Campaign contributions and innovation in a fully-endogenous quality-ladder model

Economic Analysis of Law, Politics, and Regions
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Abstract

This paper examines the effect of campaign contributions on patent policy, welfare, and innovation using a fully-endogenous quality-ladder model. Assuming two types of households, where one type holds patents and the other does not, we analyze political conflicts between the two groups. Our analysis shows campaign contributions increase the rate of innovation to an excessive level from the viewpoint of social welfare when the innovation-maximizing patent policy is sufficiently strong. This result is important because it implies that the rate of innovation distorted by campaign contributions can be too high from the viewpoint of social welfare.

Keywords

Campaign contribution Patent protection Innovation Political conflict Quality-ladder model 

JEL Classification

O31 O34 O41 

1 Introduction

Modern economies have become knowledge-intensive, and intellectual property is considered a main source of competitiveness and economic growth. In fact, companies and research institutes in developed countries dominate world research and development (hereafter R&D), and these countries experienced rapid economic growth after World War II.1 As a result, intellectual property rights protection has been one of the most important policy issues, because many policymakers presume that strong and comprehensive patent policies increase the profitability of R&D and promote economic growth. Consequently, patent policies in many countries have been strengthened since the 1980s.

The fact that strong patent protection increases the profitability of R&D leads to the following hypothesis: firms or industries dependent on their research divisions have incentives for carrying out political activities to achieve stronger patent policies. Political activities by firms usually take the form of campaign contributions and government lobbying. As evidence that supports this hypothesis, Fig. 1 indicates the lobbying expenditures by industry in 2015. According to the Center for Responsive Politics (2015), the total amount spent on lobbying in the United States in 2015 reached 32.2 billion dollars. Figure 1 suggests that, at 509 million dollars for lobbying activities, the health industry, including pharmaceutical and medical companies, spent the largest amount of all industries on lobbying. Moreover, communications/electronics is also involved in political activities, spending 385 million dollars for lobbying. Because these industries are R&D intensive, they have strong incentives to use political tools to influence the government’s policy-making process.
Fig. 1

Lobbying expenditure by industry in 2015.

[Source: Center for Responsive Politics (2015)]

Many researchers discuss the effect of patent policies on development and economic growth. Among the theoretical studies, Judd (1985) uses an exogenous growth model to examine how patent length affects the market equilibrium path. After the development of the endogenous growth theory by Romer (1990), Grossman and Helpman (1991), and Aghion and Howitt (1992), numerous studies have investigated the relationship between patent policies and economic growth. Specifically, Iwaisako and Futagami (2003) use a variety-expansion growth model based on Romer (1990) to show that the optimal patent length to maximize social welfare is finite. Lai and Qiu (2003) and Grossman and Lai (2004) develop the North–South trade model to analyze the international effects of patent protection, and show that the governments of developed countries choose stronger patent protection than developing countries. Their analyses, however, assume that governments maximize household utility as a measure of social welfare, and ignore incentives for firms to engage in political activities. In contrast, Eicher and García-Peñalosa (2008) shows how private incentives to protect patent rights affect economic growth. They show that multiple steady states can emerge in their model; one steady state is high-growth equilibrium characterized by stronger patent protection, and the other is no-growth equilibrium in the absence of patent protection. However, they do not consider the role of government explicitly. This implies that they cannot examine the interaction between patent holders and government.

In the literature of political influence on patent protection, Chu (2008) is also important because his model examines interdependence between patent holders’ incentives to engage in political activities and the government policy-making process. His model succeeds in explaining the distortion of policymaking due to political activities. However, he uses a semi-endogenous growth model. In semi-endogenous growth models, economic growth depends on the rate of population growth, not on policy variables. This means that patent policy has no effect on economic growth in his model. Therefore, his framework cannot explain how political activities affect innovation and economic growth. This drawback is unsatisfactory because many researchers have already studied the relationship between patent policies and economic growth.

The purpose of this study is to clarify how patent protection distorted by campaign contributions affects innovation and economic growth. To achieve this goal, we provide a theoretical analysis based on the fully-endogenous quality-ladder model developed by Grossman and Helpman (1991, Chap. 4). As many researchers point out, the fully-endogenous growth model causes scale effects. Nevertheless, we use this model because it provides richer implications for policy analysis than a semi-endogenous growth model does. Using the fully-endogenous growth model enables us to examine the influence of politically determined patent policy not only on welfare, but also on the growth rate. As a result, our model succeeds in analyzing broader interactions among policy-making, welfare, and economic growth than Chu (2008) does.

Our main results are as follows. First, we determine the existence of a patent policy that maximizes innovation intensity under some mild assumptions. The existence of innovation-maximizing patent protection reflects two effects of patent policy on research activity: the profit magnification and patent cost effects. Second, we show that campaign contributions by patent holders result in a stronger patent policy than welfare-maximizing policy (using the average utility of patent holders and non-holders). This result is basically the same as Chu (2008). Third, we prove the following results. If the innovation-maximizing patent policy is greater than the optimum patent protection for patent holders, campaign contributions raise the rate of innovation to an excessive level from the viewpoint of social welfare. On the other hand, if the innovation-maximizing patent policy is smaller than the patent policy that maximizes social welfare, campaign contributions decrease not only social welfare, but also the rate of innovation.

The third result is important because it cannot be derived from Chu (2008), which uses a semi-endogenous growth model. This result implies that political distortion caused by campaign contributions can affect not only welfare, but also innovation. Importantly, our analysis shows that the rate of innovation distorted by campaign contributions is too high from the viewpoint of social welfare.

This paper is organized as follows. Section 2 describes the basic elements of the model and identifies equilibrium conditions. Section 3 establishes the unique steady-state equilibrium and explicit solutions for the endogenous variables. Section 4 examines how the government determines patent policy. Section 4.1 considers the case in which the government is benevolent and maximizes social welfare, while Sect. 4.2 examines how campaign contributions by patent holders affect patent policy and innovation. Section 5 provides concluding remarks.

