CAES Working Paper Series

Inbound Demand in Fear of Pandemic Disease:
Consumption with Inventories Perspective
Junmin Wan

Faculty of Economics, Fukuoka University, Japan

March 22, 2021

WP-2021-005

Center for Advanced Economic Study
Fukuoka University(CAES)
8-19-1 Nanakuma, Jonan-ku, Fukuoka,
JAPAN 814-0180+81-92-871-6631

Inbound Demand in Fear of Pandemic Disease:
Consumption with Inventories Perspective
Junmin Wan 1
Faculty of Economics, Fukuoka University
March 22, 2021

1The author thanks Shinji Miyake and Masao Ogaki for their useful comments on the earlier version of this manuscript, entitled “ATheory of Consumption with Inventories,” which was presented at conferences held in 2017 by the Japanese Economic Associationand Japanese Association of Applied Economics (JAAE). The author also thanks Kenichiro Ikeshita and Tomoya Sakagami for theircomments when this manuscript was presented at the Meeting of West Branch of JAAE on March 20, 2021. Any errors are theauthor’s own. Correspondence address: Nakanuma 8-19-1, Jonan Ward, Fukuoka City, Fukuoka 8140180, Japan; e-mail:wan@fukuoka-u.ac.jp; telephone: +81-92-871-6631 (ext.4208); fax: +81-92-864-2904.

1

Abstract
I used a continuous-time framework to model the effects of fixed costs, such as
those associated with the coronavirus disease 2019 (COVID-19) pandemic and thephenomenon whereby one type of storable product can have two different prices, on thephenomenon “inbound-outbound tourism demand.” Using closed-form solutions, I
showed that travel frequency, fixed costs, lifetime consumption behavior, and inventory
costs are simultaneously determined. Although the optimal path of consumption showsa smooth curve in the absence of inventory costs, it demonstrates a “saw-toothed” shapewhen time-dependent inventory costs and repeated “explosive purchasing” behavior areconsidered. The COVID-19 pandemic is the most important influence on the recoveryof the global tourism industry, and is likely to also be important for the domestic
market.

JEL classification: D11, E21

Keywords: Inbound, outbound, price difference, inventory, saw-toothed consumption,
storable goods, pandemic, COVID-192

1 Introduction
A reduction of approximately 4% in global gross domestic product in 2020,compared to 2019, was estimated by the International Monetary Fund (IMF, 2020) andWord Bank (2021). On December 17, 2020, the United Nations World Tourism
Organization (UNWTO) declared, “Tourism is back to 1990 levels as arrivals fall bymore than 7%.” A loss of approximately 2 trillion US dollars (approximately 2% of
global gross domestic product) was attributed to this massive drop in tourism, due to thecoronavirus disease 2019 (COVID-19) pandemic. Thus, recovery of the global tourism
industry is essential for full recovery of the global economy. To stimulate tourism, theconcept of “inbound-outbound tourism demand” must first be understood at both micro
and macro levels.
In recent years, China has provided the largest number of outbound tourists
worldwide (UNWTO, 2020). Chinese tourists also tend to spend more money than thosefrom other countries. “Bakugai,” which roughly translates to “explosive purchasing,”
was voted as one of the most popular Japanese words in 2015 in reference to Chinese
tourists. As shown in Table 1, it was reported that the mostly expected item of China
tourist was the shopping at Japan before arriving Japan (Japan Tourism Agency, 2014).
However, the total expenditure of Chinese tourists visiting Japan in 2020 decreased by3

43% relative to 2019, because of the COVID-19 pandemic (UNWTO 2020). This drop
has been the subject of considerable debate in the literature.
[Insert Table 1 here.]
To my knowledge, there has been minimal research concerning the explosive
purchasing behavior of Chinese tourists. In this study, I regard this type of consumer
behavior as rational, and analyze the impact of the COVID-19 pandemic on inboundoutbound tourism demand. I use a continuous-time framework to model the effects of
the COVID-19 pandemic, as well as the effect of international differences in the price ofsortable goods on consumer purchasing behavior. I show that fixed costs, consumption
trends, and the frequency and quantity of purchases are simultaneously determined.
Moreover, the optimal path of consumption has a “saw-toothed” shape. The results
indicate that the COVID-19 pandemic is the most important influence on the recoveryof the global tourism industry, and presumably on domestic markets.
The rest of the paper is organized as follows. Section 2 presents the theoreticalframework. Section 3 presents the results of analyses of inbound-outbound tourism
demand at the macro level, and discusses the policy implications of the pandemic, suchas increased taxation to recoup costs. Section 4 discusses the applicability of the results
to purchasing behavior in convenience stores and shopping malls. Section 5 presents the4conclusions.

