Model Uncertainty, Limited Market Participation

and Asset Prices

H. Henry Cao, Tan Wang and Harold H. Zhang∗

This Version: July 14, 2003

∗H. Henry Cao and Harold H. Zhang are with the Kenan-Flagler Business School, Universityof North Carolina, Chapel Hill, NC 27599-3490, USA, caoh@kenan-flagler.unc.edu, (919) 962-5913,zhangha@kenan-flagler.unc.edu, (919) 843-8340; and Tan Wang is with the Faculty of Commerce and Busi-ness Administration, University of British Columbia, Vancouver, British Columbia, Canada, V6T 1Z2,tan.wang@commerce.ubc.ca, (604) 822-9414. The authors are grateful to seminar participants at the eco-nomics department of the University of North Carolina at Chapel Hill, the finance group of Cheung-KongGraduate School of Business, Cornell University, Duke University, Hong Kong University of Science andTechnology, National University of Singapore, New York University, Rice University, University of Arizona,University of California at Irvine, University of California at Los Angeles, University of California at River-side, University of Georgia at Athens, University of Minnesota, University of North Carolina at Chapel Hill,University of North Carolina at Charlotte, and University of Texas at Austin, for helpful comments. TanWang thanks the Social Sciences and Humanities Research Council of Canada for financial support.

Model Uncertainty, Limited Market Participation

and Asset Prices

Abstract

We demonstrate that limited participation can arise endogenously in the presence

of model uncertainty and heterogeneous uncertainty averse investors. Our model gen-

erates novel predictions on how limited participation relates to equity premium and

diversification discount. When the dispersion in investors’ model uncertainty is small,

full participation prevails in equilibrium. In this case, equity premium is unrelated to

uncertainty dispersion and a conglomerate trades at a price equal to the sum of its

single segment counterparts. When uncertainty dispersion is large, however, investors

with relatively high uncertainty optimally choose to stay sidelined in equilibrium. In

this case, equity premium can decrease with uncertainty dispersion. Our results sug-

gest that the understanding in the existing literature that limited participation leads

to higher equity premium is not robust. Moreover, when limited participation occurs,

a conglomerate trades at a discount relative to its single segment counterparts. The

discount increases in uncertainty dispersion and is positively related to the proportion

of investors not participating in the markets.

1 Introduction

It has been well documented that a significant proportion of the U.S. households does not

participate in the stock markets. For example, based on the 1984 Panel Study of Income

Dynamics (PSID) data, Mankiw and Zeldes (1991) find that among a sample of 2998 families,

only 27.6% of the households own stocks. For families with liquid assets of $100,000 or

more, only 47.7% own stocks. More recent surveys show that even during the 1990s with

the tremendous growth of the U.S. stock markets, such limited market participation still

exists. For instance, the 1998 Survey of Consumer Finances shows that less than 50% of

U.S. households own stocks and/or stock mutual funds (including holdings in their retirement

accounts).

The purpose of this paper is to examine how limited participation may arise in an

equilibrium and its implications for equilibrium asset prices. Specifically, we demonstrate

that (1) limited participation may arise due to investors’ uncertainty about the distribution

of asset payoffs; (2) equity premium could be lower in the economy in which there is limited

participation than in the economy in which there is full participation; and (3) when there

is limited participation, a merged firm is traded at a value lower than the sum of values

of its component firms traded separately, i.e., there can be a diversification (conglomerate)

discount.

Several models have been proposed to explain why limited market participation may

exist. Allen and Gale (1994), Williamson (1994), Vissing-Jorgensen (1999), and Yaron and

Zhang (2000) focus on how entry costs and/or liquidity needs create limited market partici-

pation. However, it is difficult for these models to explain why families with substantial liquid

assets (over $100,000) may not participate in stock markets. Haliassos and Bertaut (1995)

also investigate possible explanations for limited market participation and demonstrate that

risk aversion, heterogeneous beliefs, habit persistence, time-nonseparability, borrowing con-

straints, differential borrowing and lending rates, and minimum-investment requirements are

unable to account for the phenomenon. They argue informally that fixed cost and departure

from expected utility maximization are more promising explanations.1

1In partial equilibrium models, Dow and Werlang (1992) provides an explanation why some investorsmay not participate in the stock market based on uncertainty aversion while Ang, Bekaert and Liu (2002)

1

Our study offers an explanation for limited market participation based on investors’

heterogeneity in model uncertainty. Compared to models based on fixed costs, our model

can also explain why families with substantial liquid assets do not participate in the stock

markets. The theoretical prediction of our model on market participation is consistent with

the empirical evidence documented in Blume and Zeldes (1994) and Haliassos and Bertaut

(1995).

The implications of limited participation on equity premium has been examined in sev-

eral studies. For instance, Mankiw and Zeldes (1991), Vissing-Jorgensen (1999), and Brav,

Constantinides, and Geczy (2002) empirically demonstrate that if the sub-sample of stock-

holders’ observations is used, the standard asset pricing models such as Lucas (1978) and

Breeden (1979) can generate higher equity premium with a reasonable risk aversion coeffi-

cient or can match the observed equity premium using a much lower risk aversion coefficient

than the one used in Mehra and Prescott (1985), thus helps to resolve the equity premium

puzzle. Basak and Cuoco (1998) show theoretically that equity premium increases as more

investors are excluded from the risky asset markets.2 The limited market participation in

these studies, however, is exogenously specified. Moreover, these studies are all based on

standard expected utility maximization models which abstract from model uncertainty. In

our framework, investors face both risk and model uncertainty. The equilibrium equity pre-

mium can be decomposed into the risk premium and the uncertainty premium. When limited

market participation arises endogenously, the reduced market participation could drive un-

certainty premium down more than it drives risk premium up, leading to a lower total equity

premium. Our results suggest that the conventional view that limited participation leads to

higher equity premium is not robust when market participation is endogenous.

In the corporate finance literature, the diversification discount phenomenon has been

documented in a number of studies. For example, Lang and Stulz (1994), Berger and Ofek

(1995), Rajan, Servaes, and Zingales (2000) show that conglomerates exhibit on average

negative excess value of 10-15%.3 As the flip side of the conglomerate discount, Hite and

argue that disappointment aversion can produce similar results.2Shapiro (2002) extends Basak and Cuoco’s model to multiple stocks.3Currently, there is a debate as to how significant diversification discount is. For example, a recent study

by Bevelander (2002) finds that most diversified firms have Tobin’s qs comparable to those of focused firmswhen firm age is explicitly accounted for. Villanoga (2002) documents that diversified firms actually trade

2

Owers (1983), Miles and Rosenfeld (1983) and Schipper and Smith (1983) document a 2-3%

increase in shareholder value around the corporate spin-offs. Rajan, Servaes, and Zingales

(2000) and Scharfstein and Stein (2000) argue that diversification discount can be attributed

to inefficient internal capital markets and/or inefficient resource allocation.4 We show that

diversification discount may arise due to investor heterogeneity in uncertainty about firms’

asset returns and is closely associated with limited participation. Our study establishes

relations among the diversification discount, investor heterogeneity, and limited participation

from an asset pricing perspective.5

We perform our analysis in a tractable single period mean-variance framework with het-

erogeneous agents. The added feature of our study is that, in addition to the risk associated

with asset payoffs, investors are uncertain about the distribution of stock payoffs. That is,

there is model uncertainty. In his seminal work, Knight (1921) argues that risk, a situa-

tion where the probability distribution is known, is fundamentally different from uncertainty

where the distribution is unknown. Such difference is now well recognized in both the aca-

demic literature (Ellsberg (1961)) and the investment industry. As one wall street journalist

put it (Lowenstein (2000)), “there is a key difference between a share of IBM ... and a

pair of dice. With dice, there is risk—you could, after all, roll snake eyes—but there is no

uncertainty, because you know (for certain) the chances of getting a 7 and every other result.

Investing confronts us with both risk and uncertainty. There is a risk that the price of a

share of IBM will fall, and there is uncertainty about how likely it is to do so.”

Our study follows the Knightian approach to model uncertainty which differs from the

more common Bayesian approach. In the Bayesian approach, when facing uncertainty about

model specifications, the investor uses a prior distribution over a set of models.6 Thus

the investor’s preference is represented by a Savage (subjective) expected utility. In the

at a significant premium using the Business Information Tracking Series (BITS) data which covers the wholeU.S. economy at the establishment level.

4Recent studies by Graham, Lemmon and Wolf (2002) and Maksimovic and Phillips (2002) did not findempirical evidence that would suggest inefficient resource allocation across divisions of a conglomerate.

5There are alternative theories about the diversification discount and spin-off premium. For example,Miller (1977) argues informally that short-sale constraint and differences of opinion can explain the diver-sification discount. Naik, Habib and Johnsen (1999) offer an explanation of spin-off premium based onhigher incentive to collect information associated with spin-offs which reduces the risk premium. Gomes andLivdan (2002) develop a dynamic model of firm diversification focusing on the role of diminishing marginalproductivity of technology and synergies of different activities in reducing fixed cost of production.

6Wang (2002) shows that an investor’s portfolio decision is highly sensitive to the choice of prior.

3

Knightian approach to uncertainty,7 while investors may still use von-Neumann-Morgenstern

expected utility when facing risk, they no longer use a single prior when facing uncertainty.

