ISSN (Print) 0473-453X

Discussion Paper No. 1204

ISSN (Online) 2435-0982

ECONOMIC STIMULUS EFFECTS
OF PRODUCT INNOVATIONUNDER DEMAND STAGNATION

Daisuke Matsuzaki
Yoshiyasu Ono

March 2023

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

March 2023

Economic stimulus effects of product innovation
under demand stagnation*
byDaisuke Matsuzaki† and Yoshiyasu Ono‡
Abstract
When faced with economic stagnation, innovation, product innovation in particular, isoften cited as an effective stimulus because it is thought to encourage household consumption
and lead to higher demand. Using a secular stagnation model with wealth preference, we
examine the effects of product innovation on employment and consumption. Two types of
product innovation are examined: quantity-augmenting-like innovation and addictive
innovation. The former works as if a larger quantity were consumed although the actual
quantity remains the same. The latter reduces the elasticity of the marginal utility of
consumption. We find that the former reduces both consumption and employment whereas
the latter expands them.
JEL classification: O33, E31, E24
Keywords: demand stagnation, product innovation, consumer price index, economic stimulus*)

The authors wish to thank M. Akesaka, K. Hashimoto, N. Hidano, N. Kikuchi and R. Mikami for helpfulcomments. This work was financially supported by the JSPS KAKENHI Grants (JP20H05631 and
JP20K01661), the program of the Joint Usage/Research Center for “Behavioral Economics” at ISER of Osaka
University, and the Asahi Glass Foundation. The authors are solely responsible for any errors.
†

Faculty of Economics, Toyo University, 5-20-28 Hakusan, Bunkyo, Tokyo 112-8606, JAPAN. E-mail:matsuzaki@toyo.jp‡

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
Many advanced countries have experienced long stagnation with aggregate demand
shortages, and innovation is often emphasized as an important economic stimulus along withfiscal and monetary policies.1 There are two kinds of innovation: process innovation andproduct innovation. The former improves production efficiency whereas the latter improvescommodity quality and utility. With respect to process innovation, Ono (1994, 2001)
proposed a model of secular stagnation owing to wealth preference,2 and showed that in thepresence of aggregate demand shortages, increases in productivity arising from process
innovation decrease labor demand, worsen deflation and consequently reduce employment
and consumption. In contrast, product innovation increases the utility of consumption; thus
it seems to enhance consumption and employment. However, we find that this is notnecessarily the case.
This study uses Ono’s secular stagnation model with liquidity (or wealth) preference,
mentioned above, to examine the effects of product innovation on consumption andemployment. It finds that there are different types of product innovation and that they may
increase or decrease consumption and employment because they increase not only the valueof consumption, which encourages households to consume more, but also the value of money(financial assets), which encourages households to hold money.

1

For example, action plan for a “New Form of Capitalism” in Japan, creation of a new Department for
Science Innovation and Technology in Britain, CHIPS and Science Act in the USA, “France 2030”, and creationof the German Agency for Transfer and Innovation (DATI).
2

Recently, Michau (2018) and Michaillat and Saez (2022) have also treated a model with wealth preference
and found that secular demand stagnation occurs because the marginal utility of wealth stays strictly positive.
Illing, Ono and Schlegl (2018) and Mian, Straub and Sufi (2021) have considered wealth preference andanalyzed depressed aggregate demand.

1

Product innovation is an important issue for economic growth, but almost no studies
have examined the effect of product innovation on aggregate demand under demandstagnation. 3 As economic stimulus policies, fiscal and monetary policies are much more
broadly discussed. To fill this void between theoretical research and policymaking, we study
the effects of product innovations that directly influence the utility of consumption on
aggregate demand and employment.
After providing a general formula for the effects of product innovation on employment
and real consumption, we specify two types of product innovation. One type of productinnovation makes households feel as if they were consuming a larger quantity of a given
commodity despite actually consuming the same quantity. We call this quantity-augmentinglike product innovation. The other reduces the elasticity of the marginal utility of
consumption, which we describe as addictive innovation. By applying utility functions that
include these innovations in the general formula, we explore the effects of the two productinnovations on employment and consumption.

2. A single-commodity economy
First, we consider an economy with a single commodity 𝑥, where 𝑥 is its quantity, andproduct innovation. A representative household’s utility of consumption is represented by𝑈𝑢 𝑥; 𝛾 ; 𝑢 ≡

0, 𝑢

3

0, 𝑢

0,

(1)

Since Krugman (1980), product innovation has often been discussed in the context of “love of variety”
and treated as a factor influencing firms’ location choices and economic growth. Recent examples are Baldwin
and Martin (2004), Davis and Hashimoto (2016), Iwaisako and Tanaka (2017) and Kane and Peretto (2020).
However, it is not regarded as a demand stimulus. Johdo (2008) may be the only exception. While he considers
the love of variety and examines the effect of an increase in product variety on aggregate demand, we focus on
product innovation that enhances the utility of an existing product.

