ISSN (Print) 0473-453X

Discussion Paper No. 1204

ISSN (Online) 2435-0982

ECONOMIC STIMULUS EFFECTS
OF PRODUCT INNOVATIONUNDER DEMAND STAGNATION

Daisuke Matsuzaki
Yoshiyasu Ono

Revised July 2023
March 2023

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

March 2023
Revised July 2023

Economic stimulus effects of product innovation
under demand stagnation*
Daisuke Matsuzaki† and Yoshiyasu Ono‡
Abstract
When confronting economic stagnation, innovation (product innovation in particular) isoften cited as an effective stimulus because it is assumed to encourage household
consumption and lead to higher demand. Using a secular stagnation model with wealthpreference, we examine the effects of product innovation on employment and consumption.
This study examines three types of product innovation, including quantity-augmenting-likeinnovation, addictive innovation, and variety expansion. The first works as if a larger quantity
were consumed although the actual quantity remains the same, the second reduces theelasticity of the marginal utility of consumption, and the third increases the variety of
consumption commodities. We find that the first and third reduce both consumption andemployment, whereas the second expands them. It suggests that policy makers shouldcarefully choose the type of product innovation to promote as an economic stimulus:
addictive innovation stimulates business activity whereas quantity-augmenting-likeinnovation and variety expansion worsen stagnation.
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, H. Ishise, N. Kikuchi, T. Matsumura,
J.-B. Michau, R. Mikami, and participants of OSSIP lunch seminar at Osaka University and of the seminar oncontemporary economy at Okayama University for helpful comments. This work was financially supported by
the JSPS KAKENHI Grants (JP20H05631 and JP20K01661), the program of the Joint Usage/Research Centerfor “Behavioral Economics” at ISER of Osaka University, and the Asahi Glass Foundation. The authors aresolely 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 nations have experienced long periods of stagnation with aggregate
demand shortages, and innovation is often emphasized as an important economic stimulus
along with fiscal and monetary policies.1 The two kinds of innovation include process andproduct innovation. The former improves production efficiency, whereas the latter improvescommodity quality and utility. Regarding process innovation, Ono (1994, 2001) proposed amodel of secular stagnation related to wealth preference, 2 demonstrating that whenconfronting 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, andappears 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 toexamine the effects of product innovation on consumption and employment, determining thatdifferent types of product innovation may increase or decrease consumption and employment
because they increase not only the value of consumption, which encourages households toconsume more, but also the value of money (financial assets), which encourages households
to hold money.

1

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

Michau (2018) and Michaillat and Saez (2022) also treated models with wealth preference and determinedthat secular demand stagnation occurs because the marginal utility of wealth remains strictly positive. Illing,
Ono, and Schlegl (2018); Mian, Straub, and Sufi (2021); and Hashimoto, Ono, and Schlegl (2023) consideredwealth preference to analyze depressed aggregate demand.

1

Product innovation is an important issue in the literature regarding economic growth, but
it has not been considered as a potential demand stimulus under demand stagnation.3 Fiscal
and monetary policies are much more broadly analyzed as demand stimulus policies. To fillthis gap between theoretical research and policymaking, we examine the effects of productinnovations that directly influence the utility of consumption on aggregate demand andemployment.
We specify three types of product innovation. The first makes households feel as if they
were consuming a larger quantity of a given commodity despite actually consuming the same
quantity, which we refer to as quantity-augmenting-like product innovation. The second
reduces the elasticity of the marginal utility of consumption, which we describe as addictive
product innovation. The third is variety expansion, which enables households to consume awider variation of commodities. Naturally, each form of innovation increases realconsumption when full employment is achieved; however, we determine that only addictive
product innovation increases real consumption and employment under demand stagnation.
Variety expansion and quantity-augmenting-like product innovation exert the same influence
as process innovation, decreasing real consumption and employment.

3

Following 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 have included
Baldwin and Martin (2004), Davis and Hashimoto (2016), Iwaisako and Tanaka (2017), and Kane and Peretto(2020). Nevertheless, product innovation has not been considered as demand stimulus.

22. A single-commodity economy
We first consider an economy with a single commodity 𝑥 , where 𝑥 represents thecommodity’s quantity. A representative household’s utility of consumption is represented asfollows:𝑈

𝑢 𝑥; 𝛾 ; 𝑢 ≡

0, 𝑢

0, 𝑢

0,

(1)

where 𝛾 is a product innovation parameter. Innovations typically require development costs
and resources, which affect economic activity through changes in aggregate supply anddemand, exerting a crowding-out effect without demand shortages and a demand-stimulus
effect in the presence of demand shortages. However, 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. The nominal wage is 𝑊, the nominal price of 𝑥 is 𝑃 , and the firm will
infinitely expand employment and output if 𝑃 𝜃

𝑊 but will cease production if 𝑃 𝜃𝑊;therefore, to achieve equilibrium in the commodity market, the following must always hold:𝑃𝜃𝑊.

