Innovation Driven Economic Growth in Multiple Regions and Taxation

Abstract

We provide the first theoretical analysis of the effects of alternate forms of taxation on economic growth in a dynamic model with multiple regions. The regions are heterogeneous, but, in each region, consumers have constant relative risk aversion preferences, there is no growth in the stock of human capital, and there are three kinds of manufacturing activities involving the production of blueprints for inputs or machines, the inputs or machines themselves, and a single consumption good. Our analysis generates four salient findings. First, we define the multiregion equilibrium. Second, we characterize the multiregion equilibrium and show that in this equilibrium, each region grows at a constant rate starting at time t = 0. Third, we show that except in knife-edge cases, output in each region grows at a different long-run rate. Finally, we determine the effects of asset, profit, and investment taxes on the economic growth rates of the regions under study.

Keywords

economic growth human capital innovation multiple regions taxation

Introduction

In contemporary times, there is little dispute about the point that innovation has a fundamental role to play in promoting the economic growth of regions and nations. In this regard, consider the recent work of Audretsch and Aldridge (2009), Fischer and Nijkamp (2009), Baumol (2010), and Batabyal and Nijkamp (2012a, 2012b). These researchers have all underscored the powerful nexuses between innovative activities on one hand and regional economic growth and development on the other hand. In the words of Fischer and Nijkamp (2009, 185), “entrepreneurial innovation is the essence of capitalism and its process of creative destruction, embodied in new products, new production processes and new forms of organisation.” A key conclusion emanating from this line of research is that regional growth and development are very closely related to the activities of innovative entrepreneurs.

Researchers in general and theorists in particular in regional science have certainly recognized the fundamental connections between innovative activities and regional economic growth. This notwithstanding, there are two gaps in the extant literature on this subject. The first gap stems from the fact that until recently, most theoretical analyses have concentrated on economic growth and other relevant matters within a single region. Therefore, there are very few studies that have analyzed innovation driven economic growth and other growth-related issues in multiple regions. The second gap relates to the fact that there is a great dearth of studies that have examined the effects of alternate forms of taxation on innovation driven economic growth in a multiregional world.

Given this state of affairs, we now briefly review the extant literature that is related to the two lacunae mentioned in the previous paragraph. Zhang (1997) studies the equilibria that arise in a two-region model in which there are interactions between capital accumulation, knowledge growth, and the regional distribution of capital and labor. Liew and Siriwardana (2002) analyze the general equilibrium effects of tariff changes in a multiregional model in which governments behave strategically. They show that by “engaging in competitive money creation, governments are able to cushion the impact of tariff changes on national and regional absorption” (Liew and Siriwardana, 2002, 1). Fujita and Gokan (2005) use a two-region model to show that when information technology reduces communication and trade costs between management and production facilities within firms, the result is firm fragmentation. Zhang (2007) focuses on economic growth in a multiregional small open economy. His analysis shows that an increase in the preference for large cities may “accelerate agglomeration of the population and economic activities into a region with high productivity” (Zhang, 2007, 515). N’Diaye, Zhang, and Zhang (2010) use a multiregional empirical model of the United States, the Euro area, and Japan to show that there are substantial benefits from reforms that stimulate domestic demand in the various regions under study. Finally, Batabyal (2012) studies a game-theoretic model in which a representative entrepreneur in a regional economy has a project that gives rise to a random cash flow and that needs investment that this entrepreneur raises from a competitive market.

Moving on to the subject of taxation, Monchuk et al. (2007) point out that state and local tax burdens are an important determinant of economic growth in counties in the United States. Poulson and Kaplan (2008, 53) use the framework of an endogenous economic growth model to show that there is a “significant negative impact of higher marginal tax rates on economic growth” in the various states of the United States. The changing degree to which individuals and firms respond to state tax policy in the United States is the focus of a recent article by Deskins and Hill (2010). These authors note that the degree to which higher tax burdens decrease economic growth has diminished substantially in the United States. Shifting the focus to Japan, Doi (2010) uses a dynamic model to demonstrate that land allocation tax (LAT) grants lower regional income and are also responsible for creating what the author calls “poverty traps.” Finally, Skidmore and Tosun (2011) use county data in Michigan to study the effects of a cap in the growth of property values for tax purposes. They show that this assessment cap has given rise to a differential in tax prices between potential new property owners and long-time property owners, and it has also “reduced in-migration” (Skidmore and Tosun, 2011, 256).

