DP2022-28

Innovation to Keep or to Sell andTax Incentives
Colin DAVIS
Laixun ZHAO

Revised November 21, 2022

Innovation to Keep or to Sell and Tax Incentives
Colin Davis
Doshisha University*

Laixun Zhao
Kobe University†

November 2022

Abstract
We study how tax policy affects economic growth through entrepreneurs’ choice of commercialization mode. Introducing both heterogeneous quality jumps and a leapfrog versus
sell choice into the quality-ladders model, we show that entrepreneurs use high-quality innovations to leapfrog incumbent firms and become new market leaders, but sell low quality
ones to incumbents. Tax incentives that promote leapfrogging slow the rate of innovation.
A numerical analysis concludes that subsidies to product design improve welfare. Corporate
taxes, capital gains taxes, and subsidies to market entry all harm welfare.
Key Words: Innovation-based growth, heterogenous quality improvements, innovation sales,
corporate tax, capital gains tax, market entry subsidy, product design subsidy
JEL Classifications: O31; O33; O43* The Institute for the Liberal Arts, Doshisha University, Karasuma-Higashi-iru, Imadegawa-dori, Kamigyo, Ky-

oto, Japan, 602-8580, cdavis@mail.doshisha.ac.jp.
† Corresponding Author. Research Institute for Economics & Business, Kobe University, 2-1 Rokkodai Machi,
Nada-ku, Kobe, Japan, 657-8501, zhao@rieb.kobe-u.ac.jp, Fax: 81-78-803-7006.
Acknowledgements: The authors are grateful for helpful comments from participants of workshops at China Agricultural University, Dongnan University, Kobe University and Nankai University. They also acknowledge financial
support from JSPS through Grants-in-Aid for Scientific Research: #19K01638, #19K01606 and #19H01484. Allremaining errors are their own.

1

1 Introduction
In the innovation sector, “the-winner-taking-all” is a common phenomenon. In other words,
many innovations are not directly commercialized as final products sold to consumers, simply
because they are too small and trivial. Instead, they are sold to the large incumbent firms whichbecome the winners, while the small innovators exit with the money, and possibly start newrounds of innovation. Sufficiently large innovations, however, enable innovators to leapfrog
incumbents, forcing them to exit instead. Indeed, there are abundant stories and stylized facts
documenting the entrepreneur’s leapfrog/sell choice, especially in the high-tech industries.1
A key question then arises regarding which mode of commercialization best promotes economic growth and welfare. And, more practically, how policy makers should shape fiscal incentives for entrepreneurs investing in research and development (R&D)? In this paper, we tackle
these issues by investigating the entrepreneur’s choice between using a new innovation to start a
new firm, thereby displacing (i.e., leapfrogging) the incumbent market leader, or selling the new
innovation to the incumbent firm.
Specifically, we set up a framework for innovation-based endogenous growth in which entrepreneurs invest in the development of new product designs that generate heterogenous quality improvements over existing product lines. In each industry, we characterize an equilibrium
pattern of entrepreneurial behavior, whereby, once the quality of a new design is known, theentrepreneur makes a leapfrog versus sell decision. High-quality designs are used to leapfrog
incumbent leaders, while low-quality designs are sold to the incumbents.
The long-run evolution of product quality in the average industry can be characterized withtwo variables: a threshold design quality for which entrepreneurs are indifferent between leapfrogging vs. selling, and the product quality of the incumbent leader. These variables are intrinsicallylinked as a lower threshold expands the share of designs used to leapfrog, reducing the frequency
1 Large firms such as Google, Facebook and Microsoft buy thousands of applications to improve on their own

platforms. The relationship between Google and Yahoo is especially interesting. In 1998, Google’s Larry Page andSergey Brin reportedly approached Yahoo to sell their PageRank system for as little as $1 million, because theywanted to focus on their studies at Stanford University, but Yahoo did not value the system much and declined.

2with which incumbents update their product lines by purchasing new designs.
The interaction between the entrepreneur’s leapfrog/sell choice and the rate of innovation
depends on how the labor and investment markets adjust. In the labor market, an increase in thethreshold quality lowers the average cost of introducing a new product as incumbents are moreproductive at commercializing new designs; and in the investment market, the increases in thethreshold quality and average incumbent quality improve the threshold profits from leapfrogging
over selling, raising the return to investment in product development. Together, these channels
generate a positive relationship between the threshold quality required for leapfrogging and theinnovation rate.
We then use the framework to study a number of policy initiatives: corporate taxes on operating profits, capital gains taxes on the sale of new designs, subsidies to entrepreneurs developing
new product designs, and subsidies to entrepreneurs entering production with new designs. The
direct effects of these policies are standard, with lower taxation and higher subsidy rates promoting investment in product development. However, because entrepreneurs face a choice between
leapfrogging vs. selling, the overall policy impacts on the rate of innovation are not obvious.
Indeed, we find that when firms receive tax deductions for R&D costs, the relationship between
tax policy and investment in innovation depends on the balance between opposing profit andtax exemption effects. Surprisingly, tax incentives that increase the likelihood of leapfrogging
rather than selling, tend to slow the rate of innovation, through an indirect effect–the positive link
between the threshold quality required to leapfrog and the innovation rate. Because either the
direct or the indirect channel may dominate when their directions do not align, the overall impact of each policy on the rate of innovation is generally ambiguous. For a benchmark parameter
set, however, numerical analysis suggests that subsidies to product design increase investment
in innovation, raising the rate of economic growth and improving welfare. Corporate taxes on
operating profits, capital gains taxes on the sale of product designs, and subsidies to market entry
slow the rate of innovation, thereby reducing the rate of economic growth and harming welfare.
In the literature, Akcigit et al. (2016) study issues similar to ours, but using a Lancaster3

varieties’ approach, where they show that incumbent firms keep or buy innovations of varieties
close to their existing business but sell off faraway ones. Akcigit and Kerr (2018) examine
heterogenous quality improvements, but they assume incumbents invest in internal innovations toimprove their existing products, while new entrants invest in external innovations to acquire new
product lines. Cunningham et al. (2020) investigate ”killer acquistions” in the pharmaceuticalindustry and find that acquired drug projects are less likely to be developed when they overlap
with the acquirer’s existing product portfolio, especially when the acquirer’s market power islarge because of weak competition or distant patent expiration. In contrast, the decision to sellor leapfrog in our model depends on the endogenous significance of the innovation rather thanon exogenous technical closeness and overlap.
Also, in Klette and Kortum (2004), innovation is concentrated in incumbents and more productive incumbents innovate more, whereas in Dinopoulos and Unel (2011), the variety expansion approach is used to show that high-quality firms select to export while low-quality ones
sell domestically. In earlier models of endogenous growth, such as Segerstrom et al. (1990),
Grossman and Helpman (1991) and Agion and Howitt (1992), quality improvements are constant. While recent studies have extended the quality-ladders framework to consider heterogenous quality improvements (Minniti et al., 2013; Acemoglu and Cao, 2015; Chu et al., 2017;
Parello, 2018; Iwaisako and Ohki, 2019), they do not consider the entrepreneur’s option to sell
to an incumbent firm.
Finally, our framework is related to a literature that adopts a game-theoretic approach to study
how acquisition and licensing between startups and incumbent firms in the commercialization of
new technologies affects investment in R&D (Gans and Stern, 2000; Henkel et al., 2015). In
particular, Haufler et al. (2014) investigate how tax incentives affect the entrepreneur’s choice
between market entry and selling to an incumbent, and show that entrepreneurs may choose R&D
projects with too little risks when entering the market, and hence tax incentives aimed at promoting market entry may lead to welfare losses. In contrast, our macroeconomic model explores
the links between the entrepreneur’s choice of commercialization mode, economic growth, and4welfare, within a dynamic and quantitative rather than game-theoretic framework.
The paper proceeds as follows. In the next section, we introduce our framework in which
entrepreneurs choose between leapfrogging and selling based on heterogenous quality improvements. Then, Section 3 provides a characterization of the long-run equilibrium. In Section 4,
we investigate how changes in economic policy affect the entrepreneur’s leapfrog/sell choice,economic growth and social welfare. Section 5 concludes. All mathematical derivations arerelegated to the appendices.

2 ModelWe analyze the choice between leapfrogging and selling in a quality ladders framework ofinnovation-based endogenous growth (e.g., Grossman and Helpman, 1991). A unit mass ofindustries employs labor in the production of final goods for household consumption. Entry into
a representative industry requires a new product design that introduces a quality improvement
to the existing product line. New designs are created in a competitive innovation sector whereentrepreneurs develop designs with heterogeneous quality improvements. Depending on the sizeof the quality improvement, entrepreneurs choose between either using the designs to leapfrog
incumbent market leaders to become leaders themselves (leapfrogging), or selling the designs toincumbent leaders (selling). Labor is the sole factor of production.

2.1 Household Preferences
The demand side of the economy consists of a fixed population of dynastic households thatmaximize utility over an infinite time horizon. The intertemporal preferences of a household aredescribed byU=Z ∞e−ρ t ln D(t)dt,(1)0where ρ is the subjective discount rate. The instantaneous utility derived from consumption
follows a quality-augmented index with unitary elasticity of substitution across the products

5

produced within a unit mass of industries indexed by i ∈ [0, 1]:

ln D(t) =

Z 1
0

j(i)

ln

!

