Search, money and capital in an overlapping generations model Ryoji Hiraguchi

y

June 11, 2010

Abstract We introduce capital in Zhu ("An overlapping generations model with search", Journal of Economic Theory 2008) and show that the Friedman rule is not optimal for some parameter values. Capital is required for the production of the general good. A deviation from the Friedman rule through monetary expansion increases the opportunity cost of holding money and then it makes agents replace some of their money with capital. Although the deviation lowers the payo¤ from the decentralized market where money is essential, it can increase the payo¤ from the centralized market if capital is under-accumulated.

Keywords:search; money; overlapping generations model JEL classi…cation: E5;

Faculty of Economics, Ritsumeikan university. 1-1-1, Noji-higashi, Kusatsu, Shiga, Japan. Tel: 81-77561-4837. Fax: 81-77-561-4837. y Email: [email protected]

1

1

Introduction

Search-theoretic models of monetary exchange have been intensively investigated in modern macroeconomics. Prominent examples are the in…nite horizons models of Kiyotaki and Wright (1989) and Lagos and Wright (2005). A recent paper of Zhu (2008) constructs a monetary overlapping generations model with search and derives the optimal money growth rate. We introduce capital into Zhu (2008). The agents have two assets with possibly di¤erent rates of returns, money and capital. Capital, as well as labor, is needed to produce the general good traded in the centralized market. Money is essential since we can only use money at the decentralized market. We focus on the degenerate steady state and prove that for some parameter values, the Friedman rule of setting nominal interest rates to zero is not optimal. Optimality of the Friedman rule in overlapping generations models have been investigated by many authors including Gahvari (2008). However, as Lagos and Wright (2005) criticize, these are reduced-form and make assumptions, such as putting money in the utility functions or imposing a cash-in-advance constraint on the second period consumption. The problem is that, in their models, a role of money is not explicit. We study the same problem in a micro founded model, in which we consider a transactions role of money using the pairwise meetings. In this note, money is not neutral. A deviation from the Friedman rule through monetary expansion increases costs of holding money and agents begin to replace some of their holding with capital. Consequently, more capital is accumulated at the equilibrium. Although the deviation always reduces the (expected) payo¤ from the decentralized market, it can raise the payo¤ from the centralized market if capital is under-accumulated. Hence the Friedman rule may not be optimal when the payo¤ increase from the centralized market is large.

2

Aruoba and Wright (2005) introduce capital in Lagos and Wright (2005) and they show that money and capital are dichotomized and that the Friedman rule is the optimal monetary policy. As Zhu (2008) argues, the neutrality crucially depends on the quasi-linearity of the utility function. However, some authors cast doubt on the empirical plausibility of the monetary superneutrality. For example, Rapach (2005) analyzes the e¤ects of a change in in‡ation on the long-run real interest rate in 14 industrialized countries by using a structural vector autoregression framework. He …nds that long-run monetary superneutrality is rejected for all countries using annual data. Recently, Aruoba et al. (2008) break the dichotomy in Aruoba and Wright (2005) by assuming that capital accumulation at the centralized market reduces the cost functions of sellers at the decentralized markets. In our OLG model, such a direct connection between the centralized market and the decentralized market does not exists, but we still observe the monetary nonneutrality. One of the main …ndings of Zhu (2008) is that when the agents are risk-averse, the optimal money growth rate can be positive. Then he conjectures that if capital is incorporated into his framework, money creation may enhance more capital accumulation (see Section 4 of Zhu (2008)). The results in this note are consistent with his conjecture. Zhu (2008) also argues that when the quasi-linearity is replaced with strict concavity, the Friedman rule may not be optimal. However, in his OLG framework, the strict concavity is actually not needed. This note assumes that the utility function of the old agent is linear, not strictly concave, but monetary expansion still induce capital accumulation and the optimal monetary policy can deviate from the Friedman rule. In the following, Section 2 describes the model. Section 3 characterizes the competitive equilibrium. Section 4 studies the optimality of the Friedman rule. Section 5 concludes. Proofs are in the Appendix.

3

2

Environment

This section sets up the model which is based on Zhu(2008).