The studies discussed in the preceding two paragraphs have advanced our understanding of some aspects of economic growth in models with multiple regions and the effects of taxation on regional economic growth. This notwithstanding, the central point of the second paragraph in this section that there are virtually no multiregional, innovation driven models of economic growth that also analyze the effects of alternate forms of taxation on the growth rate of regions, remains valid.

Given the conclusion in the preceding paragraph, in our article, we provide a theoretical analysis of the effects of alternate forms of taxation on innovation driven economic growth in a dynamic model with multiple closed regions.¹ The regions are heterogeneous, but, in each region, consumers have constant relative risk aversion (CRRA) preferences, there is no growth in the stock of human capital, and there are three kinds of manufacturing activities involving the production of blueprints for inputs or machines, the inputs or machines themselves, and a single consumption good. Our analysis generates four noteworthy findings. First, we define the multiregion equilibrium. Second, we characterize the multiregion equilibrium and show that in this equilibrium, each region grows at a constant rate starting at time t = 0. Third, we show that except in knife-edge cases, output in each region grows at a different long-run rate. Finally, we determine the effects of asset, profit, and investment taxes on the economic growth rates of the various regions under study.

The rest of this article is organized as follows. The section titled “Preliminaries” delineates the basic or single-region theoretical model. This basic model is adapted from the prior work of Rivera-Batiz and Romer (1991) and Acemoglu (2009, 433–444). The section titled “Machine Invention and Production” describes how inputs or machines are first invented and then produced. The section titled “The BGP Equilibrium” focuses on the balanced growth path (BGP) equilibrium. The following “Preliminaries” section first generalizes the single-region model of the earlier sections titled “Preliminaries,” “Machines Invention and Production,” and “The BGP Equilibrium” and then defines the world or multiregion equilibrium in which every region is in a BGP equilibrium.² The section titled “Description of the Multiregion Equilibrium” describes the world equilibrium in which each region grows at a constant rate starting at time t = 0. The “Output Growth in the Various Regions” section points out that except in knife-edge cases, output in each region grows at a different long-run rate. The sections titled “Asset Tax,” “Profit Tax,” and “Investment Tax” analyze the effects of asset, profit, and investment taxes on the BGP growth rate in the various regions under study. The final section concludes and then discusses potential extensions of the research delineated in this article.

The Theoretical Model With a Single Region

Preliminaries

We begin by briefly explicating the details of a model presented in Acemoglu (2009, 433–444). We shall then generalize this discussion to the case of multiple regions. To this end, consider a stylized, infinite horizon regional economy in which the representative consumer displays CRRA and this CRRA preferences are given by $\int_{0}^{\infty} exp (- α t) [{C (t)^{1 - β} - 1} / (1 - β)] d t$ , where β ≠ 1, C(t) is consumption at time t, α is the constant time discount rate, and β ≥ 0 is the constant coefficient of relative risk aversion.³ There is no growth in the stock of human capital in our regional economy and the total available human capital H is supplied inelastically.⁴

The single final good for consumption is produced competitively with the production function

Y (t) = \frac{1}{1 - γ} [\int_{0}^{N (t)} x (v, t)^{1 - γ} d v] H^{γ}

where H is the aggregate human capital input, N(t) denotes the different number of the varieties of machines that are used to produce the final good at time t, x (v, t) is the total amount of the machine of variety v that is used at time t, and γ∊(0, 1) is a parameter of the production function. We assume that the x(.,.)′s depreciate fully after they have been used. This assumption is made for two reasons. First, with this assumption, we can think of these x(.,.)′s as generic inputs. Second, because these x(.,.)′s depreciate immediately, in our subsequent mathematical analysis, we will not have to work with additional state variables. Note that for a given number of varieties of machines or N(t) the production function in equation (1) exhibits constant returns to scale. In addition, we normalize the price of the final consumption good at all time points to equal unity. Our next task is to discuss how new machines in our regional economy are invented and then produced.