2-2. Households
Real total consumption 𝑐 is obtained as follows:

3

(2)

𝑐

𝑝 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 from equation (1) as follows:
𝑈

𝑢

;𝛾 .

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

𝑃 /𝑃 such that the level of real consumption 𝑐 that yields

the same utility remains unchanged. 4 Mathematically, a rise in 𝛾 alters 𝑝 so that 𝑈 isunchanged for a given 𝑐𝑝 𝑥 ; hence, using the properties of 𝑢 𝑥; 𝛾 from equation (1),

we obtain the following:
0

𝑢 𝑥

𝑢

⟹

0.

(4)

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

𝑃 /𝑃 rises.

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

0, 𝑣

0, 𝑣

𝑣 𝑚

𝑣

∞

0, 𝑣

𝑎 𝑒

𝑑𝑡,
0,

(5)

where 𝜌 is the subjective discount rate, 𝑣 𝑚 is the utility of money 𝑚

𝑀/𝑃 for the𝑈 𝑐; 𝛾 ≡ 𝑢transaction motive, and 𝑣

0, 𝑣

;𝛾 ,

𝑎 is the utility of holding assets 𝑎 related to wealth preference.

Assets 𝑎 include real money 𝑚 and interest-bearing asset 𝑏 with the nominal interest rate𝑅𝑟4

𝜋 , where 𝑟 is the real interest rate and 𝜋𝑃⁄𝑃 is the CPI inflation rate.

This aligns with the standard definition of CPI used in Samuelson and Swamy (1974), for example.

4

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

𝑈 𝑐; 𝛾

𝑣 𝑚

𝑣

𝑎

𝜆 𝑟𝑎

𝑤𝑛

𝑅𝑚

𝑐

𝑧,

where 𝜆 is the co-state variable, and 𝑝 is invariant over time before and after 𝛾
unanticipatedly changes and makes 𝑝 jump as shown in equation (4). Given that 𝑥

𝑐/𝑝 ,the first-order optimal condition is
𝜌

𝜉

𝜋

;

;

𝑅

;

,
(7)

;

where 𝑈 𝑐; 𝛾

, 𝜉≡

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 as follows:
𝑥

𝜃𝑛.

5

(9)

Given that 𝑝 is constant over time, we obtain the following from equations (2) and (3):≡𝜋When full employment (𝑛1) prevails, 𝑥.

𝜃, and 𝑊 moves in parallel with 𝑃. In contrast,

when confronting demand shortages, 𝑊 falls slowly, depending on the deflationary gap, and𝑃 moves in parallel with 𝑊 as follows:
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, regardless of 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,
represented as follows:
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,𝜇Considering that 𝑎

0.

𝑚 , the previously noted dynamics of 𝑚 and the dynamics of 𝑐

represented by (7) formulate autonomous dynamics with respect to 𝑚 and 𝑐 as follows:

6

;

𝜋

𝜌

where 𝑣 𝑚 ≡ 𝑣 𝑚

𝑣

𝑚 ,
(11)

𝜇

𝜋.

3. Product innovation
Having established the model, we now examine the effects of product innovation. Before
introducing 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
The full employment mentioned above may not be reached, as mentioned below. In thefull employment steady state, where 𝑛1, 𝑐0, and 𝑚

0, from (7), (9) and (11), thefollowing must hold:
𝜌

𝜇

;

;

where 𝑈 𝑝 𝜃; 𝛾

.

(12)

However, if the marginal utility of asset holding has a positive lower bound 𝛽, namely,𝑣 ∞ ≡𝑣 ∞where 𝑣 ∞

𝑣

∞

𝑣

∞

0 from (5), and if 𝛽 satisfies the following:

7

𝛽

0,

𝜌
where 𝑝

𝜌

𝜇

;

;

for any 𝑃 ,;1 because 𝑃 is initially set to equal 𝑃 , there is no 𝑃

(13)

𝑃 that satisfies (12),implying that a full-employment steady state is not achievable.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 fall in a sluggish manner, represented by (10), and 𝑣′ 𝑚eventually reaches 𝛽. From (7), (9), (10), and (11), in the stagnation steady state, 𝑛 satisfies
the following:6
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 𝑥latter when 𝑛𝜃 , in order for the solution of 𝑛 ∈ 0,1 to exist, the former is larger than the
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)

5

See Ono (1994, 2001) and Michau (2018), who determined the condition for the stagnation steady state tooccur. Using parametric and non-parametric approaches, Ono, Ogawa and Yoshida (2014) empiricallydemonstrated that the utility of financial wealth is insatiable. Ono and Yamada (2018) used survey data andempirically validated a status preference for money that makes the marginal utility of money constant and causes
secular demand stagnation to occur.
6

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

8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 to𝜇;,where 𝑛 is the solution of (14).
Totally differentiating (14) and applying (4), (9), and (15) to the result yields 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 1: 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 functions 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. As noted previously, the former isquantity-augmenting-like innovation, whereas the latter is termed addictive innovation. We7𝜌 𝑝 𝛽/𝑢 𝜃; 𝛾𝜌 𝜇 , both the full-employment and stagnation
When 𝜇 𝑝 𝛽/𝑢 𝜃𝑛; 𝛾
steady states exist. See Ono and Ishida (2014) for details on the conditions for the two steady states to emerge.