2 Model

2.1 Households

In this economy, there are two types of households. These households consume homogeneous goods and quality-enhancing goods. Households try to maximize the following lifetime utility:

$$ U\left( i \right) = \int\limits_{0}^{\infty } {\left[ {c_{h,t} \left( i \right) + \alpha \log c_{q,t} \left( i \right)} \right]e^{ - \rho t} } {\text{d}}t, $$
(1)
where \( i = I,II \) represents the type of household, \( c_{h,t} \left( i \right) \) is the per capita consumption of homogeneous goods, \( c_{q,t} \left( i \right) \) is the per capita consumption of quality-enhancing goods, \( \alpha \) is a positive parameter which describes preference, and \( \rho > 0 \) is the rate of time preference. Equation (1) shows that the two types of households have the same quasi-linear preference.2 The total number of households is \( L, \) and each household supplies one unit of labor inelastically.

Next, the budget constraint is given as,

$$ \dot{a}_{t} \left( i \right) = r_{t} a_{t} \left( i \right) + w_{t} - e_{t} \left( i \right), $$
(2)
where \( a_{t} \left( i \right) \) is the amount of financial assets of each household, \( r_{t} \) is the real rate of return on financial assets, \( w_{t} \) is the wage, and \( e_{t} \left( i \right) \) is the expenditure. \( \dot{a}_{t} \left( i \right) \) is the derivative of \( a_{t} \left( i \right) \) with respect to time, that is, \( \dot{a}_{t} \left( i \right) = {{{\text{d}}a_{t} \left( i \right)} \mathord{\left/ {\vphantom {{{\text{d}}a_{t} \left( i \right)} {{\text{d}}t}}} \right. \kern-0pt} {{\text{d}}t}} \). As in Chu (2008), we assume that these two types of households face different investment opportunities. We suppose that while Type-I households can freely access the financial markets, Type-II households cannot participate in the financial markets. We will see that this assumption generates a political conflict between patent owners (Type-I households), who have patent rights as financial assets, and other consumers (Type-II households), who do not. We assume that the fractions of Type-I and Type-II households are \( s \) and \( 1 - s, \) respectively.

Type-I households face intertemporal optimization. Solving this intertemporal utility maximization gives us the following Euler equation,

$$ \frac{{\dot{e}_{t} \left( I \right)}}{{e_{t} \left( I \right)}} = r_{t} - \rho , $$
(3)
where \( \dot{e}\left( I \right) = {{{\text{d}}e\left( I \right)} \mathord{\left/ {\vphantom {{{\text{d}}e\left( I \right)} {{\text{d}}t}}} \right. \kern-0pt} {{\text{d}}t}} \). Therefore, \( {{\dot{e}\left( I \right)} \mathord{\left/ {\vphantom {{\dot{e}\left( I \right)} {e\left( I \right)}}} \right. \kern-0pt} {e\left( I \right)}} \) corresponds to the growth rate of expenditure of Type-I households. On the other hand, Type-II households have no choice, but to spend their entire wage income because they do not have access to the financial sector. This implies.
$$ e_{t} \left( {II} \right) = w_{t} . $$
(4)
We consider the way households allocate their expenditure across homogeneous goods and quality-enhancing goods. Homogeneous goods are chosen as the numéraire, and \( P_{q,t} \) is the price index of the quality-enhancing goods. The intratemporal budget constraint is given by.
$$ c_{h,t} \left( i \right) + P_{q,t} c_{q,t} \left( i \right) = e_{t} \left( i \right). $$
(5)
Because the two types of households have the same quasi-linear preference, the amount of spending on quality-enhancing goods is determined as.
$$ P_{q,t} c_{q,t} \left( i \right) = \alpha . $$
(6)
Equation (6) shows us that both types of households consume the same amount of quality-enhancing goods. On the other hand, the consumption of homogeneous goods differs between the two. From (5) and (6), the amount of homogeneous goods consumed by Type-I households is given as
$$ c_{h,t} \left( I \right) = e_{t} \left( I \right) - \alpha . $$
(7)
On the other hand (4) implies that the amount of homogeneous goods consumed by Type-II households is
$$ c_{h,t} \left( {II} \right) = w_{t} - \alpha . $$
(8)

Next, we introduce the consumption index of quality-enhancing goods and explicitly derive the demand function of each quality-enhancing good. The consumption index is described as a Cobb–Douglas aggregator of differentiated quality-enhancing goods \( j \in \left[ {0,1} \right] \), which is given by

$$ \log c_{q,t} \left( i \right) = \int\limits_{0}^{1} {\log \left[ {\mathop \sum \limits_{m} \lambda^{m} x_{t} \left( {j,m} \right)} \right]} {\text{d}}j, $$
(9)
where \( \lambda > 1 \) is the exogenous size of quality improvement, \( m \) is the number of successful innovations that have occurred, and \( x_{t} \left( {j,m} \right) \) is the demand for quality-enhancing goods with quality \( \lambda^{m} \).
Given this consumption index, the consumer’s static optimization problem is solved in two steps. First, consumers allocate their budgets evenly across product lines. Second, consumers allocate their budget to the lowest quality-adjusted price products in each product line because vertically differentiated products in each industry provide perfect substitutes. We will see later that the highest quality goods have the lowest quality-adjusted prices as a result of Bertrand competition. This implies that consumers buy only the highest quality products within each industry. When \( p_{t} \left( j \right) \) denotes the price of the highest quality product, (6) implies that each household’s demand for the highest quality product in each industry in \( j \) becomes \( x_{t} \left( j \right) = {\alpha \mathord{\left/ {\vphantom {\alpha {p_{t} \left( j \right)}}} \right. \kern-0pt} {p_{t} \left( j \right)}} \).3 Multiplying this demand by \( L \), the aggregate demand for highest quality becomes
$$ X_{t} \left( j \right) = \frac{\alpha L}{{p_{t} \left( j \right)}}. $$
(10)

2.2 Product markets

First, we describe the markets of homogeneous goods. We assume that labor is the only production factor. We suppose that one unit of labor produces one unit of homogeneous or quality-enhancing goods. The homogeneous goods markets are perfectly competitive. Since the price of homogeneous goods is normalized to unity, the wage (i.e., marginal cost of production) also becomes unity.