2where 𝛾 is a product innovation parameter. Innovations typically require development costs
and resources, which affect economic activity through changes in both aggregate supply anddemand: a crowding-out effect without demand shortages, and a demand-stimulus effect in
the presence of demand shortages. However, in order to focus on the pure effect of productinnovation on economic activity, we ignore these cost effects and analyze the effect of an
exogenous rise in 𝛾.

2-1. Firms
A representative firm is competitive and produces 𝑥 using only labor with constantproductivity 𝜃. Given that the nominal wage is 𝑊 and that the nominal price of 𝑥 is 𝑃 , thefirm will infinitely expand employment and output if 𝑃 𝜃𝑃𝜃

𝑊 but will cease production if

𝑊. Therefore, to achieve equilibrium in the commodity market, it must always hold

that
𝑃𝜃

𝑊.

(2)

2-2. Households
Real total consumption 𝑐 is given by𝑐𝑝 𝑥, 𝑝,(3)

where 𝑝 is the real price of 𝑥 and 𝑃 is the consumer price index (CPI). Using (3) we rewrite
consumption utility 𝑈 given by (1) as follows:
𝑈

𝑢

;𝛾 .

3

When product innovation occurs and 𝛾 changes such that 𝑢 increases, the consumer priceindex 𝑃 must fall and change the real commodity price 𝑝

𝑃 /𝑃 such that the level of realconsumption 𝑐 that yields the same utility remains unchanged.4 Mathematically, a rise in 𝛾
varies 𝑝 so that 𝑈 is unchanged for a given 𝑐𝑝 𝑥 ; hence, using the properties of 𝑢 𝑥; 𝛾

given in (1) we obtain0

𝑢 𝑥

𝑢 𝑑𝛾 ⟹

0.

(4)

Intuitively, product innovation that increases utility raises the value of 𝑥 so that the CPI(which is 𝑃 falls if the nominal price 𝑃 is the same, and consequently, 𝑝

𝑃 /𝑃 rises.

The household’s lifetime utility is then represented as
𝑈 𝑐; 𝛾
𝑣

0, 𝑣

0, 𝑣

𝑣 𝑚

𝑣

∞

0, 𝑣

𝑈 𝑐; 𝛾 ≡ 𝑢

𝑎 𝑒

𝑑𝑡,
0, 𝑣

0,

;𝛾 ,

where 𝜌 is the subjective discount rate, 𝑣 𝑚 is the utility of money 𝑚transaction motive and 𝑣(5)

𝑀/𝑃 for the𝑎 is the utility of holding assets 𝑎 owing to wealth preference.

Assets 𝑎 consist of real money 𝑚 and interest-bearing asset 𝑏 with the nominal interest rate𝑅𝑟𝜋, where 𝑟 is the real interest rate and 𝜋𝑃⁄𝑃 is the CPI inflation rate. The

household’s labor endowment is normalized to unity but unemployment may appear, andthen the realized labor supply (= demand) 𝑛 is less than 1. Therefore, the flow budget
equation and asset constraint are

4

This approach is consistent with the hedonic approach. The hedonic hypothesis presented by Rosen (1974)
is used for the impact of product innovation on real prices. Feldstein (2017) shows that it is used to estimate the
CPI.

4

𝑎

𝑟𝑎

𝑤𝑛

𝑅𝑚

𝑐

𝑧,

𝑚

𝑏,

𝑛

1,
(6)

𝑎

where 𝑤 is the real wage and 𝑧 is government transfers.
The Hamiltonian function 𝐻 of the household behavior mentioned above is𝐻

𝑈 𝑐; 𝛾

𝑣 𝑚

𝑣

𝑎

𝜆 𝑟𝑎

𝑤𝑛

𝑅𝑚

𝑐

where 𝜆 is the co-state variable and 𝑝 is invariant over time. Given that 𝑥𝑧,
𝑐/𝑝 , the first-

order optimal condition is
𝜌

𝜉

𝜋

;
;

where 𝑈 𝑐; 𝛾

𝑅

;

, 𝜉≡

;

,
(7)
0.

The left-hand side is the intertemporal marginal rate of substitution in consumption 𝑐, themiddle is the marginal benefit of holding money 𝑚 and the right-hand side is the marginal
benefit of holding interest-bearing asset 𝑏. The transversality condition islim 𝜆𝑎 exp
→

𝜌𝑡

0.

(8)

2-3. Markets
The commodity market equilibrium is
𝜃𝑛.

𝑥

(9)

Given that 𝑝 is constant over time, from (2) and (3), we have
≡𝜋

5

.

