The Insurance Role of Rosca in the Presence of

Credit Markets: Theory and Evidence∗

Hanming Fang† Rongzhu Ke‡

November 23, 2006

Abstract

Rotating Savings and Credit Associations (Roscas) is an important informal financial insti-tution in many parts of the world. Existing models of Roscas assume that their participants donot have access to formal financial markets and predict that the implicit interest rates in bid-ding Roscas should be declining over their lives. Evidence from survey and field data sets fromWenzhou of Zhejiang Province in Southeastern China shows instead that Roscas are prevalenteven in the presence of formal financial markets and more puzzlingly a large fraction of Roscaparticipants reported borrowing from the formal credit market to fulfil their Rosca obligationsand saving their extra Rosca winnings to the formal credit market. Moreover, we find thatthe implicit interest rates observed in a unique Rosca bidding data set are not monotonicallydeclining over its life.We develop a sequential auction model of risk averse Rosca participants facing income risks

to investigate the interaction between formal and informal financial institutions to provide apossible explanation of the above two phenomenon in the Chinese data. In this model Roscaprovides insurance to its participants even in the presence of the formal credit markets. Theintuition is simple: while the formal credit market allows individuals to smooth their intertem-poral income risks by borrowing and saving, Rosca provides an additional instrument for itsparticipants to share contemporaneous income shocks amongst them. We also show that in thismodel it is possible that the implicit interest rate in equilibrium may not be declining over thelife of a Rosca.

Keywords: Roscas, Informal Credit Institutions, Sequential Auction

JEL Classificatioin Numbers: D44; G21; O16; O17

∗Preliminary and Incomplete, please do not circulate without permisson. All comments are welcome.†Department of Economics, Yale University, P.O. Box 208264, New Haven, CT 06520-8264. Email:

hanming.fang@yale.edu‡Department of Economics, MIT, 50 Memorial Drive, Cambridge, MA 02142. Email: rogerke@mit.edu

1 Introduction

Rotating savings and credit associations (Roscas) are an important informal financial institution

around the world. They are most common in developing countries, but immigrant groups in the

United States also used them.1 All forms of Roscas share the following common feature: a group

of individuals, mostly connected in some close social networks, commit to contribute a fixed sum

of money into a “pot” in each of the equally-spaced periods of the life of the Rosca; in each period,

the “pot” is then allocated to one member of the Rosca through some mechanisms. The mechanism

through which the “pot” is allocated is one of the key dimensions of the many variations of how

Roscas operates around the world, as documented by the classic anthropological studies of Roscas

by Geertz (1962) and Ardener (1964). Two main varieties are random Rosca and bidding Rosca. In

a random Rosca, the pot in each period is allocated to one of the members determined by the draw

of lots, with past winners excluded from later draws until each member of the Rosca has received

the pot once. In a bidding Rosca, auctions are used to determine the priority of assessing the pot.

In each period, the individual who bid more than the competitors in the form of a pledge of higher

contributions to the Rosca is chosen to access the pot. As in the Random Rosca, the past winners

are excluded from latter auctions.2

A seminal paper by Besley, Coate and Loury (1993) provide rigorous comparisons of random and

bidding Roscas in an environment where individuals save for an indivisible durable good purchase

where formal credit markets are not available. Besley, Coate and Loury (1994) further compared

the efficiency properties of allocations achieved by random and bidding Roscas and those achieved

with a credit market and with efficient allocations more generally. They found that neither form

of Rosca is efficient.

Our paper is motivated by two observations about the existing theoretical literature on Rosca

and some empirical facts about bidding Roscas in Southeastern China (see Section 2 for more

details).

First, almost all papers on Rosca assumed that Rosca participants do not have access to formal

financial markets. Indeed, it is a somewhat accepted wisdom in the literature that Roscas, as an

informal financial institution, exist because of the lack of formal financial markets; moreover, it is

expected that the development of formal credit markets will lead to the eventual disappearance of

Roscas. It is undoubtedly true that Roscas are more commonly observed in developing countries

with underdeveloped formal financial markets, results from a recent large scale survey in Southeast-

ern China painted a puzzling picture. In that survey, more than 50 percent of Rosca participants

1Roscas are called by many different names around the world. They are called Taihui in China, chit funds in

India, Susu in Ghana, tontines in Senegal, cheetu in Sri Lanka, to name a few examples. Light (1972) and Bonnett

(1981) respectively documented the use of Roscas among Korean and West India immigrants to the United States.2Other forms of alloction mechanisms are described in Ardener (1964), but bidding and drawing lots are the two

most common mechanisms.

1

experienced liquidity difficulties at least once and were unable to pay their pledged contribution to

the Roscas, and as a result, about 70 percent of them borrowed from formal credit markets to fulfil

their Rosca obligations; Moreover, 40 percent of Rosca participants reported lending some of their

Rosca winnings to the formal market (Zhang 2001). This observation raises some important ques-

tions. How can we rationalize active Rosca operations with the presence of formal credit market?

What is the economic role of Roscas when individuals can lend or borrow in a credit market?

Second, the existing models of bidding Roscas as a saving mechanism for durable good con-

sumption (Besley, Coate and Loury 1993, 1994) predict that the implicit interest rate implied by

the winning bid should (weakly) decline over the life of a Rosca. In Besley, Coate and Loury

(1993) model, if the Rosca participants have heterogenous preferences, the order of the receipt of

the pot will be determined by the individuals’ valuation of early consumption of the durable goods,

and thus the implicit interest rates across adjacent periods will decline over the life of the Rosca

deterministically. This prediction is also true in Rosca as a form of savings for lumpy investment

(Kovsted and Lyk-Jesen 1999) where the intuition is that an agent with the highest rate of return

from investment will offers the highest bid, followed by the one with the second highest rate of re-

turn and so on. In Section 2, we examine a large Rosca bidding data set fromWenzhou of Zhejiang

Province in Southeastern China and found that the implicit interest rates over the lives of Roscas

fluctuate and are systematically non-monotonic.3

In this paper, we develop a sequential auction model of risk averse Rosca participants facing

income risks to investigate the interaction between formal and informal financial institutions to

provide a possible explanation of the above two phenomenon in the Chinese data. In this model

Rosca provides insurance to its participants even in the presence of the formal credit markets. The

intuition is simple: while the formal credit market allows individuals to smooth their intertemporal

income risks by borrowing and saving, Rosca provides an additional instrument for its participants

to share contemporaneous income shocks amongst them. We also show that in this model it is

possible that the implicit interest rate in equilibrium may not be declining over the life of a Rosca.

The idea that Rosca may serve as a form of insurance for its participants is not new. Calomiris

and Rajaraman (1998) made the forceful argument for the insurance role of Rosca, but no models are

developed in their paper. Klonner (2000) introduces a model of Rosca as an insurance mechanism

where individuals privately observe their future incomes, but his model does not allow borrowing

and saving via the formal credit market. He showed that individuals will prefer bidding Roscas to

random Roscas if the temporal risk aversion is not greater than static risk aversion.

There are also a few papers that analyzes Rosca bidding games with private information. Kuo

(1993) analyzes the equilibrium of a bidding Rosca where participants’ private information is their

discount factors and moreover he assumes that their discount factors are independently redrawn

3Calomiris and Rajaraman (1998) also made this observation with the bidding history of one Rosca. They used

this case study to support their argument of Rosca as insurance mechanism.

2

every period. This effectively reduces the Rosca bidding to a sequence of independent auctions.

Kovsted and Lyk-Jenson (1999) studied the role of Roscas in financing indivisible investment ex-

penditures for indivisible investment where individuals differ in their privately observed rate of

returns. They only allow individuals to access the outside credit markets in order to fill the gap

between the Rosca winning and the necessary lump investment. As we mentioned above, their

model predicts that the implicit interest rates should decline over the life of a Rosca.

While we highlight insurance role of Rosca in our model, we do believe that many Roscas

in the real world are motivated by saving for durable good purchases and indivisible investment,

as supported by some empirical studies reported in Levenson and Besley (1996) and Handa and

Kirton (1999). Most likely the exact roles of Roscas may change with economic development.

In Southeastern China in the presence of credit markets, the savings role of Roscas may be less

important than its insurance role.

Finally, our theoretical auction model of Rosca bidding in the presence of credit markets also

contributes to the auction literature. Three important features of Rosca bidding in this paper differ

from those of the standard auction literature.4 First, in Rosca bidding there is no ex ante “buyers”

and “sellers,” the roles of “sellers” (or “lenders”) and “buyers” (or “borrowers”) are determined

endogenously. Second, in standard private value auctions a bidder cares about the bids of others

only to the extent that the opponents’ bids affect her probability of winning, in contrast, in Rosca

bidding, the opponents’ bid will affect the size of the pot in subsequent periods and thus directly

affect the bidder’s continuation value even if she loses in the current period bidding. Third, because

Rosca participants can access outside credit markets every period, in this model bidders’ valuations

are endogenously derived from the value function of an optimal consumption/saving problem.

The remainder of the paper is structured as follows. Section 2 provides more details about the

two key empirical findings about Roscas in China that motivate our study. Section 3 presents and

analyzes a simple two period Rosca where we show that Rosca can indeed be welfare improving

as an insurance mechanism even in the presence of perfect credit markets. Section 4 extends the

two period model to multi-periods and in particular we characterize the bidding equilibrium of a

three-period Rosca. We show that in this model the winning bids, and as a result the implicit

interest rates, may not be declining over time. Section 5 concludes and provides some additional

discussions. All proofs are contained in the Appendix.

2 Motivating Evidence

In this section, we provide more details of two empirical facts that motivate our theoretical

investigation. First, Roscas participation may persist in the presence of formal credit markets;

second, the implicit interest rates in bidding Roscas may not be declining over their lives.

4The first two features also appear in Klonner (2003), but the third feature is unique in this paper.

3

2.1 Rosca Operations in Wenzhou

Roscas had a long history in China with the earliest documented Roscas, called “she” at the

time, dating back to 850s AD in the Tang Dynasty.5. Roscas in China are often called Hui, Tai

Hui or Yao Hui are thriving again since the start of the Chinese economic reforms in late 1970s.

Wenzhou, a city in Zhejiang province in Southeastern China, is famous for its vibrant private

enterprises and thriving private financial activities since China’s economic reforms. Roscas are very

prevalent in Wenzhou. Zhang et al. (1993) reported that in 1985, more than 100,000 individuals,

both in urban and rural areas, participated in one or more Roscas. A more recent survey conducted

in 2002 by a county branch of China’s Central Bank – People’s Bank of China – reveals that in

rural areas of Wenzhou, almost every household attended 1 or 2 Roscas on average, and households

contributed 1000-3000 Yuan quarterly to Roscas.6,7

2.1.1 Fact 1: Roscas Are Active in the Presence of Formal Financial Markets

Because of a highly developed private sector and relatively high per capita income, private

borrowing and lending in Wenzhou are active and sizeable (Tsai, 2002). Private entrepreneurs not

only have access to state banks though with some restrictions, they can also borrow from a well

developed private credit market.8 A strong indicator for a developed credit market in Wenzhou

is that the interest rates of private lending across different subregions in Wenzhou, as well as

between private and official interest rates, more or less converged in recent years. Zhang (1997)

documented that in 1997 the monthly lending rate in private markets was stable at 2.0-2.5 percent,

with negligible divergence across sub-regions withinWenzhou. The private market monthly interest

rates declined to about 1.0-1.2 percent in 2002, which is comparable to official lending rate at State

banks, where the base annual interest rate is around 5.8 percent, but can float as high as 30 percent

for small and medium-sized firms.9

More importantly, a recent survey by Zhang (2001) inWenzhou and its vicinity showed a couple

of interesting facts.10 First, Rosca participation is very prevalent inWenzhou, with more than half

of the respondents reported that they attended at least two Roscas, and Rosca members are mostly

5See the biographies of Wei Zhou, who is credited in the establishment of the “she” (meaning association) in The

New Volume of Tang Dynasty (Vol. 197, p. 563). Some earlier historical and sociological studies documented and

analyzed the role and functioning of Roscas in China (see e.g., Wang, 1935; Fei **, and Yang, 1952).6See Research group of Pingyang Branch of the People’s Bank of China (2002).7The per capita income for rural households in Wenzhou in 2002 was about 5100 Yuan (Wenzhou Statistical

Yearbook, 2003).8Zhang (1997) provides statistics that in Wenzhou private credit accounted for about 40 percent of total credits,

while loans from State banks and other credit cooperatives accounted for about 20 percent.9See Research group of Pingyang Branch of the People’s Bank of China (2002).10The sample size is 287 and the survey was conducted in 2001.