∑ ∏ λ (h)q(i, j;t) di,j(i) h=1(2)where q(i, j) is the quantity demanded of product vintage j, with j(i) indicating the number ofquality innovations (product generations) that have been introduced to date in industry i. Thej(i)product quality of the jth generation in industry i is captured by ∏h=1 λ (h), with λ (h) > 1
measuring the heterogeneous quality improvement associated with each generation.
With each household supplying one unit of labor inelastically, intertemporal optimization requires that the representative household select an optimal path for expenditure with the objectiveof maximizing lifetime utility (1) subject to the flow budget constraint:Ȧ(t) = r(t)A(t) + w(t) − E(t) − T,(3)

where A is household asset wealth, r is the interest rate, w is the wage rate, E is household
expenditure, T is a lump-sum tax on household income, and a dot over a variable denotes timedifferentiation. The optimal expenditure path is described by the Euler condition:Ė(t)= r(t) − ρ .
E(t)

(4)

Henceforth, setting household expenditure as the model numeraire, E = 1, we have r = ρ at all
moments in time. Time notation is suppressed hereafter for simplification.
Given the unitary elasticity of substitution across industries, at each moment in time thehousehold sets an even allocation of expenditure across the unit mass of product lines. The housej(i)hold then purchases only the product with the lowest quality-adjusted price px (i, j)/ ∏h=1 λ (h)from each product line, with px (i, j) being the product price, and sets its demand for all remaining products to zero. Defining the state-of-the-art as the product with the lowest quality-adjusted

6

price and denoting it by j(i) = J(i) in industry i, the total demand for product j(i) isx(i, j) =q(i, j)L =L
px (i, j)0f or j(i) = J(i),(5)

otherwise,

where we have aggregated across the total population of households L and used E = 1. Whenever the consumer is indifferent between two products with the same quality-adjusted price, we
assume that he/she always buys the higher quality (newer generation) product.

2.2 Production
The production sector features firms that compete via Bertrand pricing strategies. Each firmholds a patent protecting its right as the sole producer of a given generation of the industry’s
product line. All firms have access to the same production technology

x(i) = LX (i),(6)

where x(i) and LX (i) are firm-level output and labor employment in production respectively. The
common marginal cost of production across firms is the wage rate, w.
The optimal pricing strategy of the industry leader, who with a patent for the state-of-the-art
product design of the highest quality level, is to set the quality-adjusted price just equal to theclosest rival firm’s marginal cost, with the aim of forcing the rival out of the market. Given
the production technology (6), this limit price is px (i) = wλ (i), where λ (i) is the difference inquality between the industry leader over the closest rival firm in industry i.
Following Grossman and Helpman (1991), we rewrite δ (i) ≡ 1/λ (i) ∈ (0, 1) as the inversequality-gap between the state-of-the-art product and the closest vintage product. In other words,
the lower the value of δ , the greater the difference in quality between the products of the market
leader and the closest rival firm. Applying the limit pricing rule, we can write the gross profit of7the market leader as

π (i) = px (i)x(i) − wLX (i) = (1 − δ (i)) L,(7)

where demand condition (5) has been used. An increase in the quality improvement λ (i.e., adecrease in δ ) clearly raises the profit of the market leader by expanding the price-cost markup.

2.3 Innovation
New product designs are created by entrepreneurs working independently in the competitive
innovation sector. Each new product design adds a heterogeneous quality improvement to thecurrent state-of-the-art in its respective industry, while adopting all of the quality improvementsthat have been introduced to date. At the completion of the design process, the entrepreneur applies for a patent to protect the design from imitation. Before issuing a patent, however, the patentoffice reviews the application to ensure that the design represents a sufficiently large improvement over the current state-of-the-art in the industry. If awarded, the patent protects the productdesign indefinitely. Entrepreneurs that have successfully obtained a patent choose whether to sell
the patented design to the incumbent market leader, or even better, to use the design to leapfrog
the incumbent firm to become the new market leader. Further, because leapfrogging requires anadditional fixed cost for the entrepreneur, the sell/leapfrog choice is intrinsically linked with thesize of the quality improvement generated by the new design.
More formally, an entrepreneur in industry i employs ι (i)α aL units of labor at each momentin time to generate an instantaneous probability ι (i) of creating a new product design, wherea > 0 is a constant parameter, α ∈ (0, 1] is the share of product development in innovation costs,
and L captures a dilution effect in which the product development cost increases in market size.2
The inverse-quality gap (δ ) is randomly drawn from a continuous, strictly increasing probability
distribution function F(δ ), with density f (δ ) and support [0, 1]. The strength of patent regulation
is captured by δ ∗ ∈ (0, 1), with a patent application rejected for δ > δ ∗ , when the patent examiner
judges that the quality improvement associated with the product design is not sufficient to be
2 See Laincz and Peretto (2006) for a discussion of the dilution effect.

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granted a patent. Therefore, an entrepreneur employing ι (i)α aL units of labor in industry iinvents a patentable product design with an instantaneous probability of ι (i)F(δ ∗ )dt.
Once a patent has been successfully obtained, the entrepreneur chooses between selling andleapfrogging. If the entrepreneur decides to leapfrog the incumbent, an additional (1 − α )aLunits of labor are required to market the product. Alternatively, if the entrepreneur decides tosell the design, the incumbent firm employs an additional (1 − ξ )(1 − α )aL units of labor tomarket the product, where ξ ∈ (0, 1) regulates the degree to which the incumbent is more proficient at commercializing new product designs, given its existing production capacity and supply
networks.
In addition to the patent regulation (δ ∗ ) introduced above, the government implements thefollowing policies: (i) a corporate tax rate of τ π ∈ (0, 1) on per-period operating profits; (ii) a
capital gains tax of τ V ∈ (0, 1) on the sale of designs; (iii) a subsidy of sE ∈ (0, 1) for initial
product design; and (iv) a subsidy of sM ∈ (0, 1) for market entry through leapfrogging. All
R&D investment costs are fully deductible from taxable income. We assume that firms survive
in the market long enough to use the full tax deduction by carrying forward investment costs.
In the sections below, we characterize each of the entrepreneur’s choices and derive a thresholdinverse-quality level that determines whether an entrepreneur chooses selling or leapfrogging.
We then consider how policy changes affect the entrepreneur’s choice and study the implications
for growth and welfare.

2.4 Leapfrogging the Incumbent
When a new product design is used to leapfrog the incumbent, creating a new market leader, thedesign generates an after-tax profit stream with a present value equal to

VM (i) =

Z ∞
0

Rt

′

∗

′

e− 0 (ρ +ι (i,t )F(δ ))dt π M (i)dt,

9(8)

where the subscript M denotes variables associated with leapfrogging, and the gross operating
profit is π M (i) = (1 − δ (i))L. The discount factor is ρ + ι (i,t ′)F(δ ∗ ) with ι (i,t ′)F(δ ∗ ) capturing
the risk of potentially losing the market to a rival firm entering with a patentable superior productdesign at a future date t ′ .
Accounting for the additional labor cost incurred when marketing the product, the profit witha new design of inverse-quality δ (i) isΠM (i) =(1 − τ π )VM (i) −CM waL, f or δ (i) ≤ δ 0 ,


0,

(9)

f or δ (i) > δ 0 ,where we summarize the cost parameters associated with leapfrogging using CM ≡ (1 − τ π )((1 −sE )α + (1 − sM )(1 − α )), with sE ∈ (0, 1) and sM ∈ (0, 1) denoting the government subsidies
to initial product design and market entry through leapfrogging. The profit associated withleapfrogging is illustrated by the ΠM (i) curve in Figure 1. Importantly, the leapfrogging valueof a design is increasing in product quality (i.e., ∂ VM (i)/∂ δ (i) < 0). As such, the entrepreneur
can only use the design to leapfrog the incumbent if δ ≤ δ 0 , where we define δ 0 as the thresholdinverse-quality gap at which ΠM (i) = 0, signaling that the entrepreneur is indifferent between
entering or not entering the market through leapfrogging. If δ (i) > δ 0 , because leapfrogging is
unprofitable, the entrepreneur will not enter the market, but may instead have an incentive to sell
the design to the incumbent market leader, to which we turn next.

2.5 Selling to the Incumbent
When an entrepreneur sells a new product design to the incumbent market leader, the incumbent’s product quality improves from λ I (i) ≡ 1/δ I (i) to λ I (i)λ (i) ≡ 1/(δ I (i)δ (i)), allowing theincumbent’s product price to rise to px (i) = w/(δ I (i)δ (i)) after the design has been adopted.

10Prior to purchasing the design, the incumbent’s gross value is

VI (i) =

Z ∞
0

Rt

′

∗

′

e− 0 (ρ +ι (i,t )F(δ ))dt π I (i)dt,(10)

where the subscript I denotes variables associated with incumbent firms, and referencing (7) wehave π I (i) = (1 − δ I (i))L. But after purchasing and adopting the new design, the incumbent’s
gross value becomes
VS (i) =

Z ∞
0

Rt

′

∗

′

e− 0 (ρ +ι (i,t )F(δ ))dt π S (i)dt,(11)

where the subscript S indicates variables associated with selling, and π S (i) = (1 − δ I (i)δ (i))L.
Also, the incumbent firm’s per-period operating profit rises by π S (i) − π I (i) = δ I (i)(1 −δ (i))L > 0, and firms with a higher δ I (i) benefit more from the adoption of a new design purchase. Denoting the design price by PV (i), the change in the incumbent’s net value after purchasing a new design is ΠB (i) = (1 − τ π )(VS (i) − VI (i) − PV (i) − (1 − ξ )(1 − α )waL). We assume
that the entrepreneur makes a take-it-or-leave-it offer that just sets the change in the net value ofthe incumbent to zero: PV (i) = VS (i) −VI (i) − (1 − ξ )(1 − α )waL.
Then entrepreneur’s net profit from selling the design isΠS (i) = (1 − τ V )(VS (i) −VI (i)) −CSwaL,(12)

where we summarize the cost parameters associated with selling using CS ≡ (1 − τ V )((1 −sE )α + (1 − ξ )(1 − α )). The net profit from selling is an increasing function of product quality(i.e., ∂ VS (i)/∂ δ (i) < 0), and becomes negative if the quality of the design is too low (δ (i) > δ S ),as illustrated by the ΠS (i) curve in Figure 1. Importantly, low incumbent product quality (i.e.,a higher δ I (i)) is associated with a higher design price, and greater net profit from selling (i.e.,∂ (VS (i) −VI (i))/∂ δ I (i) > 0). Thus, a rise (fall) in δ I induces an upward (downward) shift in theΠS (i) curve.

