Impact of inter-country corruption differences on wages and economic growth

Abstract

Theoretically and numerically, this paper attempts to examine the macroeconomic effects of corruption by using the two-country directed technical change model. At a single-country level, an increase in corruption levels in one country leads to an intra-country decrease in the demand for labor and wages and a permanent slowdown of technological-knowledge progress and economic growth. At the inter-country level, a country-specific increase in corruption enlarges inter-country wage and technological-knowledge gaps. Overall, higher corruption levels in one country are detrimental to global economic growth. Through calibration, it is shown that when the differences between the corruption of non-corrupt and corrupt countries increase: (1) economic growth is mainly stimulated in the corrupt countries India, Mexico, and Brazil; (2) the lowest wage inequality compared to non-corrupt countries is observed in the corrupt countries Greece, Portugal, and Spain.

1 Introduction

By engaging in corruption, government officials use their authority for private gain, thus using their power for self-enrichment, as Rose-Ackerman (1978, 1997), Murphy et al. (1993), Tanzi (1997) have rightly pointed out.

This paper analyzes static and dynamic macroeconomic effects associated with different levels of corruption across countries, particularly the effects on wage inequality between skilled and unskilled workers located in corrupt and non-corrupt countries, as well as the effects on the overall economic growth rate. For this purpose, following, among others, Acemoglu (2002) and Afonso (2012), a Direct Technical Change (DTC) model is extended. Static effects arise in the aftermath of initial differences between countries, including in the level of corruption. Dynamic effects are driven by the interaction between countries. This interaction leads to an overall steady state and is reflected in the progress and direction of technological knowledge, which in turn impacts the variables of interest.

In the model, there is the worldwide composite final good which is competitively produced from each country-specific aggregate final good. The worldwide composite final good is the numeraire and is used to consume, produce intermediate goods, and conduct R&D. Each country-specific aggregate final good is competitively produced from a unit continuum of tasks. Each task is competitively produced from the specific labor (the skilled labor or the unskilled labor) and a continuum of specific non-durable quality-adjusted intermediate goods. The producer of each task chooses to produce it either with unskilled labor or skilled labor. Hence, the range of task production with skilled and unskilled labor is endogenously determined. Each intermediate good is quality-adjusted and is monopolistically produced from the worldwide composite final good and the quality of intermediate goods improves as a result of vertical R&D. The R&D sector exhibits free entry. Moreover, two channels affected by corruption are considered: (1) the intermediate goods sector, in which firms pay the cost of business corruption with the final world composite good. According to our vision, corruption penalizes the production costs of intermediate goods similarly in both countries, but corruption levels are different across them. This aligns with Murphy et al. (1991, 1993), Mauro (1995), and Afonso and Longras (2022) since intermediate goods producers must obtain government goods, such as licenses, and permits underlying intellectual property rights to start producing. Therefore, corruption penalizes the proper allocation of resources, creating permanent distortions from which only a few individuals benefit; and (2) the skilled labor productivity, which is negatively impacted by an increase in this variable. In agreement with, for example, Mandal and Marjit (2010), corruption also particularly affects skilled workers as it decreases the stock of skills and depresses the returns to education, lowering growth levels. This reduction in skills impacts wage inequality and shrinks welfare levels, as proposed by Dimant et al. (2013) and Cooray and Schneider (2016).

Since there are different levels of corruption between countries, there will also be a technological-knowledge bias in favor of the country with the lowest level of corruption, affecting countries’ wages differently. Indeed, a country-specific increase in the corruption level enlarges inter-country wage inequality and technological-knowledge gap. Furthermore, expanding corruption levels in each country also hinders economic growth.

Additionally, to measure the magnitude of the impacts of corruption, calibration of the theoretical model has been conducted to examine the extent of the effect of corruption levels on economic growth and wage inequality (unskilled and skilled) between non-corrupt countries and corrupt countries.¹

Looking at the existing literature, it is observed that there are excellent seminal reviews on the economic impact of corruption—e.g., Rose-Ackerman (1997, 1999), Tanzi (1998), Wei (1999), and Trabelsi (2024). In these reviews, which emerged in the 1990s, the literature explores the implications of corruption on growth, investment and innovation (e.g., Mauro 1995, 1998; Knack and Keefer 1996; Murphy et al. 1993; D’Agostino et al. 2016; Cieślik and Goczek 2018) and on income and development of countries (Saha and Gounder 2013), the efficient distribution of resources affecting the rate of human-capital formation (e.g., Mauro 1998), the composition of government spending (e.g., Tanzi and Davoodi 1997; Mauro 1998), and the allocation of foreign direct investment (e.g., Wei 1997). In particular, Mauro (1995) was the first to analyze how corruption affects growth in a sample of countries, concluding that corruption causes slower growth. In turn, Murphy et al. (1991) observed that higher rent-seeking is associated with a lower growth rate. Knack and Keefer (1995) revealed that the quality of government institutions, including the degree of corruption, affects investment and growth as much as other variables in the political economy. More recently, Cieślik and Goczek (2018) predict that corruption, viewed as a burden on a nation’s wealth, hinders the volume of international investment, leading to a slowdown in the growth rate. Finally, in a similar vein, Gründler and Potrafke (2019) conducted an empirical study that supports the idea that corruption reduces economic growth, especially in countries with low investment rates and poor governance.

While the negative relationship between corruption and economic growth is relatively consensual in the literature, particularly in emerging economies, as Simo-Kengne and Bitterhout (2023) pointed out, some theoretical and empirical studies suggest that globalization may mitigate it. On the one hand, researchers such as Gould (1991) and Leiken (1997) have argued that globalization encourages corrupt practices, as the resulting business relationships place government systems and firms in scenarios that promote corruption. On the other hand, Sung and Chu (2003) and Das and DiRienzo (2009) argue that globalization can decrease levels of corruption, primarily because international organizations (e.g., International Monetary Fund, World Bank, and World Trade Organization) enforce anti-corruption policies and a higher level of transparency in trade, noting, however, that the magnitude of the impact depends on the level of globalization of countries (e.g., Das and DiRienzo 2009). In turn, in this context of globalization, Egger and Winner (2006), in their empirical model evaluating the impact of corruption on bilateral Foreign Direct Investment (FDI) stocks, highlight a negative relationship between the level of corruption and FDI in host countries. Moreover, Sharma and Mitra (2019) report a positive impact on trade among countries in all income groups, while a recent approach conducted by Aïssaoui and Fabian (2021) showed that globalization is an important source of welfare stimulus, especially in lower-income countries, because it fosters economic efficiency and leads to decreased corrupt practices and a considerable increase in GDP.

Therefore, although the literature already emphasizes the corruption-growth relationship, this relationship has always been considered in the intra-country context. Moreover, the distributional consequences of inter-country corruption have so far been ignored. It is true that Tanzi (1995) and Rose-Ackerman (1997, 1999), for example, address the issue of income distribution, but they do so in the intra-country context.

This paper can contribute to the literature on corruption in five ways. First, the proposed setup differs from existing analysis frameworks by introducing a DTC model to derive a set of implications for the effects of corruption on inter-country wage inequality and overall economic growth. Second, it examines, for the first time, the relationship between corruption and inter-country wage inequality. Third, it contributes to the robustness of the relationship between economic growth and corruption by proposing a dynamic endogenous growth model. Fourth, it allows the effects of corruption to depend on the differentiated size of corruption levels across countries. Finally, it extends the DTC model in two ways: by taking into account a dual DTC, between two representative countries and at the level between skilled and unskilled tasks and by introducing corruption into the process. While the base DTC model generates specialized and supply-biased endogenous technological-knowledge bias—i.e., technological knowledge bias responds to changes in labor supply (e.g., Acemoglu 2002)—the DTC model now proposed generates endogenous technological-knowledge bias due to corruption.

The recent work by Afonso and Longras (2022) can be considered the closest in terms of setup. However, there are significant differences. In Afonso and Longras (2022), workers in each country are undifferentiated in terms of qualifications and each country produces a set of final goods (or tasks) to analyze whether, due to the effects of corruption, institutional quality and offshoring affect comparative advantage, inter-country wage inequality, and economic growth. In the present work, in each country, each final good (or task) can be produced by skilled or unskilled workers, and the impacts can also be studied on premiums for unskilled and skilled workers. For example, an increase in a country’s corruption level affects intra- and inter-country wage inequality and slows the progress of technological knowledge and economic growth.

The rest of the paper is organized as follows. Section 2 presents the development of the model, explaining the setup. Section 3 endogenizes the technological knowledge, determines the dynamic equilibrium, and contains the main theoretical results. Section 4 describes the data and calibration used and illustrates the quantitative results. Section 5 concludes the paper.

2 Benchmark model

This section describes the economic setup of the world economy composed of two countries $s=\{1,2\}$ in which infinitely-lived households inelastically supply labor, maximize the utility of consumption from the aggregate worldwide composite final good, and invest in a firm’s equity. The inputs of the aggregate worldwide composite final good, Y; i.e., the numeraire, are two country-specific aggregate final goods, $Y_{1}$ produced in country 1 and $Y_{2}$ produced in country 2, each one composed by many competitive firms that produce a continuum of tasks, indexed by $v_{s}\in \left[ 0,1\right]$. Each country is endowed with a positive fixed level of internationally immobile unskilled and skilled labor; i.e., $L_{s}$ and $H_{s}$, respectively, and thus produces a certain number of unskilled and skilled tasks; moreover, it is assumed that country 1 (2) is relatively abundant in unskilled (skilled) labor, $\frac{H_{1}}{L_{1}}<\frac{H_{2}}{L_{2}}$. The range of tasks within each country employs, alongside specialized labor, a continuous set of specific non-durable quality-adjusted intermediate goods. Each sector for intermediate goods, in its turn, comprises a continuous array of industries, $j\in (0,J_{s}]$, and is characterized by monopolistic competition: the monopolist in industry j uses a design (technological knowledge), sold by the R&D sector and protected by a patent, and numeraire to produce at a price that maximizes profits.²

Considering the exploitative nature of corruption, we integrate it into the production of intermediate goods and acknowledge its impact on the absolute productivity of labor. In alignment with existing literature (e.g., Mandal and Marjit 2010), we posit that corruption disproportionately affects more skilled labor. Building on seminal works by Hall and Jones (1999), Olson et al. (2000), and Lambsdorff (2003), a negative correlation between corruption and productivity is established, leading to variations in the economic growth trajectories of different countries. Effective government policies and institutional quality play pivotal roles in enhancing productivity by fostering an environment conducive to the adoption of innovative ideas and more efficient technologies. This, in turn, encourages skill development among workers (Hall and Jones 1999), reduces preferential treatment for those involved in bribery, thereby promoting employee efficiency, and curtails the diversion of resources toward private or large side-payment projects (Lambsdorff 2003). In the realm of the R&D sector, entry is unrestricted, and each potential participant allocates resources to invent successful vertical designs for supply to a new monopolistic intermediate goods firm/industry. Essentially, R&D contributes to elevating the quality of intermediate goods and, consequently, technological knowledge, complementing both unskilled and skilled labor.

2.1 Preferences

Each country is composed of a set of households with an infinite life expectancy. Within each country, there is a representative household which, at time $t=0$, aims to maximize utility, considering the flow budget constraint. This optimization implies perfect anticipation of the evolution of technological knowledge over time. Thus, in country s, the discounted lifetime utility of a household is $U_{C_{s}}=\int _{0}^{\infty }\left( \frac{C_{s}(t)^{1-\theta }-1}{1-\theta }\right) e^{-\rho t}\hbox {d}t$, where $C_{s}(t)$ is the consumption of the aggregate commodity, $\rho >0$ is the individual time preference rate, and $\theta >0$ is the inverse of the inter-temporal elasticity of substitution. In country s, households accrue income through two different sources: investments in financial assets and labor supply. The flow budget constraint is given by,

$$\begin{aligned} \dot{a_{s}}(t)=r_{s}(t)\cdot a_{s}(t)+w_{L_{s}}(t)\cdot L_{s}+w_{H_{s}}(t)\cdot H_{s}-C_{s}(t)+\tau _{s}(t), \end{aligned}$$

(1)

where $a_{s}(t)$ is the household’s real financial assets/wealth holdings; $r_{s}$ is the real interest rate; $w_{L_{s}}$ and $w_{H_{s}}$ are the real wages paid for unskilled and skilled labor, respectively; $L_{s}$ and $H_{s}$ are, respectively, unskilled and skilled labor inelastically supplied by households; $\tau _{s}$ denotes a lump-sum gain of government officials from using their authority for private benefit; i.e., the benefits of corruption accrue to those individuals. From standard dynamic optimization, we derive the consumption Euler equation for region s:

$$\begin{aligned} \dot{C_{s}}(t)=\frac{1}{\theta }\cdot \left( r_{s}(t)-\rho \right) \cdot C_{s}(t). \end{aligned}$$

(2)

The corresponding transversality condition is $\lim _{t\rightarrow +\infty }e^{-\rho t}\cdot C_{s}(t)^{-\theta }\cdot a_{s}(t)=0$.

2.2 Technology, output and prices

2.2.1 Worldwide economy

Country 1 and country 2, competitively, produce the aggregate output Y using a CES aggregate production function:

$$\begin{aligned} \begin{array}{ccc} Y(t)&=\left[ \sum \limits _{s=1,2}\chi _{s}\cdot Y_{s}(t)^{\frac{\varepsilon -1}{\varepsilon }}\right] ^{\frac{\varepsilon }{\varepsilon -1}}&,\end{array}\;\varepsilon \in \left( 0,+\infty \right) , \end{aligned}$$

(3)

where $Y_{1}$ and $Y_{2}$ designate the total outputs produced by country 1 and country 2, respectively; $\chi _{1}$ and $\chi _{2}$, with $\sum _{s=1,2}\chi _{s}=1$, represent the distribution parameters, gauging the relevance/importance of the nations; $\varepsilon \ge 0$ characterize the elasticity of substitution between the two countries.³ The hypothesis that firms in each country produce their final goods competitively gives rise to the following maximization problem: $\max _{Y_{1},Y_{2}}\,\Pi _{Y}=P_{Y}\cdot Y-\sum _{s=1,2}P_{s}\cdot Y_{s}$, which through the first-order condition lead us to attain the inverse demand for $Y_{s}$, $s=\{1,2\}$:⁴

$$\begin{aligned} \frac{P_{s}}{P_{Y}}=\chi _{s}\left( \frac{Y}{Y_{s}}\right) ^{\frac{1}{\varepsilon }}\Leftrightarrow Y_{s}=\left( \frac{P_{s}}{P_{Y}\cdot \chi _{s}}\right) ^{-\varepsilon }Y. \end{aligned}$$

(4)

Therefore, we can attain the relative demand for output of country 2, 

$$\begin{aligned} \frac{Y_{2}}{Y_{1}}=\left( \frac{\chi _{2}}{\chi _{1}}\frac{P_{1}}{P_{2}}\right) ^{\epsilon }, \end{aligned}$$

(5)

Substituting (4) in (3) we achieve $P_{Y}=\left[ \sum _{s=1,2}\chi _{s}^{\varepsilon }\cdot P_{s}^{1-\varepsilon }\right] ^{\frac{1}{1-\varepsilon }}$, where $P_{1}$ and $P_{2}$ are the output prices of, respectively, the nations 1 and 2. Moreover, the right-hand side of the equation give us the unit cost of production. Bearing in mind (4) we reach $P_{s}\cdot Y_{s}=P_{Y}\cdot Y^{\frac{1}{\varepsilon }}+\chi _{s}\cdot Y_{s}^{\frac{\varepsilon -1}{\varepsilon }}$ and $P_{Y}\cdot Y=\sum _{s=1,2}P_{s}\cdot Y_{s}$ when all countries are summed up.