9apply these to (16) to examine their effects on total employment 𝑛 and real consumption 𝑐(or utility 𝑈).
We first assume the following utility function:ℎ 𝛾𝑥 ; ℎ 𝛾𝑥

𝑢 𝑥; 𝛾0, ℎ 𝛾𝑥0,(17)This implies that innovation 𝛾 functions as if the consumption of the commodity increased
by a factor of 𝛾. From (17), we obtain the following properties:𝑥𝑢𝛾𝑢

𝛾𝑥ℎ 𝛾𝑥0,

10.

Applying these properties to (16) yields the following:
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.

Intuitively, quantity-augmenting-like innovation increases the value of the commodity
and the value of money by a factor of 𝛾 because the value of money is measured by how
much value of the commodity can be purchased. Therefore, the marginal utility of
consumption decreases if the quantity of the commodity is held constant, while the marginalutility of money is unchanged to be 𝛽. Consequently, the commodity is consumed less andtotal employment decreases, which worsens deflation and further reduces real consumption.

10

We next consider addictive innovation using the following utility function:𝑢 𝑥; 𝛾
𝑢

𝑥

𝑥

𝑐/𝑝

1);

𝑥 ln 𝑥10, 𝛾 𝑢1

𝜉We regard 𝑥(where 𝛾

𝛾

0, 𝑢

𝑥

1,

(19)

ln 𝑥.

1 as the survival level of consumption and examine the case where

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 𝜉1𝛾 .

Applying (19) to (16) yields𝑢𝜓 1; 𝛾𝑥 𝛾 𝛺𝜓 𝑥; 𝛾 ≡ 𝑥 1

0, 𝜓 ∞; 𝛾0, 𝜓 𝑥; 𝛾

𝛺

𝛾

𝛾 𝑥

𝛾 ln 𝑥11𝛺1,
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:

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.

8

This 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 applyWe examine whether addictive product innovation increases employment and realconsumption using consumption data for G7 countries. Because 𝑥

1 is regarded as thesurvival level of consumption, as noted previously, the threshold 𝑥
𝜓 𝑥 ;𝛾that satisfies

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 examine whether addictive innovation leads to higher employment and realconsumption.

Table 1: 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: 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

Source: Appendix 2 of Jolliffe and Prydz (2016). The authors 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 World Bank data, https://data.worldbank.org/indicator/NE.CON.PRVT.KN andhttps://data.worldbank.org/indicator/SP.POP.TOTL.

Table 1 presents the ratios of the threshold level to the survival level for various
elasticities 𝜉1𝛾 because various levels of elasticity of the marginal utility of12consumption are proposed in previous research, approximately 0.3 in microeconomic
approaches such as Harrison et al. (2005), Anderson and Mellor (2008), and Meissner andPfeiffer (2022) and 2 in the dynamic stochastic general equilibrium (DSGE) modeling
literature.9 Table 2 shows the ratio of average household consumption to the poverty line inG7 countries. The two tables reveal that the ratios in Table 2 are almost always smaller than𝑥 , and the only exception is the case in which we adopt 𝜉

2 and the ratio of US household

final consumption to the national poverty line. Therefore, using Lemma 1, we determine that
addictive product innovation will almost always increase total employment and realconsumption if the representative household’s consumption equals the average consumption
levels of G7 countries.

4. A two-commodity economy
We next turn to an economy with two commodities, 𝑥 and 𝑦, to examine whether the
results obtained in the single-commodity economy remain valid. We implicitly consider that𝑥 is the commodity for which product innovation occurs, while 𝑦 is the composite of all othercommodities.

4-1. The model
Firms are competitive and produce commodity 𝑗𝑥, 𝑦with constant laborproductivity 𝜃 . Their profit maximization behavior leads to

9Hall (1988) and Guvenen (2006) asserted that 1/𝜉 is much smaller than 1. Chiappori and Paiella (2011)
argued that the median of relative risk aversion 𝜉 is 2. This is because in the standard framework without wealthpreference, 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 A for details.