In a seminal paper, Gilboa and and Schmeidler (1989) operationalize Knightian uncertainty

using a multi-prior expected utility framework. In this framework, an uncertainty averse

investor evaluates an investment strategy according to the expected utility with the worst

case probability distribution in the set of priors distributions. Thus, an uncertainty averse

investor will be more conservative when he faces more uncertainty.8

In our model, investors are uncertain about the mean of risky asset payoffs and are het-

erogeneous in their levels of uncertainty.9 The crucial aspect of our framework, pertaining

to the Knightian approach in modeling investors’ imperfect knowledge of the payoff distri-

bution, is that for each uncertainty averse investor there is a region of prices over which the

investor neither buys nor sells short the stocks. In the single stock case, the nonparticipation

region of prices lies between the lowest and the highest expected payoffs of the investor’s

admissible payoff distribution set. For any price in the nonparticipation region, the investor

will not take a long position in the stock, as he uses the lowest expected payoff to evaluate

this strategy. Similarly the investor will not take a short position as he uses the highest

expected payoff in this case. Investors who are more uncertain about the mean payoff of

the stock have larger nonparticipation regions than investors with less uncertainty about the

mean payoff. As a result, when uncertainty dispersion is sufficiently high, in equilibrium,

investors with relatively high uncertainty choose not to participate in the stock market,

giving rise to limited participation.10 Further, more investors choose to stay sidelined as

uncertainty dispersion increases.

7See Dow and Werlang (1992), Epstein and Wang (1994, 1995), Anderson, Hansen and Sargent (1999),Maenhout (2000), Uppal and Wang (2001), Chen and Epstein (2001), Epstein and Miao (2003), Routledgeand Zin (2001), Liu, Pan and Wang (2002), among others.

8For alternative specifications of Knightian uncertainty, see Schmeidler (1989) and Bewley (2002). InSchmeidler (1989), an agent evaluates a choice according to its expected utility using a capacity (Choquetexpected utility with non-additive probability). Bewley (2002) considers incomplete preferences where aconsumption bundle dominates another if and only if its expected utility dominates the other using allprobabilities in a set.

9This modeling strategy of uncertainty has been used in a number of studies including Epstein and Miao(2003), Uppal and Wang (2001), Kogan and Wang (2002), Liu, Pan and Wang (2002) among others. Koganand Wang (2002) is the closest to ours in terms of modeling framework but we consider an equilibrium modelwith heterogeneous agents.

10Models based on Bayesian theory such as Williams (1977) cannot generate limited participation, unlessthere is short-sale constraint or investors who are sidelined believe that the equity premia are zero.

4

The intuition for the implication of limited market participation for equity premium

can be explained as follows. In our model, equity premium is decomposed into the risk

premium and the uncertainty premium. The risk premium corresponds to the equity pre-

mium obtained in standard expected utility models. The uncertainty premium arises due

to the uncertainty that investors have about the distribution of the stock payoff. When the

uncertainty dispersion is high, only investors with low uncertainty participate in the stock

markets.11 As a result, there are two forces affecting equity premium when limited market

participation occurs. On the one hand, since fewer investors hold the stocks, the equilib-

rium risk premium has to be higher to induce those investors to hold the stocks so that the

markets clear. On the other hand, because investors participating in the stock markets have

lower uncertainty, a lower uncertainty premium is sufficient to induce these investors to hold

the stocks. Consequently, the net effect of an increase in uncertainty dispersion on equity

premium depends on the relative dominance of these two opposing forces. We find that the

decrease in the uncertainty premium outweighs the increase in the risk premium, leading

to a lower equity premium than that in the economy with full market participation. Fur-

thermore, equity premium decreases (increases) as model uncertainty becomes more (less)

dispersed and more (less) investors stay sidelined.

The intuition behind the relation between limited participation and diversification dis-

count is also straightforward. When there is limited participation at firm level, the pre-

merger price of an individual firm is determined by investors with less uncertainty for that

firm. They therefore demand a lower uncertainty premium and are willing to pay a higher

price for the firm’s stock. After the merger investors have to buy both firms as a bundle.

An investor who is less uncertain about one firm is not necessarily less uncertain about the

other firm and is only willing to offer a lower price for the combined firm than the sum of

the prices of the two firms traded separately. Thus, when there is limited participation, a

merger results in a discount in the price of the combined firm.12

The rest of the paper is organized as follows. Section 2 presents the model. Section 3

11In standard models, risk aversion alone cannot explain why investors have zero rather than small holdingsin stocks.

12Since we want to focus on the link between limited participation and diversification discount, we haveabstracted from the production decisions of the firms. Gomes and Livdan (2003) show how a diversificationdiscount can be generated in a model with endogenous production and industrial equilibrium. They do notconsider how limited participation is related to diversification discount.

5

describes investors’ portfolio choices and their participation decisions. Section 4 analyzes

how the limited market participation relates to equity premium, while Section 5 extends the

model to allow for two stocks and investigates the relation between limited participation

and the diversification discount. Section 6 discusses the robustness of our results in a more

general setting with richer factor structure for asset payoffs. Section 7 concludes. Proofs

and some technical details are collected in the appendix.

2 The Model

To highlight the intuition, we first consider a market with one stock and one bond. The case

of multiple stocks is considered in Section 5.

2.1 The Setting

We assume a one-period heterogeneous agent economy. Investment decisions are made at

the beginning of the period while consumption takes place at the end of the period. There

are one stock and one bond in the economy. Investors are endowed with shares of the stock.

The per capita supply of the stock is denoted x > 0. The risk-free rate is set at zero. The

stock can be viewed as the market portfolio in a single factor economy in which the market

portfolio is the only factor. The payoff of the stock, denoted r, is normally distributed with

mean µ and variance σ2. We assume that investors have a precise estimate of the variance

σ2, but do not know exactly the mean of the stock payoff r. This is motivated by both the

analytical tractability and the empirical evidence on the strong predictability of the volatility

of stock returns, for instance using ARCH and GARCH models, but the unpredictability of

the level of stock returns. The imperfect knowledge of the stock payoff distribution gives

rise to model uncertainty.

2.2 The Preferences

Each agent in the economy has a CARA utility function u(W ), where W is the end of period

wealth. Due to the lack of perfect knowledge of the probability law of stock payoff, the

6

agent’s preference is represented by a multi-prior expected utility (Gilboa and Schmeidler

(1989))

minQ∈P

EQ[u(W )]

, (1)

where EQ denotes the expectation under the probability measure Q and P is a set of proba-

bility measures. The general nature and the axiomatic foundation of these preferences have

been examined in detail in Gilboa and Schmeidler (1989). The basic intuition behind this

preference is that in the real world, people do not know precisely the probability distribution

of stock payoff. They may have a reference model of the probability distribution, such as

a normal distribution with certain mean and variance, which could be the result of some

econometric analysis. However, they are not completely sure that the reference model is

the right model. For example, there may be uncertainty about the estimate of the mean

stock payoff. As a result, instead of restricting themselves to using distribution P , they

are willing to consider a set P of alternative probability distributions. These investors are

averse to (model) uncertainty. They evaluate the expected utility of a random consumption,

EQ[u(W )], under the worst case scenario in the set P.

The specification of the set P follows Kogan and Wang (2002) closely. In the interest of

focusing on the issues being studied, we provide a brief description of P and its intuition.

More details can be found in Kogan and Wang (2002). Let µ be the point estimate of the

mean stock payoff. One way to quantify the uncertainty is by using the confidence region. For

example, if P is the reference probability distribution., the distribution resulted from econo-

metric analysis, and Q is an alternative measure, then the inequality EP [− ln(dQ/dP )] < η

defines a confidence region according to the log likelihood ratio where η depends on the

investor’s uncertainty on the mean stock payoff.13 By choosing appropriate η, the region can

yield the desired level of confidence, for example, 95%. In other words, if P is defined to be

the set of all Q that satisfies the inequality, then with 95% confidence, we can reject any Q

outside P being the right model or the probability that the right model lies outside P is no

more than 5%.14

13As Kogan and Wang (2002) show, the investor’s aversion to uncertainty can also be embedded in η. Forease of exposition, we assume that all investors in our economy have the same level of uncertainty aversion.Of course, allowing for heterogeneous uncertainty aversion will strengthen our results.

14In the case of multiple risky assets, investors may have different levels of uncertainty on the mean assetpayoff and aversion to the uncertainty.

7

Kogan and Wang (2002) show that for normal distributions with a common known

variance, the confidence region can always be described by a set of quadratic inequalities.

For the specific case of this section, it takes the form of (µ+v) where v denotes the adjustment

to the estimated mean stock payoff and satisfies:

v2σ−2 ≤ φ2, (2)

where φ is a parameter that captures the investor’s uncertainty about the mean stock payoff.

We therefore define the set P as the collection of all normal distributions with mean µ + v

and variance σ2 where v satisfies (2). All investors are assumed to have the same point

estimate µ.15 Note that the higher the φ, the wider the range for the expectation of r. Thus

we call φ the investor’s level of uncertainty on the mean stock payoff.

In our economy, investors have the same risk aversion, but are heterogeneous in their

levels of uncertainty.16 Such heterogeneity can arise for a variety of reasons. For example,

the data collected by the research department of investment banks and asset management

firms are private and of different quality. Different firms are likely to have different data

sets about the same stock, which may lead to differences in estimated payoff distributions.

Different firms also use their own proprietary models to analyze the data. This may also

lead to different estimated payoff distributions.17 We abstract from the details of how such

heterogeneity arises and simply assume that the differences are captured by different φ.

Furthermore, we assume that φ is uniformly distributed among investors on the interval18

S1 = [φ− δ, φ+ δ],

where φ ≥ δ and δ measures the dispersion of the uncertainty among investors. When

φ = δ = 0, there is no model uncertainty and our model collapses to the standard expected

utility model.