2.2.2 Country economy

Following a production function characterized by constant returns to scale $Y_{s}=\exp \left( \int _{0}^{1}\ln Y_{v_{s}}dv_{s}\right)$, i.e., indication that $Y_{s}$ is a continuum of goods production generated by tasks $Y_{v_{i}}$ indexed by $v_{1}\in \left[ 0,1\right]$, for country 1,  and $v_{2}\in \left[ 0,1\right]$, for country 2, the output $Y_{s}$ of each country’s $s=\{1,2\}$ production is carried out under conditions of perfect competition. The producer of $Y_{s}$ seeks to maximize the profits defined by ${{\Pi _\text {s}}}=P_{s}\cdot Y_{s}\text { - }\int _{0}^{1}P_{v_{s}}\cdot Y_{v_{s}}dv_{s}$, under the constraint established by the mathematical expression of the production function Y. Thus, given the assumption of perfect competition, through the first-order condition, we have $Y_{v_{s}}=\frac{P_{s}\cdot Y_{s}}{P_{v_{s}}}$. Hence, bearing in mind that $P_{v_{s}}\cdot Y_{v_{s}}=P_{s}\cdot Y_{s}$ is a constant, which by substituting in the expression of the profits and in the production function leads to, respectively:

$$\begin{aligned} {{\Pi _\text {s}}}& = {} P_{s}\cdot Y_{s}-\int _{0}^{1}P_{s}\cdot Y_{s}dv_{s}=0, \end{aligned}$$

(6)

$$\begin{aligned} Y_{s}& = {} \exp \left( \int _{0}^{1}\ln \frac{P_{s}\cdot Y_{s}}{P_{v_{s}}}dv_{s}\right) \Leftrightarrow P_{s}=\exp \left( \int _{0}^{1}\ln P_{v_{s}}dv_{s}\right) . \end{aligned}$$

(7)

2.2.3 Country tasks

The task producer in each country $s=\{1,2\}$ is required to decide whether to produce tasks using unskilled labor, L, or skilled labor, H.⁵ This decision involves selecting from the following Cobb–Douglas production functions:⁶

$$\begin{aligned} Y_{v_{s}}^{L_{s}}(t)& = {} \left[ \int _{0}^{J_{s}}\left( {\widetilde{x}}_{s}^{L}(k,j,t)\right) ^{1-\alpha }dj\right] \left[ {\widetilde{L}}_{v_{s}}(t)\right] ^{\alpha }, \end{aligned}$$

(8)

$$\begin{aligned} Y_{v_{s}}^{H_{s}}(t)& = {} \left[ \int _{0}^{J_{s}}\left( {\widetilde{x}}_{s}^{H}(k,j,t)\right) ^{1-\alpha }dj\right] \left[ {\widetilde{H}}_{v_{s}}(t)\right] ^{\alpha }. \end{aligned}$$

(9)

where ${\widetilde{x}}_{v_{s}}^{L}(k,j,t)=q^{k(j,t)}{{\cdot }}x_{v_{s}}^{L_{s}}(k,j,t)$, ${\widetilde{x}}_{v_{s}}^{H}(k,j,t)=q^{k(j,t)}{{\cdot }}x_{v_{s}}^{H_{s}}(k,j,t)$, ${\widetilde{L}}_{v_{s}}(t)=\left( 1-v_{s}(t)\right) \cdot l\cdot L_{v_{s}}$, and ${\widetilde{H}}_{v_{s}}(t)=v_{s}(t)\cdot h_{s}\cdot H_{v_{s}}$. The production of each task requires the use of labor, L or H, which takes into account an income share $\alpha$ and, the use of intermediate goods, with an income of $1-\alpha$. Thus, on the first term on the right-hand side, $x_{v_{s}}^{L}(k,j,t)$ and $x_{v_{s}}^{H}(k,j,t)$ characterize the amount of intermediate goods required for task $v_{s}$ if it is designed to be carried out by L or by H, according to the same order. Moreover, each intermediate good $j\in \left[ 0,J_{s}\right]$ used in $v_{s}$ is adjusted according to quality being, $q>1$ the constant quality upgrade and k is the top-quality rung at t. In the same vein, the second term on the right-hand side, reflects, to used to produce each task $v_{s}$, the amount of unskilled labor, $L_{v_{s}}$ and skilled labor, $H_{v_{s}}$, and, also, two forms of adjustment factors to address productivity variations. Parameters l and $h_{s}$ reflect the absolute productivity of skilled over unskilled labor, with the condition $h_{s}>l=1$ (i.e., henceforth, we normalize $l=1$); the value of $h_{s}$ decreases as it approaches 1, correlating with higher levels of corruption. The second type, influenced by perspectives like Acemoglu and Zilibotti (2001) and Afonso (2012), gauges the relative productivity advantage of each labor type, denoted by the terms $(1-v_{i})$ and $v_{i}$, which implies that H is more productive in tasks with larger $v_{i}$, and vice-versa. Both choices, whether tasks employ unskilled or skilled labor, lead to distinct maximization problems. Although similar, we will illustrate for the case when the producer opts to produce with $i=\{L,H\}$:

$$\begin{aligned} \max _{x_{v_{s}}^{i_{s}}(k,j,t),i_{v_{s}}}{{\Pi _{v_{s}}}}(t)&= {} P_{v_{s}}^{i_{s}}(t) {\cdot }Y_{v_{s}}^{i_{s}}(t)-\int _{0}^{J_{s}}p(k,j,t){{\cdot }}x_{v_{s}}^{i_{s}}(k,j,t) {\cdot }dj\text { - }w_{i_{s}}(t)\cdot i_{v_{s}},\,i_{s}\\{} & {} =\{L_{s},H_{s}\},\,s=\{1,2\}, \end{aligned}$$

bearing in mind (8) and where $P_{v_{s}}^{i_{s}}(t)$ is the price of task $v_{s}$ produced by labor type i in country s at time t, p(k, j, t) denotes the price paid for the intermediate good j with quality k at time t by a producer of a task $v_{s}$ that in country s uses labor type i, and $w_{i_{s}}(t)$ is, as already stated, the price of each unit of labor type i in country s at time t—these prices are determined for the perfectly competitive task producers. The first-order conditions concerning intermediate goods enable us to derive the following:

$$\begin{aligned} x_{v_{s}}^{L_{s}}(k,j,t)& = {} \left[ \frac{P_{v_{i}}^{L_{s}}(t)\cdot (1-\alpha )}{p(k,j,t)}\right] ^{\frac{1}{\alpha }}\cdot q^{k(j,t)\frac{1-\alpha }{\alpha }}\cdot {\widetilde{L}}_{v_{s}}(t); \end{aligned}$$

(10)

$$\begin{aligned} x_{v_{s}}^{H_{s}}(k,j,t)& = {} \left[ \frac{P_{v_{i}}^{H_{s}}(t)\cdot (1-\alpha )}{p(k,j,t)}\right] ^{\frac{1}{\alpha }}\cdot q^{k(j,t)\frac{1-\alpha }{\alpha }}\cdot {\widetilde{H}}_{v_{s}}(t). \end{aligned}$$

(11)

Replacing (10) and (11) in the corresponding production functions (8) and (9), we have that:

$$\begin{aligned} Y_{v_{s}}^{L_{s}}(t)& = {} \left[ \frac{P_{v_{s}}^{L_{s}}(t)\cdot (1-\alpha )}{p(k,j,t)}\right] ^{\frac{1-\alpha }{\alpha }}\cdot Q_{s}(t)\cdot {\widetilde{L}}_{v_{s}}(t); \end{aligned}$$

(12)

$$\begin{aligned} Y_{v_{s}}^{H_{s}}(t)& = {} \left[ \frac{P_{v_{s}}^{H_{s}}(t)\cdot (1-\alpha )}{p(k,j,t)}\right] ^{\frac{1-\alpha }{\alpha }}\cdot Q_{s}(t)\cdot {\widetilde{H}}_{v_{s}}(t), \end{aligned}$$

(13)

where $Q_{s}\equiv \int _{0}^{J_{s}}q^{k(j,t)\frac{1-\alpha }{\alpha }}dj$ represents the quality level of intermediate goods utilized in country s, a variable that will be endogenously determined in Sect. 3, giving rise to the dynamic effects of the model, as will be demonstrated.

2.2.4 Country wages and threshold task

The first-order conditions with respect to labor units lead to the following:

$$\begin{aligned} w_{L_{s}}(t)& = {} \frac{\alpha \cdot P_{v_{i}}^{L_{s}}(t)\cdot Y_{v_{i}}^{L_{s}}(t)}{L_{v_{s}}}=\left[ P_{v_{s}}^{L_{s}}(t)\right] ^{\frac{1}{\alpha }}\cdot \left[ \frac{1-\alpha }{p(k,j,t)}\right] ^{\frac{1-\alpha }{\alpha }}\cdot Q_{s}(t)\cdot \left( 1-v_{s}(t)\right) ; \end{aligned}$$

(14)

$$\begin{aligned} w_{H_{s}}(t)& = {} \frac{\alpha \cdot P_{v_{i}}^{H_{s}}(t)\cdot Y_{v_{i}}^{H_{s}}(t)}{H_{v_{s}}}=\left[ P_{v_{s}}^{H_{s}}(t)\right] ^{\frac{1}{\alpha }} \cdot \left[ \frac{1-\alpha }{p(k,j,t)}\right] ^{\frac{1-\alpha }{\alpha }}\cdot Q_{s}(t)\cdot v_{s}(t)\cdot h_{s}. \end{aligned}$$

(15)

In equilibrium, there is a threshold task that ensures that each type of labor gets the same wage regardless of the task it is used for. To achieve this, we can define the following price indexes as constants:

$$\begin{aligned} \left[ P_{s}^{L_{s}}(t)\right] ^{\frac{1}{\alpha }}=\left[ P_{v_{s}}^{L_{s}}(t)\right] ^{\frac{1}{\alpha }}\cdot \left( 1-v_{s}(t)\right) \,\,\text{and}\,\,\left[ P_{s}^{H_{s}}(t)\right] ^{\frac{1}{\alpha }} =\left[ P_{v_{s}}^{H_{s}}(t)\right] ^{\frac{1}{\alpha }}\cdot v_{s}(t). \end{aligned}$$

(16)

As shown in Appendix 1.1, this implies that (1) tasks with a very low (high) $v_{s}$ have a lower price if produced by $L_{s}$ ($H_{s}$) rather than $H_{s}$ ($L_{s}$), such that perfectly competitive producers use $L_{s}$ ($H_{s}$) to avoid being driven out of the market, (2) there is a threshold task $v_{s}$ where prices are equal and is given by the following expression:

$$\begin{aligned} {\overline{v}}_{s}=\left[ 1+\left( h_{s}\frac{H_{s}}{L_{s}}\right) ^{\frac{1}{2}}\right] ^{-1}, \end{aligned}$$

(17)

where a small threshold task ${\overline{v}}_{s}$, implies that the number of tasks produced with $H_{s}$ is large when both the relative skilled-labor supply, $\frac{H_{s}}{L_{s}}$, and the absolute advantage of skilled labor, which is, negatively influenced by an increase in the corruption level, $h_{s}$, are also larger; i.e., in each country, tasks will be allocated to factors depending on their comparative advantage. In particular, an increment in $h_{s}$, caused for a smaller corruption level, raises the labor’s $H_{s}$ “comparative advantage” over $L_{s}$ and induces a decrease in ${\overline{v}}_{s}$.

2.3 Intermediate-goods: technology, output and prices

In order to develop a new design for the intermediate good j with the aim of achieving improved quality, referred to as k, it is vital to invest in R&D, with the costs being recovered if there is a future profit. To ensure that the new upgrade developed is protected, a legal right is conferred on the leader firm’s monopoly—intellectual property rights (IPRs). Simultaneously, this technological knowledge is accessible at minimal cost to other firms. That said, the firms that hold this patent for the new top quality k, developed for the intermediate good j at t provide it for all tasks, $v_{s}$, in country s.⁷ If we take into account that each unit of intermediate good $j\in \left[ 0,J_{s}\right]$ requires one unit of final output Y since its price is 1 to 1 and bearing in mind the cost of business corruption $z_{s}$ that is perceived as a monetary bribe given to state officials, $\tau _{s},$ in exchange for preferential treatment,⁸ on the production of intermediate goods, the producer of j gets profits $\pi _{s}(k,j,t)=\left[ p(k,j,t)-\left( 1+z_{s}\right) \right] \cdot x_{s}(k,j,t)$ where $x_{s}(k,j,t)=\int _{0}^{{\overline{v}}_{s}}x_{v_{s}}^{R_{s}}(k,j,t)\cdot dv_{s}+\int _{{\overline{v}}_{s}}^{1}x_{v_{s}}^{L_{s}}(k,j,t)\cdot dv_{s}$ is the demand for intermediate good j from all the producers of tasks $v_{s}$ that use such input, regardless of the labor—unskilled or skilled—used in the tasks.

Assuming that the monopolist charges the same price, p(k, j, t), for all these firms, we can find the optimal price by replacing $x_{s}(k,j,t)$ with the demand of the producer of a single task $v_{s}$ , i.e., either by $x_{v_{s}}^{L_{s}}(k,j,t)$ or by $x_{v_{s}}^{H_{s}}(k,j,t)$ and then maximizing with respect to p(k, j, t). This can be seen by $\pi _{s}(k,j,t)=\int _{0}^{1}\pi _{v_{s}}(k,j,t)\cdot dv_{s}=\int _{0}^{{\overline{v}}_{i}}\pi _{v_{s}}^{L_{s}}(k,j,t)\cdot dv_{s}+\int _{{\overline{v}}_{s}}^{1}\pi _{v_{s}}^{H_{s}}(k,j,t)\cdot dv_{s}$, where $\pi _{v_{i}}^{L_{s}}(k,j,t)$ and $\pi _{v_{s}}^{H_{s}}(k,j,t)$ denotes the profits of the producer of j for selling this intermediate good to the producer of task $v_{s}$. Therefore, we can find p(k, j, t) by solving the following maximization problem $\max _{p(k,j,t)}\left[ p(k,j,t)-\left( 1+z_{s}\right) \right] \cdot x_{v_{i}}^{s}(k,j,t)$, where $x_{v_{i}}^{s}(k,j,t)$ can be done by (10) or (11). From the first-order condition $\frac{\partial \pi _{s}(k,j,t)}{\partial p(k,j,t)}$, we attain the price of intermediate goods, which is country-specific since it depends, beyond the constant quality upgrade, q, on the country’s corruption degree measured by $z_{s}$,

$$\begin{aligned} p(k,j,t)\equiv p=\frac{1+z_{s}}{1-\alpha }=q\left( 1+z_{s}\right) , \end{aligned}$$

(18)

assuming that the limit pricing strategy is binding.⁹

Taking also into account $p=q\left( 1+z_{s}\right)$, (16), (10), and (11), the demand for the intermediate good j used in country s together with unskilled labor is,

$$\begin{aligned} x_{s}^{L_{s}}(k,j,t)=\int _{0}^{{\overline{v}}_{s}}x_{v_{s}}^{L_{s}}(k,j,t)\cdot dv_{s}=\left[ \frac{P_{s}^{L_{s}}(t)\cdot (1-\alpha )}{q\cdot \left( 1+z_{s}\right) }\right] ^{\frac{1}{\alpha }}\cdot q^{k(j,t)\frac{1-\alpha }{\alpha }}\cdot L_{s}, \end{aligned}$$

(19)

and together with skilled labor is,

$$\begin{aligned} x_{s}^{H_{s}}(k,j,t)=\int _{{\overline{v}}_{s}}^{1}x_{v_{s}}^{H_{s}}(k,j,t)\cdot dv_{s}=\left[ \frac{P_{s}^{H_{s}}(t)\cdot (1-\alpha )}{q\cdot \left( 1+z_{s}\right) }\right] ^{\frac{1}{\alpha }}\cdot q^{k(j,t)\frac{1-\alpha }{\alpha }}\cdot h_{s}\cdot H_{s}. \end{aligned}$$

(20)

Therefore, the total demand for intermediate good j used in country s is $x_{s}(k,j,t)=x_{s}^{L_{s}}(k,j,t)+x_{s}^{H_{s}}(k,j,t)$, which will be calculated in the next subsection.

