We develop an endogenous growth model with North–South interactions, monetary policy, and a multi-dimensional role for fiscal policy. To boost economic performance, the government of the less developed country subsidizes R&D and intermediate goods production. Besides, to account for the effects of excessive public debt, we introduce a negative externality on productivity and a risk premium effect. Our findings reveal that the steady-state growth rate of both economies depends positively on the subsidies in the South and negatively on the public debt’s externality on productivity and the risk premium affecting the indebted economy. Additionally, public debt externalities increase the equilibrium wage inequality, making it more significant in the South. To minimize these effects, both countries can agree to mutualize their debts to overcome the risk premium. Then, the economy’s steady-state growth rate would increase in both countries. Finally, several policy implications were retrieved.
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1 Introduction
Economic growth is a central theme in economics, with different researchers focusing on different factors. The role of foreign direct investment (Iamsiraroj 2016), financial innovations (Laeven et al. 2015), and climate change (Fankhauser and Tol 2005) are some examples of this discussion. Fiscal policy has also been addressed as a factor driving economic growth. For instance, R&D is described as a main contributor to growth, although firms in competitive markets tend to underinvest in these activities (Şener 2008). In light of this, R&D subsidies are one of the possible policies to increase the total investment in this category (Becker 2015; González and Pazó 2008).
Nevertheless, the long-run effects of fiscal policy are not necessarily positive. The accumulation of unsustainable levels of public debt may hamper growth. Although defining a threshold above which public debt leads to this adverse effect may be tricky (Ramos-Herrera and Sosvilla-Rivero 2017), Reinhart and Rogoff (2010) presented \(90\%\) as the critical ratio for public debt on GDP for developed countries. Above this ratio, negative externalities of public debt on GDP growth would arise. Belonging to a monetary union like the Euro Area would not erase the negative impact of unsustainable debt levels (Baum et al. 2013; Albu and Albu 2021; Bhimjee et al. 2020). Empirically, Sanyal and Ehlen (2017) identified these opposite fiscal policy effects with Bayesian non-parametric analysis, though they underlined the need for further research.
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Considering the potential contradictory results of fiscal policy, we defined the following research questions: (i) How does introducing public debt change the effects of R&D subsidies? (ii) What is the effect of public debt’s negative externalities on the indebted country’s output and wage premium? (iii) Do the negative effects of public debt spillover to non-indebted countries?
To answer these questions, we developed a globalized directed technical change (DTC) model, applied to the case of a monetary union and capturing the multidimensional role of fiscal policy. Although other studies can be found addressing the fiscal policy’s positive or negative impacts, we contribute to the literature by including both effects in the same model. Considering multiple effects of fiscal policy in the same model is essential to better understand the long-run results. We also contribute by distinguishing between public debt’s real and financial impacts. The spillover of the financial consequences, in terms of greater interest rates, from the government to firms is one of the novelties of our model. Additionally, apart from several policy implications, our framework also allows us to model public debt mutualization as a possible policy option in a monetary union. Finally, given that public debt and income inequality may be interlinked (Azzimonti et al. 2014; Miyashita 2023; Salti 2015; Carrera and de la Vega 2021; Arslan 2019), our approach also explores the relation between fiscal policy and the wage gap.
First, we briefly explain some of the literature on the topic, to better frame our analysis. Starting with the positive role of fiscal policy, Afonso et al. (2009) developed an endogenous growth model that considers the role of subsidies in the production of intermediate goods and R&D activities in a framework with two countries in different stages of their development. R&D activities in the most developed country would aim to be innovative, while the less developed would be focused on imitations of innovations. Then, they argued that providing the subsidies in the less developed country (that cannot be too far behind the most developed) would benefit both this economy and the most developed one because of feedback effects. Similarly, Currie et al. (1999) also found international spillovers of R&D subsidies in their model, which may motivate policy coordination. Besides, Gil and Iglésias (2020) model pointed that the positive effects of subsidies in the steady-state growth rate would be larger the smaller the share of the intermediate goods in the production of final goods. Nevertheless, they also highlighted that higher inflation rates would decrease the positive impacts of this policy. This non-neutral role of inflation in the effects of fiscal policy led us to rely on a model with monetary policy and inflation, as it will be describe further.
On the other hand, Davidson and Segerstrom (1998) defended that R&D subsidies destined for imitation would slow economic growth. However, they underlined that it would be difficult for a government to distinguish between innovative and imitative activities in the real world. Hence, R&D subsidies, in general, would promote sounder economic growth.
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Afonso and Sequeira (2022) added to this literature by building a North–South growth model with a monetary authority and linking inflation to wage inequality and specialization across countries. They concluded that an increase in long-run inflation in the South country (the least developed) would increase the wage inequality (between skilled and unskilled workers) compared to the North. Given their modeling of a common monetary policy, we use this model as our baseline. Still, as explained, we add a multi-dimensional role for fiscal policy and assess its effects on economic growth and wage inequality.
While studying the positive effects of subsidies, Afonso et al. (2009) indicated some potential adverse consequences. Providing subsidies to boost long-run economic performance could pose additional risks for the public accounts in countries under permanent financial stress. However, they assumed that those impacts would be manageable.
Nevertheless, there is some evidence that the accumulation of public debt above certain levels may hamper economic growth, as already explained. The existing literature captures these results well for endogenous growth models with public debt. Saint-Paul (1992), for example, stated that public debt would reduce the growth rate of the product in the long run, but promoting a fiscal consolidation would also not be Pareto optimal since it would hurt present generations. Still, the author suggested further analysis of the role of public debt, particularly in models with R&D. Greiner (2015) agreed that a country with permanent deficits (and thus accumulating public debt) would register a lower growth rate in the long run. Besides, an increase in the money growth rate would help to minimize these impacts, but at the expense of higher inflation and lower welfare.
On a different approach, Cheron et al. (2019), considering that public debt would also improve households’ utility, found that two different Balance Growth Paths (BGPs) could be verified as long as the public debt to GDP ratio was large enough. Interestingly, public debt would hamper growth in the lower BGP, while the opposite would happen in the higher BGP. The agent’s expectations would determine which steady state they would be in. Notwithstanding, the crucial result is that public debt does not necessarily penalize economic performance in the long run.
We can then understand that R&D subsidies are not sufficiently studied in the literature, particularly in models accounting for public debt. Besides, the multiple possible effects of public debt are usually individually considered, making a multi-dimensional approach necessary. For that reason, we introduce public debt in the Afonso and Sequeira (2022) model, following a dual approach: (i) a negative externality of debt on each country’s productivity; (ii) a risk premium that would spread across the economy, reflecting the aversion investors have to lend to financially stressed countries. To better understand the effects and their propagation, we assume that only one country (the South) has reached unsustainable debt levels.