Machine Invention and Production

Once the blueprint for a particular variety of machine has been invented, one unit of this machine can be produced at marginal cost equal to ψ > 0 units of the final consumption good. The so-called innovation possibilities frontier⁵ in our regional economy is given by

\frac{d N (t)}{d t} = \dot{N} (t) = δ Z (t),

where δ > 0 is a flow or productivity parameter and Z(t) is the total expenditure on research and development (R&D) in our regional economy. Inspecting equation (2), it is straightforward to verify that increased expenditure on R&D leads to a more rapid invention of new machines.

There is free entry into R&D activities, and this means that any individual or firm in our regional economy can spend one unit of the final consumption good at time t to create a flow rate δ of new blueprints. A firm that discovers a blueprint for a new machine, that is invents a new machine, receives a fully enforced perpetual patent.⁶ The N (0) number of initial machine varieties is supplied by monopolists with perpetual patents. Note that because many different firms in our regional economy are engaging in expenditures on R&D activities, there is no aggregate uncertainty in the innovation process and hence equation (2) holds deterministically in the model.

Given the above patent structure, a firm that invents a new machine of variety v is the monopolistic supplier of this variety and, as such, at any time t, it sets a price p^x (v,t) to maximize profits. Note that because of our assumption of full depreciation of machines, this price p^x (v,t) can also be interpreted as the user cost of this machine. The demand for a machine of variety v—on which more details are given in the next section—is given by maximizing the net total profit of the final consumption good sector.

We denote the net present discounted value of owning the blueprint of a machine of variety v by the time differentiable function V(v,t). The assumed time differentiability of the value function means that we can write this function in the form of a Hamilton-Jacobi-Bellman (HJB) equation given by⁷

r (t) V (v, t) - \dot{V} (v, t) = π (v, t),

where $π (v, t) = p^{x} (v, t) x (v, t) - ψ x (v, t)$ is the profit of the monopolist producing the machine of variety v at time t, p^x (v,t), and x(v,t) are the monopolists’ profit maximizing price and quantity choices, and $r (t)$ is the interest rate at time t. Finally, the resource constraint affecting our regional economy at time t is

C (t) + X (t) + Z (t) \leq Y (t),

where C(t), Z(t), and Y(t) have been explained previously and X(t) is total spending or investment on machines. With this background in place, the task before us now is to characterize the BGP equilibrium for our innovative regional economy.

The BGP Equilibrium

A BGP is an equilibrium in which consumption C(t) and output Y(t) grow at the same constant rate. If C(t) and Y(t) are growing at a constant rate, then it can be shown that N(t), the number of the varieties of machines that are used to produce the final good at time t, also grows at a constant rate. If we let $g_{C}^{*}$ denote the growth rate of consumption in our regional economy, then $g_{C}^{*}$ will be constant only if the interest rate is constant for all time t, that is if $r (t) = r^{*}, \forall t$ .

Now, straightforward algebra yields three results.⁸ First, we see that a constant or BGP interest rate exists and it is given by r* = γδH. Second, in a BGP, $g_{C}^{*}$ must equal the growth rate of output or g* and hence we have $g^{*} = g_{C}^{*}$ . Third, from the consumer Euler equation, we deduce that the growth rate of our regional economy is

g^{*} = \frac{1}{β} (γ δ H - α) .

Given equation (5), let us assume that

γ δ H > α a n d t h a t α > γ δ (1 - β) H .

The first assumption ensures that the regional growth rate g* > 0, and the second assumption guarantees both the finiteness of the representative consumer’s utility and the satisfaction of the so-called transversality condition.

Combining the first inequality in (6) above with the fact that the BGP interest rate r* = γδH, it is clear that for a meaningful BGP equilibrium with positive growth to exist, we must have r* ≠ α. In particular, if r* = α then from equation (5) above it is clear that g* = 0, and we would then have an uninteresting and trivial BGP equilibrium with no (zero) growth. Since we are not interested in analyzing an equilibrium with no growth in this article, we posit that the two inequalities in (6) above hold and this also clearly means that r* ≠ α. Put differently, for the reasons we have just given, in a BGP equilibrium, we cannot have the interest rate (r*) equaling the time discount rate (α).

We now state an important result concerning the existence and the uniqueness of the BGP equilibrium that is due to Acemoglu⁹ (2009, 439). We shall use this result in our subsequent analysis of the model with multiple regions.

