; kt ; ht ; ut ht gt¼0 1 is uniquely determined. Next, for any path fct ; kt ; ht ; nt gt¼0 , 1 fct ; kt ; ht ; ut g1 t¼0 ¼ fc t ; kt ; ht ; nt =ht gt¼0 is always well-defined and is uniquely determined since ht > 0. h
Using the new variable nt , we can rewrite the problem ðP2 Þ as
ð15Þ
Since dt ¼ ct , (11) and (15) are the same. This leads to the following proposition.
1 X
ðP2 Þ :
max 1
fct ;htþ1 ;nt gt¼0
s:t:
1 X
bt
t¼0
r1 c1 t ; 1r a
ktþ1 ¼ Akt n1a þ kt ; t htþ1 ¼ dðht nt Þ þ ht ; nt P 0; ht nt P 0:
As Takahashi (2008) points out, here the production function is constant returns to scale. The problem ðP 2 Þ is concave and the Lagrangian
L¼
1 X
a
bt ½uðct Þ þ kt fAkt nt1a þ kt ktþ1 ct g þ lt fdðht nt Þ þ ht
t¼0
htþ1 g þ /t nt þ gt ðht nt Þ;
4. Human-capital growth model
is also concave. Here lt , kt , /t and gt are multipliers. Therefore, the FOCs and the transversality conditions are sufficient for optimality.
4.1. Model In this section, we consider the discrete time version of the human capital models with no externality. The human-capital growth model is written as
ðP2 Þ :
max
fct ;ktþ1 ;htþ1 ;ut g1 t¼0
s:t:
1 X t¼0
bt
r1 c1 t ; 1r a
ð16Þ 1a
ktþ1 ¼ Akt ðut ht Þ þ kt ct ; htþ1 ¼ dht ð1 ut Þ þ ht ; ut P 0; 1 ut P 0;
ð17Þ ð18Þ ð19Þ ð20Þ
where kt ðht Þ is physical (human) capital, ct is consumption, ut is time for physical capital accumulation and ð1 ut Þ is time for human-capital accumulation. Here the initial capital ðk0 ; h0 Þ 2 R2þþ is given. Let uðcÞ ¼ ðc1r 1Þ=ð1 rÞ denote the instantaneous utility function. In the model ðP2 Þ, we can assume the inequality constraints a htþ1 6 dht ð1 ut Þ þ ht and ktþ1 6 Akt ðut ht Þ1a þ kt ct instead of the equality constraints (17) and (18). If a path fct ; ktþ1 ; htþ1 ; a 1a þ ut g1 t¼0 is such that htþ1 < dht ð1 ut Þ þ ht or ktþ1 < Akt ðut ht Þ kt ct for some t, then we can easily find out a feasible and wel~tþ1 ; h ~ tþ1 ; ut g1 such that h ~tþ1 ¼ dh ~t fare-improving path f~ct ; k t¼0 1a a ~ ~ ~ ~ ~ ð1 ut Þ þ ht and ktþ1 ¼ Akt ðut ht Þ þ kt ~ct for all t and that
4.2. The first order conditions In this subsection, we show that the FOCs of the original problem and of the converted concave problem are the same. The FOCs of the original problem ðP 2 Þ are
ct : ct r ¼ kt ; kt :
ð21Þ
a1 kt1 ¼ bkt ðaAkt ðut ht Þ1a þ a 1a d t ht ¼ kt ð1 aÞAkt ua t ht
1Þ;
l þ /t gt ; a ht : lt1 ¼ blt ð1 þ dð1 ut ÞÞ þ bkt ð1 aÞAkt ut ðut ht Þa : ut :
ð22Þ ð23Þ ð24Þ
On the other hand, the FOCs of the converted concave problem ðP 2 Þ are
ct : ct r ¼ kt ; kt :
l ht : l
nt :
ð25Þ
a1 kt1 ¼ bkt ðaAkt n1a þ 1Þ; t a d t ¼ ð1 aÞAkt na k t t þ /t t1 ¼ b t ð1 þ dÞ þ b t :
l
g
ð26Þ gt ;
ð27Þ ð28Þ
The following lemma shows that the non-zero constraint on nt and ut are not binding and then the multipliers on these constraints are zero.
