Monetary Policy and Endowment Risk in a Limited Participation Model∗ Choi, Hyung Sun

†

April 2009

Abstract A limited participation model is constructed to study the risk-sharing role of monetary policy. A fraction of households exchange money for interest-bearing government nominal bonds in the asset market and the government injects money through open market operations. In equilibrium, money is nonneutral and monetary policy redistributes consumption across households. Without idiosyncratic endowment risk, monetary policy becomes a perfect risk-sharing tool, but with idiosyncratic endowment risk, it is not. The Friedman rule is not optimal in general.

Key Words: monetary policy, endowment risk, limited participation, redistribution, Friedman rule JEL Classifications: E4; E5

∗

I am very grateful to my advisor, Steve Williamson, for his invaluable comments and suggestions. I also thank Gabriele Camera, Dean Corbae, James Feigenbaum, Philip Jones, Youngsik Kim, Raymond Riezman, Charles Whiteman, and two anonymous referees. All errors are mine. † Correspondence: Hyung Sun Choi, Department of Financial Institution and Regulation, Korea Insurance Research Institute, 12th Floor K.F.P.A. Bldg., Yoido-Dong, YoungdeungpoGu, Seoul, 150-606, South Korea; [email protected].

1

1

Introduction

An asset market segmentation model is constructed in this paper to study the risk-sharing role of monetary policy. Economic agents face uninsurable endowment risk when there is limited asset market participation. Since an initial money injection goes first to those who can access to the asset market, money is nonneutral and monetary policy redistributes consumption across agents. Thus, monetary policy plays a risk-sharing role which provides crude insurance. Limited participation models that capture nonneutralities of money were initially developed by Grossman and Weiss (1983) and Rotemberg (1984) and followed by Lucas (1990), Fuerst (1992), and Chatterjee and Corbae (1992). Recently, Alvarez and Atkeson (1997), Alvarez, Lucas, and Weber (2001), and Alvarez, Atkeson, and Kehoe (2002) have made important contributions, but they focus on the implications of liquidity effects, asset prices and the exchange rate instead of the risk-sharing implications of monetary policy. My model is built on Alvarez, Lucas, and Weber (2001). There are two types of households: traders who participate in the asset market and nontraders who do not. These types are determined exogenously. In each period, traders receive constant endowments, while nontraders face uninsurable endowment shocks. In the asset market, the government injects money through open market operations and traders initially receive the money injection. In equilibrium, money is nonneutral and monetary policy redistributes consumption between traders and nontraders. The government money injection increases traders’ consumption and decreases nontraders’ consumption because traders get the money injection through the asset market while nontraders suffer from inflation. If nontraders all receive the same endowment shock, then monetary policy is a perfect risk-sharing tool that can smooth out consumption across traders and nontraders. However, if nontraders receive idiosyncratic endowment shocks, then monetary policy is not enough to perfectly insure nontraders. Although monetary policy does not achieve a Pareto optimal allocation, it can mitigate the dispersion of consumption across nontraders. The optimal money growth rate can be positive or negative depending on the endowment distribution. The Friedman rule is not optimal in general. There is a set of limited participation literature where the asset market is endogenously segmented: Chatterjee and Corbae (1992), Alavarez, Atkeson, and Kehoe (2002), Chiu (2004) and Chiu and Molico (2007). While endogenous asset market segmentation is useful in some contexts, for this paper nothing is lost by assuming exogenous segmentation, and much is gained in simplicity. Models of the redistributional effects of monetary policy include Levine (1991), Shi (1999), Berentsen, Camera, and Waller (2005) and Molico (2006). The models 2