11

Figure 1: Quality Thresholds
Π(i)ΠM (i)

ΠS (i)0δ M (i)

δ0

δ∗

δS

1

δ (i)

2.6 Leapfrogging vs. Selling
Having separately described the options of the entrepreneur to either leapfrog or sell to the incumbent, we now consider the entrepreneur’s equilibrium choice. To facilitate the analysis, wefirst define a threshold inverse-quality gap δ (i) = δ M (i) that equates the profits associated withleapfrogging and selling (ΠM (δ M ) = ΠS (δ M )), as shown by the intersection of the ΠM (i) andΠS (i) curves in Figure 1. We find that ΠM (i) > ΠS (i), for δ (i) < δ M (i), and the entrepreneur
leapfrogs. Alternatively, ΠM (i) < ΠS (i) for δ (i) > δ M (i), and the entrepreneur sells the design.
Thus, referencing the effects of changes in δ I (i) on the profit from selling ΠS (i), we straightforwardly obtain
Proposition 1 There are three cases for the entrepreneur’s leapfrogging vs. selling choice:(i) For δ I (i) > (1 − τ π )/(1 − τ V ), we have δ M (i) = 0, and the entrepreneur sells the productdesign to the incumbent firm for all inverse-quality gaps δ (i) ∈ (0, δ ∗ ).
(ii) For (1− τ π )/(1− τ V ) > δ I (i) > ((1−sE )α +(1− ξ )(1− α ))/((1−sE )α +(1−sM )(1− α )),we have δ M (i) ∈ (0, δ ∗ ), and the entrepreneur leapfrogs the incumbent for δ (i) ∈ (0, δ M (i)), but
sells the design for δ (i) ∈ (δ M (i), δ ∗ ).
(iii) For ((1 − sE )α + (1 − ξ )(1 − α ))/((1 − sE )α + (1 − sM )(1 − α )) > δ I (i), we have δ M (i) =δ ∗ , and the entrepreneur leapfrogs the incumbent for all inverse-quality gaps δ (i) ∈ (0, δ ∗ ).
Proof: See Appendix A.
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The entrepreneur’s leapfrog/sell choice is closely linked with the inverse-quality gap of theincumbent through the effect of changes in δ I (i) on the net profit from selling. If the incumbent’s
product design is low-quality, δ I (i) > (1 − τ π )/(1 − τ V ), the entrepreneur always sells its design
to the incumbent, because the net profit is greater from selling than from leapfrogging for all
values of δ (i) ∈ (0, δ ∗ ). Alternatively, if the incumbent’s design is high-quality, ((1 − sE )α +(1 − ξ )(1 − α ))/((1 − sE )α + (1 − sM )(1 − α )) > δ I (i), the entrepreneur leapfrogs because thenet profit is higher from leapfrogging than from selling.
In the intermediate case presented in Figure 1, where (1 − τ π )/(1 − τ V ) > δ I (i) > ((1 −sE )α + (1 − ξ )(1 − α ))/((1 − sE )α + (1 − sM )(1 − α )), the entrepreneur’s leapfrog/sell choicedepends on the quality of its new product design. Leapfrogging arises when δ (i) ∈ (0, δ M (i)),
but selling occurs when δ (i) ∈ (δ M (i), δ ∗ ). We take the time derivative of ΠM (δ M ) = ΠS (δ M )to derive an investment condition that describes the leapfrog/sell decision:ρ + ι F(δ ∗ ) =(1 − τ π )π M (δ M ) − (1 − τ V )(π S (δ M ) − π I ) ẇ+ .
(CM −CS )waL
w

(13)

This no-arbitrage condition equates the threshold relative return to leapfrogging over sellingwith the risk-adjusted interest rate for the overall economy. Consistent with Proposition 1,
the existence of δ M (i) ∈ (0, 1) requires 1 − τ π > (1 − τ V )δ I , since no level of design quality generates a positive return to leapfrogging if (1 − τ π )π M (δ M ) − (1 − τ V )(π S (δ M ) − π I ) =((1 − τ π ) − (1 − τ V )δ I )π M (δ ) < 0.
2.7 Free Entry
Expected profits drive the entry and exit of entrepreneurs in innovation. Referencing Figure 1,
the expected profit associated with a new product design in industry i isΠE (i) =Z δ M (i)0

ΠM (δ )dF(δ ) +Z δ∗

δ M (i)ΠS (δ )dF(δ ) −

13

Z 1
δ∗

(1 − sE )α waLdF(δ ),(14)where the first term on the RHS is the expected profit from leapfrogging, the second term is thatfrom selling the design, and the third term is the expected loss from market exit when the qualityof the design is not sufficient to obtain a patent.
Free entry and exit into R&D drives the expected profit of entrepreneurs to zero, ΠE (i) = 0.
Setting (14) to zero and taking the time derivative of the result yields the following no-arbitrage
condition for investment in the development of a new product design in industry i:ρ + ι F(δ ∗ ) =π E (δ M , δ I ) ẇ+ ,CE (δ M )waL w(15)

where we have used (8), (9), (10), (11), and (12). The expected per-period profit, net of taxes
and conditional on the design being patentable, is π E (δ M , δ I ) ≡ (1 − τ π ) 0δ M π M dF(δ ) + (1 −R∗τ V ) δδM (π S − π I )dF(δ ), and the expected cost associated with product development is describedRby CE (δ M ) ≡ (1 − sE )α (1 − F(δ ∗ )) + CM F(δ M ) + CS (F(δ ∗ ) − F(δ M )). As usual, the asset
condition (15) requires that the expected after-tax return to investment equal the risk-free interest
rate plus a risk premium to account for the threat of future market entry. The return consists ofthe expected dividends from leapfrogging and selling, in addition to expected capital gains.3

2.8 Evolution of Incumbent Product Quality
We now describe the evolution of the quality gap between the incumbent market leader andits nearest rival firm. The dynamics of δ I are driven by two mechanisms. First, as the entrant replaces the incumbent when δ < δ M (i), the expected change in the inverse-quality gap
R δ (i)that results from leapfrogging is ι (i) 0 M (δ − δ I (i))dF(δ ); Second, if the product design
is sold to the incumbent under δ > δ M (i), the expected change in the inverse-quality gap is
R ∗ι (i)δ I (i) δδM (i)(δ − 1)dF(δ ). Combining the above delivers the expected change in the inverse3 In our analysis, capital gains taxes are not applied to adjustments in stock prices (ẇ/w) in secondary markets.

This assumption will not influence our policy analysis as we find that stock prices are constant in the steady state of
the model. See also Peretto (2011) for a study of the effects of capital gains taxation on stock price fluctuations in
an endogenous growth framework.

14

quality gap,δ̇ I (i) =Z δ (i)M0δ dF(δ ) + δ I (i)Z δ∗δ M (i)δ dF(δ ) − δ I (i)

Z δ∗
0



dF(δ ) ι (i),(16)

with the first term indicating the effect of leapfrogging, the second term the effect of selling, andthe third term the exit of the industry leader.

2.9 Labor Market Equilibrium
At each moment in time, households inelastically supply a total of L units of labor to production and innovation. First, combining the demand condition (5) and the limit price px = w/δ I ,employment in production is LX = δ I L/w for the average industry, where δ I = 01 δ I (i)di is theaverage inverse-quality gap associated with the average incumbent firm.

R

Second, in the average industry, entrepreneurs employ ια aL units of labor in product development and ι (1 − α )aLF(δ M ) units of labor in market entry, conditional on leapfrogging, whereδ M denotes the leapfrog/sell quality threshold associated with the average industry. Meanwhile,
incumbent firms employ ι (1 − α )(1 − ξ )aL(F(δ ∗ ) − F(δ M ))aL units of labor in market entry, conditional on purchasing new product designs. Combining these yields the average labordemand from R&D: LR = ι R(δ M )aL, where R(δ M ) ≡ α + (1 − α )(F(δ M ) + (1 − ξ )(F(δ ∗ ) −F(δ M ))) describes the productivity of labor in innovation.
Summing up the labor demands from production and R&D, we derive the average arrival rate
for new product designs that clears the labor market (L = LX + LR ):ιL =1 − δ I /w.
R(δ M )a(17)

Thus, the average innovation rate is decreasing in the fixed cost of innovation (a) and the share of
product development in innovation costs (α ), but increasing in the strength of patent regulation(δ ∗ ) and incumbent productivity in market entry (ξ ).