Therefore, given the close relationship between the technological-knowledge bias and wages, emphasized by the literature on DTC (Acemoglu 2002; Afonso 2012; Afonso and Sequeira 2022), our North–South dynamic general equilibrium growth model of endogenous DTC with international trade captures the influence of fiscal policy on the steady-state growth rate and income inequality. For this purpose, as in Afonso et al. (2009), we consider that the government may subsidize the production of intermediate goods and R&D activity. In addition, the negative effects of excessive public debt are also introduced. Thus, for the indebted country, a negative externality on productivity and higher financing costs (risk premium) for the government and firms are included.
We also consider that while the North is fiscally prudent and has not accumulated excessive public debt, the South has reached that unsustainable level and suffers from the described negative impacts. Nonetheless, we assume that the level of debt remains exogenous and constant through time. That is, we assume that the South does not accumulate any more debt, but it also does not reduce the current level since it would not be Pareto optimal (Saint-Paul 1992), and it could temporarily increase wage inequality (Röhrs and Winter 2017).
Moreover, we assume that both countries are in an economic and monetary union. Thus, the Central Bank sets an inflation target, the currency is the same for both countries (no role for the exchange rates), and there are no barriers to international trade of intermediate goods. In this framework, we consider that countries may want to mutualize their debts, a policy often suggested in the literature (Beetsma and Mavromatis 2014; Esteves and Tunçer 2016). Each country would issue guarantees on the debt of the other. With this step, the negative effect of public debt on productivity would remain. However, the risk premium would be eliminated, given that the investors’ trust in the fiscal prudent country to guarantee the debt of the other.
Once the model is derived, we conclude that the steady-state growth rate of both economies depends positively on the government subsidies in the South, similar to the conclusions already found in the literature (Afonso et al. 2009; Currie et al. 1999; Afonso and Sequeira 2022; Gil and Iglésias 2020; Davidson and Segerstrom 1998). However, we also conclude that the steady-state growth rate of both countries depends negatively on the externality of the South’s public debt on productivity and the risk premium. In fact, the risk premium also increases the equilibrium wage inequality. The possible negative effects of government subsidies (Afonso et al. 2009; Davidson and Segerstrom 1998) or of fiscal policy in general (Saint-Paul 1992; Greiner 2015; Cheron et al. 2019) have already been introduced by the literature on endogenous growth models. However, by considering the positive and negative effects of fiscal policy, our model better illustrates its multi-dimensional role. Particularly, this combined approach highlights that the gains delivered by government subsidies may be much smaller than what has previously been reported if the public debt has reached unsustainable levels.
Furthermore, in the context of a monetary union, we also show that a country may face the consequences of excessive public debt, even if its own debt is sustainable. Hence, the financial and real effects of public debt in the indebted country affect both economies’ steady-state growth rate and wage premium. So, a common effort to minimize some of these negative impacts would benefit both countries because of the feedback effects. By mutualizing their debts, the risk premium impacts could be minimized, increasing the steady-state growth rate in both economies. Yet, wage inequality would increase in the South. Then, we also contribute to the literature by showing that the negative effects of fiscal policy can be minimized without actually reducing the public debt. In this case, our model shows that government subsidies would expand their positive role in both economies. Naturally, these benefits of a common policy would have to be weighted by its costs, such as moral hazard.
Therefore, several policy implications result from this study. First, the decision to grant government subsidies must be carefully considered since if public debt reaches unsustainable levels, several negative effects will hamper the long-term economic growth rate. Second, in a monetary union, our results indicate that all countries must continuously monitor each other’s public debt, as the negative effects spillover. In fact, the monitoring of country-specific fiscal performances has been discussed in the literature (Hallett and Hougaard Jensen 2012; Juncker et al. 2015; Von Hagen 2014; Sapir and Wolff 2015). Still, its necessity and how it should be executed is far from unanimous. While Von Hagen (2014) argued that monitoring may actually worsen fiscal sustainability, Sapir and Wolff (2015) underlined the benefits of a broader monitorization, encompassing macroeconomic imbalances. Third, our findings also indicate that public debt mutualization is an effective policy to minimize (but not fully eliminate) the negative consequences of public debt. Still, if mutualization is implemented, institutional reform is required to prevent moral hazard concerns. Otherwise, reducing the debt burden may further increase the public debt stock, making this policy ineffective. Finally, the negative effects of unsustainable public debt can only be fully eliminated if the public debt is reduced.
Additionally, our model provides a framework to discuss the inflationary periods from a fiscal policy and monetary union perspective. Excessive inflation has direct adverse effects on the economy, including risk premium affecting nominal interest rates (Shen et al. 1998; Rother 2004). Moreover, fiscal policy volatility and uncertainty can crucially contribute to deepening the negative effects of inflation (Rother 2004; Sims 2011; Montes and Curi 2017). In an inflationary period, at least partially explained by a supply-side shock, governments could have political incentives to support firms by providing more subsidies (Dao et al. 2023; Ginn and Pourroy 2022; Mirza et al. 2023; Prammer and Reiss 2023). In our model, this means that the government in the South would have incentives to increase the subsidies provided to intermediate goods producers and R&D developers, eventually increasing its public debt. On top of the negative externality on productivity and the debt risk premium, that option could increase the nominal interest rate via inflation risk premium, reducing the steady-state growth rate for both economies. By stabilizing the public debt at lower levels (van Aarle et al. 2018) and thus constraining these political incentives, debt mutualization, and institutional reform could bring additional benefits in periods of high inflation.
We begin by providing a specification of the model in Sect. 2, followed by the derivation of the equilibrium and the steady-state in Sect. 3. In Sect. 4, we carry out the numerical implementation and discuss the results. In Sect. 5, we develop a counterfactual analysis, removing international trade from the model. Section 6 presents a brief conclusion.
2 Model
We develop a globalized DTC model, introducing public debt and R&D subsidies in the Afonso and Sequeira (2022) framework. Afonso and Sequeira (2022) defined an endogenous growth model with North–South interactions and monetary policy. Our work aims to capture the multi-dimensional role of fiscal policy. On the one hand, as in Afonso et al. (2009), the government may subsidize the production of intermediate goods and R&D activity. For the indebted country, the modeled consequences are a negative externality on productivity and higher financing costs (risk premium) for the government and firms.
Throughout this section, we present the specification of the model.
2.1 Households
We assume that both countries have a constant number of households that live forever, consume, and earn income from labor and investment in financial assets. Each household inelastically supplies unskilled, L, or skilled labor, H, to the firms producing the final goods.