It is important to note that the preferences represented in (1) exhibit aversion to uncer-

tainty whereas preferences represented by Savage’s expected utility are uncertainty neutral.15Our results regarding limited participation, equity premium, and diversification discount hold without

assuming the same point estimate for all investors.16In an early version, we allowed for heterogeneity in investors’ risk aversion. Because our results are not

affected, we assume the same risk aversion for all investors for the ease of exposition.17Differences in perceived familarity about the stock market may contribute to different degrees of uncer-

tainty.18In Section 6, we consider more general distribution for uncertainty among investors.

8

The simplest example that highlights the difference is the Ellsberg experiment (Ellsberg

(1961)). It can be readily checked that no matter what prior distribution is used to assess

the probability of drawing a red ball from the urn with unknown proportion of each color

(the second urn), to be consistent with the behavior exhibited in the Savage expected utility,

the predictive probabilities of drawing a red or a white must be equal. This implies that a

decision maker should be indifferent between betting red on the first urn and betting red

on the second urn. In other words, the fact that the probability of drawing a red from the

second urn is unknown does not make any difference in his decision making. Such neutrality

to model uncertainty, however, does not hold for preferences represented by (1). In fact,

decision makers with such preferences will always prefer betting red on the first urn to bet-

ting red on the second urn. This difference between the multi-prior expected utility and the

standard expected utility is critical in understanding why our findings differ from the results

of existing studies.

3 Equilibrium Market Participation

In this section, we first solve for investors’ optimal portfolio choice and then find the closed-

form solution for the equilibrium. We show that when the dispersion in investor’s uncertainty

is high, there is limited market participation in equilibrium.

3.1 Portfolio Choices

Let P denote the price of the stock, the investor’s wealth constraint is

W1i = W0i +Di(r − P ),

where W0i and W1i are investor i’s beginning of period and end of period wealth, respectively,

Di is investor i’s demand for the stock. Under the assumption of normality and the CARA

utility, investor i’s utility maximization problem becomes:

maxDi

minv

Di(µ− v − P ) − γσ2

2D2i ,

9

where γ is the investor’s risk aversion coefficient. The first term of the objective function

represents the expected excess payoff from investing in the stock with adjustment to the

mean due to uncertainty, and the last term represents adjustment due to risk. Solving the

inner minimization, the above problem reduces to

maxDi

Di[µ− sgn(Di)φiσ − P ] − γσ2

2D2i , (3)

where sgn(·) is an indicator function which takes the sign of its argument. The expression,

µ − sgn(Di)φiσ, represents the uncertainty-adjusted expected payoff under the worst case

scenario for an investment strategy of Di shares in the stock. The intuition behind (3) is

the following. If investor i takes a long position (Di > 0 and sgn(Di) = 1), the worst case

scenario for him is that the mean excess payoff is given by µ−φiσ−P . Similarly, if he takes

a short position (Di < 0 and sgn(Di) = −1), the worst case scenario is that the mean excess

payoff is µ+ φiσ − P .

The investor’s optimal holding in the stock is given by

Di =

1γσ2 (µ− φiσ − P ) if µ− P > φiσ,

0 if − φiσ ≤ µ− P ≤ φiσ,1γσ2 (µ+ φiσ − P ) if µ− P < −φiσ.

(4)

It can be seen from (4) that, due to his uncertainty about the mean of the stock payoff,

investor i buys (sells short) the stock only when the equity premium is strictly positive

(negative) in the worst case scenario, v = φiσ (v = −φiσ). When the price P falls in the

region, [µ − φiσ, µ + φiσ], the investor optimally chooses not to participate in the market.

The nonparticipation region for investor i is the interval between the investor’s lowest and

the highest expected payoffs. For any price in the region, investor i will not take a long

position because his uncertainty-adjusted expected payoff under the worse case scenario for

a long position is too low. He will not take a short position either as his uncertainty-adjusted

expected payoff under the worst case scenario for a short position is too high. This is in

sharp contrast to the standard result based on expected utility models that as long as the

equity premium is not zero (µ − P = 0), the investor will hold a nonzero position in the

stock.

10

3.2 Full Participation

In equilibrium, the market clears. The supply of the stock is equal to its aggregate demand.

As will be seen later, in equilibrium, the stock always sells at a discount, i.e., µ − P > 0.

Given this, the expression of the demand function (4) implies that no investor will hold short

positions in the stock. In the full participation equilibrium, all investors participate in the

stock market. The market clearing condition for the stock can thus be written as

x =

∫φi∈S1

(µ− σφi − P )

γσ2

1

2δdφi =

1

γσ2(µ− σφ− P ).

Solving for P , we arrive at the following equilibrium price for the stock,

P = µ− γσ2x− σφ. (5)

As long as there exists model uncertainty (φ > 0), the equilibrium stock price will be less

than the expected stock payoff, i.e., µ− P > 0.

We can rewrite Equation (5) to express the equity premium as follows

µ− P = γσ2x+ σφ. (6)

The first term on the right hand side of Equation (6) is standard. It represents the risk

premium, which is proportional to investors’ risk aversion coefficient, the variance of the stock

payoff, and per capita stock supply. The second term is new. It represents the uncertainty

premium and is proportional to the average level of uncertainty (φ) of all investors about

the mean stock payoff.

The full participation equilibrium prevails when all investors participate in the stock

market. This requires all investors, including investors with the highest uncertainty, to hold

long positions. Equation (4) thus implies

µ− (φ+ δ)σ − P > 0.

Substituting (5) into the expression, we have that in equilibrium all investors participate in

the market if

γσ2x > δσ. (7)

11

This full participation condition indicates that whether all investors participate in the stock

market depends on the dispersion of investors’ model uncertainty. When δ is small, investors

are reasonably homogeneous and all investors will participate. It also depends on the Sharpe

ratio, (γσ2x)/σ. Given the model uncertainty, the higher the Sharpe ratio, the more likely

investors with the highest uncertainty will participate in the stock market.

It is interesting to note that under full market participation, investors’ model uncertainty

dispersion has no effect on price. What matters is the average uncertainty. Investors who are

more uncertain about the mean stock payoff will hold less of the stock, while investors who

are less uncertain will hold more. In aggregate, the market price behaves as if all investors

have the average uncertainty.

3.3 Limited Participation

When the uncertainty dispersion among investors is sufficiently high such that condition (7)

is not satisfied, investors with high uncertainty optimally choose not to participate in the

stock market. Let φ∗ denote the lowest level of uncertainty at which investors do not hold

the stock. Then

µ− σφ∗ − P = 0. (8)

For an investor with uncertainty higher than φ∗, he chooses not to participate in the stock

market. Thus, the market clearing condition is given by

x =

∫φi<φ∗,φi∈S1

Di

2δdφi =

1

γσ2

(φ∗ − φ+ δ)

2δ

[µ− σ(φ∗ + φ− δ)

2− P

]. (9)

Solving for φ∗ yields

φ∗ = φ− δ + 2√γδσx. (10)

Using Equation (8), we arrive at the following equilibrium price for the stock:

P = µ− σ(φ− δ) − 2σ√γδσx. (11)

Once again, in the presence of model uncertainty, the stock is traded at a discount relative

to its expected payoff.19

19Note that φ − δ ≥ 0 by definition.

12

To gain intuition of this pricing formula, we rewrite Equation (9) as

µ− P =γσ2x

α+ φpσ, (12)

where

α =φ∗ − (φ− δ)

2δ=

√γσx

δ, (13)

is the proportion of investors holding long positions in the stock, and

φp =1

2(φ∗ + φ− δ) = φ− δ +

√γδσx, (14)

is the conditional average level of uncertainty about the mean stock payoff among market

participants. Because market participants have lower uncertainty than nonparticipants, the

conditional average uncertainty (φp) is lower than the unconditional average uncertainty (φ).

Equation (12) suggests that the equity premium can be decomposed into two components:

the risk premium and the uncertainty premium. The lower the participation rate, the higher

the risk premium. This is consistent with what is known in the literature (for example,

Basak and Cuoco (1998)). However, both the participation rate and the uncertainty pre-

mium decrease as the uncertainty dispersion (δ) increases. Consequently, lower uncertainty

premium is associated with lower participation rate.

Figure 1 provides some simple comparative static results with respect to average model

uncertainty (φ), stock supply (x), risk aversion (γ), and model uncertainty dispersion (δ).

The stock price decreases in average model uncertainty and the supply. Given the stock

payoff, higher model uncertainty and supply make the stock less attractive to investors,

which leads to a lower price. When the supply is very low (x is very small) such that there is

limited participation (condition (7) does not satisfy), the stock price becomes very sensitive

to the change in supply. This is reflected in steeper pricing function as the supply decreases.

When the supply is very large, there is full participation and the price becomes linear in

supply. The stock price also decreases in investors’ risk aversion. Intuitively, investors with

low risk aversion are more likely to buy a stock for the same future payoff than investors with

high risk aversion. As investors’ risk aversion increases, the demand for the stock decreases.

This leads to lower price for the stock.

13

As uncertainty dispersion increases, investors who are more uncertain about the mean

payoff of the stock gradually withdraw from the market, leaving those who are less uncertain

in the market. Because these remaining investors have lower uncertainty and are willing

to pay higher price for the stock, the equilibrium stock price increases. Note that when

uncertainty dispersion is low, the stock price does not vary with uncertainty dispersion.

This is because at low uncertainty dispersion there is full market participation and the

equilibrium price is determined by the average uncertainty and not affected by the dispersion

of uncertainty.