2.1

Set-up

Time is discrete (t = 0; 1; :::). There is unit mass of two-period lived agents. The economy consists of a centralized market (CM) and decentralized markets (DM). There are general good, special goods, money and capital. Agents can carry non-negative amount of money and capital. Each date has two stages, day and night and each agent lives for three stages. He is young for the …rst two stages and old at his third stage. Day market is centralized, but the night market is decentralized and buyer needs to have money to get the special good. The initial old hold the initial money M0 and the initial capital K0 . At the 1st stage (day), each agent supplies labor inelastically, receive wage and hold money and capital. At the 2nd stage (night), he have an equal probability

2 (0; 0:5] to be

a buyer and a seller. The probability of a double coincidence is zero. In the DM, his utility from consuming q is q and his disutility from producing c(q). The function c is positive, increasing, concave, and satis…es c(0) = 0 and c0 (0) = 0. At the 3rd stage (day), agents consume but do not work. The utility from consuming c is u3 (c) = c, with

> 0.

Production of the general good requires labor and capital. There is a representative …rm with a constant returns to scale production function Akt nt1 time t, kt is capital at time t and

2 (0; 1) is capital share. Assume nt = 1. Capital

is fully depreciated. The wage is wt = A(1 Rt = A k t

1

, where nt is labor at

)kt

w(kt ) and the capital rental rate is

R(kt ). Let f (k) = Ak .

The government sets the money supply so that Mt+1 = (1 + held by the old at time t is

t

t )Mt .

The real balance

= Mt =pt , where pt is the price of the general good in terms of

money. The nominal interest rates are it+1 = Rt+1 pt =pt+1

4

1 = (1 +

t )Rt+1 t = t+1

1.

2.2

Preference

Now we de…ne the preferences of the agent born at time t, say agent t. We focus on the degenerate equilibrium. Let x denote his ratio of the real balance to the current money stock Mt+1 . Also let kt+1 denote his capital holdings. We summarize his state as a vector (x; kt+1 ). Suppose that if he becomes a buyer, he consumes qb;t units of the special good and pays money to the seller by Mt+1 lb;t . Also suppose that if he becomes a seller, he produces qs;t units of the special good and receives money by Mt+1 ls;t from the buyer. Then his utility is

U = (qb;t

t+1 lb;t )

+ (

t+1 ls;t

c(qs;t )) + (

t+1 x

+ Rt+1 kt+1 ):

The budget constraint when he is young is

wt = (1 +

New money

t

t) tx

t

(1)

+ kt+1 :

is injected in the form of lump-sum transfers.

If the degenerate equilibrium exists, each agent optimally chooses x = 1 and he consumes the special good by the same amount qt . The utility is U = s (qt )+ ( s(q) = q

t+1 +Rt+1 kt+1 ),

where

c(q) is the surplus function. We assume that s is maximized when q = q > 0.

Then we have s0 (q) > 0 if q < q , s0 (q ) = 0 and s0 (q) < 0 if q > q . If x = 1, the young agent’s budget constraint becomes w(kt ) =

t +kt+1

and the utility function can be expressed

as U = s (qt ) + (f (kt+1 ) The real balance

t

kt+2 ):

(2)

works as a intergenerational transfer from young to old. Eq. (2) implies

that the total consumption ct satis…es the feasibility constraint ct + kt+1 = f (kt ).

5

2.3

Pairwise meeting at the decentralized market

Since the payo¤ of an agent who enters the 3rd stage in state (z; k),

(

t+1 z

+ Rt+1 k) is

linear, the bargaining problem between seller in state (zs ; ks ) and buyer in state (zb ; kb ) is independent of ks and kb and is:

max

q 0; 0 l zb

Here

q

t+1 l

1

is the buyer’s bargaining power. Let z(q) =

q(m) = z 1 (m) and

= z(q ) =

1

1 t+1 l)

( c(q) +

f c(q ) + (1

f c + (1

(3)

;

)qc0 g=f + (1

)c0 g,

)q g. Then we have:

Lemma 1 The function q satis…es q 0 (m) > 0 for m <

, q 0 (0) = 1, q( ) = q and

q0( ) < :

(4)

Proof. See Appendix. Let lt+1 =

=

t+1 .

The trade in the problem (3), (q; l) is

(q; l) =

(q( t+1 zb ); zb ) if zb lt+1 : if zb > lt+1 : (q ; lt+1 )

If x = lt+1 , the agent’s real balance coincides with (insu¢ cient) if it is more (less) than

. We say his real balance is su¢ cient

. We put one assumption on q( ):

Assumption 1: The function q(m) is concave if 0

m

.

We have the following lemma on the satisfaction of Assumption 1. Lemma 2 If either i)

= 1 or ii) c(q) = c0 q n+1 (c0 ; n > 0) and

Proof. See Appendix. 6

1=2, then q is concave.

3

Competitive equilibrium

This section characterizes the degenerate equilibrium where all the agents choose x = 1.