Households maximize the infinite horizon lifetime utility, U, at each time period, t, given by
where C(t) is the final-good consumption; \(\rho\) is the subjective discount rate; and \(\theta\) is the inverse of the intertemporal elasticity of substitution. Consumers choose the path of C(t) that maximizes the mentioned utility function (2.1), subject to the flow budget constraint:
where a(t) is the households’ real financial assets; m(t) is the real money balances; r(t) is the real interest rate; \(w_{L}(t)\) and \(w_{H}(t)\) are the wages paid to L and H, respectively; T(t) is a lump-sum tax; \(\pi (t)\) is the inflation rate, which represents the cost of money; i(t) is the nominal interest rate; \(\chi\) is the risk premium from the excessive level of public debt and that affects all the economy; b(t) is the amount of money firms borrowed from households; and \(D_{0}\) is the exogenous level of public debt. The Cash-in-Advance (CIA) constraint implies that \(b(t)\le m(t)\), and we assume this constraint as binding: \(b(t)=m(t)\). From expressions (2.1) and (2.2), building the Hamiltonian and applying the maximum principle conditions—see On-line Appendix 1—, it is possible to achieve (i) a no-arbitrage condition between real money holdings and real financial assets:
It is also fundamental to consider the transversality conditions: \(\mathrm{{lim}}_{t\rightarrow +\infty }e^{-\rho t}C(t)^{-\theta }a(t)=0\) and \(\mathrm{{lim}}_{t\rightarrow +\infty }e^{-\rho t}C(t)^{-\theta }m(t)=0\).
Therefore, the Fischer Eq. (2.3) contemplates the additional risk premium from the presence of public debt. When public debt reaches an excessive level, its negative effects spillover to the economy as a whole (Alesina et al. 1992). Thus, on top of the interest rate, the households demand a risk premium to lend money. Following the empirical literature (Díaz et al. 2013; Haugh et al. 2009; Liu 2021; Eminidou et al. 2023), this risk premium is then an additional revenue that households demand to be willing to lend to the firms and to the government of the indebted country. Hence, the risk premium originated from public debt spreads throughout the economy. Later, we will describe the differences in this premium across countries.
2.2 Production and prices
2.2.1 Final goods market
Following Acemoglu and Zilibotti (2001); Afonso and Sequeira (2022); Afonso (2012), in each country, each final good n, \(n\in \left[ 0,1\right]\), is produced by one of two technologies: L, which uses unskilled workers and L-specific intermediate goods, indexed by \(j\in \left[ 0,J\right]\); H, which uses skilled workers complemented with H-specific intermediate goods indexed by \(j\in \left[ J,1\right]\). Thus, the constant returns to scale production function at time t is:
where \(z_{n}(j,t)=q^{k(j,t)}X_{n}(j,t)\) and \(q>1\) represents the size of each quality upgrade resulting from R&D, and \(h>l\ge 1\) highlights the absolute productivity advantage of skilled over unskilled labor. For simplicity, from now on, we normalize \(l=1\) and thus \(h>1\). The product \(Y_{n}\) results then from the combination of intermediate goods—\(X_{n}(j,t)\) with a share of \((1-\alpha )\)—with its specific labor—L or H, with a share of \(\alpha\). Firms buy both types of intermediate goods to producers from both countries, \(\tilde{L}_{f}=\frac{L_{f}}{L_\text{N}+L_\text{S}}\) and \(\tilde{H}_{f}=\frac{H_{f}}{H_\text{N}+H_\text{S}}\), where \(f=\left\{ N,S\right\}\), is the share of L- and H-type intermediate goods production, respectively, of each country that is sold to final producers from country f.
The production of both types of goods is affected by parameter A, specific to each country, that captures the influence of the domestic specific factors (e.g., institutions, like the political regime (Acemoglu et al. 2019) or the policy choices (Fatás and Mihov 2013)) on the productivity. Unlike Afonso and Sequeira (2022), we consider that this parameter is affected by the presence of public debt in the model. The parameter \(\lambda\) constitutes the public debt’s negative externality on productivity when the public debt reaches excessive levels, as described by, for example, Woo and Kumar (2015) and Salotti and Trecroci (2016). By including this negative externality of public debt in a model that also contemplates a risk premium, we better capture the effects of fiscal policy. The negative externality on productivity constitutes an additional channel through which fiscal policy affects the real economy. This dual approach to modeling the consequences of excessive public debt is essential to highlight that some policies may reduce these effects but fail to completely eliminate them.
Then, the North is fiscally prudent, i.e., it does not allow the public debt to reach a level that may damage economic growth and, as a result, we exogenously set \(\lambda _\text{N}=1\). On the contrary, the South has made choices that conducted the public debt to levels that hurt productivity and growth and, consequently, \(\lambda _\text{S}<1\). Hence, the differences in the domestic-specific factors that affect productivity between both countries are now exacerbated by the impacts of the public debt in the South.
The relative productivity advantage is defined by the term n and \((1-n)\), which transforms the index n in an ordering index, and so the larger the n of the final good, the more intensive this is in skilled labor. Thus, a threshold \(\bar{n}(t)\) can be defined; if \(n<\bar{n}(t)\), the final good is produced with L-specific technologies, but if \(n>\bar{n}(t)\), H-technologies are used, which means that, at each moment, \(\bar{n}(t)\)L-specific and \(\left[ 1-\bar{n}(t)\right]\)H-specific final goods are being produced.
In turn, being \(p_{n}(t)\) the real price of the final good n, p(j, t) the real price of the intermediate good j, and \(X_{n}(j,t)\) the demand for the intermediate good, the representative firm (that produces the nth final good) maximizes the profit function:
Producers choose \(X_{n}\), \(L_{n}\) and \(H_{n}\) in order to maximize \(\Pi (n)\), and solving the maximization problem, as in On-line Appendix 1, it is possible to obtain the following demand functions for each intermediate good j by the final producer:
Thus, the higher \(\bar{n}\), the larger the production of the final good produced with unskilled technologies. The final goods are not subject to international trade.
2.2.2 Intermediate goods market
Apart from consumption, final goods are also used to produce intermediate goods, which are then used by domestic firms or traded internationally. Firms apply one unit of aggregate output to produce one unit of the j intermediate good, and thus, the real marginal cost is 1. Nevertheless, a CIA constraint is considered, and so part of the costs \(\left( \Omega _{m}\in \left[ 0,1\right] ,m=\left\{ L,H\right\} \right)\) are financed with money that households lend to firms, subject to the nominal interest rate i(t). Therefore, the cost functions reflect a financial and operational component. The role of fiscal policy must also be considered. However, we now assume that the government can subsidize the production of these intermediate goods by paying a fraction \(z_{X}\in \left[ 0,1\right]\) of the production cost (notice that we are also assuming that the government does not differentiate between goods produced with L- or H-technologies). Thus, the marginal cost function is given by \(\left( 1-z_{X}+\Omega _{m}\cdot (i(t)+\chi )\right)\), where \(\chi\) represents the risk premium demanded by lenders when there is an excess of public debt in that country. The risk premium affects all the sectors of the indebted economy that need financing from the families (Alesina et al. 1992). It is not a component intrinsically associated with the domestic interest rate but rather a cost that the firms and the government of the indebted country must pay to borrow, following, for instance, the empirical results of (Díaz et al. 2013; Haugh et al. 2009; Liu 2021). Nevertheless, we assume that firms only get funds from domestic households. While in the North \(\chi _\text{N}=0\) due to fiscal prudence (no risk premium affecting the economy), in the South \(\chi _\text{S}>0\). Then, apart from possible differences in the nominal interest rates, the financing costs in both countries also diverge due to the presence of a risk premium in the South.