When full employment (𝑛1) prevails, 𝑥

𝜃, and 𝑊 moves in parallel with 𝑃. In thepresence of demand shortages, in contrast, 𝑊 falls slowly, depending on the deflationary gap,
and 𝑃 moves in parallel with 𝑊:
𝜋

𝛼 𝑛

1

if 𝑛

1,

(10)

where 𝛼 is the adjustment speed of 𝑊. This is because workers do not accept rapid falls in
𝑊 but welcome any increase in 𝑊, no matter how rapid it is.
Firm equities have no value because firm profits are zero under linear technology andthe net supply of bonds is assumed to be zero. Therefore, the total supply of 𝑏 is zero:
𝑏

0, 𝑎

𝑚.

The nominal money supply is 𝑀 and its expansion rate is 𝜇

𝑀 ⁄𝑀 . The governmenttransfers seigniorage 𝜇𝑚 to households:𝑧𝜇𝑚 .

From the money market equilibrium condition, 𝑚𝑀 ⁄𝑃𝑚, we obtain 𝑚⁄𝑚𝜇𝜋. As 𝑊 ⁄𝑊 must be positive for the adjustment of 𝑊 to support the full-employment steadystate, the inflation rate must be non-negative; hence,𝜇Given that 𝑎0.

𝑚, the dynamics of 𝑚 mentioned above and the dynamics of 𝑐 representedby (7) formulate autonomous dynamics with respect to 𝑚 and 𝑐:;𝜋𝜌where 𝑣 𝑚 ≡ 𝑣 𝑚

𝑣

𝑚 ,
(11)

𝜇

6

𝜋.

3. Product innovations
Having established the model, we now examine the effects of product innovation. Before
dealing with the case of stagnation, we consider the standard case with full employment,
where 𝑛1 and 𝑥𝜃 from (9), as a benchmark. From (4), a rise in 𝛾 raises the real price𝑝 ; therefore, both real consumption 𝑐𝑝 𝜃 ) and the utility of consumption 𝑢 𝜃; 𝛾

increase.

Proposition 1: Under full employment, product innovation increases real consumption andthe utility of consumption.

3-1. Stagnation steady state
Full employment, mentioned above, may not be reached. In the full employment steadystate, where 𝑛1, 𝑐0 and 𝑚
𝜌

and 𝑝

𝜇

0, from (9) and (11) it must hold that;;

where 𝑈 𝑐; 𝛾

,

(12)

1 because 𝑃 is initially set equal to 𝑃 . However, if the marginal utility of asset

holding has a positive lower bound 𝛽, namely,𝑣 ∞ ≡𝑣 ∞where 𝑣 ∞

𝑣

∞

𝑣

∞

𝛽

0,

0 from (5), and if 𝛽 satisfies
𝜌

𝜌

𝜇

;

;

7

for any 𝑃 ,(13)there is no 𝑃

𝑃 that satisfies (12), implying that a full-employment steady state does notexist.5
If (13) is valid, the left-hand side of (12), representing the desire for consumption, isalways smaller than the right-hand side, representing the desire for wealth (or money)
accumulation, or the wealth premium, when 𝑥

𝜃 . Therefore, household demand forcommodity 𝑥 decreases below 𝜃, which leads to demand shortage and causes 𝑊 and 𝑃 to
fall. In this case, 𝑊 (and 𝑃 as well) falls in a sluggish manner, represented by (10), and 𝑣′ 𝑚eventually reaches 𝛽. From (7), (9), (10) and (11), 𝑛 in the stagnation steady state satisfies6
𝜌

𝛼 𝑛

1

;

.

(14)

Since (13) implies that the left-hand side of (14) is smaller than the right-hand side when𝑛

1 (or 𝑥𝜃 , in order for the solution of 𝑛 ∈ 0,1 to exist, the former is larger than thelatter when 𝑛

0 (or 𝑥0 . Therefore, the former is less inclined than the latter as 𝑛increases. These properties are mathematically represented as follows:
𝜌

𝛼,

𝛺

1

0.

(15)

Note that deflation continues and 𝑚 diverges to infinity in this steady state; however, thetransversality condition (8) is valid as long as 𝑚/𝑚
𝜋

𝛼 𝑛

𝜇

𝜋 is smaller than 𝜌.7 Given that

1 , from (14), the transversality condition is equivalent to5

See Ono (1994,2001) and Michau (2018), who show the condition for the stagnation steady state to occur.
Using parametric and non-parametric approaches, Ono, Ogawa and Yoshida (2014) empirically show that theutility of financial wealth is insatiable. Ono and Yamada (2018) use survey data and empirically find the validityof a status preference for money that makes the marginal utility of money constant and causes secular demandstagnation to occur.
6

See Ono (1994, 2001) for the proof of saddle stability in the neighborhood of the stagnation steady state.