studied by Levine (1991) and Mollico (2006) are particularly relevant. Levine (1991) studies an incomplete markets environment where positive money growth implemented through lump-sum transfers can be optimal. Molico (2006) studies a search environment and shows, similarly, that positive inflation with lump-sum trnasfers can be welfare-enhancing, but not at high rates of inflation. In both the Levine (1991) and Molico (2006) environments, there is an insurance role for monetary policy, much as in the environment studied in this paper. The model here has the virtue of being much more tractable than either the Levine (1991) or Molico (2006) models. Further, a novelty here is to study an insurance role for monetary policy in the context of market segmentation, where the agents who are received the first-round effects of monetary policy do not wish to share risk among themselves. The key risk-sharing problem has to do with risk sharing between asset market participants and non-participants. The remainder of the paper is organized as follows. In Section 2 the model is constructed. Section 3 discusses the equilibrium dynamics. Then, Section 4 and 5 study monetary policy implications without endowment risk and with endowment risk. Section 6 is a conclusion.

2

The Model

Time is discrete and indexed by t = 0, 1, 2, .... There is a continuum of households with unit mass. Each household is infinitely-lived and consists of a shopper and a seller. A fraction α of households are traders who participate in the asset market and the rest, 1 − α, are nontraders who never participate in the asset market. The preferences of each household are given by U({ct }∞ t=0 )

= E0

∞ X t=0

β

t



ct1−γ − 1 1−γ



,

where E0 is the expectation operator conditional on information in period 0, β ∈ (0, 1) is the discount factor, and ct represents perishable consumption. In each period, traders receive constant endowments, y, and nontraders get idiosyncratic endowments, yi,t ∈ [yl,t , yh,t] where yi,t is i.i.d. across nontraders with a distribution Ft (y) and yh,t > yl,t > 0. The aggregate output is Yt = αy + (1 − α)ytn which fluctuates depending on the realization of the nontraders’ R n average endowments, yt = yi,t dFt . At the beginning of period t ≥ 0, traders enter the period with Mr,t units of currency and Bt units of interest-bearing one-period government nominal bonds. Nontrader i enters the period with Mi,t units of currency. Then, traders receive 3

constant endowments, y, and nontraders receive idiosyncratic endowments, yi,t . Traders and nontraders cannot consume their own endowments. In the asset market, traders exchange money for interest-bearing one-period government nominal bonds. Each bond sells for qt units of money in period t and is a claim to one unit of money in period t + 1. The government controls the money stock through open market operations. In period t, the government budget constraint is ¯t − qt B ¯t+1 = M s − M s = µt M s B t+1 t t ¯t denotes nominal bonds that mature in period t; B ¯t+1 is newly issued where B nominal bonds with price qt that mature in period t + 1; µt > −1 is the net money growth rate. Note that nontraders do not go to the asset market, so the government money injection initially goes to traders. In the goods market, traders and nontraders exchange consumption goods and the cash-in-advance constraints of traders and nontrader i are, respectively, Pt cr,t ≤ Mr,t + Bt − qt Bt+1

(1)

Pt ci,t ≤ Mi,t

(2)

where cr,t is the consumption of traders; ci,t is the consumption of nontrader i. At the end of period t, everyone returns home with the revenue of sales. No further trade arises. The budget constraints of traders and nontrader i are Pt cr,t + Mr,t+1 = Mr,t + Bt − qt Bt+1 + Pt y Pt ci,t + Mi,t+1 = Mi,t + Pt yi,t . where Mr,t+1 is the cash holding of traders transferred to the next period t + 1; Mi,t+1 is the cash holding of nontrader i transferred to the next period t + 1; Pt y is the traders’ revenue from sales; Pt yi,t is the nontrader i’s revenue from sales.