15

2.10 Government
We set a relatively passive role for the government in adjusting the lump-sum tax on householdincome (T ), to balance the fiscal budget at all moments in time; that is,ι sE α waL + ι sM (1 − α )waLF(δ M ) = τ π π I (δ I ) + ιτ V

Z δ∗
δM

PV (δ )dF(δ ) + T L.

(18)

The LHS represents the cost of subsidies, and the RHS consists of the expected tax revenue. We
assume that the government budget is supported fully by tax revenue, and that the government isunable to issue bonds; i.e., budget deficits are not allowed.

3 Long-Run Equilibrium
The long-run equilibrium of the model features a constant allocation of labor between innovation and production, presupposing constant values for average incumbent quality, the innovation
rate, and the wage rate. Interestingly, the model can be reduced to two autonomous dynamic
systems, with the first governing the dynamics of product quality, and the second the dynamics
of innovation and wages.4 In the next subsection, we reduce the model to a steady-state systemwith two conditions that implicitly determine the average incumbent inverse-quality gap (δ I )and the threshold inverse-quality gap associated with the average industry (δ M ). Then, in thefollowing subsection, the steady-state innovation rate (ι ) and wage rate (w) are resolved throughequilibrium in the investment and labor markets.

3.1 Steady-State Innovation for Sale and Incumbent Firm Quality
Turning to the product quality gaps, the first condition captures combinations of the average incumbent inverse-quality gap and the threshold inverse-quality gap where the free-entry condition
4 Alternatively, the dynamics of the model can be studied within a single dynamic system that describes theevolution of the average incumbent inverse-quality gap (δ I ), the threshold inverse-quality gap (δ M ), and the wage
rate (w). The consideration of two independent dynamic systems, however, allows for a more intuitive explanationof the comparative static results associated with the policy analysis presented in Section 4.

16

Figure 2: Steady-state Product Quality Gaps
δM
δ

∗

I

δ̇ I = 0

M
1

0

δI

This stylized figure can be reproduced numerically setting f (δ ) = kδ k−1 with the following parametervalues: a = 1.0625, α = 0.671, ξ = 0.95, ρ = 0.02, k = 7.3, τ π = 0.21, τ V = 0.20, sE = 0.05, sM = 0.1,
and δ ∗ = 0.895. This parameter set generates δ I = 0.758, δ M = 0.877, δ 0 = 0.908, and δ S = 0.917.

for investment in a new product design and the threshold investment condition for leapfrogging
are simultaneously satisfied. Combining (13) and (15) gives the following investment condition:

π E (δ M , δ I ) (1 − τ π )π M (δ M ) − (1 − τ V )(π S (δ M , δ I ) − π I (δ I )).
=CE (δ M )CM −CS(19)

The LHS describes the return to product development as the ratio of the expected profit stream
from a new product design to the expected cost of product development. The RHS shows therelative return to leapfrogging over selling at the threshold inverse-quality gap (δ M ), as the ratioof the per-period profit differential to the cost differential associated with the entrepreneur’s
leapfrog/sell choice. The M-curve depicts (19) in Figure 2, and sets δ M to equate the return toproduct development with the relative return to leapfrogging over selling, for a given averageincumbent inverse-quality gap (δ I ). Thus, we find that equilibrium in the investment market
requires that the economy lie on the M-curve at all moments in time. Note that (19) is satisfied
independently of the innovation rate (ι ) and the wage rate (w).
The second condition is derived by setting the dynamics of incumbent quality (16) to zero17

for the average industry (δ̇ I = 0), and reorganizing the result:Z δM
0π M (δ )dF(δ ) +

Z δ∗
δM

π S (δ , δ I )dF(δ ) =Z δ∗0π I (δ I )dF(δ ).

(20)This expression implicitly determines the steady-state inverse-quality gap for the average incumbent firm (δ I ). In the long-run equilibrium, incumbent operating profit converges to the expected
operating profit associated with a patentable new product design. The steady-state locus captures
the positive relationship between δ M and δ I , as depicted by the I-curve in Figure 2. Intuitively,
a decrease in the share of product designs sold to incumbent firms (i.e., a rise in δ M ) reduces thefrequency with which incumbent products are updated, leading to a fall in average incumbent
product quality (i.e., a rise in δ I ).
The system includes one differential equation (20) and one side condition (19). By investigating the local dynamics around the steady state described by the intersection of the M-curveand I-curve, we have:
Lemma 1 The steady-state equilibrium with positive shares of product designs for both selling
and leapfrogging is stable.
Proof: See Appendix B.
The dynamic system can be reduced to a single differential equation for the average incumbent quality gap (δ̇ I ). As shown in Figure 2, for values to the left (right) of the I-curve, δ I rises(falls). Accordingly, the slope of the I-curve must be strictly greater than that of the M-curve, toensure that the economy converges along the M-curve to the long-run equilibrium. In Appendix
B, we show that the slope of the M-curve is strictly negative, and that the necessary slope ranking
is therefore always satisfied.

3.2 Long-Run Innovation
With the average incumbent quality gap and the threshold quality gap pinned down by (19)and (20), the long-run rate of innovation is determined together with the wage rate through18

Figure 3: Long-run Innovation Rate
ι
ιL

b

ιV
w

0

This stylized figure can be reproduced numerically setting f (δ ) = kδ k−1 with the following parameter values: a = 1.0625, α = 0.671, ξ = 0.95, ρ = 0.02, k = 7.3, τ π = 0.21, τ V = 0.20, sE = 0.05,
sM = 0.1, and δ ∗ = 0.895. This parameter set generates δ I = 0.758, δ M = 0.877, ι = 0.181, and w = 1.022.

equilibrium in the labor and investment markets. On the one hand, the labor market equilibriumis described by (17) with a strictly positive relationship between ι and w, as illustrated by theι L -curve in Figure 3. On the other hand, the investment market equilibrium is captured by the
no-arbitrage condition in new product designs (15), which can be rewritten in its steady-state
form using (19):ιV =(1 − τ π − (1 − τ V )δ I )π M (δ M )ρ
−,∗waLF (δ )(CM −CS )F(δ ∗ )

(21)

where we have set ẇ = 0. This expression captures the strictly negative relationship between
the innovation rate and the wage rate in the investment market, as depicted in Figure 3 by theι V -curve.
The local dynamics of the system around the steady state (i.e., the intersection of the ι L curve and ι V -curve) are consistent with the standard quality ladders model of innovation-based
growth (e.g., Grossman and Helpman, 1991). In Appendix B, we reduce the system to a single
differential equation in the wage rate (ẇ). And, as shown in Figure 3, for values to the left (right)of the ι V -curve the wage rate is falling (rising). Therefore, at each moment in time, the economy
must jump immediately to the steady-state equilibrium for the investment and labor markets,given the current values for the average incumbent inverse-quality gap (δ I ) and the threshold19

inverse-quality gap (δ M ). The resulting innovation rate isι=(1 − τ π − (1 − τ V )δ I )π M (δ M ) − δ I aLρ (CM −CS ).
(1 − τ π − (1 − τ V )δ I )π M (δ M )R(δ M )a + δ I aLF(δ ∗ )(CM −CS )(22)

Clearly, (1 − τ π − (1 − τ V )δ I )π M (δ M ) > δ I aLρ (CM −CS ) is required for ι ≥ 0. Hereafter, we
assume that the long-run equilibrium satisfies the necessary conditions for a positive rate ofinnovation.
In preparation for the policy analysis of Section 4, we consider how the innovation rate (22)adjusts with changes in the threshold quality gap (δ M ) associated with the leapfrog/sell decisionand the average incumbent quality gap (δ I ). The following lemma outlines the relationship thatarises between δ M , δ I , and ι in the labor and investment markets:
Lemma 1 Decreases in the average threshold quality gap for leapfrogging (i.e., a rise in δ M )and in the average incumbent quality gap (i.e., a rise in δ I ) lower the rate of innovation (ι ).
First, in the labor market (17), the average amout of labor required to introduce a new productdesign R(δ M )aL increases with a rise in the share of leapfrogging designs, as incumbent firms
require less labor in the marketing of new products. In addition, a fall in the average quality
gap of incumbent firms lowers the average product price px = w/δ I , inducing an expansion in
production as household demand rises. These two effects result in a lower level of employment
in innovation that exhibits as a downward shift in the ι L -curve in Figure 3.
Second, in the investment market (21), increases in δ M and δ I both reduce the threshold profit
from leapfrogging over selling (1 − τ π − (1 − τ V )δ I )π M (δ M ), lowering the threshold relative return to leapfrogging. This in turn decreases investment in innovation, described by a downward
shift in the ι V -curve in Figure 3. From the labor market and investment market channels, decreases in the average threshold quality gap and in the average incumbent quality gap lead to aslower rate of innovation.