Proposition 1: If the inequalities in (6) hold, then there exists a unique BGP in which consumption, output, and technology all grow at the same rate g* given in equation (5).

With this background in place, we are now in a position to generalize the single-region model that we have been discussing thus far to the case of multiple regions.

The Theoretical Model With Multiple Regions

Preliminaries

Consider a world economy consisting of j = 1,2, . . . , J regions. Consistent with the discussion in the first section and note 1, each of these regions is closed and hence they have the same production and R&D technologies as delineated in the sections titled “Preliminaries” and “Machine Invention and Production.” However, the individual regions differ from each other in terms of their human capital input H_j , the productivity of R&D δ _j , and the time discount rate α _j . In addition, to further distinguish between the different regions, we suppose that one unit of R&D expenditure costs ∊ _j units of the final consumption good in region j. Finally, note that because the J regions under study are closed, there are no technological exchanges between the various regions.

We now want to define the multiregion equilibrium in which each region is in an equilibrium of the sort described for the single region in the sections titled “Preliminaries,” “Machine Invention and Production,” and “The BGP Equilibrium.” Specifically, the multiregion equilibrium is a collection of time paths of allocations and prices for each region, which are given by

([C_{j} (t), Y_{j} (t), X_{j} (t), Z_{j} (t), N_{j} (t), {p_{j}^{x} (v, t), x_{j} (v, t)}_{v \in N_{j} (t)}, r_{j} (t), w_{j} (t)]_{t = 0}^{\infty})_{j \in {1, 2, \dots, J}},

such that in each region j, all monopolists choose ${p_{j}^{x} (v, t), x_{j} (v, t)}_{v \in [0, N_{j} (t)], t = 0}^{\infty}$ to maximize the discounted value of profits, the evolution of ${N_{j} (t)_{t = 0}^{\infty}}$ is determined by free entry, the time paths of interest rates and wage rates ${r_{j} (t), w_{j} (t)}_{t = 0}^{\infty}$ clear capital and human capital markets, and the trajectories of the aggregate allocations ${C (t), X (t), Z (t)}_{t = 0}^{\infty}$ are consistent with utility maximization by consumers. We now proceed to characterize the world equilibrium that we have just defined.

Description of the Multiregion Equilibrium

Because there are no interactions between the various regions under study, each region is described separately as a closed economy. Note that the characterization of the closed economy equilibrium for each region j is very similar to the characterization of the single region discussed in the sections titled “Preliminaries,” “Machine Invention and Production,” and “The BGP Equilibrium.” The only material difference in the present model from the one discussed in the sections titled “Preliminaries,” “Machine Invention and Production,” and “The BGP Equilibrium” is the presence of the ∊ _j parameter that controls the costs of R&D expenditures.

Since ∊ _j units of the final consumption good spent on R&D give rise to a flow rate of δ _j new machine blueprints, one unit of the final consumption good spent on R&D gives rise to a flow rate of δ _j /∊ _j blueprints. Therefore, if we define ${\hat{δ}}_{j} = δ_{j} / ε_{j}$ to be the unit productivity of R&D in region j, then the theoretical model for each region j becomes identical to the single region model discussed in the sections titled “Preliminaries,” “Machine Invention and Production,” and “The BGP Equilibrium.” Given this observation, it should be clear to the reader that if the parameters of region j satisfy the two inequalities presented in (6), then Proposition 1 in “The BGP Equilibrium” section and proposition 13.2 in Acemoglu (2009, 439) both apply to region j. Keeping this central fact in mind, we now substitute the unit productivity parameter ${\hat{δ}}_{j} = δ_{j} / ε_{j}$ into equation (5) and then simplify the resulting expression. This last step together with the reasoning in this paragraph gives us a salient result that we now state as

Proposition 2: In each region j, consumption C_j (t), output Y_j (t), and the machine varieties N_j (t) all grow at the constant rate given by $g_{j}^{*} = (1 / β) {γ H_{j} (δ_{j} / ϵ_{j}) - α_{j}}$ .

The reader should note that the region j variables mentioned in Proposition 2 grow at the rate indicated at time t = 0. In other words, there are no transitional dynamics in this more general model. This concludes our characterization of the multiregion equilibrium. We now note that except in knife-edge cases, output in each of the J regions grows at a different long-run rate.