3

Equilibrium Dynamics In equilibrium, the cash-in-advance constraint of traders binds if  ′  u (cr,t+1 ) Pt qt = βEt <1 u′ (cr,t ) Pt+1

4

(3)

which implies the price of nominal bonds should be less than one. Similarly, the cash-in-advance constraint of nontrader i binds if   ′ u (ci,t+1 ) Pt < 1. (4) βEt u′ (ci,t ) Pt+1 In the asset market, the money injection from the government to each traders is represented by µt Mt Bt − qt Bt+1 = . (5) α In the goods market, assuming both cash-in-advance constraints bind, traders and nontraders hold all of their sales of revenue in a form of cash and carry them into the next period, Mr,t+1 = Pt y

and

Mi,t+1 = Pt yi,t ,

(6)

and the aggregate money stock is n Mt+1 = αMr,t+1 + (1 − α)Mt+1 = Pt Y t

R n where Mr,t+1 represents the aggregate money stock of traders; Mt+1 = Mi,t dFt represents the aggregate money stock of nontraders; Pt Yt represents the aggregate nominal revenue. The average price level is Pt = and the inflation rate is

Mt+1 Yt

(7)

Pt Yt−1 = (1 + µt ) . Pt−1 Yt

Equations (1), (2), (5), (6), and (7) imply Pt cr,t = Pt−1 y +

µt Mt α

Pt ci,t = Pt−1 yi,t−1 and consumption of traders and nontraders is   y + µt Yt−1 /α Yt cr,t (µt ) = 1 + µt Yt−1   yi,t−1 Yt ci,t (µt ) = . 1 + µt Yt−1 5

(8) (9)

In equilibrium, due to asset market segmentation, a change in the money stock has asymmetric effects on consumption of traders, cr,t , and nontraders, cn,t , and redistributes consumption between traders and nontraders. For example, in equations (8) and (9), cr,t increases and ci,t , for all i, decreases as the result of the money injection, µt > 0. The money injection results in inflation and both traders and nontraders consume less, 1/(1 + µt ). However, a positive money injection redistributes consumption from nontraders to traders, µt Yt−1 /α. Thus, traders gain consumption while nontraders do not. On the other hand, if the money stock decreases, µt < 0, then traders consume less and nontraders consume more. The price of nominal bonds depends on the marginal rate of substitution of traders γ   1 Yt+1 cr,t , (10) qt = βEt cr,t+1 1 + µt+1 Yt and a liquidity effect exists. If the government injects money, the nominal interest rate, 1/qt , can decrease since cr,t increases. Given inequalities (3) and (4) with equation (10), the money growth rate should satisfy the following constraint: for all i, !
γ1 α Yt−1 Yt−1 1 γ−1 1 − < Y βΨ , (11) < i 1 t n (1 − α)yt−1 1 + µt yi,t−1 (Ytγ−1 βΨr ) γ

where Ψr > 0 and Ψi > 0 are constant,  γ  1 Yt+1 Ψr = Et and cr,t+1 1 + µt+1

Ψi = Et



1 ci,t+1

γ

 Yt+1 . 1 + µt+1

The government risk-sharing role becomes more transparent with inequality (11), i.e. the return on nominal bonds is positive. For example, if inequality (11) does not hold and there is no return on nominal bonds, qt = 1, which implies the Friedman rule   Yt−1 α 1 1 −  =
γ1 , FR n γ−1 (1 − α)yt−1 1 + µt Y βΨ t

r

then, traders have no incentive to hold nominal bonds and the government is not able to perform monetary policy.

4

Without Idiosyncratic Shocks

Without idiosyncratic endowment risk, monetary policy plays a perfect risksharing role by redistributing consumption between traders and nontraders and 6

in a Pareto optimum, traders and nontraders consume equally, for all i, c∗r,t = R n Yt = c∗i,t = cn∗ ci,t dFt . Suppose yi,t is identical across nontraders, t , where ct = n i.e. yi,t = yt for all i. Then, in equilibrium, monetary policy perfectly smooth out consumption across households if the money growth rate is
α n µ∗t = yt−1 −y (12) Yt−1 1

1

where inequality (11) holds (Ytγ−1 βΨr ) γ < 1 < (Ytγ−1 βΨn ) γ and Ψi is constant across nontraders, i.e. Ψi = Ψn for all i. n Depending on the discrepancy of endowments between y and yt−1 , the optimal money growth rate can be positive or negative. The Friedman rule is not optimal in general. For example, if nontraders and traders receive identical endowments, n yt−1 = y, then, everyone consumes identically and the money growth rate, µ∗t , should be zero. No redistributive monetary policy is necessary. If nontraders n have larger endowments than traders, yt−1 > y, then a money injection, µ∗t > 0, achieves perfect risk-sharing by redistributing consumption from nontraders to traders. On the other hand, if traders receive larger endowments than nontraders, n yt−1 < y, then a reduction of the money stock, µ∗t < 0, achieves perfect risksharing by redistributing consumption from traders to nontraders.