Each intermediate producer is a monopolist on that specific j intermediate good, because of the patent. Thus, the monopolist maximizes the profit flow:
where X(j, t) represents the aggregate demand function: \(X(j,t)_{|0<j<J}=\int _{0}^{\bar{n}(t)}X_{n}(j,t)\text{d}n\) and \(X(j,t)_{|J<j<1}=\int _{\bar{n}(t)}^{1}X_{n}(j,t)\text{d}n\). Because it is assumed that intermediate goods depreciate after each period t, the monopolist does not face any dynamic constraints and chooses p(j, t) to maximize the profit, resulting in \(p(j,t)_{|0<j<J}=\frac{1-z_{X}+\Omega _{L}\cdot \left( i(t)+\chi \right) }{1-\alpha }\) or \(p(j,t)_{|J<j<1}=\frac{1-z_{X}+\Omega _{H}\cdot \left( i(t)+\chi \right) }{1-\alpha }\) – see On-line Appendix 1. In practice, because \(0<\alpha <1\), the monopolist is setting a mark-up \(q=\frac{1}{1-\alpha }\) over the marginal cost, such that \(p(j,t)_{|0<j<J}=q\left[ 1-z_{X}+\Omega _{L}\cdot \left( i(t)+\chi \right) \right]\) or \(p(j,t)_{|J<j<1}=q\left[ 1-z_{X}+\Omega _{H}\cdot \left( i(t)+\chi \right) \right]\).
2.3 Authorities
2.3.1 Monetary authority
Regarding the monetary policy, Afonso and Sequeira (2022) assumed that the authority sets an inflation target and exogenously chooses the nominal interest rate. Given the Fisher Eq. (2.3), the inflation rate \(\pi (t)\) is endogenously determined. Thus, the monetary authority varies the money growth rate by whatever is necessary for the chosen nominal interest rate to prevail in each country and thus to achieve its inflation target. The seigniorage revenue that the monetary authority obtains with the change of the money growth rate is transferred to the fiscal policy authority.
2.3.2 Government
While Afonso and Sequeira (2022) do not include the role of fiscal policy, we now consider the presence of a government that can subsidize the production of intermediate goods and R&D activities, as in Afonso et al. (2009). The economic incentives that motivate these subsidies will be detailed further on. We will also assume that there is an initial exogenous level of public debt.1 However, the government still runs a balanced budget (thus, not accumulating more debt nor making an effort to reduce the absolute level of debt). Therefore, the government budget constraint:
where T(t) is a lump-sum tax (as already mentioned); \((\dot{m}(t)+\pi (t)\cdot m(t))\) is the seigniorage revenues transferred from the monetary authority; \(z_{R}\in \left[ 0,1\right]\) is the government subsidy in the case of the South for R&D activities, and it is assumed that the Southern government does not differentiate between goods produced with L- or H-technologies; R(t) corresponds to the total resources devoted to R&D activities; S(t) corresponds to the primary public spending, except the subsidies; and \(D_{f}\) therefore represents to the exogenous level of debt. Thus, in equilibrium, the lump-sum taxes and the seigniorage revenue must be sufficient to cover the public spending, the subsidies, the interests, and the risk premium—based on Bernoth et al. (2012), for instance—on the public debt.
2.4 R&D sector
Regarding the R&D sector, it is important to underline that the North innovates, expanding the technological-knowledge frontier and increasing the quality of the intermediate goods, while the South imitates the innovations of the North. Although the innovator in the North is protected with a patent, that patent is not enforced abroad. The monopolist who produces under the patent uses limit pricing to ensure monopoly. The probability of innovation for the North and the probability of imitation for the South can be represented, respectively:
where \(f=\left\{ N,S\right\}\), \(y_{f}(j,t)\) is the flow of domestic resources (measured in terms of final good) committed to R&D in intermediate good j; \(0<\beta _\text{S}<\beta _\text{N},\,k_\text{S}\le k_\text{N}\) represents the learning-by-doing positive effects of past successful innovations/imitations (the South imitates the innovations of the North, and so it has lower learning-by-doing effects); \(\varsigma _{f}^{-1}q^{^{-\alpha ^{-1}k(j,t)}},\,\varsigma _\text{N}>\varsigma _\text{S}>0\) constitutes the cost of complexity, i.e., the difficulty that results from the increased complexity of the improvements (and it is higher for the case of innovations); \(\left( \sum _{m=f}m_{f}\right) ^{-\xi _{f}}=(m_\text{N}+m_\text{S})^{-\xi _{f}}\), \(m=L\) when \(0\le j<J\) and \(m=H\) when \(J<j\le 1\), \(\xi _{f}>0\), captures the negative effect of the market size, i.e., that fact the the larger the market the more difficult it is to introduce new intermediate goods. Besides, we assume that \(\frac{\beta _\text{S}}{\varsigma {}_\text{S}}>\frac{\beta _\text{N}}{\varsigma {}_\text{N}}\) to account that imitation is less costly than innovation, regardless of the learning-by-doing effect favoring the North.
3 Equilibrium and steady-state
In the present section, we start by describing the equilibrium in all the markets, after which we compute the steady state.
3.1 Production and prices
As described above, the relative productivity advantage is defined by the terms n and \((1-n)\), and the larger the n of the final good, the more intensive this is in skilled labor. Thus, a threshold \(\bar{n}(t)\) can be defined, and if \(n<\bar{n}(t)\), the final good is produced with L-specific technologies, but if \(n>\bar{n}(t)\), H-technologies are used. Then, at each moment, \(\bar{n}(t)\) reflects the optimal choice between L- or H-technologies, derived from profit maximization (of both final and intermediate goods producers) and full-employment equilibrium in factor markets (from the profit maximization conditions of final goods production), given the labor supply and the current state of technological-knowledge. Thus, as developed in On-line Appendix 1,
where \(f=\left\{ N,S\right\}\), depending on which country we are referring to. \(G(t)\equiv \frac{Q_{H}(t)}{Q_{L}(t)}\), \(Q_{L}(t)\equiv \intop _{0}^{J}q^{k(j,t)\left[ \frac{1-\alpha }{\alpha }\right] }dj\), \(Q_{H}(t)\equiv \intop _{J}^{1}q^{k(j,t)\left[ \frac{1-\alpha }{\alpha }\right] }dj\), \(\tilde{L}_{f}=\frac{L_{f}}{L_\text{N}+L_\text{S}}\) and \(\tilde{H}_{f}=\frac{H_{f}}{H_\text{N}+H_\text{S}}\). \(Q_{L}\) and \(Q_{H}\) represent aggregate quality indexes of the technological-knowledge stocks, and G(t) is the measure of the technological-knowledge bias.