4

relatives, neighbors and colleagues. Second, a majority (55 percent) of the respondents reported

attending bidding Roscas while a very small fraction (less than 5 percent) of people attended random

Roscas. Third, more than 50 percent of Rosca participants experienced liquidity difficulties at least

once and were unable to pay their pledged contribution to the Roscas, and as a result, about 70

percent of them borrowed from formal credit markets (either from the private credit market or the

State banks) to fulfil their Rosca obligations. Moreover, 40 percent of Rosca participants reported

lending some of their Rosca winnings to the formal credit market.

2.1.2 Fact 2: The Implicit Interest Rates Implied by the Winning Bids are NotMonotonically Declining.

The second motivating fact for our theoretical investigation is that the implicit interest rates

implied by the winning bids are not monotonically declining, as predicted by the existing models of

Rosca. This empirical fact has been mentioned by Calomiris and Rajaraman (1998) using a single

case study of Rosca bidding. In this subsection, we describe a large data sets of Roscas collected from

Wenzhou and its vicinity counties in 2002 by one of the authors. The data set contained detailed

information on 93 bidding Roscas spanning from 1988 to 2002, and contains 1,358 auctions.11, 12

Our sample exhibits large variations in the size, duration and meeting patterns across individual

Roscas are very large. The largest Rosca in size involves 5,000 Yuan, and the smallest one only has

50 Yuan. The total pot size each period ranges from 500 Yuan to 28,000 Yuan. The (premium) bids

vary from 8.6 Yuan to 790 Yuan, with the mean of 135 yuan. The meeting pattern also exhibits

large disparity across Roscas. The summary statistics of our data is provided in Table 1.

[Table 1 About Here]

We now document the basic patterns of winning bids, and thus the implicit interest rate, over

the lives of Roscas in our data set. Tables 2 and 3 respectively report the results from OLS and

fixed effect regressions of the following model:

Log(bkt) = α+ β(t) + γXkt + εkt (1)

where the dependent variable is the log of bkt, the winning bid in the t-th round in the Rosca

k; and β (t) is a polynomial(up to the third order) of the round time t in Rosca k in order to

capture the potential nonlinear relationship between the winning bid and the sequence of rounds;

Xkt is a vector of control variables, such as size, year dummies, and some specific month dummies

( i.e., Chinese Spring Festival and January). Since macroeconomic situations, such as shifts in

11The majority of Roscas in the sample (70 Roscas) started operation after 1997.12This is a rather large data set in the Rosca literature. For comparison, Klonner’s (2006) data contains 23 Roscas

with 149 auctions.

5

interest rate, might affect the incentive to bid, year dummies are included to capture the effect of

certain shocks common to each Rosca. We control for Chinese Spring Festival and January dummy

simply because in these months considerable amounts of money will be diverted to festival-related

expenditures such as food, new clothes and gifts for children and relatives. The fixed effect model

also control for the effects of some, both observed and unobserved, constant factors associated with

each individual Rosca which may affect the bidding behavior.

[Tables 2-3 About Here]

Table 2 presents regression results using ordinary least squares estimation. Columns 1-3 report

results without controlling year dummies while regressions in columns 4-6 control for year dummies.

No matter whether year dummies are included, a robust result emerges: although a downward

trend in bidding over the sequence of rounds is identified in a simple regression (see column 1 and

4), the bid-round relationship is not monotonically decreasing. Rather, it is highly nonlinear, as

suggested by the statistically significant estimates of the effects of both round squared and cubic.

The estimated coefficient of round squared is negative and significant at 5 percent level in all

specifications, suggesting the concavity of bid-bound relationship. Rosca size has positive impact

on the winning bid. After controlling the Log(size) and year dummies, in column 6, we find that the

Rosca size has an interesting effect on the bid-round relationship: the estimated coefficient of the

interaction term between size and round is negative and significant at 5 percent level. This implies

that the winning bid in larger-sized Roscas is more likely to decrease. This finding accords with

the intuition that the larger-sized Roscas are more likely to be motivated by indivisible investment

than smaller-sized ones, while the latter are mostly motivated by insurance. The estimates of

Spring Festival dummy and January dummy are both negative as expected, but not significant at

conventional levels, indicating the effect of money diversion in these months is not notable in this

time period.

Table 3 reports results from fixed-effect regressions, which are qualitatively similar to what we

see in Table 2. The central result about the nonlinearity of the bid-round relationship remains robust

to different specifications when we controlled for some Rosca-specific fixed factors. In contrast with

the results in Table 2, the downward trend of bids over rounds in the simple regression (columns 1

and 4 in Table 3) is not significant under fixed-effect estimation. The Spring Festival dummy and

January dummy both have expected effects on biding, but neither of them is statistically significant

at conventional levels.

2.2 Summary and Road Map

To summarize, the two main pieces of empirical evidence we gleaned fromWenzhou City’s Rosca

data are as follows:

6

1. Rosca participants have access to formal credit market; moreover, they use the formal credit

market in conjunction with participation in Rosca;

2. The winning bids are not monotonically declining over the life of a Rosca.

The existing theoretical models of Rosca do not address Fact 1 in that they do not allow for

the interaction between formal credit market and informal Rosca; and their predictions about bids

are not consistent with Fact 2. The main purpose of the rest of the paper is to provide a sequential

auction model of Rosca as insurance to provide an explanation for both facts described above.

3 Two-Period Rosca

In this section, we describe the model of a two-period bidding Rosca, derive its bidding equi-

librium, and examine the welfare improving properties of Rosca in the presence of credit markets.

Many of the derivations in this section are naturally extended to the multi-period Rosca environ-

ment we study in Section 4.

3.1 The Model

Environment. Consider two risk averse agents indexed by i = 1, 2. Suppose that both of them

have constant absolute risk aversion (CARA) utility functions:

u (c) =1− exp (−βc)

β,

where β > 0 is the constant absolute risk aversion. The income process for each individual in each

period is modelled as follows. Each receives some risk-free income y each period; but they may

experience negative shocks, which occurs with probability 1−p ∈ (0, 1), (thus p is the probability ofno negative income shock). The level of agent i0s income shock Xi, if it occurs, is randomly drawnfrom CDF F (·) with a continuous density f (·) on the support (0, x̄] where x̄ > 0. We assume thatthe income shocks are independent across the agents and between the periods.

Credit Markets and/or Rosca. In the absence of Rosca, each individual will have access to

credit market where they can borrow and lend at interest rate r > 0. If the two agents do form

Rosca, then they need to decide its size m, which is the amount of money each has to contribute to

the pot each period. We assume that the Rosca will allocate the priority of using the pot through

a premium-bidding auction. More specifically, a premium-bidding auction is organized as follows.

Each agent, after the realization of their income in period 1, submits a bid indicating how much

premium he/she is willing to add to the mandatory size m in period 2 if he/she wins the right to

use the pot in period 1; and the one who submits the higher premium bid wins the right to use the

pot. The winner has to subsequently honor his/her bid in the second period. It is important to

emphasize that credit market is always accessible to the agents even after the formation of a Rosca.

7

Timing. The timing of the game is assumed as follows. In the beginning of period 1, before

realization of period 1 income shocks, the two agents decide whether to form a Rosca, and if so,

the size of the Rosca m. Then each agent observes his/her period-1 income shock, which is his/her

private information. If no Rosca is formed, then the two agents will only use the credit market to

insure their income risks; if a Rosca is formed, then the agents can use both the Rosca and the

credit market to insure their income risks. In a Rosca, the agents submit their premium bid after

realizing their period-1 income shock, and the winner is able to use the pot money in period 1.

Agents then make consumption and saving decisions after the Rosca bidding. Then time moves to

the second period when the period-2 income shocks are realized. The winner of the period-1 Rosca

bidding contributes m plus his/her winning premium bid to the pot, which is used by the loser of

the period-1 bidding. Both agents then make consumption decisions.

[Figure 1 About Here]

3.2 Analysis of the Two-Period Model

3.2.1 The Case with No Rosca

We first analyze the case in which the two agents do not form a Rosca, hence they will only use

the credit market to insure against their income risk. Consider the problem for agent 1 in period

1 after her income realization y1 ≡ y − x1. Her problem can be stated as:

Vn (y1; r) = max{c1,a1}

u(c1) + δEu(C2)

s.t. c1 + a1 ≤ y1, (2)

C2 ≤ Y2 + (1 + r)a1where c1 is her period-1 consumption, and a1 is her saving/debt choice; and r is the interest rate

in the credit market, and Y2 is the random variable denoting her period-2 income realization. The

following proposition characterizes the value function without Rosca:

Proposition 1 The expected value for an agent with income realization y1 in the absence of Roscais given by

Vn (y1; r) =1 + δ

β− 2 + r

β (1 + r)exp

∙−β (1 + r)

2 + ry1

¸[(1 + r) δK]

12+r (3)

where K ≡ Eexp (−βY2) is a constant.

3.2.2 Bidding Equilibrium of Rosca with size m

Now we characterize the bidding equilibrium of Rosca with size m. To this end, we first define

two indirect value functions. Without loss of generality, consider bidder 1 with period-1 income

realization y1. If he wins the right to access the pot money at the end of period 1.

8

The first indirect value function delineates bidder 1’s value from winning the auction at a

premium bid b. In this event, he would subsequently solve the following problem of consumption

and saving:

Vw (b; y1, r) ≡ max{c1,a1}

u(c1) + δEu(C2) (4)

s.t. c1 + a1 ≤ y1 +m, (5)

C2 ≤ Y2 + (1 + r)a1 −m− b (6)

where constraint (5) reflects his access to the additional amount m because he wins the auction,

and constraint (6) reflects his paying additional premium b in excess of the normal Rosca payment

m in the second period. As before, a1 is his asset (debt if negative) holdings in period 1. Analogous

to the solution to the problem we solved when there is no Rosca, Vw (b, y1; r) can be expressed as:

Vw (b, y1; r) =1 + δ

β− 2 + r

β (1 + r)[(1 + r) δK]

12+r exp

½−β [(1 + r) y1 + (mr − b)]

2 + r

¾. (7)

The second indirect value function delineates bidder 1’s value from losing the period 1 auction

while his opponent wins with a premium bid of b̂. In this event, he would subsequently solve the

following problem of consumption and saving:

Vf

³b̂; y1, r

´≡ max

{c1,a1}u(c1) + δEu(C2) (8)

s.t. c1 + a1 ≤ y1 −m, (9)

C2 ≤ Y2 + (1 + r)a1 +m+ b̂ (10)

where constraint (9) reflects the fact that bidder 1 has to contribute m to the pot because he loses

the bidding, and constraint (10) shows that he gets back m and the promised premium of b̂ from his

opponent in period 2. Analogous to the solution to Problems (2) and (4), the solution to Problem

(8) is given by:

Vf

³b̂; y1, r

´=1 + δ

β− 2 + r

β (1 + r)[(1 + r) δK]

12+r exp

⎧⎨⎩−βh(1 + r) y1 −

³mr − b̂

´i2 + r

⎫⎬⎭ . (11)

Now we are ready to characterize the bidding equilibrium of Rosca with size m. Proposition

2 characterizes the equilibrium bidding behavior under the assumption that bidders who do not

experience negative incomes shocks bid zero. We will later show that this restriction on the type-0

bidders maximizes ex ante welfare. This restriction is also enforceable because of our assumption

that whether or not one experiences an income shock is publicly observable.

Proposition 2 The symmetric Bayesian Nash Equilibrium of the bidding equilibrium for Rosca

with size m is given by:

b (x) = mr +

µ2 + r

2β

¶ln

(1−

∙1− exp

µ−2βmr2 + r

¶¸∙p

p+ (1− p)F (x)¸2)

(12)

9

for all x ≥ 0.

Given the characterization of the equilibrium bidding strategy given in (12), we can now show

the welfare consequences of Rosca in the presence of credit market. We will do this welfare analysis

in two steps: first, we provide expressions for interim welfare for agents from participating in Rosca

after the realization of their income shocks; second, we investigate the ex ante welfare before the

knowledge of their period-1 income realizations.

Proposition 3 At the interim stage, agents with type x > 0 is strictly better off from participating

in the Rosca; but type 0 agents are strictly worse off.

The reason for the discontinuity of the interim welfare at type 0 is that the type distribution

itself is discontinuous due to the probability mass of p at type 0.

Remark 1 An alternative modelling approach is to take p = 0, but introduce spread between bor-rowing interest rate rb and saving interest rate rs with rb > rs. Most of the results still go through,

but with significant analytical complication.