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R. Hiraguchi / Journal of Banking & Finance 33 (2009) 568–572
Lemma 3. For all t, nt > 0 and ut > 0 at the optimal. Therefore, /t ¼ /t ¼ 0. Proof. If nt ¼ 0 for some t, the right hand side of Eq. (27) goes to +1, since na t ¼ 1. Similarly, if ut ¼ 0 for some t, then the right hand side of (23) goes to +1. Hence, we must have nt > 0 and ut > 0 at the optimal. The multipliers satisfy /t nt ¼ 0 and /t ut ¼ 0. h
r < 1 and ð1 rÞ þ hð1 rÞ 6 1; or r P 1:
Note that when a > 0 and b > 0, the function is concave if and only if a þ b 6 1. It is easy to check that the condition (35) is exactly the same as the condition (39). Therefore, if Cysne’s condition (35) is satisfied, the Lagrangian of the problem ðP 4 Þ,
L ¼
" h ðAvt ht Þ1r 1 h ðh1h vt Þ þ ht htþ1 Þ þ gt ðdh t t 1r #
1h
a1
ðut ht Þ1a þ 1Þ; a aÞAkt ðut ht Þa ð t =ht Þ;
r ct1 ¼ bct r ðaAkt
ð29Þ
dlt ¼ kt ð1 g lt1 ¼ blt ð1 þ dÞ þ bðgt =ht Þ:
ð31Þ
ð30Þ
[FOCs of the converted problem ðP2 Þ] a1 1a nt þ 1Þ; a a d t ¼ kt ð1 aÞAkt nt t ; t1 ¼ b t ð1 þ dÞ þ b t :
r ct1 ¼ bct r ðaAkt
g
6. Human-capital model with externalities (2) In this section, we provide an example where the conversion method can be used even if Arrow’s theorem is not available. Here we consider the following human-capital model in Cysne (2006) where the production function exhibits increasing return to scale:
ð33Þ
ðP4 Þ : max
ð34Þ
s:t:
c;h;u
Note that ut ht ¼ nt by definition. If we let lt ¼ lt and g ¼ gt =ht , the FOCs of the two problems coincide. These results lead to the following proposition. t
Proposition 2. In the human-capital model, the FOCs of the original problem and the ones of the converted problem are the same whether the optimal path is interior or not.
5. Human-capital model with externalities (1) In this section, we consider the model of Parente and Prescott (1994) and Cysne (2006) where the externalities affect the production of human capital:
ðP3 Þ :
max 1
fct ;htþ1 ;ut gt¼0
1 X
bt
t¼0
ðAut ht Þ1r 1 ; 1r 1h
s:t:
htþ1 ¼ dðht Þ
max 1
fct ;htþ1 ;vt gt¼0
s:t:
1 X t¼0
hð1rÞ
bt
rh A1r ½v1 t t 1r
1
;
t Þh fðht Þ1h vt g þ ht ; htþ1 ¼ dðh
ð36Þ
ðht Þ1h vt P 0;
ð37Þ
vt P 0:
ð38Þ
The human-capital accumulation process (36) is obviously concave because 1 h 2 ð0; 1Þ. Next, the objective function is concave if and only if
h_ ¼ dhð1 uÞ; b c c þ k_ ¼ Ak ðuhÞ1b h ;
ð40Þ
where d ¼ pd. Note that if x and y satisfy y ¼ xa where a is constant, _ ¼ aðx=xÞ. _ then y=y Using z and n, we express the problem ðP4 Þ as
ut 2 ½0; 1;
ðP3 Þ :
0
c1r 1 dt; 1r
z_ h_ ¼ p ¼ dð1 uÞ; z h
s:t:
Now we show that if Cysne’s condition (35) holds, then we can rewrite the model ðP 3 Þ as a concave optimization problem. To see 1h denote a new control variable. Since this, let vt ¼ ut ht h ut ht ¼ vt ht , we can express the model ðP 3 Þ as
eqt
Cysne (2006) pointed out that Arrow’s sufficiency theorem is p p not available if c > 0. However, if we let z ¼ h and n ¼ uh ¼ uz where p ¼ ð1 b þ cÞ=ð1 bÞ is a constant, we can rewrite the model as a concave problem. To see this, first we have
c;k;z;n
ð35Þ
1
k0 > 0; h0 > 0:
t Þh ð1 ut Þ þ ht ; ðh
h rP : 1þh
Z
u 2 ½0; 1;
ðP4 Þ : max
t denotes the human-capital externality and is taken as where h exogenous. Cysne (2006) showed that in his continuous time model, Arrow’s theorem is available if
vt Þ ;
is concave, where gt and /t are the multipliers. Following Section 3.2, we can show that the FOCs in the original problem coincide with the ones in the converted problem.
ð32Þ
g
l
bt
þ/t ðht
lt1 ¼ blt ð1 þ dð1 ut ÞÞ þ bðdlt ut þ gt ut =ht Þ:
l l
1 X t¼0
Substituting Eq. (23) into (24) yields
Moreover, the multiplier on the constraint (20), gt satisfies ð1 ut Þgt ¼ 0 and then gt ¼ ut gt . Hence, the FOCs of the original problem and of the converted problems are written as follows: [FOCs of the original problem ðP2 Þ]
ð39Þ xa yb
Z
c1r 1 dt; 1r 0 z_ ¼ dðz nÞ; b c þ k_ ¼ Ak n1b ; 1
eqt
n 2 ½0; z: p Note that here zt > 0 for all t since z0 ¼ h0 > 0 and z_ P 0.