7𝜌 𝜇, both the full-employment steady state and the stagnation steady state can
When 𝜌 𝛽/𝑢 𝜃; 𝛾
hold. See Ono and Ishida (2014) for details on the conditions for the two steady states to appear.

8

𝜇

;

.

Totally differentiating (14) and applying (4), (9) and (15) to the result gives the effects
of product innovation 𝛾 on total employment 𝑛 and real consumption 𝑐 as follows:
𝛺

𝛺

1 ,
(16)

𝛺

𝛺

𝛺

1

.

The inequality in the second line of (16) implies the following property:

Lemma: Suppose that unemployment occurs because of a shortage of demand. If productinnovation increases employment, real consumption also increases.

3-2. Various product innovations
We consider two types of innovation: one works as if consumption of the commodity
increased although the actual quantity is unchanged, whereas the other decreases theelasticity of the marginal utility of the commodity. The former is quantity-augmenting-likeinnovation, whereas the latter is termed addictive innovation. We apply these to (16) andexamine their effects on total employment 𝑛 and real consumption 𝑐 (or utility 𝑈).
We first assume the following utility function:𝑢 𝑥; 𝛾ℎ 𝛾𝑥 ; ℎ 𝛾𝑥0, ℎ 𝛾𝑥0.

(17)This implies that innovation works as if the consumption of the commodity increased by afactor of 𝛾. From (17), we obtain the following two properties:𝑥𝑢𝛾𝑢

𝛾𝑥ℎ 𝛾𝑥

1

0,

9

0.

Applying these properties to (16) yields
𝛺

1,
(18)

𝛺
Noting that 𝛺𝛺0.

0 from (15), these properties give the following proposition:

Proposition 2: If unemployment occurs because of a shortage of demand, quantityaugmenting-like product innovation, which affects utility as shown in (17), worsens both total
employment and real consumption.

Quantity-augmenting-like innovation makes households feel as if they were consuming
𝛾 times as much of the commodity as before, even though they are consuming the same
quantity. If they then decrease their consumption of the commodity by a factor of 𝛾 and savethe remaining income, both the marginal utility of consumption and that of money remainunchanged because the latter is constant at 𝛽 in the stagnation steady state. Therefore, thewealth premium is the same as before and equals the time preference rate 𝜌𝜋 (representing

the desire for consumption). However, households decrease real consumption because thedemand for labor to realize the same real consumption decreases and deflation worsens,
which makes it more advantageous for households to lower real consumption and increase
money accumulation. Eventually, both total employment and real consumption decrease.
Intuitively, quantity-augmenting-like innovation increases both the value of thecommodity and the value of money by a factor of 𝛾 because the value of money is measured
by how much of the commodity can be purchased. Subsequently, the marginal utility of
consumption decreases if the quantity of the commodity is held constant, while the marginal10

utility of money is unchanged to be 𝛽, so that the commodity is consumed less and total
employment decreases, which worsens deflation and reduces real consumption.
Next, we consider addictive innovation using the following utility function:𝑢 𝑥; 𝛾
𝑢

𝑥

𝑥

1);

𝑥 ln 𝑥10, 𝛾 𝑢𝜉
We regard 𝑥(where 𝛾

1

𝛾

0, 𝑢

𝑥

1,

(19)

ln 𝑥.

1 as the survival level of consumption and examine the case where𝑐/𝑝

1 because an increase in innovation 𝛾 does not affect utility (𝑢0) when 𝑥

increases utility (𝑢0 when 𝑥

1 and

1. We describe an increase in 𝛾 as addictive productinnovation because it works in such a way that it reduces the elasticity of the marginal utilityof consumption 𝜉

1𝛾 .

Noting that initially 𝑝

1, and that 11

𝛺

0 from (15), we apply (19) to (16)and obtain𝑢𝜓 1; 𝛾𝑥 𝛾 𝛺𝜓 𝑥; 𝛾 ≡ 𝑥 1

0, 𝜓 ∞; 𝛾0, 𝜓 𝑥; 𝛾

𝛺

𝛾

𝛾 𝑥

𝛺

𝛾 ln 𝑥111,
𝛾 ln 𝑥 ;(20).

Noting that 𝜓 is positive (and 𝑑𝑛/𝑑𝛾
and 𝜓 1; 𝛾10) when 𝑥

1 is close to 1 because 𝜓 1; 𝛾

00, and negative when 𝑥 is sufficiently large,8 we summarize the above

result as follows:

8This is obvious if 𝛾

0 because lim 𝑥

l’Hôpital’s rule and find 𝜓 ∞; 𝛾1→∞. However, if 𝛾0.