Remark 2 Our results are robust when we let p to be arbitrary small.

3.3 Ex ante Efficiency and Optimal Choice of Rosca Size m

The ex ante welfare gain from participating in a Rosca with sizem relative to no Rosca, denoted

by ∆W (m) , is given by:

∆W (m) = p∆U (0) + (1− p)E [∆U(X)|X > 0] .

Note that our calculation for ∆U (0) and ∆U (x) , x > 0, above are valid for any Rosca size m. We

have the following proposition regarding ∆W (m) :

Proposition 4 (Ex Ante Welfare Gain of Rosca and Optimal Size m∗)

1. ∆W (m) is a concave function of m;

2. There exist a unique optimal Rosca size m∗ that maximizes welfare gain;

3. Optimal Rosca size m∗ is strictly increasing in the expected marginal disutility of incomeshocks µ where

µ ≡ E½exp

∙β (1 + r)

2 + rX

¸¯̄̄̄X > 0

¾; (13)

4. The optimal Rosca size m∗ strictly decreases in p.

10

The results in Proposition 4 are intuitive. Claim 3 states that the optimal Rosca size m∗

increases in µ ≡ E {exp [β (1 + r)X/ (2 + r)]|X > 0}.Note that the optimal period-1 consumptionc∗1 derived in (A2) responds to each unit of income shock X by a proportion of (1 + r) / (2 + r) ,

thus µ can be interpreted as the expected marginal disutility of income shock X. Hence, the higher

the impact of the income shock on marginal utility, the larger the optimal size of Rosca.

Claim 4 states that as the probability of negative income shock occurrence increases, the optimal

size of Rosca also increases. In fact, we can show that as p → 1 (i.e. the probability of negative

income shock occurrence 1− p goes to zero), m∗ goes to zero. To see this, note that as p→ 1, the

first order condition (A17) converges to

limp→1

"1

2

µ1

z2+ 1

¶− [(1− p)µ+ p] zp

1− (1− z2) p2

#=1

2

µ1

z2+ 1

¶= 0

which implies that the only solution is z∗ = 1, which means that m∗ = 0. That is, in the limit whenno one experiences income risk, there is no welfare improving role of Rosca. Moreover, as p → 0,

the optimal Rosca size m∗ increases but remains finite. To see this, note that as p → 0, the first

order condition (A17) for z∗ converges to

2zµ =1

z2+ 1.

This third order polynomial equation has a unique real and positive solution given by:

limp→0 z

∗ =1

6µ

∙1 +

³6√3µp1 + 27µ2 + 1 + 54µ2

´− 13+³6√3µp1 + 27µ2 + 1 + 54µ2

´13

¸.

Thus z∗ is finite and hence m∗ is finite as p→ 0.

It is useful to make a few additional remarks. First, the optimal Rosca size m∗ does notdepend on the risk-free income component y and the discount factor δ, as a result of the CARA

utility function specification. This can be seen from the fact that the first order condition (A17)

which characterizes z∗ (and thus m∗) does not contain any terms involving y. Note that the rest ofeconomic environment all affect the optimal Rosca size m∗, including the risk aversion parameterβ, interest rate r, and income risk process p and F (·).

Second, the relationship between the optimal Rosca size m∗ and the risk aversion parameter βand the interest rate r is ambiguous. Note that β and r affect m∗ through two channels. The firstchannel is that β and r affects z∗ though their effect on µ ≡ E {exp [β (1 + r)X/ (2 + r)]|X > 0} .Obviously, for a fixed income shock process, µ increases in both the risk aversion parameter β and

the interest rate. As µ increases, the first order condition (A17) which characterizes z∗ impliesthat z∗ will decrease. The second channel is more direct: the relationship between m∗ and z∗ asdescribed by (A18) implies that for a given z∗, increases in β and r lowers m∗. Thus on the onehand, increases in β and r lead to a decrease in z∗, which leads to higher m∗; on the other hand,for a given z∗, increases inβ and r lowers m∗. Thus the net effect of β and r on m∗ is ambiguous.

11

We now come to the main result of this section, which justifies the existence of Rosca in the

presence of credit markets.

Proposition 5 (Ex Ante Welfare Gains from Rosca with Optimal Size m∗)

1. For any environment, a Rosca with optimal m∗ always leads to ex ante welfare gain for itsparticipants;

2. The ex ante welfare gain from a Rosca with optimal size m∗ increases in µ;

3. The ex ante welfare gain from a Rosca with optimal size m∗ is a non-monotonic function ofp. Moreover, ∆W (m∗) increases in p when p is in the vicinity of 0, and decreases in p whenp is in the vicinity of 1.

The results summarized in Proposition 5 are intuitive. Claim 1 essentially says that a Rosca

with optimal size lead to welfare gains to its participants despite the presence of credit market. The

intuition is as follows. The credit market offers an opportunity for each individual to intertemporally

smooth her income risks by borrowing and lending. However, Rosca fulfils a distinct role of risk

sharing by allowing a select group of individuals to share contemporaneously their income shocks.

Our analysis makes it clear the distinct roles of risk sharing provided by the credit market and the

Rosca. Claim 2 states that the welfare gains from Rosca is increasing in the expected marginal

disutility of income shocks. This is so because the higher µ is, the larger the income risk to share

among the Rosca participants. Claim 3 is more subtle. When p = 0, i.e., when everyone will

be subject to negative income shocks, the presence of the credit market will impose a strict upper

bound of implicit interest rate that agents are willing to bid (no more than the interest rate charged

by the credit market), and competitive pressure will indeed force all bidders to bid that interest

rate irrespective of their income loss realization. Thus credit market effectively crowds out any

additional role of risk sharing by Rosca when p = 1. Thus the welfare gain is zero when p is zero.

When p = 1, no one is subject to income losses, thus of courses there is no need for risk sharing

either.

It is important that the welfare gains result reported in Proposition 5 crucially depends on the

Rosca size m∗ being optimally chosen. If the Rosca size m is exogenous, then participating in a

Rosca may not be ex ante efficient:

Proposition 6 (Rosca with Exogenous Size May Lead to Welfare Loss) Fix an exoge-nously given Rosca size m > 0. Then participating in a Rosca with size m will:

1. lead to welfare losses when p ∈ (0, 1) is sufficiently close to 1; but

2. lead to welfare gains when p ∈ (0, 1) is sufficiently close to 0 and µ is sufficiently large.

12

3.4 A Numerical Example

In this subsection, we work out a numerical example to illustrate our analytical results. Suppose

that the absolute risk aversion β = .15, credit market interest rate r = .02, riskless income y = 200.

We first proceed by taking the probability of no income shock p, and the expected marginal disutility

of income shocks µ as parameters, and examine how the optimal Rosca size m∗ vary with p and

Optimal Rosca Size m∗ as a Function of p and µ Figure 2 depicts the optimal Rosca size

m∗ as a function of p and µ, while holding β and r fixed. As analytically proved in Proposition 4,m∗ increases in µ and decreases in p.

[Figure 2 About Here]

Welfare Effects of Rosca with Optimal and Exogenous Sizes Figure 3 shows that the

welfare gains from Rosca with optimal size m∗ is always non-negative and is non-monotonic in p.However, if the Rosca size m is exogenous, then there always exists open neighborhoods to the left

of p = 1 for which Rosca with size m generates a welfare loss, as shown in Figure 4.

[Figures 3-4 About Here]

Percentage Welfare Gains from Rosca with Optimal Sizes. Have a graph that depicts the

welfare gains from Rosca with optimal size in percentages of the benchmark welfare without Rosca.

[To be added]

Equilibrium Bidding Function. Figure 5 depicts the equilibrium bidding function when the

probability of experiencing negative income shock is 1 − p = 0.8 and if the income shocks are

uniformly drawn from support (0, 50].

[Figure 5 About Here]

Participation Rates in the Credit Market. One can also calculate that in this numerical

example, all agents will use the credit market in the absence of credit market. In particular, some

calculations show that those with negative income x1 higher than 35.07 will borrow in the first

period. Thus with 24 percent (0.8*(50-35.07)/50) chance, an agent will be a borrower; and with

the remaining 76 percent the agent will be a saver in the absence of Rosca.

When there is a Rosca, the first period Rosca winner will always be a saver and the first period

loser will always be a borrower.

13

4 Extension to Multi-Period Rosca

In this section, we first show that the basic analysis of the two-period Rosca in Section 3.2 can

be extended to multi-period models. We first characterize the value functions from the optimal

consumption/saving decisions.

Let wt denote an agent’s wealth after the realization of period-t income shock; and let Vt (wt) de-

note the continuation value of an optimizing agent with wealth wt. Thus by principles of optimality,

we have:

Vt(wi) = max{ct}

u(ct) + δEVt+1(Wt+1)

s.t. ct + ρWt+1 ≤ wt + ρYt+1, (14)

where π ≡ 1/ (1 + r) . For convenience, we write Wt+ =PTj=t+1 ρ

n−j−1Yj as the discounted sumof incomes from period t+ 1 onwards. The following lemma summarizes the solution to the above

dynamic programming problem:

Lemma 1 (Solution to Optimal Consumption/Saving Problem)

1. The value function for problem (14) is given by:

Vs(ws) =1− δT−s+1

β(1− δ)− (1− ρT−s+1)

β(1− ρ)Qs exp

∙− β(1− ρ)

1− ρT−s+1ws

¸(15)

where

Qi = exp

⎡⎣T−iXj=1

ρj − ρT−i+1

1− ρT−i+1

µlnKi+j−1 + ln

δ

ρ

¶⎤⎦ (16)

with

Ki ≡ Eexp∙−β(1− ρ)

1− ρn−iYi+1

¸. (17)

2. The optimal consumption function is given by

c∗s (ws) =1− ρ

1− ρT−s+1ws − 1

βlnQs. (18)

3. The wealth dynamics is given by, w1 = y1 and for all s ≥ 1

ws+1 =¡1− ρT−s

¢ s+1Xj=1

yj1− ρT−j+1

+1− ρT−s

βρ

sXj=1

lnQj1− ρT−j

. (19)

14

4.1 Bidding Equilibrium: Preliminary

Before we analyzing the sequential bidding equilibrium of the multi-period Rosca, we first

introduce some additional assumptions regarding information and some additional notation. First,

in order to maintain symmetry among remaining bidders in latter rounds of auction, we assume

that:

Assumption 1: At the end of each period, only the winning bid is publicly revealed. In particular,the agents’ past losing bids and income realizations are not revealed to others.

The key implication of this assumption is that, at any latter rounds, all remaining bidders have

the same belief about other remaining bidders’ “types.” The relevant type of a bidder in round s

auction should include the private income shock realizations up to period s− 1, i.e. (x1, ..., xs−1) .However, the fact that each bidder makes optimal borrowing/saving decision every period implies

that, as shown by the wealth dynamics characterized by formula (19), we can, for all s ≥ 2, collapsethis multidimensional type (x1, ..., xs) into a single dimensional summary, denoted by χs, given by:

χs =¡1− ρT−s+1

¢ sXj=1

xj1− ρT−j+1

. (20)

As we will show below that in each stage t the bidding strategy will be monotonically increasing

in χt, Assumption 1 thus implies that in period t + 1, all remaining bidders know and only know

that their opponents’ χt was lower than a threshold χ∗t as revealed by the winning bidder’s bid.Thus in period t + 1, each remaining bidder will have a belief about her opponent’s types χt+1given by G

¡χt+1|χt ≤ χ∗t

¢. Even though it is possible to explicitly express G (·), we will not do so

here for this general setting; instead we will do so in the next section where we explicitly solve for

bidding equilibrium for a 3-period Rosca.

4.2 Bidding Equilibrium for a 3-Period Model

In this subsection, we explicitly solve for the bidding equilibrium for a 3-period model. The goal

of this section is to show that the equilibrium winning bids in our model may be non-decreasing

over time. Analytically solving the equilibrium for Roscas lasting more than three periods is a

challenging task, and we will only provide some results from numerically simulations.