In Figure 2 we plot the percentage of nonparticipation in the stock market, (1 − α), as

a function of stock payoff volatility (σ), stock supply (x), risk aversion (γ), and uncertainty

dispersion (δ). The percentage of investors not participating in the stock market decreases

as the payoff volatility, stock supply, and investors’ risk aversion increase. This is caused by

lower price associated with higher payoff volatility, stock supply, and risk aversion. On the

other hand, because the price is higher at higher dispersion of model uncertainty, investors

are increasingly staying sidelined as model uncertainty dispersion increases. This finding is

in contrast to the result in the standard expected utility model in which investors always

take some positions in the stock when there is a positive equity premium.

4 Market Participation and Equity Premium

A widely investigated issue in the literature on limited market participation is how equity

premium relates to limited market participation. Several studies (Mankiw and Zeldes (1991),

Basak and Cuoco (1998), Vissing-Jorgensen (1999), and Brav, Constantinides, and Geczy

(2002)) find that equity premium increases as fewer investors participate in stock markets.

These studies, however, assume that some investors are exogenously excluded from partic-

ipating in certain markets. In this section, we re-examine the relation between the equity

premium and limited market participation by allowing investors to optimally choose whether

or not to participate in the markets so that limited market participation arises endogenously

in our economy.

The following is the main result of this section.

14

Proposition 1 If γσx < δ, then there is limited market participation in equilibrium. Fur-

ther, in the presence of limited market participation,

(a) investors’ participation rate α decreases with uncertainty dispersion, i.e., ∂α/∂δ < 0;

(b) the average model uncertainty for investors holding positive positions in the stock de-

creases with uncertainty dispersion, i.e., ∂φp/∂δ < 0; and

(c) the equity premium decreases with uncertainty dispersion, i.e., ∂(µ − P )/∂δ < 0.

We have shown earlier that with full market participation, equity premium does not

depend upon model uncertainty dispersion among investors. With limited market participa-

tion, however, equity premium decreases with model uncertainty dispersion. The intuition is

as follows. As uncertainty dispersion increases, investors with relatively high uncertainty will

not participate in the stock market. In this case, an investor taking a long position uses the

lowest expected payoff in his payoff distribution set to evaluate his investment strategy and

an investor taking a short position uses the highest expected payoff to evaluate his invest-

ment strategy while a sidelined investor uses the expected payoff that lies between the two

extremes. When there is full market participation, every investor uses the lowest expected

payoff in his payoff distribution set to evaluate his investment strategy. Given the same

average uncertainty, the aggregate demand function for the stock is thus higher when there

is limited participation than in the economy with full participation. As a result, the equilib-

rium price is higher and the equity premium is lower under limited participation than under

the full participation. This can be readily seen from Equation (12). As uncertainty disper-

sion increases, according to Proposition 1(a), market participation rate decreases and risk

premium increases. At the same time, however, uncertainty premium decreases according to

Proposition 1(b). This is because the remaining market participants have lower uncertainty

about the mean stock payoff and are willing to accept lower uncertainty premium. More-

over, in our model, when there is limited market participation, the uncertainty premium

effect dominates the risk premium effect, leading to a lower equity premium as shown in

Proposition 1(c). Since more investors choose to stay sidelined when the uncertainty disper-

sion is higher, our results imply that lower (higher) equity premium is associated with less

(more) stock market participation.

15

There is some empirical support for the predicted positive relation between stock market

participation and stock returns. In Figure 3, we plot the stock returns against the stock

market participation for seven countries including France, Germany, Italy, the Netherlands,

Sweden, the United Kingdom, and the United States. The stock returns are total annual

stock returns in US dollars for the period between 1986 and 1997. Two measures of the

stock market participation are used: direct participation which considers traded and non-

traded stocks held directly and total participation which also includes mutual funds and

managed investment accounts.20 The stock market participation measures are based on

1998 microeconomic surveys in respective countries except for Sweden which is based on a

1999 survey. Overall, the figure suggests that stock returns are positively related to both

measures of stock market participation.

Our theoretical prediction also is consistent with some recent empirical studies on the as-

set pricing effect of the breadth of stock ownership. Specifically, if narrower stock ownership

is used as a proxy for lower participation at the firm level, our prediction that lower partici-

pation is associated with lower equity premium is consistent with the finding in Chen, Hong

and Stein (2002) that a narrower breadth of stock ownership is related to lower expected

stock returns.21

The following proposition provides some further results regarding the comparative statics

of the uncertainty premium and the risk premium with respect to the volatility of the stock

payoff, the investors’ risk aversion, average model uncertainty, and uncertainty dispersion.

Proposition 2 Both the uncertainty premium and the risk premium increase with the vari-

ance of the stock payoff (σ) and the risk aversion coefficient (γ). The risk premium is

unrelated to the average model uncertainty (φ) but increases with the uncertainty dispersion

(δ). However, the uncertainty premium increases with the average model uncertainty (φ) but

decreases with the dispersion uncertainty (δ).

To gauge the relative importance of the two components of the equity premium, we

conduct some numerical analysis. Figure 4 shows the equity premium (solid line) decomposed20The data are from Tables 4 and 11 of Guiso, Haliassos and Jappelli (2002). For Germany, only direct

participation measure is available.21Chen, Hong and Stein (2002) interpret their empirical results using Miller’s model which is based on

heterogeneous beliefs and short-sale constraint.

16

into the uncertainty premium (dashed line) and the risk premium (dash-dotted line) as a

function of the stock supply (x), the payoff volatility (σ), the investor’s risk aversion (γ),

and the uncertainty dispersion among investors (δ). Both the uncertainty premium and the

risk premium as well as the total equity premium are increasing in the stock supply and

the volatility of stock payoff. This reflects the fact that investors demand higher returns

to hold more stock and to compensate them for higher risk. Similarly, as investors’ risk

aversion increases, a higher risk premium is required to induce investors to hold the stock.

Furthermore, a higher risk aversion implies a higher conditional average uncertainty among

the market participants (φp), hence higher uncertainty premium. Notice that the uncertainty

premium initially stays flat and then decreases as the model uncertainty becomes more

dispersed among investors. This happens because there is full market participation at low

levels of uncertainty dispersion. The uncertainty premium at full market participation only

depends upon the average level of model uncertainty and is not affected by the uncertainty

dispersion. As the uncertainty dispersion increases, investors with high uncertainty stay

sidelined, which creates limited market participation. When limited market participation

occurs, the uncertainty premium decreases with the uncertainty dispersion. In fact, the

decrease in uncertainty premium outweighs the increase in the risk premium so that the

total equity premium also decreases.

5 Multiple Stocks and the Diversification Discount

In this section, we examine how, in the absence of any synergy between two firms, the

market value of the merged firm (the conglomerate) depends upon investors’ uncertainty

dispersion and how it relates to limited participation at firm level. Our study differs and

also complements studies based on production and synergy such as Gomes and Livdan (2002)

by focusing on the demand for risky assets. For our purpose, we extend the model in Section

2 to allow for two stocks. The payoffs on the two stocks follow a two factor model

r1 = fA + ρfB, (15)

r2 = ρfA + fB, (16)

17

where r1 and r2 are payoffs on stock 1 and 2, respectively, fA and fB are two (random)

systematic factors, and ρ represents the factor loading. We assume |ρ| < 1. The total

supplies of the two stocks are given by x1 and x2, with x1 = x2 = x > 0. All variables

are jointly normally distributed. Investors know that the factors, fA and fB, follow a joint

normal distribution with a non-degenerate variance-covariance matrix ΩF but do not have

perfect knowledge of the distribution of the random factors. As in Section 2, we assume

that investors have a precise estimate of the variance-covariance matrix (ΩF). Without loss

of generality, we assume that ΩF is diagonal with common element σ2. However, investors

do not know exactly the mean of fA or fB. Let µA = µB = µ be the reference level for the

mean of fA and fB. Let vA and vB be the adjustment to the mean µA and µB, respectively.

We therefore define the set P by the following constraints on vA and vB:

v2Aσ

−2 ≤ φ2A, (17)

and

v2Bσ

−2 ≤ φ2B, (18)

where φA and φB are parameter that captures the investor’s uncertainty about the mean of

factor A and B, respectively. In other words, investors believe the mean of factor fA and fB

fall into the set

(µA + vA, µB + vB) : vA and vB satisfy (17) and (18), respectively.

We assume that φ = (φA, φB) are uniformly distributed on the square22

S2 = [φ− δ, φ+ δ] × [φ− δ, φ+ δ],

where φ ≥ δ.

When there is a merger, firms combine their operations and cash flows to form a con-

glomerate denoted M . The cash flows are unaffected by the merger so that the payoff from

firm M is rM = r1 + r2 = (1 + ρ)(fA+ fB).23 After the merger, investors can no longer trade

stocks 1 and 2 and can only hold asset M .22More general distributions are considered in Section 6. Implicitly, we assume that investors may have

different degrees of uncertainty about different factors. For example, an American investor may have lessuncertainty about the US stock returns than the Japanese stock returns while a Japanese investor mayhave less uncertainty about the Japanese stock returns than the US stock returns. Perceived familiarity ofdifferent stocks may affect the degree of uncertainty about these stocks (Huberman (2001)).

23We abstract from agency problems such as inefficient internal capital markets and rent-seeking atdivision-level considered by Rajan, Servaes and Zingales (2000) and Scharfstein and Stein (2000).

18

For ease of exposition, we introduce factor portfolios tracking the two factors A and B.

Let DiA and DiB be investor i’s holdings of the factor portfolio A and B, respectively. Let

PA and PB be the prices of factor portfolio A and B, respectively. Let Pj denote the price

of the stock j, Dij denote investor i’s demand for stock j, for j = 1, 2. Let DiM be investor

i’s demand for the merged firm and PM be the market price of the merged firm. Given the

payoff structure for stocks, the payoffs for the factor portfolios A and B are given by:

fA ≡ r1 − ρr21 − ρ2

,

fB ≡ r2 − ρr11 − ρ2

,

and the demand for the factor portfolios, DiA and DiB, can be written as:

DiA = Di1 + ρDi2,

DiB = ρDi1 +Di2.