110, lim 𝑥→

0. In this case, we use

Proposition 3: If the utility of consumption is given by (19) and unemployment owing to
demand shortage appears, addictive product innovation may increase or decrease total
employment and real consumption.

Using consumption data for G7 countries, we examine whether addictive productinnovation increases employment and real consumption. Because 𝑥

1 is regarded as thesurvival level of consumption, as aforementioned, the threshold 𝑥 that satisfies 𝜓 𝑥 ; 𝛾
0 in (20) also represents the ratio of the threshold level to the survival level. By comparing
this with the ratio of household consumption per capita to the poverty line for G7 countries,
we identify whether the addictive innovation leads to higher employment and realconsumption.

Table 1: the threshold 𝑥 below which 𝑑𝑛/𝑑𝛾
𝜉

1

𝛾

𝑥

0

0.2

0.3

0.4

1.5

2

496

96.3

41.2

4.60

3.51

Table 2: the ratio of household consumption per capita to the poverty line (G7 countries)

household survey
mean/poverty line
household final
consumption/
poverty line

US
UK
Germany France(2010) (2011) (2011) (2011)

Canada(2010)

Italy(2011)

Japan(2015)

2.87

2.11

2.00

2.14

2.38

2.01

unavailable

4.27

2.78

2.33

2.21

2.56

2.71

1.93

(Data source) Appendix 2 of Jolliffe and Prydz (2016). They do not provide data for Japan; therefore, we use
the poverty line in the report of Japan's Ministry of Health, Labor and Welfare (in Japanese), p.25, given byhttps://www.mhlw.go.jp/toukei/list/dl/20-21-h28_rev2.pdf, and household final consumption expenditure percapita estimated from the World Bank data, https://data.worldbank.org/indicator/NE.CON.PRVT.KN andhttps://data.worldbank.org/indicator/SP.POP.TOTL.

12

Table 1 shows the ratios of the threshold level to the survival level for various elasticities𝜉1𝛾 because various levels of elasticity of the marginal utility of consumption areproposed in the literature: approximately 0.3 in microeconomic approaches such as Harrison
et al. (2005), Anderson and Mellor (2008) and Meissner and Pfeiffer (2022), and 2 in the
DSGE literature.9 Table 2 shows the ratios of average household consumption to the poverty
line for G7 countries. From the two tables, we find that the ratios in Table 2 are almost always
smaller than 𝑥 , and the only exception is the case where we adopt 𝜉

2 and the ratio of US

household final consumption to the national poverty line. This implies that addictive productinnovation will almost always increase total employment and then, from Lemma, realconsumption will also increase.

4. A two-commodity economy
Let us now turn to an economy with two commodities, 𝑥 and 𝑦, and see if the results
obtained in the single-commodity economy are still valid. We implicitly consider that 𝑥 is
the commodity on which product innovation takes place while 𝑦 is the composite of all othercommodities.

4-1. The model

9

Hall (1988) and Guvenen (2006) state that 1/𝜉 to be much smaller than 1. Chiappori and Paiella (2011)
also state that the median of the relative risk aversion 𝜉 is 2. This is because in the standard framework without
wealth preference a low relative risk aversion is inconsistent with a response of 𝑐/𝑐 to a change in the interest
rate. However, in the presence of wealth preference this problem does not appear. See Appendix B for details.

13Firms are competitive and produce commodity 𝑗𝑥, 𝑦with constant laborproductivity 𝜃 . Their profit maximization behavior leads to𝑃𝜃
where 𝑃 and 𝑝

𝑊 and 𝑝 𝜃

𝑤 for 𝑗𝑥, 𝑦; 𝜔 ≡constant,(21)

𝑃 /𝑃 are respectively the nominal and real prices of commodity 𝑗 .

Nominal wage 𝑊 moves as mentioned in (10). Since 𝑃 and 𝑃 satisfy (21), they move inparallel with 𝑊, and consequently 𝑃 also does, implying that 𝑤

𝑊/𝑃 is constant overtime.
The utility of consumption is assumed to be homothetic so that the CPI does not dependon total consumption. It is represented by𝑈𝑢 𝑥, 𝑦; 𝛾 .

Given real consumption 𝑐, the representative household maximizes 𝑈 subject to
𝑝 𝑥

𝑝 𝑦

𝑐,

(22)

and satisfies
, ;

, ;

, ;

, ;

𝜔 ⇒

𝜓 𝛾 ,

where 𝑢 implies the partial derivative of 𝑢 with respect to variable 𝑗(23)𝑥, 𝑦, 𝛾 and 𝜔 isconstant from (21). Therefore, we obtain𝑝 𝑥

𝛿 𝛾 𝑐,𝑝 𝑦1

𝛿 𝛾 𝑐, 𝜓 𝛾

,
(24)

𝛿 𝛾

, 𝛿′ 𝛾

𝜓′ 𝛾 ,

while 𝑈 is represented as a function of 𝑐 and 𝛾, which satisfies

14

𝑈 𝑐; 𝛾

𝑢

,

;𝛾 ,
(25)

, ;

, ;

𝑈

0,

𝑝 𝑈

𝑢 𝑥

𝑢 𝑦

0.