To solve for the bidding equilibrium of a 3-period Rosca, first consider period 2. The bidding

environment in period 2 is identical to the 2-period model we analyzed in Section 3.2. Analogous

to the steps in the derivation of VT−1 (·) as described in (A23)-(A25), we know that we can expressa bidder’s indirect utility from winning the pot at a bid of b2 by

Vw,2(b2;w2) =1 + δ

β− 1 + ρ

βQ2 exp

∙− β

1 + ρ

hw2 +

³m+ b̂1

´+ ρ (mr − b2)

i¸, (21)

15

where b̂1 is winning premium bid submitted by the period-1 auction winner, w2 is the bidder’s

private period-2 type in terms of her wealth, and Q2 is as defined by (A25). Similarly, if her

opponent wins the period-2 auction with a bid b̂2, then her indirect utility can be written as:

Vf,2(b̂2;w2) =1 + δ

β− 1

β(1 + ρ)Q2 exp

∙− β

1 + ρ

hw2 + ρ

³m+ b̂1

´− ρ

³mr − b̂2

´i¸. (22)

Suppose for the moment that the ex ante belief of the remaining bidder’s end-of-period-2 wealth

w2 is given by CDF G̃2 (·) which we assume to be piecewise differentiable. [Its derivation is pro-vided latter] Now consider a symmetric equilibrium bidding strategy in the period-2 auction given

by b2 (·) . Similar to two-period model we analyzed in Section 3.2, we can write the expected con-tinuation value of a bidder of type w2 from reporting w̃2 is given by:

U2(w2, w̃2) =h1− G̃2 (w̃2)

iVw,2 (b2 (w̃2) ;w2) +

Z w̃2

w2

Vf,2(b2 (τ) ;w2)dG2(τ ).

The first order condition

∂U2 (w2, w̃2)

∂w̃2

¯̄̄̄w̃2=w2

= −g̃2 (w2)Vw,2 (b2 (w2) ;w2) +h1− G̃2 (w̃2)

i ∂Vw,2∂b2

b02 (w2)

+ Vf,2(b2 (w2) ;w2)g̃2 (w2) = 0

implies that, in a symmetric Bayesian Nash equilibrium, b2 (·) must satisfy (after some simplifica-tion),

b02(w2) =g̃2(w2)

1− G̃2(w2)Vw,2 − Vf,2∂Vw,2/∂b2

=g̃2(w2)

1− G̃2(w2)exp

hβρ1+ρ

h³3m+ b̂1

´r − 2b2 (w2)

ii− 1

βρ/ (1 + ρ).

If we denote the type of a bidder to be χ2 instead of w2, noting that w2 is a monotonically decreasing

function of χ2, we can with some abuse of notation, rewrite the first order condition as

b02(χ2; b̂1) =g2(χ2)

G2(χ2)

exph

βρ1+ρ

h³3m+ b̂1

´r − 2b2 (χ2)

ii− 1

βρ/ (1 + ρ). (23)

Using the change of variable

h2³χ2; b̂1

´≡ βρ

1 + ρ

h³3m+ b̂1

´r − 2b2 (χ2)

i,

we can easily obtain that the solution to the differential equation is

h2³χ2; b̂1

´= − ln

⎡⎢⎣1− ω2³b̂1´⎛⎝G2

³χ2

´G2 (χ2)

⎞⎠2⎤⎥⎦

16

where ω2 is an integration constant and ω2³b̂1´= 1− exp

h−h2

³χ2; b̂1´i. Note that χ

2= 0. Also

note that given h2³χ2; b̂1

´, we have

b2(χ2; b̂1) =1

2

∙³3m+ b̂1

´r − 1 + ρ

βρh2

³χ2; b̂1

´¸.

It is important to emphasize that the second period bidding is contingent on the realization of the

first period winning bid b̂1. Thus, for every realization of b̂1, a different bidding equilibrium in the

second period is specified up to the constant ω2³b̂1

´, which effectively determines b2

³χ2; b̂1´.

Given the characterization of symmetric bidding equilibrium in period 2 auction, we can sub-

stitute it into the indirect utility function and obtain that, for type χ2 > χ2= 0,

U2(χ2,χ2; b̂1)

=1 + δ

β− 1

β(1 + ρ)Q2 exp

∙− β

1 + ρw2 (χ2)

¸×⎛⎝ G2 (χ2) exp

h− β1+ρ

h³m+ b̂1

´+ ρ (mr − b2 (χ2))

ii+R w2w2exp

h− β1+ρ

hρ³m+ b̂1

´− ρ (mr − b2 (τ))

iidG2(τ)

⎞⎠=

1 + δ

β− 1

β(1 + ρ)Q2 exp

∙− β

1 + ρw2 (χ2)

¸exp

∙−β2(m+ b̂1)

¸×(

G2 (χ2) exp

∙−12h2 (χ2)

¸+

Z w2

w2

exp

∙1

2h2 (τ)

¸dG2 (τ)

)

=1 + δ

β− 1

β(1 + ρ)Q2 exp

∙− β

1 + ρw2 (χ2)

¸exp

∙−β2(m+ b̂1)

¸r1− ω2

³b̂1

´G2 (0)

2 (24)

where the last equality follows from steps similar to those used in the derivation of U (x, x) in

(A11).

In order to fully determine the second period bidding equilibrium following first period win-

ning bid b̂1, we need to determine b2(χ2; b̂1). In what follows, we make the following important

assumption:

Assumption 2: ω2³b̂1´is chosen to maximize U2(χ2,χ2; b̂1) as described by (24), anticipating

that the choice of ω2³b̂1´will affect the first period bidding equilibrium.

Before we present the main result of this section, we describe how in principle the distribution

17

of χ2 =(1−ρ2)X11−ρ3 +X2, namely G2 (χ2) , could be explicitly derived:

G2

³χ2|b̂1

´= Pr(χ2 ≤ χ2 |X1 < x∗1 )

=Pr(χ2 ≤ χ2, X1 < x

∗1)

Pr (X1 ≤ x∗1)

=1

p+ (1− p)F (x∗1)×

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩X2=0,X1=0z }| {

Pr (X2 = 0,X1 = 0) +

X2=0,X1∈(0,x∗1)z }| {Pr

µ(1− ρ2)X11− ρ3

≤ χ2, 0 < X1 ≤ x∗1, X2 = 0¶

+

X1∈(0,x∗1),X2>0z }| {Pr

µ(1− ρ2)X11− ρ3

+X2 ≤ χ2, X2 > 0, x∗1 > X1 > 0

¶+

X2>0,X1=0z }| {Pr(0 < X2 ≤ χ2, X1 = 0)

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭where x∗1 = b

−11

³b̂1

´.

The main result of this subsection is the following proposition that characterizes the equilibrium

of the three-period sequential auction under assumptions 1 and 2:

Proposition 7 Under Assumptions 1-2, the symmetric equilibrium for the three period model is

as follows:

1. The first period equilibrium bidding function is:

b1(x1) =(2 + ρ)mr

1 + ρ+1+ ρ+ ρ2

βρ (1 + ρ)ln

∙1− ω1

[p+ (1− p)F (x1)2¸

(25)

where

ω1 =

½1− exp

∙− βρ

1 + ρ+ ρ2[(2 + ρ)mr − (1 + ρ) b1]

¸¾p2

and b1 solves

1− exp∙−βmr (2 + ρ)

1 + ρ+ βb1

¸= p2

½1− exp

∙−βρ (3m+ b1) r

1 + ρ

¸¾; (26)

2. Given period one winning bid b̂1, the second period equilibrium bidding function is:

b2³χ2; b̂1

´=

³3m+ b̂1

´r

2+1 + ρ

2βρln

⎡⎢⎣1− ω2

³b̂1

´G2

³χ2

´2G2 (χ2)

2

⎤⎥⎦ (27)

where

ω2

³b̂1

´G2

³χ2

´2= 1− exp

⎧⎨⎩−βhmr (2 + ρ)− (1 + ρ) b̂1

i1 + ρ

⎫⎬⎭ , (28)

18

and

ω2

³b̂1

´=

1− exph− βρ1+ρ+ρ2

[(2 + ρ)mr − (1 + ρ) b1]i

p2n1− exp

h− βρ1+ρ+ρ2

h(2 + ρ)mr − (1 + ρ) b̂1

iio×⎧⎨⎩1− exp

⎡⎣−βhmr (2 + ρ)− (1 + ρ) b̂1

i1 + ρ

⎤⎦⎫⎬⎭ . (29)

The equilibrium bidding functions characterized above have intuitive appeal. First, consider the

first period equilibrium bidding function b1 (x1) as described in (25). The first term (2 + ρ)mr/ (1 + ρ)

represents the outside interest rate; the second term, which is negative, can be regarded as the

amount of shading that will reflects the insurance benefits that one gets from participating in the

Rosca. Notice that the level of shading decreases with a bidder’s negative income shock x1.

Similarly, for the second period equilibrium bidding function b2³χ2; b̂1

´described in (27), the

first term³3m+ b̂1

´r/2 can again be interpreted as the implicit interest payment that a bidder is

willing to pay for assessing the pot of money at period 2 instead of period 3, if she had only access

to the outside credit market. To see this, note that the net cash flow if she accesses the pot 2m+ b̂1at period 2 with a premium bid of b2 is given by 2m+ b̂1− ρ (m+ b2) , and her net cash flow if she

accesses the pot at period 3 which will be valued at 2m+ b̂1 + b2 is given by ρ(2m+ b̂1 + b2)−m.Equating these two terms will lead to an indifference level of b2 =

³3m+ b̂1

´r/2. It can be shown

that the second term in the right hand side of (27) is negative, thus bidders will shade their

premium bids in the second period as well. Not surprisingly, the level of shading decreases with

the cumulative negative income shock χ2; moreover, as a result of Assumption 2, which requires

that b2 (·) in the second period is set optimally, one can show that second period shading decreaseswith first period winning bid b̂1.

4.3 Non-Decreasing Winning Bids in a Three-Period Model

Now we show that in this model, the winning bids may not be decreasing over the life of the

Rosca.

Proposition 8 (Potential Non-Decreasing Winning Bids)

1. If b̂1 ≤ mr, then b2³χ2; b̂1

´≤ b̂1 for all χ2;

2. If b̂1 > mr, there exist a cut-off level χ∗2 such that , b2³χ2; b̂1

´> b̂1 if and only if χ2 > χ∗2.

Proposition 8 allows us to calibrate the probability of observing period one winning bid to be

lower than the period two winning bids in a three-period auction. For the rising equilibrium bids

to occur in equilibrium, two conditions need to be satisfied: first, the highest first period shock

19

(among three bidders) must be such that the first period winning bid exceeds mr; second, the

cumulative second period shock χ2 must exceed χ∗2.In order for Proposition 8 to provide a possible rationalization for the observed non-declining

winning bids we documented in Section 2, we have to show that the event b̂1 > mr will actually

occur with positive probability. Note that the probability that first period winning bid b̂1 > mr is

given by

1− £p+ (1− p)F ¡b−11 (mr)¢¤3;

where p+ (1− p)F ¡b−11 (mr)¢is the probability that all three bidder’s bid is less than mr. Using

the formula for b1 (·) as described by (25),

F¡b−11 (mr)

¢= Pr

∙(2 + ρ)mr

1 + ρ+1+ ρ+ ρ2

βρ (1 + ρ)ln

∙1− ω1

[p+ (1− p)F (x1)2¸≤ mr

¸= Pr

∙1− ω1

[p+ (1− p)F (x1)]2≤ exp

µ− βρmr

1 + ρ+ ρ2

¶¸

= Pr

⎧⎪⎨⎪⎩F (x1) ≤ p

1− p

⎡⎢⎣vuuut1− exp

h− βρ1+ρ+ρ2

[(2 + ρ)mr − (1 + ρ) b1]i

1− exp³− βρmr1+ρ+ρ2

´ − 1

⎤⎥⎦⎫⎪⎬⎪⎭

= min

⎧⎪⎨⎪⎩ p

1− p

⎡⎢⎣vuuut1− exp

h− βρ1+ρ+ρ2

[(2 + ρ)mr − (1 + ρ) b1]i

1− exp³− βρmr1+ρ+ρ2

´ − 1

⎤⎥⎦ , 1⎫⎪⎬⎪⎭ .

Numerical examples show that F¡b−11 (mr)

¢calculated above could be strictly less than 1, indicat-

ing that the event b̂1 > mr does occur with strictly positive probability. [Add a Figure Here toShow this]

Proposition 8 establishes that it is possible,for some realizations of shocks, to observe increasing

winning bids. The next proposition provides conditions under which the expected winning bids in

the second period may be higher than the first period winning bid.

Proposition 9 (Increasing Winning Bids in Expectation) When p→ 0 or p→ 1,

Ehb2

³χ̂2; b̂1

´¯̄̄b̂1, χ̂2 is the period 2 winner

i≥ b̂1.

Even though the result is shown analytically only for limiting cases as p → 0 or p → 1, by

continuity, it must be true for p that is close to 0 or close to 1.