2 The Model
2.1 Consumption Inventories[Insert Figure 1 here.]
Let us assume that a consumer (tourist) will pay a higher price, p1, in theirhome country for storable goods (e.g., due to safety issues or market conditions), and alower price, p0, in a foreign country (e.g., due to a lack of taxes) (Fig. 1). For simplicity,
p0 is assumed to be 1. Domestically, the consumer can buy goods at p1 without any
fixed or inventory costs, but in foreign markets must pay a fixed cost, υ , because ofhealth risks posed by the COVID-19 pandemic, as well as a fee for traveling abroad.
There is also an inventory cost (i.e., the opportunity cost associated with buying at p0).
According to Wan (2014), the consumer chooses the inventory period, τ, based on
the price difference for a given product. The marginal cost of storable goods at time 0,p0, will increase with the interest rate, r , and thus becomes
p0 e rτat time τ. This cost should be lower than p1:5(1)p0 e rτ ≤ p1 .

(2)

By assuming that p0 = 1, I now consider a consumer facing the problem
articulated by Ramsey (1928) and Wan (2014), where∞max ∑ ∫ctn =0( n +1)τ

nτ

ln ct e − rt dt

(3)

s.t.
∞

( n +1)τ

∑ e τ  ∫ τ−nn =0n∞
ct dt
dt + υ  = ∫ e − rt ydt
ydt + A00
.

(4)

Under a constant discount rate of r (> 0) , according to Wan (2021), aconsumer chooses consumption behavior ct based on their budget constraints, namely
initial wealth A0 , when there is a constant wage income, y (≥ 0) , and the discount rateis r (> 0) at time t . In this case, the Lagrangian function becomes∞L{ct ,λ } = ∑ ∫n =0( n +1)τ

nτ∞( n +1)τ
 ∞ln ct e − rt dt + λ  ∫ e − rt ydt + A0 − ∑ e − nτ  ∫ct dt + υ  0nτ
 .
n =0


(5)

I then find the optimal solution for λ and ct :[]−1λ* (τ ) = r ( A0 + y / r − υ (1 − e − rτ ) −1 ) ,6(6)ct* (τ ) = r ( A0 + y / r − υ (1 − e − rτ )e − rt + nτ , forn = 0,1,2,...∞ .

(7)

The optimal quantity, I 0 , of goods purchased isτ

I 0* (τ ) = ∫ ct* (τ )dt0,(8)= ( A0 + y / r )(1 − e − rτ ) − υ .

(9)The optimal inventory at time t is

I t* (τ ) = ∫( n +1)τ

nτ + tct* (τ )dt, for n = 0,1,2,...∞(10)= ( A0 + y / r − υ (1 − e − rτ ))(e − r (t + nτ ) − e − r ( n +1)τ )e nτ ,for n = 0,1,2,...∞ .

(11)

The optimal closed-form solutions for purchases, consumption behavior, andinventories are shown in Figure 1. Their paths demonstrate a saw-toothed shape, rather
than a smooth shape.

2.2 Maximum and Optimal Inventory Periods7

2.2.1 Maximum Inventory Period
Based on Eq. (2), the maximum (but not optimal) inventory period, τmax, isτ max = r −1 (ln p1 − ln p0 ) = r −1 ln p1 .

(12)

2.2 Optimal Inventory Period
The consumer’s consumption behavior is transformed to yield∞U (τ ) = max ∑ ∫τ

n =0( n +1)τ

nτ

ln ct* (τ )e − rt dt

ln(r ( A0 + y / r − υ (1 − e − rτ ) −1 ))
= r −1 rτ−1
− 1 + τ (e − 1)
 .

,(13)

(14)

I can then obtain,

Theorem 1:
The optimal inventory period (τ*) is unique, and 0 < τ* < τmax.

The appendix provides a proof.
For example, it can be set that τ* =θτmax where 0<θ<1. When p0=1, A0=200,8υ =5, θ≈0.6 for p1=1.5, θ≈0.2 for p1=3, and θ≈0.1 for p1=10.

2.3 Necessary and Sufficient Conditions
The fixed cost, υ , of a product should be positive according to Arrow et al.
(1951). Without this cost, the consumer could buy the goods at price p0 at any time,and there would be no inventory cost. To determine the sufficient conditions, I compare
the consumer’s total utility with and without inventory:τ*τ*00− rt− rτ −1− rt∫ e ln(r ( A + y / r − υ (1 − e ) ) / p0 )dt ≥ ∫ e ln(r ( A + y / r ) / p1 )dt ,*00(15)

The right and left sides of Eq. (15) are the utility with and without inventory,respectively. From this, I obtain

Theorem 2:
Inbound (outbound) tourism occurs if and only if the following sufficientcondition is satisfied (see appendix for proof):−2( p1 − 1) p1 ( p1 − e − r )( A0 + y / r ) ≥ υ > 0 .
9(16)The necessary condition is p1 − e −r > 0 .