Similarly, we can rewrite the demand for the stocks in terms of the demands for the factor

portfolios:

Di1 =DiA − ρDiB

1 − ρ2,

Di2 =DiB − ρDiA

1 − ρ2.

The supply of each factor portfolio is x ≡ (1 + ρ)x. Since the two factors are orthogonal

to each other and investors have CARA utility function, the prices of the factors (PA and

PB) can be determined similarly as in the case with a single stock in Section 3. The prices

of the stocks when traded separately can be obtained using the relation P1 = PA + ρPB and

P2 = ρPA + PB.

5.1 Full Participation

We first consider the case of full participation. The following proposition summarizes the

main results under full participation.

19

Proposition 3 When (1 + ρ)xγσ ≥ δ, there is full participation in the economy with or

without merger. Moreover, the prices of the two stocks in the economy without merger are:

P1 = P2 = (1 + ρ)[µ− σφ− γ(1 + ρ)xσ2]. (19)

and the price for the conglomerate in the economy after the merger is given by

PM = P1 + P2 = 2(1 + ρ)[µ− σφ− γ(1 + ρ)xσ2]. (20)

When (1 + ρ)xγσ ≥ δ, there is full participation in both the economy with separate

trading of the two stocks and the economy with only the merged firm being traded. Equation

(20) indicates that the price of the merged firm is simply the sum of the market values of

the two firms traded separately, i.e., PM = P1 + P2. Investors who would have preferred to

buy more of factor portfolio A than portfolio B now buy both portfolios with equal weight.

Similarly, investors who would like to buy more of portfolio B than portfolio A now buy

both at equal weight. Nevertheless, because of full participation, the price is determined

by the average investor who has model uncertainty φ for both factors A and B. With or

without merger, the average investor will hold the market portfolio. As a result, there is no

diversification discount.

5.2 Limited Participation

Now for the case of limited participation, the result is summarized in the following proposi-

tion.

Proposition 4 If (1 + ρ)xγσ < δ, there is limited participation in equilibrium and

P1 = P2 = (1 + ρ)[µ − σ(φ− δ) − 2σ√γ(1 + ρ)xσδ],

PM = 2(1 + ρ)µ− σg−1[γ(1 + ρ)xσ]/2,where

g(y) =(y − 2φ+ 2δ)3 − 2[(y − 2φ)+]3

/(48δ2),

and PM < P1 + P2, i.e., there is a diversification discount.

20

This proposition states that, with limited participation, the price of a conglomerate is

less than the sum of the prices of its components traded separately. Intuitively, when an

investor has low uncertainty on factor A, he is likely to participate in the market for factor

A portfolio when there is separate trading. However, the same investor may not have a low

uncertainty on factor B and may choose to stay away from factor B portfolio under separate

trading. When the two stocks are combined and traded as a bundle only, the investor can

no longer trade the stock for which he has a low uncertainty only and has to buy both factor

portfolios with equal weight. If there is no diversification discount, i.e., PM = P1 + P2, the

investor may choose not to hold the bundled asset. As a result, the price of the conglomerate

has to be reduced to attract investors.24

Corollary 1 Keeping the average uncertainty φ fixed, when there is limited participation,

diversification discount increases with uncertainty dispersion (δ).

Proposition 4 and Corollary 1 together predict that there exists a conglomerate discount

when there is limited participation. Moreover, since uncertainty dispersion is positively cor-

related with the fraction of investors not participating in the stock markets, diversification

discount is positively correlated with the nonparticipation rate. While diversification dis-

count for conglomerates has been well documented in the literature (Lang and Stulz (1992),

Berger and Ofek (1995), and Rajan, Servaes, Zingales (2000), Hite and Owers (1983), Miles

and Rosenfeld (1983) and Schipper and Smith (1983)), our study is the first to establish the

relation between diversification discount and limited participation.

While existing empirical studies have yet to provide any guidance on the relation between

diversification discount and limited participation, our theoretical prediction suggests some

possible empirical tests. For example, surveys can be conducted to collect information on

investors’ uncertainty measures and participation status by asking a randomly selected group

of investors their range of expected payoffs (returns) for stocks of both single segment firms

and conglomerates as well as the investors’ ownership status on these stocks. The survey

information will allow for empirical analysis on how the diversification discount and limited

24Our results are obtained in a two-risky asset economy. It is straightforward to extend the analysis to amultiple-risky asset economy. However, to match the magnitude of diversification discount documented inexisting empirical studies, some form of market imperfection such as short sale constraint may need to beimposed.

21

participation are related to investor’s uncertainty on the mean payoffs of these assets. For

another example, if some existing databases contain information on companies which are

merged at some point in time and information on investors’ holdings of the stocks of these

companies, then the information would allow empirical researchers to explore the association

between diversification discount and limited participation.

In Figure 5 we plot the diversification discount as a function of the model uncertainty

dispersion among investors (δ). Both the discounts measured in level and in percentage show

the same features. Initially, at low levels of the model uncertainty dispersion, there is full

participation and the diversification discount is zero. As the uncertainty dispersion becomes

sufficiently large, limited participation occurs and the diversification discount becomes pos-

itive. As uncertainty dispersion increases, the diversification discount also increases. Both

panels in the figure suggest that the discount can be quite sizable.

Corollary 2 When there is limited participation, a conglomerate merger will further reduce

participation.

Figure 6 plots the nonparticipation rate for both the case where the two stocks are traded

separately (solid line) and the case in which the two stocks are merged into one conglomerate

(dashed line). The figure illustrates that at low levels of uncertainty dispersion, we observe

full participation. When the uncertainty dispersion becomes sufficiently large, investors with

high uncertainty optimally choose to stay sidelined. As the uncertainty dispersion further

increases, the fraction of investors withdrawn from the market increases. Moreover, the

percentage of investors staying sidelined is always higher when investors can only trade the

conglomerate than when they can trade the two individual stocks separately.

6 Generalization

In this section we extend the model in Section 5 to allow for a more general factor structure

and demonstrate that our basic results are robust in the more general setting. Specifically,

22

the payoffs of stocks 1 and 2 are now given by

(r1r2

)= β

(fAfB

)+

(ε1ε2

), (21)

where

β =

(β1A β2A

β1B β2B

). (22)

The matrix β represents the factor loadings and is assumed to be invertible. The factors fA

and fB are the systematic factors as before. The random shock ε1 and ε2 are the idiosyncratic

factor for stock 1 and 2, respectively.

All variables are jointly normally distributed. The idiosyncratic factors are uncorre-

lated with each other and are uncorrelated with the systematic factors. As in Section 5,

investors do not have perfect knowledge on the stock payoffs. They know that the factors

(fA, fB, ε1, ε2) follow a joint normal distribution. The risk of stock payoffs is summarized by

the non-degenerate variance-covariance matrix ΩF. We assume that investors have a precise

estimate of ΩF. Without loss of generality, we assume that fA and fB have equal variance

σ2, ε1 and ε2 have variance σ21 and σ2

2, respectively. However, investors do not know exactly

the mean of fA or fB.25 Investors’ uncertainty is represented by the constraints

v2iAσ

−2 ≤ φ2iA, v2

iBσ−2 ≤ φ2

iB.

Similar to Section 5, investors’ heterogeneity is characterized by φi = (φiA, φiB). We

assume that φiA and φiB have a joint distribution defined on S3 ≡ [φlA, φuA]× [φlB , φuB] with

a continuous probability density function h(φiA, φiB), where φlj and φuj, j = A,B, are the

lower and upper bound, respectively. The average of φi is denoted by φ = (φA, φB). Let

x1 and x2 be the supplies of stock 1 and 2, respectively. The corresponding factor portfolio

supplies are given by

x ≡[xAxB

]= β

[x1

x2

].

25By assumption, the idiosyncratic factors have zero mean.

23

6.1 Portfolio Choice and Market Participation

Under the assumption of normality and the CARA utility, investor i’s maximization problem

is reduced to:

maxDi1,Di2

minviA,viB

−Di1P1 −Di2P2 + (β1ADi1 + β2ADi2)(µA − viA) + (β1BDi1 + β2BDi2)(µB − viB)

−γ2

[(β1ADi1 + β2ADi2)

2σ2 + (β1BDi1 + β2BDi2)2σ2 +D2

i1σ21 +D2

i2σ22

]. (23)

The first line in Equation (23) represents the expected payoffs adjusted for uncertainty

for investor i. The second line derives from investors’ risk aversion. The demand for factor

portfolios are given by: (DiA

DiB

)≡ β

(Di1

Di2

). (24)

Since the matrix β is invertible, there is a one-to-one correspondence between the holdings

of factor portfolios and those of the stocks 1 and 2. In particular, if an investor does not

participate in the factor portfolio markets, he does not participate in the stock markets.

The first–order condition for investor i’s optimal holdings of the factor portfolios is given

by

[µA − φiAσ + λiAµB − φiBσ + λiB

]− P = γ

([σ2 00 σ2

]+ (β−1)

[σ2

1 00 σ2

2

]β−1

) [DiA

DiB

],

where λiA and λiB are nonnegative numbers such that 0 ≤ λiA ≤ 2σφiA and 0 ≤ λiB ≤ 2σφiB,

and P ≡ (PA, PB) = (β−1)(P1, P2). These constraints imply that when the investor holds

a long position in factorA, investor i takes the lowest expectation in his payoff distribution set

(λiA = 0). Similarly, if investor i holds a short position, he takes the highest expectation in his

payoff distributions set and thus λiA = 2σφiA. When he does not hold any position in factor

A, he takes the expectation between the two extremes, and in this case, 0 < λiA < 2σφiA.