Next, we explain the market structure of quality-enhancing goods. This is closely related to the patent policy enforced by the government. Although there are several ways to formulate patent policies in models of economic growth, we explain two representative policy instruments in this study. The first is patent length. Patent length represents how long the patentee can have the exclusive right to produce and sell its product. The second is patent breadth. Patent breadth represents the scope of products that the patentee can prevent other firms from producing and selling.4 Judd (1985), Iwaisako and Futagami (2003), and Futagami and Iwaisako (2007) employ patent length as an instrument of patent policy. In general, however, using finite patent length as a policy instrument causes a quite complicated equilibrium path. Under the assumption of finite patent length, the equilibrium rate of innovation cannot be derived as an explicit function of patent policy.5 For this reason, we assume that the patent length is infinite and that the government uses only patent breadth as a policy instrument. Although this assumption seems to be very strong, it is quite useful because the equilibrium rate of innovation is expressed as a simple function of patent policy. Patent breadth is often used in endogenous growth literature because of this advantage. For example, Goh and Olivier (2002), O’Donoghue and Zweimüller (2004), and Iwaisako and Futagami (2013) examine how a change in patent breadth affects innovation and welfare.

In this study, while a firm capable of producing the highest-quality product has the exclusive right to produce it, the production or sale of imitated products depends on the effectiveness of patent policy. For example, if the government strictly enforces its patent policy, and infringement of the patent holder’s right is strictly forbidden, imitations cannot be sold on the market and the patent holder can avoid competition. This allows patent holders to set higher prices. In contrast, if the government chooses a weak patent policy, competitors can easily infringe on the patent holders’ rights and sell imitations. This implies that the price of a patented product is pushed down to the competitive level.

When a firm capable of producing the highest-quality product can ignore price competition with imitators, it could charge \( p_{t} \left( j \right) = \lambda > 1 \) due to the Bertrand competition with lower quality products. On the other hand, competition with similar products prevents patent holders from charging monopoly prices. Considering these arguments, the price of a quality-enhancing good is assumed to be.
$$ p_{t} \left( j \right) = 1 + \beta \left( {\lambda - 1} \right), $$
(11)
where \( \beta \in \left[ {0,1} \right] \), and \( \beta \) expresses the degree of patent enforcement.

\( \beta \left( {\lambda - 1} \right) \) can be interpreted as a constant per unit cost of imitation that potential competitors face. In this setting, the Bertrand competition implies that firms capable of producing the highest-quality product can capture all the industry demand by setting the limit price given in (11). Moreover, we can consider the patent licensing agreement between a patentee (that is, a firm capable of producing the highest-quality product) and potential licensees. In this case, if the license fee is higher than the imitation cost given by \( \beta \left( {\lambda - 1} \right) \), the licensees will not enter into the patent licensing agreement, but will choose to imitate the product because the cost for production in the licensing contract is higher than the cost of imitation. This discussion suggests that \( \beta \left( {\lambda - 1} \right) \) can be interpreted as the maximum licensing fee that the patentee can impose on licensees.

Equation (11) implies that the price of the quality-enhancing goods is the same across all product lines. The firm can charge a price that is close to the monopoly price when \( \beta \) is close to one. This implies that a higher \( \beta \) corresponds to stronger patent protection.

Substituting (11) into the aggregate demand function (10) determines the quantity of demand for the quality-enhancing goods, which is given by
$$ X_{t} \left( j \right) = \frac{\alpha L}{{1 + \beta \left( {\lambda - 1} \right)}}. $$
(12)

Equation (12) clearly shows that an increase in \( \beta \) reduces the quantity of demand because the firm can charge a higher price for its product. Moreover, it is represented as a combination of parameters, which means (12) is constant over time. When we consider the market equilibrium of each quality-enhancing good, \( X_{t} \left( j \right) \) is considered the quantity of production. Under the assumption that the marginal cost of production (i.e., wage) is unity, profit is expressed as \( \pi_{t} \left( j \right) = \left[ {p_{t} \left( j \right) - 1} \right]X_{t} \left( j \right) \). Substituting (11) and (12) into this expression yields.

$$ \pi_{t} \left( j \right) = \left( {1 - \frac{1}{{1 + \beta \left( {\lambda - 1} \right)}}} \right)\alpha L, $$
(13)
which is also constant over time. Equation (13) shows that an increase in \( \beta \) raises the profitability of the firm that produces the quality-enhancing goods.

Next, we consider the demand for homogeneous goods. The number of Type-I households is \( sL \) and the amount of homogeneous goods consumed by Type-I households is given by (7). These imply that the aggregate demand for homogeneous goods by Type-I households becomes \( s\left[ {e_{t} \left( I \right) - \alpha } \right]L \). Similarly, the number of Type-II households is \( \left( {1 - s} \right)L \), and the amount of homogeneous goods consumed by Type-I households is given by (8). The aggregate demand for homogeneous goods by Type-II households becomes \( \left( {1 - s} \right)\left( {1 - \alpha } \right)L \). In equilibrium, the sum of demand for homogeneous goods by the two types of households is equivalent to the quantity of production.

2.3 R&D activity

The flow of profits provides a strong incentive for firms to engage in innovative activity. In this subsection, following the standard practice of the literature on the quality-ladder model developed by Grossman and Helpman (1991), we specify the technology for product improvement.

Any firm that engages in this activity at intensity \( \iota_{t} \) for the time interval of length \( {\text{d}}t \) succeeds in inventing the next-generation product with probability \( \iota_{t} {\text{d}}t \). To generate this probability, this firm must invest \( \iota_{t} \delta \left( \beta \right){\text{d}}t \) units of labor. In this setting, \( \iota_{t} \) is interpreted as a flow of probability or a frequency of successful innovation. \( \delta \left( \beta \right) \) corresponds to the unit cost of innovation. Unlike Chu (2008), we assume that this unit cost depends on the level of patent policy. In general, it is likely that the cost of R&D increases as patent policy provides greater protection to existing patented ideas. The wider the breadth of patent protection, the more difficult it is to develop new products without infringing on existing patents. Moreover, as the intellectual property system is improved, the number of employees in the legal departments of firms increases, which also raises R&D expenses. Considering this background, it is reasonable to assume that \( \delta \) is increasing with \( \beta \); that is, \( \delta^{\prime}\left( \beta \right) > 0 \).