Lemma 4. A one-to-one correspondence between a path ðc; k; h; uÞ in the model ðP 4 Þ and a path ðc; k; z; nÞ in the model ðP 4 Þ exists and then ðP 4 Þ and ðP4 Þ are equivalent. p
p
Proof. If ðc; k; h; uÞ is fixed, then ðc; k; z; nÞ ¼ ðc; k; h ; uh Þ is uniquely determined. Next, if ðc; k; z; nÞ is fixed, then ðc; k; h; uÞ ¼ ðc; k; z1=p ; n=z1=p Þ is also uniquely determined. Note that z > 0 for all t. h Since ðP 4 Þ is concave, the FOCs and the transversality conditions are sufficient for optimality. One limitation of our approach is that here the concavification is available only if time is continuous. Although there is a linear _ relationship between z_ =z and h=h (see (40)), if time is discrete, ztþ1 =zt and htþ1 =ht have no such simple relationship. The Hamiltonian of the original problem ðP 4 Þ is
H¼
c1r 1 b c þ kðAk ðuhÞ1b h cÞ þ udhð1 uÞ þ gð1 uÞ; 1r
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R. Hiraguchi / Journal of Banking & Finance 33 (2009) 568–572
where k, u and g are the multipliers.2 On the other hand, the Hamiltonian of the converted problem ðP 4 Þ is
H ¼
c1r 1 b þ k ðAk n1b cÞ þ u pdðz nÞ þ g ðz nÞ; 1r
where k , u and g are the multipliers. The Hamiltonian H is clearly concave. As in the previous subsections, we can easily find out a one-to-one correspondence between ðk; u; gÞ and ðk ; u ; g Þ. Hence, the FOCs are exactly the same. 7. Conclusion This note provides an approach to solve the non-concavity problems in the shopping-time models and the human-capital models. The advantage of our approach is that it is applicable to the discrete time models as well as the continuous time models. As a future study, we hope to obtain weaker parametric conditions to ensure the concavity, as Cysne (2008) recently did in the continuous time shopping-time model. We also want to incorporate the productivity shocks into the models and investigate the
2 As we showed in the previous subsection, the constraint u P 0 is not binding here. It is because the marginal productivity of labor, which appears in the first order conditions, goes to þ1 when u ¼ 0. Therefore, we only consider the constraint 1 u P 0.
non-concavity problems in a stochastic framework. It is interesting to see how the conversion method is applicable to the stochastic optimal control problems in Stain (2007) and Huang and Milevsky (2008). References Benhabib, J., Perli, R., 1994. Uniqueness and indeterminacy: on the dynamics of endogenous growth. Journal of Economic Theory 63, 113–142. Cysne, Rubens P., 2006. A note on the non-convexity problem in some shoppingtime and human-capital models. Journal of Banking and Finance 30 (10), 2737– 2745. Cysne, Rubens P., 2008. A note on ‘‘inflation and welfare”. Journal of Banking and Finance 32 (9), 1984–1987. Huang, H., Milevsky, M.A., 2008. Portfolio choice and mortality-contingent claims: the general HARA case. Journal of Banking and Finance 32, 2444–2452. Ladrón-de-Guevara, A., Ortigueira, S., Santos, M.S., 1999. A two-sector model of endogenous growth with leisure. Review of Economic Studies 66, 609–631. Lucas Jr., R.E., 2000. Inflation and welfare. Econometrica 68, 247–274. Parente, S.L., Prescott, E.C., 1994. Barriers to technology adoption and development. The Journal of Political Economy 102, 298–321. Stein, Jerome L., 2007. United States current account deficits: a stochastic optimal control analysis. Journal of Banking and Finance 31, 1321–1350. Takahashi, H., 2008. Optimal balanced growth in a general multi-sector endogenous growth model with constant returns. Economic Theory 37 (1), 31–49.
Contents lists available at ScienceDirect
Journal of Banking & Finance journal homepage: www.elsevier.com/locate/jbf
Non-concavity problems in the dynamic macroeconomic models: A note Ryoji Hiraguchi *,1 Institute of Economic Research, Yoshida-honmachi, Sakyo-ko, Kyoto University, Kyoto, Japan
a r t i c l e
i n f o
Article history: Received 2 June 2008 Accepted 1 September 2008 Available online 9 September 2008 JEL classification: E21 O40 Keywords: Non-concavity Shopping-time model Human capital
a b s t r a c t This note provides a method to convert the dynamic models in Cysne [Cysne, Rubens P., 2006. A note on the non-convexity problem in some shopping-time and human-capital models. Journal of Banking and Finance 30 (10), 2737–2745] and in Cysne [Cysne, Rubens P., 2008. A note on ‘‘inflation and welfare”. Journal of Banking and Finance 32 (9), 1984–1987] to concave optimization problems. We do this by introducing new control and state variables in the models. Cysne (2006, 2008) restrict attention to continuous time models and derive parametric conditions to use Arrow’s sufficiency theorem. When the sufficient conditions presented in Cysne (2006) are satisfied (but not under the sharper sufficient conditions presented in Cysne (2008)) we can rewrite these models as concave optimization problems even if time is discrete. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction In the dynamic macroeconomic models, the first order conditions (FOCs) play a crucial role in characterizing the equilibrium paths. However, some of the models have the non-concavity (or non-convexity) problems. In these models, the FOCs may not be sufficient for optimality. Cysne (2006) investigated two continuous time non-concave models (the shopping-time models and the human-capital models) and found that under some parametric conditions, Arrow’s theorem is available. Hence, if Cysne’s conditions hold, a path satisfying the FOCs and the transversality conditions is still optimal. Recently Cysne (2008) reinvestigated the shopping-time model in Cysne (2006) and found the sharper parametric conditions to use Arrow’s theorem. In this note, we propose another approach to solve the problems, which is to introduce new control and state variables in the non-concave models for making them concave. We consider the two non-concave models in Cysne (2006) and show that if the parametric conditions in Cysne (2006) hold, we can rewrite these models as concave optimization problems. Ladrón-deGuevara et al. (1999) and Takahashi (2008) also introduced new variables in their human-capital growth models. Here we show that a similar approach is available in the shopping-time model. We also show that the FOCs of the original problem and of the converted concave problems are the same. * Tel.: +81 75 753 7133; fax: +81 75 753 7193. E-mail address: [email protected] 1 We thank an anonymous referee and Editor for the detailed comments. 0378-4266/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jbankfin.2008.09.001
One advantage of our approach is that the conversion method is available even if time is discrete, as long as the model satisfies the parametric conditions in Cysne (2006). Therefore, we can use the Lagrangian to characterize the optimal path. To use Arrow’s theorem, Cysne (2006) had to restrict his attention to the continuous time models, although many of the recent monetary models are discrete (see Lucas (2000)). Another advantage is that the conversion method is sometimes possible in models where Arrow’s theorem is unavailable. As an example, we investigate the growth model in which human-capital accumulation exhibits increasing returns to scale. In this model, we cannot use Arrow’s theorem. However, by introducing new variables, we can still rewrite the model as a concave optimization problem. The conversion method has one limitation which needs to be noted. As we mentioned above, Cysne (2008) reinvestigated the shopping-time model and found a sharper sufficient condition for optimality regarding the coefficient of the relative risk aversion r. However, the conversion method cannot obtain the same conditions. Cysne (2008) showed that if r > 0:0085, the Hamiltonian is concave and Arrow’s theorem is available, while our paper and Cysne (2006) require a more strict condition r P 1=2. Hence, our approach cannot dominate the Arrow-theorem method in the shopping-time model. In the human capital model, on the other hand, the parametric conditions to use Arrow’s theorem and the conditions to use the conversion method coincide. The rest of the paper is organized as follows. Section 2 uses a simple example to explain the concavification. Section 3 describes the shopping-time models. Section 4 describes the human-capital
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growth model. Sections 5 and 6 investigates the human-capital model with externalities. Conclusion is outlined in Section 7.
ðP 1 Þ :
max
fdt ;st ;mtþ1 g1 t¼0
s:t: 2. Conversion method In this section, we use an example to explain the conversion method. The objective of the method is to introduce new variables in the non-concave models for expressing them only with concave functions. Let us consider the following two dimensional optimization problem:
ðPNC Þ : maxðx1=2 þ y3=2 Þ s:t: x þ y3 6 2: x;y
Clearly the problem ðPNC Þ is non-concave. Now let z ¼ y3 . Using x and z, we can express the optimization problem ðPNC Þ as
pffiffiffi pffiffiffi ðPC Þ : maxð x þ zÞ s:t: x þ z 6 2: x;z
The new problem ðP C Þ is concave and ðx; zÞ ¼ ð1; 1Þ is the unique solution. Hence, ðx; yÞ ¼ ð1; 1Þ is the unique solution to the original non-concave problem ðPNC Þ. Below we call the problem ðP C Þ as the converted concave problem. The two problems ðP NC Þ and ðP C Þ are equivalent simply because there exists a one-to-one correspondence between y and z. By investigating the strictly concave problem ðPC Þ, we easily see that the solution to the non-concave problem ðPNC Þ exists, is interior and is unique. Although this example is simple, such a conversion method is sometimes available in the complicated dynamic models with many variables.