2.4 Relative prices, output, and wages between sectors

Having established the threshold task and the price of intermediate goods for each country s, we can now proceed to ascertain, for a given set of factor/input levels: (1) the price indices of each country, $P_{1}$ and $P_{2}$; (2) the production output that each type of labor generates in each country, $Y_{s}^{L_{s}}$ and $Y_{s}^{H_{s}}$; (3) the absolute output of each country, $Y_{1}$ and $Y_{2}$, and in relative terms, $\frac{Y_{2}}{Y_{1}}$; (4) the relative price of countries, $\frac{P_{2}}{P_{1}}$; (5) the relative value of the output produced by country 2, $\frac{P_{2}Y_{2}}{P_{1}Y_{1}}$; (4) the wages and wage differences between types of labor in various contexts. This analysis is conducted without adjusting inter-country technological knowledge, allowing for static results that exclude impacts on technological-knowledge progress

We can begin by establishing absolute values for price indexes. To derive this, we utilize the definition of the output price implied by the profit maximization problem of the output producer in country s, $Y_{s}$, which means $P_{s}=\exp \left( \int _{0}^{1}\ln P_{v_{s}}dv_{s}\right)$—see (7). We make also the use of the result that the value of each task, $P_{v_{s}}Y_{v_{s}}$, is a constant for all $v_{s}$, and we also use (16) and (17) to have $P_{s}^{H_{s}}=\left( \frac{{\overline{v}}_{s}}{1-{\overline{v}}_{s}}\right) ^{\alpha }P_{s}^{L_{s}}$. From this analysis, we derive the subsequent expressions—see Appendix 1.2:

$$\begin{aligned} P_{s}^{L_{s}}& = {} P_{s}\cdot \exp \left( -\alpha \right) \cdot {\overline{v}}_{s}^{-\alpha }\,and\ :P_{s}^{H_{s}}=P_{s}\cdot \exp \left( -\alpha \right) \cdot \left( 1-{\overline{v}}_{s}\right) ^{-\alpha }\Rightarrow \nonumber \\ \frac{P_{s}^{H_{s}}}{P_{s}^{L_{s}}}& = {} \left( h_{s}\frac{H_{s}}{L_{s}}\right) ^{-\frac{\alpha }{2}}, \end{aligned}$$

(21)

where $P_{1}$ and $P_{2}$ are also determined in Appendix 1.2. An increase in the labor level of country s has market size on the demand for intermediate goods through the term ${\overline{v}}_{s}$. However, by affecting ${\overline{v}}_{s}$, the same effect has, in addition, a price effect since increases the supply of output of country s that induces a decrease in the absolute price of this output, $P_{s}$, and, therefore, a decrease in the price index of tasks produced in this country, $P_{s}^{L_{s}}$ or $P_{s}^{H_{s}}$. This decreases the output of each task, which decreases demand for intermediate goods in this country—see (19) and (20). Moreover, the relative price index of tasks, $\frac{P_{s}^{H_{s}}}{P_{s}^{L_{s}}}$, performed with skilled labor is higher the lower the effective skilled-labor level in the country; i.e., it is higher the higher the corruption since it decreases $h_{s}$.

From the profit maximization problem of the producer of Y, and given that in each country, a portion of the tasks is executed by unskilled labor while another part is undertaken by skilled labor, the overall output is as follows: $P_{s}Y_{s}=\int _{0}^{1}P_{v_{s}}Y_{v_{s}}dv_{s}$ $=\int _{0}^{{\overline{v}}_{s}}P_{v_{s}}^{L_{s}}Y_{v_{s}}^{L_{s}}dv_{s} +\int _{{\overline{v}}_{s}}^{1}P_{v_{s}}^{H_{s}}Y_{v_{s}}^{H_{s}}dv_{s}$ $=P_{s}Y_{s}^{L_{s}}+P_{s}Y_{s}^{H_{s}}$, where $P_{s}Y_{s}^{L_{s}}=\int _{0}^{{\overline{v}}_{s}}P_{v_{s}}^{L_{s}}Y_{v_{s}}^{L_{s}}dv_{s}$ and $P_{s}Y_{s}^{H_{s}}=\int _{{\overline{v}}_{s}}^{1}P_{v_{s}}^{H_{s}}Y_{v_{s}}^{H_{s}}dv_{s}$. Based on these definitions and taking into account (12), (13), (16), and (21), the outputs in country s performed by unskilled labor, $Y_{s}^{L_{s}}$, and skilled labor, $Y_{s}^{H_{s}}$, are:

$$\begin{aligned} Y_{s}^{L_{s}}=\exp \left( -1\right) \cdot \left[ \frac{P_{s}\cdot (1-\alpha )}{q\cdot \left( 1+z_{s}\right) }\right] ^{\frac{1-\alpha }{\alpha }}\cdot Q_{s}(t)\cdot \frac{L_{s}}{{\overline{v}}_{s}}, \end{aligned}$$

(22)

,

$$\begin{aligned} Y_{s}^{H_{s}}=\exp \left( -1\right) \cdot \left[ \frac{P_{s}\cdot (1-\alpha )}{q\cdot \left( 1+z_{s}\right) }\right] ^{\frac{1-\alpha }{\alpha }}\cdot Q_{s}(t)\cdot \frac{h_{s}\cdot H_{s}}{1-{\overline{v}}_{s}}. \end{aligned}$$

(23)

Therefore, from $P_{s}Y_{s}=P_{s}Y_{s}^{L_{s}}+P_{s}Y_{s}^{H_{s}}$, the output of each sector is:

$$\begin{aligned} Y_{s}=\exp \left( -1\right) \cdot \left[ \frac{P_{s}\cdot (1-\alpha )}{q\cdot \left( 1+z_{s}\right) }\right] ^{\frac{1-\alpha }{\alpha }}\cdot Q_{s}(t)\cdot M_{s}, \end{aligned}$$

(24)

where, bearing in mind (17), $M_{s}=\frac{L_{s}}{{\overline{v}}_{s}}+\frac{hH_{s}}{1-{\overline{v}}_{s}}=$ $\left[ \left( L_{s}\right) ^{\frac{1}{2}}+\left( h_{s}\cdot H_{s}\right) ^{\frac{1}{2}}\right] ^{2}$ evaluates the market size. The output performed by each type of labor in each country, $Y_{s}^{L_{s}}$ and $Y_{s}^{H_{s}}$, and the output of each country, $Y_{s}$, are affected by the (1) current technological knowledge in the country, $Q_{s}(t)$, measured by the aggregate quality index; hence, the country’s technological-knowledge progress will be responsible for economic growth; (2) absolute productivity of skilled labor, $h_{s}$, which depends on the corruption levels; (3) labor levels. An increase in labor levels has the following impacts: (1) scale effects, which increase directly the amount of output produced and thereby the relative demand for intermediate goods through terms $L_{s}$ or $h_{s}H_{s}$ in (22) and (23); (2) price effect, which decreases indirectly the demand for intermediate goods, by increasing the relative advantage. As equilibrium implies a rise in the share of tasks produced by the expanded type of labor, this leads to a reduction in the absolute level of the price index in the corresponding sector, as influenced by the terms $\left( {\overline{v}}_{i}\right) ^{-1}$ and $(1-{\overline{v}}_{i})^{-1}$ in (22) and (23) that affects accordingly the absolute price level of all tasks produced in each country. This results in a reduction of the output for each task, leading to a decrease in the demand for intermediate goods and profits within the country.

In turn, the inter-country output ratio,

$$\begin{aligned} \frac{Y_{2}}{Y_{1}}=\left( \frac{\chi _{2}}{\chi _{1}}\right) ^{\frac{\epsilon \left( 1-\alpha \right) }{\alpha (\epsilon -1)+1}}\left( \frac{1+z_{2}}{1+z_{1}}\right) ^ {-\frac{\left( 1-\alpha \right) \epsilon }{\alpha (\epsilon -1)+1}}\left( \frac{M_{2}}{M_{1}}\frac{Q_{2}}{Q_{1}}\right) ^{\frac{\epsilon \cdot \alpha }{\alpha (\epsilon -1)+1}}, \end{aligned}$$

(25)

demonstrates that the country experiencing the most pronounced progress in technological knowledge undergoes greater growth than the other country. Additionally, the relative production level between countries essentially relies on the bias of technological knowledge toward a particular country and on the country’s factor allocations.

Moreover, from (5), $\frac{P_{2}}{P_{1}}=\frac{\chi _{2}}{\chi _{1}}$ $\left( \frac{Y_{2}}{Y_{1}}\right) ^{-\frac{1}{\epsilon }}$ and then considering (24), the relative price of country 2 is:

$$\begin{aligned} \frac{P_{2}}{P_{1}}=\left( \frac{\chi _{2}}{\chi _{1}}\right) ^{\frac{\epsilon \cdot \alpha }{\alpha (\epsilon -1)+1}} \left( \frac{1+z_{2}}{1+z_{1}}\right) ^{\frac{\left( 1-\alpha \right) }{\alpha (\epsilon -1)+1}} \left( \frac{M_{2}}{M_{1}}\frac{Q_{2}}{Q_{1}}\right) ^{-\frac{\alpha }{\alpha (\epsilon -1)+1}}, \end{aligned}$$

(26)

The intuition behind (26) can be grasped by taking into account that an increase in the technological-knowledge bias, $\frac{Q_{2}}{Q_{1}}$, or the relative size of the market of country 2, $\frac{M_{2}}{M_{1}}$, which can be stimulated by a skilled-labor level variation caused by more corruption since it contributes positively to $h_{s}$. Therefore, these two channels amplify the provision of relative output via (25), resulting in a decline in the relative prices through (5). On the one hand, an increase in the relative relevance of country 2 in the production of the aggregate final good, $\frac{\chi _{2}}{\chi _{1}}$, increases the relative demand for output in this country that leads to an increase in relative prices. On the other hand, an increase in the relative corruption of country 2, $\frac{1+z_{2}}{1+z1}$, causes a decrease in the inter-country output ratio that increases the relative prices of country 2. Hence, equation (26) shows that if either the technological-knowledge is highly country 2 biased, $\frac{Q_{2}}{Q_{1}}$, or if there is a large relative supply of labor in country 2, $\frac{M_{2}}{M_{1}}$, the fraction of final goods produced in country 2 is large—see (25)—, which implies a low relative price of tasks produced in this country. In this case, the demand for the country’s 2 intermediate goods is low, discouraging R&D activities aimed to improve their quality, as shown below. Thus, labor structure influences the trajectory of R&D via the price mechanism, a concept explored in several works by Acemoglu (e.g., 2002), although typically overshadowed by the market-size channel. In our case, this latter channel is removed and, consequently, becomes absent.

Lastly, in this subsection, the matter of wages and the potential wage differentials that may arise still requires consideration. Bearing in mind (16), the wages in (14) and (15) can be rewritten in the form:

$$\begin{aligned} w_{L_{s}}(t)& = {} \left[ P_{s}^{L_{s}}(t)\right] ^{\frac{1}{\alpha }}\cdot \left( \frac{1-\alpha }{q\cdot \left( 1+z_{s}\right) }\right) ^{\frac{1-\alpha }{\alpha }}\cdot Q_{s}(t); \end{aligned}$$

(27)

$$\begin{aligned} w_{H_{s}}(t)& = {} \left[ P_{s}^{H_{s}}(t)\right] ^{\frac{1}{\alpha }}\cdot \left( \frac{1-\alpha }{q\cdot \left( 1+z_{s}\right) }\right) ^{\frac{1-\alpha }{\alpha }}\cdot h\cdot Q_{s}(t). \end{aligned}$$

(28)

In view of our proposed setup, more corruption, materialized in the increase of $z_{s}$, reduces the wages of all workers in $s=1$ and $s=2$ for a given technological-knowledge level.

From (27), (28), and (21), we can obtain the wage differentials between types of labor in each country s,

$$\begin{aligned} \frac{w_{H_{s}}}{w_{L_{s}}}=\left( \frac{P_{s}^{H_{s}}}{P_{s}^{L_{s}}}\right) ^{\frac{1}{\alpha }}\cdot h_{s}=\left( h_{s}\frac{L_{s}}{H_{s}}\right) ^{\frac{1}{2}}. \end{aligned}$$

(29)

The inter-country unskilled-premium is given by

$$\begin{aligned} \frac{w_{L_{2}}}{w_{L_{1}}}& = {} \left( \frac{P_{2}^{L_{2}}}{P_{1}^{L_{1}}}\right) ^{\frac{1}{\alpha }}\cdot \left( \frac{1-z_{1}}{1-z_{2}}\right) ^{\frac{1-\alpha }{\alpha }}\cdot \frac{Q_{2}}{Q_{1}}= \left( \frac{\chi _{2}}{\chi _{1}}\right) ^{\frac{\epsilon }{\alpha (\epsilon -1)+1}}\nonumber \\{} & {} \left( \frac{L_{1}}{L_{2}}\right) ^{\frac{1}{2}}\left( \frac{M_{2}}{M_{1}}\right) ^{\frac{\alpha (\epsilon -1)-1}{2\cdot [\alpha (\epsilon -1)+1]}} \left( \frac{1+z_{2}}{1+z_{1}}\right) ^{1-\frac{\epsilon }{\alpha (\epsilon -1)+1}} \left( \frac{Q_{2}}{Q_{1}}\right) ^{\frac{\alpha \left( \epsilon \cdot -1\right) }{\alpha (\epsilon -1)+1\cdot }}, \end{aligned}$$

(30)

the inter-country skilled-premium is defined as

$$\begin{aligned} \frac{w_{H_{2}}}{w_{H_{1}}}& = {} \left( \frac{P_{2}^{H_{2}}}{P_{1}^{H_{1}}}\right) ^ {\frac{1}{\alpha }}\cdot \left( \frac{1-z_{1}}{1-z_{2}}\right) ^{\frac{1-\alpha }{\alpha }} \cdot \frac{Q_{2}}{Q_{1}}=\left( \frac{\chi _{2}}{\chi _{1}}\right) ^{\frac{\epsilon }{\alpha (\epsilon -1)+1}}\nonumber \\{} & {} \left( \frac{H_{1}}{H_{2}}\right) ^{\frac{1}{2}}\left( \frac{M_{2}}{M_{1}}\right) ^ {\frac{\alpha (\epsilon -1)-1}{2\cdot [\alpha (\epsilon -1)+1]}}\left( \frac{1+z_{2}}{1+z_{1}}\right) ^ {1-\frac{\epsilon }{\alpha (\epsilon -1)+1}}\left( \frac{Q_{2}}{Q_{1}}\right) ^{\frac{\alpha \left( \epsilon \cdot -1\right) }{\alpha (\epsilon -1)+1\cdot }}, \end{aligned}$$

