CIGS Working Paper Series No. 21-005E

Expectations-driven productivity in the layered
markets

Keiichiro Kobayashi (Keio University / The Canon Institute for Global Studies)2021.9※Opinions expressed or implied in the CIGS Working Paper Series are solely those of theauthor, and do not necessarily represent the views of the CIGS or its sponsor.
※CIGS Working Paper Series is circulated in order to stimulate lively discussion and comments.
※Copyright belongs to the author(s) of each paper unless stated otherwise.

General Incorporated Foundation

The Canon Institute for Global Studies
一般財団法人 キヤノングローバル戦略研究所

Phone: +81-3-6213-0550

http://www.canon-igs.org

Expectations-driven productivity in the layered markets(Preliminary and incomplete.)

Keiichiro Kobayashi∗This version: July 08, 2021 (First version: May 2021.)

Abstract
This paper explores a hypothesis that the macroeconomic expectations may affect the aggregate productivity, even in the business cycle frequencies. The economy
consists of the layered markets, in which firms engage in monopolistic competition
with free entry. The firms form the division of labor and produce varieties of goods,
while the number of varieties determines the productivity. The number of varieties
in one market is determined in equilibrium, given the expectations on the numberof varieties in another market. Coordination of the expectations between the layered
markets generates multiple equilibria, corresponding to high and low productivity. A
policy that works on the expectations may change economic organizations and theobserved state of technology.
Key words: Division of labor, monopolistic competition, multiple equilibria.
JEL Classification: E23, E30, O40.

1Introduction
The division of labor is limited by theextent of the market.
Adam Smith (1776) Wealth of Nations

What are the sources of productivity shocks in business cycle frequencies? Can pessimistic
and optimistic expectations affect the productivity by changing actions and coordinationof economic agents? We study a theoretical hypothesis of endogenous changes in theaggregate productivity in response to macroeconomic expectations. We consider the modelof the Dixit-Stiglitz monopolistic competition, where the productivity is determined by
the number of the varieties of goods that can vary in accordance with the number of∗I thank Tomohiro Hirano, Ryuichiro Izumi, and Yuta Takahashi for their insightful comments. Thisis an incomplete version and all errors are mine.

1firms’ entries. Multiple equilibria can emerge in the economy with layered markets ofmonopolistic competition.
The number of varieties can be interpreted as a degree of specialization or the divisionof labor. The extent of the market in the well-known theorem of Adam Smith quoted in
the above epigraph is also represented by the number of varieties in the demand market.
In this paper, we formalize Smith’s theorem as a two-way interaction that the numberof varieties in the demand market (i.e., the extent of the market) affects the number ofvarieties in the supply market (the division of labor) and vice versa. This interaction thus
affects the aggregate productivity.
The key innovation of the model is the layered-market structure in which several markets with monopolistic competition are layered and agents in one market take the prices
and the numbers of varieties in other markets as given. In the rational expectations equilibrium in the layered markets, multiple equilibria can emerge. In one equilibrium, alarge number of firms enter, given the expectations that the number of varieties in theother market is large, because an increase in the number of varieties in the other market
raises the profits of entrants in this market under the layered production. As a result, thenumbers of varieties are large in both markets and the aggregate productivity is high. In
another equilibrium, the numbers of varieties are small in both markets and the productivity is low. Thus, fluctuations in the macroeconomic expectations in the layered markets
can possibly change the observed productivity frequently over time.
Our theory demonstrates that the equilibrium switching between the multiple equilibria may cause the changes in productivity, and it sheds some new lights on the relationship between the expectations and the productivity: the optimistic expectations on theprospects of the economy may raise the aggregate productivity, whereas the pessimism
may have the opposite effect.
The rest of the paper is organized as follows. The literature is reviewed in Section
2. The setup of the model is provided in Section 3. The multiplicity and stability of theequilibria are characterized in Section 4. Section 5 provides the concluding remarks.

2

Literature

Our work is related to the literature of endogenous changes in productivity in the mediumterm business cycles that is analyzed by Evans, Honkapohja, and Romer (1998) and Cominand Gertler (2006). Evans et al. use the Dixit-Stiglitz model of monopolistic competition
to formulate that an increase in the number of varieties of capital goods increases the aggregate productivity. Multiple equilibria arise from the intertemporal expectations on the
future returns on investments in their model, while in our model multiplicity arises fromthe simultaneous expectations on the contemporaneous actions of other agents. “Animal2

spirits” affect the investment decisions in Evans et al., whereas the optimistic or pessimistic expectations affect simultaneous division of labor in our model. While Evans etal. explain the medium-term shift of the trend growth rate as a shift between high and low
growth equilibria, the negative productivity shocks in the short-run are left unexplained.
In our model, a decrease in the productivity in the short-run can be expressed as a shift
from the high to low equilibrium. Comin and Gertler also use the model of Dixit-Stiglitz
monopolistic competition to show that endogenous R&D investment, driven by exogenous
shocks on wage markups, changes the number of varieties, leading to an endogenous productivity fluctuations in the medium-term. In their model, productivity is not driven by
the expectations, but by the fundamental shocks, though it is non-technical, while theproductivity is expectations-driven in our model. Though our model does not allow the(wage) markups to vary as in Comin and Gertler (2006), it would be easily extended so
as to replicate the countercyclical markups.1
The vast literature on the monopolistic competition and the aggregate demand externality (e.g., Blanchard and Kiyotaki [1987], Farhi and Werning [2016]) is related to thispaper, while the literature focus on the external effect of the nominal rigidities on the aggregate demand, and does not focus on the productivity. Our model is a real model that
does not include nominal rigidities, and focuses on the the external effect of the market
size on the entry decisions of firms in the other markets.
The literature on coordination failure and multiple equilibria is closely related (Cooper
and John [1988]). There is a strategic complementarity between the numbers of varieties
in the layered markets in our model. In our model, the complementarity in the firm entry
generates multiplicity in productivity, while multiple equilibria in the existing literature,e.g., Kiyotaki (1988) and Gali (1996), correspond to the multiplicity in investments, notin productivity.
Finally, our model is related to the long tradition of the literature on the division oflabor and the extent of the market (e.g., Stigler [1951]). The original theorem of Adam
Smith cited in the epigraph is an impressive statement that the size of the demand determines the productivity of the supply, represented by the division of labor, which must be arecurrent observation in economic progress in all times and places. The recent literature,e.g., Becker and Murphy (1992), however, emphasize the importance of coordination costs
and general knowledge in forming the division of labor, implying that the nature of the
supply side is crucial. Our model integrates both insights: the size of the demand affects
1

As Comin and Gertler (2006) did, we can assume that the elasticity of substitutions among theintermediate goods is decreasing in the number of varieties. In this case, the markup is low (high) in thehigh (low) productivity equilibrium, which is associated with a large (small) number of varieties. Thus, inthe extended model, the markup would be countercyclical. We could introduce the wage markup in thesimilar manner to show it is also countercyclical.

3the division of labor in the supply market, whereas the productivity in the supply market, which can be interpreted as reduction in coordination costs or an increase in generalknowledge, raises the division of labor in the demand market.

3

Model

We consider a static model of the following layered production of Dixit-Stiglitz technology.
Given the expectations on prices and the numbers of varieties in the layered markets, firms
decide whether to enter the markets. For clarity of analysis, we assume that there are threelayers of production in the economy: L1, L2, and L3.
L1 is the layer for production of the final goods, Y , from the downstream intermediate
inputs, yi , i ∈ [0, N ], where N is the number of varieties of the downstream intermediate
goods, which is endogenously determined in equilibrium. L2 is the layer for productionof the downstream intermediate goods, yi , from the upstream intermediate inputs, zji ,j ∈ [0, M ], where M is the number of varieties, which is also endogenously determined inequilibrium. L3 is the layer for production of the upstream intermediate goods, zji , fromlabor input, lji . The consumer’s problem is
max ln C + γ ln(1 − L),(1)s.t. C ≤ wL + Π + c,(2)

where C is consumption, w is the wage rate, L is the labor supply, Π is the total dividend
from the firms, and c is an endowment, which is given exogenously. In equilibrium,
C = Y + c,
Z NZ Mlji didj,
L=
0
0
Z N
Z M
Π=
πyi di +πzj dj,00where πyi is the profit of a producer of the downstream goods, while πzj is that of the
upstream goods.

3.1

Final goods production (L1)

Production technology is
Z N
Y =
0

ε−1
ε

yi

ε
 ε−1
di
.

Cost minimization problem is
Z N
min
yi

Z N
pi yi di,

s.t.

0

0

4

ε−1
ε

yi

ε
 ε−1
di
≥ Y,

where pi is the price of yi . The solution is p −ε
i
yi =
Y,
P
1
Z N
 1−ε
1−ε
where P ≡
pi di
.
0

1

By definition, P = N − ε−1 p in the symmetric equilibrium.

3.2

Downstream intermediate goods production (L2)

Production technology for yi is
Z M
yi =

zji dj0

Cost minimization problem is
Z M
qj zji dj,
min
zji

σ
 σ−1

σ−1
σ

Z M
s.t.

0

.

σ−1
σ

σ
 σ−1

zji dj0≥ yi ,

where qj is the price of the j-th goods, zji . The solution is
 −σ
qj
yi ,
zji =
Q
1
Z M
 1−σ
1−σ
where Q ≡
qj dj
.
0

In the symmetric equilibrium, Q = M

1
− σ−1

q by definition, and the profit maximization

for a downstream intermediate producer isZ M
πyi = max pi yi −qj zji dj −σq= max py(p) − M q
y(p)
p
Q


Y
q 1−σ −ε1−ε= max ε p
− M −σ p
,
p
P
Q
pi

0

which can be denoted as πy by omitting the subscript i in the symmetric equilibrium. The
solution is

p=

ε
ε−1




qM

q
Q

−σ
.

(3)

The equilibrium value of N is determined by the free entry condition:
πy = κ,where κ is the fixed cost for establishing a new firm, in terms of the final goods. We
assume for simplicity that the fixed cost κ is not a dead-weight loss, but a transfer to thehoueshold, so that the total output Y is not affected by the total payment of the fixed
cost κN in equilibrium.
5

3.3

Upstream intermediate goods production (L3)

Production technology is
zji = lji ,where lji is the labor input. Given the demand zji = (qj /Q)−σ yi , an upstream intermediate
producer solves the following problem in the symmetric equilibrium:
Z N
Z N
lji dizji di − w
πzj = max qj
qj
0
0
Z N
  −σ
q
= max
yi di
(q − w),qQ0which can be denoted as πz by omitting the subscript j in the symmetric equilibrium. Theσsolution in the symmetric equilibrium is q = σ−1w, which implies
1σw,
Q = M − σ−1
σ−1
 −σ  
σ
ε
q
p −ε
zji = lji =Y = M − σ−1 N − ε−1 Y.
QPSimilar to the downstream market, the equilibrium value of M is determined by the freeentry condition:πz = κ,where κ is not a dead-weight loss, but is a transfer from the firm to the households.

4

Equilibrium
1

1

In the symmetric equilibrium, we have p = P N ε−1 , and q = QM σ−1 . Given the solution
σw, the solution for the downstream firm (3) implies thatfor the upstream firm q = σ−1

the price of downstream intermediate good is



1
ε
σ
p=
wM − σ−1 ,ε−1
σ−1which then implies

P =

ε
ε−1



σ
σ−1



1

1

wM − σ−1 N − ε−1 .

The profit of the upstream intermediate producer is
πz =

wY
1

σ

.

(σ − 1)N ε−1 M σ−1

The profit of the downstream intermediate producer is
πy =

σwY
ε

1

(ε − 1)(σ − 1)N ε−1 M σ−1
6

.

(4)

The total labor supply is
Z NZ ML=0

1

1

lji djdi = M − σ−1 N − ε−1 Y.

0Therefore, the aggregate production can be written as
1

1

Y = M σ−1 N ε−1 L,
1

1

1

=w

(5)

implying that the aggregate productivity is M σ−1 N ε−1 . Consumer’s utility maximization
impliesγ(Y + c)1

1 − M − σ−1 N − ε−1 Y(6)Since P is the unit cost of Y , we impose the restriction:
P = 1,(7)and use Y as the numeraire. Then, (4) implies that



1
1
ε−1
σ−1
w=
M σ−1 N ε−1 .
ε
σ

(8)

Thus, (6) implies
1

1

ξ≡

φ
,
φ+γ

Y = ξM σ−1 N ε−1 − η,

(9)

where

φ≡

ε−1
ε



σ−1
σ


,

and

η≡

γc
.
φ+γ

The profit of upstream producer can be rewritten as
πz =

2−σ
1
φξ
ηφ
N ε−1 M σ−1 −M −1 ,
σ−1
σ−1

(10)

and also the profit of downstream producer can be rewritten as
πy =

2−ε
1
σφξ
σηφ
N ε−1 M σ−1 −N −1 .
(ε − 1)(σ − 1)
(ε − 1)(σ − 1)(11)

The equilibrium values of the varieties, (M, N ), are determined by the free entry condition:πz = κ and πy = κ.

4.1

Multiple equilibria

We set the parameter restrictions on the elasticity of substitution, ε and σ:ε > 2,

σ > 2,(ε − 2)(σ − 2) − 1 > 0.

and

7

(12)

This restriction is plausible as the elasticity of substitution among intermediate goods is
set at around 6 in the New Keynesian literature (Nutahara [2021], Khan, Phaneuf andVictor [2020]). This value makes the markup rate around 20% in the New Keynesianmodels, and the micro evidence shows that the industry-level markup rates are actuallyaround 20% (De Loecker and Warzynski [2012]).
For simplicity of exposition, we change the variables and define1m ≡ M σ−1 ,1n ≡ N ε−1 .
Note that the aggregate productivity is mn, as the aggregate production can be written
as Y = mnL. Free entry conditions, πz = κ and πy = κ, are rewritten asπz (m, n) ≡ anm2−σ − bm1−σ = κ,σπy (m, n) ≡[amn2−ε − bn1−ε ] = κ,
ε−1

(13)
(14)

where
a≡

φξ
,
σ−1

and

b≡

ηφ
γφc
=
.
σ−1
(σ − 1)(φ + γ)

The competitive equilibrium in this economy is the set of prices (P = 1, Q, p, q, w) andquantities (Y, y, z, l, m, n) such that the quantities solve for the consumer’s problem andthe profit maximization for the firms in the upstream and downstream markets, and all
markets clear. The equilibrium values of all variables are determined, once the pair (m, n)
is given. Thus, for brevity, we call (m, n) the equilibrium in what follows.
Definition 1. The equilibrium is the pair (m, n) that solves (13) and (14).
We have to note that the free entry condition implies that if πz < κ, then no firms
enter the upstream market and m becomes zero, and that if πy < κ, then no firms enter
the downstream market and n becomes zero. Thus, there may exist a singular equilibrium, which corresponds to the corner solution, i.e., m = n = 0. To avoid unnecessary
complications in our analysis, we simply exclude the corner solution from the equilibriumby assuming the following.
Assumption 1. There exists a shared belief in this economy that in equilibrium there
exist non-zero entries both in the upstream and downstream markets, i.e.,m > 0,

andn > 0.

The following proposition demonstrates the multiplicity of the equilibria.

8

(15)

Proposition 1. Suppose that the parameters satisfy the condition:H(m̃) < 0,

(16)

where

 1
ε−1
ε+σ−2
σ
b (ε − 2)(σ − 2) − 1
H(m) ≡ −m ε−1 ,a(ε − 1)(σ − 1)
ε−1
ε−1


 1  (ε−2)(σ−2)−1
ε−1
ε+σ−2
σ
a
ε−1
 ε−1
m̃ = 
.
(σ − 1)κ

(17)

(18)

Then, there exist two equilibria, (ml , nl ) and (mh , nh ), where ml < mh and nl < nh . If
H(m̃) = 0, the equilibrium is unique. If H(m̃) > 0 there exists no equilibrium.
The condition for multiplicity (16) is satisfied if the outside endowment, c, is sufficiently
small. As c becomes small, the parameter b becomes small proportionately, while all otherparameter values are fixed, (16) is obviously satisfied.
All proofs of propositions are given in the Appendix. The outline of the proof is
the following. In the Appendix, we define a continuously differentiable function F (m),where the solution to F (m) = 0 equals the solution to (13) and (14). F (m) satisfies that
F (0) = ab > 0 and limm→∞ F (m) = ∞. In the Appendix, it is shown that F 0 (m) = 0
has two solutions m = 0, m̃, and F 0 (m) < 0 for 0 < m < m̃ and F 0 (m) > 0 form > m̃. Thus we can say that F (m̃) is the minimum value of F (m). It is also shownthat H(m̃) = F (m̃). Thus, the condition H(m̃) < 0 means that the minimum value ofF (m) at m̃ > 0 is negative, which implies, together with the fact that F (0) > 0 andlimm→∞ F (m) = ∞, that F (m) = 0 has two solutions.
Intuitive explanation:

When the productivity in the downstream market (n) is ex-pected to be small, then the demand for the upstream goods is expected to be small.
Thus, fewer firms enter the upstream market because the expected profit πz is small, anddivision of labor is established to a lesser degree (i.e., m is small). On the other hand,
given the expectations of the low productivity in the upstream market (m) that implies
the scarcity of the upstream goods, fewer firms enter the downstream market (i.e., n issmall), because the expected profit πy is small. Thus, the equilibrium with small m andn is supported by the rational expectations. As the output Y is produced from labor Lby the aggregate production function (5), i.e., Y = mnL, the aggregate productivity ofthe economy (mn) is low in this equilibrium. There exists another equilibrium, which is
described as follows. Given the productivity in the downstream market (n) and hence thesize of the demand is expected to be large, more firms enter the upstream goods market(i.e., m is large), whereas given the expectations of the high productivity in the upstream
9market (m) and hence the abundance of the upstream goods, more firms enter the downstream market (i.e., n is large). Thus, the equilibrium with large m and n is also supportedby the rational expectations. The aggregate productivity is high in this equilibrium.

4.2

Nash stability of the equilibria

For conveniece of notation, we denote an equilibrium by (me , ne ). We have shown themultiplicity of the equilibria, (me , ne ) = (ml , nl ) and (me , ne ) = (mh , lh ). Next, we needto examine the stability of them. We define the following concept of the Nash stability.
∂
πz (me , ne ) <Definition 2. The equilibrium, (me , ne ), is Nash stable, if the inequalities, ∂m∂
πy (me , ne ) < 0, are both satisfied.
0 and ∂nThis condition means that the firms have no incentive to increase or decrease thenumber of firms in the upstream market m from me by changing their entry decisions,
given the number of firms in the downstream market ne , and vice versa. Of course the Nashstability defined above is connected with the concept of the Nash equilibrium. Although mor n are not directly chosen by the firms, their choice whether to enter the market decides
m and n indirectly. In this sense, we can say that the firms in the upstream (downstream)
market choose m (n). The intuition of the Nash stability can be explained as follows: The“equilibrium response” of the upstream (downstream) firms is to choose m = me (n = ne ),in response to the downstream (upstream) firms’ choice ne (me ). To clarify this intuition,
we define the equilibrium response of the firms as follows.
The equilibrium response of the firms:or (13), in the (m, n)-space.

10

Figure 1 shows the graph of πz (m, n) = κ,nπz (m, n) = κ
Rm(n’)n’m0

Figure 1
We define in what follows the equilibrium response m = Rm (n0 ) of the upstream firms
to the fixed value of n = n0 . We first define lm (n0 ) and hm (n0 ) as the two values of m thatsolve πz (m, n0 ) = κ, where lm (n0 ) ≤ hm (n0 ). We show that Rm (n0 ) = hm (n0 ) as follows.
Since πz (m, n) is continuously differentiable, we have the following lemma:
Lemma 2. Given a fixed value of n0 , the equation πz (m, n0 ) = κ has at most two solutions, lm (n0 ) and hm (n0 ), with lm (n0 ) ≤ hm (n0 ), while there is no solution if κ is large
enough. πz (m, n0 ) satisfies that πz (m, n0 ) < κ for m < lm (n0 ) and m > hm (n0 ), and∂∂πz (m, n0 ) satisfies that ∂mπz (lm (n0 ), n0 ) > 0πz (m, n0 ) > κ for lm (n0 ) < m < hm (n0 ). ∂m∂∂and ∂mπz (hm (n0 ), n0 ) < 0 if lm (n0 ) < hm (n0 ), and ∂mπz (hm (n0 ), n0 ) = 0 if lm (n0 ) =hm (n0 ).
Given n0 , the free entry condition implies that only m = lm (n0 ) or m = hm (n0 ) can be theequilibrium response of the firms in the upstream market. Under the (implicit) assumptionthat a firm is an atomic agent, a firm’s deviation of the entry decision does not affect thevalue of m at all and, therefore, both lm (n0 ) and hm (n0 ) can be the equilibrium values ofm. Here, to consider the stability in an more realistic environment, we add the following
assumption that a firm’s entry decision infinitesimally affects the value of m:
Assumption 2. If a firm enters the upstream market, the number of firms changes fromm to m + δm , where δm > 0 is an infinitesimally small number. If a firm enters the11downstream market, the number of firms changes from n to n + δn , where δn > 0 is an
infinitesimally small number.
Under this assumption, m = lm (n0 ) is unstable because new entry of firms inevitably occurs
at m = lm (n0 ) as an entry of a new firm gives the entrant a positive profit: πz (lm (n0 ) +∂δm , n0 ) − κ ≈ ∂mπz (lm (n0 ), n0 ) × δm > 0. On the other hand, m = hm (n0 ) is stable because

no new entry occurs at this point as a new entry of a firm would give the entrant a negative∂
πz (hm (n0 ), n0 )×δm < 0. Thus, the equilibrium response
profit: πz (hm (n0 )+δm , n0 )−κ ≈ ∂mof m to the given value of n0 is only hm (n0 ), under Assumption 2. We have shown thatRm (n0 ) = hm (n0 ),if there exists the solution to πz (m, n0 ) = κ, and Rm (n0 ) does not exist otherwise. Notethat the free entry implies that there would exist the corner solution that corresponds toπz < κ, i.e., Rm (n0 ) = 0, which we eliminate by Assumption 1. Similar argument holds
for the downstream market. Figure 2 shows the graph of πy (m, n) = κ, i.e., (14).

nπy (m, n) = κRn(m’)

0m’mFigure 2
Lemma 3. Given a fixed value of m0 , the equation πy (m0 , n) = κ has at most two solutions, ln (m0 ) and hn (m0 ), with ln (m0 ) ≤ hn (m0 ), while there is no solution if κ is large
enough. πy (m0 , n) satisfies that πy (m0 , n) < κ for n < ln (m0 ) and n > hn (m0 ), andπy (m0 , n) > κ for ln (m0 ) < n < hn (m0 ).
and

∂
∂
0
0
0
∂n πy (m , n) satisfies that ∂n πy (m , ln (m )) > 0
∂
∂
0
0
0
0
0
0
0
∂n πz (m , hn (m )) < 0 if ln (m ) < hn (m ), and ∂n πy (m , hn (m )) = 0 if ln (m ) =hn (m0 ).
12Under Assumption 2, the equilibrium response n of the downstream firms to the fixedvalue of m0 , Rn (m0 ), is given byRn (m0 ) = hn (m0 ),if there exists the solution to πy (m0 , n) = κ, and Rn (m0 ) does not exist otherwise. Figure
3 combines the graphs of (13) and (14) to show m = Rm (n) and n = Rn (m). There are
two intersections of πz (m, n) = κ and πy (m, n) = κ, which are the equilibria, (me , ne ) =(ml , nl ) and (mh , nh ).

nπz (m, n) = κπy (m, n) = κ(mh, nh)(ml, nl)0m

Figure 3
The above definition of the Nash stability can be rewritten as follows:
Definition 20 . An equilibrium (me , ne ) is Nash stable, if the equilibrium responses in theboth markets satisfy that Rm (ne ) = me and Rn (me ) = ne .
Proposition 4. In the case of multiple equilibria, both (ml , nl ) and (mh , nh ) are Nashstable, if the parameters satisfy the following condition:ε+σ−2(σ − 1)b

 1 ,
ε−1
σ
(σ − 2) ε−1
a

(19)

ε+σ−2

(ε − 1)b 1 .
σ−1(ε − 2) ε−1
a
σ

(20)

(ml ) ε−1 >(nl ) ε−1 >Can these conditions, (19) and (20), be satisfied for a reasonable range of parameters?
The answer is yes, and we construct an example of a range of parameters as follows.
13

We consider various values for parameter c, the outside endowment, given that all otherparameter values are fixed. Suppose that c is chosen so that the value of parameter bsatisfies H(m̃) = 0. Note that the value of c affects the value of b only, and does not affectany other parameters. Denote the value of b that satisfies H(m̃) = 0 by bm . For b < bm ,condition (16) is satisfied and the multiple equilibria emerge. We also consider the valueof b (and c) that satisfies (19) and (20), given that the values of all other parameters are
fixed. As we show in the Appendix, there exist bz and by such that (19) is satisfied forb > bz and (20) is satisfied for b > by . We can prove the following lemma.
Proposition 5. There exists positive real numbers, bm , by , and bz , that satisfy 0 <max{bz , by } < bm , and for b ∈ (max {bz , by }, bm ), there exist two Nash stable equilibria, (ml , nl ) and (mh , nh ). For b ∈ (0, max{bz , by }), there exists only one Nash stableequilibrium (mh , nh ).
This proposition can be graphically interpreted in the (m, n)-space as follows: For b ∈(max{bz , by }, bm ), the functions m = Rm (n) and n = Rn (m) have two intersections,which are (ml , nl ) and (mh , nh ), and both of them are Nash stable (Figure 3); For b ∈(0, max{bz , by }), the functions m = Rm (n) and n = Rn (m) have only one intersection,
which is (mh , nh ).
As the variables (m, n) are flow variables such as consumption in the growth model,
any positive values can be chosen and the economy instantaneously jumps to one of thepossible equilibria. Thus, under Assumption 2, the Nash stable equilibrium is the only
equilibrium that can be realized in the model, and Proposition 5 says that there exist two
equilibria for a range of the parameter values.
Numerical example:

The case where ε = σ = 6 is shown in the Appendix. In thiscase,max{by , bz }= 0.882.
bm
For the outside endowment c that makes b satisfy 0.882 × bm < b < bm , there exist two
Nash-stable equilibria that correspond to high and low productivities.

5

Conclusion

We have theoretically shown that the macroeconomic expectations may affect the aggregate productivity. Changes in economic organizations that determine the productivity
such as the division of labor in one market are induced by changes in the expectations on
the extent of the other markets. These changes can be modeled as switching among themultiple equilibria due to sunspot shocks that may occur in the business cycle frequencies.

14

The economy consists of layered markets, in which firms engage in monopolistic competition with free entry. They form the division of labor that determines the productivity,
which is indicated by the number of varieties of the intermediate goods in the market. The
number of varieties in one market is determined in equilibrium, given the expectations on
the number of varieties in another market. As a result of coordination of the expectations
on the numbers of varieties between the layered markets, there emerge multiple equilibria,
corresponding to high and low productivity, respectively: More (Less) firms engage in thedivision of labor and the number of varieties in the market becomes large (small), given
the expectations that the number of varieties in the other market is large (small). Thisis because under the layered production, the profits of entrants in the market is larger(smaller), as the number of varieties in the neighboring market is larger (smaller).
It is confirmed that the multiplicity is robustly observed for a wide range of parametervalues. This model provides a theoretical hypothesis that the technology shocks in thebusiness cycles may be manifestation of the equilibrium switching due to changes in themacroeconomic expectations. Policy implications could be broad, since our result implies
that a policy that works on the expectations may change the economic fundamentals andthe observed state of technology. That is, the optimistic or pessimistic expectations maychange the aggregate productivity in a positive or negative directions. Awaiting future
research is to further clarify the nature of the expectations-driven productivity and better
control the business cycles and economic growth.

References (incomplete)
Becker, Gary S., and Kevin M. Murphy. (1992) “The Division of Labor, Coordination
Costs, and Knowledge.” Quarterly Journal of Economics 107, no. 4: 1137-160.
Farhi, Emmanuel, and Ivan Werning. (2016) “A Theory of Macroprudential Policies in
the Presence of Nominal Rigidities.” Econometrica 84 (5): 1645-1704.

Appendix
Proofs of propositions are given in this Appendix.

15

Proof of Proposition 1
What is to show is that there exits two solutions to simultaneous equations (13) and (14).
ε−1 , and (13) can be rewritten as F (m) = 0, where
These equations imply mσ−1 = ε−1σ nκF (m) ≡ mσ−1 −
a



σ
ε−1

 1

ε−1

ε+σ−2

m ε−1 +ba

The parameter restriction (12) implies that F 0 (m) = 0 has two solutions m = 0, and m̃,and that F 0 (m) < 0 for 0 < m < m̃ and F 0 (m) > 0 for m > m̃. Therefore, F (m̃) =minm≥0 F (m). The necessary and sufficient condition for the existence of two solutionsfor (13) and (14) is F (m̃) < 0. The first-order condition F 0 (m̃) = 0 that holds at m = m̃implies that F (m̃) = H(m̃).

Proof of Lemma 2
Since∂
πz (m, n0 ) = m−σ [(σ − 1)b − (σ − 2)an0 m],∂m(σ−1)b(σ−1)bπz (m, n0 )−κ is increasing in m for m ∈ (0, (σ−2)an0 ), and decreasing in m for m > (σ−2)an0 .

The lemma follows from this fact straightforwardly.

Proof of Lemma 3
Since
σ
∂
πy (m0 , n) =[(ε − 1)b − (ε − 2)am0 n],∂n(ε − 1)nε(ε−1)b(ε−1)bπy (m0 , n) − κ is increasing in n for n ∈ (0, (ε−2)am0 ), and decreasing in n for n > (ε−2)am0 .

The lemma follows from this fact straightforwardly.

Proof of Proposition 4∂∂
The condition for the Nash stability is that both ∂mπz (me , ne ) < 0 and ∂nπy (me , ne ) < 0

are satisfied at the equilibrium values of (me , ne ). The definitions of πz (m, n) and πy (m, n)ε−1
in (13) and (14), together with meσ−1 = ε−1
σ ne , imply that these inequalities can be

rewritten asε+σ−2(σ − 1)b

 1 ,
ε−1
σ
(σ − 2) ε−1
a

ε+σ−2

(ε − 1)b 1 .
σ−1(ε − 2) ε−1aσ(me ) ε−1 >(ne ) ε−1 >As we know that ml < mh and nl < nh , it is straightforward that (ml , nl ) and (mh , nh )are both Nash stable if (19) and (20) are satisfied.
16

Proof of Proposition 5
Here we regard b as a variable and all other parameters are fixed. We redefine F (m) as afunction of m and b, and denote it by F (m, b):
κF (m, b) ≡ mσ−1 −
a



σ
ε−1

 1

ε−1

ε+σ−2

m ε−1 +ba

The condition (16) can be rewritten asm̃ > m∗ (b),
where
 ε−1


∗


m (b) ≡ ε+σ−2

(ε − 1)(σ − 1)b


 1 
ε−1
σ
[(ε − 2)(σ − 2) − 1] ε−1a.

This condition for b can be rewritten as b < bm , where bm is the solution to m̃ = m∗ (b),that is,[(ε − 2)(σ − 2) − 1]
bm =



σ
ε−1

 1

ε−1

a

ε+σ−2

m̃ ε−1 .