3.1

Utility as a function of state

We derive the expected utility of the agent given the state (k; x), by assuming that the states ~ x~) satisfy x~ = 1. We treat case with l of all the other agents (k; t+1 separately. First assume lt+1

1 (i.e.

~ 1), the trade is (q; l) = (q( in state (k; Note that his real balance

t+1 x

t+1 ). t+1 x); x)

If he becomes a buyer and meets a seller

for x

(q(

t+1 x)

t+1 x)

(q

>

t+1 ); 1).

The expected utility U1 is

t+1

c q( t+1 ) ) + ( t+1 x + Rt+1 k) for x lt+1 ; c q( t+1 ) ) + ( t+1 x + Rt+1 k) for x > lt+1 :

t+1 ).

When the agent in state (k; x) becomes a seller and

+ (

)+ (

Next assume lt+1 > 1(i.e.

lt+1 and (q; l) = (q ; lt+1 ) for x > lt+1 .

is insu¢ cient if x < lt+1 . On the other hand, if he becomes

~ 1), (q; l) = (q( a seller and meets a buyer in state (k;

U1 =

1 and case with lt+1 > 1

t+1

meets buyer in state (k; 1), (q; l) = (q ; lt+1 ) and the trade is e¢ cient. The case where he becomes buyer is the same as the one with lt+1 < 1. The expected utility U2 is (q(

U2 =

t+1 x)

(q

t+1 x)

+ (

)+ (

c(q )) + ( t+1 x + Rt+1 k) for x lt+1 ; c(q )) + ( t+1 x + Rt+1 k) for x > lt+1 :

From Eq. (1), the second term of U satis…es Rt+1 k + Rt+1 (wt +

t)

=

it

t

t+1 x

where

t

=

is a constant. Hence we can express Uj as a function of x, uj (x):

uj (x) =

Here B 1 = f

t+1 x

t+1

equilibrium, it+1

c(q(

(q( t+1 x) (q t+1 ))g

+

t+1 x)

)

t

it+1

it+1 t+1 x + B j for x lt+1 ; j for x > lt+1 : t+1 x + B :

and B 2 = f

c(q )g +

t

(5)

are constant. At the

0 must hold. Otherwise uj (1) = +1 and the optimal x does not exist.

7

3.2

Equilibrium condition

Consider the problem of agents who optimally chooses x. If we can show that the agent with utility function uj optimally chooses x = 1, then the degenerate equilibrium exists. The function uj (j = 1; 2) satis…es i) u0j (0) = +1 (* q 0 (0) = 1). ii) u0j lt+1 iii) u0j (x) =

0 =

t+1 [

it+1

t+1

fq 0 ( )

g

it+1 ] < 0 (see Eq. (4)).

< 0 if x > lt+1 .

Hence the optimal x is interior (i.e. x 2 (0; lt+1 )) and satis…es the …rst order condition (FOC) u0j (x) = q 0 (

( + it+1 ) = 0. Under Assumption 1, u00 < 0 and then the

t+1 x)

agent chooses x = 1 i¤ lt+1 > 1 and it+1 = (q 0 ( the FOC implies q 0 (

t+1 )

t+1 )

). Note that q 00 < 0 and if it+1

0,

. Then from Eq. (4), the condition lt+1 > 1 is automatically

satis…ed. Therefore a sequence f t ; kt g1 t=0 is the competitive equilibrium if and only if

t

(q 0 ( t )=

= w(kt )

(6)

kt+1 ;

1) = (it+1 =)

t (1

+

t )R(kt+1 )

1

(7)

0:

t+1

Eq (6) shows the budget constraint and Eq. (7) shows the FOCs in the DM. The trade at DM is qt = q( t ) and the utility of agent t is s(qt ) + (f (kt+1 )

kt+2 ).

The steady state equilibrium ( ; k) with constant money growth rate

= w(k) (q 0 ( )=

is de…ned as

(8)

k;

1) = (i =) (1 + )R(k)

1

0:

(9)

The amount of trade in the DM is q( ) and the utility is s(q)+ (f (k) k). The second term of the utility, f (k) k is maximized when k is the golden rule level of capital kG = ( A)1=(1 satisfying f 0 (kG ) = 1. 8

)

4

Non-optimality of the Friedman rule

This section investigates the optimality of the Friedman rule.