3 Calculation of Inbound-Outbound Tourism Demand
3.1 Demand at a Given Point in TimeLet us also assume that there are n persons in the economy differing withrespect to income y. The income is distributed from 0 to ymax. The first and nth person
earn zero and have the highest possible income, respectively. Thus, I obtain the grossdomestic product (GDP) using 0.5nymax. For simplicity, wealth is also assumed to bezero.
[Insert Figure 2 here.]
At some point in time, as shown in Figure 2, the mth person’s income isassumed to satisfy the necessary condition (Eq. (16)), and I obtain the aggregate
number of inbound-outbound tourists asm-n(17)

The aggregate purchase amount for m-n persons is
I0(m-n) = ∑𝑚𝑚
𝑖𝑖=𝑛𝑛(𝐼𝐼0𝑖𝑖 ).

10(18)

3.2 Demand During a Fixed Period
Purchase frequency (1/τ*i) differs between people due to income differences(see Theorem 1), such that for a fixed period T (e.g., 1 year), the ith person shouldhave T/τ*i or 1/τ*i (for T=1) times of outbound demands. I then obtain

Theorem 3:
Outbound (or inbound) tourism demand in a fixed period can be expressed as1I(m-n) = ∑𝑚𝑚
𝑖𝑖=𝑛𝑛(𝐼𝐼0𝑖𝑖 ) ・(𝜏𝜏∗ ).
𝑖𝑖

(19)

3.3 Policy Analysis
I assume that the government raises the money to develop a new vaccine ormedicine for COVID-19 by collecting taxes. Within the framework presented here, theeffectiveness of the new vaccine or medicine is assumed to reduce fixed cost υ , andthe production function for the new vaccine or medicine is assumed to be υ = υ(income tax). This decreases with income tax ( υ ’ < 0, υ ’’ > 0, υ (maximum incometax) > 0). To maximize the utility of all households in the economy, the government
could use the above model to find the optimal income tax rate with an endogenous
fixed cost υ (although this fixed cost is exogenous at the individual household level).
11

4 Purchasing Behavior at Convenience Stores and Shopping Malls
Domestic sales occur frequently, and consumers will often make large
purchases during sales. Hendel and Navo (2006) analyzed supermarket sales andconsumer inventories, and tested their model using data concerning soft drink
purchases. Feenstra and Shapiro (2001) posited that the consumer price index cannot
be calculated precisely if inventory behavior is not considered. They tested thissupposition using data regarding canned tuna purchases.
Generally, a consumer is likely to encounter two prices for the same product.
For example, a beverage will likely be more expensive at a convenience store than a(less convenient) shopping mall. The theory laid out in this paper could be used tomaximize the utility of the consumer in this context.

5 Conclusions
I constructed a model to explain inbound-outbound tourism demand, and used
it to explain the large and sudden decline in tourism in 2020 associated with theCOVID-19 pandemic. According to the theory presented herein, the COVID-19
pandemic is the most important influence on the recovery of the global tourism
12industry.
The model presented here can also explain purchase and consumption behavior
in the context of a dichotomous choice between a convenience store and shopping mall.
Under conditions of a price difference for storable goods between such outlets,purchasing and consumption behavior were modelled. I have shown that inboundoutbound tourism demand, fixed costs, and lifetime consumption behavior are
determined simultaneously. The optimal path of consumption is saw-toothed in shapebecause of both repeat purchases and time-dependent inventory costs. According to thetheory presented herein, effective treatments for COVID-19, visa exemptions, tax-freeproducts, and general growth of income and assets would likely stimulate tourism
demand while also promoting tourist welfare.
In future research, modeling multiple goods would be useful for understanding
consumption and inventory behavior. Furthermore, the general influence of the supply
sector and pricing behavior should be considered.

13References
Arrow, K. J., T. Harris, and J. Marschak (1951): “Optimal Inventory Policy,”
Econometrica, 19(3), 250-272.
Feenstra, R. C., and M. D. Shapiro (2001): “High-Frequency Substitution and theMeasurement of Price Indexes,” NBER Working Paper Series. 8176.
Hendel, I., and A. Nevo (2006): “Measuring The Implications of Sales and Consumer
Inventory Behavior, ” Econometrica, 74(6), 1637-1673.
International Monetary Fund (2020): “World Economic Outlook Reports, October
2020: A Long and Difficult Ascent,” October 7. https://www.imf.org//media/Files/Publications/WEO/2020/October/English/text.ashx
Japan Tourism Agency (2014): “Consumption Trend Survey on Foreign Visitors inJapan.” https://www.mlit.go.jp/common/001084270.xls
Ramsey, F. P. (1928): “A Mathematical Theory of Saving,” The Economic Journal, 38,543-559.
United Nations World Tourism Organization (2020): “News Release.”https://webunwto.s3.eu-west-1.amazonaws.com/s3fs-public/2021-01/210117executive-council-en.pdf?SkeUfB1u581pV_yC.OLC8Nm0P3B.O.Dd
United Nations World Tourism Organization (2020): “World Tourism Barometer.”14