Given that the consumer price index 𝑃 must be defined such that a change in 𝛾 does notaffect 𝑈 in (25) for a given 𝑐using (23), we find that 𝑝

𝐶/𝑃 , by totally differentiating (25) where 𝑐 is fixed and
𝑃 /𝑃 satisfies
𝛿

Note that if 𝛿

0.

(26)

1 (i.e., the single-commodity case), (26) is equivalent to (4). The

household’s intertemporal behavior is the same as that in the single-commodity economy
represented by (5) and (6) except that 𝑈 𝑐; 𝛾 is given by (25). Thus, the first-order optimal
condition is represented by (7) where 𝑈 is given by (25). From (21), (24) and the market
equilibria of the two commodities:𝑥𝑛 𝜃 , 𝑦we find that each sector’s employment 𝑛 (𝑗
𝑛

𝛿𝑛, 𝑛

𝑛 𝜃 ,

𝑥, 𝑦 and total employment 𝑛
1

𝛿 𝑛; 𝑛

.

𝑛

𝑛

are

(27)

Before analyzing the case of unemployment, we consider the case of full employment(𝑛1) as a benchmark. From (25) and (27), 𝑐 and 𝑈 are
𝑐

𝑝 𝜃

𝑝 𝜃 ,

𝑈

𝑢 𝜃 𝛿 𝛾 ,𝜃 1

𝛿 𝛾 ;𝛾 .

Therefore, using (21), (23) and (26), we obtain
0,

15

𝑢

0.

(28)

These properties imply that Proposition 1, which holds in the single-commodity economy,
also holds in the two-commodity economy; that is, product innovation increases realconsumption if full employment is achieved in the steady state.

4-2. Stagnation steady state in the two-commodity economy
In the full-employment steady state, we still have (12) where 𝑈 is given by (25) and 𝑐satisfies (28). Therefore, if (13) holds, the full-employment steady state does not exist andthe stagnation steady state represented by (14) where 𝑈 is given by (25) is reached. Then,
using (27), we obtain
𝜌

𝛼 𝑛

1

,

, 𝑈 𝑐, 𝛾

,

;

(29)

.

For the same reason that (15) is derived from (14), we have
𝜌

𝛼,

𝛺

1

0.

(30)

Note that these conditions are the same as in (15) if only commodity 𝑥 exists (i.e., 𝑦0).

From (26), (27), (29) and (30), we obtain
𝛺

𝛺

1 ,
(31)

𝛺

𝛺

1

where the derivation is given in Appendix A. Note that, if 𝑦,0, the two equations in (31)are the same as those in (16), which are valid in the single-commodity economy. From (26),(30) and (31), we find 𝑑𝑐/𝑐 /𝑑𝛾

𝑑𝑛/𝑛 /𝑑𝛾, implying that Lemma also holds in thetwo-commodity economy.
First, we consider quantity-augmenting-like product innovation by assuming thefollowing utility function:16

𝑢 𝑥, 𝑦; 𝛾ℎ 𝛾𝑥, 𝑦 .

𝛾𝑥ℎ 𝛾𝑥, 𝑦0, 𝑢It satisfies
𝑥𝑢

𝛾𝑢

1
where ℎ 𝑖ℎ 𝛾𝑥, 𝑦

, 𝛾𝑢

𝑥𝑢

0,

𝛾𝑥ℎ ,

1,2 is the partial derivative with respect to the 𝑖-th factor of ℎ 𝛾𝑥, 𝑦 . By

applying these properties to (31) and noting that 𝑢

and 𝑈 in (25) are negative while 𝛺 in(30) is positive, we obtain
𝛺

𝛿 1

𝛿 1

𝛺

0.

0,

They imply that Proposition 2 also holds in the two-commodity economy: quantityaugmenting-like product innovation worsens both total employment and real consumption.
We next examine addictive product innovation using the following utility function:𝑢 𝑥, 𝑦; 𝛾(where 𝛾𝜀

𝑢
𝑢

𝑥𝑢

,
, 𝑢

0, 𝛾
𝑦𝑢

𝜀

𝑥 𝑦 ln 𝑥,
1

,

1);

𝛾

(32)
𝜀 ,

where addictive product innovation is represented by an increase in 𝛾. By substituting thevalues in (32) into (31), we obtain

17

𝑢

𝑥 𝛾

𝜀 𝛺

𝜓 𝑥; 𝛾, 𝜀 ,𝜓 𝑥; 𝛾, 𝜀 ≡ 𝑥
𝜓 𝑥; 𝛾, 𝜀

1
𝛾

1

𝜀 𝑥

𝛾

𝜀

1

ln 𝑥
1

𝛺

𝛺

1,

𝛾

𝜀

ln 𝑥 ;(33).