4.1

Competitive equilibrium under the Friedman rule

If it = 0, from Eq. (7), a real balance and the trade in the DM, qF = q(

F

F)

= q 0(

F

1)

( ) = z 0 ( ) is time-independent constant

is also constant. The equilibrium f

= q 0(

1)

( ) = w(kF;t )

F)

satis…es:

(10)

kF;t+1 :

The money growth rate implementing the Friedman rule is Eq. (4), one has q 0 (

1 F ; kF;t gt=0

> q 0 ( ). Since (q 0 )0 < 0, we have

F;t F

= 1=R(kF;t+1 )

1. From

and then qF < q . The

<

Friedman rule cannot achieve the e¢ ciency in the DM. Now consider the steady states. For a inverted-U shaped function (k) = w(k) the steady state capital k satis…es [ (1

)A]1=(1

)

F

(k). The function

=

and the maximum is

= A1=(1

implement the Friedman rule, we must have

with

A1=(1

)

while EF0 is unstable, because

0

F ; kF )

and EF0 = (

(kF ) < 0 and

0

Proof. See Appendix. 9

2

)1

1

. Hence, to

(11)

0 F ; kF ).

F

= (k) and then there are

The equilibrium EF is stable

(kF0 ) > 0. All the nonstationary equilibria

following the Friedman rule converges to EF . We have Lemma 3 Welfare at EF is higher than at EF0 .

= (1

:

If Eq. (11) hold, two roots kF and kF0 (kF0 < k < kF ) solves two steady state equilibria EF = (

is maximized k = k =

or equivalently,

F

z0( )

)

k,

4.2

Capital accumulation through money growth

This subsection shows that for some parameter values, the Friedman rule is not optimal. As in Zhu (2009), optimality here is maximization of the steady-state expected utility. First we show that monetary expansion enhances capital accumulation. We focus on the equilibrium with higher welfare EF . From Eq. (9), the money growth rate and the capital accumulation satis…es = Let

F

1

(12)

(k):

= (kF ) denote the money growth rate that implements the Friedman rule. We have

(q 0 )0 < 0 by assumption, kF . If

(q 0 ( (k))= 1) + 1 R(k)

rises from

F,

0

(kF ) < 0 since kF > k and R0 (k) < 0. Hence

0

(k) > 0 if k is near

capital k increases, the real balance (k) is reduced and the nominal

interest rates i = (q 0 ( (k))=

1) become positive. Then we have

Proposition 1 When the money growth rate rises from

F

which implements the Friedman

rule, capital increases. Money is not neutral. The result di¤ers from Aruoba and Wright (2005) who shows the neutrality. They consider the quasi-linear utility function u(c)

l and the saving technology f (k) in the CM,

and obtain the FOC f 0 (kt ) = 1 for every t, where l is labor and

is the discount factor. If

instead f 0 (kj+1 ) > 1 for some j, consider the following variational arguments in the CM: Date j: The agent supplies 1 unit of labor and save 1 unit of capital additionally. Date j +1: He enjoys f 0 (kj+1 ) units of additional leisure. (Consumptions are unchanged.) The utility change

j

( 1 + f 0 (kj+1 )) is positive but this contradicts the optimality. Hence

the FOC holds and money cannot a¤ect capital. The point is that the bargaining problem in the DM is independent of the changes in the CM, since the utility function is quasi-linear. Similar arguments are not applicable and monetary policy has real e¤ects here.

10

4.3

Welfare improvement

The steady state welfare is written as a function of k, v(k) = s(q( (k))) + (f (k) Di¤erentiating v by k yields v 0 (k) =

0

(k)q 0 ( )s0 (q( )) + (f 0 (k)

k).

1). From Proposition 2,

the money growth rate raises the welfare v(k) if v 0 (kF ) > 0. Eq. (10) implies

1 0

v (kF ) =

The term

0

0

(kF )s0 (qF ) + (f 0 (kF )

(kF )s0 (qF ) is negative because

0

(13)

1):

(kF ) < 0 and qF < q . This negativity implies

that monetary expansion away from the Friedman rule lowers the expected surplus in the DM. Hence v 0 (kF ) > 0 i¤ the probability parameter

0<

with

= (f 0 (kF )

1)=(

0

satis…es

(14)

< ;

(kF )s0 (qF )). Eq. (14), requires that

is positive and hence

f 0 (kF ) > 1. In other words, capital must be under-accumulated. We have Lemma 4 We have f 0 (kF ) > 1 i¤

1A

1=(1

)

< z 0 ( ) with

1

= (1

)

=(1

)

.