December, Vol.18(7). Print ISSN: 1728-9246.
https://www.e-unwto.org/doi/epdf/10.18111/wtobarometereng.2020.18.1.7
Wan, J. (2014): “Rational Addiction with an Optimal Inventory: Theory and Evidence
from Japanese Daily and Monthly Purchases,” Chapter 2 of “Consumer Casualties:
Exploring the Economics of Habit, Information, and Uncertainty in Japan,”
Palgrave Macmillan, New York.
Wan, J. (2021): “US-China Trade War: Speculative Saving Perspective,” The Chinese
Economy, Vol. 54(2), pp. 107-123.
https://doi.org/10.1080/10971475.2020.1830230

World Bank (2021): “Global Economic Prospects,” January.
https://openknowledge.worldbank.org/bitstream/handle/10986/34710/97814648161
23.pdf15

Appendix
Proof of Theorem 1:
The first order condition of Eq. (4) is as follows,υ1𝜏𝜏 = r [𝐴𝐴0 − (1 − 𝑒𝑒 −𝑟𝑟𝑟𝑟 )−1 ] + r [1 − 𝑒𝑒 −𝑟𝑟𝑟𝑟 ].

(A.1)

υ (1 − 𝑒𝑒 −𝑟𝑟𝑟𝑟 )2 𝑒𝑒 −𝑟𝑟𝑟𝑟 + 𝑒𝑒 −𝑟𝑟𝑟𝑟 > 0.

(A.2)

To differentiate the right hand side of Eq. (A.1) with respect to τ, I obtain,

For the right hand side of Eq. (A.1),
υ

1[𝐴𝐴0 − (1 − 𝑒𝑒 −𝑟𝑟𝑟𝑟 )−1 ] + r [1 − 𝑒𝑒 −𝑟𝑟𝑟𝑟 ] → −∞ for τ→0+ ,r(A.3)

andυ

1

[𝐴𝐴0 − (1 − 𝑒𝑒 −𝑟𝑟𝑟𝑟 )−1 ] + r [1 − 𝑒𝑒 −𝑟𝑟𝑟𝑟 ] > 𝜏𝜏 𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑟𝑟 −1 ln𝑝𝑝1 for τ = 𝜏𝜏 𝑚𝑚𝑚𝑚𝑚𝑚 ,r(A.4)

then the unique τ*, 0 < τ* < τmax can be found via fixed point theorem.

Q.E.D.

16

Figure 1: (a) Consumption behavior according to product price, and (b) purchase andinventory data over time.
Price(a)p1p0oτoτ2τTime(b)172τ TimeFigure 2: Income distribution and impact of changes in fixed costs associated with theCOVID-19 pandemic.
In 2019,

=α(y)

1/nυo

In 2020,α(ymax)=α(y)1/noα(ymax) in 202018

υ in 2020

Table 1: Expected vs. practiced items of China tourists to Japan before and after arriving in Japan
Items

Expected items before arriving Expected items before arriving
in Japanin Japan(Single choice, %)(Multiple choices, %)Practiced items after arriving
in Japan(Multiple choices, %)Shopping
Eating Japanese foods
Visiting
Walking at street
Enjoying spring water
Staying at hotel
Drinking Japanese wine
Japanese history and culture
Theme park

24.3
20.4
21.5
3.6
8.9
0.9
0.5
3.3
6.0

76.9
78.2
66.3
52.4
53.4
36.3
18.6
16.1
20.7

87.9
94.1
76.4
71.7
60.4
65.0
28.2
15.7
22.9

Enjoying seasonal beauty (Cherry-blossom
viewing, Autumn leaves, snow, etc.)

2.6

14.6

11.1

0.9
1.8
1.4
0.4
1.6
0.9
0.20.1
0.1
0.5
100.0

10.9
8.7
7.2
5.4
5.1
4.3
1.3
0.9
0.8
2.0
480.1

12.4
7.8
4.2
4.9
3.5
3.3
0.3
0.3
0.7
0.1
570.8

MuseumMorden culture (fashion, anime, etc.)
Visiting film or anime locations
Tour on natural, agriculture and fishing villages
Ski and snowboat
Stage appreciation (Kabuki, theater, music, etc.)
Treatment or health examination
Watching sports (Sumo, football, etc.)
Golf
Others
Total

Source: Consumption Trend Survey on Foreign Vistors in Japan in 2014 (calendar year) by Japan Tourism Agency.

19