20

3.3 Social Welfare
Due to the random quality improvements of new product designs in each industry and the choice
between leapfrogging and selling, a general analysis of welfare is not tractable. As an alternative,
we follow Minniti et al. (2013) in assuming that the random quality improvements are governedby a Pareto distribution with a probability density function of f −1 (λ ) = k/λ k+1 , where k > 1 isa shape parameter. Then, the random inverse-quality gaps are generated by f (δ ) = kδ k−1 .
As households only consume the state-of-the-art product in each industry, instantaneous utility (2) reduces to log D(t) = log(x(t)/L) + I(t)E[log λ ], with the first term describing the utility stemming from the average quantity of consumption (x(t) =R10 x(t, i)di), and the second

term capturing the utility derived from the average quality of products consumed. Importantly,
I(t) = 0t ι (t ′)dt ′ = ι t measures the expected number of new product designs introduced beforeRktime t, under a constant steady-state rate of innovation, and E[log λ ] = (1 − k log δ ∗ )δ ∗ /k describes the average size of the quality improvements associated with new product designs.
With constant steady-state employment in production and innovation, the time derivative of
instantaneous utility yields the long-run rate of growth as g ≡ ι E[log λ ], where we have used thefact that only designs satisfying patent regulation update product quality (i.e., λ ≥ λ ∗ ≡ 1/δ ∗ ).
Combining the growth rate with lifetime utility (1), we can calculate steady-state welfare askU=log(1 − ι R(δ M )a) ι (1 − k log δ ∗ )δ ∗
+
,
ρ
kρ 2

(23)

where R(δ M ) ≡ α + (1 − α )ξ δ kM + (1 − α )(1 − ξ )δ ∗ . The first term on the RHS shows thekwelfare derived from the steady-state level of consumption, and the second term describes thatderived from future growth in product quality. Thus, we find that there are three possible channels
through which economic policy may affect household welfare. First, an increase in the frequencyof leapfrogging (i.e., a rise in δ M ) lowers welfare by increasing the labor required for market
entry, thus contracting the production of goods for consumption. Second, an increase in theinnovation rate similarly has a negative welfare effect as it pulls labor away from production.

21

Third, an increase in the rate of innovation improves welfare by accelerating growth in productquality. As such, the net welfare changes depend on both the type of policy and its magnitude.

4 Policy Analysis
This section studies the effects of economic policy on the entrepreneur’s leapfrog/sell choice, therate of innovation, and social welfare. We focus on several alternative policy instruments that areavailable to the government: the corporate tax rate (τ π ), the capital gains tax (τ V ), the productdesign subsidy (sE ), and the leapfrog subsidy (sM ).
Our policy analysis includes numerical evaluations of steady-state welfare. As the numericalresults are sensitive to changes in parameter values, we select the benchmark parameters based on
three documented facts. First, Serrano (2010) reports that 13.5% of patents are sold (transferred)at least once in the United States. Second, empirical evidence suggests that the average price-cost
markup lies between 1.2 and 1.6 in the United States (De Loecker et al., 2020). Third, we target
at a rate of economic growth of 2% as the benchmark for our policy analysis to approximate theaverage rate of growth observed historically for the United States (Jones, 2005). We also use avalue of ρ = 0.02 for the intertemporal discount rate following Jones et al. (1993). Given thesemotivations, we set the corporate tax rate to τ π = 0.21 and the capital gains tax rate to τ V = 0.20
to match with the current rates in the United States.5 The product design subsidy and the leapfrog
subsidy are respectively set to sE = 0.05 and sM = 0.1. The share of R&D costs incurred before
patent application is α = 0.671, the relative productivity of incumbent firms in marketing new
product designs is ξ = 0.95, and the scale parameter for innovation costs is a = 1.0625. Finally,
we let k = 7.3 and δ ∗ = 0.895 for the shape parameter of the density function and the strengthof patent regulation. This benchmark parameter set produces the following equilibrium values:δ I = 0.758, δ M = 0.877, ι = 0.184, and w = 1.022, x/L = 0.899, and U = 42.45. The pricecost markup is therefore px /w = 1/δ I = 1.31. The rate of innovation is substantially lower than
5 The US 2017 Tax Cuts and Jobs Act revised the definition of capital assets to exclude patents, thereby raising

the tax rate imposed on primary patent sales from the capital gains tax rate of 20% to the ordinary income tax rateof 37% for individuals in the highest income tax bracket.

22the rate of economic growth, as in the real world only 44.5% of new product designs have asufficient quality level to obtain patent protection.6 , and 13.5% of patented product designs are
sold to incumbent firms.

4.1 Corporate TaxesFirst, we consider the effects of changes in the corporate tax rate (τ π ). The relationships between
the corporate tax rate, the leapfrog/sell decision (δ M ), and average incumbent quality (δ I ) canbe uncovered by taking the partial derivative of the investment condition (19) with respect to τ π ,revealing two opposing effects between leapfrogging and selling:(1 − τ V )−(1 − τ π )R ∗δ I δδM π M dF(δ ) δ I π M (δ M )+CE
CM −CS!

+((1 − sE )α (1 − F(δ ∗ )) +CS F(δ ∗ ))CM π E.
(1 − τ π )(CM −CS )CE2

The first term is the negative profit effect of higher corporate taxation, showing that an increasein τ π reduces the incentive for entrepreneurs to leapfrog, pushing the M-curve downwards in
Figure 2. The second term is the positive tax exemption effect, stemming from an increase in the
tax allowance for R&D expenditures associated with leapfrogging. A rise in the tax exemption
tends to shift the M-curve upwards, as entrepreneurs have more incentive to leapfrog.
If the profit effect dominates, the share of designs used to leapfrog falls, (i.e., δ M falls),
and average incumbent quality rises as the product designs of market leaders are updated more
frequently (i.e., δ I rises). Otherwise, if the tax exemption effect dominates, the design quality
required to induce leapfrogging decreases (i.e., δ M rises), leading to lower average incumbent
quality (i.e., δ I rises). These results and their implications for the long-run rate of innovation aresummarized as:
Proposition 2 (i). If the positive tax exemption effect dominates, an increase in the corporate
tax rate (τ π ) lowers the share of product designs sold to incumbent firms (i.e, a rise in δ M ), while
6 Carley et al. (2015) estimate that only 55.8% of applications filed with the US Patent and Trademark Office

were granted patents between 1996 and 2013. The lower share of new designs awarded patents in our analysis canbe rationalized by considering that many inventors recognize that their product designs do not include innovations
that are insufficient to be awarded a patent and therefore do not initiate a patent application.

23

reducing the average product quality of market leaders (i.e., a rise in δ I ). The rate of innovation(ι ) falls. (ii). If the negative profit effect dominates, however, an increase in τ π lowers δ M andδ I . The effect on the innovation rate is ambiguous.
Proof: See Appendices C and D.
An increase in the corporate tax rate (τ π ) affects the innovation rate through adjustments inthe labor and investment markets. First, referring to (21), we find that the negative profit andpositive tax exemption effects have a direct impact on the threshold return to leapfrogging overselling in the investment market. However, the direct negative profit effect always dominates
the direct tax exemption effect, slowing the rate of innovation as the ι V -curve shifts downwards
in Figure 3 (i.e., ∂ ι /∂ τ π < 0). Next, following the results of Lemma 1, the increase in τ πalso affects innovation indirectly through the adjustments in the labor and innovation markets
that coincide with changes in the average threshold inverse-quality gap (δ M ) and the averageincumbent inverse-quality gap (δ I ). On the one hand, if the positive tax exemption effect dominates, δ M and δ I increase, generating downward shifts in the ι L -curve and the ι V -curve (i.e.,∂ ι /∂ δ M · ∂ δ M /∂ τ π < 0 and ∂ ι /∂ δ I · ∂ δ I /∂ τ π < 0). In this case, the directions of the direct andindirect effects align, and an increase in the corporate tax rate slows the rate of innovation. On
the other hand, if the negative profit effect dominates, δ M and δ I decrease, causing the ι L -curveand ι V -curve to shift upwards (i.e., ∂ ι /∂ δ M · ∂ δ M /∂ τ π > 0 and ∂ ι /∂ δ I · ∂ δ I /∂ τ π > 0). Then,in this case, the balance between the direct and indirect effects of a change in τ π determines therelationship between the corporate tax rate and the rate of innovation.
Figure 4 presents a numerical analysis of the effects of changes in the corporate tax rate on
the share of designs sold to incumbent firms (δ M ), average incumbent firm quality (δ I ), the rateof economic growth (g), household consumption (x/L), and welfare (U ) over the policy rangeτ π ∈ (0, 0.4). The plots suggest that the tax exemption effect dominates when the corporate tax
rate is low, with an increase in τ π reducing the share of designs sold to incumbents, and lowering
average incumbent product quality, while the profit effect dominates when the corporate tax rate
is high, with an increase in τ π lowering δ M and δ I . Turning to innovation, we find that under24

Figure 4: Corporate Tax
0.022

0.9

0.02150.85
0.021
0.0205

0.8

0.02
0.75
0.0195
0.7
0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.019
0.05

0.4

0.86

46

0.855

45

0.85

44

0.845

43

0.84

42

0.835

41

0.83
0.05

0.1

0.15

0.2

0.25

0.3

0.35

40
0.05

0.4

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.1

0.15

0.2

0.25

0.3

0.35

0.4

These figures are produced using f (δ ) = kδ k−1 with the parameter values a = 1.0625, α = 0.671,
ξ = 0.95, ρ = 0.02, k = 7.3, sE = 0.05, sM = 0.1, τ V = 0.2, and δ ∗ = 0.895. This benchmark parameterset yields δ I = 0.758, δ M = 0.877, w = 1.022, ι = 0.184, g = 0.02, x/L = 0.899, and U = 42.45.

the assumed parameter set the direct effect of changes in the corporate tax rate dominates acrossthe policy range, and an increase in τ π reduces employment in innovation and slows the rate of
economic growth. Adjustments in the price-cost markup depend on the balance of the profit andtax exemption effects, but a decrease in the wage rate is the key factor behind a fall in the productprice of the average industry (px = w/δ I ) that generates an expansion in household consumption(x/L). Ultimately, household welfare follows the rate of economic growth, with rising corporate
taxes reducing steady-state utility (U ).