In this study, patent protection policy has two contradicting effects. First, it raises the profitability of innovation. This effect strengthens the incentive for R&D activity. Second, it also raises the cost of innovation. This weakens the incentive for R&D. From these two different effects, we can show that there is innovation-maximizing patent protection in this model; this point will be discussed later.

Let \( v_{t} \) denote the value of a patent at time \( t \). We assume that firms can engage in R&D activity freely. At the cost of \( \iota_{t} \delta \left( \beta \right){\text{d}}t \), a firm can succeed in developing a new-generation product and attain value \( v_{t} \) with the probability \( \iota_{t} {\text{d}}t \). The firm can finance the cost of R&D by issuing equities. In this case, the firm chooses intensity \( \iota_{t} \) to maximize the expected profit given by \( \left( {v_{t} \iota_{t} {\text{d}}t - \delta \left( \beta \right)\iota_{t} {\text{d}}t} \right) \). The free-entry condition implies that \( \iota_{t} = 0 \) whenever \( v_{t} < \delta \left( \beta \right) \) and positive and finite R&D effort can take place only when \( v_{t} = \delta \left( \beta \right) \). Therefore, we can conclude that
$$ v_{t} \le \delta \left( \beta \right), $$
(14)
with equality whenever \( \iota_{t} \) is positive.
Next, we turn to the valuation of patents. A no-arbitrage condition relates the expected patent return to the interest on a riskless bond. Patent holders receive profit \( \pi_{t} \) as a dividend and enjoy capital gain \( \dot{v}_{t} \). In contrast, patent holders face the risk of other firms succeeding in research activities. In this case patent holders lose their markets and suffer capital losses. The size of capital loss is \( \iota_{t} v_{t} \). Therefore, we can express the expected return on the patent as \( \pi_{t} + \dot{v}_{t} - \iota_{t} v_{t} \). This must equal the return on an equal size investment in a riskless bond. This implies
$$ r_{t} v_{t} = \pi_{t} + \dot{v}_{t} - \iota_{t} v_{t} . $$
(15)

2.4 Labor market

Finally, we close our model by considering the equilibrium of the labor market. When a firm engages in R&D activity at intensity \( \iota_{t} \), \( \delta \left( \beta \right)\iota_{t} \) units of labor are allocated to this activity. Research intensity \( \iota_{t} \) is the same across product lines, and the measure of product lines is unity. These imply that total employment in R&D equals \( \delta \left( \beta \right)\iota_{t} \). The demand for quality-enhancing goods is given by (12). The demand for homogeneous goods by Type-I and Type-II households are \( s\left[ {e_{t} \left( I \right) - \alpha } \right]L \) and \( \left( {1 - s} \right)\left( {1 - \alpha } \right)L, \) respectively. These demands for goods are equivalent to labor demand because the unit labor requirement is unity in this economy. On the other hand, the labor supply is constant and exogenously given by \( L \). Therefore, the labor market equilibrium is given as
$$ \left\{ {s\left[ {e_{t} \left( I \right) - \alpha } \right] + \left( {1 - s} \right)\left( {1 - \alpha } \right)} \right\}L + \frac{\alpha L}{{1 + \lambda \left( {\beta - 1} \right)}} + \delta \left( \beta \right)\iota_{t} = L. $$
(16)
Using (13) and (16), we derive the following equation.
$$ \left\{ {se_{t} \left( I \right) + \left( {1 - s} \right)} \right\}L + \delta \left( \beta \right)\iota_{t} = L + \pi_{t} . $$
(17)

The left side of the equation implies total expenditure (consumption for goods plus R&D investment). The right side is total income (labor income plus dividends from equity). Therefore, (17) corresponds to the equilibrium condition of national income accounting.6

3 Steady-state equilibrium

3.1 Steady-state research intensity

We consider steady-state equilibrium in which research intensity \( \iota \) is constant over time. The labor market equilibrium given in (16) implies that \( e_{t} \left( I \right) \) is constant over time if \( \iota \) is constant. Because of these considerations and (3), \( r_{t} = \rho \) must hold. Dividing both sides of (15) by \( v_{t} \) and substituting (13), (14), and \( r_{t} = \rho \), we find the steady-state value of \( \iota \) to be as follows,
$$ \iota = \frac{1}{\delta \left( \beta \right)}\left( {1 - \frac{1}{{1 + \beta \left( {\lambda - 1} \right)}}} \right)\alpha L - \rho . $$
(18)

In Chu (2008), a semi-endogenous growth model is used, and the growth rate is proportional to population growth and independent from patent protection. On the other hand, we use a standard endogenous growth model. Therefore, our analysis succeeds in identifying how patent policy can possibly affect the growth rate. This is a large difference between Chu (2008) and our model.

Next, we explain some of the implications of (18). First, (18) shows that the steady-state value of research intensity is higher when the economy has a larger population \( L \). In other words, (18) represents scale effect. This result is a natural consequence of using the quality-ladder model developed by Grossman and Helpman (1991). Second, (18) does not include the parameter \( s \), which implies that the steady-state research intensity is independent from the fraction of Type-I households in the total population. This is because both types of households consume the same amount of quality-enhancing goods, and the profit given by (13) does not depend on this fraction.

Moreover, the steady-state research intensity given in (18) depends on the degree of patent enforcement \( \beta \). \( \beta \) has two different effects on \( \iota \). First, stronger patent policy raises the profitability of innovation; this effect is called the profit magnification effect. Second, stronger patent policy increases the cost of R&D and weakens the profitability of innovation; this effect is called the patent cost effect. Since research intensity \( \iota \) is a function of the degree of patent enforcement \( \beta \), we can express this relationship as \( \iota = I\left( \beta \right) \). Differentiating \( I\left( \beta \right) \) with respect to \( \beta \) yields.
$$ \frac{{{\text{d}}I\left( \beta \right)}}{{{\text{d}}\beta }} = \frac{\lambda - 1}{\delta \left( \beta \right)}\frac{\alpha L}{{1 + \beta \left( {\lambda - 1} \right)}}\left[ {\frac{1}{{1 + \beta \left( {\lambda - 1} \right)}} - \frac{{\delta^{\prime}\left( \beta \right)\beta }}{\delta \left( \beta \right)}} \right]. $$
(19)

The first term in the square brackets represents the profit magnification effect, which implies that stronger patent protection raises the profitability of the firm and enhances the incentive of innovation. In contrast, the second term corresponds to the patent cost effect, which implies that stronger patent protection also raises R&D cost. We find that the second effect depends on the elasticity of R&D cost with respect to \( \beta \). How patent policy affects research intensity depends on the relationship between the two effects.