Besides, from the computation of the threshold final good in On-line Appendix 1, we also retrieved the ratio of index prices of goods used with H- and L- technologies. Recalling that \(p_{n}(t)\) is the real price of the final good n and considering that at \(\bar{n}(t)\), the firms using each type of technology should break even, we get the ratio of price indices:
The lower the threshold final good \(\bar{n}\), the lower the relative price of final goods produced with H-technology. From the derived expressions of the demand for intermediate goods, we can conclude that the lower the threshold, the lower the relative price of the final good and, hence, the lower the demand for H-intermediate goods, discouraging R&D activities. Each price index is defined further on.
On the other hand, substituting the demand functions of the intermediate goods (2.7) and (2.8) in the constant returns to scale production function (2.5):
where \(f=\left\{ N,S\right\}\), depending on which country we are referring to. Moreover, \(\tilde{L}_{f}=\frac{L_{f}}{L_\text{N}+L_\text{S}}, \tilde{H}_{f}=\frac{H_{f}}{H_\text{N}+H_\text{S}}\). Thus, the production of the nth final good results from the joint use of skilled and unskilled labor and low and high-skilled specific intermediate goods. We should also recall that there is international trade of intermediate goods but not final goods. Then, firms buy the intermediate goods to the North, if the South has not been able to imitate yet. It buys to the South if imitation has already occurred, to minimize the costs. Nevertheless, both countries produce several different varieties for each type of intermediate goods. Therefore, firms buy low and high skilled-specific goods from both countries.
Normalizing \(P_{Y}\) to 1 (by imposing that \(P_{Y}=\exp \int _{0}^{1}\ln p_{n}dn=1\)), and recalling the threshold \(\bar{n}\)—expression (3.1) – and the ratio of index prices—expression (3.2)—, we get the real price index of L-technology final good, \(p_{L}(t)=p_{n}\cdot (1-n)^{\alpha }=\exp \left( -\alpha \right) \cdot \bar{n}(t)^{-\alpha }\), and of H-technology final good, \(p_{H}(t)=p_{n}\cdot n^{\alpha }=\exp \left( -\alpha \right) \cdot (1-\bar{n}(t))^{-\alpha }\) —see On-line Appendix 1. Following (Acemoglu and Zilibotti 2001) and considering that the composite final good is given by \(Y=\int _{0}^{1}p_{n}\cdot Y_{n}dn\), we get—see On-line appendix 1:
where \(f=\left\{ N,S\right\}\) depending on which country we are referring to. We can then conclude that the economic growth rate, i.e., the growth rate of the production of the final good, is determined by technological-knowledge progress. The composite final good is then affected by the subsidies granted to produce intermediate goods and the negative effects of public debt. While granting subsidies requires a discretionary decision from the government of the South, the adverse effects of debt occur when the country has an excessive level of public debt.
Proposition 1
The level of the composite final good is affected by the multiple impacts of fiscal policy. Ceteris paribus: (i) subsidies to the production of intermediate goods increase the level of output; (ii) if public debt reaches an excessive level, the negative externality of public debt on productivity, \(\lambda\), reduces the level of composite final good; (iii) also when public debt reaches an excessive level, the risk premium that spreads through the economy reduces the level of output.
If the government in the South decides to increase the subsidies for the production of intermediate goods, from expression (3.4) we conclude that the level of the composite final good increases. Intuitively, these subsidies reduce the unitary cost of producing intermediate goods in the South, which is passed on to the prices by the domestic intermediate goods producers. Then, because of the lower price, final good producers increase the demand for intermediate goods, as derived in Sect. 2.2.1, resulting in a higher final output level for each final good producer. Then, the overall level of the composite final good is greater after introducing (or increasing) the subsidies. Notice that the level of final output rises in both countries in response to subsidies in the South. This effect is only possible due to the international trade of intermediate goods. Once final producers use intermediate goods produced in both countries, any variation in the price of these products in one country affects final production in both countries.
The risk premium has a similar underlined economic mechanism. If the risk premium increases, the unitary costs of the intermediate good producers (that face the CIA constraint) increase. Higher prices must be set to counterbalance the cost increase, leading to a lower demand for intermediate goods by the final producers, and lower level of the composite final good in both countries. As it will be demonstrated further on, international trade of intermediate goods is fundamental for this result. If there was no trade, the negative externality of public debt in the South would not have any consequences for the North’s composite final good. The absence of international trade for intermediate goods would isolate the economies from shocks in the foreign’s intermediate goods production.
On the other hand, a positive exogenous shock in the South’s public debt would increase the negative externality of public debt on productivity (lower \(\lambda\)). The productivity of the domestic final good producers would decrease, reducing the production of the final good obtained with the same amount of intermediate goods. Then, the domestic final producers no longer want to maintain the previous output level, resulting in lower intermediate-goods demand and, naturally, a lower level of composite final good.
Finally, since full employment in the labor market is guaranteed—result already mentioned to determined \(\bar{n}(t)\)—and since the marginal productivity of labor (of each type) equals the marginal cost, we can derive the equilibrium skill premium—see On-line Appendix 1:
This skill premium captures the wage inequality and is greater: (i) the more G is biased toward skilled labor (in this case the greater the absolute productivity advantage of this type of labour); (ii) the lower the availability of skilled labor; (iii) the smaller the impact of CIA constraints on the production of intermediate goods produced with H-technology. How the subsidies, the nominal interest rate, and the risk premium affect wage inequality depends on the relation between \(\Omega _{L}\) and \(\Omega _{H}\). If the CIA constraint affecting the production of unskilled intermediate goods is greater (lower) from the one affecting the intermediate goods produced with skilled technology, the intermediate-good production subsidies have a positive (negative) impact on the wage premium.
Proposition 2
If the CIA constraints affecting L-type intermediate goods are greater, \(\Omega _{L}>\Omega _{H}\), (lower, \(\Omega _{L}<\Omega _{H}\)) than the ones affecting H-type intermediate goods in both countries, the intermediate good production subsidies, \(z_{X}\), increases (reduces) the equilibrium skilled premium. On the other hand, if both CIA constraints are identical, the subsidies do not affect wage inequality. Even if the CIA constraint applies to only one economy, if the constraints affecting L-type intermediate goods are greater (lower), the intermediate good production subsidies, \(z_{X}\) increases (reduces) the equilibrium skilled premium.