From (32) and (33) we derive the minimum 𝑥 that holds when 𝑐 equals the survival level, thethreshold 𝑥 above which an increase in 𝛾 leads to greater employment and real consumption,and the ratio of these two.
First, we obtain the minimum 𝑥. Applying 𝑢 and 𝑢 in (32) to (22) and (23) yields
𝑝 𝑥

𝑐,

𝑝 𝑦

𝑐,

𝑦

𝑥.

(34)

As mentioned at the outset of this section, 𝑥 represents the product on which productinnovation takes place while 𝑦 represents the composite of all other products. By aligning
the quantity units of 𝑥 and 𝑦 in such a way that the relative price 𝜔 is equal to one,10 we canregard 𝑦/𝑥𝜀/ 𝛾𝜔

as the quantity ratio of the composite to the innovative product thathouseholds rationally determine. This is naturally larger than 1:
1,

𝜔

1.

(35)

By applying the third equation in (34) to 𝑢 given in (32) and using (35), we obtain𝛾

𝜀 𝑢𝜑 𝑥; 𝛾, 𝜀 ≡ 𝑥

𝜑 1; 𝛾, 𝜀

10

ln 𝑥

1

0, 𝜑 ∞; 𝛾, 𝜀

1

1

0,

0,

In other words, the unit is the quantity of each commodity that is produced by a unit of labor.

18

𝜑 𝑥; 𝛾, 𝜀

𝛾

𝜀

𝑥

ln 𝑥

0.

Therefore, 𝑥 and its minimum 𝑥 satisfy
1𝑥𝑥 where 𝜑 𝑥 ; 𝛾, 𝜀We next obtain the threshold 𝑥
threshold 𝑥 at which 𝜓 𝑥 ; 𝛾, 𝜀𝜓 𝑥 ; 𝛾, 𝜀

1
𝑥

𝛾

0).

(36)

. From (33), (35) and (36), 𝜓 𝑥; 𝛾, 𝜀 , the0, and the minimum 𝑥𝑥

𝜓 𝑥 ; 𝛾, 𝜀𝑥0 (⇔ 𝑢

1

𝛾

𝜀

ln 𝑥

𝜀

𝑥

given in (36) satisfy

ln 𝑥
0,

1

𝜓 ∞; 𝛾, 𝜀0.
0,which implies
𝜓⋛0 ⇔

⋛0

for 𝑥 ⋚ 𝑥

where 𝑥𝑥 .

(37)

From (37) and Lemma, we find that Proposition 3, which holds in the single-commodity
economy, also holds in the two-commodity economy: addictive product innovation may
increase or decrease total employment and real consumption.
Under homothetic utility, the ratio 𝑥 /𝑥 is equal to the consumption ratio of thethreshold and survival levels. Table 3 lists the ratios calculated for different pairs of 𝛾, 𝜀 .
Note that 𝛾 and 𝜀 satisfy (35) and that the elasticity of the marginal utility of realconsumption 𝜉 is 1
𝑈 𝑐; 𝛾, 𝜀𝛾

𝜀 because substituting them into 𝑢 𝑥, 𝑦; 𝛾 in (32) yields
, 𝜉≡

1

𝛾

𝜀

0.

The parameter values in Table 3 satisfy these properties.1111

We do not treat the case where the elasticity of the marginal utility of consumption 1
𝛾 𝜀 is larger
than 1 in the two-commodity case. This is because the effect of a rise in 𝛾 on the demand ratio of the two

19

Table 3: the survival consumption level and the threshold (the two-commodity case)

𝜉

1

𝛾

𝜀

0.2

0.3

0.4

𝜀/𝛾

1

10

100

1

10

100

1

10

100

𝑥 /𝑥

496

162

150

96.3

30.8

28.5

41.2

13.4

12.4

By comparing 𝑥 /𝑥 in Table 3 with the G7 countries’ data in Table 2, we find that thethreshold ratios presented in Table 3 are all higher than the ratio of household consumption
per capita to the poverty line for G7 countries in Table 2, which implies 𝑑𝑛/𝑑𝛾

0 for G7

countries. Therefore, using Lemma, we find that addictive innovation increases both
employment and real consumption in the two-commodity economy as well as in the singlecommodity economy.