An even stronger form of comparison is the comparison of

Eχ̂2,b̂1

hb2³χ̂2; b̂1

´¯̄̄b̂1, χ̂2 is the period 2 winner

iand Eb̂1. But for the limiting case when p → 0 or p → 1, this comparison is the same as the

comparison in Proposition 9 because in these limits, b̂1 is converging to a degenerate distribution.

20

It is interesting to provide some intuition for the non-declining winning bid results in this

section. In a dynamic auction with uncertainty, there are several competing effects. First, there

are two types of information effect. On the one hand, bidders in the latter rounds learns that

their opponents did not receive the highest income shock in the previous round, thus their belief

about their opponent’s previous round income shock is right-truncated. This learning effect tends to

drive bid down. On the other hand, bidders’ income shocks cumulate over time, thus the cumulative

income shocks of their opponent in latter rounds may be more severe than in earlier rounds. This

effect tends to drive bid higher.

The second set of effects is related to the stake of competition and the number of competitors.

The number of bidders in latter rounds, by definition, is lower than in the earlier rounds. This

tends to drive down the bids. But, the stakes are higher in latter rounds because in each period,

the winning premium bid of the previous round is added to the total size of the pot. This increase

in stakes tends to drive up the bids. Whether the winning bids in latter rounds are higher or lower

than those in the earlier rounds are as a result mixed depending on the strength of the various

forces.

4.4 Geneal T -Period Rosca

[Add simulation results]

5 Conclusion

In this paper, we document two empirical facts from Roscas in Wenzhou, China. First, Roscas

are prevalent even in the presence of formal financial markets and a large fraction of Rosca partici-

pants reported borrowing from the formal credit market to fulfil their Rosca obligations and saving

their extra Rosca winnings to the formal credit market; second, the implicit interest rates observed

in a unique Rosca bidding data set are not monotonically declining over its life.

We develop a sequential auction model of risk averse Rosca participants facing income risks to

investigate the interaction between formal and informal financial institutions to provide a possible

explanation of the above two phenomenon in the Chinese data. In this model Rosca provides

insurance to its participants even in the presence of the formal credit markets. The intuition

is simple: while the formal credit market allows individuals to smooth their intertemporal income

risks by borrowing and saving, Rosca provides an additional instrument for its participants to share

contemporaneous income shocks amongst them. We also show that in this model it is possible that

the implicit interest rate in equilibrium may not be declining over the life of a Rosca.

[To be completed]

21

References

[1] Adams, D. W., and Canavesi de Sahonero, M. L. (1989), “Rotating Saving and Credit Asso-

ciations in Bolivia,” Saving and Development, Vol. 13, 219-236.

[2] Aleem, I. (1990), “Imperfect Information, Screening, and the Costs of Information Lending:

A Study of a Rural Credit Market in Pakistan,” World Bank Economic Review, Vol. 4, No. 3,

329-349.

[3] Ardener, S. (1964), “The Comparative Study of Rotating Credit Associations,” Royal Anthro-

pological Institute of Great Britain and Ireland, Vol. 94, 201-229.

[4] Besley, Timothy J., Stephen Coate and Glenn Loury (1992), “On the Allocation of Rotating

Saving and Credit Associations,” Research Program in Development Studies at Princeton

University Discussion Paper, No. 163.

[5] Besley, Timothy J., Stephen Coate and Glenn Loury (1993), “The Economics of Rotating

Savings and Credit Associations,” American Economic Review, Vol. 83, 792-810.

[6] Besley, Timothy, Coate, Stephen., and Loury, Glenn., 1994, Rotating Savings and Credit

Associations, credit market and efficiency, Review of Economic Studies, 61:701-719.

[7] Besley, Timothy, and Alec R Levenson, 1996, The role of information finance in household

capital accumulation evidence from Taiwan. The Economic Journal. 106 January, 39-59.

[8] Besley, Timothy and Levenson, Alec R., 1996, The anatomy of an information financial market:

Rosca participation in Taiwan. Journal of Development Economics. Vol. 51 45-68.

[9] Bonnett, Aubrey W. (1981). Institutional Adaptation of West Indian Immigrants to America:

An Analysis of Rotating Credit Associations. Washington, DC: University Press of America.

[10] Calomiris, Charles and Indira Rajaraman, 1998, The role of ROSCAs: lumpy durables or event

insurance? Journal of Development Economics, Vol. 56, 207-216.

[11] Crawford, V. 1987, International lending, long-term credit relationships, and dynamic contract

theory. Princeton Studies in International Finance, 59.

[12] Geertz, Clifford, 1962, The Rotating Credit Association: A "Middle Rung" in Development,

Economic Development and Culture Change., April, 10, 241-63.

[13] Grossman, Richard, 1992, Deposit insurance, regulation, and moral hazard in the thrift indus-

try: Evidence from the 1930’ s. American Economic Review, September 82, 800-22.

22

[14] Handa, Sudhanshu, and Kirton Claremont, 1999, The economics of rotating saving and credit

associations: evidence from the Jamaican ’Partner’. Journal of Development Economics. Vol.

60, 173-194.

[15] Kimuyu, Peter- Kiko, 1999, Rotating Savings and Credit Associations in rural East Africa,

World Development, Vol. 27, No. 7 1299-1308.

[16] Klonner, Stefan., 2000, Rotating Savings and Credit Associations as Insurance., Sudasien

Institut der University Heldeberg., Working paper.

[17] Klonner, Stefan (2003), “Rotating Savings and Credit Associations when Participants are Risk

Averse,” International Economic Review, Vol. 44, No. 3, 979-1005

[18] Klonner, Stefan (2006). “Private Information and Altruism in Bidding Roscas.” mimeo, Cornell

University.

[19] Kovsted, Jens and Peter Lyk-Jensen, 1999, Rotating Savings and Credit Associations: the

choice between random and bidding allocation of funds, Journal of Development Economics,

Vol. 60, 143-172.

[20] Kuo, P., 1993, Loans, bidding strategies and equilibrium in the discount-bid rotating credit

association. Academia Economic Papers, 21(2):261-303.

[21] Kuo, P., 1996, Lending and bidding in the bidding Rotating Credit Associations: Discount-bid

vs. Premium-bid.Working paper, The Institute of Economics, Academia Sinica, April 26, 1996.

[22] Kuo, P., 1996, Defaults, bidding and participation in Rotating Credit Associations: An equi-

librium analysis. Working paper, The Institute of Economics, Academia Sinica, October 3,

1996.

[23] Jinzheng, Li, 2000, The Debit-Credit Relationship and Rural Changes: The Studies of Debit-

Credit Relationship and Rural North China Republican Era, Hebei University Press.

[24] Light, Ivan (1972). Ethnic Enterprises in America. Berkeley: University of California Press.

[25] Kellee S, Tsai 2000, Banquet banking: Gender and Rotating Savings and Credit Associations

in South China, China Quarterly, Mar, Issue, 161, Chart 1, 142-171.

[26] Verhoef, Grietjie, 2001, Informal financial service institutions for survival: African women and

Stokvels in urban South Africa, 1930-1998. Enterprise & Society, Jun2001, Vol. 2 Issue 2

[27] Ven den Brink, and R. Chavas, 1997, The microeconomics of an indigenous African Institutions:

the Rotating Savings and Credit Associations. Economic Development and Culture Change,

45 (4).

23

[28] Zhiduo Wu, 1935, The inquiry on borrowing and lending between peasants in the first district

of North Tongxiang County (Bei Tongxiang Di Yi Qu Ping Min Jie Dai Zhuang Kuang Zhi

YanJiu), Yenching University Press.

[29] Yang, Lien-Sheng, 1952, Money and Credit in China. Harvard Yenching Institute Monographs

12. Harvard University Press.

[30] Zhang, Jun, 1997, Informal banking sector in post-reform rural china: the case of Wenzhou,

Chinese Social Science Quarterly (HongKong) Vol. 20, 22-35.

[31] Zhang, Weikung 2001, A survey on participation of Roscas in Taizhou City, Unpublished memo,

Normal College of Taizhou.

24

A Appendix: Proofs

Proof of Proposition 1:Proof. Substituting C2 = Y2 + (1 + r) (y1 − c1) into the objective function and taking thederivative of the objective function with respect to c1, we obtain the first order condition:

u0(c1) = δ (1 + r) Eu0(C2). (A1)

Since u0 (c) = exp (−βc) , the first order condition can be simplified as:

exp (−βc1) = δ (1 + r) exp [−β(1 + r)(y1 − c1)] E exp (−βY2) .

With some algebra, we have:

c∗1 =1 + r

2 + ry1 − ln [(1 + r) δK]

β(2 + r), (A2)

where K ≡ Eexp (−βY2) is a constant that equals to expected marginal utility of the secondperiod’s income. Thus,

Vn(y1; r) = u(c∗1) + δEu(C2)

=1− exp (−βc∗1)

β+

δ [1− Eexp (−βC∗2 )]β

=1 + δ

β−µ1 +

1

1 + r

¶exp (−βc∗1)

β

=1 + δ

β− 2 + r

β (1 + r)

2 + r

β (1 + r)exp

∙−β (1 + r)

2 + ry1

¸[(1 + r) δK]

12+r .

where the third equality follows from (A1).

Proof of Proposition 2:Proof. Without loss of generality, consider bidder 1. Suppose that his opponent followsbidding strategy b (·) that satisfies b0 > 0. Consider the revelation mechanism for bidder 1with type x > 0. The expected utility for bidder 1 of type-x from reporting type x̃ is givenby:

U(x, x̃) = {Pr (X2 = 0) + [1− Pr(X2 = 0)] Pr (0 < X2 ≤ x̃)}Vw (b(x̃); y + x, r)+ [1− Pr(X2 = 0)] Pr (X2 > x̃) E[Vf(b (X2) ; y + x, r)|X2 > x̃]. (A3)

To understand this expression, note that the first term is the expected utility for a type-xbidder 1 from reporting x̃. Such misreporting will lead the mechanism to submit a bid ofb (x̃) for him and this bid will win against the opponent when her type is between [0, x̃] ,

25

which occurs with total probability Pr (X2 = 0) + [1− Pr(X2 = 0)] Pr (0 < X2 ≤ x̃) , andwinning with a premium bud of b (x̃) will yield a continuation value of Vw (b(x̃); y + x, r) . Thesecond term can be similarly understood: with probability[1− Pr(X2 = 0)] Pr (X2 > x̃) theopponent’s type will be drawn from (x̃, x] and bidder 1 will lose the auction when submittingthe bid b (x̃) . However, bidder 1’s continuation value from losing the auction depends on thepremium bid submitted by her opponent b (X2) , thus bidder 1’s expected continuation valuefrom losing is given by E[Vf(b (X2) ; y + x, r)|X2 > x̃]. Using the expressions for Vw and Vfas derived in (7) and (11), we can rewrite U (x, x̃) as:

U(x, x̃) =1 + δ

β− (2 + r) [(1 + r) δK]

12+r

β (1 + r)exp

∙−β (1 + r) (y − x)

2 + r

¸×½

[p+ (1− p)F (x̃)] exp∙−β [mr − b (x̃)]

2 + r

¸+ (1− p)

Z x̄

x̃

exp

∙β [mr − b(x2)]

2 + r

¸dF (x2)

¾.(A4)

The first order condition for truth telling equilibrium requires that

∂U (x, x̃)

∂x̃

¯̄̄̄x̃=x

= 0,

which, after some simplification, yields:

[p+ (1− p)F (x)] exp∙−β [mr − b (x)]

2 + r

¸βb0(x)2 + r

+ (1− p) exp∙−β [mr − b (x)]

2 + r

¸f (x)

− (1− p) exp∙β [mr − b (x)]

2 + r

¸f(x) = 0.

