This paper examines the effect of demand uncertainty on the properties of the first period contract between a lender and the incumbent, when there is a threat of entry. The main findings are that unlike the cost uncertainty case, entry has no effect on the incumbent's incentives and it leads the lender to learn less.

Neelam Jain

English

Research Papers in Economics

     
 
 
  
  
Volume 29, Issue 1 
  
Lender learning and entry under demand uncertainty 
  
 
 
Neelam Jain  
Department of Economics, Northern Illinois University 
Abstract 
This paper examines the effect of demand uncertainty on the properties of the first period contract between a lender 
and the incumbent, when there is a threat of entry. The main findings are that unlike the cost uncertainty case, entry 
has no effect on the incumbent's incentives and it leads the lender to learn less.
 
Citation: Neelam Jain, (2009) ''Lender learning and entry under demand uncertainty'', Economics Bulletin, Vol. 29 no.1 pp. 100-107. 
Submitted: Dec 12 2008.   Published: February 15, 2009. 
 
     1 Introduction
The purpose of this paper is to consider the impact of demand uncertainty rather than
cost uncertainty on the results of Jain, Jeitschko and Mirman 2002, henceforth JJM. They
examine the interaction between the contracting relationship between an incumbent and a
lender and the classic entry-deterrence game (see Milgrom and Roberts, 1983), when the
incumbent’s marginal cost is its private information. In their case, the agency problem is
that the incumbent wants to signal high cost to the lender (to reduce repayment) but low
cost to the entrant (to show the entrant that entry is unproﬁtable). In contrast, when the
demand intercept is only known to the incumbent, it has the incentive to signal low intercept
to the lender as well as to the entrant. In this paper, I examine the impact of this change in
incentives on the probability of entry, learning and the properties of the equilibrium contract
between the incumbent and the lender.1
I ﬁnd several interesting results that diﬀer from JJM, focusing on the case that is analo-
gous to theirs, namely, when entry occurs if and only if demand is high2. First, the analysis
shows that in contrast to JJM, the threat of entry does not beneﬁt the lender because it
does not weaken the incumbent’s incentives to misrepresent demand to the lender in the ﬁrst
period. Further, with demand uncertainty, in contrast to JJM, probability of entry increases
in learning and thus, the lender chooses to learn less. Thus, the eﬀect of entry on learning
and incumbent incentives depends crucially on the source of uncertainty.
2M o d e l w i t h o u t E n t r y
Following JJM, the inverse demand for the good is given by,
p = a − q +  ,
where a is the source of private information. That is, the true intercept value is only known
to the incumbent. The lender believes that a ∈ {a ,a},a< a, with probability of a = a
being ρ. The slope is normalized to be 1 for simplicity. I similarly use upper bars and
lower bars on prices and quantities to denote these variables under high demand and low
demand respectively. The demand shock   is a random, unobservable term that is distributed
uniformly on the interval [−η,η],w h e r eη>0. There are two time periods with identical
inverse demand functions. Further, the random component of demand is assumed to be i.i.d.
over the two time periods. Costs are assumed to be zero, for simplicity.
1Jain and Mirman, 2002, also study asymmetric information about demand, rather than costs. However,
they only study a static model and thus, their focus is on the eﬀects of diﬀe r e n tm a r k e ts t r u c t u r e sa n d
informational assumptions on contracts, rather than learning and entry.
2In their case, entry occurs if and only if the incumbent’s cost is not low.
1The incumbent chooses quantity q and thereafter   and p are realized. The price p is