Output Growth in the Various Regions

From the expression for the growth rate of each region j in Proposition 2, it follows that different regions grow at different rates except in knife-edge cases. An implication of this finding is that in the model of this third section, small changes in preferences, population, or the R&D technology of regions will lead to large differences in the levels of consumption and output in the various regions in the long run. We now move on to analyze the effects of asset, profit, and investment taxes on the BGP growth rate in the various regions under study.

Taxation and its Effect on the BGP Growth Rate

Asset Tax

We begin by pointing out that the government of any one of the J regions under study can choose to tax one or more aspects of the underlying economy in several ways. Therefore, given that our article is about innovation driven regional economic growth, we have chosen to focus on three of the most pertinent kinds of taxation. To this end, let us first consider the case of asset taxes. The reader will note that this case can also be thought of as the taxation of capital income.

Suppose the government of region j taxes the returns on assets linearly at the rate $τ_{j}^{A}$ It then distributes the proceeds of this tax back to consumers in this jth region in a lump-sum or nondistortionary manner. When this nondistortionary manner of distribution holds, it can be shown that the consumer Euler equation in region j is

\frac{{\dot{C}}_{j} (t)}{C_{j} (t)} = \frac{1}{β} [r_{j} (t) {1 - τ_{j}^{A}} - α_{j}],

where ${\dot{C}}_{j} (t) = d C_{j} (t) / d t .$ Adapting equation 13.11 in Acemoglu (2009, 436) to the taxation problem of this section, it is clear that the monopolistic machine producer’s per period profit in region j is given by $π_{j} (v, t) = γ H_{j}$ . Some thought tells us that the value function along the BGP—the section titled “Machine Invention and Production” —is now given by $V_{j} (v, t) = (γ H_{j}) / r_{j}^{*} .$ Finally, because there is free entry into research in our model, the BGP interest rate can be determined definitively, and it is given by $r_{j}^{*} = ({γ δ}_{j} H_{j}) / ε_{j} .$ Using this last expression for $r_{j}^{*}$ in equation (8) above gives us the BGP growth rate of the economy in region j. Specifically, we get

g_{j}^{*} = \frac{1}{β} \{\frac{{γ δ}_{j} H_{j}}{ϵ_{j}} (1 - τ_{j}^{A}) - α_{j}\} .

The allocation in region j delineated by equations (8) and (9) constitutes an equilibrium if the two inequalities in (6) are satisfied. When this happens, consumption, output, and technology in region j all grow at the rate given in equation (9), starting at time t = 0. In other words, there are no transitional dynamics in the model. Most importantly, note that a linear asset (capital income) tax reduces economic growth in region j because it lowers the incentives of the representative consumer to save. Because R&D investments in our model are financed with the savings of the representative consumer, a reduction in these savings slows down innovation and ultimately economic growth in region j. We now proceed to study the profit tax.

Profit Tax

In this case, we suppose that the government in region j taxes the profit of the monopolistic machine producers at a linear rate given by $τ_{j}^{π}$ . As in the section on the “Asset Tax,” the government distributes the tax revenues to consumers in this jth region in a lump-sum (nondistortionary) manner. Applying the logic of the “Asset Tax” section to this case, the monopolistic machine producer’s per period profit is given by $π_{j} (v, t) = γ H_{j} (1 - τ_{j}^{π})$ . Similarly, the value function on the BGP is given by $V_{j} (v, t) = {γ H_{j} (1 - τ_{j}^{π})} / r_{j}^{*} .$ Finally, as in the “Asset Tax” section, the BGP interest rate can be ascertained definitively and, in this case, it is given by $r_{j}^{*} = {{γ δ}_{j} H_{j} (1 - τ_{j}^{π})} / ε_{j} .$ Using this last expression for $r_{j}^{*}$ in the relevant consumer Euler equation gives us region js growth rate and that rate is given by

g_{j}^{*} = \frac{1}{β} \{\frac{{γ δ}_{j} H_{j}}{ϵ_{j}} (1 - τ_{j}^{π}) - α_{j}\} .