Proof. See Appendix. The condition on the capital under-accumulation in Lemma 3 and Eq. (11) which is the necessary condition to implement the Friedman rule as the steady state equilibrium are combined as 1

< A1=(

1) 0

z( )

:

(15)

From Proposition 3, we get the following conclusion. Proposition 2 If Eqs. (14) and (15) hold, the Friedman rule is not optimal. If A1=(

1) 0

<

z ( ), the Friedman rule cannot be implemented and the equilibrium nominal interest

rates must be positive. 11

The set of parameters satisfying Eqs. (14) and (15) are easily shown to be nonempty.

1

When the money growth rate rises, money becomes costly and the agents begin to replace some of their money with capital. It is true that the expansionary monetary policy reduces the (expected) payo¤ from the DM where money is essential. However, it increases the payo¤ from the CM if capital is under-accumulated. If the payo¤ gains in the CM dominates the loss in the DM, then the Friedman rule is not optimal.

5

Conclusion

In this note, we incorporate capital accumulation into monetary OLG model with search in Zhu (2008). We prove that the Friedman rule may not be optimal and that in‡ation raises the capital level. These results are di¤erent from the recent in…nite horizons monetary search model of Aruoba and Wright (2005). Acknowledgement We thank seminar participants at Kyushu university and Otaru university of commerce for their valuable comments.

Appendix

A

Proof of Lemma 1

The function z satis…es z(0) = 0 and Moreover,

z 0 (q) = c0 =f + (1

z 0 (0) = 0 and z 0 (q) > 0 if s(q)

= z(q ). Hence q(0) = z 1 (0) = 0 and q( ) = q .

)c0 g + (1

)c00 s=f + (1

)c0 g2 with s = s(q). Hence

0. Since 1 = c0 (q ) and s(q ) > 0, z 0 (q ) > 1.

1

In Eq. (15), 1 , 2 , and z 0 ( ) are independent of the paremeters A and . Moreover, in Eq. (14), is independent of since kF and the functions s( ), q( ), k( ) and f ( ) are not related to . Finally, = A1=(1 ) is the maximum of . Hence A1=(1 ) > (kG ) = 1 A1=(1 ) and > 1 .

12

B

Proof of Lemma 2

It is su¢ cient to show that z 00 (q) Proof of i) For

= 1, z(q) =

1

0 for q

q .

c(q) is clearly convex.

Proof of ii) If c(q) = c0 q n+1 , z(q) = z0 q n+1 =(a + q n ) with z0 > 0 and a = =f(1

)c0 (n + 1)g,

and z0 1 z 0 (q) = 1+a((n 1)q n a)=(a+q n )2 . If we let y = q n and g(y) = ((n 1)y a)=(a+y)2 , z is convex i¤ g 0 (y)

0. Since c0 (q) = nc0 y, we have:

(n + 1)a (n g (y) = (a + y)3 0

Hence g 0

0 for q such that c0 (q)

1)y

(n + 1) (n 1)(1 )c0 (q) = (1 )c0 (n + 1)(a + y)3

1 if (n + 1)

(n

1)(1

). This holds for any n if

1=2.

C

Proof of Proposition 2

The steady state welfare is expressed as s(q( )) + ( + R(k)k). The term R(k)k = A k is an increasing function of k. Hence the welfare is higher at the equilibrium with higher capital when the real balance

D

is the same.

Proof of Lemma 3

We have f 0 (kF ) > 1 i¤ kF < kG . Since kF , kG > k and F

= (kF ) > (kG ) =

1A

1=(1

)

0

(k) < 0 for all k

.

References [1] B. Aruoba, C. Waller, R. Wright, 2008. Money and capital. mimeo.

13

k, kF < kG i¤

[2] B. Aruoba, R. Wright, Search, money, and capital, a neoclassical dichotomy, J. Money, Credit Banking 35 (2003) 1085–1105. [3] F. Gahvari, The Friedman rule: Old and new, Journal of Monetary Economics, 54 (2007) 581-589. [4] N. Kiyotaki, R. Wright. On money as a medium of exchange, Journal of Political Economy, 97(1989) 927–954. [5] R. Lagos, R. Wright, A uni…ed framework for monetary theory and policy analysis, J. Polit. Economy 113 (2005) 463–484. [6] D. Rapach, International evidence on the long-run impact of in‡ation, Journal of Money, Credit, and Banking 35 (2003), 23-48. [7] T. Zhu, An overlapping generations model with search, Journal of Economic Theory 142 (2008), 318 –331.

14

Search, money and capital in an overlapping ...

Jun 11, 2010 - Fax: 81$77$561$4837. †Email: [email protected] ... At the 1st stage (day), each agent supplies labor inelastically, receive wage and hold.

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