publicly observable, but the quantity, q, is the private information of the producing ﬁrm(s)
and unveriﬁable to others. The incumbent requires outside funding in the amount of F
in each period. Thus, contracts are assumed to be short term. A ﬁnancial intermediary,
i.e., a bank, provides these funds in exchange for later repayment of R.F o rs i m p l i c i t y ,a n d
without loss of generality, F is normalized to be zero. The contract between the bank and
the incumbent takes the form of a repayment schedule that maps the observed market price
p into an amount R(p) that the incumbent must pay to the bank. As in JJM, I only analyze
the expected repayment, however.
The parameters of the model satisfy the following assumptions:3
Assumption 1: Belief ρ is such that the incumbent produces a strictly positive output in
each state of demand.
Assumption 2: For any given output, the range of possible prices is large, that is, η is
large enough to generate full learning as well as no learning.
2.1 The Benchmark Second Period
After observing the ﬁrst period price and updating its beliefs about demand, there are only
two possible scenarios for the lender to consider because of the uniform distribution of the
demand shock: either it learns the true demand intercept or learns nothing. I consider the
two cases in turn.
2.1.1 The Bank is Sure about demand
In this case, given full information, the lender simply calculates the “ﬁrst best” level of proﬁt
for each state of demand, and sets this amount as repayment in return for lending funds F
in the second period. Letting ub denote the bank’s second period expected payoﬀ,Io b t a i n ,
ub =
a2
4
. (1)
for a ∈ {a ,a}. If the bank’s beliefs are correct, and in equilibrium they are correct, the
payoﬀ of the ﬁrm is its reservation level of utility, namely 0. It is assumed that this is
enforced through a forcing contract in the second period with substantial penalties for out—
of—equilibrium price observations. If the bank incorrectly believes demand to be low when
it actually is high, the incumbent’s expected payoﬀ is,
u =( a − a)(a − q
∗). (2)
3These assumptions are standard to ensure a non-trivial analysis. See Jeitschko, Mirman and Salgueiro,
2002.
2That is, the ﬁrm’s payoﬀ is its demand advantage over the low demand type multiplied by
the ﬁrst—best level of price of a low demand ﬁrm, p∗ = a − q∗.4
2.1.2 The Bank is Unsure about demand
In this case, the bank maximizes its second period proﬁt, (1 − ρ)R + ρR,w h e r eR is the
expected repayment received when demand is low and R when demand is high, subject to
the incumbent’s individual rationality and incentive compatibility constraints respectively:
R =( a − q)q, (3)
R =( a − q)q − (a − a)(a − q)
Thus, the lender chooses q and q, to maximize,
ρ
¡
(a − q)q − (a − a)(a − q)
¢
+( 1− ρ)(a − q)q.
The ﬁrst—order—conditions are suﬃcient and yield,
q =
a
2
+
ρ
1 − ρ
a − a
2
,
q =
a
2
.
ub =
a2
4
+
ρ
1 − ρ
(a − a)2
4
, (4)
Thus, the contract yields the standard features. Further, a result similar to JJM holds:
b p ≡ ρp +( 1− ρ)p =
a
2
≡ p
∗. (5)
T h i sh a si m p l i c a t i o n sf o rt h ee n t r a n t .
2.2 The Benchmark First Period
In the ﬁrst period, the bank maximizes,
(1 − ρ)R + ρR + Eub, (6)
subject to the two individual rationality and incentive compatibility constraints of the ﬁrst
period5. In order to calculate expected proﬁts, the following Lemma is needed.
4Note that the high incumbent cannot target the low incumbent’s target output because that could lead
to severe penalties. Instead, it is the price target that is mimicked. In JJM, targeting outputs and prices is
equivalent.
5For convenience, the same notation is used here for expected repayments as for the second period.
3Lemma 1 The posterior belief ρ2 is given by,
ρ2(p1|p,p)=
⎧
⎪ ⎨
⎪ ⎩
0, if p1 ∈ (−∞,p − η),
ρ, if p1 ∈ [p − η,p+ η],
1, if p1 ∈ (p + η,∞).
Pr{ρ2 =0 } =( 1− ρ)
(p − p)
2η
,
Pr{ρ2 = ρ} =
2η + p − p
2η
,
Pr{ρ2 =1 } = ρ
(p − p)
2η
.