Given that we will assume that \(\Omega _{L}>\Omega _{H}\) for both countries,2 introducing intermediate good production subsidies reduces the unitary costs for all the intermediate good producers. However, because of the difference in the CIA constraints, the costs for those using L-type technology are greater than for those using H-type technology. Then, the introduction or increase of the subsidies makes the H-type intermediate goods relatively cheaper compared to the L-type, increasing the bias to H-type technology. Hence, final good firms will produce more final goods with H-technologies and demand more skilled labor, increasing the wage premium between skilled and unskilled workers.
Nevertheless, it is also important to notice that for the North \(\chi _\text{N}=0\) (it is fiscal prudent, and so there is no risk premium affecting the economy), while for the South \(\chi _\text{S}>0\). Hence, the public debt in the South reduces the equilibrium skilled premium via the effects on the costs of financing, as defined in Proposition 3.
Proposition 3
The impact of public debt on the skilled premium depends on whether or not there is an excessive level of public debt (in which case there is a negative externality on productivity and a risk premium that spreads through the economy and so \(\chi >0\)). If the country keeps public debt within sustainable levels, no risk premium affects the costs of financing (\(\chi =0\)) and no negative externality on productivity, and so there are no effects on skilled premium. If and only if there is an excessive public debt, if the CIA constraints affecting L-type intermediate goods are different from the ones affecting H-type intermediate goods (\(\Omega _{L}\ne \Omega _{H}\)), at least for one of the countries, the presence of a risk premium reduces the equilibrium skilled premium. If in each country the CIA constraints for the different types of intermediate goods are similar, the risk premium does not affect the wage inequality.
An increase in the risk premium makes the financing costs higher in the South for all intermediate goods producers. Still, given \(\Omega _{L}>\Omega _{H}\), the intermediate goods producers using the L-type technologies are most harmed by this increase, given that they are more dependent on money borrowed from the households. At the same time, the unitary cost of H-type intermediate goods also increases, though less. One could think this last type of goods would become more attractive, increasing the wage premium. However, the monopolist passes the higher costs to the final good producers, via price. To face this undesired effect, the final firms reduce the production of the goods that require the more expensive labor: those using H-type intermediate goods. The lower demand for skilled labor reduces the wage premium.
3.2 R&D sector
Following the aforementioned specification of the probability of innovation, for the North, and the probability of imitation, for the South, the free-entry equilibrium in the R&D sector implies that the expected revenue must be equal to the total amount of resources devoted to such activity. Similarly to Afonso and Sequeira (2022), we consider that the CIA constraint also affects the R&D sector. Thus, part of the costs, \(\gamma _{0}\), must be financed with money borrowed from the households. However, as already mentioned, we also consider that the South’s government grants subsidies to R&D activities. Thus, joining all these components, the amount of domestic private resources devoted to R&D is given by \(y_\text{S}(j,t)\cdot \left[ 1-z_{R}+\gamma _\text{S}\cdot (i_\text{S}(t)+\chi )\right]\) in the South and \(y_\text{N}(j,t)\cdot \left[ 1+\gamma _\text{N}\cdot i_\text{N}(t)\right]\) in the North, with \(\gamma _{f}\in \left[ 0,1\right] ,f=\left\{ {N}, {S}\right\}\) being the share of the R&D costs financed with money borrowed from the households. The previous expressions show that part of the domestic resources devoted to R&D are not directly applied to these activities. The liquidity constraint increases the costs of promoting R&D by introducing the role of interests. Given that the expected revenue should equal the amount invested, the CIA constraint reduces the overall amount devoted to R&D activities. If the promoters had the financial capacity to support the projects without external funding, the probability of success would be greater.
Regarding the revenues, these must be weighted by the probability of success—from expression (2.12) –, and thus they are given by \(I_{f}(j,t)\cdot y_{f}(j,t),\,f=\left\{ N,S\right\}\). Considering that the probability of success of R&D depends on the amount of resources actually devoted to R&D, the CIA constraint also affects the expected revenues.
Then, the described dual effects of the CIA constraint make the R&D sector’s equilibrium different from the one that would arise without liquidity problems. Moreover, by increasing the interest rate, public debt worsens these consequences. Therefore, the government has the incentive to introduce subsidies, reducing the overall negative impacts of the CIA constraint and public debt externalities.
Hence, the free-entry equilibrium in the South can be represented as:
where \(V_\text{S}(j,t)\) constitutes the value of the leading-edge patent for the producer of an intermediate good j at time t. The value of each patent corresponds to the present value of the profits generated by the intermediate good producer under that patent. However, each period’s profits must be weighted by the probability of successful innovation in the other country. If that innovation occurs, an upgraded intermediate good enters the market and the patent is no longer useful. Therefore, and as in Afonso and Sequeira (2022), \(V_{s}(j,k,t)=\frac{\Pi _\text{S}(j,k,t)}{r_\text{S}(t)+I_\text{N}(j,t)}\), in equilibrium. Removing the market-size scale effects (\(\xi _\text{S}=1\)), we can get the equilibrium probability of successful innovation in a H-specific intermediate good in the North—see On-line Appendix 1:
where \(D_{H}(t)=\tilde{H}_\text{S}\left[ A_\text{S}^{\lambda }\cdot p_{H,S}(t) \right] ^{\frac{1}{\alpha }}+\tilde{H}_\text{N}\left[ A_\text{N}\cdot p_{H,N}(t)\right] ^{\frac{1}{\alpha }}\), \(\tilde{H}_{f}=\frac{H_{f}}{H_\text{N}+H_\text{S}}\), \(A_\text{N}>A_\text{S}^{\lambda }\) and \(H_\text{N}>H_\text{S}\).
We can immediately conclude that both countries influence each other due to the international trade of intermediate goods. The expected revenues in the South are greater the lower the probability of innovation in the North. This result is intuitive if we recall that the less developed economy (the South) can imitate at a lower cost. Similarly, the value of innovations in the North is greater the lower the likelihood of imitation by the South. These cross-border interactions make the equilibrium probability of successful innovation in an H-specific intermediate good in the North dependent on the parameters of the South. Naturally, free trade and international competition are fundamental assumptions for this result. If countries were isolated, intermediate goods producers would be protected, and the R&D sector outcome would not depend on the neighboring country’s parameters.