For more detailed analyses, we must know more about \( \delta \left( \beta \right) \). We could consider several functional forms of \( \delta \left( \beta \right) \). To simplify our analysis, we would like to assume \( \delta \left( \beta \right) = \beta^{\theta } \) where \( \theta > 0 \). \( \theta \) is a parameter which characterizes the marginal cost of R&D activity. In this case, the elasticity of R&D cost is given as \( \theta \). From (19), we find that \( {{{\text{d}}I\left( \beta \right)} \mathord{\left/ {\vphantom {{{\text{d}}I\left( \beta \right)} {{\text{d}}\beta }}} \right. \kern-0pt} {{\text{d}}\beta }} < 0 \) for any \( \beta \) if \( \theta \ge 1 \). On the other hand, \( {{{\text{d}}I\left( \beta \right)} \mathord{\left/ {\vphantom {{{\text{d}}I\left( \beta \right)} {{\text{d}}\beta }}} \right. \kern-0pt} {{\text{d}}\beta }} > 0 \) for any \( \beta \) if \( 0 < \theta \le 1/\lambda \). Finally, there exists a unique value of \( \beta \) that maximizes research intensity if \( 1/\lambda \le \theta < 1 \). This relationship is clearly shown in Fig. 2. In Fig. 2, \( {1 \mathord{\left/ {\vphantom {1 {\left[ {1 + \beta \left( {\lambda - 1} \right)} \right]}}} \right. \kern-0pt} {\left[ {1 + \beta \left( {\lambda - 1} \right)} \right]}} \) is decreasing curve with \( \beta \) and \( {{\delta^{\prime}\left( \beta \right)\beta } \mathord{\left/ {\vphantom {{\delta^{\prime}\left( \beta \right)\beta } {\delta \left( \beta \right)}}} \right. \kern-0pt} {\delta \left( \beta \right)}} = \theta \) is a horizontal line. The intersection of these two lines corresponds to the value of \( \beta \) that maximizes research intensity.
Fig. 2

Growth-maximizing patent protection

Hereafter, we will focus on the last case where an innovation-maximizing patent policy exists. From (19), we can attain a value of \( \beta \) that maximizes research intensity as follows,
$$ \beta^{\text{g}} = \frac{1}{\lambda - 1}\left( {\frac{1}{\theta } - 1} \right). $$
(20)

3.2 Steady-state consumption of homogeneous goods

Next, we consider the steady-state value of the consumption of homogeneous goods. By substituting (18) into (16), the expenditure of a Type-I household is obtained as
$$ e_{t} \left( I \right) = 1 + \frac{\delta \left( \beta \right)}{sL}. $$
(21)
Using (7) and (21), we calculate the steady-state consumption of homogeneous goods by a Type-I household as
$$ c_{\text{h}} \left( I \right) = 1 - \alpha + \frac{\delta \left( \beta \right)}{sL}. $$
(22)
On the other hand, (8) and \( w_{t} = 1 \) yield the steady-state consumption of homogeneous goods by a Type-II household as
$$ c_{\text{h}} \left( {II} \right) = 1 - \alpha . $$
(23)

We can derive some interesting results from (22) and (23). First, these two equations imply that \( c_{\text{h}} \left( I \right) \) is larger than \( c_{\text{h}} \left( {II} \right) \). Second, (22) shows that the steady-state consumption of homogeneous goods by a Type-I household positively depends on patent policy because Type-I households can have patents as assets. On the other hand, Type-II households cannot be patent holders, and their consumption of homogeneous goods is independent of patent policy. Third, (22) shows that \( c_{\text{h}} \left( I \right) \) negatively depends on the fraction of Type-I households in the total population, \( s \). Intuitively, this implies that a small number of Type-I households hold all the assets in this economy and enjoy greater consumption of quality-enhancing goods.

3.3 Welfare

In this subsection, we focus on the lifetime utility of each household. Steady-state consumption of quality-enhancing goods is the same between the two types of households, and its logarithmic value is given by7
$$ \log c_{q,t} = \iota t\log \lambda + \log \alpha - \log \left[ {1 + \beta \left( {\lambda - 1} \right)} \right]. $$
(24)

Equation (24) clearly shows that \( \beta \) has two effects on the consumption of quality-enhancing goods. First, patent protection policy affects research intensity. This effect is shown in the first term of (24). Second, it also affects the price of a quality-enhancing good. Since stronger patent protection rises the price of that good, consumption will decrease. This effect is included in the third term of (24).

We have already derived the steady-state consumption of homogeneous and quality-enhancing goods. Substituting these consumptions into lifetime utility yields.
$$ U\left( {i,\beta } \right) = \frac{1}{\rho }\left\{ {c_{\text{h}} \left( i \right) + \alpha \log \alpha - \alpha \log \left[ {1 + \beta \left( {\lambda - 1} \right)} \right]} \right\} + \frac{\alpha }{{\rho^{2} }}I\left( \beta \right)\log \lambda . $$
(25)
By differentiating (25) with respect to \( \beta \), we find how patent policy affects each household’s utility.
$$ \frac{{\partial U\left( {I,\beta } \right)}}{\partial \beta } = \frac{1}{\rho }\left\{ {\frac{{\delta^{\prime}\left( \beta \right)\rho }}{sL} - \frac{{\alpha \left( {\lambda - 1} \right)}}{{1 + \beta \left( {\lambda - 1} \right)}}} \right\} + \frac{\alpha \log \lambda }{{\rho^{2} }}\frac{{{\text{d}}I\left( \beta \right)}}{{{\text{d}}\beta }}, $$
(26)
$$ \frac{{\partial U\left( {II,\beta } \right)}}{\partial \beta } = \frac{1}{\rho }\left\{ { - \frac{{\alpha \left( {\lambda - 1} \right)}}{{1 + \beta \left( {\lambda - 1} \right)}}} \right\} + \frac{\alpha \log \lambda }{{\rho^{2} }}\frac{{{\text{d}}I\left( \beta \right)}}{{{\text{d}}\beta }}. $$
(27)

Equations (26) and (27) show that there are three effects of patent policy on households. First, stronger patent protection raises not only the cost of research activity but also the value of patents. This increases the consumption of homogeneous goods and the welfare of Type-I households because they are patent holders. Second, stronger patent protection raises the prices of quality-enhancing goods and reduces the welfare of both types of households. Third, stronger patent protection enhances the incentive for innovation, and the quality of goods increases rapidly. This increases the welfare of both types of households. As Type-I households are patent holders, the first effect appears in (26). In contrast, it does not appear in (27) because Type-II households cannot hold patents as assets. This difference will create a conflict of interest over patent protection.