Inspecting equation (10), we see that taxing the profit of the monopolistic machine producers in region j reduces the value of the innovated varieties of machines. In turn, this reduction has the effect of lowering both innovation and economic growth in this jth region. We now move on to analyze the effects of an investment tax.

Investment Tax

A lot has been said and written about creating the apposite incentives so that R&D activities in innovative regional economies of the sort studied in this article are optimal. Therefore, our focus in this section will be on R&D. Specifically, suppose that the government of region j taxes R&D investment linearly at the rate $τ_{j}^{R} .$ As in the sections titled “Asset Tax” and “Profit Tax,” we suppose that the tax revenues are returned to consumers in this jth region in a lump-sum (nondistortionary) fashion.¹⁰ With this linear R&D investment tax $τ_{j}^{R},$ , the value function on a BGP is given by $δ_{j} V_{j} (v, t) = ε_{j} / (1 - τ_{j}^{R}) .$ Therefore, the BGP interest rate is now given by $r_{j} (t) = {{γ δ}_{j} H_{j} (1 - τ_{j}^{R})} / ε_{j} .$ Using these last two expressions, we deduce that the economy in region j will grow at the rate given by

g_{j}^{*} = \frac{1}{β} \{\frac{{γ δ}_{j} H_{j}}{ϵ_{j}} (1 - τ_{j}^{R}) - α_{j}\} .

Equation (11) tells us that the linear R&D investment tax reduces the value of innovation and this, once again, lowers the rate at which the economy in region j grows. Inspecting equations (9), (10), and (11) carefully, we are led to an interesting result concerning the impact of taxes on regional economic growth rates. We now state this result as

Proposition 3: A linear tax of the same magnitude, whether levied on assets (capital income), or profit, or R&D investment, will lead to the same economic growth rate and equilibrium path in region j.

Our analysis thus far shows that taxes that deter savings, or private sector profits, or innovation tend to reduce the growth rate of any of the J regional economies under study. In addition, it is also clear that if any two regions in our set of J regions have different tax policies or different time discount rates, then not only will they have different growth rates but their income per capita levels will also diverge greatly. A clear implication then of the multiple regions’ model of this article is that small differences in policies across various regions can lead to large income differences across these same regions. In this regard, it is very likely that if we were to permit interactions between the various regions in goods, financial, and R&D markets, then there would exist stabilizing influences that would result in the various regions growing at very similar rates. This concludes our discussion of the effects of alternate forms of taxation on the growth rates of the J regional economies under study.

Conclusions

In this article, we conducted, to the best of our knowledge for the first time, a theoretical analysis of the effects of alternate forms of taxation on innovation driven economic growth in a dynamic model with multiple closed regions. First, we discussed the key attributes of the single-region model upon which our subsequent and more general analysis was built. Second, we defined the multiregion equilibrium. Third, we described the multiregion equilibrium and showed that in this equilibrium, each region grows at a constant rate starting at time t = 0 Fourth, we showed that except in knife-edge cases, output in each region grows at a different long-run rate. Finally, we ascertained the effects of asset, profit, and investment taxes on the economic growth rates of the various regions under study.

The analysis in this article can be extended in a number of directions. In this regard, note that because we were interested in highlighting the effects of alternate forms of taxation on the economic growth rate of the various regions in a stark manner, we studied the case of closed regions. However and as noted in the last paragraph of the “Investment Tax” section, this feature of the model gave rise to rather dramatic results, suggesting significant eventual differences in key variables such as per capita incomes across the various regions. Therefore, following the work of Oladi and Beladi (2008), it would be useful to introduce trade into the model and then study the effects of interregional trade on the economic growth rates of the various regions under study.

In addition, it would also be instructive to shed light on political economy issues by ascertaining the impact that dissimilar governmental objective functions have on innovation driven regional growth rates in a dynamic model with multiple regions. Studies that incorporate these features of the problem into the analysis will increase our understanding of the ways in which tax policies and governmental objectives interact to either enhance or retard the growth rates of dynamic regional economies.

Footnotes

Acknowledgments

We thank the Editor-in-Chief Sergio J. Rey and three anonymous referees for their helpful comments on a previous version of this article. Batabyal acknowledges financial support from the Gosnell endowment at RIT. The usual disclaimer applies.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Notes

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