4.2 Capital Gains Taxes
Second, we investigate the effects of changes in the capital gains tax (τ V ). Partially differentiating
the investment condition (19) yields:
R ∗δ I δδM π M dF(δ ) δ I π M (δ M )+CE
CM −CS!

((1 − sE )α (1 − F(δ ∗ )) +CM F(δ ∗ ))CS π E.
−(1 − τ V )(CM −CS )CE2
25

The first term is the positive profit effect, indicating that an increase in τ V makes entrepreneurs
more likely to leapfrog incumbents, causing an upward shift in the M-curve. The second termis the negative tax exemption effect, showing that an increase in the capital gains tax associatedwith selling product designs tends to shift the M-curve downwards.
When the positive profit effect dominates, the share of designs for leapfrogging increases(i.e., δ M rises), and average incumbent product quality falls, as their product designs are updated
less frequently (i.e., δ I rises). Alternatively, when the negative tax exemption effect dominates,
the design quality required to leapfrog rises (i.e., δ M falls), leading to more frequent updating ofincumbent designs, and higher average incumbent quality (i.e., δ I falls). We summarize theseresults and their implications for the rate of innovation as:
Proposition 3 (i). When the positive tax effect dominates, an increase in the capital gains tax
rate (τ V ) decreases the share of product designs sold to ithe ncumbent firms (i.e., a rise in δ M ),while lowering the average product quality of market leaders (i.e., a rise in δ I ). The rate ofinnovation (ι ) slows down. (ii). When the negative tax exemption effect dominates, an increasein τ V lowers δ M and δ I . The effect on the innovation rate (ι ) is ambiguous.
Proof: See Appendices C and D.
An increase in the capital gains tax (τ V ) also affects the rate of innovation (ι ) via adjustmentsin the labor and investment markets. First, the direct effects of raising τ V are a negative effect on
the profit from selling designs and a positive effect on the tax allowances for R&D expendituresassociated with selling. The direct negative profit effect dominates, causing a downward shift inthe ι V -curve in Figure 3 (i.e., ∂ ι /∂ τ V < 0). Recalling Lemma 1, the increase in τ V also affects
innovation indirectly through adjustments in the labor and investment markets that coincide withchanges in the threshold inverse-quality gap (δ M ) and the average incumbent inverse-quality gap(δ I ). If the positive profit effect dominates, δ M and δ I increase, causing the ι L -curve and theι V -curve to shift downwards (i.e., ∂ ι /∂ δ M · ∂ δ M /∂ τ V < 0 and ∂ ι /∂ δ I · ∂ δ I /∂ τ V < 0). Hence,
because the directions of the direct and indirect effects match, increasing the capital gains tax
slows the rate of innovation. Alternatively, if the negative tax exemption effect dominates, δ M
26

Figure 5: Capital Gains Tax
0.9

0.022
0.0215

0.85
0.021
0.8

0.02050.02

0.75
0.0195
0.019

0.7
0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.855

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

46

0.85

44

0.845
42
0.84
40

0.835
0.83

38
0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

These figures are produced using f (δ ) = kδ k−1 with the parameter values a = 1.0625, α = 0.671,
ξ = 0.95, ρ = 0.02, k = 7.3, sE = 0.05, sM = 0.1, τ π = 0.21, and δ ∗ = 0.895. This benchmark parameterset yields δ I = 0.758, δ M = 0.877, w = 1.022, ι = 0.184, g = 0.02, x/L = 0.899, and U = 42.45.

and δ I decrease, shifting the ι L -curve and the ι V -curve upwards (i.e., ∂ ι /∂ δ M · ∂ δ M /∂ τ V > 0and ∂ ι /∂ δ I · ∂ δ I /∂ τ V > 0). The balance of the direct and indirect effects then determines therelationship between the capital gains tax and the innovation rate.
Figure 5 plots numerical results for the effects of a change in the capital gains tax rate (τ V ) on
the leapfrog/sell choice (δ M ), the average incumbent inverse-quality gap (δ I ), the rate of growth(g), household consumption (x/L), and welfare (U ) over the policy range τ V ∈ (0, 0.4). Fromthe plots, we infer that the positive profit effect dominates, with an increase in the capital gains
tax reducing the share of designs sold to incumbents and lowering the average product qualityof market leaders, given the assumed parameter set. The indirect effects align with the negative
direct effect of capital gains taxation on employment in innovation, resulting in a slower rate of
growth. Despite a rising wage rate, a fall in the price-cost markup ensures that the product priceof the average industry falls (px = w/δ I ), allowing for an expansion in household consumption(x/L). Adjustments in the rate of economic growth drive the direction of changes in household

27

utility, however, with an increase in the capital gains tax rate hurting welfare (U ).

4.3 Product Design Subsidy
Next, we study the effects of changes in the subsidy to product design (sE ). Partially differentiating (19) with respect to sE yields

α (1 − F(δ ∗ ) + (1 − τ π )F(δ M ) + (1 − τ V )(F(δ ∗ ) − F(δ M )))π E α (τ π − τ V )π E+.
(CM −CS )CE
CE2
The first term is the positive effect of the product design subsidy on the expected return to productdevelopment that results from a reduction in the expected cost (CE ). This effect shifts the Mcurve downwards in Figure 2. The second term captures the effect of the subsidy on the cost ofleapfrogging relative to that of selling (CM −CS ), the sign of which depends on the balance of thecorporate and capital gains tax rates. This effect shifts the M-curve downwards for τ π > τ V , andupwards for τ π < τ V . Overall, we find an increase in sE shifts the M-curve downwards, causing
the share of product designs sold to incumbents to rise (i.e., δ M falls). The higher frequency withwhich incumbent product designs are updated then results in higher average incumbent quality(i.e., δ I falls). We summarize these results as:
Proposition 4 An increase in the subsidy (sE ) to product design (i.e., initial R&D) increases
the share of product designs sold to incumbent firms (i.e., δ M falls), while raising the average
product quality of market leaders (i.e., δ I falls). The effect of a change in the subsidy to productdesign on the innovation rate (ι ) is generally ambiguous.
Proof: See Appendices C and D.
Changes in the product design subsidy (sE ) affect the innovation rate (ι ) directly through theinvestment market. When τ π > τ V , an increase in sE lowers the cost of selling more than thatof leapfrogging, shifting the ι V -curve upwards in Figure 3. Alternatively, when τ π < τ V , there
is a greater decrease in the cost of selling, and the ι V -curve shifts downwards. From Lemma 1,
the design subsidy also has indirect effects on the innovation rate, with decreases in δ M and δ I
28Figure 6: Product Design Subsidy
0.9

0.0215

0.85

0.021

0.8

0.0205

0.02

0.75
0

0.05

0.1

0.15

0

0.05

0.1

0.15

0

0.05

0.1

0.15

44.5

0.848
0.846

44

0.844

43.5

0.842

43

0.84

42.5

0.838

42

0.836

41.5
0

0.05

0.1

0.15

These figures are produced using f (δ ) = kδ k−1 with the parameter values a = 1.0625, α = 0.671,
ξ = 0.95, ρ = 0.02, k = 7.3, sM = 0.1, τ π = 0.21, τ V = 0.2, and δ ∗ = 0.895. This benchmark parameterset yields δ I = 0.758, δ M = 0.877, w = 1.022, ι = 0.184, g = 0.02, x/L = 0.899, and U = 42.45.

shifting both the ι L -curve and the ι V -curve upwards in Figure 3. (i.e., ∂ ι /∂ δ M · ∂ δ M /∂ sE > 0and ∂ ι /∂ δ I · ∂ δ I /∂ sE > 0). Summarizing, the net effect of the design subsidy on innovation isdetermined by whether the direct effect or indirect effect dominates.
In Figure 6, we present a numerical analysis of the effects of the design subsidy (sE ) on theleapfrog/sell choice (δ M ), the average incumbent inverse-quality gap (δ I ), the rate of growth (g),household consumption (x/L), and welfare (U ) over the range sE ∈ (0, 0.15). Essentially, thetheoretical results of Proposition 4 are reproduced numerically, with an increase in sE expanding
the share of designs sold to incumbents, and raising the average quality of market leaders. For thebenchmark parameter set, the positive indirect effects of investment and labor market adjustments
dominate, ensuring that the design subsidy encourages investment in innovation, and causing an
increase in the growth rate. A rise in the price-cost markup of the average industry and a higher
wage rate raise product price (px = w/δ I ), leading to a contraction in household consumption(x/L). Overall, we find that introducing a subsidy to product design improves household welfare29

(U ) as the positive effect of faster growth dominates the negative effect of lower consumption forthe benchmark parameter set.

