Information, Price, and Welfare in a Large Economy Mei Dongy

Janet Hua Jiangz Feb 2012

Abstract We study the role of information transparency - measured as the fraction of informed buyers who can observe prices posted by sellers - in a directed search model where buyers incur ex ante investment to participate in the exchange process. Conventional wisdom suggests that information transparency increases competition among sellers and helps to reduce equilibrium prices. Using a directed search model with capacity constraint and …nite number of agents, Lester (2011) shows that prices may increase, decrease or stay the same as information becomes more transparent. In this paper, we add ex ante investment and …nd that the results in Lester hold without the assumptions of capacity constraint and …nite number of agents. In the presence of ex ante investment by buyers, information transparency has two opposing e¤ects on prices. On one hand, information transparency increases competition among sellers driving the price downward. On the other hand, it induces more buyers to enter the market, and competition among buyers exerts an upward pressure on the price. Which e¤ect is stronger depends on the fraction of informed buyers. In particular, there exist two threshold values. When the fraction of informed buyers is below the lower threshold value, having more informed buyers leads to an increase in prices. As there are more informed buyers, prices start to decrease until the fraction of informed buyers reaches the higher threshold. Further increase in the fraction of informed buyers does not a¤ect prices any more. Keywords: Information; Prices; Search Externality JEL: D4; D8 The authors thank comments and suggestions from participants at the Canadian Economic Association Meetings and the Chicago Fed Workshop on Money, Banking, Payments and Finance. The views expressed in this paper are those of the authors. No responsibility should be attributed to the Bank of Canada. y Bank of Canada; [email protected]. z Bank of Canada and University of Manitoba; [email protected]

1

1

Introduction

Conventional wisdom suggests that information transparency increases competition among sellers and helps to reduce equilibrium prices. Most recently, Lester (2011) shows that in a search model with a …nite number of agents, prices may increase, decrease or stay the same as information becomes more transparent. The extent of information transparency is measured as the fraction of buyers who are informed about the seller’s price posting and therefore can direct their search towards sellers who post the best terms of trade. In this paper, we show that if buyers have to pay a positive entry fee to participate in the exchange process, the result in Lester (2011) may also appear in a large economy with a continuum of buyers and sellers. In the presence of the entry fee, information transparency has two opposing e¤ects on the price. On one hand, information transparency increases competition among sellers and drives the price downward. On the other hand, it induces more buyers to enter the market, and competition among buyers exerts an upward pressure on the price. Which e¤ect is stronger depends on the fraction of informed buyers. In particular, there exist two threshold values. When the fraction of informed buyers is below the lower threshold value, the second e¤ect dominates and having more informed buyers leads to an increase in price. As there are more informed buyers, the …rst e¤ect starts to dominate and prices start to decrease until the fraction of informed buyers reaches the higher threshold. Further increase in the fraction of informed buyers does not a¤ect prices any more. We also investigate the e¤ect of information transparency on welfare in each situation, which is not the focus in Lester. The paper proceeds as follows. Section 2 describes the environment. We solve the equilibrium in Section 3. In Section 4, we examine the e¤ect of information transparency on price and welfare. Section 5 concludes.

2

2

Environment

The economy is populated with sellers of measure 1 and buyers of measure b > 0. Each seller is endowed with one unit of indivisible good. Buyers derive one unit of utility per unit of consumption. Sellers and buyers play a three-stage sequential game. At stage one, sellers post and commit to a price p. A fraction 0

1 of buyers have perfect information

about both the prices and locations of all sellers and we call them informed buyers. The remaining fraction 1

of buyers cannot observe the prices posted by any particular seller

and we call them uninformed buyers. At stage two, buyers decide whether to pay an entry fee 0

k < 1 to participate in the market. At stage three, buyers choose a seller to visit.

Informed buyers conduct directed search and choose the seller promising the maximum expected utility. Uninformed buyers engage in undirected search and choose sellers in a purely random manner. The matching probability depends on the queue length (denoted as Q) at the seller’s location according to the urn ball matching function. The matching probability for the seller is (Q) = 1

e

Q

; and the buyer’s matching probability is (Q) =

(Q)=Q: Once matched, the pair of matched buyer and seller trade according to the posted price p. The seller derives surplus p and the buyer derives surplus 1

3

p from the trade.

The Equilibrium

The equilibrium concept that we use is the subgame perfect equilibrium. We can solve the game backwards in three steps. The …rst step is to solve the buyer’s choice about which seller to visit at stage three. For uninformed buyers, the choice is trivial: they simply choose a seller at random. Informed buyers will choose the seller that promises the maximum expected utility, i.e., they will visit a seller with price p0 and expected queue length Q0 only if the expected payo¤, (Q0 )(1

p0 )

is at least as large as the maximal expected payo¤ from going elsewhere, denoted as VI (the 3

subscript "I" stands for "informed"). The second step is to solve the buyer’s entry decision at stage two. A buyer decides to pay the entry fee and participate in the market if and only if their expected trading surplus exceeds the …xed entry fee. Let VI and VU denote the expected trading surplus for informed and uninformed buyers, respectively. Denote the fraction of informed and uninformed that decide to enter by

I

and

U,

respectively.

The third step is to solve the seller’s price posting decision at stage one. As shown in Lester (2011), sellers either post a low price pL < 1 and serve both informed and uninformed buyers, or post a high price pH = 1 and serve only uniformed buyers. The expected pro…t of serving only uninformed buyers is given by

H

=1

QH

e

where QH =

)b. Sellers

U (1

who choose to serve both types of buyers take the market value of informed buyers VI as given and choose the price pL and queue length QL to maximize expected pro…t

e

QL

The …rst-order condition is VI = e

QL

L

L

<

H

= max pL (1 pL ;QL

) subject to

e QL (1 QL

1

pL ) = VI .

. Sellers choose to post pH if

and randomize between pH and pL if

H

=

L.

Let

H

>

L,

post pL if

denote the fraction of sellers

who choose pL . We can summarize the results above and de…ne the equilibrium as follows. De…nition 1 The equilibrium is a mapping from ( ; b; k) to a pro…le of prices and allocations (VI ;

U ; VU ;

U ; pH ; QH ;

H ; pL ; QL ;

L;

) that satis…es the following equations:

pH = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pH ;seller maximize serving only uninformed VI = e

QL

. . . . . . . . . . . . . . . . . . . pL ;seller maximize serving both informed and uninformed

H

= pH (1

e

QH

L

= PL (1

e

QL

) .............................

)............................... 4

H ;pro…t

for sellers posting high price

H ;pro…t

for sellers posting low price

H; =

L

if

< 1...............................

pL ) 1

VI = (1

e QL QL

, seller choose to post high or low

. . . . . . . . . . . . . . . . . . . . . . . . . . VI , expected trading surplus for informed

VU = VI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VU , expected trading surplus for uninformed VI

k; = if

I

VU

k; = if

U

QH = QL =

4

U (1 I

b

+

< 1 ........................................ < 1 ....................................

U,

I,

informed entry decision

uninformed entry decision

)b . . . . . . . . . . . . . . . . . . . . QH ;queue length consistent with agents’decisions U (1

)b . . . . . . . . . . . . . . QL ;queue length consistent with agents’decisions

The E¤ect of Information Transparency

In this section, we analyze the equilibrium and the e¤ect of information transparency on price and welfare. We conduct analysis in three situations depending on the values of the entry fee, k, and the measure of buyers, b. Situation A is characterized by zero entry fee (k = 0). In this case, all buyers (informed and uninformed) choose to enter (

I

=

U

= 1).

The model reduces to the large economy in Lester. In situation B, there is a moderate entry fee (0 < k < e b ). All informed buyers enter the market while uninformed buyers may or may not enter. In situation C, the entry fee is so large (k

e b ) that even informed buyers

may not enter the market.