(31)

and then, the inter-country comparison of both premiums is specified as

$$\begin{aligned} \frac{\frac{w_{H_{2}}}{w_{H_{1}}}}{\frac{w_{L_{2}}}{w_{L_{1}}}}=\left( \frac{H_{1}}{H_{2}} \frac{L_{2}}{L_{1}}\right) ^{\frac{1}{2}}. \end{aligned}$$

(32)

Finally, from (19) and (20), and by accessing (21), we can determine, respectively, the total demand and profits for the intermediate good j:

$$x_{s} (k,j,t) = \exp ( - 1) \cdot \left[ {\frac{{P_{s} \cdot (1 - \alpha )}}{{q \cdot \left( {1 + z_{s} } \right)}}} \right]^{{\frac{1}{\alpha }}} \cdot q^{{k(j,t)\left( {\frac{{1 - \alpha }}{\alpha }} \right)}} \cdot M_{s} ,{\text{ }}$$

(33)

$$\pi _{s} (k,j,t) = \left( {q - 1} \right)\left( {1 + z_{s} } \right) \cdot \exp ( - 1) \cdot \left[ {\frac{{P_{s} \cdot (1 - \alpha )}}{{q \cdot \left( {1 + z_{s} } \right)}}} \right]^{{\frac{1}{\alpha }}} \cdot q^{{k(j,t)\left( {\frac{{1 - \alpha }}{\alpha }} \right)}} \cdot M_{s} .{\text{ }}$$

(34)

And we can derive an expression for the intermediate goods produced for each sector s:

$$X_{s} (t) = \int_{0}^{{J_{s} }} {x_{s} } (k,j,t) \cdot dj = \exp ( - 1) \cdot \left[ {\frac{{P_{s} \cdot (1 - \alpha )}}{{q \cdot \left( {1 + z_{s} } \right)}}} \right]^{{\frac{1}{\alpha }}} \cdot Q_{s} (t) \cdot M_{s} ,$$

(35)

where the aggregate resources devoted to intermediate-goods production in sector s, $X_{s}$, is also expressible as a function of the currently given technological knowledge in sector s.

2.5 R&D activity: technology, patent value, and expenditures

As mentioned in Sect. 2.3, to ensure the protection of the leading firm’s monopoly in the creation of a new quality of intermediate goods, patents are granted to facilitate this process. To establish a monopoly, the firm responsible for designing this patent establishes limit price schemes, whereby the value of the leading-edge patent is dependent on the profits earned and on the duration of its monopoly. In its turn, the length of this duration hinges on the probability of a new innovation, which in turn has a crucial role to perform in the dynamics of R&D activities (e.g., Aghion and Howitt 1992; Grossman and Helpman 1991, ch. 12, and Barroand Sala-i-Martin 2004, ch. 7), disrupting the existing cutting-edge project. Hence, the instantaneous probability at time t in country $s=\{1,2\}$ of successful innovation in the next higher quality $\left[ k(j,t)+1\right]$ in j bearing in mind the current rung quality k can be defined by:

$$\begin{aligned} {\mathcal {I}}_{s}(k,j,t)=e_{s}(k,j,t)\cdot \beta q^{k(j,t)}\cdot \zeta ^{-1}q^{-\alpha ^{-1}k(j,t)}\cdot M_{s}^{-\xi }, \end{aligned}$$

(36)

where $e_{s}(k,j,t)$ characterizes the attribution of domestic final-good resources to R&D in j in $s=\{1,2\}$ (Afonso 2012); $\beta$ $q^{k(j,t)}$, $\beta >0$, indicates the learning impact of knowledge attained from previous successful domestic R&D (Grossman and Helpman 1991, ch. 12; Afonso 2012); $\zeta ^{-1}q^{-\alpha ^{-1}k(j,t)}$, $\zeta >0$, represents the cost of complexity (Afonso 2012);¹⁰$M_{s}^{-\xi }=\left[ \left( L_{s}\right) ^{\frac{1}{2}}+\left( h_{s}H_{s}\right) ^{\frac{1}{2}}\right] ^{-2\cdot \xi }$ with $\xi \ge 0$¹¹, stands for the negative effect of market size in the introduction of new intermediate goods. In other words, it reflects the difficulty of successfully bringing in new products and replacing existing ones, which partially depends on the size of the market under study in each country.

Our aim is not to determine who is driving R&D. Therefore, we consider that R&D is undertaken by entrants since we assume that the probability of innovation presented in (36) is similar for incumbents and entrants—see Appendix 1.3. The patent owned by the innovator, that developed the new quality of an intermediate good j from k to $k+1$, $q^{k\left( j\right) +1}$ is valid from time t to $\iota$, being the market value of the monopolistic firm or the patent value, that, in its turn, depends on sum of profits earned, $\pi _{s}(k,j,t)$, given by $V_{s}\left( k,j,t\right)$, within this period—see Appendix 1.4 for a detailed deduction of the patent value following Aghion and Howitt (1992), Barro and Sala-i-Martin (2004), and Gil et al. (2013).

$$\begin{aligned} V_{s}\left( k+1,j,t,\iota \right) =\int _{t}^{\iota }\pi _{s}(k+1,j,v)\cdot \exp \left[ -\int _{t}^{\omega } r_{s}\left( \omega \right) d\omega \right] dv. \end{aligned}$$

(37)

As, in equilibrium, the interest rate between t and $\iota$ is constant, the expected value of $V_{s}\left( k+1,j,t\right)$ is— see Appendix 1.5:

$$\begin{aligned} V_{s}\left( k+1,j,t\right)& = {} E\left[ V_{s}\left( k+1,j,t,\iota \right) \right] \nonumber \\& = {} \int _{t}^{\infty }V_{s}(k+1,j,t,\iota ) \cdot {\mathcal {I}}_{s}\left( k,j,\iota \right) \cdot \exp \left[ -{\mathcal {I}}_{s}\left( k,j,\iota \right) \cdot \left( \iota -t\right) \right] \cdot d\iota =\frac{\pi _{s}(k+1,j,t)}{r_{s}\left( t\right) +{\mathcal {I}}_{s}\left( k,j,t\right) }. \end{aligned}$$

(38)

We can identify the expression (38) as the no-arbitrage condition. Thus, the profit flow $\pi _{s}(k,j,t)$ can be equalized to the sum of anticipated earnings attained from a successful quality innovation k at time t, given by $V_{s}\left( k,j,t\right) \cdot r_{s}\left( t\right)$, and the anticipated capital loss, $V_{s}\left( k,j,t\right) \cdot {\mathcal {I}}_{s}\left( k,j,\tau \right) .$

Lastly, bearing in mind the instantaneous probability of successful innovation in the next higher quality, (36), we achieve that:

$$\begin{aligned} E_{s}(t)=\int _{0}^{J_{s}}e_{s}(k,j,t)={\mathcal {I}}_{s}(k,j,t)\cdot \frac{\zeta }{\beta }\cdot Q_{s}\cdot M_{s}^{\xi }, \end{aligned}$$

(39)

since, by definition, ${\mathcal {I}}_{s}(k,j,t)$ does not distinguish the various intermediate goods originating in the same country.

3 General equilibrium

Throughout this section we have characterized the general equilibrium, given that the country’s economic structure has been defined for particular states of technological knowledge embodied by the $Q_{1}$ and $Q_{2}$. To this end, we first deduce the equilibrium, of attaining higher quality indexes. Here, both households and firms act rationally, the conditions for free-entry into R&D are fulfilled and the markets clear is assured. Subsequently, we outline the overall resource constraints of the economy and illustrate that all variables depend on the evolutionary dynamics of technological-knowledge levels. Finally, our approach involves deriving the law of motion for each quality index, followed by an assessment of the steady-state growth of the model.

3.1 R&D equilibrium and law of motion of technological knowledge

According to the free-entry condition, the anticipated return generated by developing a new intermediate good j, ${\mathcal {I}}_{s}(k,j,t)\cdot E\left[ V_{s}\left( k+1,j,t,\tau \right) \right]$, where ${\mathcal {I}}_{s}(k,j,t)$ and $E\left[ V_{s}\left( k+1,j,t,\tau \right) \right]$ are given by, considering the same order, (36) and (38), must equalize the expenditure on R&D aimed at upgrading j, $e_{s}(k,j,t)$. Thus, we can attain the equilibrium probability function:

$$\begin{aligned} {\mathcal {I}}_{s}(k,j,t)={\mathcal {I}}_{s}(j,t)=\frac{\beta }{\zeta }\cdot \left( \frac{q-1}{q}\right) \cdot \exp \left( -1\right) \cdot \left( \frac{1-\alpha }{1+z_{s}}\right) ^{\frac{1}{\alpha }}\cdot P_{s}^{\frac{1}{\alpha }}(t)\cdot M_{s}^{1-\xi }-r_{s}(t). \end{aligned}$$

(40)

Therefore, ${\mathcal {I}}_{s}(k,j,t)$=${\mathcal {I}}_{s}(t)$ since${\mathcal {I}}_{s}(k,j,t)$ remains unaffected by the top-quality rung k and j.

Furthermore, if a new quality of intermediate good j would be invented, the variation rate of the country’s quality index would be given by, $\varDelta Q_{s}=Q_{s}\left( k+1,t\right) -Q_{s}(k,t)=\int _{0}^{J_{s}}q^{\left[ k(j,t)+1\right] \left( \frac{1-\alpha }{\alpha }\right) }-\int _{0}^{J_{s}}q^{k(j,t)\left( \frac{1-\alpha }{\alpha }\right) }$ and thus $\frac{\varDelta Q_{s}}{Q_{s}}=\left[ q^{\left( \frac{1-\alpha }{\alpha }\right) }-1\right]$. Considering that the likelihood of this taking place per unit of time is defined by ${\mathcal {I}}_{s}(t)$, then:

$$\begin{aligned} \frac{{\dot{Q}}_{s}(t)}{Q_{s}(t)}={\mathcal {I}}_{s}(t)\cdot \left[ q^{\left( \frac{1-\alpha }{\alpha }\right) }-1\right] . \end{aligned}$$

(41)

3.2 Transitional dynamics and steady-state results

Throughout this subsection, we can prove that, under equilibrium conditions, the aggregate flow constraint of households can be given by $Y=C+X+E$—see Appendix 1.6—since that the total spending on the final goods are expressed by $Y=\sum _{s=1,2}P_{s}\cdot Y_{s}$, that the total expenditure on intermediate goods and R&D projects are the sums of the aggregates in both nations and that assets are the current value of the patents held by all producers of intermediate goods. Thus, since Y, X and E depend on the technological-knowledge levels$Q_{1}$ and $Q_{2}$, the aggregate flow restriction also entails that consumption C depends on quality indexes. Hence, this suggests that the evolution of all the key variables outside of the steady state, are conditioned by the differential equation that determines the technological-knowledge bias toward country 2, i.e., $\frac{\dot{{\mathcal {Q}}}(t)}{{\mathcal {Q}}(t)}=\frac{\dot{Q_{2}}(t)}{Q_{2}(t)}-\frac{\dot{Q_{1}}(t)}{Q_{1}(t)}$. Thus, using (41), the technological-knowledge bias is expressed as:

$$\begin{aligned} \frac{\dot{{\mathcal {Q}}}(t)}{{\mathcal {Q}}(t)}& = {} \left[ q^{\left( \frac{1-\alpha }{\alpha }\right) }-1\right] \cdot \left\{ \frac{\beta }{\zeta }\cdot \frac{q-1}{q}\cdot \exp \left( -1\right) \cdot (1-\alpha )^{\frac{1}{\alpha }}\cdot \left[ \left( \frac{P_{2}}{1+z_{2}}\right) ^{\frac{1}{\alpha }}\cdot \right. \right. \nonumber \\{} & {} \left. \left. M_{2}^{1-\xi }-\left( \frac{P_{1}}{1+z_{1}}\right) ^{\frac{1}{\alpha }}\cdot M_{1}^{1-\xi }\right] -r_{2}(t)+r_{1}(t)\right\} . \end{aligned}$$

(42)

At the conclusion of transitional dynamics, the economy converges to a unique and stable steady state, with all relevant macroeconomic variables growing at the same constant rate—the steady-state worldwide growth rate, $g^{*}$, is:

$$\begin{aligned} g^{*}\equiv \left( \frac{\dot{Q_{1}}}{Q_{1}}\right) ^{*}=\left( \frac{\dot{Q_{2}}}{Q_{2}}\right) ^{*}= \left( \frac{{\dot{Y}}}{Y}\right) ^{*}=\left( \frac{{\dot{X}}}{X}\right) ^{*}=\left( \frac{{\dot{E}}}{E}\right) ^{*} =\left( \frac{{\dot{C}}}{C}\right) ^{*}=\frac{r^{*}-\rho }{\theta }. \end{aligned}$$

(43)

Bearing in mind (41) and (43), we first determine the stable and unique steady-state worldwide interest rate since $r^{*}=r_{1}^{*}=r_{2}^{*}$ and then we insert that expression in 43 to determine the steady-state worldwide economic growth rate:

$$\begin{aligned} g^{*}=\frac{\frac{\beta }{\zeta }\cdot \left( \frac{q-1}{q}\right) \cdot \exp \left( -1\right) \cdot \left( \frac{1-\alpha }{1+z_{s}}\right) ^{\frac{1}{\alpha }}\cdot P_{s}^{*^{\frac{1}{\alpha }}}(t)\cdot M_{s}^{1-\xi }-\rho }{\left( q^{\frac{1-\alpha }{\alpha }}-1\right) ^{-1}+\theta }\text {,} \end{aligned}$$

(44)

where $P_{1}^{*}$ and $P_{2}^{*}$ are determined in Appendix 1.7 where is also proved the uniqueness of $g^{*}$.

Proposition 1

From (44), the worldwide economic growth rate increases when the corruption levels decrease through the cost of corruption, $z_{s}$, and the absolute productivity of skilled workers, $h_{s}$. Nevertheless, the negative effect driven by corruption can be partially offset by $P_{s}$.

Proof

Results directly from (44), and (62) and (63). $\square$

Considering (42) and (26) the steady-state value of the technological-knowledge bias is unique and globally stable and given by the following expression,

$${\mathcal{Q}}^{*} = \left( {\frac{{\chi _{2} }}{{\chi _{1} }}} \right)^{\epsilon } \left( {\frac{{1 + z_{2} }}{{1 + z_{1} }}} \right)^{{ - \alpha }} \left( {\frac{{M_{2} }}{{M_{1} }}} \right)^{{\left( {1 - \xi } \right)\left[ {\alpha (\epsilon - 1) + 1} \right] - 1}} .$$

(45)

Proposition 2

The steady-state technological-knowledge bias toward country 2 depends negatively on the relative relevance of corruption in country 2,$\frac{1+z_{2}}{1+z_{1}}$, and positively or negatively, according to the values adopted by the parameters, on the relative labor levels in country 2, $\frac{M_{2}}{M_{1}}$, unless there is complete removal of scale effects; in this case, it becomes independent of the labor levels.