By setting (26) and (27) to zero, we can obtain the optimum protection level for each type of household. Let \( \beta_{I} \) and \( \beta_{II} \) denote the optimal patent policies for Type-I and Type-II, respectively. Obviously, \( \beta_{I} \) is higher than \( \beta_{II} \) because Type-I households can enjoy greater patent value. This relationship is clearly shown in Fig. 3. Moreover, it is clear the optimal patent protection for Type-II households, \( \beta_{II} \), is smaller than the research maximizing patent protection \( \beta^{\text{g}} \) because they cannot access financial markets and only suffer from higher prices of quality-enhancing goods. We summarize these results as Proposition 1.
Fig. 3

Utility-maximizing patent protection

Proposition 1

The optimum patent protection for Type-I households is greater than that for Type-II ( \( \beta_{II} < \beta_{I} \) ). Moreover, innovation-maximizing patent protection is greater than the optimum patent protection for Type-II households ( \( \beta_{II} < \beta^{\text{g}} \) ).

In contrast, we cannot determine whether the optimum protection for Type-I households \( \beta_{I} \) is larger than \( \beta^{\text{g}} \) because we do not know whether the sign of the first term of (26) is positive or negative. However, the first term of (26) tends to be positive and \( \beta^{\text{g}} < \beta_{I} \) when the fraction of Type-I households, \( s \), is small enough. When \( s \) is sufficiently small, a small number of patent holders hold all the assets of this economy. In this case, each patent holder possesses a large amount of assets. This implies that Type-I households have an incentive to choose stronger patent policy.8

4 Determination of patent policy

4.1 Benevolent government

In the last section, we show that Type-I households desire stronger patent protection than Type-II households do. On the other hand, the government must consider both types of households when it determines optimal patent policy. In this section, we examine how governments determine patent protection.

First, we examine the case in which the government is benevolent and chooses its patent policy to maximize the average utility of the two types of households. In this economy, the fraction of Type-I households is \( s, \) and the fraction of Type-II households is \( 1 - s \). Therefore, the government’s social welfare objective function is given by the following,
$$ W\left( \beta \right) = sU\left( {I,\beta } \right) + \left( {1 - s} \right)U\left( {II,\beta } \right). $$
(28)
Differentiating (28) with respect to \( \beta \) yields
$$ \frac{\partial W\left( \beta \right)}{\partial \beta } = \frac{1}{\rho }\left\{ {\frac{{\delta^{\prime}\left( \beta \right)\rho }}{L} - \frac{{\alpha \left( {\lambda - 1} \right)}}{{1 + \beta \left( {\lambda - 1} \right)}}} \right\} + \frac{\alpha \log \lambda }{{\rho^{2} }}\frac{{{\text{d}}I\left( \beta \right)}}{{{\text{d}}\beta }}. $$
(29)

By setting (28) to zero, it is possible to obtain the welfare-maximizing patent protection for the government. Let \( \beta^{*} \) denote the optimal patent policy derived from (29). In this case, it is obvious that \( \beta^{*} \) is between \( \beta_{I} \) and \( \beta_{II} \) because \( s \) is between zero and one. Moreover, (29) implies that \( \beta^{*} \) does not depend on \( s \).9 We summarize these results as Proposition 2.

Proposition 2

If the government is benevolent and pays attention to average utility, the optimal patent protection for the government is between the optimal policy for Type-I and that for Type-II ( \( \beta_{II} < \beta^{*} < \beta_{I} \) ).

4.2 Campaign contributions

In this subsection, we model the campaign contributions for legislative influence. In addition to social welfare, we assume that the government values campaign contributions. Let \( C \) denote the value of contribution; \( C \) is measured in units of homogenous goods. The objective function of the government is given as a weighted average of \( W\left( \beta \right) \) and \( C \) given by

$$ \widetilde{W}\left( {\beta ,C} \right) = \left( {1 - \zeta } \right)W\left( \beta \right) + \zeta C, $$
(30)
where \( 0 < \zeta < 1 \) is the weight that the government places on campaign contributions.10 In this study, we assume that Type-I households engage in political activity and make campaign contributions. The optimal patent protection for the government when its objective function is given by (28) is lower than that desired by Type-I households (\( \beta^{*} < \beta_{I} \)). This implies that Type–I households have an incentive to make political contributions to politicians or to engage in lobbying activities to raise their utility.
For simplicity, we assume that Type-I households and the government engage in efficient bargaining and commit to the bargaining outcome on patent policy and campaign contributions. In other words, the two agents first negotiate and reach an agreement on \( \beta \) and \( C \). Then, the Type-I households make \( C \) units of campaign contribution while the government chooses the strength of the patent policy \( \beta \). All these decisions are made in the initial period and are not changed subsequently.11 To derive the equilibrium of efficient bargaining, we employ a relatively simple approach proposed by Grossman and Helpman (2001, Chap. 7) and Chu (2008). In this case, the efficient bargaining outcome can be derived by maximizing (30) subject to \( sU\left( {I,\beta } \right) - C \ge U_{0} \) for some value of \( U_{0} \). In the case of efficient bargaining, this inequality constraint must be binding. This implies that the constraint is rewritten as
$$ C = sU\left( {I,\beta } \right) - U_{0} . $$
(31)
By substituting (31) into (30), the government’s objective function becomes12
$$ \widetilde{W}\left( \beta \right) = sU\left( {I,\beta } \right) + \left( {1 - \zeta } \right)\left( {1 - s} \right)U\left( {II,\beta } \right) - \zeta U_{0} . $$
(32)
Equation (32) implies that when the government values campaign contributions, Type-I households can change the weight on household utility. By comparing (28) with (32), it is obvious that the weight on Type-I household utility becomes larger. This fact clearly shows that the government values the interests of Type-I households, who are patent holders. Differentiating (32) with respect to \( \beta \) yields
$$ \frac{{\partial \widetilde{W}\left( \beta \right)}}{\partial \beta } = \frac{1}{\rho }\left\{ {\frac{{\delta^{\prime}\left( \beta \right)\rho }}{L} - \frac{{\alpha \left( {\lambda - 1} \right)\left[ {s + \left( {1 - \zeta } \right)\left( {1 - s} \right)} \right]}}{{1 + \beta \left( {\lambda - 1} \right)}}} \right\} + \frac{{\alpha \left[ {s + \left( {1 - \zeta } \right)\left( {1 - s} \right)} \right]\log \lambda }}{{\rho^{2} }}\frac{{{\text{d}}I\left( \beta \right)}}{{{\text{d}}\beta }}. $$
(33)