Similar arguments hold for the constraint on λiB.

Let Di = (DiA, DiB), λi = (λiA, λiB), it follows that the demand for factor portfolios

is given by

Di = (γΩ)−1 (µ− σφi + λi − P ) , (25)

24

where

Ω =

[σ2 00 σ2

]+ (β−1)

[σ2

1 00 σ2

2

]β−1

is the variance-covariance matrix of the factor portfolios. Without loss of generality, we

assume that Ω is positive.26 The demand for factor portfolios can be rewritten as

Di = (γΩ)−1 [µ− P ] − (γΩ)−1 [σφi − λi] . (26)

The first term on the right hand side is the standard mean-variance demand, and the second

term is the adjustment to the mean-variance demand due to model uncertainty. Given the

demand for the factor portfolios, the demand for individual stocks can be determined by

(Di1, Di2) = β−1Di. (27)

We summarize the result on participation in the general setting as follows.

Theorem 1 If minσφuA, σφuB > M ≡ max|µA − PA|, |µB − PB|, then investors with

uncertainty level in S = [M,φuA] × [M,φuB] will take no position in the stock markets.

6.2 Equilibrium and Market Participation

In the general setting of this section, we do not have a closed-form solution for the equilib-

rium. However, the existence of equilibrium is guaranteed by the following theorem.

Theorem 2 There exists an equilibrium. In addition, if the supply of both factor portfolios

are strictly positive, the equilibrium is unique. Moreover, both factor portfolios are traded at

a discount, i.e., µA − PA > 0 and µB − PB > 0.

Note that when the supply of a factor portfolio is zero, there could exist multiple equilib-

rium prices for that factor. Multiple equilibria occur because there could exist a continuum of

prices for that factor portfolio such that investors choose not to participate at these prices.

The aggregate demand for the factor portfolio is thus zero and the market for the factor

portfolio clears.

26Otherwise, we can redefine a new set of factors f ′A = −fA, f ′

B = fB such that Ω is positive.

25

6.3 Equity Premium under Limited Participation

In equilibrium, the asset markets must clear, or equivalently, the factor portfolio markets

must clear, i.e.,

xA =

∫ ∫DiA =0,φi∈S3

DiA h(φiA, φiB)dφiAdφiB, (28)

xB =

∫ ∫DiB =0,φi∈S3

DiB h(φiA, φiB)dφiAdφiB. (29)

We say that there is limited participation in factor portfolio j, j = A,B, when the

measure of nonparticipants in factor portfolio j is positive. We have the following results

regarding the relation between limited participation and asset prices.

Theorem 3 Let ωij denote the (i, j)th component of (γΩ)−1.

(a) If γx ≥ σδ(ω11 − ω12, ω22 − ω12), then all investors hold both factor A and factor B

portfolios. In addition, the full participation equilibrium prices for the factor portfolios,

P f ≡ (P fA, P

fB), are given by

P f = µ− σφ− γΩx. (30)

(b) If there is limited participation in factor portfolio A (B) but full participation in factor

portfolio B (A), then PA > P fA and PB = P f

B (PA = P fA and PB > P f

B);

(c) If there is limited participation in both factor portfolios, PA > P fA and PB > P f

B.

Market participation clearly depends on the dispersion of investors’ uncertainty. When

investors are relatively homogeneous, φuA−φlA and φuB −φlB are small, a full participation

equilibrium prevails. When investors are relatively homogeneous with respect to one factor

but not the other, there could be limited participation with respect to one of the factor

portfolios. Finally, when investors’ uncertainty is very heterogeneous with respect to both

factors, some investors do not participate in either factor portfolio and we have limited

participation in both markets.

26

6.4 Diversification Discount

In this subsection, we show that our results on the diversification discount still go through in

the general setting. Let the two firms merge to form a conglomerate. The payoffs from the

conglomerate is now rM = x1r1 + x2r2. Without loss of generality, we normalize the share

supply of the conglomerate to one. We summarize the results in the following theorem.

Theorem 4 When there is full participation, there is no diversification discount, i.e., PM =

x1P1 +x2P2. When there is limited participation, there is diversification discount, i.e., PM ≤x1P1+x2P2. Moreover, the diversification discount is strictly positive when a positive measure

of investors participate only in one of the factor portfolios.

7 Conclusion

We have studied the relation among three well-documented empirical phenomena: limited

participation, the equity premium, and the diversification discount in a general equilib-

rium framework with model uncertainty. We find that the phenomena are fundamentally

connected in the heterogeneity of investors’ uncertainty about the distribution of stock pay-

offs.27 Specifically, the presence of model uncertainty leads to higher equity premium than

that obtained in standard expected utility framework without model uncertainty. Large dis-

persion in model uncertainty among investors results in limited participation. Merged firms

are traded at a discount when limited participation occurs in equilibrium.28 Diversification

discount can then cause further decrease in market participation.

Interestingly, we find that when there is limited market participation, caused by high

dispersion in uncertainty, equity premium can be lower than that in an economy with full

market participation. This is in contrast to the understanding in the literature that limited

participation tends to lead to higher equity premium.

27It is conceivable that other forms of heterogeneity such as disappointment aversion, fixed pariticipationcost also can generate similar results.

28More generally, our results imply that sufficient dispersion in uncertainty makes packaging of securitiesmatter. Thus our model can be used to explain the close-end fund discount when there are short-saleconstraints and why tracking stocks exist.

27

With respect to the diversification discount, our model predicts that diversification dis-

count and spin-off premium are larger when there is more uncertainty associated with the

stocks and investors are more heterogeneous. Such prediction is novel and can be empirically

tested in future research.

Overall, our study provides an integrated framework which allows us to analyze limited

participation, the equity premium, and the diversification discount simultaneouly. Our re-

sults suggest that model uncertainty and investors aversion to uncertainty are important

factors in understanding some of the well-documented asset pricing anomalies.

28

Appendix

Proof of Proposition 3: The proof for the case without merger is very similar to the single

stock case. Let x = (1 + ρ)x. When γxσ ≥ δ, there is full market participation. The prices

for the factor portfolios are given by

PA = PB = µ− φ− γxσ2.

As a result we have

P1 = PA + ρPB = PB + ρPA = P2 = (1 + ρ)[µ − φ− γxσ2].

In the economy with merger, the maximization problem is

maxDiM

DiM [2(1 + ρ)µ− (1 + ρ)sgn(DiM)(φiA + φiB)σ − PM ] − γD2iM(1 + ρ)2σ2.

When γxσ ≥ δ, we have

DiM =2(1 + ρ)µ− (1 + ρ)(φiA + φiB)σ − PM

2(1 + ρ)2γσ2.

Again there is full market participation. Aggregating the demands across all investors and

equal the aggregate demand to aggregate supply, we arrive at

x =2(1 + ρ)µ− 2(1 + ρ)φσ − PM

2(1 + ρ)2γσ2,

which implies

PM = 2(1 + ρ)[µ− φ− γxσ2] = (1 + ρ)(PA + PB) = P1 + P2,

and there is no diversification discount.

Proof of Proposition 4: When γxσ < δ, in the economy without merger, there is limited

participation. Following the results in Proposition 2, we get

PA = PB = µ− σ(φ− δ) − 2σ√γxδσ.

29

As a result we have

P1 = PA + ρPB = PB + ρPA = P2 = (1 + ρ)[µ− σ(φ− δ) − 2σ√γxδσ]. (A1)

When γxσ < δ, in the economy with merger, there is also limited participation. Thus,

there exists a φ such that when φiA+φiB ≥ φ, investor i will not invest in the conglomerate.

φ is determined by,

[2(1 + ρ)µ− (1 + ρ)φσ − PM ] = 0. (A2)

Notice that no investor will sell short the conglomerate and thus DiM > 0 for investors par-

ticipating in the market. Consequently, sgn(DiM) = 1 for participating investors. Summing

up investors’ demand, we have

x =

∫ ∫φiA+φiB<φ,φi∈S2

1

8(1 + ρ)2γσ2δ2[2(1 + ρ)µ− (1 + ρ)(φiA + φiB)σ − PM ]dφiAdφiB.

Utilizing Equation(A2), we arrive at the following market clearing condition

(1 + ρ)γσx =

∫ ∫φiA+φiB<φ,φi∈S2

[φ− (φiA + φiB)]

8δ2dφiAdφiB = g(φ),

where

g(φ) ≡ (φ− 2φ+ 2δ)3 − 2[(φ− 2φ)+]3

48δ2. (A3)

We claim that g is an increasing function of φ. To show this, define z = φ − 2φ. Then we

must have z ≤ 2φ + 2δ − 2φ = 2δ. It is clear that g is an increasing function in φ when

z ≤ 0. When 2δ ≥ z > 0,

dg

dz=

2δ2 − (z − δ)2

16δ2> 0.

Thus, g is also strictly increasing in φ when 2δ ≥ z > 0. The equilibrium price of the

conglomerate is thus given by

PM = 2(1 + ρ)[µ− σg−1(γxσ)/2]. (A4)

30

It is readily verified that PM < P1 + P2. Later in the proof of Theorem 4, we provide the

proof for a more general case.