Proof

From (42), if $\left( \frac{P_{2}}{1+z_{2}}\right) ^{\frac{1}{\alpha }}\cdot M_{2}^{1-\xi }>\left( \frac{P_{1}}{1+z_{1}}\right) ^{\frac{1}{\alpha }}\cdot M_{1}^{1-\xi }$, producers of intermediate goods have more incentives to invest in country 2 than in country 1, causing a positive growth of technological-knowledge bias, $\frac{\dot{{\mathcal {Q}}}}{{\mathcal {Q}}}>0$. This implies a decrease in the price of output in country 2 and an increase in the price of output in country 1. This decreases relative incentives to invest in country 2, which, in turn, reduces the growth of technological-knowledge bias. The steady-state equilibrium is globally stable since a similar process would occur in the opposite scenario. The steady-state value is found by solving $\frac{\dot{{\mathcal {Q}}}}{{\mathcal {Q}}}=0$, which requires that $\left( \frac{1+z_{1}}{1+z_{2}}\right) ^{\frac{1}{\alpha }}\left( \frac{P_{2}}{P_{1}} \right) ^{\frac{1}{\alpha }}=\left( \frac{M_{1}}{M_{2}}\right) ^{1-\xi }$, to which, bearing in mind (26), there is a unique solution given by (45). $\square$

From (25) and (45), we have that the relative output of the country 2:

$$\begin{aligned} \left( \frac{Y_{2}}{Y_{1}}\right) ^{*}=\left( \frac{\chi _{2}}{\chi _{1}}\right) ^{\epsilon } \left( \frac{1+z_{2}}{1+z_{1}}\right) ^{\frac{-\left[ \epsilon -\epsilon \alpha +\epsilon \alpha ^{2}\right] }{\alpha (\epsilon -1)+1}}\left( \frac{M_{2}}{M_{1}}\right) ^{\left( 1-\xi \right) \epsilon \alpha }. \end{aligned}$$

(46)

Proposition 3

The relative output of country 2 depends positively or negatively, according to the values adopted by the parameters, on the relative relevance of corruption in country 2, $\frac{1+z_{2}}{1+z_{1}}$ and on the relative labor levels in country 2, $\frac{M_{2}}{M_{1}}$, unless there is complete removal of scale effects; in this case, it becomes independent of the labor levels.

Proof

Results directly from 46. $\square$

Taking into account (26) and (45) we have that the relative price of the skilled sector is:

$$\begin{aligned} \left( \frac{P_{2}}{P_{1}}\right) ^{*}=\left( \frac{1+z_{2}}{1+z_{1}}\right) ^{\frac{\alpha \left( \alpha -1\right) +1}{\alpha (\epsilon -1)+1}}\left( \frac{M_{2}}{M_{1}}\right) ^{-\alpha \left( 1-\xi \right) }. \end{aligned}$$

(47)

Proposition 4

The relative price of country 2 depends positively on the relative relevance of corruption in country 2, $\frac{1+z_{2}}{1+z_{1}}$ and negatively on the relatively level of labor levels in country 2, $\frac{M_{2}}{M_{1}}$, unless there is a complete removal of scale effects.

Proof

Results directly from (47). $\square$

Finally, from (30) and (45) we have the steady-state inter-country unskilled premium:

$$\begin{aligned} \left( \frac{w_{L_{2}}}{w_{L_{1}}}\right) ^{*}& = {} \left( \frac{P_{2}^{L_{2}}}{P_{1}^{L_{1}}}\right) ^{\frac{1}{\alpha }}\cdot \frac{Q_{2}}{Q_{1}}=\left( \frac{\chi _{2}}{\chi _{1}}\right) ^{\epsilon } \left( \frac{1+z_{2}}{1+z_{1}}\right) ^{1-\frac{\left[ \epsilon +\alpha ^{2}\epsilon -\alpha ^{2}\right] }{\alpha (\epsilon -1)+1}}\nonumber \\{} & {} \left( \frac{L_{1}}{L_{2}}\right) ^{\frac{1}{2}} \left( \frac{M_{2}}{M_{1}}\right) ^{-\frac{1}{2}+\alpha \left( \epsilon -1\right) \left( 1-\xi \right) }, \end{aligned}$$

(48)

while from (31) and (45) we have the steady-state inter-country skilled premium:

$$\begin{aligned} \left( \frac{w_{H_{2}}}{w_{H_{1}}}\right) ^{*}& = {} \left( \frac{P_{2}^{H_{2}}}{P_{1}^{H_{1}}}\right) ^{\frac{1}{\alpha }}\cdot \frac{Q_{2}}{Q_{1}}=\left( \frac{\chi _{2}}{\chi _{1}}\right) ^{\epsilon } \left( \frac{1+z_{2}}{1+z_{1}}\right) ^{1-\frac{\left[ \epsilon +\alpha ^{2}\epsilon -\alpha ^{2}\right] }{\alpha (\epsilon -1)+1}}\nonumber \\{} & {} \left( \frac{H_{1}}{H_{2}}\right) ^{\frac{1}{2}}\left( \frac{M_{2}}{M_{1}} \right) ^{-\frac{1}{2}+\alpha \left( \epsilon -1\right) \left( 1-\xi \right) }, \end{aligned}$$

(49)

Proposition 5

The steady-state inter-country unskilled and skilled premiums depend positively or negatively, according to the values adopted by the parameters, on the relative relevance of corruption in country 2, $\frac{1+z_{2}}{1+z_{1}}$, and on the relative labor levels in country 2, $\frac{M_{2}}{M_{1}}$.

Proof

Result directly from (30) and (31). $\square$

The economic intuition underlying the propositions set out is essentially as follows: given the presence of scale effects, a rise in the corruption levels in country 2 has a positive direct impact on the cost of business corruption and a negative indirect effect on the absolute productivity of skilled labor. Thus, these variations will have different repercussions on (1) technological-knowledge bias and (2) the market size, which will impact the remaining variables:

• 1. if the negative effect of corruption on productivity, and hence on the size of tfhe market, is offset by the greater relevance of corruption, then it will cause technological knowledge to become biased toward country 2, which results in an increment in the relative supply of labor, driving, consequently, to a decrease in relative prices toward that country and, therefore, to a drop in workers’s wages;

• 2. if the negative effect of corruption on productivity, and hence on market size, is smaller than the positive impact experienced on the relevance of corruption, then it will lead to technological knowledge becoming biased toward country 1, resulting in a reduction in the relative supply of labor, leading, therefore, to an increase in relative prices toward that country and, thus, to an improvement in workers’s wages.

Taking into account the information provided, it is clear that if we are facing case 1, a drop in demand for the country’s intermediate goods can be expected, discouraging R&D activities and consequently leading to a decline in the global economic growth rate. In turn, the opposite occurs if we deal with case 2—see Table 1.a.

Given the absence of scale effects, an increase in the corruption levels in country 2 has a positive direct impact on the cost of business corruption and has no effect on the absolute productivity of high-skilled workers, which leads to technological knowledge becoming highly biased for country 1. Hence, there is a decline in the relative supply of labor, causing a relative price increase which increases the workers’ wages, on the condition that $(\epsilon \alpha )^{2}-\epsilon \alpha ^{2}+\epsilon <\alpha (\epsilon -1)+1$. Moreover, considering that there is an increase in the relative prices performed by country 2, this indicates that there will be an increase in the demand for its intermediate goods, which will stimulate R&D activities and, consequently, lead to an improvement in world economic growth—see Table 1.b.

Table 1 Effect of an increase in corruption parameters of country 2 on the steady-state value of the main variables

4 Quantitative results

4.1 Calibration and data

For the quantitative results, it is required to calibrate several parameters and exogenous variables, previously exhibited on (44), (30) and (31). Thus, to calibrate the income share, we considered the standard values adopted by Jones et al. (1993), $\alpha =0.64$ and, for the learning-by-past domestic R&D, measured by the obsolescence of (past) R&D investments, we followed Afonso (2012), $\beta =2$. For the cost of complexity, we considered the work conducted by Hummels and Klenow (2005) and collected the values obtained for Sweden, Canada, Australia, Germany, the USA, $s=1$—characterized as non-corrupt countries—and Spain, Portugal, Greece, Brazil, Mexico, and India—identified as corrupt countries, $s=2$—calculating, then, an average, where $\zeta =1.13$. The constant quality upgrade was obtained given the following function $\frac{1}{1-\alpha },$ where $q=2.78$, and for the elasticity of substitution we followed Willman (2002) where $\epsilon =1.05$. In its turn, the rate time of preference, $\rho$, and the inverse of the inter-temporal elasticity of substitution, $\theta$, was chosen based on Arrow (1999) and De la Croix and Delavallade (2009), respectively, where $\rho =0.015$ and $\theta =0.5$. Lastly, the values for the scale benefits on profits were assumed, where $\xi =0$, $\xi =0.5$, and $\xi =1$. Each parameter with the respective value, description and source is presented in Table 2.

Table 2 Parameters calibrated

Bearing in mind the variables, we collected annual data for both non-corrupt and corrupt countries, where the distinction between the countries stems from the values assumed by the Corruption Perception Index (CPI) proposed by Transparency International Organization. Therefore, we considered the value of 65 as the threshold for differentiating between country types, where all those with a score below this number were classified as corrupt. Therefore, for the country’s corruption degree, $z_{s},$ the proxy used was the following,

$$\begin{aligned} z_{s}=\frac{100-\hbox {CPI}_{s}}{100}, \end{aligned}$$

(50)

where the $\hbox {CPI}_{s}$ evaluates the CPI in s for the period 2000–2020. Concerning the relative importance of each type of country, $\chi _{s},$ and the absolute productivity advantage of skilled labor, $h_{s}$, proxies used were, respectively,

$$\begin{aligned} \chi _{s}=\frac{\hbox {GDP}\,\hbox {per}\,\hbox {capita}_{s}\cdot (1-z_{s})}{\hbox {GDP}\,\hbox {per}\,\hbox {capita}_{1}\cdot (1-z_{1}) +\hbox {GDP}\,\hbox {per}\,\hbox {capita}_{2}\cdot (1-z_{2})}, \end{aligned}$$

(51)

and

$$\begin{aligned} h_{s}=\frac{\hbox {Productivity}\,\hbox {per}\,\hbox {hour}_{s}}{100}+(1-z_{s}), \end{aligned}$$

(52)

where the GDP per capita was attained from the World Bank Database for the 2000–2019 interval. The productivity per hour, measured by $\frac{\hbox {Output-side}\,\hbox {real}\,\hbox {GDP}\,\hbox {at}\,\hbox {chained}\,\hbox {PPPs}}{\hbox {Number}\,\hbox {of}\,\hbox {persons}\,\hbox {engaged}\,\cdot \, \hbox {Average}\,\hbox {annual}\,\hbox {hours}\,\hbox {worked}\,\hbox {by}\,\hbox {persons}\,\hbox {engaged}}$, was collected from the Penn World Table (version 9.1) for 2000–2017. Lastly, to measure the variables linked to the unskilled, $L_{s}$, and skilled, $H_{s}$, labor levels, we used the annual indicator of employment by education (in thousands) based on the International Standard Classification of Education (ISCED), provided from the International Labor Organization database (ILOSTAT). The data on employment by educational level are characterized according to the comparable versions provided by ISCED—ISCED-97 and ISCED-11—see Appendix 2.1. For the number of unskilled employees, $L_{1}$and $L_{2}$, we considered the sum of the statistics illustrated between the categories one to four. In turn, the number of skilled workers, $H_{1}$ and $H_{2}$, we take into account the sum of the statistics available between the categories five to eight and five to six in the case of ISCED-11 and ISCED-97, respectively. The values of these variables were normalized by the population in the country.¹² The respective value, description, and source associated with $z_{s}$, $h_{s}$, $L_{s}$, and $H_{s}$ are exhibited in Table 3. Following the same line, in Table 4, the results, description, and source for $\chi _{s}$ are stated.

Table 3 Exogenous variables $z_{s}$, $h_{s}$, $L_{s}$ and $H_{s}$

Table 4 Exogenous variable $\chi _{s}$

4.2 Corruption and economic growth

Throughout this subsection, we intend to evaluate the effects of corruption on the economic growth rate, g, disposed on (44). Thus, we propose two approaches:

• 1. Determine the theoretical economic growth for each pair of corrupt and non-corrupt countries, bearing in mind exclusively the scale effect of $\xi =0.5$—see Fig. 1;¹³

• 2. Assess how the average of economic growth, for each region pair, reacts by a range of values—illustrated in Table 5—for the corruption degree ratio,$\frac{z_{2}}{z_{1}}$, and for the absolute productivity advantage of skilled labor,$\frac{h_{2}}{h_{1}}$, considering $\xi =0.5$. To compute these ratios, we assumed two standard deviations from the mean of each variable, $z_{s}$ and $h_{s}$, and then, we considered that $z_{2}$ acquires decreasing values as $z_{1}$ takes increasing values and, in its turn, that $h_{2}$ attains increasing values as $h_{1}$ gets decreasing values—see Fig. 2.

According to Fig. 1, we conclude that, overall, the economic growth rate decreases considerably as $z_{s}$ and $h_{s}$ between countries converge. As stated by Aïssaoui and Fabian (2021), globalization essentially boosts the welfare of low-income countries, particularly regarding GDP and corruption. The increase in welfare and wealth occurs substantially due to the economic efficiency induced by high-income countries that affect less developed regions through diffusion mechanisms. Thus, this process has an impact on reducing corruption levels, which consequently stimulates economic growth. Regarding each subfigure, India, Mexico, and Brazil exhibit, as predicted, the highest economic growth rates. Despite being the regions with the greatest and lowest levels of corruption and productivity, they are the nations that benefit the most from the mechanism of diffusion induced by globalization. In turn, Greece, Portugal, and Spain illustrate the lowest growth rates despite their low levels of corruption and high production values. This occurs because these nations cannot obtain identical benefits from the diffusion effects caused by globalization, as countries with higher corruption rates since they are already highly developed economically.

Fig. 1

Economic growth rate and corruption degree when $\xi =0.5$

Table 5 Values for $\frac{z_{2}}{z_{1}}$ and $\frac{h_{2}}{h_{1}}$

Through each subfigure disposed in Fig. 2, we observe that the economic growth rate between each pair of countries decreases significantly as the inequality between the relative levels of corruption, $z_{2}$ and $z_{1}$, and the absolute productivity of skilled labor, $h_{2}$ and $h_{1}$, decreases. When analyzing each case, we notice that each corrupt country, in cooperation with Sweden, reaches the highest economic growth rate. When cooperating with Germany and the USA, the values are considerably lower. This arises since, in the latter case, the benefits diffused to the corrupt countries are lower. The pattern of economic growth rates is similar to the one observed in Fig. 1, as expected.

Fig. 2

Economic growth rate according to different values of $z_{s}$ and $h_{s}$ when $\xi =0.5$

Furthermore, to examine the results obtained in each subfigure, we calculated the absolute change in the dependent variable for each pair of countries—see Table 6.

Table 6 Absolute variation (in percentual points) of the economic growth rate for each corrupt and non-corrupt country

According to the previous table, India, Mexico, and Brazil are the regions that experience a more significant drop in growth rate as the levels of corruption between the two types of countries converge. This occurs because as the degree of corruption in a non-corrupt (corrupt) country increases (decreases), the advantages associated with the diffusion mechanism decrease substantially. In turn, Greece, Portugal, and Spain, as the regions least affected by the advantages linked with globalization regardless of their level of corruption, experience a smaller decrease in this rate as the corruption levels between non-corrupt and corrupt countries get closer.