13

𝑃𝜃
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 does also, implying that 𝑤

𝑊/𝑃 is constant overtime.
The utility of consumption is𝑈𝑢 𝑥, 𝑦; 𝛾 ,which is assumed to be homothetic. Given real consumption 𝑐, the representative household
maximizes 𝑈 subject to
𝑝 𝑥

𝑝 𝑦

𝑐,

(22)

and satisfies
, ;

, ;

, ;

, ;

𝜔

⇒

𝜁 𝛾 ,

(23)

because of homotheticity, where 𝑢 implies the partial derivative of 𝑢 with respect to variable𝑗𝑥, 𝑦, 𝛾 and 𝜔 is constant from (21). Therefore, we obtain
𝑝 𝑥

𝛿𝑐,

𝑝 𝑦

1

𝛿 𝑐, 𝜁 𝛾

,
(24)

𝛿

, 𝛿′ 𝛾

𝛿 𝛾

𝜁′ 𝛾 ,

where 𝛿 is the expenditure share of commodity 𝑥. 𝑈 is then represented as a function of 𝑐and 𝛾, which satisfies
𝑈 𝑐; 𝛾

𝑢

,

;𝛾 ,
(25)

𝑈

, ;

, ;

0,

𝑝 𝑈

14

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 𝑐
we find that 𝑝

𝐶/𝑃 , by differentiating (25) where 𝑐 is fixed and using (23),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 intertemporal
optimal condition is represented by (7) where 𝑈 is given by (25). From (21), (24) and themarket equilibria of the two commodities, we obtain the following:𝑥𝑛 𝜃 , 𝑦and find that each sector’s employment 𝑛 (𝑗𝑛 𝜃 ,𝑥, 𝑦 and total employment 𝑛

𝑛

𝑛

are
𝑛

𝛿𝑛, 𝑛

1

𝛿 𝑛; 𝑛.

(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 ,(28)

where 𝛿 𝛾 is given by (24). Therefore, using (21), (23) and (26), we obtain
0,

𝑢

0.

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.

15

4-2. Stagnation steady state in the two-commodity economy
In the full-employment steady state, we still have (12) in which 𝑈 is given by (25) and𝑐 satisfies (28). Therefore, if (13) holds, the full-employment steady state does not exist andthe stagnation steady state represented by (14) in which 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

the derivation for which is presented in Appendix B. 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 1 also holds inthe two-commodity economy.
First, we consider quantity-augmenting-like product innovation for commodity 𝑥 byassuming the following utility function:𝑢 𝑥, 𝑦; 𝛾ℎ 𝛾𝑥, 𝑦 ,which satisfies the following:

16

𝑥𝑢

𝛾𝑥ℎ 𝛾𝑥, 𝑦𝛾𝑢1where ℎ 𝑖0, 𝑢

ℎ 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

0,

𝛺

0.

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

𝑥𝑢

,
, 𝑢

𝑢

0, 𝛾
𝑦𝑢

𝜀

1);

𝑥 𝑦 ln 𝑥,
1

,

𝛾

(32)
𝜀 ,

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

𝑥 𝛾

𝜀 𝛺

𝜓 𝑥; 𝛾, 𝜀 ,𝜓 𝑥; 𝛾, 𝜀 ≡ 𝑥
𝜓 𝑥; 𝛾, 𝜀

𝛺

1
𝛾

1

𝜀 𝑥

𝛾
1

𝛺

𝜀

ln 𝑥
1

𝛾

.

17

1,
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 the 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 for 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, which is naturally larger than 1; hence,
1,

𝜔

1.

(35)

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

𝜀 𝑢𝜑 𝑥; 𝛾, 𝜀 ≡ 𝑥

𝜑 1; 𝛾, 𝜀
𝜑 𝑥; 𝛾, 𝜀

ln 𝑥

1

1

0, 𝜑 ∞; 𝛾, 𝜀
𝛾

𝜀

𝑥

1

0,

0,

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, 𝜀10𝑥0 (⇔ 𝑢0).

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

𝑥given in (36) satisfy

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

18𝜓 𝑥 ; 𝛾, 𝜀𝑥𝜓 𝑥 ; 𝛾, 𝜀

1
𝑥

𝛾

1

𝛾

𝜀

ln 𝑥

𝜀

ln 𝑥
0,

1

0,

𝜓 ∞; 𝛾, 𝜀0,which implies
𝜓⋛0 ⇔

⋛0

for 𝑥 ⋚ 𝑥

where 𝑥𝑥 .

(37)

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

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

𝑈 𝑐; 𝛾, 𝜀

1

𝛾

𝜀

0.

The parameter values in Table 3 satisfy these properties.11

Table 3: Survival consumption level and 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

11

We do not examine the case in which the elasticity of the marginal utility of consumption 1
𝛾 𝜀 islarger than 1 in the two-commodity case. This is because the effect of a rise in 𝛾 on the demand ratio of the two
commodities is quite unrealistic. See Appendix A for details.