Thus, the expected value function of the lender is,
Eub =
a2
4
(1 − ρ)
(a − a) − (q − q)
2η
+
a2
4
ρ
(a − a) − (q − q)
2η
+
+
µ
a2
4
+
ρ
1 − ρ
(a − a)2
4
¶
2η − (a − a)+( q − q)
2η
= ρ
µ
(a)2
4
−
(a)2
4
+
(a − a)2
(1 − ρ)4
¶
(q − q)
2η
+ A (7)
where A does not depend on p or p.
I now consider the constraints. Since the low demand ﬁrm’s second period payoﬀ is zero,
regardless of the bank’s beliefs, the individual rationality constraint of this type is as before
and given by Equation 3.
Suppose now that demand is high. Since its individual rationality constraint is slack,
one needs only consider the high-demand ﬁrm’s incentive compatibility constraint. This
constraint diﬀers from the static second period constraint in that the high demand ﬁrm must
be paid up—front its discounted potential gain from deception, equal to (a−a)p∗ =( a−a)
a
2,
see Equation 2. Thus,
R =( a − q)q − (a − a)(a − q) − (a − a)
a
2
(a − a) − (q − q)
2η
. (8)
Substituting the bank’s future expected payoﬀ (Equation 7), and the two binding con-
straints (Equations 3 and 8), into the bank’s maximization problem (Equation 6), the ﬁrst
order conditions of the bank’s problem yield,
q =
a
2
+
ρ
1 − ρ
µ
a − a
2
+ B
1
4η
¶
,
q =
a
2
− B
1
4η
,
4where B =
(a)2
4 −
(a−a)2
(1−ρ)4 −
(a−a)(a)
2 −
(a)2
4 < 0.
Since B<0, the following result follows:
Proposition 1 T h eb a n ks e t st h eﬁrst period outputs further apart (expected ﬁrst period
prices closer together) compared to the ﬁrst period outputs in a static model.
As in JJM, this result derives from two underlying factors: one is the ratchet eﬀect
payment (Equation 2) that the high-demand incumbent requires in order to reveal its type
and the second is the experimentation eﬀect. The probability of future gain for the high
demand incumbent increases as learning increases. To reduce this gain, the lender sets the
price targets closer together. On the other hand, the lender has an incentive to learn so that
its future expected proﬁts increase.6 The Proposition shows that the experimentation eﬀect
is dominated by the ratchet eﬀect.
Note that this experimentation result is diﬀerent from the monopoly experimentation
result of MSU where a monopolist is unable to learn when demand curves are parallel.
There, the reason is that a change in q simply shifts both distributions equally, and thus
reveals no information. Here, the contracting relationship makes it possible for the lender to
learn because the lender can choose two diﬀerent price targets and therefore, inﬂuence the
distribution of posterior beliefs.
3M o d e l w i t h E n t r y
The potential entrant in the second period has the same information as the lender and
therefore, the same updated beliefs. Based on these beliefs, the entrant decides whether or
not to enter the market. As in JJM, ﬁrms compete in quantities if entry occurs.
The entrant incurs a ﬁxed cost of entry, denoted by K,t h a ti tc a nﬁnance internally.
There are no other costs, as is the case with the incumbent. The entrant is also risk—neutral
and maximizes expected proﬁts. The size of ﬁxed cost K determines the entry rule. There
are only two possibilities for entry even though there are three states. Equation 5 shows
that the entry decision must be the same in the two states {0,ρ} because entrant’s proﬁts
are b p2. The expected price needs to be derived for the Cournot game. Since the analysis is
similar to JJM and JM, I simply report the results:
qe =
a
3
,q =
a
2
−
a
6
,q=
a
3
+
ρ
1 − ρ
a − a
2
,
b q =
b a
2
−
a
6
+ ρ
a − a
2
,
b p = b a − b q − qe =
a
3
.
6To see this, note that, by Assumption 1, dEub
d(q−q) = ρ
³
(a)2
4 −
(a)2
4 +
(a−a)2
(1−ρ)4
´
1
2η < 0.
5I limit the analysis to the case when entry occurs if and only if demand is high, so that
probability of entry is inﬂuenced by ﬁrst period decisions.
Proposition 2 The probability of entry increases in learning.
The proof is straightforward from Lemma 1.
This result is the opposite of JJM. Recall that there, entry occurs if and only if cost is