4.1

Situation A

In situation A, since there is no entry fee, all buyers choose to enter the market. As shown in Lester, we can analyze the economy in two regions of Case 1: when

is small ( <

1)

divided by

1

=

ln(1+b) b

< 1:

or the fraction of informed buyers is small, some sellers

5

can make a living by serving only the uninformed buyers and charging them the rip-o¤ price pH = 1. As a result, only a fraction of sellers post the low price pL , i.e., all sellers make the same expected pro…ts

H

< 1. In this case,

= . The equilibrium is characterized

=

L

I

=1

by:

U

e

QL

=

= VI

pL (1

e

QL

e QL (1 QL

1

VI =

) = 1

VU =

QH

e

pL )

=

VI

QH = (1 QL =

)b

b + QH

As more buyers become informed ( "), there will be less opportunities to rip o¤ uninformed buyers. In response to this, more sellers choose to post the low price pL (

")

causing the buyer/seller ratio to decrease (QL #). The increased competition among sellers puts downward pressure on the price (pL #). Buyers bene…t and sellers lose (VI and VU ", #). This is the conventional wisdom. Proposition 1 Case 1: as

Proof. From e

QL

";

"; QL #; QH #; pL #; VI "; VU ";

= VI ;

@V =@QL = @QL =@VI =

e 1 VI

6

QL

=

< 0:

VI < 0;

#:

From e

QL

= VI and VI =

1 e QL (1 QL

pL );

VI ln VI ! 1 VI 1 = [(ln VI + 1)(1 VI ) + VI ln VI ] (1 VI )2 1 = (ln VI + 1 VI ) (1 VI )2

pL = 1 + @pL @VI

VI . Since G0 (VI ) =

De…ne G(VI ) = ln VI + 1

1 VI

1 > 0 (note that VI < 1) and G(VI =

1) = 0; we know G(VI < 1) < 0, which implies

@pL =@VI < 0:

From e

QL

= VI and pL (1

QL

e

)=1

e

QH

pL =

Using pL = 1 +

VI ln VI 1 VI

=

1 VI

1

VI + VI ln VI = @ @VI @VI @

e

QH

VI

, we get

1

From 1

!

=

!

= ln VI < 0; =

1 < 0: ln VI

QH

> 0;

= , we have @ @QH @QH @

= e

= eQH > 0:

To sum up, QL ; pL ; ; QH all move in opposite direction to VI :

7

From QH = (1

)b; @QH = @

It then follows that From QL =

@VI @

> 0 and

+ (1

@QL @pL @ ,@ , @ @

b < 0:

< 0:

) b;

= @ @

QL b

2

= > 0

=

1+ 1

4Q

L b

1

+1

1 QL b

32 5

1

!

+1

1 @QL b @

( QbL

1)

2

From VU = VI ; @VU @

=

@VI @ + VI @ @

> 0:

Here let us also investigate how the total number of trade, N , and the total welfare, W , are a¤ected by the extent of information transparency. The total number of trade can be calculated by adding up the number of trade by sellers posting the low price and the number of trade by sellers posting the low price.

N =

(1

e

QL

=

[1

e

( +1

) + (1 )b

)[1

] + (1

e )[1

QH

] e

(1

)b

]:

The total welfare is the di¤erence between the total trading surplus (which is equal to the

8

number of trade) and the total entry fee (remember all buyers enter in situation A):

W =N

We can show that

@N @

and

@W @

> 0 when

bk.

is small and < 0 when

is big. I.e., as more

buyers become informed about prices, the number of trade and total surplus …rst decrease and then increase. The intuition is as follows. In situation A; the region for case 1 starts with = 0 and ends with

=

ln(1+b) . b

When

is small or is very close to the lower bound of the

case 1 region, most buyers and sellers go to the same market, the market with the high price PH . The concentration of agents on the same market results in a proper buyer/seller ratio (the queue length Q is close to b) so that the number of trade is large. It can be shown that the number of trade is maximized when Q = b. As

becomes large in the sense that it gets

very close to the upper bound of the case 1 region (think about the case where

=

ln(1+b) ), b

most agents gather on the same market (the market with pL ) again which results in e¢ cient matching as well. When

is in the intermediate region, agents spread unevenly across the

two types of markets (in particular, QL > b > QH ) which results in lower number of total trade. Case 2: when

is big enough (

1 ),

sellers earn more pro…t by posting the low price.