19Comparing 𝑥 /𝑥 in Table 3 with the G7 countries’ data in Table 2 reveals 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 thosevalues of 𝑥 /𝑥 . Therefore, using Lemma 1, we find that addictive innovation increases both
employment and real consumption in the two-commodity economy as well as in the singlecommodity economy if the representative household’s consumption equals the average
consumption levels of G7 countries.

5. Variety expansion
Finally, we consider the effect of product innovation that expands the variety of
consumption commodities on employment and real consumption.

5-1. Constant elasticity of substitution utility
We first treat the case where the utility of consumption is one of constant elasticity of
substitution (CES) functions as follows:𝑈𝑢 𝑥, 𝛾

ℎ

𝑥

𝑑𝑗

,

𝜎

1; ℎ

0, ℎ

0,

(38)

where 𝛾 represents the variety of commodities, which is exogenous, 𝜎 is the elasticity of
substitution, and 𝑥 is the quantity of commodity 𝑗 ∈ 0, 𝛾 .
By assuming common labor productivity 𝜃 for all 𝑗 and symmetric preference for 𝑥 , wefind that real consumption 𝑐 is𝑐𝑝 𝑥 𝑑𝑗

𝛾𝑝 𝑥,

𝑝 ,𝑥

20

𝑝 ,𝑥

for 𝑗 ∈ 0, 𝛾 ,(39)where 𝑝 is the real price of commodity 𝑗 while 𝑝 and 𝑥 are respectively the common real
price and quantity. Therefore, from (38) and (39), 𝑈 satisfies
𝑈𝑢 𝑥; 𝛾

ℎ 𝛾

𝑥

ℎ

𝑐 ,

Comparing (40) with (17) and replacing 𝛾 /

𝑈

ℎ′

𝑐 .

(40)

in (40) by 𝛾 in (17) suggests that variety

expansion 𝛾 should function in the same way as quantity-augmenting-like innovation,
discussed in subsection 3-1, by the magnitude of 𝛾 /.12

To see this, we first note that 𝑈 must be invariant for the same 𝑐 before and after a jumpof 𝛾 if 𝑝 is properly defined. Therefore, from (40), we obtain

𝑈

𝑐ℎ′

𝑐

0 ⇒

0, 𝑈

(41)

0.

Furthermore, the household’s intertemporal behavior is the same as that in the singlecommodity economy represented by (5) and (6), except that 𝑈 𝑐; 𝛾 is given by (40). Thus,
the intertemporal optimal condition is represented by (7) where 𝑈 is presented by (40).
Given that labor productivity is 𝜃 for all 𝑗 ∈ 0, 𝛾 , using (39), we find that each variety’s
employment 𝑛 and total employment 𝑛𝑛
If full employment (𝑛

𝑛

.

(42)

1) prevails, from (40) and (42) 𝑐 and 𝑈 satisfy𝑐Noting that ∂𝑝 /𝜕𝛾,𝛾𝑛 satisfy

𝑝 𝜃,

𝑈

ℎ 𝜃𝛾

.

0 from (41), we find that the above equations imply𝛾 ↑ ⇒ 𝑐 ↑, 𝑈 ↑ under full employment.

In particular, if 𝜎 → ∞, implying that all varieties are perfect substitutes, 𝛾 /expansion naturally becomes the same as quantity-augmenting-like innovation.
12

21𝛾; hence, the variety

In this state, real money 𝑚 must satisfy (12); hence,
𝜌

𝛽

𝜇

;

;

where 𝑈 𝑝 𝜃; 𝛾

ℎ′ 𝜃𝛾

.

Therefore, if (13) in which is valid, there is no full-employment steady state and aggregate
demand stagnation emerges in the steady state.
In the stagnation steady state, we have condition (14) in which 𝑈 is given by (40).
Therefore, from (42), the following is obtained:
𝜌

𝛼 𝑛

1

where 𝑛

,

.

(43)

From (41), an increase in 𝛾 does not vary 𝑈 , implying that the wealth premium 𝛽/𝑈 givenby the right-hand side is unaffected by a change in 𝛾 if 𝑐 is the same. However, lessemployment is required (i.e., 𝑛𝑐/ 𝑝 𝜃 decreases) to achieve the same 𝑐 because anincrease in 𝛾 raises 𝑝 , as shown in (41). Therefore, deflation worsens, which lowers thedesire for consumption 𝜌

𝛼 𝑛

1 given by the left-hand side, and real consumption andemployment decrease. From (43), it is also apparent that an increase in 𝛾, which raises 𝑝 ,works in the same way as process innovation represented by an increase in labor productivity
𝜃 . Thus, this variety expansion works in the same way as quantity-augmenting-likeinnovation examined in subsection 3-2.
In summary, we obtain the following proposition:

Proposition 4: If the utility of consumption is of the CES type given by (38), productinnovation that expands the variety 𝛾 increases real consumption under full employment. In
the presence of aggregate demand shortage, however, it decreases employment and realconsumption.