Collecting terms we obtain:

βb0(x)2 + r

=(1− p) f (x)

nexp

h2β[mr−b(x)]

2+r

i− 1o

p+ (1− p)F (x) . (A5)

With the following change of variables:

h(x) ≡ 2β [mr − b (x)]2 + r

, (A6)

we can rewrite the differentiation equation (A5) as:

h0 (x) =2 (1− p) f (x)p+ (1− p)F (x) {1− exp [h (x)]} . (A7)

The general solution to this differentiation equation is given by:

h (x) = − ln(1− ω

∙p

p+ (1− p)F (x)¸2)

, (A8)

26

where ω is an integration constant. The specific solution that we focus on is the one thatsatisfies b (0) = 0, which requires that

h (0) =2βmr

2 + r, (A9)

and thus

ω = 1− exp [−h (0)] = 1− expµ−2βmr2 + r

¶. (A10)

We can then solve b (x) from (A8) to obtain (12).13

Now we verify that the second order condition for equilibrium is satisfied. To this end,note that given the candidate equilibrium bidding strategy b (·) as described by (12) andusing formula (A4), we have that for any x > 0,

U(x, x)− U(x, x̃) ∝ [p+ (1− p)F (x̃)]½exp

∙−h (x̃)

2

¸− exp

∙−h (x)

2

¸¾+(1− p)

∙Z x̄

x̃

exp

∙h (x2)

2

¸dF (x2)−

Z x̄

x

exp

∙h (x2)

2

¸dF (x2)

¸= −p

2

Z x̃

x

exp

∙−h (x2)

2

¸h0(x2)dx2

+(1− p)Z x̃

x

exp

∙−h (x2)

2

¸ ∙f (x2)− F (x2) h

0(x2)2

¸dx2

− (1− p)Z x̃

x

exp

∙h (x2)

2

¸f (x2) dx2

= −12

Z x̃

x

[p + (1− p)F (x2)] exp∙−h (x2)

2

¸h0(x2)dx2

− (1− p)Z x̃

x

exp

∙−h (x2)

2

¸exp [h (x2)− 1] f (x2) dx2.

Noticing from the restated first order condition (A7) that

[p+ (1− p)F (x2)]h0 (x2) = 2 (1− p) f (x2) {1− exp [h (x2)]}for all x2, the above expression can be rewritten as

U(x, x)− U(x, x̃) ∝ (1− p)Z x̃

x

exp

∙−h (x2)

2

¸[exp [h (x2)]− 1] f (x2) dx2

− (1− p)Z x̃

x

exp

∙−h (x2)

2

¸exp [h (x2)− 1] f (x2) dx2

= 0.

13Clearly b (x) < mr for all x as long as p > 0.

27

Thus the second order condition is satisfied. Finally, for type-0 bidders, because of ourassumption that x = 0 is observable, we can implement b (0) for type-0 bidders.

Proof of Proposition 3:Proof. Replacing x̃ by x in expression (A4) and using the change of variable (A6), we have

U(x, x) =1 + δ

β− (2 + r) [(1 + r) δK]

12+r

β (1 + r)exp

∙−β (1 + r) (y − x)

2 + r

¸×½

[p+ (1− p)F (x)] exp∙−h (x)

2

¸+ (1− p)

Z x̄

x

exp

∙h (x2)

2

¸dF (x2)

¾.

Plugging the equilibrium bidding function as given implicitly by (A8), we have

exp

∙−h (x)

2

¸=

s1− ω

∙p

p+ (1− p)F (x)¸2.

Thus,

U(x, x) =1 + δ

β− (2 + r) [(1 + r) δK]

12+r

β (1 + r)exp

∙−β (1 + r) (y − x)

2 + r

¸×⎧⎪⎪⎨⎪⎪⎩

q[p+ (1− p)F (x)]2 − ωp2 +

Z x̄

x

d[p+ (1− p)F (x2)]r1− ω

hp

p+(1−p)F (x2)i2⎫⎪⎪⎬⎪⎪⎭

=1 + δ

β− (2 + r) [(1 + r) δK]

12+r

β (1 + r)exp

∙−β (1 + r) (y − x)

2 + r

¸×⎧⎨⎩

q[p+ (1− p)F (x)]2 − ωp2 +

Z x̄

x

d[p+ (1− p)F (x2)]2

2q[p+ (1− p)F (x2)]2 − ωp2

⎫⎬⎭ .Now using the change of variable z ≡ [p + (1 − p)F (x2)]2, the above expression can berewritten as:

28

1 + δ

β− (2 + r) [(1 + r) δK]

12+r

β (1 + r)exp

∙−β (1 + r) (y − x)

2 + r

¸×(q

[p+ (1− p)F (x)]2 − ωp2 +

Z 1

[p+(1−p)F (x)]2dz

2pz − ωp2

)(A11)

=1 + δ

β− (2 + r) [(1 + r) δK]

12+r

β (1 + r)exp

∙−β (1 + r) (y − x)

2 + r

¸×½q

[p+ (1− p)F (x)]2 − ωp2 +pz − ωp2

¯̄̄1[p+(1−p)F (x)]2

¾=

1 + δ

β− (2 + r) [(1 + r) δK]

12+r

β (1 + r)exp

∙−β (1 + r) (y − x)

2 + r

¸×p1− ωp2.

Using expression for the expected utility for an agent in the absence of Rosca as given by(3), we have that for all x > 0,

∆U(x) = U(x, x)− Vn (y − x; r)=

2 + r

β (1 + r)

2 + r

β (1 + r)exp

∙−β (1 + r) (y − x)

2 + r

¸[(1 + r) δK]

12+r

h1−

p1− ωp2

i(A12)

> 0.

The interim welfare calculation for type-0 bidders are slightly different. In equilibriumthe expected utility for type-0 bidder from participating in a Rosca is given by

U (0, 0) =1

2Pr (X2 = 0) [Vw (b(0); y, r) + Vf (b (0) ; y, r)]

+ [1− Pr(X2 = 0)] E[Vf(b (X2) ; y, r)|X2 > 0]

where the first term reflects the probability of tying when her opponent is also of type0 [occurring with probability Pr (X2 = 0)] in which case she will obtain Vw (b(0); y, r) andVf (b (0) ; y, r) with equal probability; and the second term is the expected payoff in eventswhen her opponent’s type is higher than 0. Using the expressions for Vw and Vf as givenrespectively by (7) and (11), and using the fact that b (0) = 0 in equilibrium, we have:

U (0, 0) =1 + δ

β− 2 + r

β (1 + r)[(1 + r) δK]

12+r exp

∙−β (1 + r) y

2 + r

¸×½

p

2

∙exp

µ−h (0)

2

¶+ exp

µh (0)

2

¶¸+ (1− p)

Z x̄

0

exp

∙h (x2)

2

¸dF (x2)

¾.

29

Thus

∆U (0) = U (0, 0)− Vn (y; r)= − 2 + r

β (1 + r)[(1 + r) δK]

12+r exp

∙−β (1 + r) y

2 + r

¸×½

p

2

∙exp

µ−h (0)

2

¶+ exp

µh (0)

2

¶¸(A13)

+(1− p)Z x̄

0

exp

∙h (x2)

2

¸dF (x2)− 1

¾.

Since exp (z) + exp (−z) is a convex function with a minimum of 2 achieved at z = 0, wehave:

exp

µ−h (0)

2

¶+ exp

µh (0)

2

¶> 2;

moreover because h (x2) > 0 for all x2, we haveZ x̄

0

exp

∙h (x2)

2

¸dF (x2) > 1.

Thus

∆U (0) < − 2 + r

β (1 + r)[(1 + r) δK]

12+r exp

∙−β (1 + r) y

2 + r

¸× [p+ (1− p)− 1] = 0.

Proof of Proposition 4:Proof. (Claim 1:) To show Claim (1), note that with expressions for ∆U (0) and ∆U (x)

as in (A13) and (A12), we have:

∆W (m) = p∆U (0) + (1− p)E [∆U(X)|X > 0]

∝ (1− p)h1−

p1− ωp2

iE

½exp

∙β (1 + r)

2 + rX

¸¯̄̄̄X > 0

¾−p½p

2

∙exp

µ−h (0)

2

¶+ exp

µh (0)

2

¶¸+ (1− p)

Z x̄

0

exp

∙h (x2)

2

¸dF (x2)− 1

¾Using the same change of variable as in the derivation of U (x, x) as shown in (A11), wehave:

(1− p)Z x̄

0

exp

∙h (x2)

2

¸dF (x2) =

p1− ωp2 − p exp

∙−h (0)

2

¸.

30

Together with the definition of µ as defined in (13), we can write

∆W (m) ∝ (1− p)h1−

p1− ωp2

iµ

−p½p

2

∙exp

µ−h (0)

2

¶+ exp

µh (0)

2

¶¸+

∙p1− ωp2 − p exp

µ−h (0)

2

¶¸− 1¾

=

Π(m;p,µ)z }| {[(1− p)µ+ p]

h1−

p1− ωp2

i− p

2

2

∙exp

µh (0)

2

¶− exp

µ−h (0)

2

¶¸where it is important to note that the proportionality is independent of m. Note that maffects ∆W (m) only through its effect on h (0) and ω which we recall from (A9) and (A10)are both functions of m.Now write z ≡ exp [−h (0) /2] , using the fact that ω = 1− exp [−h (0)] = 1− z2, we have

G (z; p, µ) = [(1− p)µ+ p]h1−

p1− (1− z2) p2

i− p

2

2

¡z−1 − z¢

and

Π (m; p, µ) = G

µexp

µ−h (0)

2

¶; p, µ

¶= G

µexp

µ− βmr

2 + r

¶; p, µ

¶. (A14)

Now,

∂G

∂z= − p2zp

1− (1− z2) p2 [(1− p)µ+ p]−p2

2

µ− 1z2− 1¶

= p2

"1

2

µ1

z2+ 1

¶− [(1− p)µ+ p] zp

1− (1− z2) p2

#(A15)

∂2G

∂z2= p2

"− 1z3− [(1− p)µ+ p]

p1− (1− z2) p2

1− (1− z2) p2 +p2 [(1− p)µ+ p] z2[1− (1− z2) p2] 32

#

= −p2(1

z3+[(1− p)µ+ p] (1− p2)[1− (1− z2) p2] 32

)< 0. (A16)

Because of the relationship between G (·) and Π (·) as described by (A14), we have

∂2Π

∂m2=

∂2G

∂z2

µ∂z

∂m

¶2+

∂G

∂z

∂2z

∂m2.

31

After plugging in the expressions for ∂2G/∂z2 and ∂G/∂z, we obtain

∂2Π

∂m2= −p2

(1

z3+[(1− p)µ+ p] (1− p2)[1− (1− z2) p2] 32

)z2µ

βr

2 + r

¶2+p2

"1

2

µ1

z2+ 1

¶− [(1− p)µ+ p] zp

1− (1− z2) p2

#z

µβr

2 + r

¶2= p2

µβr

2 + r

¶2z

(− [(1− p)µ+ p] zp

1− (1− z2) p2∙1 +

1− p21− (1− z2) p2

¸− 12

µ1

z2− 1¶)

< 0,

where the last inequality follows from the fact that z ≡ exp [−h (0) /2] ≤ 0.Because ∆W (m) is proportional to Π (m; p, µ) and the proportionality does not depend

on m, we conclude that ∆W (m) is concave in m.(Claim 2:) Since we have established that ∆W is concave in m and moreover,

∆W

∂m∝ ∂Π

∂m=

∂G

∂z

∂z

∂m

= −p2"1

2

µ1

z2+ 1

¶− [(1− p)µ+ p] zp

1− (1− z2) p2

#βrz

2 + r.

Note that, if m is sufficiently close to 0, z ≡ exp [−βmr/ (2 + r)] will be sufficiently close to1. Because

limz→1

(−p2

"1

2

µ1

z2+ 1

¶− [(1− p)µ+ p] zp

1− (1− z2) p2

#βrz

2 + r

)= −p2 {1− [(1− p)µ+ p]} βr

2 + r> 0,

where the inequality follows from µ > 1, we conclude that ∂∆W (m) /∂m > 0 when mis close to zero. This, together with the result we established in Claim 1 that ∆W (m) isconcave in m, implies that there exists a unique m∗ that maximizes ∆W (m) .

Moreover, the optimal Rosca size m∗ is determined by the solution of z∗ in the first ordercondition

1

2

µ1

z2+ 1

¶− [(1− p)µ+ p] zp

1− (1− z2) p2 = 0 (A17)

where

m∗ = −(2 + r) ln z∗

βr. (A18)

(Claim 3:) Since m∗ satisfies the first order condition

∂∆W (m∗)∂m

= 0,

32

the implicit function theorem gives us:

∂m∗

∂µ= −∂2∆W/∂m∂µ

∂2∆W/∂m2.

Because ∂2∆W/∂m2 < 0 as shown in Claim 1, we have:

sign

µ∂m∗

∂µ

¶= sign

µ∂2∆W

∂m∂µ

¶= sign

µ∂2Π

∂m∂µ

¶= −sign

µ∂2G

∂z∂µ

¶> 0

where the inequality follows from the fact that, taking derivative of ∂G/∂z as given by (A15)with respect to µ gives us

∂2G

∂z∂µ= − p2 (1− p) zp

1− (1− z2) p2 < 0.