We can derive the political-equilibrium patent policy denoted by \( \beta^{C} \) by setting (33) to zero. Comparing (29) and (33), we can show \( \beta^{C} > \beta^{*} \). In other words, because the government values the interests of Type-I households, it chooses its patent policy close to the level that maximizes patent holder benefits. Therefore, we can state the following proposition:

Proposition 3

If the government values not only household utility but also campaign contributions, it chooses a stronger patent policy over a welfare-maximizing policy. ( \( \beta^{C} > \beta^{*} \) ).

We analyze the effects of two parameters. First, we consider how the weight on campaign contributions, \( \zeta \) affects patent policy. From (33), we can confirm that \( \beta^{C} \) is higher when \( \zeta \) is higher. Moreover, we can easily show that \( \beta^{C} \to \beta_{I} \) when \( \zeta \to 1 \), and that \( \beta^{C} \to \beta^{*} \) when \( \zeta \to 0 \). Second, we consider how the fraction of Type-I households, \( s \), affects patent policy. Equation (33) implies that \( \beta^{C} \) is higher when the fraction, \( s \) is smaller. Intuitively, when \( s \) is sufficiently small, Type-I households have an incentive to choose stronger patent policy because each patent holder can possess a large amount of assets. In addition, remember that the weight on Type-I household utility becomes larger in the case of campaign contributions. For these reasons, \( \beta^{C} \) becomes higher when the fraction of Type-I households is smaller.

Moreover, campaign contributions by Type-I households can affect innovation and economic growth. First, we focus on the case in which \( \beta^{g} \) is greater than the optimum patent protection for Type-I households, \( \beta_{I} \). If \( \beta_{I} < \beta^{g} \) is satisfied, we can easily show \( \beta^{*} < \beta^{C} < \beta^{g} \) and \( I\left( {\beta^{*} } \right) < I\left( {\beta^{C} } \right) \). These results imply that campaign contributions can increase the rate of innovation. However, the rate of innovation distorted by campaign contributions is too high from the viewpoint of social welfare. Second, we focus on the case in which \( \beta^{g} < \beta^{*} \). In this case, \( \beta^{g} < \beta^{*} < \beta^{C} \) and \( I\left( {\beta^{*} } \right) > I\left( {\beta^{C} } \right) \). These suggest that campaign contributions decrease not only social welfare but the rate of innovation. We summarize these results as the following proposition.

Proposition 4

(a) If the innovation-maximizing patent policy is greater than the optimum patent protection for Type-I households ( \( \beta_{I} < \beta^{g} \) ), campaign contributions raise the rate of innovation to an excessive level from the viewpoint of social welfare ( \( I\left( {\beta^{*} } \right) < I\left( {\beta^{C} } \right) \) ). (b) If the innovation-maximizing patent policy is smaller than the welfare-maximizing patent policy ( \( \beta^{g} < \beta^{*} \) ), campaign contributions decrease not only social welfare but also the rate of innovation ( \( I\left( {\beta^{*} } \right) > I\left( {\beta^{C} } \right) \) ).

The results shown in Proposition 4 are the main contributions of this study, and cannot be derived by Chu (2008), which uses a semi-endogenous growth model. In semi-endogenous growth models, innovation and economic growth depend on population growth, and patent policy has no effect on economic growth. In contrast, our model shows the possibility that patent policy affects innovation. Therefore, political distortion caused by campaign contributions also affects innovation and welfare. Specifically, our result shows that the rate of innovation distorted by campaign contributions is too high from the viewpoint of social welfare.

These results are likely to hold in general policy settings, such as in case of patent length, as we find that strengthening patent policies is advantageous to Type-I households, even when assuming any policy instruments. For example, Chu (2008) shows that patent protection can be distorted favorably for the patent holders under the assumption of finite patent length.

Finally, we explain how the amount of contributions is determined. For simplicity, we treat two extreme cases.13 First, we assume that the government has the first-mover advantage and makes a take-or-leave offer to Type-I households. Under these assumptions, if the bargaining between the government and Type-I households is broken down, \( \widetilde{W}\left( \beta \right) = \left( {1 - \zeta } \right)W\left( \beta \right) \) would be maximized because the government cannot receive any campaign contributions. In this case, Type-I households can attain \( U\left( {I,\beta^{*} } \right) \) units of utility, which is the reservation utility when negotiation begins. This implies that \( U_{0} = sU\left( {I,\beta^{*} } \right) \). Combining this and (31), we can calculate the amount of contributions, \( \bar{C} \) as
$$ \overline{C} = s\left[ {U\left( {I,\beta^{C} } \right) - U\left( {I,\beta^{*} } \right)} \right]. $$
(34)

In this setting, since the government has very strong bargaining power, it can extract all the surplus from patent holders.14

Second, we can assume that Type-I households have the first-mover advantage and make a take-or-leave it offer to the government. Under these assumptions, we can easily calculate the amount of contributions, \( \underline{C} \) as15
$$ \underline{C} = \frac{1 - \zeta }{\zeta }\left[ {W\left( {\beta^{*} } \right) - W\left( {\beta^{C} } \right)} \right]. $$
(35)

In this setting, since Type-I households have all the bargaining power, they can contribute the smallest amount so that the government is indifferent between \( \beta^{*} \) and \( \beta^{C} \).16

5 Concluding remarks

In this study, we build a fully-endogenous quality-ladder model and examine how campaign contributions affect innovation and economic growth. One advantage of our study is that our growth model enables us to examine the influence of a politically determined patent policy not only on welfare, but also on the growth rate. As a result, our model can reveal broader interactions among policy-making, welfare, and economic growth than Chu (2008) does.