4.4 Leapfrog Subsidy
Finally, we consider the effects of changes in the subsidy to market entry through leapfrogging(sM ). An increase in sM reduces the cost to entrepreneurs of marketing a new product design(∂ CM /∂ sM = −(1 − τ π )(1 − α ) < 0), raising the threshold return to leapfrogging and shifting
the M-curve upwards in Figure 2. Consequently, a lower design quality is required to entice
entrepreneurs to leapfrog (i.e., δ M rises). And, as the share of designs sold to incumbents contracts, average incumbent quality falls (i.e., δ I rises). These results and the subsequent effect on
the innovation rate are summarized as:
Proposition 5 An increase in the subsidy (sM ) to market entry through leapfrogging decreases
the share of product designs sold to incumbent firms (i.e., δ M rises), while lowering the average
product quality of market leaders (i.e., δ I rises). The effect on the innovation rate (ι ) is generally
ambiguous.
Proof: See Appendices C and D.
The direct effect of the leapfrog subsidy (sM ) on the innovation rate (ι ) is an increase in
the threshold return to leapfrogging over selling that shifts the ι V -curve upwards in Figure 3(i.e., ∂ ι /∂ sM > 0), as the R&D costs associated with leapfrogging fall. From Lemma 1, however, the market entry subsidy also has indirect effects on the labor and investment markets,with the increases in δ M and δ I shifting both the ι L -curve and the ι V -curve downwards (i.e.,∂ ι /∂ δ M · ∂ δ M /∂ sM < 0 and ∂ ι /∂ δ I · ∂ δ I /∂ sM < 0). Accordingly, the relationship between
the leapfrog subsidy and the innovation rate depends on whether the direct effect or the indirect
effect dominates.
Figure 7 provides a numerical analysis of the effects of changes in the subsidy to market
entry (s) on the leapfrog/sell choice (δ M ), the average incumbent inverse-quality gap (δ I ), the30

Figure 7: Leapfrog Subsidy
0.90.020350.020348

0.85

0.020346
0.020344

0.8

0.020342

0.75

0.020338

0.02034

0

0.05

0.1

0.15

0.2

0.25

0.3

0.845

0

0.05

0.1

0.15

0.2

0.25

0.3

0

0.05

0.1

0.15

0.2

0.25

0.3

42.45
42.4

0.844

42.35
0.843
42.3
0.842
42.25
0.841

42.2

0.84

42.15
0

0.05

0.1

0.15

0.2

0.25

0.3

These figures are produced using f (δ ) = kδ k−1 with the parameter values a = 1.0625, α = 0.671,
ξ = 0.95, ρ = 0.02, k = 7.3, sE = 0.05, τ π = 0.21, τ V = 0.2, and δ ∗ = 0.895. This benchmark parameterset yields δ I = 0.758, δ M = 0.877, w = 1.022, ι = 0.184, g = 0.02, x/L = 0.899, and U = 42.45.

growth rate (g), household consumption (x/L), and welfare (U ) over the range sM ∈ (0, 0.3). The
plots demonstrate the theoretical results of Proposition 5, with an increase in the leapfrog subsidy
reducing the share of designs sold to incumbents, and lowering the average product quality ofmarket leaders. Although the adjustments in economic growth are small, under the assumed
parameter set, our numerical analysis suggests that initially the positive direct effect dominates,
hastening innovation and accelerating growth. Beyond a certain level, however, the negative
indirect effects of investment and labor market adjustments dominate, and further increases in
the leapfrog subsidy slow innovation, thereby reducing the growth rate. While the rise in δ I
reduces the price-cost markup in the average industry, the wage rate increases leading to a higher
product price (px = w/δ I ). As a result, household consumption (x/L) contracts. Overall, we findthat introducing a leapfrog subsidy hurts household welfare (U ) for the parameter set that wehave considered here.

31

5 Conclusion
This paper studies how tax policy affects economic growth through its influence on entrepreneurs’
choice of commercialization mode for new innovations. In particular, we introduce a model ofinnovation-based economic growth in which entrepreneurs invest in new product designs that
improve the quality of vintage product lines. The size of quality improvements are randomly
drawn, generating heterogeneous values for product designs. After the quality of a design isrevealed, an entrepreneur decides whether to use the design to leapfrog the incumbent firm andbecome the market leader or sell the design to the incumbent. We show that high-quality designs
are used to leapfrog and low-quality designs are sold.
We use the framework to study how tax incentives affect the entrepreneur’s leapfrog/sell
choice, and consider the implications for economic growth and social welfare. Characterizing
an average industry using incumbent product quality and the threshold design quality required toleapfrog, we find that tax incentives influence the rate of innovation through two channels. The
first is the standard direct channel where tax incentives reduce expected R&D costs and promote
investment in innovation. The second indirect channel operates through the positive relationshipthat arises between the threshold design quality required for leapfrogging and the innovation
rate. Tax incentives that increase the likelihood of leapfrogging rather than selling, tend to slow
the rate of innovation. Because either the direct or the indirect channel may dominate when the
directions of the channels do not align, the overall impact of tax policy on the rate of innovationis generally ambiguous. Further, numerical analysis suggests that although subsidies to productdesign raise the growth rate and improve welfare, corporate taxes on operating profits, capitalgains taxes on the sale of product designs and subsidies to market entry all slow the rate of
economic growth and harm welfare.

32

Appendix A
This appendix derives the existence conditions for δ M (i) ∈ (0, δ ∗ ). First, we obtain a condition
for the required horizontal intercept ranking δ S > δ 0 . Referencing (9) and (12), and reorganizingΠM (δ 0 ) = 0 and ΠS (δ S ) = 0 yields waL = (1 − τ π )VM (δ 0 )/CM = (1 − τ V )(VM (δ s ) − VI )/CS ,which implies (1 − τ π )(1 − δ 0 )/CM = (1 − τ V )(1 − δ S )δ I /CS . Rearranging givesδ S − δ 0 = (1 − δ 0 ) 1 −
(1 − sE )α + (1 − ξ )(1 − α ),((1 − sE )α + (1 − sM )(1 − α ))δ I (i)and thus δ I > ((1 − sE )α + (1 − ξ )(1 − α ))/((1 − sE )α + (1 − sM )(1 − α )) is required for δ S >δ 0 . Second, we derive a condition for the required slope ranking ∂ ΠM (i)/∂ δ (i) < ∂ ΠS (i)/∂ δ (i).
Taking the derivatives of (9) and (12) yields
Rt∞′′dΠM (δ (i)) dΠS (δ (i))−=−
e− 0 (ρ +ι (i,t ))dt ((1 − τ π ) − (1 − τ V )δ I (i)) Ldt < 0.
d δ (i)d δ (i)0Z

Therefore, (1 − τ π )/(1 − τ V ) > δ I (i) is required for dΠM (i)/d δ (i) − dΠS (i)/d δ (i) < 0. Combining the above results, the threshold quality level δ M (i) ∈ (0, δ ∗ ) exists for (1− τ π )/(1− τ V ) >δ I (i) > ((1 − sE )α + (1 − ξ )(1 − α ))/((1 − sE )α + (1 − sM )(1 − α )). Finally, noting that at
δ (i) = 1, we have VM (i) = 0 and VS (i) − VI (i) = 0. Thus, we require CM > CS for δ S > δ 0 , or
(1 − τ π )/(1 − τ V ) > ((1 − sE )α + (1 − ξ )(1 − α ))/((1 − sE )α + (1 − sM )(1 − α )). This completes the proof of Proposition 1.

Appendix B: Stability of Long-Run Equilibrium
We first consider the stability of the steady state illustrated in Figure 2. A Taylor expansion ofδ̇ I =Z δ
0M

δ dF(δ ) + δ I

Z δ∗
δM

δ dF(δ ) − δ I

33

Z δ∗
0



dF(δ ) ι ,evaluating around (19) and δ̇ I = 0 yield∂ δ̇ I= −(1 − δ I )δ M f (δ M )ι
∂δI



dδ M
dδ M
−
d δ I I−curve d δ I M−curve



< 0,

where
δM
δ dF(δ )/δ I
dδ M
> 0,
= 0
dδ I I(1 − δ I )δ M f (δ M )Rdδ M(1 − τ V )(CM −CS )=−dδ I M((1 − τ π ) − (1 − τ V )δ I )LR δ∗

δ M π M (δ )dF(δ )CE (δ M )π M (δ M )+CM −CS!

< 0,

and the expected entry cost is denoted by CE (δ M ) ≡ (1 − sE )α (1 − F(δ ∗ )) + (CM −CS )F(δ M ) +CS F(δ ∗ ). As δ I is a state variable, stability requires ∂ δ̇ I /∂ δ I < 0.
Second, we examine the stability of the steady state depicted in Figure 3. Using (15) and(17), the system can be reduced to a single differential equation describing the dynamics of thewage rate:ẇ = wρ +(w − δ I )F(δ ∗ ) (1 − τ π )π M (δ M ) − (1 − τ V )(π S (δ M ) − π I )−.
CE (δ M )a
(CM −CS )aL

Setting w as a control variable, a Taylor expansion delivers ∂ ẇ/∂ w = ρ +F(δ ∗ )/(CE (δ M )a) > 0,
indicating that the system is saddle-path stable, with the wage rate jumping immediately andpermanently to its steady-state level.

Appendix C: Comparative Statics for Product Quality
Appendix C calculates the effects of policy changes on the entrepreneur’s leapfrog/sell choice
and incumbent quality. To facilitate the analysis, we rewrite (19) and (20) in implicit form:Ω1 =π E (δ M , δ I ) ((1 − τ π ) − (1 − τ V )δ I )π M (δ M ),−CE (δ M )CM −CS

Ω2 =

Z δM
0

δ dF(δ ) + δ I

Z δ∗
δM

34

δ dF(δ ) − δ IZ δ∗0dF(δ ),

where∂ Ω1((1 − τ π ) − (1 − τ V )δ I )L
> 0,
=
∂δM
CM −CSR ∗(1 − τ V ) δδM π M (δ )dF(δ ) (1 − τ V )π M (δ M )
∂ Ω1
> 0,
=
+
∂δI
CE (δ M )CM −CS∂ Ω2= (1 − δ I )δ M f (δ M ) > 0,
∂δM
Z
∂ Ω2
1 δM
δ dF(δ ) < 0,
=−
∂δI
δI 0
and thus we have |J1 | = (∂ Ω1 /∂ δ M )(∂ Ω2 /∂ δ I ) − (∂ Ω1 /∂ δ I )(∂ Ω2 /∂ δ M ) < 0.
Corporate Tax: Taking the total derivatives of Ω1 and Ω2 with respect to τ π yields∂ Ω1 ∂ Ω2 1
dδ M
≷ 0,
=−
dτ π
∂ τ π ∂ δ I |J1 |∂ Ω1 ∂ Ω2 1
dδ I
≷ 0,
=
dτ π
∂ τ π ∂ δ M |J1 |where ∂ Ω2 /∂ δ M > 0, ∂ Ω2 /∂ δ I < 0, |J1 | < 0, and∗δ I δδM π M dF(δ ) δ I π M (δ M )+CE
CM −CSR∂ Ω1 (1 − τ V )=∂τπ(1 − τ π )!