22

Note that the above result is valid, whether under monopolistic competition or under perfectcompetition.13
Using the same CES utility function of consumption and utility of money, Johdo (2008)
analyzed the effects of a production subsidy on employment and real consumption, assumingthat the market is monopolistically competitive with free entry and that firms are symmetricand require both variable and fixed production costs. In this setup, Johdo concluded that a
production subsidy expands variety and increases employment and real consumption,
contradicting Proposition 4, if the elasticity of substitution 𝜎 is sufficiently large. Thisfinding is valid not because the variety expands but because new entrant firms require fixed
costs, which functions in the same manner as government purchases and stimulates
employment. If there is no fixed cost and the variety is exogenously determined, variety
expansion decreases employment and real consumption, as stated in Proposition 4. Variety
expansion itself worsens employment and real consumption, regardless of the magnitude of𝜎

1 .

5-2. A general case
Quantity-augmenting-like innovation in the two-commodity case, which is analyzed in
section 4-2, can be regarded as an introduction of commodity 𝑥 (i.e., variety expansion) toan economy with only 𝑦 . Given that the consumption utility function is 𝑢 𝑥, 𝑦; 𝛾ℎ 𝛾𝑥, 𝑦 , the case of 𝛾

0 implies that 𝑥 does not exist, and an increase in 𝛾 from zero

indicates that commodity 𝑥 is newly introduced. Noting that Proposition 2 holds regardless13

The nominal price of each commodity 𝑃
𝑃 equals the marginal cost 𝑊/𝜃 under perfect competition
whereas it equals 𝜎/ 𝜎 1 𝑊/𝜃 under monopolistic competition. However, this difference does not affectthe present analysis because the variety is exogenously given. .

23

of the level of 𝛾, we find that an introduction of commodity 𝑥 into an economy with only 𝑦lowers real consumption and employment.
Analogously, using the following homothetic function of consumption utility with 𝑁
commodities:𝑈ℎ 𝑥 ,𝑥 ,…,𝑥

, 𝛾𝑥 ,

this subsection examines the effects of an increase in 𝛾 on real consumption and employment
and finds that Proposition 2 is valid in the 𝑁-commodity economy; hence, the introduction(i.e., variety expansion) of commodity 𝑁 reduces real consumption and total employment.
This means that regardless of whether the consumption utility function is of the CES type,variety expansion always worsens business activity under homothetic utility.
For given real prices 𝑝 for 𝑖 ∈ 1, 𝑁 and real consumption 𝑐, the intratemporal rational
behavior of a representative household satisfiesℎ 𝑥 ,𝑥 ,…,𝑥
ℎ

, 𝛾𝑥

𝜆𝑝

𝑥 ,𝑥 ,…,𝑥𝑐

for 𝑖 ∈ 1, 𝑁

, 𝛾𝑥

∑

𝜆

,

1,
(44)

𝑝𝑥,

where ℎ denotes the partial derivative of the utility function ℎ with respect to the 𝑖-th factorand 𝜆 is the Lagrange multiplier of 𝑐 . Given that 𝛿 denotes the expenditure share ofcommodity 𝑖, utility 𝑈 is rewritten as follows:𝑝𝑥

𝛿𝑐for 𝑖 ∈ 1, 𝑁 ,

𝑈

ℎ

,…,

,𝛾

,

∑

𝛿

1, ∑

𝑑𝛿

0.

Firms are competitive and labor productivity 𝜃
therefore, all the relative prices are constant and satisfy

24

𝑥 /𝑛

(45)

for 𝑖 ∈ 1, 𝑁 is constant;𝑝𝜃𝑝 𝜃 andfor any 𝑖, 𝑗 ∈ 1, 𝑁 .

(46)

From (45) and (46), employment 𝑛 in sector 𝑖 and total employment 𝑛 are
𝑛

,

𝑛

.

(47)

CPI is defined so that a change in 𝛾 does not vary 𝑈 for given 𝑐. Therefore, partiallydifferentiating the second equation of (45) with respect to 𝛾, and applying (44), the propertiesof 𝛿 and 𝛿 ′ given by (45), and (46) to the result yields
𝑈

∑

ℎ

/

ℎ

which implies that 𝑝

(48)

/

𝜆𝑐

/

ℎ
0,

𝑃 /𝑃 for any 𝑖 ∈ 1, 𝑁 satisfies
𝛿

0 for any 𝑖 ∈ 1, 𝑁 .