(Claim 4:) Analogous to the proof of Claim 3, we have:

sign

µ∂m∗

∂p

¶= sign

µ∂2∆W

∂m∂p

¶= sign

µ∂2Π

∂m∂p

¶= −sign

µ∂2G

∂z∂p

¶.

Note that:

∂2G

∂z∂p= 2p

Term Az }| {"1

2

µ1

z2+ 1

¶− [(1− p)µ+ p] zp

1− (1− z2) p2

#− p2 (1− µ) zp

1− (1− z2) p2 −[(1− p)µ+ p] (1− z2) p3z

[1− (1− z2) p2] 32

= − p2 (1− µ) zp1− (1− z2) p2 −

[(1− p)µ+ p] (1− z2) p3z[1− (1− z2) p2] 32

= − p2z

[1− (1− z2) p2] 32©(1− µ) £1− ¡1− z2¢ p2¤+ [(1− p)µ+ p] ¡1− z2¢ pª

= − p2z

[1− (1− z2) p2] 32©1− £1− p ¡1− z2¢¤µª (A19)

where the second equality uses the fact that Term A is equal to zero because of (A17). Nowwe show that ∂2G/∂z∂p|z=z∗ > 0 by proving that 1 − [1− p (1− z∗2)]µ < 0. To see this,note again, the first order condition (A17) which characterizes z∗ implies that:

[(1− p)µ+ p] z∗p1− (1− z∗2) p2 =

1

2

µ1

z∗2+ 1

¶≥ 1

33

since z∗ = exp (−h (0)) ≤ 1. Thus,

z∗2 ≥ 1 + p

[(1− p)µ+ 2p]µ.

Hence,

1− £1− p ¡1− z∗2¢¤µ≤ 1− µ+

Term Bz }| {pµ− (1 + p) p

(1− p)µ+ 2p.

Consider Term B above. Its derivative with respect to p yields:

∂

∂p

∙pµ− (1 + p) p

(1− p)µ+ 2p¸

= µ− (1 + 2p) [(1− p)µ+ 2p]− (1 + p) p (2− µ)[(1− p)µ+ 2p]2

= µ− µ (1 + 2p− p2) + 2p2

[(1− p)µ+ 2p]2

=

Term Cz }| {(µ− 1)2 £(µ− 2) p2 − 2µp¤− µ+ µ3

[(1− p)µ+ 2p]2 .

Focus on the the numerator of the above expression, denoted as Term C. If µ ≤ 2, Term C isobviously strictly decreasing in p, hence it achieves a minimum at p = 1. If µ > 2, Term C isthen a strictly convex function of p. If p were unconstrained, it would have reached minimumat p = µ/ (µ− 2) which is larger than 1. Since p ∈ (0, 1) , Term C also reaches its minion atthe boundary p = 1. Thus regardless of the value of µ, Term C reaches its minimum whenp = 1, hence

(µ− 1)2 £(µ− 2) p2 − 2µp¤− µ+ µ3≥ (µ− 1)2 [(µ− 2)− 2µ]− µ+ µ3= 2 (µ− 1) > 0.

Thus, for all µ,∂

∂p

∙pµ− (1 + p) p

(1− p)µ+ 2p¸> 0.

But, ∙pµ− (1 + p) p

(1− p)µ+ 2p¸¯̄̄̄p=1

= µ− 1,

34

hence1− £1− p ¡1− z∗2¢¤µ < 1− µ+ (µ− 1) = 0.

Thus, for all µ, we have shown that∂m∗

∂p< 0.

Proof of Proposition 5:Proof. (Claim 1:) From the proof of Proposition 4, we know that

∆W (m) ∝ Π (m; p, µ)

= [(1− p)µ+ p]h1−

p1− ωp2

i− p

2

2

∙exp

µh (0)

2

¶− exp

µ−h (0)

2

¶¸Thus

∆W (m∗) ≥ ∆W (0) ∝ Π (0; p, µ) = 0.

Moreover, we know that limp→1m∗ = 0 and, for any p ∈ (0, 1) , ∂m∗/∂p < 0 [Claim 4 inProposition 4], thus m∗ > 0 for all p ∈ (0, 1) . Thus for interior values of p, ∆W (m∗) > 0.(Claim 2:) Because ∆W (m) ∝ Π (m; p, µ) and the proportionality does not depend

on µ, it suffices to show that dΠ (m∗; p, µ) /dµ > 0. To see this, note that by the EnvelopeTheorem,

dΠ(m∗; p, µ)dµ

=∂Π(m∗; p, µ)

∂µ= (1− p)

h1−

p1− ωp2

i> 0.

(Claim 3:) To see that ∆W (m∗) is non-monotonic in p, note that at as p → 0, z∗

converges to a finite number; and as p→ 1, z∗ → 1. Thus,

limp&0

Π (m∗; p, µ) = limp%1

Π (m∗; p, µ) = 0.

Because ∆W (m∗) is proportional to Π (m∗; p, µ) , we thus know that

limp&0

∆W (m∗) = limp%1

∆W (m∗) = 0.

Together with the fact ∆W (m∗) is strictly positive for p ∈ (0, 1) , we conclude that∆W (m∗)must be non-monotonic in p.We now show that in the (right) vicinity of p = 0, ∆W (m∗) is increasing in p; but

in the (left) vicinity of p = 1, ∆W (m∗) is decreasing in p. First note again that because∆W (m) ∝ Π (m; p, µ) and the proportionality does not depend on p, it suffices to examine

35

dΠ (m∗; p, µ) /dp. Note that by the Envelope Theorem,

dΠ(m∗; p, µ)dp

=∂Π(m∗; p, µ)

∂p

= (1− µ)h1−

p1− (1− z∗2) p2

i+[(1− p)µ+ p] p (1− z∗2)p

1− (1− z∗2) p2 − p ¡z∗−1 − z∗¢= (1− µ)

h1−

p1− (1− z∗2) p2

i+

1

2z∗

µ1

z∗2+ 1

¶p¡1− z∗2¢− p ¡z∗−1 − z∗¢

=

Term Dz }| {(1− µ)

h1−

p1− (1− z∗2) p2

i+(1− z∗2)2 p

2z∗3

where the third equality follows from the first order condition (A17) which characterizes z∗.Apparently, we have

limp&0

dΠ(m∗; p, µ)dp

= limp%1

dΠ(m∗; p, µ)dp

= 0

because as we showed earlier, when p→ 0, z∗ is finite and when p→ 1, z∗ → 1.

Taking the derivative of Term D above with respect to p, we obtain:

d2Π(m∗; p, µ)dp2

= (1− µ) p(1− z∗2)− pz∗ (∂z∗/∂p)p1− (1− z∗2) p2 − (1− z

∗2) (z∗2 + 3) p2z∗4

∂z∗

∂p+(1− z∗2)22z∗3

.

Since as p→ 0, z∗ is finite, we have:

limp&0

d2Π(m∗; p, µ)dp2

=(1− z∗2)22z∗3

> 0.

Thus in the vicinity of p = 0, Π (m∗; p, µ) must be increasing in p. Similarly, since as p→ 1,

z∗ → 1, we have:

limp%1

d2Π(m∗; p, µ)dp2

= (µ− 1) ∂z∗

∂p> 0, (A20)

where the inequality follows from the fact that µ > 1, and ∂z∗/∂p > 0 (due to Claim 3 inProposition 4 that ∂m∗/∂p < 0). Because we already know that limp%1 dΠ(m∗; p, µ)/dp = 0,(A20) must imply that in the left vicinity of 1, Π (m∗; p, µ) is decreasing in p.

Proof of Proposition 6:Proof. We know that the welfare gains ∆W (m) is proportional to Π (m; p, µ) where

Π (m; p, µ) = [(1− p)µ+ p]h1−

p1− (1− z2) p2

i−p

2

2

¡z−1 − z¢ with z = expµ− βmr

2 + r

¶.

36

Thus, for a given m > 0,

limp→0

Π (m; 0, µ) = 0, (A21)

limp→1

Π (m; 1, µ) = 1− 12

¡z + z−1

¢< 0. (A22)

Now Claim 1 follows from (A22) by continuity, i.e., there exists values of p sufficiently closeto 1 such that Π (m; p, µ) < 0.To prove Claim 1, we show that the derivative of Π (m; p, µ) with respect to p is positive

at the vicinity of p = 0. To see this, note:

limp→0

∂Π(m; p, µ)

∂p= (1− µ)

h1−

p1− (1− z2) p2

i+[(1− p)µ+ p] p (1− z2)p

1− (1− z2) p2 − p ¡z−1 − z¢¯̄̄̄¯p=0

= 0,

Note that

limp→0

∂2Π(m; p, µ)

∂p2=

[3 (1− µ) p+ µ] (1− z2)p1− (1− z2) p2 − ¡z−1 − z¢+ [(1− p)µ+ p] p2 (1− z2)2

1− (1− z2) p2¯̄̄̄¯p=0

=¡µ− z−1¢ ¡1− z2¢ ,

thus

limp→0

∂2Π(m; p, µ)

∂p2> 0 if µ > z−1 = exp

µβmr

2 + r

¶.

Hence, for any m > 0, if µ > exp [βmr/ (2 + r)] , ∂Π (m; p, µ) /∂p > 0 in the vicinity ofp = 0. Therefore when p ∈ (0, 1) is sufficiently small, Π (m; p, µ) is strictly positive.

Proof of Lemma 1:Proof. We will prove the lemma by induction. First consider period T − 1. The problem inperiod-(T − 1) is identical to the two-period problem we considered in Section 3.2, thus wecan appeal to Proposition 1 and obtain

VT−1 (wT−1) =1 + δ

β− 2 + r

β (1 + r)exp

∙−β (1 + r)

2 + rwT−1

¸{(1 + r) δ [E exp (−βYT )]}

12+r .

(A23)Exploiting the shorthand notation of ρ = 1/ (1 + r) , VT−1 (wT−1) can be written as

VT−1 (wT−1) =1 + δ

β− 1 + ρ

βQT−1 exp

∙− β

1 + ρwT−1

¸(A24)

37

where

QT−1 =µδKT−1

ρ

¶ ρ1+ρ

with KT−1 = Eexp (−βYT ) , (A25)

which indeed conforms to the form of (15) for t = T − 1.Suppose that (15) is true for all t = s+ 1, where s+ 1 ≤ T − 1. That is,

Vs+1(ws+1) =

PT−s−1t=0 δt

β−PT−s−1

t=0 ρt

βQs+1 exp

"− βPT−s−1

t=0 ρtws+1

#Consider period s.Given Vs+1 (·) , the first order condition for the optimization problem of period s can be

written as:

exp [−βcs] = δPT−s−1

t=0 ρt

βQs+1E

(exp

"− βPT−s−1

t=0 ρtWs+1

#βPT−s−1

t=0 ρt∂Ws+1

∂cs

)

=δ

ρQs+1Eexp

"− βPT−s−1

t=0 ρt

µws − cs

ρ+ Ys+1

¶#(A26)

Collecting terms involving cs, we obtain:

exp

"−βcs

Ã1 +

1

ρPT−s−1

t=0 ρt

!#=

δ

ρQs+1 exp

"− βws

ρPT−s−1

t=0 ρt

# ≡Ksz }| {Eexp

"− βPT−s−1

t=0 ρtYs+1

#This is,

exp (−βcs) =µδ

ρQs+1Ks

¶ρ T−s−1t=0 ρt

T−st=0 ρt

exp

"− βwsPT−s

t=0 ρt

#= Qs exp

"− βwsPT−s

t=0 ρt

#. (A27)

Claim: We now show that the solution of Qs, s = 1, ..., T − 1 implied by the differenceequation:

Qs ≡µδ

ρQs+1Ks

¶ ρ T−s−1t=0 ρt

T−st=0 ρt

, (A28)

with

QT−1 =µδKT−1

ρ

¶ ρ1+ρ

. (A29)

will take the form as give by (16). First, note that for i = T − 1, the expression (16) is

38

equivalent to (A29). Now, taking logs on both sides of (A28), we obtain:

lnQs =ρPT−s−1

t=0 ρtPT−st=0 ρt

∙lnQs+1 + lnKs + ln

µδ

ρ

¶¸=

ρ¡1− ρT−s

¢1− ρT−s+1

∙lnQs+1 + lnKs + ln

µδ

ρ

¶¸=

ρ¡1− ρT−s

¢1− ρT−s+1

(ρ¡1− ρT−s−1

¢1− ρT−s

µlnQs+2 + lnKs+1 + ln

δ

ρ

¶+

µlnKs + ln

δ

ρ

¶)

=ρT−s(1− ρ)

1− ρT−s+1lnQT +

ρ− ρT−s+1

1− ρT−s+1

µlnKs + ln

δ

ρ

¶+

ρ2 − ρT−s+1

1− ρT−s+1

µlnKs+1 + ln

δ

ρ

¶+...+

ρT−s − ρT−s+1

1− ρT−s+1

µlnKT−1 + ln

δ

ρ

¶=

T−sXj=1

ρj − ρT−s+1

1− ρT−s+1

µlnKs+j−1 + ln

δ

ρ

¶.