The equilibrium allocation is given by:

U

=

I

=

VI = VU = pL (1

e

QL

) =

L

QH = (1 QL = b e

b

= V

9

>

=1 1 H

)b

e b (1 b = (1

pL ) e

QH

)

In case 2, all sellers post pL (

= 1). As more buyers become informed (and less are

uninformed) ( "), there is no new force of sellers to join the market with PL . Buyer/seller ratio is constant, the extent of competition is constant, so price and welfare are also constant. Lester (2011) shows that in an economy with no entry fee and with a …nite number of agents, when the number of informed buyers is large relative to the number of uninformed buyers and sellers, sellers all post pL < 1 which may increase with the the number of informed buyers. I.e., Lester acquires the counter-conventional wisdom in case 2 of a …nite economy. In the following, we show that when there is a positive entry fee (0 < k < e b ), the same result may appear in a large economy where there is a continuum of buyers and sellers.

4.2

Situation B

When entry is costly, buyers face a non-trivial entry decision. Since informed buyers can observe price postings and choose sellers with the best terms of trade, they enjoy higher expected trading surplus than uninformed buyers. When the entry fee is small and more speci…cally 0 < k < e b , all informed buyers choose to enter (

I

= 1). The uniformed buyers

may or may not enter. In this case, we can analyze the model in three cases de…ned by two thresholds values of : 0 <

0

<

1

=

ln(1+b) . b

We can analyze the economy in situation B

in three cases. Case 0: if

2 (0;

0 ),

only a fraction of uninformed enter (

U

< 1) because not enough

sellers post PL . Case 1: if

2 [ 0;

Case 2: if

2 [ 1 ; 1], all sellers post pL ( = 1).

1 ),

all buyers enter (

U

= 1) and a fraction of sellers post pL ( < 1)

The discussion of cases 1 and 2 is the same as in situation A except that the e¤ect on the number of trade and total welfare in case 1 is a but di¤erent. In situation A; we show

10

that

@N @

and

@W @

to be …rst < 0 then > 0 in the region of case 1. In situation B; we have

the same result if the entry fee k is small. As k becomes larger, we can show that @W @

@N @

and

> 0 in the whole region of case 1. In case 0; the equilibrium (QL ; VI ; a;

I

e

QL

= 1 = VI

VI = pL (1

e

QL

is characterized by:

U ; pL )

e QL (1 QL

1

) = 1

VU =

e

QH

U (1

QL =

)b

b + QH

In this case, all informed buyers enter the market ( U

=

VI = k

QH =

market (

pL )

I

= 1), not all uninformed enter the

< 1) and only a fraction of sellers choose to post PL ( < 1). As more buyers

become informed ( "), more sellers choose to post the non-rip-o¤ price ( "). As a result, uninformed buyers are less likely to be ripped o¤ so more uninformed buyers choose to enter (

U

") the market. The buyer base expands, the queue length increase (Q "), and

increased competition among buyers result in a higher price (pL "), lower welfare for buyers (VI #; VU = k) and higher welfare for sellers ( "). Proposition 2 Case 0: when

<

0 ;as

Proof. Same as in case 1, we have 1

"; ;

U ; pL

and QL ", and VI # :

VI + VI ln VI = , and QL ; pL ; ; QH all move in

opposite direction as VI : From VI = k ! @ =@VI =

k VI2

< 0.

The equation that characterizes VI as a function of ( ; b; k) is (using pL = 1 + 11

VI ln VI 1 VI

=

1 VI

)

1

e

VI + VI ln VI

=

e

QH

(QL

b)

VI + VI ln VI

= 0!

VI e k VI

VI + VI ln VI

= 0!

1

b

e k VI

ln VI

= 0!

@VI @

From QH =

@QL @ , @ @

L , @p , @

)!

U (1

e

QH

!