If full employment prevails (i.e., 𝑛(49)1), from (45), (47), (48) and (49), we obtain

𝑈

𝑈

𝜃

0.

Proposition 1 is valid and variety expansion always increases real consumption andconsumption utility.
If aggregate demand stagnation occurs in the steady state, the household’s intertemporal
optimization is the same as that in equations (14), (29), and (43), which is given as follows:
𝜌

𝛼 𝑛

1

,

𝑈

∑

𝛿

𝛾𝛿

𝜆,

(50)

where total employment 𝑛 is given by (47) and the second equation is from (44) and (45).
Furthermore, 𝑈

𝜆 satisfies the following property:

Lemma 2: If the utility of consumption is homothetic, then

25

𝑈

0.

Proof. See Appendix C.

From Lemma 2, an increase in 𝛾 does not affect the right-hand side of the first equationin (50) while it raises 𝑝 from (49), decreases total employment 𝑛 given by (47) for the same𝑐, and lowers the left-hand side of the first equation in (50). Specifically, the right-hand side
represents the desire for wealth accumulation, which is unaffected, and the left-hand side
represents the desire for consumption, which decreases. Consequently, real consumption 𝑐decreases, which results in a reduction in total employment 𝑛

𝑐/ 𝑝 𝜃

, where 𝑝 rises,

as shown by (49). This result is formally restated as follows:

Proposition 5: In the presence of aggregate demand shortage, quantity-augmenting-likeinnovation on a commodity decreases real consumption and total employment in the Ncommodity economy. This also implies that an introduction of a new commodity (i.e., variety
expansion) decreases real consumption and total employment.

Therefore, under full employment, variety expansion increases real consumption, whereas in
the presence of aggregate demand shortage, variety expansion generally diminishes businessactivity as long as the consumption utility is homothetic. This holds true, regardless ofwhether the commodities are substitutes or complements to one another.

26

6. Concluding remarks
Product innovation raises the utility of consumption and is expected to increase
consumption. Under full employment this assumption holds; therefore, product innovation isoften cited as an economic stimulus even when a country is confronting unemployment due
to a shortage of aggregate demand. However, it increases not only the value of consumption
but also the value of money, which is measured by the commodities it can buy. The former
stimulates the desire for consumption and the latter urges people to hold money if they have
wealth preference. Consequently, product innovation may increase or decrease realconsumption and employment under aggregate demand stagnation, depending on the relative
magnitude of the two.
We consider three types of product innovation, including quantity-augmenting-likeproduct innovation, addictive product innovation, and variety expansion. The first works as
if households were consuming a larger quantity of the commodity for which the innovation
takes place, although they actually consume the same quantity. The second decreases theelasticity of the marginal utility of consumption. The third expands the variety of
consumption commodities.
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, decreasing realconsumption. Employment also declines because the product innovation decreases thequantity of 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 same27

amount of money can buy the commodity that gives higher utility. Therefore, it is generally
ambiguous which raises its own marginal utility more, consumption or money, implying thatthe 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.
Finally, variety expansion functions in the same manner as quantity-augmenting-likeproduct innovation. Therefore, it reduces both real consumption and total employment.
In conclusion, addictive product innovation should be encouraged as an economic
stimulus, while quantity-augmenting-like product innovation or variety expansion should not
be encouraged, because addictive product innovation stimulates aggregate demand andexpands employment while the remaining two aggravate stagnation.

Appendices
Appendix A: The elasticity of the marginal utility of consumption
Without wealth preferences, as assumed in the standard model, the Ramsey equation in(7) reduces to
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. However, in the presence of wealth preference, theRamsey equation (7) is as follows:
𝜌

𝜉

𝜋

𝑅

;

.

Therefore, regardless of the value of 𝜉, a change in 𝑅 can be absorbed by a change in thewealth premium 𝑣

𝑎 /𝑈 𝑐; 𝛾 without affecting 𝑐/𝑐.
28

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 highly unrealistic, as shown below. Under this utility function, 𝑥and 𝑦 satisfy the third equation in (34); hence,.
This demonstrates that in the case where 𝛾0, 𝜀0; thus, 𝜉

1

𝛾

𝜀

1, which we

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

1

𝛾 from (32), both of which are due to a rise in 𝛾, naturally raise demandfor 𝑥. However, in the case where 𝛾0, 𝜀0; thus, 𝜉

1

𝛾

𝜀

1, a rise in 𝛾 (orequivalently, a decrease in |𝛾|) reduces demand for 𝑥. Intuitively, a rise in 𝛾 increases theutility of 𝑥 and decreases the elasticity (or the curvature) of the marginal utility of 𝑥, andnevertheless, it decreases demand for 𝑥, which is unrealistic. Therefore, we ignore the casewhere 𝛾0, 𝜀0 and 𝜉1 in the text.