high (ρ2 =0 ) . Since entry decision must be the same for ρ2 ∈ {0,ρ}, probability of entry
decreases in learning in their setting.
Since entry only occurs in the good state, the second period outcomes for ρ2 ∈ {0,ρ}
continue to be given by equations 1 and 4 in the benchmark model. When ρ2 =1 , the
expected payoﬀ of the lender is the Cournot duopoly proﬁt:
ub =
(a)2
9b
.
As in the benchmark model, I obtain,
Eub = ρ
µ
(a)2
4
−
(a)2
9
+
(a − a)2
(1 − ρ)4
¶
(q − q)
2η
+ A
0, (10)
where A0 does not depend on q or q.
Consider now the constraints of the two types of incumbent. It turns out that unlike
the cost uncertainty case, the incentive compatibility constraint of the incumbent ﬁrm with
high demand does not change due to entry. This is because entry does not occur if the high-
demand incumbent pretends to be low-demand. Thus, both binding constraints continue to
be given by the benchmark model and the only change induced by potential entry is in the
expected proﬁts of the bank, given by Equation 10.
The ﬁrst order conditions of the bank’s two-period maximization problem are suﬃcient
and yield the following outputs for the ﬁrst period:
q =
a
2
+
ρ
1 − ρ
µ
a − a
2
+ B
0 1
4η
¶
,
q =
a
2
− B
0 1
4η
,
where B0 =
(a)2
9 −
(a−a)2
(1−ρ)4 −
(a−a)(a)
2 −
(a)2
4 <B<0.
The following result follows:
Proposition 3 The ﬁrst period outputs are set further apart implying less learning in equi-
librium with entry than without.
6This result is signiﬁc a n ts i n c ei ts h o w st h a tt h et h r e a to fe n t r yl e a d st h eb a n kt ol e a r n
less in contrast to JJM. The intuition is straightforward. The signal dampening eﬀect is
unchanged since low demand does not invite entry. At the same time, information is less
valuable since the probability of entry increases as learning increases. Thus the overall eﬀect
of the threat of entry is to decrease learning and thus reduce the probability of entry.7
4C o n c l u s i o n
This paper shows that the source of asymmetric information matters in determining the
eﬀect of entry on learning by a lender. When it is the marginal cost parameter that is
private information of the incumbent, the threat of entry beneﬁts the lender by weakening
the incentive problem. Further, probability of entry decreases in learning, leading the lender
to learn more. When the demand intercept is private information of the incumbent, there is
no beneﬁt to the lender from potential entry and probability of entry increases in learning.
Thus, the lender learns less.
References
[1] Jain, N. Th. D. Jeitschko, and L. J. Mirman (2002) “Strategic Experimentation
in Financial Intermediation with Threat of Entry,” Annals of Operations Research 114
(1-4), 203-227.
[2] J a i n ,N .a n dL .J .M i r m a n(2002) “Agency Games," in Essays in Economic Theory,
Growth and Labor Markets: A Festschrift in Honor of E. Drandakis by George Bitros
and Yannis Katsoulacos, eds., Edward Elgar, 223-242
[3] Jeitschko, Th. D., L. J. Mirman, and Egas M. D. S. Salgueiro (2002) “Experi-
mentation and Signal Dampening in Agency–The Ratchet Eﬀect Revisited,” Economic
Theory 19(3),5 4 9 - 5 7 0 .
[4] Milgrom, R. and J. Roberts (1981) “Limit Pricing under Incomplete Information,”
Econometrica 50, 443—60.
7If ﬁxed cost of entry K is such that entry occurs surely, a similar result obtains overall, that is the lender
learns less.
7

Experimentation and Signal Dampening

i n ,N .a n dL .J .M i r m a n(2002) “Agency Games,&quot;

Limit Pricing under Incomplete Information,” Econometrica 50, 443—60. 7If ﬁxed cost of entry K is such that entry occurs surely, a similar result obtains overall, that is the lender learns less.

Strategic Experimentation in

Lender learning and entry under demand uncertainty

http://www.accessecon.com/Pubs/EB/2009/Volume29/EB-09-V29-I1-P9.pdf

Lender learning and entry under demand uncertainty

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics