Discussion Paper No. 1054

IMMIGRATION
AND SECULAR STAGNATION

Kaz Miyagiwa
Yoshiyasu Ono

Revised June 2019
May 2019

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Immigration and Secular Stagnation*

by
Kaz Miyagiwa** and Yoshiyasu Ono†

May 9, 2019
Revised June 4, 2019

Abstract
We examine the effect of immigration on the host country in the dynamic model that can deal
with secular unemployment. Immigration has contrasting effects, depending on the economic
state of the host country. If there is unemployment, immigration worsens unemployment anddecreases consumption by native residents whereas if full employment prevails, immigration
has the effect of boosting consumption while keeping full employment. However, an influx of
too many immigrants can turn the host country into stagnation. We also find that immigrants’remittances are harmful to the host country under full employment but beneficial under secularstagnation.
JEL: F16, F22, F41
Keywords: Immigration, secular stagnation, unemployment, aggregate demand deficiency,
remittances*

The authors thank seminar participants at ISER, Osaka University, and Wayne State University, for theirvaluable comments. Ono acknowledges financial support from the Joint Usage/Research Center at ISER, Osaka
University, and the Grants-in-Aid for Scientific Research (#15H05728), the Japan Society for the Promotion of
Science (JSPS).
**

Department of Economics, Florida International University, Miami, FL33199, U.S.A. E-mail:kmiyagiwa@gmail.com†

Institute of Social and Economic Research, Osaka University, 6-1, Mihogaoka, Ibaraki, Osaka 567-0047,
JAPAN. E-mail: ono@iser.osaka-u.ac.jp.

1. Introduction
In recent years immigration has once again become a contentious issue in the United Statesand other developed economies. It is estimated that more than one million immigrants haveentered the U.S. every year since 2000.1 The situation is not different in the E.U. In 2017, forexample, 2.4 million immigrants entered the EU from non-EU countries.2 Canada, Australia,
and New Zeeland have faced similar situations in recent years. Even Japan is no longer
completely immune to illegal immigration.
Motivated by such resurgences of immigration, we present a new model of immigrationfrom the perspectives of host countries. Although there already exists extensive literatureinvestigating the impact of immigration, most of this, to be reviewed below, has focused on the“real side” of the economy, applying standard microeconomic tools. Departing from thistradition, this paper presents the dynamic model in which the real and the monetary side of theeconomy play key roles. The attractive feature of this model is that we can explain both full
employment and unemployment from the agent’s intertemporal welfare-maximizing behavior
without introducing additional constraints.
In the next two sections we present the benchmark model of a small open country(ignoring immigration). This benchmark model is an extension of the closed-country models of
Ono (1994, 2001), Ono and Ishida (2014) and Michau (2018) to an international setting, where
the host country is open to the world capital and equity markets. All agents are infinitely-lived
and maximize intertemporal utility with respect to consumption of the aggregate good and realmoney balances, given the initial endowments of internationally traded assets.
The model shares two salient features with Ono (1994, 2001). One is that the marginalutility of real balances is bounded away from zero. The other is that, although the labor market
1

https://cis.org/Report/Record-445-Million-Immigrants-20172https://ec.europa.eu/eurostat/statistics.../Migration_and_migrant_population_statistics

1

adjusts according to the Walrasian mechanism, adjustment is not instantaneous. It is worthemphasizing however that it is not sluggish wage adjustment but bounded marginal utility ofreal balances that causes chronic unemployment in our model. Since the nominal wage andprice keep falling as long as there is unemployment, if expanding real money balances candrive their marginal utility so low as to make consumption more valuable than hoarding
money, then consumers start spending, which boosts aggregate demand, creates jobs andstimulates more spending – a process that continues until full employment is restored.
However, if the marginal utility of real balances is bounded away from zero, expanding realbalances can fail to initiate spending, and as a consequence the above propitious process isstymied, condemning the economy to secular stagnation.
The benchmark model yields the following results. The country enjoys full employment if
it holds internationally traded equities below some threshold level, to be specified later.
Otherwise, it suffers from chronic unemployment.
We then turn to our main question: how an influx of immigrants can affect the hostcountry. Immigrants are assumed to have the same preferences as host-country natives but
differ in two respects. First, immigrants enter the host country with a given number ofinternationally traded assets but no host-country currency. Thus, immigrants exchange part ofthe international assets they hold for host-country currency to satisfy their demands for realbalances. Second, immigrants remit part of their earnings to families and relatives back home,whereas natives have no such obligations.
Our key findings can now be stated. (1) If natives are fully employed before immigration,then after immigration they not only remain fully employed but consume a greater quantity of
the good. This result however is predicated on immigrants being neither too many nor too rich(in the sense defined later). An influx of immigrants who are too many or too rich can turn thehost country from full employment into stagnation. (2) If the host country suffers

2unemployment initially, an influx of immigrants always reduces consumption and worsens
unemployment. (3) Immigrants’ remittances also have contrasting effects, depending on theeconomic state of the host country. If full employment prevails, remittances lower
consumption whereas if there is unemployment, they boost both consumption andemployment.
We now review the relevant literature. There exists an extensive volume of literature, bothformal and descriptive, investigating various aspects of immigration and immigration policies.
To save space, we restrict our review to formal studies only. Early literature has treatedimmigration as a case of international factor mobility within standard factor-endowment trademodels.3 Although this approach has yielded many valuable insights, subsequent research hascome to emphasize the distinction between immigration and capital movements. Pioneering inthis line of research, Ethier (1985) has focused on temporary migration – motivated byguest-worker programs administered in West Germany and elsewhere at that time. However,since then most temporary immigrants in Europe have opted to stay permanently in their hostcountries. Also, the majority of todays’ immigrants appear to be permanent settlers rather than
temporary job-seekers. In this paper, therefore, we study the effect of permanent immigration.
More recent work on immigration has turned attention to the presence of unemployment inhost countries, investigating how immigrants and host-country immigration policies can affect
employment of the native labor force.4 To model unemployment, this strand of research has
typically adopted search-theoretic approaches, where unemployment arises as an equilibrium
phenomenon.5 For example, Liu (2010), Chassamboulli and Palivos (2014) and Battisti et al.

3

See e.g., Berry and Soligo (1969), Dixit and Norman (1980) and Markusen (1983).

4

Ethier (1986) is the first to have highlighted host-country unemploymnet in his study of illegal immigration.
Subsequent work on illegal immigration, with and without unemployment, includes Bonds and Chen (1987),
Djaji ́ (1997), Carter (1999), Woodland and Yoshida (2006), Liu (2010), Mangin and Zenou (2016), andMiyagiwa and Sato (2019), among others.
5

Pissarides (2000) is the standard reference for equilibrium unemployment.

3

(2018) have utilized such an approach to examine how an influx of immigrants affects natives’wages and welfare, while Ortega (2000), Miyagiwa and Sato (2019) have studied the effect ofhost country immigration policy under endogenous immigration flows. The present paper is acontribution to this line of research. Departing from these precursory studies, however, thepresent work is unique in that it deals with involuntary unemployment instead of frictional
unemployment.6
The remainder of this paper is organized in 6 sections. The next section describes thegeneral environment of the model. Section 3 presents the benchmark model of a small open
economy and studies its properties in the absence of immigration. Section 4 details how thebenchmark model is adapted in the presence of immigration. Section 5 studies the effect ofimmigration when the host country enjoys full employment prior to immigration. Section 6
extends the analysis to the case of secular stagnation. Section 7 concludes.

2. Environment
Consider a small open host country in a continuous infinite-time horizon. The country
produces the aggregate product with labor and capital according to the neoclassical production
function,, whererespectively. (The time index
functionandare quantities of capital and labor used at time ,is suppressed below, unless ambiguities arise.) Since theexhibits constant returns to scale, we can rewrite it as
,

,

where
≡ / .

6

See Ono (2010) for a formal analysis of Japan's economic slump since the 1990s.

4The host country is open to the world capital market and takes the world equity rate as given.
Capital moves freely across borders to instantaneously adjust the host country’s capital stock
so as to keep the domestic real equity rate locked at .
Perfect competition prevails everywhere. We ignore investment, to keep things simple.
–

Then firms carry no state variables and they maximize momentary profits
at each instant, taking and(the real wage) as given. The first-order conditions are:
,

(1)
.

(2)

Under diminsing returns to factors, these equation uniquely determinethat the equilibriumandWe use the subscripts
All individuals

,

–

and

, given . Note

are independent of time.
and to denote host-country natives and immigrants, respectively.

are endowed with one unit of labor. They derive no utility fromleisure, preferring to supply their entire labor endowments to the labor market. However, they
may not be able to do so due to a demand shortage. To incorporate such possibilities into ourmodel, letdenote the realized rate of employment, with range
0

1.

The typical individual’s realized real labor income is given by. When1 we say there is

full employment; otherwise, unemployment prevails.
All individuals have identical preferences, deriving momentary utility
from consuming

units of the aggregate good and holding real money balances

at time . The subutility functions are assumed to satisfy50,Assumption 1: (a) For all

is strictly increasing, strictly concave and twice

continuously differentiable, and satisfies the Inada conditions; i.e., lim
→

lim

∞ and

0.

→

(b) For all0,

is twice continuously differentiable with positive first derivatives
0 such that ′and weakly concave. Specifically, there isβand

0 for allis strictly decreasing for all.

It is to be demonstrated in the next section that the presence of the lower bound β′

0 on

is crucial for the existence of unemployment as discussed by Ono (1994, 2001) and

Illing et al. (2018).
The representative individual maximizes the utility functional
exp
where

,

(3)

denotes the subjective discount rate, subject to the stock budget constraint and theflow budget constraint:
,

(4)
.

(4) shows that agent can hold his real assets(5)

in two forms: real money balances and real

equities (or bonds). The latter are traded in the world markets and yield the real return perunit (henceforth they are simply referred to as “bonds” and denoted bymoney balances, denoted by
the agent holding.) By contrast, real

, are neither internationally traded nor yield any interest. Thus,

receives the real interest income equal tois the nominal interest rate and≡, whereis the rate of inflation (deflation if negative). (5)describes how real asset holdings change over time (the “dot” over the variable denotes its time6

derivative; e.g.,≡/). The first term on the right is the agent’s labor income. The

second term is the real interest income as shown above. These income terms increase the valueof his assets, while consumption

0 diminish it.

and remittances

The Hamiltonian of the utility maximization problem is given by,
where

is the co-state variable. The first-order conditions are
′

,

′

,
,

which combine to yield the optimality condition that

andmust fulfill at each instant:

,
where

≡

′′ / ′(6)

0. Condition (6) has the intuitive explanation. The left-hand side

represents the intertemporal marginal rate of substitution in consumption, i.e., the desire to
consume (now instead of later). If this desire is less than the nominal interest rate , theindividual would decrease consumption. On the other hand, the right-hand side of (6) measures
the intratemporal marginal rate of substitution between real balances and consumption, i.e., thedesire to hold real balances or the “liquidity premium.” If this desire is greater than , theindividual would sell bonds to increase real balances. Thus, (6) guarantees that no one has the
incentive to change his consumption level and real money balances. In addition to (6), theoptimalandmust fulfill the transversality condition:
lim
→

exp

0.

(7)

Turning to the money market, we assume that the host-country monetary authority keeps
the country’s money supply fixed at(This assumption is slightly modified when discussing

the effect of immigration.) Equilibrium in the money market requires
7

,
where

(8)

denotes the economy-wide real balances andis the nominal price. As we assume

perfect flexibility of , time differentiation of (8) yields.

(9)

The small host country takes the international nominal priceof the aggregate product asgiven. Since the product is traded freely in the international market, the exchange rate adjusts
instantaneously to satisfy.

(10)Further, we make this assumption, common in all standard small-open economy models:

Assumption 2:.

This assumption allows us to use

andinterchangeably. Settingin (6)

yields0,
implying that under assumption 2

is constant over time. This enables us to rewrite (6) as
for

, .

(11)

Finally, we adopt the conventional Walrasian wage adjustment mechanism in the labormarket. More specifically, when there is unemployment, the nominal wage
time according to
1 ,8declines over

where the parameter0 represents the speed of adjustment. 7 When there is full

employment upward wage adjustment is assumed to occur instantaneously. Full employment
or not, equations (1) and (2), together with assumption 2, imply thatto keep the real wage constant (/andmove in tandem. Given that the money supply

does not expand

over time, we can summarize the above price adjustment process as follows:8
1

0

0

for

1,

for
1.

(12)

3. The model without immigration (benchmark)This section presents the model without immigration. To keep the analysis simple, wenormalize the native population to one. With this normalization, the budget constraints (4) and(5) apply to the whole country, with( stands for natives). Substituting from (4) andapplying (9) enables us to rewrite the flow budget constraint (5) as.

(13)

We prove, in the appendix, that the host-country economy is always in a steady state sothat its current account is always balanced. Therefore, if
holdings, taken as exogenous, we havedenotes the native’s initial bond0

. Substituting this in (13) and letting

yields.

(14)

The right-hand side represents the native’s total income, comprising the wage income and theinterest income from his bond holdings. (14) shows that the native consumes all his income.

7
Ono and Ishida (2014) present the microfoundations of wage adjustment mechanism that converges to suchadjustment.
8

Schmitt-Grohé and Uribe (2016, 2017) also assume a similar wage adjustment mechanism.

93.1. The benchmark model with full employment
1 , the benchmark model can be solved

If the host country has full employment (1 in (14) pins down the consumption level:

recursively. First, setting≡.9

Next, since the nominal price is constant under full employment, setting

0 in (11) yields.
Substituting the above

≡

(15)

into (15) determines

under the normalizationof the native population). Then, the money market-clearing condition (8) determines thenominal price , given the money supply. The exchange rate then adjusts to satisfy (10).

Having solved the benchmark model, we ask under what conditions the host country has
full employment. To that end, define by.

(16)

Combining (15) and (16) yields.
Since

by assumption 1(b), the above equation implies that
≡

.

(17)

That is, consumption is bounded from above by . In the next subsection we show that theconverse of this result holds. Thus, condition (17) is both necessary and sufficient for the
existence of an equilibrium with full employment. Moreover, this equilibrium is unique.

Proposition 1: Under assumptions 1 and 2 the model admits a unique equilibrium with full
employment if and only if

9

We assume≡/ to ensure

.

0.

10

Since the native spends all his income on consumption, if he is endowed with a large sumof bonds, his total income (can exceed the limit in violation of condition (17). In

such a case, there cannot be an equilibrium with full employment by proposition 1. We thus
explore the possibility of an equilibrium with unemployment.

3.2. The benchmark model with unemployment
We first characterize an equilibrium with unemployment (0 by (12), the nominal pricebeyond the threshold level1). Since

continuously falls, increasing the real balancessuch that1/

holds. Therefore the optimality condition(11) becomes1.

To keep the left-hand side of (18) positive for any(18)∈ 0,1 , we assume;
that is, the speed of wage adjustment is not too fast.
Since the current account (13) must always be balanced, equation (14) holds despitedeflation. Solving (14) and (18) simultaneously, we can determine the equilibrium
consumption level and employment rate. In figure 1 we depict (18) by the monotone-increasing
curve and (14) by the straight line.10 The intersection point A gives us the equilibrium values,
denoted by

10

∗

and

∗

.

The graph of (18) is strictly upward-sloping but not necessarily concave as drawn in figure 1.

11

We next investigate the conditions for the existence of equilibrium with unemployment.
∗

From figure 1 there is unemployment (0curve at1 and strictly below it at1) if the straight line is strictly above the0.11 The first condition is satisfied if

≡

/ ,

(19)

while the second implies/
Inverting condition (19) gives us

⁄

.

(20)

. This inequality says that, under full

employment, the desire to consume, , is less than the desire to hold real balances, ⁄.

Thus, when (19) holds there cannot be full employment.12 On the other hand, condition (20) is
more technical and rules out the possibility of zero employment.

≡

∗

A

0

∗

1

Figure 1:
Equilibrium with unemployment (benchmark)Figure 1 depicts the case in which
0. That is just for the sake of presentation. If
0, the straightline cuts the -axis at
0. This however does not affect our analysis, given that condition (20) holds.
11

12This proves that (17) is also sufficient for the existence of an equilibrium with full employment, as alluded
to in the paragraph leading to proposition 1.

12

The above discussion suggests that both conditions (19) and (20) are necessary for the
existence of an equilibrium with unemployment. They are also sufficient because, if they bothhold, an appeal to the intermediate-value theorem proves the existence of an equilibrium with
01.
Moreover, if the equilibrium in figure 1 is unique, the straight line must be steeper than the

curve at the intersection point. A little algebra expresses this condition as
≡

0.

(21)

For the remainder of our analysis we assume the equilibrium to be unique so condition (21)holds in its neighborhood.
Finally, we show that the transversality condition (7) is satisfied even though real balances
1 , we can write

keep expanding. Since the price falls at the rate
0 exp

. Furthermore, from (18) we have1

0 .

Substituting these into the first expression below yields
lim→

Becauseexp

stays constant at
lim
→

exp

lim

0.

0 exp

→

, the above implies that
lim

exp

→

0,

proving that the transversality condition (7) is satisfied.
The next proposition summarizes our findings of this subsection so far.

Proposition 2: Under assumptions 1 and 2, there is an equilibrium with unemployment if andonly ifsatisfies conditions (19) and (20). If it is unique, condition (21) holds in its

neighborhood.
13

To understand what causes unemployment, suppose that as
instead of→ ∞, ′

approaches zero

0. Then the optimality condition (18) is replaced by1

As expanding real balances drives ′.

down toward zero,(22)
must keep increasing to holdcondition (22). This continuous rise in consumption creates jobs, raising the employment rateuntil full employment is achieved and the nominal price halts its descent. This shows thatunemployment cannot occur without the boundedness of marginal utility of real balances awayfrom zero.
We now offer an intuitive explanation of Proposition 2. Suppose that the native’s income
happens to equal . Then the line and the curve in figure 1 meet only at≡

the native consuming

and holding real balances

1, with. There is full

employment supported by the optimality condition:.
If the native has a greater income, however, full employment cannot be

maintained. To show this, recall that consumption cannot exceed the limit ; cf. (17).
Consumption cannot be equal to , either, because if so the native’s spending falls short of his
income by

0. If he spends this difference to buy foreign assets, the hostcountry runs a perpetual current account surplus, calling for appreciation of the host-country
currency. Currency appreciation however makes host-country firms less competitive comparedwith foreign firms, reducing employment and hence the income. Actually, because the foreign
and home goods are assumed homogeneous, currency appreciation does not materialize.
Instead, adjustment occurs through changes in the employment rate . To balance the current
account,

must fall enough to bring down the native’s income equal to his actual consumption
14

level; i.e.,. This new consumption

is less than by (16) and (18) because

1.
The above discussion implies thatis the native’s maximum endowment ofbonds consistent with full employment. If he holds fewer bonds, there is full employment by
proposition 1. If he holds more bonds, there is unemployment by proposition 2. We record thisresult in

Corollary 1: (a) If(b) If, the host country has full employment.
, the host country suffers from unemployment.

4. Immigration: an overview
We now extend the benchmark model to study the effect of immigration. To that end weassume the following. At some time (, a given number, say,of immigrants enter thehost country (the subscript denotes immigrants). Immigrants are endowed with one unit oflabor and maximize the utility functional given in (3). As mentioned in the introduction,
however, immigrants differ in two respects. First, the typical immigrant arrives withunits ofinternationally-traded bonds but with no host-country currency. Thus, upon entry into the hostcountry, the immigrant sells bonds (or borrow against their future incomes) in exchange for
local money to satisfy his demand for real balances. If each immigrant acquires

units of thehost-country currency from the monetary authority, the latter holds international bonds totaling. We assume that the monetary authority rebates the interest earnings from these bondholdings evenly to all natives.13 At the end of the day, therefore, it is as if each native has

13

Because natives’ income increases, they too want to increase money holdings. To meet this money demand,the monetary authority purchases bonds with new money, further increasing the country’s money stock.
Simultaneously, it also rebates the interest earnings on the newly acquired bonds to natives. Such adjustment is15

increased his bond holdings to(while each immigrant’s bond holdings has fallen to).
The second way immigrants differ from natives is with respect to remittances. It is
0 units of the aggregate good back home whereas

assumed that each immigrant remits
natives make no such remittances

0 . We takeas given and investigate its effect.
0 asWith the above changes, we can write the native’s flow asset constraint at,(23)

and the immigrant’s as.

(24)Adding up these equations over all the host-country residents (i.e., natives and immigrants
combined) gives us the aggregate flow budget constraint:
1
where

≡

,

(the variables without subscripts henceforth denote the aggregatedvalues). Substituting from (9) and rearranging terms, we can rewrite the above constraint as
0 .

1

(25)

As shown in the appendix, the right-hand side of (25) equals zero because the host country’s
current account must always be balanced

0 .

5. Full employment
Suppose that we have full employment in a post-immigration equilibrium. Since0

under full employment, the optimal consumption and real balances satisfy,(26)

instantaneously completed the moment the host country takes immigrants in. Thereafter, the money stocks
remains constant.

16

where the last equality follows (16). Since, (26) implies that. As in thebenchmark model, the consumption levels are bounded from above by . With the nominal
price constant, the individual asset holdings do not change over time, either. Thus, we set
0 in the flow budget constraints (23) and (24) to obtain
,

(27)
.

(28)

Equations (26), (27) and (28) can be solved for determine the equilibrium levels of
which we denote by ̃ and

and

,

, ).

(

(26) is consistent with two types of equilibria, depending on whether the immigrant’s realbalances exceed. We begin with the case in which. Then (26) can be arranged toyield̃
Substituting for

≡

̃

for

;

′ ∙

0,

.

(29)

from (29) into (28) yields
̃̃

Given the monotonicity ofimmigrant’s labor income.

, (30) determines a unique ̃ , which clearly depends on the, interest income

into (29) to determine the real balances↑ ⇒ ̃ ↑,Let us call the term ((30)

and remittances. We then substitute ̃̃ . Since

0 by (29), (30) implies

↑

.

for

(31)

the immigrant’s “net worth.” Then by (31), when the immigrant

has a greater net worth, he consumes more and demands more real balances. It is
straightforward to show that there is a unique net worth given by≡such that the immigrant consumes ̃,and holds.

17. Then, (31) holds only forIf the immigrant is so rich that his net worth exceeds
equilibrium, in which the immigrant consumes exactly, we have a second type of
while holding. Sincemarginal utility of real balances is bounded by , consuming in excess of puts the desire to
hold real balances above the desire to consume. Thus, the immigrant sells bonds for money,
which reduces his interest income. In the end, his income is brought down to equal

such that.
This last equation determines the immigrant’s money balances:.

(32)

It is evident from the last two equations that
↑ ⇒ ̃

,

↑ for

.

(33)

(31) and (33) demonstrate that an increase in the immigrant’s net worth always raises his realbalances, whetherexceeds

or not.

In section 3 we showed that when natives are rich enough to hold real balances above
full employment fails to exist (proposition 2). In contrast, when the immigrant holds,,

full employment still prevails. This difference can be understood as follows. As already
explicated, immigrants can sell as many bonds as needed to bring their income down to equal. By contrast, natives lack such latitude because, if they sell bonds to reduce their income, themonetary authority would give them back the interest income on the acquired bonds. Thus,
their income remains unchanged whether they sell bonds or not. As detailed in section 3, theirincome is reduced only by a fall in the employment rate.
The preceding discussion showed that full employment persists after immigration.
However, this claim is subject to the qualification that the monetary authority does not acquire
too many bonds from immigrants, i.e.,should not be too large. Otherwise, the monetary

authority would rebate so much interest income to natives that their total income could top the18consumption limit , violating the condition for full employment. Since ̃, (27) gives us

the following result:

Proposition 3: There exists an equilibrium with full employment after immigration only if
,
where

(34)

is determined by (29) and (30) or by (32).

We complete the analysis by calculating for the equilibrium consumption and real money
balances for natives. Plugging the

given above into (26) and (27) yields:
̃

≡

̃ ,

̃

.

The second equation says that ̃ increases with immigrants’ real balances. This is evident

because natives receive more interest income when immigrants exchange more bonds formoney. Moreover, the immigrant with a greater net worth

holds more money; cf.

(31) and (33). These facts establish

Proposition 4: Suppose that (34) holds so that there is full employment before and after
immigration. Then:(a) Immigration increases natives’ consumption.
(b) The richer are immigrants (the more bonds they hold before immigration), the greater isnatives’ consumption.
(c) The greater remittances immigrants send home, the smaller is natives’ consumption.

Proposition 4 has the intuitive explanation. Immigrants exchange international bonds for local
currency. While the host-country currency is worthless in international transactions, the19acquisition of internationally traded bonds allows natives to import and consume more.
Because immigrants convert more international bonds when richer, natives’ consumption rises
with immigrant’s net worth, provided that condition (34) holds.
Notice that (34) implies (17) and hence full employment exists in a post-immigration state
only if there is full employment before immigration. However, this converse does not hold. If
the native is rich enough to consume close to

before immigration, his post-immigration

income may exceed this consumption limit due to the additional interest income

hereceives from the monetary authority. Even if the native is not as rich, an influx of too many or
too rich immigrants can makeso large to produce the same effect. In such cases, if (34)

is violated, we can only have unemployment after immigration, the topic to which we turn inthe next section.

6. Unemployment
We begin this section by characterizing a post-immigration equilibrium with
unemployment. In the presence of unemployment (falling according to (12) so thatthreshold

1 , the nominal wage and price keep

0. The implication is that natives’ real balances exceed the, implying the optimality condition:
1

.
for

The inequality, due to (16), implies that(35)

, . Given the fact that the price iscontinuously falling, (35) suggests two possible scenarios, depending on the size of theimmigrant’s real money balances
the other. In one scenario the immigrant holds

keeps increasing beyond.

20, while in. Then,Consider the case in whichstays constant, implying that the
0 in (24) toimmigrant’s asset holding is not expanding despite deflation. Thus we set
obtain10.

(36)Equations (35) and (36) determine the immigrant’s optimal real balances and consumption interms ofand his net worth; algebra shows that
;

;

0 for

,

,

0 for

,

(37)
.

(38)

The host-country’s current account also remains balanced despite deflation, validating(25). We can substitute from (36), (37) and (38) into (25) to obtain, after arranging,
1

;

.

(39)

This equation and the optimality condition from (35)
1

,

(40)

can be solved simultaneously for the native’s consumption ̂ and the unemployment rate . In
figure 2, the broken curve through point B represents (39) while the solid curve depicts
equation (40). The intersection point B specifies ̂ and . Note that (40) is identical to (18) ininto (37) and (38) gives us the consumption ̂ and real balances

section 3. Substituting thefor the immigrant.

Having solved the model, we can appeal to figure 2 to derive the conditions guaranteeing
the existence of an equilibrium with unemployment. To ensure that

1 than the solid curve does. This requirement is fulfilled if

must take a greater value at
|
where

|

1, the broken curve≡is the value of

1;
1. Sincein (39) for

21,
0, this condition necessarily

holds if≡.

i.e., the host country has unemployment before immigration; cf. corollary 1. Further, to havein 0, 1 requires that
|

≡

0;

,
0.14

i.e., the broken curve takes a smaller value than the solid curve at

↑,

↑

≡

∗

A
̂

B

|

0

∗

1

Figure 2:
Post- and pre-immigration equilibria with unemployment(scenario 1)

It is easy to check from (37) and (39) that an increase in the immigrant’s net worth
shifts the dotted curve up in figure 2. Since (40) is unaffected, the solid curve through A

14

Given condition (20), this condition holds if

is not too large.

22and ̂ decrease as the immigrant’s net worth

remains intact. Hence, we conclude that bothincreases.
Manipulation of (39) and (40) yields,.
Combining these equations and using the definition of

0 given in (21), we find that
0,0.15 Thus, the immigrant’s real balanceswhere the inequality follows fromincreases with his net worth (, reaching

net worth grows still, he must holdat some cutoff net worth. If the immigrant’. In such a case, (37) no longer applies and we turnto the second scenario.
The second scenario occurs only if the immigrant is rich enough to hold real balances
greater thanso that ′. Substituting this marginal utility turns (35) into
1

with

,

(41)

that is, in the second senario the immigrant and the native consume exactly the same amount.
Settingin (25) enables us to rewrite the host-country current account constraint as
1

0.

(42)

The middle expression vanishes as shown because the current account must be in balance, (42)can be arranged to yield.

15

This is readily verified in figure 2.

23

(43)

(41) and (43) jointly determine the equilibrium values of and , which we denote by ̂̂̂ and .
In figure 3, the broken line through point C represents equation (43) whereas the solid

curve traces equation (41). The intersection at C indicates the equilibrium values, ̂. To have 0̂ and1 requires that
,

|

(44a)

.

|

(44b)

These conditions ensure that the broken line lies above the curve near

1 and below it near

0 as in figure 3, which guarantees the existence of an equilibrium with unemployment.

↑,
≡

∗

A
̂

C

|

0

∗

1

Figure 3:Pre- and post-immigration equilibria exhibiting
unemployment (scenario 2)

24↑

To understand the two conditions in (44), notice that the immigrant consumes, where the second equality is due to (14). On the other hand, the immigrant sells
enough bonds to acquire real balances greater thanso his income is at most equal to. Since individuals consume their entire incomes, we have,
which simplifies to.
(45) implies that|in (44) increases with(45)

. Further, since we assume unemployment

before immigration, (19) and (20) hold. Therefore, (44a) necessarily holds while (44b) holds
for a sufficiently small.

Although we have seen two types of equilibria with unemployment, in both cases
immigration worsens the unemployment and reduces natives’ consumption level. To see this,note that as→ 0, both equations (39) and (43) converge to equation (14). In term of figure 2,

this convergence is equivalent to the broken curve approaching the solid line.16 Similarly infigure 3, because (45) holds, the broken line converges from above to the solid line. Since point
A marks the pre-immigration values

∗

and

∗

, taken from figure 1, the above convergence

results imply that when the host country has unemployment initially, an influx of immigrants
moves the equilibrium from A to B or C, thereby decreasing the employment rate and natives’consumption level.
It is easy to see that the immigrant’s net worth also has similar effects in both types of
equilibria. An increase in (shifts up the broken curve in figure 2 and the broken line

in figure 3.17 Thus, in both cases, the employment rate and the native’s consumption fall when

16This guarantees the existence of a unique equilibrium for sufficiently small

17This is evident from (43).

25.

the immigrant’s net worth increases. The next proposition records these results, which stand in
sharp contrast seriatim with those in proposition 4.

Proposition 5: Suppose that (17) is violated so that the host country suffers fromunemployment before and after immigration. Then:(a) Immigration always decreases the rate of employment and the native’s consumption.
(b) The richer are immigrants (in terms of international bond holdings), the lower are theemployment rate and the consumption level by natives.
(c) The more remittances immigrants make, the higher are the employment rate and theconsumption level by natives.

7. Concluding remarks
In this paper we develop a dynamic model of a small open country, where agents
maximize life-time welfare over consumption of the aggregate good and real balances theyhold. The model has two salient features: lower boundedness of marginal utility of realbalances and sluggish nominal wage adjustment. We find the following. (1) In the absence ofimmigration, the host country has full employment if it holds a quantity of international
interest-earning assets below some threshold level. Otherwise, it has unemployment. (2) If thehost country has full employment, an influx of immigrants boosts the native’s consumption
level, provided that immigrants are neither too rich nor too numerous. An influx of too rich or
too many immigrants can give rise to unemployment, however. (3) If the host country suffersfrom unemployment, immigration always worsens the unemployment rate and reduces thenative’s consumption. (4) Remittances by immigrants reduce natives’ consumption when thehost country has full employment. By contrast, when the host country has unemployment,
remittances increase natives’ consumption as well as their employment.

26

Several extensions manifest themselves. First, although we assumed native and immigrant
workers homogeneous, some studies have explored the implications of skill differences
between them.18 If there is a single aggregate good, one straightforward way to introduce theinferiority of immigrant labor into our model is as follows. Assume that the immigrantpossesses only a fraction of (effective) labor compared with the native so that the immigrant
earns a lower wage than the native. In this setting, we expect our results to remain unaffected
qualitatively. Second, our model can be applied to study the effect of emigration on the source
country. More challenging is an extension to the case of two large countries and labor
movement between them. This necessarily introduces interdependence both on the real and themonetary side of the two economies.19 We hope to address these issues in our future research.

18

See, e.g., Liu (2010), Chassambouilli and Palivos (2014) and Battisti et al. (2018), who have used
two-sector models.
19

Ono (2014, 2018) have studied policy interdependence between large countries without consideration of
immigration issues.

27Appendix: Stability
The stability around the full-employment steady state is standard so we focus on the casewith unemployment. As mentioned in deriving (11),stays constant over time in all cases.

Having this property in mind, we first examine the benchmark model with unemployment.
From (18) we have1
Substituting this

⟹

.

(A1)

to (13) gives us
,
≡

where

0,

is given in (21). These two equations indicate that the dynamics ofimplying thatkeeping

0 and stay there,

immediately jump to the levels that makeandat the initial levelis unstable,.

Turning next to the post-immigration steady state with unemployment, consider thesecond scenario from the text, in which both natives and immigrants hold real balances above; i.e.,

for, . In this case, the post-immigration dynamics is given by (42)and the analysis goes through as above, mutatis mutandis, with
/ . Consider next the first scenario, in whichbeing replaced bywhile. In this case,is given by (A1) while (35) yields
1
Applying these

and

,

⟹

,

.

to the dynamics of the immigrant’s asset holdings in (24)and the current account in (25), we obtain
1
1

,

,
,

28

whereandstay constant over time. If they jump so that

and/orare non-zero, either thefeasibility condition or the non-Ponzi game condition is violated. Thus, they initially jump to
the levels that makeandzero and stay invariant thereafter, i.e., (35) and (36) hold.

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32