Proof of Corollary 1: We prove the case for ρ = 0, γ = 1, x = 1, σ = 1. The general

case can be dealt with similarly. When δ = 6, we have from the proof of Proposition 4 that

g(φ) = 1 = δ/6 = g(2φ). When δ ≥ 6, since g(·) is an increasing function, we must have

φ ≤ 2φ. Consequently, g(φ) = 1 implies that φ = 23√

6δ2 + 2(φ− δ). By the expressions for

Pi (Equation (A1)) and PM (Equation (A4)), the diversification discount is 2(3√

6δ2 − 2√δ),

which is increasing in δ.

In the case in which 1 < δ < 6, we have φ > 2φ. The diversification discount is

z = φ− 2φ+ 2δ − 4√δ. Substituting this back into g(φ) = 1, we have

F (z, δ) = (z + 4√δ)3 − 2(z − 2δ + 4

√δ)3 − 48δ2 = 0.

Define

ψ ≡√δ, ω ≡ z + 4ψ

2ψ2.

Since 0 < φ− 2φ < 2δ, 2δ < z + 4ψ < 4δ and 1 < ω < 2. Substituting into F , we have

G(ω, ψ) ≡ ω3 − 2(ω − 1)3 − 6

ψ2= 0.

Since Gω(ω, ψ) = −3(ω−2)2 +6, Gω > 0 for 1 < ω < 2. Thus G(ω, ψ) = 0 implicitly defines

a function ψ → ω for 1 < ψ <√

6 and 1 < ω < 2. From the definition of ω, we have

dz

dψ= 4(ψω − 1) + 2ψ2dω

dψ= 4(ψω − 1) − 2ψ2Gψ

Gω

= 4[ψω − 1 − 2

ψ(−ω2 + 4ω − 2)].

With some algebraic derivation, the inequality dz/dψ > 0 is equivalent to

ω − ω3 − 2(ω − 1)3

3(−ω2 + 4ω − 2)>

√ω3 − 2(ω − 1)3

6. (A5)

From (A5) we get

[ω − ω3 − 2(ω − 1)3

3(−ω2 + 4ω − 2)]2 >

ω3 − 2(ω − 1)3

6. (A6)

31

Using one to subtract both sides of (A6), we get

[1 − ω +ω3 − 2(ω − 1)3

3(−ω2 + 4ω − 2)][1 + ω − ω3 − 2(ω − 1)3

3(−ω2 + 4ω − 2)] >

6 − ω3 + 2(ω − 1)3

6. (A7)

Multiplying both sides of Equation (A7) by 18/(2 − ω), we get

a(ω)b(ω)

c(ω)< d(ω), (A8)

where

a(ω) = −2ω3 + 3ω2 + 12ω − 8,

b(ω) =9

4− (2ω − 5

2)2,

c(ω) = [2 − (2 − ω)2]2,

d(ω) = 3(6 − (2 − ω))2.

It is easy to verify that a, c, d are increasing functions of ω for 1 < ω < 2. The function b(ω)

is increasing for 1 < ω < 5/4 but decreasing for 5/4 ≤ ω < 2 and maximized at ω = 5/4

with value 9/4. Therefore, for 1 < ω ≤ 9/8 we have

a(ω)b(ω)

c(ω)<a(9/8)b(9/8)

c(1)< d(1) < d(ω),

and, for 9/8 < ω ≤ 3/2 we have

a(ω)b(ω)

c(ω)<a(3/2)b(5/4)

c(9/8)< d(9/8) < d(ω).

Finally, for 3/2 < ω ≤ 2, we have

a(ω)b(ω)

c(ω)<a(2)b(3/2)

c(3/2)< d(3/2) < d(ω).

Consequently, z is an increasing function of ψ and thus an increasing function of δ.

32

Proof of Corollary 2: We prove the case for ρ = 0, γ = 1, x = 1, σ = 1. The more general

case can be treated similarly. First, consider δ ≥ 6. In this case φ ≤ 2φ. From expression

(13), the proportion of nonparticipants is

∫ ∫φiA+φiB≥φ,φi∈S2

dφiAdφiB4δ2

= [1 − φ∗ − φ+ δ

2δ]2 = (1 −

√1/δ)2,

in the economy in which the two stocks are traded separately. For the economy with the

conglomerate, investors with φiA + φiB ≥ φ will not participate. In this case, since δ ≥ 6,

from expression (A3), we have g(φ) = 1 ≥ 6/δ = g(2φ). Since g is an increasing function,

we must have φ ≥ 2φ. The proportion of nonparticipants is

∫ ∫φiA+φiB≥φ,φi∈S2

dφiAdφiB4δ2

= 1 − (φ− 2φ+ 2δ)2

8δ2= 1 − 3

√9

2δ2,

where the second equality follows from expression (A3). Let x = 6√

1/δ.

To prove the claim, we need to show that

(1 − x3)2 < 1 − x4 3√

9/2,

which reduces to

−2 + x3 + x 3√

9/2 < 0. (A9)

The LHS of (A9) is an increasing function of x and when x = 6√

1/6, the expression holds.

Therefore the corollary holds for the case in which δ ≥ 6.

Next, we consider the case in which 1 < δ < 6. In this case, it can be similarly shown

that φ < 2φ since g(φ) = 1 < 6/δ = g(2φ) and g is an increasing function. In the economy

with the conglomerate, the proportion of nonparticipants is

∫ ∫φiA+φiB≥φ,φi∈S2

dφiAdφiB4δ2

=(2φ+ 2δ − φ)2

8δ2.

We need to show that

(2φ+ 2δ − φ)2

8δ2> [1 − φ∗ − φ+ δ

2δ]2 = (1 −

√1/δ)2,

33

which reduces to

φ < φ ≡ 2φ+ 2δ − 2√

2δ + 2√

2δ.

To prove φ < φ, notice that g(φ) is an increasing function, and we need to prove that

g(φ) > g(φ) = 1. (A10)

Substituting the expression of φ into inequality (A10), we get

(4δ − 2√

2δ + 2√

2δ)3 − 2(2δ − 2√

2δ + 2√

2δ)3

48δ2> 1. (A11)

Let y =√δ, with some algebra, inequality (A11) reduces to

y3 + 3(√

2 + 1)y − (4 + 3√

2) > 0, (A12)

which holds since y > 1.

Proof of Theorem 1: The following information about λiA and λiB will be useful. λiA = 0,

if DiA > 0, λiA = 2σφiA if DiA < 0 and 0 ≤ λiA ≤ 2σφiA if DiA = 0. Left multiply both

sides of equation (26) by (µ − P + λi − φiσ). Then the RHS is positive and the LHS is

negative. Thus µ− P + λi − φiσ must be zero and Di = 0.

Proof of Theorem 2: Notice that 0 ≤ λiB ≤ 2σφiB. For existence, define the aggregate

demand by the right side of equations (28) and (29) and denote it by D(P ). Using the

Maximum Theorem, it is readily shown that after the inner minimization, the objective

function in (23) is continuous and strictly concave in (DA, DB). As a result, the demand

function D(P ) is continuous in P . Consider the following mapping

M(P ) = P + γΩ(D(P ) − x) = [µ− σφ+ λ(P )] − γΩx,

where

λ(P ) =

∫ ∫φi∈S3

λi(P ) h(φiA, φiB)dφiAdφiB.

This mapping is continuous in P . Let

S = [µ− σφ+ λ] − γΩx, 0 ≤ λ ≤ 2σ(φ+ δ)(1, 1).

34

Clearly, M(S) ∈ S. Applying the Brouwer’s fixed point theorem, there exists a fixed point

in which M(P ) = P .

To prove uniqueness, first notice that investors’ demand for portfolio A is decreasing in

PA and increasing in PB. Suppose that there exists two sets of equilibrium prices, P = P ′.

Without loss of generality we assume that P ′A > PA. Let P ∗

B = PB − ω11(P′A − PA)/ω12.

Notice that ω12 is off-diagonal element of (γΩ)−1 and is negative by assumption. Thus we

must have P ∗B > PB. We show that at price vector P ∗ = (P ∗

A, P∗B) = (P ′

A, P∗B) the demand for

factor portfolio A will be less than the supply. Suppose, to the contrary, thatD∗A > xA = DA.

Then for some i, D∗iA > DiA. It follows that λiA ≥ λ∗iA (This follows because D∗

iA > 0 implies

λ∗iA = 0; D∗iA = 0 implies DiA < 0 which implies that λiA = 2σφiA; and finally D∗

iA < 0

implies DiA < 0 which implies λiA = 2σφiA ≥ λ∗iA). Noting that

0 > DiA −D∗iA = ω11(P

∗A − PA + λA − λ∗iA) + ω12(P

∗B − PB + λiB − λ∗iB), (A13)

either we have a contradiction in the case when ω12 = 0, or

λiB > λ∗iB,

in the case ω12 < 0 (note that ω12 ≤ 0). In the latter case,

DiB ≤ 0, D∗iB ≥ 0,

hold. It follows that

0 ≥ DiB −D∗iB,

and hence

−ω21(P∗A − PA + λA − λ∗iA) > ω22(P

∗B − PB + λB − λ∗B) > 0. (A14)

Noting that P ∗A > PA and the two terms in the brackets of Equation(A13) and Equation

(A14) are all positive, we have

ω21ω12 > ω11ω22,

which is a contradiction. Consequently, at price vector P ∗ = (P ∗A, P

∗B) = (P ′

A, P∗B) the

demand for factor portfolio A will be less than the supply.

Since the supply is nonzero, with positive measure, some investors’ demand curve for

portfolio A must be strictly increasing with PB. As a result we must have P ′B − PB >

35

P ∗B − PB = −ω11(P

′A − PA)/ω12 > 0. Similarly, we can show that P ′

A − PA > −ω22(P′B −

PB)/ω12 > 0. These two inequalities together violates the positive definiteness of Ω−1. Thus

the equilibrium must be unique.