Appendix B: Proof of equation (31)
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; 𝛾

, ;

29

, ;

𝑢

, 1; 𝛾

,

, ;

, ;

𝑢

, ;

, ;

, ;

𝑥, 𝑦; 𝛾

𝑢

, ;

𝑢

, ;

, 1; 𝛾, ;𝜁𝑢, 1; 𝛾,, 1; 𝛾 ,from which we find𝑢𝑥, 𝑦; 𝛾

, ;

, ;

𝑢

𝑥, 𝑦; 𝛾.

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

Appendix C: Proof of Lemma 2
Given that function ℎ is homothetic, the first-order optimal conditions (44) lead to
,…,

,

/

,…,

/

,

,…,

,

/

,…,

/

,

𝛾𝑥 .

, 𝑋

(A3)

Partial derivatives of the first equation in (A3) with respect to 𝑋 and 𝑥 for 𝑗 ∈ 1, 𝑁

1

are
∑

,
,

where ℎfor 𝑗 ∈ 1, 𝑁

ℎ 𝑥 /𝑋 , … , 𝑥

30/𝑋 , 1 .

1(A4)

From (45) and (A3),for 𝑗 ∈ 1, 𝑁

1.

Using this equation and (A4), we find
∑

𝑈

𝑈

𝛿

0,
(A5)

, 𝑈

where 𝑈⁄

, 𝑈

Note that this property is obtained because ℎ 𝑥 , … , 𝑋.

⁄

is homothetic.

Partially differentiating (44) with respect to 𝛾 and applying (49) to the result produces
the following:

𝜆

⎛ ⋮ ⎞
⎜
⎟
⎟
𝐻⎜
⎜
⎟
⎜
⎟
⎝

where 𝐻

⎛
⎜
⎜
⎜

⎞
⎟
⎟
⎟

⎝

0

⎠

ℎ
⋮

⋯

ℎ

ℎ

⋯
⋯

ℎ
ℎ

ℎ
ℎ

⎠

⎛
⎜ℎ
⎜ ℎ
⎝

𝛿 𝑝
⋮
⎛
𝛿 𝑝
⎜
1
𝛿
⎝
0

⋮

𝑝

𝜆

⋯

⎞
⎟,
⎠
(A6)

𝑝

𝑝
⋮
⎞
𝑝 ⎟
.
⎟
0

⎠

By applying Cramer's rule to (A6), we obtain 𝛾𝜕𝜆/𝜆𝜕𝛾:,
where we rewrite det 𝐻

using the definition of 𝑈 in (A5) as follows:

31

(A7)

ℎ
⋮

ℎ
ℎ

det 𝐻

𝑝

⋯

ℎ

ℎ

⋯
⋯

ℎ
ℎ

ℎ
ℎ

⋯

𝑝

where Γ

𝑈
⋮
𝑈
𝑈
1

𝛿 𝑝
⋮
𝛿 𝑝
𝛿
1

∏

𝑝 Γ,

0
⋯

𝑈

𝑈

⋯
⋯
⋯

𝑈
𝑈

𝑈
𝑈
1

Subtracting column 𝑁 from columns 1, … , 𝑁

𝛿
⋮
𝛿

.

𝛿

1
0

1

1 in Γ and multiplying row 𝑗 by 𝛿 for 𝑗 ∈1, 𝑁 yields
𝑈
⋮
𝑈
𝑈
0

𝑈
Γ
∏

𝑈

𝛿

𝑈

Adding rows 1, … , 𝑁
𝑈
∏

𝑈

𝑈
⋮
𝑈
0
0

𝛿

⋯

𝛿
𝛿

⋯
⋯
⋯

𝑈

𝑈

𝛿

𝑈

𝑈
𝑈
0

𝛿
𝛿

𝑈

1 to row 𝑁 and applying (A5) and ∑
𝛿

⋯
𝛿

⋯
⋯
⋯

Note that zeros in columns 1, … , 𝑁
while the zero in column 𝑁

𝑈
𝑈

𝑈 𝛿
𝑈

𝛿

𝑈

𝛿

𝑈
0
0

𝛿

𝛿
𝑈 𝛿
1

𝛿 𝛿
⋮
.
𝛿 𝛿
𝛿
1 𝛿
0

1 to the result produces
𝑈 𝛿
𝑈
∑

𝛿
𝑈 𝛿
1

𝛿 𝛿
⋮
𝛿 𝛿
0
0

.

1 of row 𝑁 are obtained because of homotheticity,

1 of row 𝑁 is due to quantity-augmenting-like innovation.

Finally, multiplying the last row by ∑values of row 𝑁 equal 0, implying Γ𝑈 𝛿 and adding the result to row 𝑁 makes all

0 and det 𝐻

0. Therefore, we obtain
0.

32

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