5

With Idiosyncratic Shocks

This section discusses why monetary policy cannot play a perfect risk-sharing role in the presence of idiosyncratic endowment risk. If yi,t is idiosyncratic across nontraders, then, from equation (9), nontraders consume differently depending on their realization of yi,t−1 . Monetary policy cannot smooth out the dispersion of consumption across nontraders since it redistributes consumption as groups. Although monetary policy does not provide perfect risk-sharing insurance, the government can determine the money growth rate by maximizing welfare as an alternative:   Z Wt = max αu(cr,t) + (1 − α) u(ci,t )dFt µt

given equations (8) and (9). Welfare reaches a maximum when the money growth rate is ( )  γ1 n yt−1 α µ ˆt = −y , (13) n Yt−1 yet−1 R where yetn = (yi,t )1−γ dFt and µ ˆt should satisfy inequality (11). Given equations (8), (9), and (13), consumption of traders and nontraders is, for all i, 7

Yt cˆr,t = At where


1−γ ! γ1 n yt−1 n yet−1

At = 1 + α

  

and




1−γ n yt−1 n yet−1

Yt cˆi,t = At ! γ1

−1

  



yi,t−1 n yt−1



,

(14)

≥1

and clearly, traders and nontraders do not consume equally. Monetary policy is effective to shuffle individual nontrader’s consumption to some extent. The government determines µ ˆ t considering not only the redistribun tional effects between traders’ endowments, y, and nontraders’ endowments, yt−1 , 1
n n γ where but also the individual effects on nontraders’ consumption, yt−1 /e yt−1 n yet−1 reflects the spread of individual utility given the distribution of yi,t−1 for all i. For example, suppose the endowment spread of nontraders, Ft (y), changes n n and yet−1 increases while yt−1 stays same. Then, in equation (13), the government can mitigate the dispersion of across nontraders by redistributing consumption between traders and nontraders.
γ1 n n −y. Next, µ ˆt can be positive or negative depending on the sign of yt−1 /e yt−1
1/γ n n The Friedman rule is not optimal in general. If yt−1 /e yt−1 > y, then the government needs to inject money, µ ˆt > 0, to redistribute consumption from nontraders to traders since nontraders receive relatively larger endowments. On the
1/γ n n other hand, µ ˆt becomes negative if yt−1 /e yt−1 < y. The government extracts money to redistribute consumption from traders to nontraders. Furthermore, when the economy is inefficient, by Jensen’s inequality, µ ˆt in equation (13) is greater than µ∗t in equation (12), µ ˆt ≥ µ∗t . The money growth rate can be at the optimum level, µ ˆt = µ∗t , with γ = 1, but it cannot still smooth out consumption across nontraders:   yi,t−1 Yt . cˆi,t (µt ) = n yt−1

6

Conclusion

In this paper, monetary policy can provide crude insurance to economic individuals when the asset market is exogenously segmented. In the asset market, the government injects money through open market operations and traders initially receive it while nontraders cannot. In equilibrium, an asset market segmentation creates the nonneutrality of money and monetary policy has redistributional effects on consumption between traders and nontraders. Without idiosyncratic 8

endowment risk, monetary policy can perfectly smooth out consumption across households. However, with idiosyncratic endowment risk, monetary policy cannot smooth out consumption across households. The money growth rate can be positive or negative depending on the distribution of endowments, and the Friedman rule is not optimal in general.

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