Next we show that µA > PA and µB > PB. Since the supply of the factor portfolios are

positive, there must be some investors who hold strictly long positions in factor portfolio A.

Let i be such an investor. There are several cases. Suppose first that investor i also holds a

strictly long position in factor portfolio B, we have

µ− P = γΩDi + σφi > 0.

Thus µA − PA > 0 and µB − PB > 0.

Secondly, suppose that investor i holds positive amount in factor portfolio A and zero in

factor portfolio B. Then the equity premium for A must be positive as

µA − PA = γΩ11DiA + σφiA > 0.

In the third scenario, suppose that investors who hold strictly long positions in factor

portfolio A will only hold short positions in factor portfolio B. It follows that investors who

hold long positions in factor portfolio B must hold non-positive positions in factor portfolio

A. Suppose that there exists an investor j who hold strictly long positions in factor portfolio

B but zero position in factor portfolio A, then we must have µB−PB = γΩ22DjB +φjB > 0.

Moreover, we have µA − PA > σfφiA > 0. Otherwise, we must have µA − PA − σfφiA ≤ 0,

and

(µA − PA − σfφiA, µB − PB + σfφiB)Di =

(µA − PA − σfφiA, µB − PB + σfφiB)(γΩ)−1(µA − PA − σfφiA, µB − PB + σfφiB > 0,

and the RHS is strictly positive while the LHS is clearly negative.

Since investor i in the three cases above is arbitrary, we are left with the case where

investors who hold strictly long positions in factor portfolio A hold short positions in factor

portfolio B, and investors who hold long positions in factor portfolio B hold short positions

in factor portfolio A. Let i be an investor who hold positive in A, negative in B and let j be

36

an investor who hold positive in B and negative in A. We have

(−φiA−φjA, φiB+φjB)(Di−Dj) = (−φiA−φjA, φiB+φjB)(γΩ)−1σ(−φiA−φjA, φiB+φjB).

The LHS is negative while RHS is strictly positive. This is a contradiction. Therefore the

last case is impossible. This finishes the proof that µA − P > 0. Similarly, it can be shown

that µB − PB > 0.

Proof of Theorem 3: It is easy to show that DiA and DiB are continuous functions of

PA, PB, φiA, and φiB. Thus, λiA and λiB are also continuous functions of PA, PB, φiA, and

φiB. We first prove that, in equilibrium, if almost all investors hold positions in both factor

portfolios, then almost all investors will hold long positions in the factor portfolios. Suppose,

without loss of generality, that for a positive measure of investors, the demand for factor

portfolio A is negative and the demand for factor portfolioB is nonzero. Fix such an investor.

Let his uncertainty about factor portfolio A be given by φiA. Let φB = supφjB, DjA < 0.Then,

D(φiA, φB) = ω11(µA − PA − σφiA + λiA) + ω12(µB − PB − σφB) ≤ 0.

Thus for all φjB > φiB, we have DjA > 0, and for all φjB < φiB, we have DjA < 0 because

ω12 < 0. Thus λjA = 0 for φjB > φB and λjA = 2σφjA, for φjB < φB. This implies that λiA

is not continuous in φB. This is a contradiction.

Using the result just proved, it can be verified that, if investors fully participate, the

equilibrium prices for the factor portfolios are given by (30). Given positive supply and

the conjectured equilibrium price, it can also be verified that investors have strictly positive

demand (equation (26)) for both factor portfolios and markets clear. This proves part (a).

For (b) and (c), suppose that some investors do not participate in at least one of the

factor portfolios. Multiplying (26) by γΩ, integrating and using (30), we get

P − P f = λ ≡∫ ∫

φi∈S3

λih(φiA, φiB)dφiAdφiB ≥ 0,

with at least one inequality strictly holds because otherwise we get full participation. It is

then easy to see that the statements in the proposition hold.

37

Proof of Theorem 4: Let x = (xA, xB), P = (PA, PB). Let

[rArB

]=

(β)−1

[r1r2

],

which is the payoff vector of the factor portfolios. Clearly, x1r1 + x2r2 = xArA + xBrB

and x1P1 + x2P2 = xAPA + xBPB. So we simply view the merged firm as having payoff

xArA + xBrB and give the proof in terms of factor portfolios.

Denote µM ≡ xµ. We deal with the full participation case first. When the firms are

merged,

1 =

∫ ∫φi∈S3

DiMdφiAdφiB =

∫ ∫φi∈S3

(xΩx)−1[µM − P fM − σxφi + λiM ]dφiAdφiB.

As in the case of single stock, λiM = 0 in equilibrium. When the two stocks are traded

separately, as shown in the proof of Theorem 3, λiA = λiB = 0, and hence

γΩx = µ− P − σφ, γxΩx = µM − PM − σxφ.

It is easy to check that our equilibrium demand satisfies the market clearing condition.

With limited market participation, we have P = P f + λ and PM = P fM + λM , where

λM ≡ ∫∫φi∈S3

λiMh(φiA, φiB)dφiAdφiB. We will show that at price PM = xP the market

demand for the bundled stock is less than one. Since the demand curve is downward sloping,

in equilibrium, PM < PM = xP .

We first prove that for all investors, the following condition holds:

λiM ≤ xλi. (A16)

First of all, expression (A16) must hold when the demand for the merged firm is positive

for investor i as λiM = 0. Now consider the case where investor i holds zero positions in

both factor portfolios. In this case, the investor must hold zero positions in the merged firm

and

0 = x(γΩ)Di = µM − PM − σxφi + xλi = µM − PM − σxφi + λiM .

38

Thus λiM = xλi, and condition (A16) holds too.

We are left with cases in which investors hold zero positions in the merged firm but

nonzero positions when the stocks are traded separately. In the economy without the merger,

since the excess payoffs of the two factor portfolios are both positive (Theorem 2), it is

impossible for any investor to hold negative positions in both factor portfolios or zero in

one factor portfolio and negative in the other (recall that each component of Ω is positive).

Moreover, it is not possible for investor i’s optimal demand for the two factor portfolios to

be both positive. Otherwise, we have

DiM = (xΩx)−1[µM − P fM − σxφi + λiM ] > 0,

which contradicts with the assumption that DiM = 0. Thus, the remaining case is that

investor i’s holding is strictly positive for at least one factor portfolio and nonpositive for the

other. Without loss of generality, suppose that investor i’s for factor portfolio B is positive.

Then, we must have µB−PB−σφiB > 0. Otherwise, multiplying both sides of equation (26)

by µ − P − σφi + λi would lead to a contradiction. Now suppose that investor i’s demand

for factor portfolio A is negative. Then λiB = 0, λiA = 2σφiA. Since DiM = 0, we have

λiM = −x(µ− P − σφi) < xA(σφiA − (µA − PA)) < xAσφiA < 2σφiAxA = xλi. Finally, if

investor i holds zero position in factor portfolio A, then

γΩ12DiB = µA − PA − σφiA + λiA,

γΩ22DiB = µB − PB − σφiB.

Dividing yields λiA = −(µA − PA − σφiA) + Ω12(µB − PB − σφiB)/Ω22. Hence

xAλiA + xiBλiB = −xA(µA − PA − σφiA) + xAΩ12(µB − PB − σφiB)/Ω22

> −xA(µA − PA − σφiA) > −x(µ− P − σφi) = λiM .

To summarize, condition (A16) holds for all cases. Consequently, the aggregate demand

39

for the merged firm at price PM is:

DM =

∫ ∫φi∈S3

DiMh(φiA, φiB)dφiAdφiB = (xΩx)−1[x(µ− P − σφ) + λM)

≤ (xΩx)−1[x(µ− P − σφ+ λ)

= (xΩx)−1(xΩx) = 1. (A17)

Thus when there is limited participation, the market demand is less than 1 for the conglom-

erate at price PM = xP . For the market to clear, the price of the conglomerate must be

larger than xP . Moreover, the inequality holds strictly when a positive measure of investors

holds long positions in one factor portfolio but not the other.

40

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[52] Vissing-Jorgensen, A. (1999): “Limited Stock Market Participation and the EquityPremium Puzzle,” working paper, University of Chicago.

44

Figure 1: Asset Price under Limited Market Participation

0 0.2 0.4 0.6 0.8 10.95

1

1.05

1.1

1.15

1.2

1.25

1.3

Average model uncertainty

Pric

e

0 0.1 0.2 0.3 0.41.05

1.1

1.15

1.2

1.25

Supply of asset

Pric

e

0 1 2 3 4 50.95

1

1.05

1.1

1.15

1.2

1.25

Risk aversion

Pric

e

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1.05

1.1

1.15

1.2

Model uncertainty dispersion

Pric

e

Asset price as a function of average model uncertainty (φ), stock supply (x), risk aversion(γ), and model uncertainty dispersion (δ). The baseline parameter values are set atx = 0.25, µ = 1.2, σ = 0.3, γ = 1, φ = 0.5, and δ = 0.4.

45

Figure 2: Proportion of Nonparticipation in the Asset Markets

0.1 0.2 0.3 0.4 0.50.5

0.55

0.6

0.65

0.7

0.75

Asset payoff volatility

Pro

port

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Pro

port

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1

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Pro

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rtic

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0.2

0.4

0.6

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Model uncertainty dispersion

Pro

port

ion

of n

onpa

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Proportion of investors not participating the asset markets as a function of asset payoffvolatility (σ), stock supply (x), risk aversion (γ), and model uncertainty dispersion (δ). Thebaseline parameter values are set at x = 0.25, µ = 1.2, σ = 0.3, γ = 1, φ = 0.5, and δ = 0.4.

46

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