Then, in equilibrium, the probability of successful innovation in an H-specific intermediate good is influenced by the characteristics of the production side of the economy (technology channel), a price channel captured by \(D_{H},\) and a CIA constraint channel. Once again, it becomes clear that the CIA constraint negatively affects the outcome in the R&D sector. The competitive market allocation becomes impossible to achieve, ceteris paribus. Government subsidies act in the opposite direction and may or may not compensate for the overall impact of the liquidity constraint, depending on the authority’s intentions.
Having defined the equilibrium probability of successful innovation in an H-specific intermediate good, and once that probability determines the speed of technological-knowledge progress, equilibrium is defined by the path of this progress in the North, from which the South also benefits via free trade of intermediate goods. With this relation, we get the equilibrium growth rate of H-specific technological-knowledge—see On-line Appendix 1:
From the previous two expressions, it becomes clear that there are feedback effects. Access to the latest intermediate goods increases production, increasing the resources available to R&D in the South (to conduct imitations). This feeds back to the North, increasing its technological-knowledge through creative destruction.
3.3 Steady-state equilibrium
Given that we are assuming free trade of intermediate goods, both countries have access to the state-of-the-art of these goods and the same production technology.3 Thus, the steady-state growth rate must be the same for both economies. Taking into consideration the Euler Eq. (2.4) derived in the households’ maximization problem, this means that interest rates must also be equal between North and South in the steady-state.
The aggregate final good is divided between consumption and savings, that are distributed between production of intermediate goods and R&D, which results in the aggregate resources constraint \(Y_{f}(t)=C_{f}(t)+X_{f}(t)+R_{f}(t),\,f=\left\{ N,S\right\}\), where \(Y_{f}(t)\) is the composite final good, \(C_{f}(t)=\int _{0}^{1}c_{f}(i,t)\text{d}i\) is the aggregate consumption, \(X_{f}(t)=\int _{0}^{1}\int _{0}^{1}X_{n,f}(j,t)\text{d}n\text{d}j\) is the aggregate intermediate goods, and \(R_{f}(t)=\int _{0}^{1}y_{f}(j,t)\text{d}j\) is the total resources devoted to R&D. Hence, the growth rate of each one of these variables in steady-state must be the same in both countries and equal to the North’s technological-knowledge progress.
Given the constant returns to scale in the production of the composite final good and the Euler equation, we obtain the steady-state growth rate for the North and South:
This expression results in constant steady-state levels of threshold final goods, final and intermediate goods price indexes, wage premium, and North–South technological-knowledge gap in both types of technologies.
As shown in On-line Appendix 1, with some algebra and relating to the previous expressions we get that \(r^{*}\)and \(g^{*}\) are given by:
From these expressions, we can conclude that both the positive effects of subsidies and the negative effects of public debt feedback to the North. Thus, the excessive level of public debt in the South reduces the steady-state growth rate in both economies.
Proposition 4
The steady-state growth rate in both North and South depends positively on the subsidies (to the production of intermediate goods and R&D) granted in the South and negatively on the externality of public debt in productivity and the risk premium affecting the Southern indebted economy. The larger the level of public debt in the South when compared to the North, the smaller is \(\lambda\), and so the smaller the growth rate. None of the signs of the described effects on the steady-state growth rate depend on the sign of the CIA constraint.
For a given level of private expenditures with R&D, granting subsidies increases the probability of imitation in the South and innovation in the North, due to feedback effects. If imitation in the South is now more likely, firms in the North must make an additional effort to reduce the likelihood that a successful imitation captures their demand. Then, the equilibrium growth rate of H-specific technological-knowledge increases, as demonstrated in Sect. 3.2. If the economy as a whole is accumulating technological-knowledge faster, it can also consume and produce faster, leading to a greater steady-state growth rate in both the North and South.
On the other hand, subsidies to the production of intermediate goods make each imitation more valuable by reducing the unitary costs of producing the intermediate good (a reduction that can then be passed to prices, increasing the demand for these goods). If the expected return of imitations increases, the firms in the South are willing to increase their R&D expenditures, increasing the likelihood of a successful imitation. As previously described, the feedback effects, particularly that a successful imitation can replace the old good in the market due to the lower price, induce a greater effort by the firms in the North, and increase the likelihood of successful innovations. Thus, the equilibrium growth rate of H-specific technological-knowledge and the steady-state growth rate of the economy increases. Thus, following previous results, subsidies boost the economic growth rate in both countries (Afonso et al. 2009; Celli et al. 2021; Currie et al. 1999).
Notwithstanding, there is extended literature on how the effects of fiscal stimulus are hampered when public debt is high (Bi et al. 2016; Huidrom et al. 2020; Ilzetzki 2011; Mountford and Uhlig 2009; Occhino 2023). So, the decision to grant government subsidies has to be carefully considered by the government in the South since it may contribute to increasing the already unsustainable levels of public debt, reducing even the long-run economic growth rate. In our model, high debt levels affect growth through the effects of the risk premium and externality on productivity.
The risk premium follows a similar but opposite mechanism compared to the subsidies. For the same previous level of private expenditures with R&D, the risk premium reduces the probability of fruitful imitation (now, a higher share of the effort is committed to paying the interest rates, including the risk premium, rather than pursuing R&D activities). Besides, it also increases the unitary costs of producing the intermediate goods, which must be compensated in the prices, diminishing the overall demand for these goods. Differently, the firms in the North can relax their R&D efforts due to the lower risk of successful imitations, resulting in a lower probability of innovation. Then, the accumulation of H-specific technological-knowledge is slower, leading to a slower steady-state growth rate for both economies.
The negative externality of public debt on productivity does not influence the production costs or the share of the R&D expenditures devoted to the actual R&D activities. Notwithstanding, it does reduce the productivity of final goods producers, incentivizing them to reduce the output level, resulting in lower demand for intermediate goods. Thus, intermediate goods become less profitable, reducing the expected return of a successful imitation. As before, the feedback effects also reduce the probability of innovations in the North, decreasing the output steady-state growth rate for both economies.
These negative effects of excessive public debt match previous findings in theoretical studies (Greiner 2015; Saint-Paul 1992) or in the empirical literature (Albu and Albu 2021; Baum et al. 2013; Bhimjee et al. 2020; Checherita-Westphal and Rother 2012). Checherita-Westphal and Rother (2012) estimated a negative link between excessive public debt and economic growth. With data gathered for Euro Area countries, they provide empirical evidence that when public debt rises above a given threshold, the GDP growth rate significantly reduces. Moreover, they also estimated the channels through which public debt affects growth. Three statistically significant channels were found in the Euro Area: (i) private savings; (ii) public investment; (iii) total factor productivity. When public debt rises above certain levels, it carries negative consequences to productivity, affecting economic growth. Then, Checherita-Westphal and Rother (2012) work confirms our option to introduce an externality on productivity and the results it produces.