Consequently, our theoretical framework provides the following main results. First, we find the existence of a patent policy that maximizes innovation intensity. The existence of innovation-maximizing patent protection reflects two effects of patent policy on research activity: a profit magnification effect and a patent cost effect. Second, our welfare analysis shows that campaign contributions by patent holders result in a stronger patent policy than a welfare-maximizing policy. Third, we find that campaign contributions increase the rate of innovation to an excessive level from the viewpoint of social welfare when the innovation-maximizing patent policy is sufficiently strong. On the other hand, campaign contributions decrease not only social welfare but also the rate of innovation when the innovation-maximizing patent policy is sufficiently weak. The third result is important because it implies that the rate of innovation distorted by campaign contributions can be too high from the viewpoint of social welfare. We can conclude that our theoretical framework succeeds in finding a new relationship between campaign contributions and economic growth.

Some issues remain for future research. First, in this study, we focus on the steady-state equilibrium. Our analyses do not include transition dynamics, and the stability of the steady state is not guaranteed.17 However, we recognize the importance of transition dynamics, because the transition path to steady-state equilibrium must be stable for our policy analyses to be useful. Therefore, our future research must make an effort to include transition dynamics in the analyses. Second, we assume that there are two types of households, where Type-I households have access to financial markets while Type-II households do not. This assumption is very useful for generating a political conflict between patent owners and other consumers. However, it may be too strong for deriving our main result. One way to relax this setting is to assume that Type-I households have a lower discount rate than Type-II households. Under this assumption, we can show that Type-I households can be the owners of financial assets in equilibrium. Therefore, relaxing the setting of household types can be an interesting challenge for future research.

Footnotes

  1. 1.

    According to the OECD (2013), the proportions of research and development throughout the world are 33.5% in the United States, 14.6% in China, 11.6% in Japan, 7.1% in Germany, 4.4% in Korea, 4.1% in France, and 3.7% in the United Kingdom.

  2. 2.

    The quasi-linear preference suggests that the utility of households is measured in units of homogenous goods. Although this is a relatively strong assumption, it allows us to derive the political-equilibrium patent policy with ease. We will examine this point in Sect. 4.2.

  3. 3.
    Considering the consumption index given by (9), the price index of quality-enhancing goods is expressed as
    $$ P_{q,t} = \exp \left( {\int\limits_{0}^{1} {\log p\left( j \right)} {\text{d}}j} \right). $$
  4. 4.

    For more information about patent breadth, see Gilbert and Shapiro (1990) and Klemperer (1990).

  5. 5.

    In the case of finite patent length, the relationship between the rate of innovation and patent policy is derived as an implicit function. This consequence makes the analysis more complicated. As for the difficulty caused by the assumption of finite patent length, see Iwaisako and Futagami (2013).

  6. 6.

    From Walras’s law, at least one of (16) and (17) is necessary to close our model.

  7. 7.

    For details of derivation, see Grossman and Helpman (1991, Chap.4).

  8. 8.

    In contrast, the first term of (26) tends to be negative when the economy has a larger population, \( L \). In this case, each Type-I household holds a small amount of assets. Therefore, they do not prefer stronger patent protection.

  9. 9.

    By comparing (26) and (29), we can easily confirm that \( \beta_{I} \) approaches \( \beta^{*} \) as \( s \) goes to one.

  10. 10.

    The objective function of the government is also assumed to be quasi-linear. This implies that the utility of the government is measured in units of homogenous goods.

  11. 11.

    It may be more realistic that Type-I households negotiate with the government in every period to make political contributions. However, this setting makes it difficult to formulate the negotiation process.

  12. 12.
    We could derive the efficient bargaining outcome by solving a different constrained maximization problem,
    $$ \hbox{max} \;sU\left( {I,\beta } \right) - C\;{\text{subject}}\;{\text{to}}\;\left( {1 - \zeta } \right)W\left( \beta \right) + \zeta C \ge W_{0} , $$
    for some value of \( W_{0} \). The first order condition of this problem is exactly the same as (33).
  13. 13.

    To derive a more general solution, we must add some assumptions and calculate the total surplus of the government and patent holders. See Chu’s (2008) Appendix A for details.

  14. 14.

    In this case, we can easily confirm \( \widetilde{W}\left( {\beta^{C} ,\overline{C} } \right) \ge \widetilde{W}\left( {\beta^{*} ,0} \right) = \left( {1 - \zeta } \right)W\left( {\beta^{*} } \right) \), which implies that the government’s gain when the two agents engage in efficient bargaining is larger than the gain without bargaining. Therefore, this inequality represents that the government has an incentive to strengthen its patent protection as a result of the bargaining outcome.

  15. 15.

    We use the constraint given in footnote 12 and \( W_{0} = \left( {1 - \zeta } \right)W\left( {\beta^{*} } \right) \) to derive (35).

  16. 16.

    Similar to footnote 14, we can confirm \( sU\left( {I,\beta^{C} } \right) - \underline{C} \ge sU\left( {I,\beta^{*} } \right) \), which represents that Type-I households have an incentive to contribute as a result of the bargaining outcome.

  17. 17.

    Our model is based on the quality-ladder model developed by Grossman and Helpman (1991, Chap. 4). Their model has no transitional dynamics, which implies that an economy jumps immediately to its steady state in their model.

Notes

Acknowledgements

I thank the Editor-in-Chief Yoshiro Higano, anonymous reviewers, and audiences at several seminars for their helpful comments. In addition, this work was supported by JSPS KAKENHI Grant Number 25380290.

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Copyright information

© The Japan Section of the Regional Science Association International 2018

Authors and Affiliations

  1. 1.Faculty of EconomicsKyushu UniversityFukuokaJapan

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