(ε − 1)(σ − 1)

The condition (19) can be rewritten as
ml (b) > m∗∗ (b),where ml (b) is the smaller solution to F (m, b) = 0, and
 ε−1



m∗∗ (b) ≡ 

ε+σ−2(σ − 1)b


 1 
ε−1
σ
(σ − 2) ε−1a.

Obviously, m∗∗ (b) < m∗ (b) for all b > 0. For b = bm , the equation F (m, bm ) = 0 has aunique solution and ml (bm ) = mh (bm ) = m∗ (bm ) = m̃ and m∗∗ (bm ) < m∗ (bm ) = m̃. Since
F (m̃, b) is the minimum value of F (m, b), given any b, and F (m̃, bm ) = 0, we know
F (m∗∗ (bm ), bm ) > 0.

(21)

Now we define bz as the solution to F (m∗∗ (b), b) = 0. Thus,




σ
ε−1

−(ε+σ−2) (σ − 2)bz = [(σ − 2)κ] (ε−2)(σ−2)−1 
σ−1

 1

ε−1

(ε−1)(σ−1)
 (ε−2)(σ−2)−1

a


.

The parameter restriction (12) implies that
F (m∗∗ (b), b) > 0⇐⇒
17b > bz .

(22)

The conditions (21) and (22) implies thatbz < bm .

(23)

For b ∈ (bz , bm ), we have F (m∗∗ (b), b) > 0 and m∗∗ (b) < m∗∗ (bm ) < m∗ (bm ) = m̃. Since
F (m̃, b) < 0 is the minimum value of F (m, b), the fact that F (m∗∗ (b), b) > 0 implies thatm∗∗ (b) < ml (b) < m̃. Thus conditions (16) and (19) are satisfied for b ∈ (bz , bm ).
ε−1 , can be rewritten as
The condition (20), together with mσ−1 = ε−1
σ nml (b) > m∗∗∗ (b),
where
 ε−1



m∗∗∗ (b) ≡ 

ε+σ−2(ε − 1)b


 1 
ε−1
σ
a
(ε − 2) ε−1.

Obviously, m∗∗∗ (b) < m∗ (b) for all b > 0. Thus, for b = bm , m∗∗∗ (bm ) < m∗ (bm ) = m̃.
Since F (m̃, b) is the minimum value of F (m, b), given any b, and F (m̃, bm ) = 0, we know
F (m∗∗∗ (bm ), bm ) > 0.

(24)

Now we define by as the solution to F (m∗∗∗ (b), b) = 0. Thus,


σ
by =
(ε − 1)(ε − 2)κ



(ε+σ−2)(ε−2)(σ−2)−1




ε−1
σ

(ε − 2)
σ−1

 1

σ−1

a

 (ε−1)(σ−1)(ε−2)(σ−2)−1

.

The parameter restriction (12) implies that
F (m∗∗∗ (b), b) > 0⇐⇒

b > by .

(25)

The conditions (24) and (25) implies thatby < bm .

(26)

For b ∈ (by , bm ), we have F (m∗∗∗ (b), b) > 0 and m∗∗∗ (b) < m∗∗∗ (bm ) < m∗ (bm ) = m̃. Since
F (m̃, b) < 0 is the minimum value of F (m, b), the fact that F (m∗∗∗ (b), b) > 0 implies thatm∗∗∗ (b) < ml (b) < m̃. Thus conditions (16) and (20) are satisfied for b ∈ (by , bm ).
Therefore, the open interval, (max{bz , by }, bm ), is not a null set, and the condition
for multiplicity (16) and for the Nash stability for both equilibria, (19) and (20), are allsatisfied for b ∈ (max{bz , by }, bm ).

18

Numerical example: In the case where ε = σ = 6, we have thatbz
4=
2 = 0.8399,
bm
3 × 23
2 2
4 × 53 3 65 5
by=
= 0.8817.
bm
3
Thus,

max{by ,bz }= 0.8817.
bm

Finally, we confirm that the high equilibrium (mh , nh ) is always Nash stable. Forb that satisfies b < bm , F (m̃) < 0 and there exists the high equilibrium (mh , nh ) thatsolves F (mh ) = 0 and satisfies mh ≥ m̃. As m∗ (b) > max{m∗∗ (b), m∗∗∗ (b)} for all b andm̃ > m∗ (b) for b < bm , it is the case that mh > max{m∗∗ (b), m∗∗∗ (b)} for b < bm . This∂∂inequality implies that both ∂mπz (mh , nh ) < 0 and ∂nπy (mh , nh ) < 0 are satisfied forb < bm , meaning that the high equilibrium (mh , nh ) is always Nash stable if it exists.

19