3. Shopping-time model In this section, we show that we can rewrite the shopping-time model as a concave optimization problem under several parametric restrictions. 3.1. Model Here we consider the following model of Lucas (2000):
ðP1 Þ :
1 X
max
fct ;st ;mtþ1 g1 t¼0
s:t:
t¼0
bt
r1 c1 t ; 1r
1 ðmt ct st mt Þ; 1 þ ptþ1 ct 6 mt f ðst Þ ¼ jmt st ;
mtþ1 6
ð1Þ ð2Þ ð3Þ
where b is a discount factor, ct is consumption, r is the coefficient of the relative risk aversion, mt is real money balance, pt is money growth rate, st is time devoted to transaction, mt is the lump-sum tax, f ðsÞ ¼ js is the linear transacting technology and j > 0 is constant. The initial amount of money m0 is assumed to be given. In Eq. (3), the function mf ðsÞ ¼ jms is not concave. Therefore, the Lagrangian
L¼ 1 X t¼0
" t
b
ð1rÞ
ct
1
1r
# mt ct st mt þ kt mtþ1 þ /t ðjmt st ct Þ ; 1 þ ptþ1
is also non-concave. Here kt P 0 and /t P 0 are the multipliers on the budget constraint and on the transacting technology constraint, respectively. pffiffiffiffi Now if we define a new control variable dt ¼ ct , the problem ðP1 Þ is written as
1 X t¼0
bt
ðdt Þ2ð1rÞ 1 ; 1r
1 ½mt ðdt Þ2 st mt ; mtþ1 6 1 þ ptþ1 pffiffiffiffiffiffiffiffiffiffiffiffiffi dt 6 jmt st :
ð4Þ ð5Þ
We have the following lemma. Lemma 1. There exists a one-to-one correspondence between a path 1 fct ; st ; mtþ1 g1 t¼0 in the model ðP 1 Þ and a path fdt ; st ; mtþ1 gt¼0 in the model ðP1 Þ. Hence, the model ðP1 Þ is equivalent to the model ðP1 Þ. pffiffiffiffi 1 Proof. First, for any path fct ; st ; mtþ1 g1 t¼0 , a path f c t ; st ; mtþ1 gt¼0 is 1 uniquely determined. Next, for any path fdt ; st ; mtþ1 gt¼0 , a path 2 fdt ; st ; mtþ1 g1 t¼0 is also uniquely determined. Therefore, there exists and a one-to-one correspondence between fct ; st ; mtþ1 g1 t¼0 h fdt ; st ; mtþ1 g1 t¼0 . Note that Lemma 1 holds even if the path is boundary solution such that ct ¼ 0 for some t. Now let us derive a condition to ensure that the problem ðP 1 Þ is 2 concave. First, in Eq. (4), the function mt dt st is concave in dt , st pffiffiffiffiffiffiffiffiffiffi and mt . Next, the function jms in Eq. (5) is also concave. Finally, if
r P 1=2;
ð6Þ 2ð1rÞ
then 2ð1 rÞ 6 1 and the utility function ðd 1Þ=ð1 rÞ is concave. Therefore, if the condition (6) holds, then ðP1 Þ is concave and the following Lagrangian
" 2ð1rÞ ! 2 dt 1 mt dt st mt þ kt mtþ1 1r 1 þ ptþ1 t¼0 # pffiffiffiffiffiffiffiffiffiffiffiffiffi þ/t jmt st dt ;
L ¼
1 X
bt
ð7Þ
is also concave. Here kt and /t are the multipliers on the budget constraint and on the transacting-technology constraint, respectively. In this set-up, the FOCs and the transversality condition are sufficient for optimality. Note that (6) coincides with the condition to ensure Arrow’s theorem in Cysne (2006). 2 Here the Lagrangian (7) includes the quadratic term dt . We can find a similar functional form in the optimal investment models with quadratic adjustment costs. As we pointed out in Introduction, this method has one limitation. Cysne (2008) recently obtained a sharper sufficient condition for optimality regarding the risk aversion parameter r by using Arrow’s theorem. He found that if r > 0:0085, then the Hamiltonian is concave, while our paper and Cysne (2006) require r P 1=2. Therefore, our approach cannot dominate the Arrow-theorem method in Cysne (2008). Here the main advantage of the conversion method is that it is applicable to both the discrete time models and the continuous time models. 3.2. The first order conditions In this subsection, we show that the FOCs of the original problem and of the converted concave problem are the same. Differentiating the Lagrangian L with respect to c, m and s yields the following the FOCs: [FOCs of the original problem ðP 1 Þ]
kt þ /t ; 1 þ ptþ1 ktþ1 þ b/tþ1 jstþ1 ; mtþ1 : kt ¼ b 1 þ ptþ2 kt ¼ /t jmt : st : 1 þ ptþ1 ct : ct r ¼
ð8Þ ð9Þ ð10Þ
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R. Hiraguchi / Journal of Banking & Finance 33 (2009) 568–572
r ¼ Substituting (10) into Eqs. (8) and (9) yields ð1 þ ptþ1 Þc t kt ð1 þ 1=ðjmt ÞÞ and ð1 þ ptþ2 Þkt ¼ bktþ1 ð1 þ stþ1 =mtþ1 Þ. We then get the consumption Euler equation:
ð1 þ ptþ1 Þ
r ctþ1 ct r stþ1 ¼b 1þ : 1 þ 1=ðjmt Þ 1 þ 1=ðjmtþ1 Þ mtþ1
kt dt þ /t ; 1 þ ptþ1 rffiffiffiffiffiffiffiffiffiffiffi ktþ1 b jstþ1 mtþ1 : kt ¼ b þ /tþ1 ; 1 þ ptþ2 2 mtþ1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi kt 1 jmtþ1 st : ¼ / : 1 þ ptþ1 2 t stþ1 2ð1rÞ1
¼
L¼
ð12Þ ð13Þ ð14Þ 2r
Substituting (14) into (12) and (13) yields ð1 þ ptþ1 Þdt ¼ kt ð1 þ 1=ðjmt ÞÞ and ð1 þ ptþ2 Þkt ¼ bktþ1 ð1 þ stþ1 =mtþ1 Þ. Therefore, we obtain:
ð1 þ ptþ1 Þ
2r 2r dtþ1 dt stþ1 : ¼b 1þ 1 þ 1=ðjmt Þ 1 þ 1=ðjmtþ1 Þ mtþ1
t ~ t¼0 b uðc t Þ
ð11Þ
In the Euler Eq. (11), the marginal utility of consumption is discounted by the term on the transaction cost, 1 þ 1=ðjmt Þ. Next we derive the FOCs of the converted concave problem. [FOCs of the converted problem ðP1 Þ]
dt : 2dt
P t > 1 t¼0 b uðct Þ. Therefore, without loss of generality, we can restrict our attention on the case with equality constraints. As Benhabib and Perli (1994) and Cysne (2006) pointed out, this problem is non-concave and the Lagrangian
P1
2
Proposition 1. In the shopping-time model, the FOCs of the original problem and the ones of the converted problem are exactly the same. This is true whether the optimal path is interior or not. pffiffiffiffi In fact, if we let ðkt ; /t Þ ¼ ðkt ; /t = ct Þ, the FOCs of ðP1 Þ, (8)–(10) coincide with the ones of ðP1 Þ, (12)–(14). This is not a coincidence. Here the one-to-one correspondence between fct ; st ; mtþ1 g1 t¼0 and fdt ; st ; mtþ1 g1 t¼0 exists. Hence, the utility maximization by choosing 1 fct ; st ; mtþ1 g1 t¼0 and the one by choosing fdt ; st ; mtþ1 gt¼0 are equivalent and the FOCs must coincide with each other.
a
bt ½uðct Þ þ kt fAkt ðut ht Þ1a þ kt ktþ1 ct g þ lt fdht ð1 ut Þ
t¼0
þ ht htþ1 g þ /t ut þ gt ð1 ut Þ; is also non-concave. However, we can construct a concave problem. Following Ladrón-de-Guevara et al. (1999) and Takahashi (2008), define a new control variable nt ¼ ut ht . Note that ht > 0 for all t, since htþ1 P ht and h0 > 0. We have the following lemma. Lemma 2. There exists a one-to-one correspondence between a path 1 fct ; kt ; ht ; ut g1 t¼0 and a path fc t ; kt ; ht ; nt gt¼0 , whether these paths are interior or not. 1 Proof. First, for any path fct ; kt ; ht ; ut g1 t¼0 , a path fc t ; kt ; ht ; ut ht gt¼0 1 is uniquely determined. Next, for any path fct ; kt ; ht ; nt gt¼0 , 1 fct ; kt ; ht ; ut g1 t¼0 ¼ fc t ; kt ; ht ; nt =ht gt¼0 is always well-defined and is uniquely determined since ht > 0. h
Using the new variable nt , we can rewrite the problem ðP2 Þ as
ð15Þ
Since dt ¼ ct , (11) and (15) are the same. This leads to the following proposition.
1 X
ðP2 Þ :
max 1
fct ;htþ1 ;nt gt¼0
s:t:
1 X
bt
t¼0
r1 c1 t ; 1r a
ktþ1 ¼ Akt n1a þ kt ; t htþ1 ¼ dðht nt Þ þ ht ; nt P 0; ht nt P 0:
As Takahashi (2008) points out, here the production function is constant returns to scale. The problem ðP 2 Þ is concave and the Lagrangian
L¼
1 X
a
bt ½uðct Þ þ kt fAkt nt1a þ kt ktþ1 ct g þ lt fdðht nt Þ þ ht
t¼0
htþ1 g þ /t nt þ gt ðht nt Þ;
4. Human-capital growth model
is also concave. Here lt , kt , /t and gt are multipliers. Therefore, the FOCs and the transversality conditions are sufficient for optimality.
4.1. Model In this section, we consider the discrete time version of the human capital models with no externality. The human-capital growth model is written as
ðP2 Þ :
max
fct ;ktþ1 ;htþ1 ;ut g1 t¼0
s:t:
1 X t¼0
bt
r1 c1 t ; 1r a
ð16Þ 1a
ktþ1 ¼ Akt ðut ht Þ þ kt ct ; htþ1 ¼ dht ð1 ut Þ þ ht ; ut P 0; 1 ut P 0;
ð17Þ ð18Þ ð19Þ ð20Þ
where kt ðht Þ is physical (human) capital, ct is consumption, ut is time for physical capital accumulation and ð1 ut Þ is time for human-capital accumulation. Here the initial capital ðk0 ; h0 Þ 2 R2þþ is given. Let uðcÞ ¼ ðc1r 1Þ=ð1 rÞ denote the instantaneous utility function. In the model ðP2 Þ, we can assume the inequality constraints a htþ1 6 dht ð1 ut Þ þ ht and ktþ1 6 Akt ðut ht Þ1a þ kt ct instead of the equality constraints (17) and (18). If a path fct ; ktþ1 ; htþ1 ; a 1a þ ut g1 t¼0 is such that htþ1 < dht ð1 ut Þ þ ht or ktþ1 < Akt ðut ht Þ kt ct for some t, then we can easily find out a feasible and wel~tþ1 ; h ~ tþ1 ; ut g1 such that h ~tþ1 ¼ dh ~t fare-improving path f~ct ; k t¼0 1a a ~ ~ ~ ~ ~ ð1 ut Þ þ ht and ktþ1 ¼ Akt ðut ht Þ þ kt ~ct for all t and that
4.2. The first order conditions In this subsection, we show that the FOCs of the original problem and of the converted concave problem are the same. The FOCs of the original problem ðP 2 Þ are
ct : ct r ¼ kt ; kt :
ð21Þ
a1 kt1 ¼ bkt ðaAkt ðut ht Þ1a þ a 1a d t ht ¼ kt ð1 aÞAkt ua t ht
1Þ;
l þ /t gt ; a ht : lt1 ¼ blt ð1 þ dð1 ut ÞÞ þ bkt ð1 aÞAkt ut ðut ht Þa : ut :
ð22Þ ð23Þ ð24Þ
On the other hand, the FOCs of the converted concave problem ðP 2 Þ are
ct : ct r ¼ kt ; kt :
l ht : l
nt :
ð25Þ
a1 kt1 ¼ bkt ðaAkt n1a þ 1Þ; t a d t ¼ ð1 aÞAkt na k t t þ /t t1 ¼ b t ð1 þ dÞ þ b t :
l
g
ð26Þ gt ;
ð27Þ ð28Þ
The following lemma shows that the non-zero constraint on nt and ut are not binding and then the multipliers on these constraints are zero.
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R. Hiraguchi / Journal of Banking & Finance 33 (2009) 568–572
Lemma 3. For all t, nt > 0 and ut > 0 at the optimal. Therefore, /t ¼ /t ¼ 0. Proof. If nt ¼ 0 for some t, the right hand side of Eq. (27) goes to +1, since na t ¼ 1. Similarly, if ut ¼ 0 for some t, then the right hand side of (23) goes to +1. Hence, we must have nt > 0 and ut > 0 at the optimal. The multipliers satisfy /t nt ¼ 0 and /t ut ¼ 0. h
r < 1 and ð1 rÞ þ hð1 rÞ 6 1; or r P 1:
Note that when a > 0 and b > 0, the function is concave if and only if a þ b 6 1. It is easy to check that the condition (35) is exactly the same as the condition (39). Therefore, if Cysne’s condition (35) is satisfied, the Lagrangian of the problem ðP 4 Þ,
L ¼
" h ðAvt ht Þ1r 1 h ðh1h vt Þ þ ht htþ1 Þ þ gt ðdh t t 1r #
1h
a1
ðut ht Þ1a þ 1Þ; a aÞAkt ðut ht Þa ð t =ht Þ;
r ct1 ¼ bct r ðaAkt
ð29Þ
dlt ¼ kt ð1 g lt1 ¼ blt ð1 þ dÞ þ bðgt =ht Þ:
ð31Þ
ð30Þ
[FOCs of the converted problem ðP2 Þ] a1 1a nt þ 1Þ; a a d t ¼ kt ð1 aÞAkt nt t ; t1 ¼ b t ð1 þ dÞ þ b t :
r ct1 ¼ bct r ðaAkt
g
6. Human-capital model with externalities (2) In this section, we provide an example where the conversion method can be used even if Arrow’s theorem is not available. Here we consider the following human-capital model in Cysne (2006) where the production function exhibits increasing return to scale:
ð33Þ
ðP4 Þ : max
ð34Þ
s:t:
c;h;u
Note that ut ht ¼ nt by definition. If we let lt ¼ lt and g ¼ gt =ht , the FOCs of the two problems coincide. These results lead to the following proposition. t
Proposition 2. In the human-capital model, the FOCs of the original problem and the ones of the converted problem are the same whether the optimal path is interior or not.
5. Human-capital model with externalities (1) In this section, we consider the model of Parente and Prescott (1994) and Cysne (2006) where the externalities affect the production of human capital:
ðP3 Þ :
max 1
fct ;htþ1 ;ut gt¼0
1 X
bt
t¼0
ðAut ht Þ1r 1 ; 1r 1h
s:t:
htþ1 ¼ dðht Þ
max 1
fct ;htþ1 ;vt gt¼0
s:t:
1 X t¼0
hð1rÞ
bt
rh A1r ½v1 t t 1r
1
;
t Þh fðht Þ1h vt g þ ht ; htþ1 ¼ dðh
ð36Þ
ðht Þ1h vt P 0;
ð37Þ
vt P 0:
ð38Þ
The human-capital accumulation process (36) is obviously concave because 1 h 2 ð0; 1Þ. Next, the objective function is concave if and only if
h_ ¼ dhð1 uÞ; b c c þ k_ ¼ Ak ðuhÞ1b h ;
ð40Þ
where d ¼ pd. Note that if x and y satisfy y ¼ xa where a is constant, _ ¼ aðx=xÞ. _ then y=y Using z and n, we express the problem ðP4 Þ as
ut 2 ½0; 1;
ðP3 Þ :
0
c1r 1 dt; 1r
z_ h_ ¼ p ¼ dð1 uÞ; z h
s:t:
Now we show that if Cysne’s condition (35) holds, then we can rewrite the model ðP 3 Þ as a concave optimization problem. To see 1h denote a new control variable. Since this, let vt ¼ ut ht h ut ht ¼ vt ht , we can express the model ðP 3 Þ as
eqt
Cysne (2006) pointed out that Arrow’s sufficiency theorem is p p not available if c > 0. However, if we let z ¼ h and n ¼ uh ¼ uz where p ¼ ð1 b þ cÞ=ð1 bÞ is a constant, we can rewrite the model as a concave problem. To see this, first we have
c;k;z;n
ð35Þ
1
k0 > 0; h0 > 0:
t Þh ð1 ut Þ þ ht ; ðh
h rP : 1þh
Z
u 2 ½0; 1;
ðP4 Þ : max
t denotes the human-capital externality and is taken as where h exogenous. Cysne (2006) showed that in his continuous time model, Arrow’s theorem is available if
vt Þ ;
is concave, where gt and /t are the multipliers. Following Section 3.2, we can show that the FOCs in the original problem coincide with the ones in the converted problem.
ð32Þ
g
l
bt
þ/t ðht
lt1 ¼ blt ð1 þ dð1 ut ÞÞ þ bðdlt ut þ gt ut =ht Þ:
l l
1 X t¼0
Substituting Eq. (23) into (24) yields
Moreover, the multiplier on the constraint (20), gt satisfies ð1 ut Þgt ¼ 0 and then gt ¼ ut gt . Hence, the FOCs of the original problem and of the converted problems are written as follows: [FOCs of the original problem ðP2 Þ]
ð39Þ xa yb
Z
c1r 1 dt; 1r 0 z_ ¼ dðz nÞ; b c þ k_ ¼ Ak n1b ; 1
eqt
n 2 ½0; z: p Note that here zt > 0 for all t since z0 ¼ h0 > 0 and z_ P 0.
Lemma 4. A one-to-one correspondence between a path ðc; k; h; uÞ in the model ðP 4 Þ and a path ðc; k; z; nÞ in the model ðP 4 Þ exists and then ðP 4 Þ and ðP4 Þ are equivalent. p
p
Proof. If ðc; k; h; uÞ is fixed, then ðc; k; z; nÞ ¼ ðc; k; h ; uh Þ is uniquely determined. Next, if ðc; k; z; nÞ is fixed, then ðc; k; h; uÞ ¼ ðc; k; z1=p ; n=z1=p Þ is also uniquely determined. Note that z > 0 for all t. h Since ðP 4 Þ is concave, the FOCs and the transversality conditions are sufficient for optimality. One limitation of our approach is that here the concavification is available only if time is continuous. Although there is a linear _ relationship between z_ =z and h=h (see (40)), if time is discrete, ztþ1 =zt and htþ1 =ht have no such simple relationship. The Hamiltonian of the original problem ðP 4 Þ is
H¼
c1r 1 b c þ kðAk ðuhÞ1b h cÞ þ udhð1 uÞ þ gð1 uÞ; 1r
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R. Hiraguchi / Journal of Banking & Finance 33 (2009) 568–572
where k, u and g are the multipliers.2 On the other hand, the Hamiltonian of the converted problem ðP 4 Þ is
H ¼
c1r 1 b þ k ðAk n1b cÞ þ u pdðz nÞ þ g ðz nÞ; 1r
where k , u and g are the multipliers. The Hamiltonian H is clearly concave. As in the previous subsections, we can easily find out a one-to-one correspondence between ðk; u; gÞ and ðk ; u ; g Þ. Hence, the FOCs are exactly the same. 7. Conclusion This note provides an approach to solve the non-concavity problems in the shopping-time models and the human-capital models. The advantage of our approach is that it is applicable to the discrete time models as well as the continuous time models. As a future study, we hope to obtain weaker parametric conditions to ensure the concavity, as Cysne (2008) recently did in the continuous time shopping-time model. We also want to incorporate the productivity shocks into the models and investigate the
2 As we showed in the previous subsection, the constraint u P 0 is not binding here. It is because the marginal productivity of labor, which appears in the first order conditions, goes to þ1 when u ¼ 0. Therefore, we only consider the constraint 1 u P 0.
non-concavity problems in a stochastic framework. It is interesting to see how the conversion method is applicable to the stochastic optimal control problems in Stain (2007) and Huang and Milevsky (2008). References Benhabib, J., Perli, R., 1994. Uniqueness and indeterminacy: on the dynamics of endogenous growth. Journal of Economic Theory 63, 113–142. Cysne, Rubens P., 2006. A note on the non-convexity problem in some shoppingtime and human-capital models. Journal of Banking and Finance 30 (10), 2737– 2745. Cysne, Rubens P., 2008. A note on ‘‘inflation and welfare”. Journal of Banking and Finance 32 (9), 1984–1987. Huang, H., Milevsky, M.A., 2008. Portfolio choice and mortality-contingent claims: the general HARA case. Journal of Banking and Finance 32, 2444–2452. Ladrón-de-Guevara, A., Ortigueira, S., Santos, M.S., 1999. A two-sector model of endogenous growth with leisure. Review of Economic Studies 66, 609–631. Lucas Jr., R.E., 2000. Inflation and welfare. Econometrica 68, 247–274. Parente, S.L., Prescott, E.C., 1994. Barriers to technology adoption and development. The Journal of Political Economy 102, 298–321. Stein, Jerome L., 2007. United States current account deficits: a stochastic optimal control analysis. Journal of Banking and Finance 31, 1321–1350. Takahashi, H., 2008. Optimal balanced growth in a general multi-sector endogenous growth model with constant returns. Economic Theory 37 (1), 31–49.