4.3 Corruption and inter-country unskilled and skilled premiums

In this subsection, to evaluate the impacts of corruption on the inter-country unskilled and skilled premiums, illustrated on (48) and (49) respectively, we intend to conduct two distinct procedures:

• 1. Determine the theoretical inter-country unskilled and skilled premium measure for each pair of corrupt versus non-corrupt countries, taking into account, exclusively, the scale effect of $\xi =0.5$—see Figs. 3 and 4;

• 2. Examine how the inter-country unskilled (skilled) premium, $\frac{w_{L_{2}}}{w_{L_{1}}}\left( \frac{w_{H_{2}}}{w_{H_{1}}}\right) ,$ behaves according to different values—illustrated in Table 5—of corruption degree ratio,$\frac{z_{2}}{z_{1}}$, and absolute productivity advantage of skilled labor,$\frac{h_{2}}{h_{1}}$, bearing in mind $\xi =0.5$—see Figs. 5 and 6.

According to Figs. 3 and 4, the inter-country unskilled and skilled premiums, $\frac{w_{L_{2}}}{w_{L_{1}}}$ and $\frac{w_{H_{2}}}{w_{H_{1}}},$ is greater (lower) when the corruption degree ratio, $\frac{z_{2}}{z_{1}}$, and, consequently, the absolute productivity advantage of skilled labor ratio, $\frac{h_{2}}{h_{1}}$, between (corrupt and non-corrupt) countries is, respectively, lower (higher) and higher (lower). In other words, a smaller (greater) $\frac{z_{2}}{z_{1}}$ and a greater (smaller) $\frac{h_{2}}{h_{1}}$ lead to a higher (lower) unskilled and skilled wages for workers in corrupt countries.

Analyzing each subfigure, disposed in Fig. 3, we perceive that a pattern is evident. Regarding the estimates obtained for the wages of unskilled workers, we note that, regardless of the non-corrupt country selected, India, Mexico, and Brazil present the lowest unskilled wages and, consequently, a wider wage disparity in comparison to the non-corrupt countries. Thus, based on our results, their earnings are about $2-3\%$, $9-13\%$, and $8-13\%$ of the unskilled wages practiced in non-corrupt nations, respectively. In turn, although Greece, Portugal, and Spain also report low wages compared to non-corrupt regions, the values obtained are more reasonable, being approximately $27-20\%$, $29-42\%$, and $47-68\%$ of the wages established in non-corrupt countries, respectively.

Fig. 3

Inter-country unskilled premium and corruption degree when $\xi =0.5$

The pattern of non-corrupt countries’ skilled wages, followed in each subfigure of Fig. 4, is similar to the pattern found in Fig. 3. On the one hand, India, Mexico, and Brazil have the lowest skilled wages, which are around $3-7\%$, $14-25\%$, and $16-30\%$ of skilled earnings acquired in non-corrupt countries, according to the same order. On the other hand, Greece, Portugal, and Spain, based on our findings, exhibit skilled wages around 32–56%, 42–78%, and 46–79% of the skilled wages practiced in the different non-corrupt countries, respectively.

Fig. 4

Inter-country skilled premium and corruption degree when $\xi =0.5$

As expected, India is the most affected by corruption, not only due to the high values of $z_{2}$ but mainly due to the considerable low value that $h_{2}$ entails. Concerning Mexico and Brazil, both countries present marginal wage differences depending on the relative level of corruption and productivity of skilled labor to be slightly lower and greater, respectively, in Brazil—the remaining variables illustrate very similar values. In turn, Greece presents larger inter-country unskilled and skilled premiums compared to the last three countries mentioned but lower than when compared to Portugal, as predicted, due to its higher and smaller value of $z_{2}$ and $h_{2}$. Lastly, Spain indicates the highest inter-country unskilled and skilled premiums due to its lowest $z_{2}$ and largest $h_{2}$ compared to the other non-corrupt countries. Therefore, we can observe that as the levels of relative corruption and skilled productivity labor are smaller and greater, the wage ratio between the two types of countries increases.

Figures 5 and 6 show that the inter-country unskilled and skilled premiums increase considerably as the inequality between relative corruption levels and absolute productivity of skilled labor decreases. In other words, as this disparity declines, the (unskilled and skilled) wages between corrupt and non-corrupt countries become closer. More particularly, we perceive that Greece, Portugal, and Spain are the regions that exhibit the highest unskilled and skilled wages when compared to India, Mexico, and Brazil, as the levels of relative corruption and skilled productivity between corrupt and non-corrupt countries converge, as predicted.

Fig. 5

Inter-country unskilled premium according to different values of $z_{s}$ and $h_{s}$ when $\xi =0.5$ for a corrupt versus non-corrupt country

Fig. 6

Inter-country skilled premium according to different values of $z_{s}$ and $h_{s}$ when $\xi =0.5$ for a corrupt versus non-corrupt country

Furthermore, to analyze, in detail, the results provided by each subfigure, we calculated the absolute and the relative variation of the inter-country unskilled and skilled premiums for each corrupt vs non-corrupt country—see Tables 7 and 8 for both results.

Table 7 Absolute variation (in percentual points) of the inter-country unskilled and skilled premiums for each corrupt versus non-corrupt country

Given Table 7, we conclude that, except for Spain vs Sweden, Spain vs Germany, and Greece vs Germany, the variations obtained show a greater impact on the skilled wage ratio in terms of magnitude. Nevertheless, the unskilled wage ratio also reveals a considerable increment as relative corruption and skilled productivity levels between corrupt and non-corrupt countries get closer. These results align with the findings obtained by Mandal and Marjit (2010) and Cooray and Schneider (2016). Thus, in agreement with Mandal and Marjit (2010), a less corrupt society tends to increase more skilled wages, which leads to a more significant wage gap between skilled and unskilled workers. Also, Cooray and Schneider (2016), with a slightly different approach, show that an increase in corruption leads to a considerable increase in the emigration of skilled workers. Therefore, a decrease in corruption levels and a greater increase in absolute values in the wages of this type of worker are necessary to reduce emigration rates substantially. According to our results, as the corruption values between corrupt and non-corrupt countries converge, India, by far, presents the lowest absolute variation in unskilled wages, as expected, regardless of which non-corrupt country we are comparing it to. In its turn, Spain exhibits the highest absolute variation concerning unskilled wages. India illustrates the smallest absolute fluctuation in skilled workers’ earnings, and Greece shows the largest change. Lastly, considering the differences between unskilled and skilled absolute variations, we note that, on the one hand, Brazil is, without exception, regardless of which non-corrupt country we compare it to, the region with the largest differences in fluctuations. However, Mexico, Greece, and Portugal also show significant distinctions. India and Spain, on the other hand, are the countries that present the smallest differences concerning absolute wage variations between workers.

Table 8 Relative variation (in percentage) of the inter-country unskilled and skilled premiums for each corrupt versus non-corrupt country

Regarding Table 8, we notice that a change in the relative relevance of corruption and in the absolute productivity of skilled workers leads to a considerable fluctuation in unskilled and skilled wages, essentially, in India, Mexico, and Brazil. In turn, Greece and, mainly, Portugal and Spain are the countries with the lowest wage growth rate. These results align with the literature proposed by Cooray e Dzhumashev (2018) and Dreher and Schneider (2010). Both argue that high levels of corruption tend to distort the labor market, consequently leading societies to operate under a “shadow” system. That is, due to the regulatory intensity, poor public sector services, and high taxes and social contributions, workers tend to operate outside of the formal sector to improve their level of welfare. Therefore, as the levels of corruption decrease, wages increase, and workers tend to return to the formal market. Although this type of problem is not very present in developed countries, developing regions, due to a lack of financial and technical resources and due to the weak negotiating position of regulators, tend to have ineffective regulation, which leads to a large margin for corrupt practices (Mudambi et al. 2013).

5 Concluding remarks

The practice of corruption by government officials guarantees them a private benefit at the expense of collective resources, thus creating permanent distortions from which only a few use. These distortions affect the correct allocation of resources, generating inefficiencies in productive activity.

Although the negative relationship between corruption and economic growth is more or less consensual in the literature, the distributional consequences of corruption across countries have been completely ignored. At the level of income redistribution, it makes sense to compare wages between corrupt and non-corrupt countries. To cover these gaps, this paper analyzed the macroeconomic effects of different levels of corruption across countries. For this purpose, a dynamic model of DTC was proposed with two representative countries, one corrupt and one non-corrupt, in which each country has skilled and unskilled labor.

The proposed analytical framework allowed for the analysis of the static and dynamic effects of different levels of corruption across countries. Since corruption equally affects the production of all intermediate goods in each country, it does not impact intra-country wage inequality between skilled and unskilled workers. However, since there are different levels of corruption between countries, there is a technological-knowledge bias in favor of the country with the lowest level of corruption, in addition to corruption weakening the technological-knowledge progress. Thus, corruption will eventually affect countries’ wages differently: It decreases intra-country wages and increases wage inequality between countries. However, it also negatively affects the overall growth rate in the long run.

To sum up, at a single-country level, an increase in corruption levels in one country leads to an intra-country decrease in the demand for labor and wages and a permanent slowdown of technological-knowledge change and economic growth. Moreover, a country-specific increase in corruption at the inter-country level enlarges inter-country wage and technological-knowledge gaps. Overall, higher corruption levels in one country are detrimental to global growth.

Subsequently, to assess the magnitude of the impacts on economic growth and wage inequality across countries, the model was calibrated to obtain quantitative results, considering, given the available data, different non-corrupt countries (Sweden, Canada, Australia, Germany, and the USA) and corrupt countries (India, Mexico, Brazil, Greece, Portugal, and Spain). Regarding economic growth, we see a considerable decrease as the values of the corruption parameters between corrupt and non-corrupt countries converge due to the reduction of the advantages associated with the diffusion mechanism driven by globalization. Thus, according to the results, India, Mexico, and Brazil experienced the highest growth rate, particularly when combined with countries with low levels of corruption, such as Sweden, although they experienced a steeper decline in absolute terms compared to the other corrupt nations. Regarding wage inequality between countries at the level of unskilled and skilled workers, the results reveal that the gap between non-corrupt and corrupt nations is more significant as the differences in corruption parameters between both types of countries increase.

Due to their high corruption, India, Mexico, and Brazil have lower incomes (at the level of unskilled and skilled workers) and, therefore, a higher wage disparity compared to the non-corrupt countries. In turn, Greece, Portugal, and Spain have the highest wages and, consequently, a smaller wage disparity than non-corrupt countries. Furthermore, when analyzing the absolute variation of the premiums across countries at the level of unskilled and skilled workers for each pair of corrupt and non-corrupt countries, it is observed that most combinations, except for Spain vs Sweden, Spain vs Germany, and Greece vs Germany, exhibit a higher wage fluctuation when unskilled workers are concerned. Finally, regarding the results obtained for the wage growth rate, as the values of the parameters linked to corruption converge, we find that India, Mexico, and Brazil, unlike Greece, Portugal, and Spain, exhibit considerably high growth rates.

The results, therefore, suggest the need for policies that reduce levels of corruption. Better country governance will undoubtedly improve the performance of institutions and make them more efficient, leading to higher tax revenues and, consequently, the financing of better development projects, which will stimulate economic growth. Therefore, following Rose-Ackerman and Truex (2012), to tackle high levels of corruption and achieve this previously described objective, it will be essential to draw up and enforce anti-corruption laws and adopt measures associated with better monitoring, supervision and severe legal sanctions for individuals who engage in misconduct. Moreover, strengthening transparency in public administration by offering more accessible access to information, such as publishing budgets online and government contracts accompanied by penalties if irregularities are detected, can be seen as powerful mechanisms to foster corruption. Additionally, reforms in the public sector associated with internal controls and bureaucratic incentives, such as an increase in the salary of civil servants, since underpaid officials have a stronger motivation to embezzle funds and seek bribes, can boost efficiency, and mitigate this issue. Furthermore, the enforcement of anti-corruption education and awareness programs that lead to a better allocation of resources, in addition to enhancing the quality of educational services, will lead to better employment opportunities, which will improve regulatory quality and reduce the size of the shadow economy, causing wage increases in the formal sector and the consequent return of workers to this market.

Appendices

Appendix 1

1.1 Appendix 1.1: Threshold task and labor units

From the definition of price indexes in (16), we can have that:

$$\begin{aligned} \frac{P_{s}^{H_{s}}}{P_{s}^{L_{s}}}=\frac{P_{v_{s}}^{H_{s}}}{P_{v_{s}}^{L_{s}}} \left( \frac{v_{s}}{1-v_{s}}\right) ^{\alpha }\Leftrightarrow \frac{P_{v_{s}}^{H_{s}}}{P_{v_{s}}^{L_{s}}} =\frac{P_{s}^{H_{s}}}{P_{s}^{L_{s}}}\left( \frac{1-v_{s}}{v_{s}}\right) ^{\alpha }. \end{aligned}$$

(53)

We have that: (1) $\frac{P_{v_{s}}^{H_{s}}}{P_{v_{s}}^{L_{s}}}$ is a continuous function of $v_{s}$; (2) Since $\frac{P_{s}^{H_{s}}}{P_{s}^{L_{s}}}$ is assumed to be a positive constant, $\frac{P_{v_{s}}^{H_{s}}}{P_{v_{s}}^{L_{s}}}$ varies negatively with $v_{s}$, ceteris paribus; (3) $\lim _{v_{s}\rightarrow 1}\frac{P_{v_{s}}^{H_{s}}}{P_{v_{s}}^{L_{s}}}=0$; and (4) $\lim _{v_{s}\rightarrow 0}\frac{P_{v_{s}}^{H_{s}}}{P_{v_{s}}^{L_{s}}}=\infty$. Using (1)–(4) by the Intermediate Value Theorem there is a ${\overline{v}}_{s}\in \left[ 0,1\right]$ such that $\frac{P_{v_{s}}^{H_{s}}}{P_{v_{s}}^{L_{s}}}=1\Leftrightarrow P_{v_{s}}^{H_{s}}=P_{v_{s}}^{L_{s}}$. Moreover by (1) for $v_{s}>{\overline{v}}_{s}$, $P_{v_{s}}^{H_{s}}<P_{v_{s}}^{L_{s}}$ and for $v_{s}<{\overline{v}}_{s}$ we have that $P_{v_{s}}^{H_{s}}>P_{v_{s}}^{L_{s}}$. Since the output of each variety $v_{s}$ is produced in perfect competition, firms opt for producing task $v_{s}$ with the lowest price. Therefore, for $v_{s}={\overline{v}}_{s}$ they are indifferent between labor types, but for $v_{s}<{\overline{v}}_{s}$ $\left( v_{s}>{\overline{v}}_{s}\right)$ they choose country $L_{s}$ $\left( H_{s}\right)$. From here, we can also establish that:

$$\begin{aligned} \frac{P_{s}^{H_{s}}}{P_{s}^{L_{s}}}=\left( \frac{{\overline{v}}_{s}}{1-{\overline{v}}_{s}}\right) ^{\alpha }. \end{aligned}$$

(54)

From the profit maximization problems of the producers of output in country s and task $v_{s}$ we have that—see (12) and (13)—$P_{v_{i}}^{L_{s}}(t)\cdot Y_{v_{i}}^{L_{s}}(t)=\left( P_{v_{s}}^{L_{s}}(t)\right) ^{\frac{1}{\alpha }}\cdot \left[ \frac{1-\alpha }{p(k,j,t)} \right] ^{\frac{1-\alpha }{\alpha }}\cdot Q_{s}(t)\cdot {\widetilde{L}}_{v_{s}}(t)$ and $P_{v_{i}}^{H_{s}}(t)\cdot Y_{v_{i}}^{H_{s}}(t)=\left( P_{v_{s}}^{H_{s}}(t)\right) ^{\frac{1}{\alpha }}\cdot \left[ \frac{1-\alpha }{p(k,j,t)} \right] ^{\frac{1-\alpha }{\alpha }}\cdot Q_{s}(t)\cdot {\widetilde{H}}_{v_{s}}(t)$, which bearing in mind (16) allow us to write $L_{v_{s}}=\frac{P_{v_{i}}^{L_{s}}(t)\cdot Y_{v_{i}}^{L_{s}}(t)}{\left( P_{s}^{L_{s}}(t)\right) ^{\frac{1}{\alpha }}\cdot \left[ \frac{1-\alpha }{p(k,j,t)} \right] ^{\frac{1-\alpha }{\alpha }}\cdot Q_{s}(t)}$ for $v_{s}\in \left[ 0,{\overline{v}}_{s}\right)$ and $H_{v_{s}}=\frac{P_{v_{i}}^{H_{s}}(t)\cdot Y_{v_{i}}^{H_{s}}(t)}{\left( P_{s}^{H_{s}}(t)\right) ^{\frac{1}{\alpha }}\cdot \left[ \frac{1-\alpha }{p(k,j,t)} \right] ^{\frac{1-\alpha }{\alpha }}\cdot h\cdot Q_{s}(t)}$ for $v_{s}\in \left( {\overline{v}}_{s},1\right]$. Since $P_{v_{i}}^{i_{s}}(t)\cdot Y_{v_{i}}^{i_{s}}(t)$, $i_{s}=\{L_{s},H_{s}\}$, is constants for all $v_{s}\in \left[ 0,1\right]$, $p(k,j,t)=p(k,j,t)=q\left( 1-z_{s}\right)$—as it will be shown in Section 2.3—, and $\left( P_{s}^{L_{s}}\right) ^{\frac{1}{\alpha }}$ and $\left( P_{s}^{H_{s}}\right) ^{\frac{1}{\alpha }}$ are also constants, it becomes clear that both $L_{v_{s}}$ and $H_{v_{s}}$ are constants, implying that:

$$\begin{aligned} L_{s}\equiv \int _{0}^{{\overline{v}}_{i}}L_{v_{s}}\cdot dv_{s}=L_{v_{s}}\cdot {\overline{v}}_{s}\;and\;H_{s}\equiv \int _{{\overline{v}}_{s}}^{1}H_{v_{s}}\cdot dv_{s}=H_{v_{s}}\cdot \left( 1-{\overline{v}}_{s}\right) . \end{aligned}$$

(55)

Finally, using (54) and (55), we can determine ${\overline{v}}_{s}$, by solving the equation $P_{{\overline{v}}_{s}}^{L_{s}}\cdot Y_{{\overline{v}}_{s}}^{L_{s}}=P_{{\overline{v}}_{s}}^{H_{s}}\cdot Y_{{\overline{v}}_{s}}^{H_{s}}$, from which we obtain (17); i.e., ${\overline{v}}_{s}=\left[ 1+\left( h_{s}\frac{H_{s}}{L_{s}}\right) ^{\frac{1}{2}}\right] ^{-1}.$ Further, developing the expression for ${\overline{v}}_{s}$, we have that ${\overline{v}}_{s}=\frac{\left( L_{s}\right) ^{\frac{1}{2}}}{\left( L_{s}\right) ^{\frac{1}{2}}+\left( h_{s}H_{s}\right) ^{\frac{1}{2}}}$. Therefore, the threshold task can be interpreted as the weight of effective unskilled labor units in total effective labor units used in country s.

1.2 Appendix 1.2: Price indexes of tasks and price of the output in each sector

In this appendix we determine the values for price indexes of tasks produced with each type of labor. We start from $P_{s}=\exp \left( \int _{0}^{1}\ln P_{v_{s}}dv_{s}\right)$, to write $\ln P_{s}=\int _{0}^{{\overline{v}}_{s}}\ln P_{v_{s}}^{L_{s}}\cdot dv_{s}+\int _{{\overline{v}}_{s}}^{1}\ln P_{v_{s}}^{H_{s}}\cdot dv_{s}$, which from (16) results that $\ln P_{s}=\int _{0}^{{\overline{v}}_{s}}\ln \left[ P_{s}^{L_{s}}\left( 1-v_{s}\right) ^{-\alpha }\right] \cdot dv_{s}+\int _{{\overline{v}}_{s}}^{1}\ln \left[ P_{s}^{H_{s}}v_{s}^{-\alpha }\right] \cdot dv_{s}$ or, in other words, $\ln P_{s}={\overline{v}}_{s}\ln P_{s}^{L_{s}}+\left( 1-{\overline{v}}_{s}\right) \ln P_{s}^{H_{s}}-\alpha \left[ \int _{0}^{{\overline{v}}_{s}}\ln \left( 1-v_{s}\right) \cdot dv_{s}+\int _{{\overline{v}}_{s}}^{1}\ln v_{s}\cdot dv_{s}\right]$. Now, since $\int _{0}^{{\overline{v}}_{s}}\ln \left( 1-v_{s}\right) \cdot dv_{s}=\left( {\overline{v}}_{s}-1\right) \ln \left( 1-{\overline{v}}_{s}\right) -{\overline{v}}_{s}$, $\int _{{\overline{v}}_{s}}^{1}\ln v_{s}\cdot dv_{s}=-1-{\overline{v}}_{s}\ln {\overline{v}}_{s}+{\overline{v}}_{s}$, and from the definition of price indexes $P_{s}^{H_{s}}=\left( \frac{{\overline{v}}_{s}}{1-{\overline{v}}_{s}}\right) ^{\alpha }P_{s}^{L_{s}}$, we have that

$$\begin{aligned} P_{s}^{L_{s}}=P_{s}\cdot \exp \left( -\alpha \right) \cdot {\overline{v}}_{s}^{-\alpha }, \end{aligned}$$

and, replacing in the relation between price indexes, we also have:

$$\begin{aligned} P_{s}^{H_{s}}=P_{s}\cdot \exp \left( -\alpha \right) \cdot \left( 1-{\overline{v}}_{s}\right) ^{-\alpha }. \end{aligned}$$

Moreover, from the maximization problem of the producer of Y, $P_{Y}=\left[ \sum _{s=U,H}\chi _{s}^{\varepsilon }\cdot P_{s}^{1-\varepsilon }\right] ^{\frac{1}{1-\varepsilon }}$ and thus

$$\begin{aligned} P_{2}=\left[ \frac{P_{Y}^{1-\varepsilon }-\chi _{1}^{\varepsilon }\cdot P_{1}^{1-\varepsilon }}{\chi _{2}^{\varepsilon }}\right] ^{\frac{1}{1-\varepsilon }}, \end{aligned}$$

(56)

which replaced in the expression of the relative price of the country 2 (26) allows to obtain

$$\begin{aligned} \frac{\left[ \frac{P_{Y}^{1-\varepsilon }-\chi _{1}^{\varepsilon }\cdot P_{1}^{1-\varepsilon }}{\chi _{2}^{\varepsilon }}\right] ^{\frac{1}{1-\varepsilon }}}{P_{1}}= \left( \frac{\chi _{2}}{\chi _{1}}\right) ^{\frac{\epsilon \cdot \alpha }{\epsilon \cdot \alpha +1}} \left( \frac{M_{2}}{M_{1}}\frac{Q_{2}}{Q_{1}}\right) ^{-\frac{\epsilon }{\epsilon \cdot \alpha +1}}, \end{aligned}$$

that is equivalent to $P_{Y}^{1-\varepsilon }=P_{1}^{1-\epsilon }\left[ \chi _{1}^{\varepsilon }+\chi _{2}^{\varepsilon } \left( \frac{\chi _{2}}{\chi _{1}}\right) ^{\frac{(1-\epsilon )\alpha \epsilon }{\alpha (\epsilon -1)+1}} \left( \frac{1+z_{2}}{1+z_{1}}\right) ^{\frac{(1-\epsilon )(1-\alpha )}{\alpha (\epsilon -1)+1}} \left( \frac{M_{2}}{M_{1}}\frac{Q_{2}}{Q_{1}}\right) ^{-\frac{(1-\epsilon )\alpha }{\alpha (\epsilon -1)+1}}\right]$. Solving the last expression in order $P_{1}$ gives:

$$\begin{aligned} P_{1}=\left[ \chi _{1}^{\varepsilon }+\chi _{2}^{\varepsilon }\left( \frac{\chi _{2}}{\chi _{1}}\right) ^{\frac{(1-\epsilon )\alpha \epsilon }{\alpha (\epsilon -1)+1}}\left( \frac{1+z_{2}}{1+z_{1}}\right) ^{\frac{(1-\epsilon )(1-\alpha )}{\alpha (\epsilon -1)+1}}\left( \frac{M_{2}}{M_{1}}\frac{Q_{2}}{Q_{1}}\right) ^{-\frac{(1-\epsilon )\alpha }{\alpha (\epsilon -1)+1}}\right] ^{-\frac{1}{1-\epsilon }}P_{Y}. \end{aligned}$$

(57)

We can now use the price of the output in country 1 to find the price of the output in country 2. For this purpose, it is sufficient to conjugate (56) and (57) to obtain:

$$\begin{aligned} P_{2}=\left[ \frac{\left( \frac{\chi _{2}}{\chi _{1}}\right) ^{\frac{(1-\epsilon )\alpha \epsilon }{\alpha (\epsilon -1)+1}}\left( \frac{1+z_{2}}{1+z_{1}}\right) ^{\frac{(1-\epsilon )(1-\alpha )}{\alpha (\epsilon -1)+1}} \left( \frac{M_{2}}{M_{1}}\frac{Q_{2}}{Q_{1}}\right) ^{-\frac{(1-\epsilon )\alpha }{\alpha (\epsilon -1)+1}}}{\chi _{1}^{\varepsilon }+\chi _{2}^{\varepsilon }\left( \frac{\chi _{2}}{\chi _{1}}\right) ^{\frac{(1-\epsilon ) \alpha \epsilon }{\alpha (\epsilon -1)+1}}\left( \frac{1+z_{2}}{1+z_{1}}\right) ^{\frac{(1-\epsilon ) (1-\alpha )}{\alpha (\epsilon -1)+1}}\left( \frac{M_{2}}{M_{1}}\frac{Q_{2}}{Q_{1}}\right) ^{-\frac{(1-\epsilon )\alpha }{\alpha (\epsilon -1)+1}}}\right] ^{\frac{1}{1-\epsilon }}P_{Y}. \end{aligned}$$

(58)

1.3 Appendix 1.3: Incentives to innovate

Improvements in quality can be achieved either by the incumbent firm or by a new entrant. In the first case, the incumbent firm produced an intermediate good j with quality $k-1$ and practiced a price q. By improving the quality level to k, the incumbent also changes prices to $q^{2}$. Therefore, the change in profits for the incumbent is:

$$\begin{aligned} \Delta \Pi _{Incumbent}(j)& = {} \Pi \left( k,j\right) -\Pi \left( k-1,j\right) =\\& = {} \left( q-1\right) \cdot \left( 1+z_{s}\right) \cdot \exp \left( -1\right) \cdot \left( \frac{1-\alpha }{q\left( 1+z_{s}\right) }\right) ^{\frac{1}{\alpha }}\cdot q^{k(j,t)\left( \frac{1-\alpha }{\alpha }\right) }\cdot M_{s}\cdot \left[ \left( q+1\right) \cdot q^{-\frac{1}{\alpha }}-q^{\frac{\alpha }{1-\alpha }}\right] . \end{aligned}$$

In the second case, the incumbent firm begins producing an intermediate good j with quality k and practices the price q since it is new to the market. Therefore, the change in profits for the entrants is:

$$\begin{aligned} \Delta \Pi _{Entrants}(j)=\Pi \left( k,j\right) =\left( q-1\right) \cdot \left( 1+z_{s}\right) \cdot \exp \left( -1\right) \cdot \left( \frac{1-\alpha }{q\left( 1+z_{s}\right) }\right) ^{\frac{1}{\alpha }}\cdot q^{k(j,t)\left( \frac{1-\alpha }{\alpha }\right) }\cdot M_{s}. \end{aligned}$$

Comparing both we have that $\Delta \Pi _{Incumbent}(j)=\left[ \left( q+1\right) \cdot q^{-\frac{1}{\alpha }}-q^{\frac{\alpha }{1-\alpha }}\right] \Delta \Pi _{Entrants}(j)$. Since $0<\alpha <1$ and $q=\frac{1}{1-\alpha }$, we have that $\left[ \left( q+1\right) \cdot q^{-\frac{1}{\alpha }}-q^{\frac{\alpha }{1-\alpha }}\right] <1$ and, therefore, $\Delta \Pi _{Incumbent}(j)<\Delta \Pi _{Entrants}(j)$, implying that the innovation effort will be carried out by the new entrant.

1.4 Appendix 1.4: Derivation of the patent value

Each moment in time in country $s=1,2$ there is a probability ${\mathcal {I}}_{s}(j,t)dt$ that the quality level improves by 1, i.e., $k(j,t+dt)-k(j,t)=1$, and a probability $\left( 1-{\mathcal {I}}_{s}(j,t)\right) dt$ that there is no improvement in the quality level, i.e., $k(j,t+dt)-k(j,t)=0$. Bearing this in mind, if we consider each moment in time as a random experiment that can result in success with probability ${\mathcal {I}}_{s}(j,t)$, we can characterize the time derivative of k(j, t) as a random variable that follows a Binomial Distribution with an expected value of ${\mathcal {I}}_{s}(j,t)$, i.e., ${\dot{k}}(j,t)\sim B(1,{\mathcal {I}}_{s}(j,t))$. Therefore, although k(j, t) assumes only integer values, k(j, t) and all the variables that depend on it can be differentiated in relation to time. However, as a result of the derivative being stochastic, they are also random variables.

Since k(j, t) is a random variable, $\iota$ is also a random variable with a probability distribution that is equal to ${I_{2}(\iota )=\left( 1-\int _{0}^{\iota }I(\iota )d\iota \right) \cdot {\mathcal {I}}_{1}(j,t+\iota )}$. The intuition behind this formula is that the probability of no quality improvement of an intermediate good j with quality level k being exactly equal to $\iota$ since time t, the time in which the monopoly was initiated, is the probability of no improvement occurring before $t+\iota$, times the probability of successful innovation at time $t+\iota$ (Barro and Sala-i-Martin 2004) occurring in the non-corrupt countries, which is the potential challenger. In the case of the value of a patented innovation, $V_{1}$, the challenge comes from both a new non-corrupt country innovation and a corrupt country imitation. Assuming that ${{\mathcal {I}}_{1}(j,t+\iota )={\mathcal {I}}_{1}(j,t)}$, and the ${I_{2}(\iota =0)=0}$, we have that ${I_{2}(\iota )={\mathcal {I}}_{1}(j,t)\cdot \exp \left( -{\mathcal {I}}_{1}(j,t)\cdot \iota \right) }$. Since ${V_{2}(j,k,t,\iota )}$ depends on $\iota$, this is also a random variable with the same probability density function of $\iota$, ${I_{2}(\iota )}$. Assuming that the investors are risk-neutral implies that they only care about the expected value of ${V_{S}(j,k,t,T(k))}$ (Gil et al. 2013), which is equal to the following expression:

$$\begin{aligned} {V_{s}(j,k,t)}&{=\int _{0}^{\infty }\Pi (j,s)\exp \left( -\left( \int _{t}^{v}r(w)+{\mathcal {I}}_{s}(j,t)\right) {\text{d}}w\right) {\text{d}}v} \end{aligned}$$

Assuming that all the prices and quantities are fixed during the time in which there are no quality improvements (e.g., Aghion and Howitt 1992; Barro and Sala-i-Martin 2004; Gil et al. 2013), then we have that:

$$\begin{aligned} V_{s}(j,k,t)=\frac{\Pi _{s}(j,k,t)}{r_{s}(t)+I_{s}(j,t)}. \end{aligned}$$

1.5 Appendix 1.5: Market value of patents

As stated in (38), the expected market value of a patent is a random variable that depends on the uncertainty of the period of the next successful innovation $\tau$, which then is the following: ${{V_{s}\left( k+1,j,t\right) =E\left[ V_{s}\left( k+1,j,t,\tau \right) \right] =}\int _{t}^{\infty }b\left( \tau \right) \cdot V_{s}(k+1,j,t,\tau )\cdot d\tau }$, where $b(\tau )$ is the probability density function of $\tau$, i.e., the probability that the patent lasts exactly until $\tau$, which occurs if at that period some firm is able to obtain a higher quality rung for the intermediate good j. $b(\tau )$ is determined in the following way: $b(\tau )=\left[ 1-B\left( \tau \right) \right] \cdot {\mathcal {I}}_{s}\left( k,j,\tau \right)$, where $B\left( \tau \right)$ is the cumulative density function of $\tau$ and ${\mathcal {I}}_{s}\left( k,j,\tau \right)$ is the probability of a successful quality upgrade at time $\tau$. Therefore, we have a first-order differential equation, which can be solved in the following way: $\frac{{\text{d}}B\left( \tau \right) }{{\text{d}}\tau }=\left[ 1-B\left( \tau \right) \right] \cdot {\mathcal {I}}_{s}\left( k,j,\tau \right)$. Assuming that $B\left( 0\right)$, we determine $b(\tau )={\mathcal {I}}_{s}\left( k,j,\tau \right) \cdot \exp \left[ -{\mathcal {I}}_{s}\left( k,j ,\tau \right) \cdot \left( \tau -t\right) \right]$. Thus, also bearing in mind (37), ${{V_{s}\left( k+1,j,t\right) =}\int _{t}^{\infty }b\left( \tau \right) \cdot V_{s}(k+1,j,t,\tau )\cdot d\tau }$ can be written as:

$$\begin{aligned} V_{s}\left( k+1,j,t\right)&=\int _{t}^{\infty }{\mathcal {I}}_{s}\left( k,j,\tau \right) \cdot \exp \left[ -{\mathcal {I}}_{s}\left( k,j,\tau \right) \cdot \left( \tau -t\right) \right] \cdot \left[ \int _{t}^{\tau }\pi _{s}(k+1,j,v)\cdot \exp \left[ -\int _{t}^{\omega }r\left( \omega \right) d\omega \right] dv\right] \cdot d\tau \\&=\frac{\pi _{s}(k+1,j,t)}{r(t)+{\mathcal {I}}_{s}\left( k,j,t\right) }, \end{aligned}$$

taking into account that $\lim _{T\rightarrow \infty }\int _{0}^{T}\exp \left[ -{\mathcal {I}}_{s}\left( k,j,\tau \right) \tau \right] \cdot d\tau =\lim _{T\rightarrow \infty }\left[ \frac{\exp \left[ -{\mathcal {I}}_{s}\left( k,j,\tau \right) \cdot \tau \right] }{{\mathcal {I}}_{s}\left( k,j,\tau \right) }\right] _{0}^{T}=\frac{1}{{\mathcal {I}}_{s}\left( k,j,t\right) }$ and $\lim _{T\rightarrow \infty }\int _{0}^{T}\exp \left[ -\left( r(t)+{\mathcal {I}}_{s}\left( k,j,\tau \right) \right) \cdot \tau \right] \cdot d\tau =\lim _{T\rightarrow \infty }\left[ \frac{\exp \left[ -\left( r(t)+{\mathcal {I}}_{s}\left( k,j, \tau \right) \right) \cdot \tau \right] }{-\left( r(t)+{\mathcal {I}}_{s}\left( k,j,\tau \right) \right) } \right] _{0}^{T}=\frac{1}{r(t)+{\mathcal {I}}_{s}\left( k,j,t\right) }$.

1.6 Appendix 1.6: Aggregate resources constraint

Let $a_{s}(t)=\int _{0}^{J_{s}}E\left[ V_{s}\left( k,j,t\right) \right] dj$ be the total market value of all the firms that produce intermediate goods j at time t, as it was implicit in Sect. 2.1. From the definition of market value of a firm and taking into account that in equilibrium ${\mathcal {I}}_{s}(k,j,t)={\mathcal {I}}_{s}(t)$, we can write that $E\left[ V_{s}\left( k,j,t\right) \right] =\frac{\pi _{s}(k,j,t)}{r\left( t\right) +{\mathcal {I}}_{s}\left( k-1,j,t\right) }$, which is equivalent to $r(t)\cdot E\left[ V_{s}\left( k,j,t\right) \right] =\left( q-1\right) \cdot x_{s}(k,j,t)-{\mathcal {I}}_{s}\left( t\right) \cdot E\left[ V_{s}\left( k,j,t\right) \right]$ since $\pi _{s}(k,j,t)=\left( q-1\right) \cdot x_{s}(k,j,t)$. Moreover, from the free-entry condition we have that ${\mathcal {I}}_{s}\left( t\right) \cdot E\left[ V_{s}\left( k+1,j,t\right) \right] =e_{s}(k,j,t)$, i.e., $e_{s}(k-1,j,t)={\mathcal {I}}_{s}\left( t\right) \cdot E\left[ V_{s}\left( k,j,t\right) \right]$ and from (36) $e_{s}(k-1,j,t)={\mathcal {I}}_{s}\left( t\right) \cdot \frac{\beta }{\zeta }\cdot q^{\left[ k(j,t)-1\right] \left( \frac{\alpha -1}{\alpha }\right) }\cdot M_{s}^{\xi }$ and, thus, $e_{s}(k-1,j,t)=q^{\left( \frac{\alpha -1}{\alpha }\right) }\cdot e_{s}(k,j,t)$. Using the prior information and integrating over j we have that $\int _{0}^{J_{s}}r(t)\cdot E\left[ V_{s}\left( k,j,t\right) \right] dj=\int _{0}^{J_{s}}\left( q-1\right) \cdot \left( 1+z_{s}\right) \cdot x_{s}(k,j,t)\cdot dj-\int _{0}^{J_{s}}e_{s}(k-1,j,t)$, which is equivalent to $r(t)\cdot a_{s}\left( t\right) =\left( q-1\right) \cdot \left( 1+z_{s}\right) \cdot X_{s}-q^{\left( \frac{\alpha -1}{\alpha }\right) }\int _{0}^{J_{s}}e_{s}(k,j,t)\cdot dj$ and, therefore, $r(t)\cdot a_{s}\left( t\right) =\left( q-1\right) \cdot \left( 1+z_{s}\right) \cdot X_{s}-q^{\left( \frac{\alpha -1}{\alpha }\right) }\cdot E_{s}(t)$. In turn, from (24) and (35) we have $q\cdot \left( 1+z_{s}\right) \cdot X_{s}=\left( 1-\alpha \right) \cdot P_{s}\cdot Y_{s}$ and therefore $r(t)\cdot a_{s}\left( t\right) =\left( 1-\alpha \right) \cdot P_{s}\cdot Y_{s}-\left( 1+z_{s}\right) \cdot X_{s}-q^{\left( \frac{\alpha -1}{\alpha }\right) }\cdot E_{s}(t)$.

Considering both countries of the worldwide economy, the previous analysis—considering, for simplification, the steady state; i.e., $r=r_{1}=r_{2}$—can be summarized in the expression $r\cdot \left( a_{1}+a_{2}\right) =\left( 1-\alpha \right) \cdot \left( P_{1}\cdot Y_{1}+P_{2}\cdot Y_{2}\right) -\left( X_{1}+X_{2}\right) -\left( z_{1}\cdot X_{1}+z_{2}\cdot X_{2}\right) -q^{\left( \frac{\alpha -1}{\alpha }\right) }\cdot \left( E_{1}+E_{2}\right)$; i.e.,

$$\begin{aligned} r\cdot a=\left( 1-\alpha \right) \cdot Y-X-zX-q^{\left( \frac{\alpha -1}{\alpha }\right) }\cdot E. \end{aligned}$$

(59)

From (14) and (15) we have, respectively, $w_{L_{s}}=\frac{\alpha \cdot P_{v_{i}}^{L_{s}}\cdot Y_{v_{i}}^{L_{s}}}{L_{v_{s}}}=\frac{\alpha \cdot \cdot P_{s}\cdot Y_{s}^{L_{s}}}{L_{s}}$ and $w_{H_{s}}=\frac{\alpha \cdot P_{v_{i}}^{H_{s}}\cdot Y_{v_{i}}^{H_{s}}}{H_{v_{s}}}=\frac{\alpha \cdot \cdot P_{s}\cdot Y_{s}^{H_{s}}}{H_{s}}$. Therefore, since

$$\begin{aligned} \sum _{s=1,2}w_{H_{s}}\cdot H_{s}+\sum _{s=1,2}w_{L_{s}}\cdot L_{s}&=w_{H_{1}}\cdot H_{1} +w_{H_{2}}\cdot H_{2}+w_{L_{1}}\cdot L_{1}+w_{L_{2}}\cdot L_{2}=\\&=\alpha \cdot Y, \end{aligned}$$

which replaced together with (59) in the flow budget constraint (1), we have the following

$$\begin{aligned} {\dot{a}}&=r\cdot a+\sum _{s=1,2}w_{H_{s}}\cdot H_{s}+\sum _{s=1,2}w_{L_{s}}\cdot L_{s}-C+\sum _{s=1,2}\tau _{s}\nonumber \\&=Y-X-zX-q^{\left( \frac{\alpha -1}{\alpha }\right) }\cdot E-C+\tau ,\nonumber \\&=Y-X-q^{\left( \frac{\alpha -1}{\alpha }\right) }\cdot E-C, \end{aligned}$$

(60)

since $\sum _{s=1,2}\tau _{s}=\tau =zX$. Returning to the definition of market value of firms, $E\left[ V_{s}\left( k,j,t\right) \right] =\frac{\pi _{s}(k,j,t)}{r\left( t\right) +{\mathcal {I}}_{s}\left( t\right) }$, and bearing in mind (34) and (36) we have that $E\left[ V_{s}\left( k,j,t\right) \right] =\frac{\zeta }{\beta }\cdot q^{\left( \frac{\alpha -1}{\alpha }\right) }\cdot q^{k(j,t)\left( \frac{1-\alpha }{\alpha }\right) }\cdot M_{s}^{\xi }$. Therefore, the time derivative assets of producers of intermediate goods used in sector s are $\dot{a_{s}}=E\left[ V_{s}\left( k,j,t\right) \right] =\frac{\zeta }{\beta }\cdot q^{\left( \frac{\alpha -1}{\alpha }\right) }\cdot M_{s}^{\xi }\cdot {\dot{Q}}_{s}$. Therefore, the time variation of total assets is as follows—bearing also in mind (41):

$$\begin{aligned} {\dot{a}}&={\dot{a}}_{1}+{\dot{a}}_{2}=\frac{\zeta }{\beta }\cdot q^{\left( \frac{\alpha -1}{\alpha } \right) }\cdot \left( M_{1}^{\xi }\cdot {\dot{Q}}_{1}+M_{2}^{\xi }\cdot {\dot{Q}}_{2}\right) =\frac{\zeta }{\beta }\cdot q^{\left( \frac{\alpha -1}{\alpha }\right) }\cdot \left\{ M_{1}^ {\xi }\cdot {\mathcal {I}}_{1}\cdot \left[ q^{\left( \frac{1-\alpha }{\alpha }\right) }-1\right] +M_{2}^{\xi } \cdot {\mathcal {I}}_{2}\cdot \left[ q^{\left( \frac{1-\alpha }{\alpha }\right) }-1\right] \right\} \nonumber \\&=\left[ 1-q^{\left( \frac{\alpha -1}{\alpha }\right) }\right] \cdot \left\{ \frac{\zeta }{\beta }\cdot M_{1}^{\xi }\cdot {\mathcal {I}}_{1}\cdot Q_{1}+\frac{\zeta }{\beta }\cdot M_{2}^{\xi }\cdot {\mathcal {I}}_{2}\cdot Q_{2}\right\} =\left[ 1-q^{\left( \frac{\alpha -1}{\alpha }\right) }\right] \cdot \left( E_{1}+E_{2}\right) = \left[ 1-q^{\left( \frac{\alpha -1}{\alpha }\right) }\right] \cdot E. \end{aligned}$$

(61)

Finally, replacing (61) in the flow budget constraint (60) from the households, we have that $Y=C+X+E$.

1.7 Appendix 1.7: Steady-state price of the output in each sector

From (57) and (45), the steady-state price of the output in country 1 is

$$\begin{aligned} P_{1}^{*}=\left[ \chi _{1}^{\varepsilon }+\chi _{2}^{\varepsilon }\left( \frac{1+z_{2}}{1+z_{1}}\right) ^{1-\epsilon } \left( \frac{M_{2}}{M_{1}}\right) ^{\left( 1-\xi \right) (\epsilon -1)\alpha }\right] ^{-\frac{1}{1-\epsilon }}P_{Y}. \end{aligned}$$

(62)

From (58) and (45), the steady-state price of the output in country 2 is

$$\begin{aligned} P_{2}^{*}=\left[ \frac{\left( \frac{1+z_{2}}{1+z_{1}}\right) ^{1-\epsilon } \left( \frac{M_{2}}{M_{1}}\right) ^{\left( 1-\xi \right) (\epsilon -1)\alpha }}{\chi _{1}^{\varepsilon } +\chi _{2}^{\varepsilon }\left( \frac{1+z_{2}}{1+z_{1}}\right) ^{1-\epsilon }\left( \frac{M_{2}}{M_{1}}\right) ^{\left( 1-\xi \right) (\epsilon -1)\alpha }}\right] ^{\frac{1}{1-\epsilon }}P_{Y}. \end{aligned}$$

(63)

Moreover, the uniqueness of the steady-state economic growth rate (44) requires that the growth rate calculated for $s=1$ and for $s=2$ is the same. This implies that:

$$\begin{aligned} \left( \frac{P_{2}^{*}}{1+z_{2}}\right) ^{\frac{1}{\alpha }}\cdot M_{2}^{1-\xi }=\left( \frac{P_{1}^{*}}{1+z_{1}}\right) ^{\frac{1}{\alpha }}\cdot M_{1}^{1-\xi }, \end{aligned}$$

which, by using (62) and (63), occurs.

Appendix 2

1.1 Appendix 2.1: International classification of education (ISCED)—ISCED-11 and ISCED-97

See Table 9.

Table 9 ISCED-11 and ISCED-97