5. Concluding Remarks
Product innovation increases the utility of consumption and is expected to stimulate
consumption. Therefore, it is often cited as an economic stimulus when a country faces
secular stagnation because of a lack of aggregate demand. However, it increases not only thevalue of consumption but also the value of money because the value of money is measured
by the commodities it can buy. Thus, depending on how it affects utility, it may increase or
decrease real consumption and employment.

commodities is quite unrealistic. See Appendix B for details.

20

We consider two types of product innovation: quantity-augmenting-like and addictive
product innovation. The former acts as if households were consuming a larger quantity of thecommodity on which the innovation takes place although they actually consume the same
quantity. The latter decreases the elasticity of the marginal utility of consumption.
Quantity-augmenting-like product innovation proportionally increases the value of thecommodity and the value of money. Consequently, the marginal utility of consumption
becomes smaller than that of wealth because the elasticity of the marginal utility of
consumption is greater than that of the marginal utility of wealth, and real consumption
decreases. Employment also declines because the product innovation decreases the quantityof the commodity that yields the same real consumption.
Addictive product innovation directly increases both the utility and marginal utility of
consumption, of which the former also increases the value of money because the sameamount of money can buy the commodity that gives higher utility. Therefore, it is generally
ambiguous which of consumption and money raises its own marginal utility more, implyingthat the effects on real consumption and employment are also ambiguous. However, for aplausible value of the elasticity of the marginal utility of consumption, addictive innovation
increases both employment and real consumption.
In conclusion, as an economic stimulus, addictive product innovation should be
encouraged, while quantity-augmenting-like product innovation should not be encouraged,
because the former stimulates aggregate demand and expands employment while the latteraggravates stagnation.

Appendices
Appendix A: Proof of (31)21

From (24) and (29), we obtain
𝛺

𝛿𝑢

𝑥𝑢

𝑢

𝑥

.

(A1)

Partially differentiating (23) with respect to 𝑥 and 𝛾 yields
, ;

, ;

, ;

, ;

, ;

, ;

, ;

, ;

𝑢

𝑥, 𝑦; 𝛾𝑢𝑥, 𝑦; 𝛾𝑢

, 1; 𝛾, ;𝑢, 1; 𝛾,𝑢, 1; 𝛾

,

, ;

, ;

, ;

, ;

, ;

𝜓

𝑢

, 1; 𝛾 ,𝑥, 𝑦; 𝛾.

from which we find𝑢𝑥, 𝑦; 𝛾

, ;

, ;

𝑢

(A2)

Applying 𝛿 in (24) and (A2) to (A1) and gives𝛺Given that 1
Since 𝑐

𝛿 𝑥/𝜔

𝛿𝑢

𝛿𝑥𝑢

.

𝛿𝑦 from (27), the above equation implies the first equation in (31).

𝑝 𝜃 𝑛 from the last equation in (27), 𝑑𝑝 /𝑑𝛾 /𝑝 is given by (26), and 𝛺

is given in (30), from the first equation in (31) we obtain the second equation in (31).

Appendix B: The elasticity of the marginal utility of consumption
Without wealth preferences, as assumed in the standard model, the Ramsey equation in(7) reduces to

22

𝜌

𝜉

𝜋

𝑅.

;

Therefore, a change in 𝑅 will vastly vary 𝑐/𝑐 if 𝜉 is small. This is why the DSGE literatureinsists that 𝜉 must be sufficiently large. In the presence of wealth preference, however, theRamsey equation (7) is
𝜌

𝜉

𝜋

𝑅

;

.

Therefore, a change in 𝑅 can be absorbed by a change in the wealth premium𝑣

𝑎 /𝑈 𝑐; 𝛾 without affecting 𝑐/𝑐 regardless of the value of 𝜉.
Furthermore, when the utility function is given by (32) and 𝛾

0, the effect of a rise in

𝛾 on the demand for 𝑥 is quite unrealistic, as shown below: Under this utility function, 𝑥 and𝑦 satisfy the third equation in (34); hence,.
This shows that in the case where 𝛾0, 𝜀0 and thus 𝜉

1

𝛾

𝜀

1, which wetreat in the text, a rise in 𝛾 increases demand for 𝑥. Intuitively, an increase in the utility of 𝑥
and a decrease in the elasticity (or the curvature) of the marginal utility of 𝑥, 𝑢 𝑥/𝑢1

𝛾 from (32), both due to a rise in 𝛾, naturally ‘increase’ demand for 𝑥. In the case where

𝛾

0, 𝜀

0 and thus 𝜉

1

𝛾

𝜀

1, however, a rise in 𝛾 (or equivalently, a decreasein |𝛾|) reduces demand for 𝑥. Intuitively, a rise in 𝛾 increases the utility of 𝑥 and decreases
the elasticity (or the curvature) of the marginal utility of 𝑥, and nevertheless, it ‘decreases’
demand for 𝑥, which is unrealistic. Therefore, we avoid the case where 𝛾
1 in the text.

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