Now we characterize the expression for Vs (ws) . Note that

Vs(ws) =1− exp (−βcs)

β+ δEVs+1 (Ws+1)

=1− exp (−βcs)

β+ δ

½1− δT−s

β(1− δ)− (1− ρT−s)

β(1− ρ)Qs+1Eexp

∙−β (1− ρ)

1− ρT−sWs+1

¸¾=

1− exp (−βcs)β

+ δ

½1− δT−s

β(1− δ)− (1− ρT−s)

β(1− ρ)Qs+1Eexp

∙−β (1− ρ)

1− ρT−s

µws − cs

ρ+ Ys+1

¶¸¾Using (A26) to substitute the term Qs+1Eexp

h−β (1− ρ)

³ws−cs

ρ+ Ys+1

´/¡1− ρT−s

¢i, we

obtain:

Vs(ws) =1− δT−s+1

β(1− δ)− exp (−βcs)

β

"1 +

ρ¡1− ρT−s

¢1− ρ

#

=1− δT−s+1

β(1− δ)− 1− ρT−s+1

1− ρQs exp

"− βwsPT−s

t=0 ρt

#.

Now Claim 2 follows from taking logs on both sides of (A27). To verify Claim 3, note

39

that w1 = y1, for all s ≥ 1, we have the following recursion:

ws+1 = ys +ws − c∗s

ρ

= ys +1

ρ

∙1− 1− ρ

1− ρT−s+1

¸"¡1− ρT−s+1

¢ sXj=1

yj1− ρT−j+1

+1− ρT−s+1

βρ

s−1Xj=1

lnQj1− ρT−j

#+lnQsβρ

= ys +¡1− ρT−s

¢ sXj=1

yj1− ρT−j+1

+1

βρ

"lnQs +

¡1− ρT−s

¢ s−1Xj=1

lnQj1− ρT−j

#

=¡1− ρT−s

¢ s+1Xj=1

yj1− ρT−j+1

+1− ρT−s

βρ

sXj=1

lnQj1− ρT−j

.

Proof of Proposition 7:[To Be Completed]

Proof of Proposition 8:Proof. (Claim 1) From the characterization of second period equilibrium bidding functionb2

³·; b̂1

´described in (27), we know that b2

³·; b̂1

´increases in the cumulative income shock

χ2. Note that, after some simplification, we have

b2(χ̄2; b̂1) =

³3m+ b̂1

´r

2+1 + ρ

2βρln

∙1− ω2

³b̂1´G2³χ2

´2¸

=

³3m+ b̂1

´r

2− 1

2ρ

h(2 + ρ)mr − (1 + ρ) b̂1

i= b̂1 + r

³b̂1 −mr

´.

Thus, if b̂1 ≤ mr, then b2³χ̄2; b̂1

´≤ b̂1. Claim 1 follows.

(Claim 2) If b̂1 > mr, then we know that b2(χ̄2; b̂1) > b̂1. Plugging the expression of

ω2

³b̂1

´G2

³χ2

´2as given by (28) into the expression of b2

³·; b̂1

´as described by (27), and

solving the equation b2³χ2; b̂1

´= b̂1, we have that the threshold level χ∗2 must satisfy

G2 (χ∗2) =

vuuut1− expn− β1+ρ

h(2 + ρ)mr − (1 + ρ) b̂1

io1− exp

n− β1+ρ

ρh³3m+ b̂1

´r − 2b̂1

io . (A30)

40

Because ½− β

1 + ρ

h(2 + ρ)mr − (1 + ρ) b̂1

i¾−½− β

1 + ρρh³3m+ b̂1

´r − 2b̂1

i¾=

β

1 + ρ

nρh³3m+ b̂1

´r − 2b̂1

i−h(2 + ρ)mr − (1 + ρ) b̂1

io=

β

1 + ρ

n2 (ρ− 1)mr + (1− ρ+ ρr) b̂1

o= − β

1 + ρ

2r

1 + r

³mr − b̂1

´> 0,

we know that G2 (χ∗2) as expressed in (A30) is strictly less than 1. Claim 2 then follows fromthe fact that b2(χ2; b̂1) is monotonically increasing in χ2.

Proof of Proposition 9:

Proof. To calculate Ehb2

³χ̂2; b̂1

´¯̄̄b̂1, χ̂2 is the period 2 winner

i, we first calculate

Ehh2

³χ̂2; b̂1

´¯̄̄b̂1, χ̂2 is the period 2 winner

ibecause the former is just a liner function of the later. Note:

Ehh2³χ̂2; b̂1

´¯̄̄b̂1, χ̂2 is the period 2 winner

i=

Z χ̄2

χ2

− ln

⎡⎢⎣1− ω2

³b̂1

´⎛⎝G2³χ2

´G2 (χ2)

⎞⎠2⎤⎥⎦ dG2 (χ2)2

where the term G2 (χ2)2 appears because this is the CDF of the first order statistic of χ2 for

the remaining two bidders. Using integration by parts and a few simplifications, the aboveexpectation can be written as:

41

Ehh2³χ̂2; b̂1

´¯̄̄b̂1, χ̂2 is the period 2 winner

i= −

⎧⎪⎨⎪⎩G2 (χ2)2 ln⎡⎢⎣1− ω2

³b̂1

´⎛⎝G2³χ2

´G2 (χ2)

⎞⎠2⎤⎥⎦⎫⎪⎬⎪⎭¯̄̄̄¯̄̄χ̄2

χ2

+ω2

³b̂1

´³G2

³χ2

´´2 ∙ln

µG2 (χ2)

2 − ω2

³b̂1

´³G2

³χ2

´´2¶¸¯̄̄̄χ̄2χ2

= − ln∙1− ω2

³b̂1´G2³χ2

´2¸+G2

³χ2

´2lnh1− ω2

³b̂1´i

+ω2³b̂1´³G2³χ2

´´2×∙ln

µ1− ω2

³b̂1

´³G2

³χ2

´´2¶− ln

³1− ω2

³b̂1

´´− lnG2

³χ2

´2¸= −

∙1− ω2

³b̂1

´³G2

³χ2

´´2¸ln

µ1− ω2

³b̂1

´³G2

³χ2

´´2¶+G2

³χ2

´2 h1− ω2

³b̂1

´iln³1− ω2

³b̂1

´´−ω2

³b̂1

´³G2

³χ2

´´2lnG2

³χ2

´2. (A31)

Note that limp→0G2³χ2

´2= 0. Thus, expression (A31) implies that

limp→0

Ehh2³χ̂2; b̂1

´¯̄̄b̂1, χ̂2 is the period 2 winner

i= 0.

Thus,

limp→0

Ehb2³χ̂2; b̂1

´¯̄̄b̂1, χ̂2 is the period 2 winner

i=

³3m+ b̂1

´r

2

Moreover, (25) implies that,

limp→0

b1 (x1) =(2 + ρ)mr

1 + ρfor all x1 > 0.

Because ³3m+ b̂1

´r

2≥ b̂1

holds for all r ≥ 0 with the inequality being strict when r > 0, we have

limp→0

Ehb2³χ̂2; b̂1

´¯̄̄b̂1, χ̂2 is the period 2 winner

i≥ b̂1.

42

Similarly, when p→ 1, we have

limp→1

b̂1 = b1 = ρmr2,

limp→1

G2³χ2

´2= 1,

limp→1

ω2³b̂1´= 1− exp

µ−1 + ρ+ ρ2

1 + ρβρmr

¶.

Thus,limp→1

Ehh2

³χ̂2; b̂1

´¯̄̄b̂1, χ̂2 is the period 2 winner

i= 0,

and hence,

limp→1

Ehb2³χ̂2; b̂1

´¯̄̄b̂1, χ̂2 is the period 2 winner

i=

³3m+ b̂1

´r

2

> limp→1

b̂1 = ρmr2.

43

Max Min Mean Median St. Dev.

Bid (Yuan) 790 8.6 135 76.9 140

Size (Yuan) 2000 50 524 300 542

Total Amount — n*Size (Yuan) 28000 500 6965 5000 6379

Number of Meetings 28 10 14.6 13 5.4

Time Length (Month) 42 10 24.7 24 7.0

Time Intervals b/w Meetings (Month) 3 1 1.8 2 0.74

Table 1: Summary Statistics.

44

Dependent Variable: Log bid

(1) (2) (3) (4) (5) (6)

Round (t)-.0017

(-0.68)

.0262

(1.60)

.0164*

(1.71)

-0064***

(-2.45)

.0252*

(1.66)

.00874**

(2.54)

Round squared (t2)-.0040**

(-2.32)

-.0042**

(-2.43)

-.0037**

(-2.34)

-.0043***

(-2.70)

Round cubic (t3).0001***

(2.87)

.0001***

(2.98)

.0001**

(2.48)

.0001***

(2.77)

Log (Size).9849***

(76.79)

.9833***

(76.78)

.9832***

(76.84)

1.026***

(76.50)

1.024***

(76.61)

1.066***

(40.75)

Round×Size (t×size) -.0067**

(-2.04)

Spring festival dummy-0.0564

(-1.45)

-.0472

(1.35)

January dummy-.0311

(-.69)

-.0166

(-.40)

Constant-3.855***

(-33.54)

-3.889***

(-32.73)

-3.887***

(-32.74)

-4.354***

(-31.10)

-4.393***

(-30.68)

4.398***

(-30.69)

Year dummies No No No Yes Yes Yes

R-squared .8499 .8509 .8511 .8787 .8791 .8798

Sample size 1160 1160 1160 1160 1160 1160

Note: The numbers in parentheses are t-statistics based on robust standard errors. *, **, and *** denote significance levels of

10%, 5% and 1% respectively.

Table 2: The trend of winning bids over rounds: OLS Regressions.

45

Dependent variable: Log bid

(1) (2) (3) (4) (5) (6)

Round-.0022

(-1.62)

.0183**

(2.04)

.0191**

(2.12)

-.0019

(-0.80)

.0209**

(2.18)

-.0230**

(2.35)

Round squared—.0021**

(-2.11)

-.0022**

(-2.16)

-.0010*

(-1.80)

-0.020*

(-1.91)

Round cubic5.73e-05*

(1.83)

5.80e-05*

(1.85)

4.85e-05

(1.50)

5.12e-5

(1.58)

Spring festival dummy-.0174

(-0.81)

-.346

(-1.46)

January dummy-.0167

(-.71)

Constant4.441***

(383.4)

4.396***

(200.67)

4.397***

(200.67)

4.463***

(37.93)

4.408***

(36.41)

4.391***

(34.37)

Year dummies No No No Yes Yes Yes

R-squared .0264 .0316 .0275 .0249 .0238 .0256

Sample size 1160 1160 1160 1160 1160 1160

Note: The numbers in parentheses are t-statistics based on robust standard errors. *, **, and *** denote significance levels of

10%, 5% and 1% respectively.

Table 3: The trend of winning bids over rounds: Fixed Effect Regressions.

46

-

Agents Decidewhether toForm Rosca

Income ShocksRealized

BiddingOptimal Savingor Borrowing

- Next Period

Figure 1: The Time Line of the Model

0

0.2

0.4

0.6

0.8

1

p

5

10

15

20

µ

0

200

400

600

800

m∗

0

0.2

0.4

0.6

0.8

1

p

Figure 2: Optimal Rosca size m∗ increases in µ and decreases in p : β = .15, r = .02.

47

0.2 0.4 0.6 0.8 1p

0.1

0.2

0.3

0.4

0.5

ΠHm∗;p,µL

Figure 3: The welfare gains under the optimal Rosca size is proportional to Π (m∗; p, µ) .

0.2 0.4 0.6 0.8 1p

-0.4

-0.2

0.2

0.4

ΠHm=600;p,µL

Figure 4: Rosca with exogenous size may lead to welfare losses when p is close to 1 : m = 600.

48

10 20 30 40 50x

2

4

6

8

10

12

bHxL

Figure 5: Equilibrium Bidding Function: p = 0.2,β = 0.15, r = 0.02,m∗ = 628.

49