VI + VI ln VI !

b

It then follows that

=1

U

@ @

@ @

=

@QH @

,

QH 1

U

=

=

b b V e k VI k I 1 b kb VI e k VI

< 0:

> 0:

! @QH 1 @ 1

+

QH )2

(1

> 0:

Let us now investigate how the total number of trade and total welfare are a¤ected by increased information transparency in case 0. Remember the total number of trade can be calculated as N = (1

e

The derivative of N with respect to @N @

= (1 =

e

e

QL

)

QL @QL

@

@ +( e @ + (1

QL

) + [1

e

QH

](1

):

is given by @QL @ (1 e QH ) + (1 @ @ @QH @ )e QH + (e QH e QL ) @ @ QL

)

)e

QH

@QH @

> 0:

I.e., as information becomes more widespread, the total number of trade increases. 12

The total welfare W is the total trading surplus (which is equal to N ) minus the total entry cost (the measure of entering buyers multiplied by the entry fee per entrant):

W = (1

e

QL

) + [1

e

QH

](1

)

[ +

= (1

e

QL

) + [1

e

QH

](1

)

[ QL

= (1

e

QL

QL k) + [1

e

QH

U (1

)]bk

QH + QH ]k )!

QH k](1

It can be shown that the total welfare increases with the extent of information transparency.

@W @

= (e

QL

= (e

QL

= (e

QL

= (e

QL

@QL @ @QL k) @ @QL k) @ @QL k) @ k)

+ (e

QH

k)(1

+ (e

QH

k)(1

+ (e

QH

k)(1

+ (e

QH

k)(1

@QH @ @QH ) @ @QH ) @ @QH ) @ )

QH

+ [e + [(e + [e +e

+ QH k

QH

QL

QL

e

Q

e

QL

QL k]

) + (QH

@ @

QL ) e

QL + (QH

QL ) e

[ QH + (1

)QL ]

QL

]

Q

]

@ @

@ @

@ @

> 0:

4.3

Situation C

If the entry cost exceeds a certain value, i.e., if k > e b , even the informed buyers may choose not to participate in the market. In this situation, all sellers must post low price to induce even informed buyers to enter ( = 1). As a result, the uninformed receive the same treatment as informed: VU = VI = V = k and transparency is similar to that in case 2:

I

=

U

=

< 1 . The e¤ect of information

does not a¤ect the economy. The equilibrium is

13

characterized by:

= 1 V

b

= e

Q = VI =

=k

b 1

e Q

= p(1

Q

(1 e

Q

p)

)

It turns out that in this situation, case 0 and case 1 do not exist. The intuition is as follows. In case 0 and case 1, only a fraction of sellers post, which means that some uninformed should enter. If informed earn VI = k and not all sellers post, it must be that uninformed buyers earn negative net surplus so uninformed buyers will not enter, which contradicts the initial statement that some uninformed should enter.

5

Conclusion

In this paper, we show that contrary to the conditional wisdom, more widespread information can cause the price to increase. Lester has a similar result by restricting attention to a small …nite economy. Our model shows that when there is a positive entry fee, the result may also occur in a large economy with a continuum of buyers and sellers. When buyers have to pay an entry fee to participate in the market, information transparency has two o¤setting e¤ects on the price. Information transparency induces sellers to compete among themselves by cutting prices. At the same time, information transparency also induces more buyers to enter the market, which raises competition on the demand side and causes the price to rise. The e¤ect of information transparency on price is not monotone and may be positive, negative or zero. When the fraction of informed buyers is low, the 14

second e¤ect dominates and having more informed buyers leads to an increase in the price. Total welfare, however, increases even with the higher price. When the fraction of informed buyers lies in an intermediate region, the …rst e¤ect dominates and more widespread of information causes the price to decrease. The total welfare …rst increases and then decreases with the extent of information transparency in this region. When the fraction of informed buyers is high, a further increase in information transparency has no e¤ect on the price level or total welfare.

References Lester, Benjamin. 2011. "Information and Prices with Capacity Constraints." American Economic Review, 101(4): 1591–1600.

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