((1 − sE )α (1 − F(δ ∗ )) +CS F(δ ∗ ))CM π E≷ 0.
−(1 − τ π )(CM −CS )CE2

Capital Gains Tax: Taking the total derivatives of Ω1 and Ω2 with respect to τ V givesdδ M∂ Ω1 ∂ Ω2 1
≷ 0,
=−
dτV
∂ τ V ∂ δ I |J1 |dδ I∂ Ω1 ∂ Ω2 1
≷ 0,
=
dτV
∂ τ V ∂ δ M |J1 |where ∂ Ω2 /∂ δ M > 0, ∂ Ω2 /∂ δ I < 0, |J1 | < 0, and∗δ I δδM π M dF(δ ) δ I π M (δ M ) ((1 − sE )α (1 − F(δ ∗ )) +CM F(δ ∗ ))CS π E
∂ Ω1
≷ 0.
+
=−
−
∂ τV
CE
CM −CS(1 − τ V )(CM −CS )CE2
R

Product Design Subsidy: Taking the total derivatives of Ω1 and Ω2 with respect to sE generates∂ Ω1 ∂ Ω2 1
dδ M
=−
< 0,
dsE
∂ sE ∂ δ I |J1 |dδ I∂ Ω1 ∂ Ω2 1
=
< 0,
dsE
∂ sE ∂ δ M |J1|

35

where ∂ Ω2 /∂ δ M > 0, ∂ Ω2 /∂ δ I < 0, |J1 | < 0, and∂ Ω1 ((1 − τ π )(1 − sM )(1 − τ V F(δ ∗ )) − (1 − τ V )(1 − ξ )(1 − τ π F(δ ∗ )))α (1 − α )π E
> 0.
=
∂ sE
(CM −CS )CE2
To show ∂ Ω/∂ sE > 0, we compare the minimum corporate tax rate that satisfies (1 − τ π )(1 −sM )(1 − τ V F(δ ∗ )) ≥ (1 − τ V )(1 − ξ )(1 − τ π F(δ ∗ )); that is,τ π1 ≡ 1 −(1 − τ V )(1 − ξ )(1 − F(δ ∗ )),(1 − sM )(1 − F(δ ∗ )) + (ξ − sM )(1 − τ V )F(δ ∗ )with the minimum corporate tax rate required to ensure the existence of an interior equilibrium
for both leapfrogging and selling,τ π2 ≡ 1 −(1 − τ V )((1 − sE )α + (1 − ξ )(1 − α )),(1 − sE )α + (1 − sM )(1 − α )as outlined in case (ii) of Proposition 1. Comparing these minimum tax rates yields

τ π1 − τ π2 =(1 − τ π )(1 − τ V )(ξ − sM )((1 − sE )α (1 − F(δ ∗ ) +CS F(δ ∗ ))> 0,
CM ((1 − sM )(1 − F(δ ∗ )) + (ξ − sM )(1 − τ V )F(δ ∗ ))where ξ > sM is also required for the existence of an interior equilibrium. Thus, as τ π 1 > τ π 2 ,we find that when the corporate tax rate is sufficiently low, ∂ Ω/∂ sE > 0 holds.
Leapfrog Subsidy: Taking the total derivatives of Ω1 and Ω2 with respect to sM leads todδ M∂ Ω1 ∂ Ω2 1
=−
> 0,
dsM
∂ sM ∂ δ I |J1 |∂ Ω1 ∂ Ω2 1
dδ I
=
> 0,
dsM
∂ sM ∂ δ M |J1|

where ∂ Ω2 /∂ δ M > 0, ∂ Ω2 /∂ δ I < 0, |J1 | < 0, and∂ Ω1(1 − τ π )(1 − α )((1 − sE )α (1 − F(δ ∗ )) +CS F(δ ∗ ))π E
< 0.
=−
∂ sM
(CM −CS )CE2

36

Appendix D: Comparative Statics for Innovation
This appendix derives the policy effects on the long-run innovation rate, using the following
implicit functions to describe equilibrium in the labor and investment markets:

δI
1
−
,
R(δ M )a R(δ M )wa(1 − τ π − (1 − τ V )δ I )π M (δ M )
ρ
ιV =
−
,
∗
waLF(δ )(CM −CS )F(δ ∗ )
ιL =where we have referenced (17) and (21), and we define |J2 | = π E (δ M , δ I )/(CE (δ M )aLF(δ ∗ )) +

δ I /(R(δ M )a) > 0.
Corporate Tax: Taking the total derivatives of ι L and ι V with respect to τ π , we have


1
dι
πE
∂ ι L δ I ∂ ιV
≷ 0,
=
+
∗
dτ π
CE aLF(δ ) ∂ τ π Ra ∂ τ π |J2 |where ∂ δ M /∂ τ π ≷ 0, ∂ δ I /∂ τ π ≷ 0, |J2 | = ρ /(wF(δ ∗ )) + 1/(w(α + β F(δ M ))) > 0, and(1 − α )ξ f (δ M ) d δ M
1 dδ I
∂ιL
=−
−
≷ 0,
2
∂ τπ
R a
d τ π wRa d τ π((1 − τ V )δ I CM − (1 − τ π )CS )π M (δ M )
∂ ιV
=−
∂ τπ
(1 − τ π )(CM −CS )2 waLF (δ ∗ )(1 − τ π − (1 − τ V )δ I ) d δ M(1 − τ V )π M (δ M ) d δ I
−
−
≷ 0.
∗
(CM −CS )waF(δ ) d τ π (CM −CS )waLF(δ ∗ ) d τ πCapital Gains Tax: Taking the total derivatives of ι L and ι V with respect to τ V gives
dι
=
dτV



πE
∂ ι L δ I ∂ ιV+∗
CE aLF(δ ) ∂ τ V Ra ∂ τ V

37



1
≷ 0,
|J2|

where ∂ δ M /∂ τ V ≷ 0, ∂ δ I /∂ τ V ≷ 0, |J2 | = ρ /(wF(δ ∗ )) + 1/(w(α + β F(δ M ))) > 0, and∂ ιL(1 − α )ξ f (δ M ) d δ M
1 dδ I
=−
−
≷ 0,
∂ τV
R2 a
dτV
wRa d τ V
∂ ιV((1 − τ V )δ I CM − (1 − τ π )CS )π M (δ M )=−∂ τV(1 − τ V )(CM −CS )2 waLF(δ ∗ )(1 − τ V )π M (δ M ) d δ I(1 − τ π − (1 − τ V )δ I ) d δ M
−
≷ 0.
−
∗
(CM −CS )waF(δ ) d τ V(CM −CS )waLF(δ ∗ ) d τ V
Product Design Subsidy: Taking the total derivatives of ι L and ι V with respect to sE yields
dι
=
dsE



πE
∂ ι L δ I ∂ ιV+∗
CE aLF(δ ) ∂ sE Ra ∂ sE



1
≷ 0,
|J2 |

where ∂ δ M /∂ sE < 0, ∂ δ I /∂ sE < 0, |J2 | > 0, and∂ ιL(1 − α )ξ f (δ M ) d δ M
1 dδ I
=−
−
> 0,
2
∂ sE
R a
dsE
wRa dsER ∗(1 − τ V ) δδM π M dF(δ ) d δ I α (τ π − τ V )
∂ ιV
=
+
≷ 0,
∂ sE
CE waLF(δ ∗ )dsE
CM −CS
where we have used (∂ Ω1 /∂ δ M )(d δ M /dsE ) + (∂ Ω1 /∂ δ I )(d δ I /dsE ) = −∂ Ω1 /∂ sE .
Leapfrog Subsidy: Taking the total derivatives of ι L and ι V with respect to sM yields
dι
=
dsM



πE
∂ ι L δ I ∂ ιV+∗
CE aLF(δ ) ∂ sM Ra ∂ sM



1
≷ 0,
|J2 |

where ∂ δ M /∂ sM > 0, ∂ δ I /∂ sM > 0, |J2 | > 0, and∂ιL(1 − α )ξ f (δ M ) d δ M
1 dδ I
=−
−
< 0,
2
∂ sM
R a
dsM wRa dsM
R ∗(1 − τ V ) δδM π M dF(δ ) d δ I (1 − τ π )(1 − α )F(δ M )π E
∂ ιV
=
+
> 0,
∂ sM
dsM
CE waLF(δ ∗ )CE2 waLF (δ ∗ )where we have used (∂ Ω1 /∂ δ M )(d δ M /dsM ) + (∂ Ω1 /∂ δ I )(d δ I /dsM ) = −∂ Ω1 /∂ sM .

38

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