However, the channels of private saving, public investment or even private investment (this last one is considered as a possibility but not empirically confirmed) estimated by Checherita-Westphal and Rother (2012) are not directly present in our model. Instead, we considered a negative externality of public debt on interest rates. Although this broader channel is not aligned with Checherita-Westphal and Rother (2012) findings (they did not find empirical evidence connecting public debt and interest rates), it is widely supported by the empirical literature (Afonso et al. 2012; Baum et al. 2013; Costantini et al. 2014; De Santis 2014; Kempa and Khan 2017; Maltritz 2012). Even the expectation of a future increase in public debt may be enough to increase the risk premium (Afonso et al. 2012). Hence, the empirical literature on the topic confirms the validity and the results of introducing an externality of public debt on the indebted country’s interest rate. Furthermore, the effects of the interest rate externality are not limited to the indebted country’s economy. According to the empirical literature for the Euro Area, the increase of the risk premium in the indebted economies spillovers to countries with sustainable levels of public debt (Afonso et al. 2012; Antonakakis and Vergos 2013; De Santis 2014; Kempa and Khan 2017; Faini 2006). Thus, the conclusion that the risk premium in the South hampers the growth rate in both economies is supported by the empirical literature, as the increase of the risk premium in the South spillovers to the North.
Additionally, from expression (3.5) and assuming \(\Omega _{L}>\Omega _{H}\), we can distinguish the steady-state skill premium in both countries (recalling that in the North both subsidies to the production of intermediate goods and risk premium are absent):
Thus, given the assumptions underlined in the model, the steady-state skill premium is larger in the South. Given that we are assuming \(\Omega _{L}>\Omega _{H}\) in both countries, \(\tilde{H}_\text{N}>\tilde{H}_\text{S}\), and \(\tilde{L}_\text{S}>\tilde{L}_\text{N}\), the subsidies to the production of intermediate goods exacerbate the difference between wage inequalities, for a given \(G^{*}\). The risk premium contributes to reducing this difference. Then, our results confirm the connection between wage inequality and public debt found in previous literature (Azzimonti et al. 2014; Miyashita 2023; Salti 2015; Carrera and de la Vega 2021; Arslan 2019).
The impact of the steady-state technology-knowledge bias \(G^{*}\) on the wage premium is further analyzed with the help of numerical tools. The previous expression shows that when this bias increases, the wage premium also increases.
3.3.1 Steady-state under mutualization
From the analysis of Proposition 4, it is clear that the adverse effects of public debt in the South feedback to the North, resulting in a lower steady-state growth rate in both economies. Although the North is growing at a higher output level, it could be growing at a higher rate if it was not affected by the South’s public debt externalities. Thus, with both countries belonging to the same monetary union, one possible policy to implement is public debt mutualization (Ando et al. 2023; Beetsma and Mavromatis 2014; Esteves and Tunçer 2016; van Aarle et al. 2018; Steinberg and Vermeiren 2016). For instance, each country could guarantee the public debt of the other. Considering the North’s fiscal prudency reputation, we can expect that the risk premium in the South would be significantly reduced.
We can consider this approach in our model by assuming that, under mutualization, the debt levels of each country would remain the same (thus, the negative externality on productivity, \(\lambda\), would still be present), but the prudent country would guarantee the debt of the South. Thus, no risk premium would arise (\(\chi _\text{S}=\chi _\text{N}=0\)). Hence, eliminating the risk premium, the steady-state growth rate would be:
and so, assuming that mutual guarantees would eliminate the risk premium, the steady-state growth rate would increase. Besides, the new steady-state skill premium in the South would be given by:
As already demonstrated, the wage premium in the South would increase in the absence of the risk premium—comparing to expression (3.13) –, making even greater than the wage inequality in the North:
Assuming that mutual guarantees of the public debt between the North and South would reduce or eliminate the risk premium in the South, the steady-state growth rate would increase in both economies. The negative effects of public debt in the South would still be present and would still influence this rate due to the negative externality on productivity. For a given technology-knowledge bias, this policy would still increase the wage inequality in the South, increasing the difference to the North, to which the increase would be smaller. If both CIA constraints are identical, the risk premium only affects wage inequality by affecting \(G^{*}\).
As explained in Proposition 4, the risk premium makes the probability of successful imitation and innovation smaller and increases the unitary costs of producing intermediate goods in the South. Then, it reduces the steady-state growth rate for both economies. When we consider the mutualization hypothesis, investors trust in the fiscal prudency of the North and no longer demand a risk premium to the South. For a given level of expenditures with R&D, the share committed to financing costs reduces in the South, and more resources are devoted to actual R&D activities, increasing the possibility of successful imitations. Besides, the unitary production costs of intermediate goods decrease, which should be passed to prices. The price reduction increases the demand for these goods, making the imitations more profitable. Thus, intermediate goods producers are encouraged to devote more effort to R&D, increasing the probability of imitation. Now, the firms in the North must compensate and increase the resources committed to R&D, increasing the probability of innovations. Overall, mutualization, by eliminating the risk premium, increases the steady-state growth rate of both economies.
We should highlight that assuming that mutualization eliminates the risk premium is unnecessary to describe the positive effects (only determines the size of those effects). As long as mutualization allows to reduce the risk premium, it induces a greater growth rate. Naturally, in practice, the more this policy can reduce the risk premium (and increase the growth rate), the more both countries benefit from it, and so the more likely it is that the North is willing assume the other country’s debt risk. Additionally, even if the risk premium is completely eliminated, the public debt’s externality on productivity still affects the steady-state growth rate of both countries. This is why Ando et al. (2023), for instance, described the help to steer the country into a sustainable path as one of the advantages of mutualization. Once mutualization is implemented, it is up to the indebted country to build on the common effort and reduce its public debt level. Thus, while mutualization can be useful to minimize the consequences of excessive public debt, it does not fully eliminate them. In fact, some authors (Alcidi and Gros 2015; Amato et al. 2022; Bilbiie et al. 2021; De Grauwe 2012) have highlighted that, by reducing the debt burden, mutualization could incentive further reckless policies from the indebted countries, increasing the public debt stock. If that is the case, implementing debt mutualization in a monetary union without adequate institutional reform would lead to much lower effects than what we have described. Eventually, if investors believe that the risk has risen, they might demand a risk premium again, neutralizing the impacts of the policy. Considering this in mind, when De Grauwe (2012) advocated debt mutualization, it also underlined the necessity to reform European institutions, tackling the moral hazard problem.
On the other hand, as previously described, fiscal policy can amplify the already negative effects of excessive inflation (Rother 2004; Sims 2011; Montes and Curi 2017). During an inflationary period, partially explained by a supply-side shock and the substantial cost increase it represents to firms, governments could face political pressures to implement support programs to help families and firms. In light of our model, this would be represented by additional subsidies to intermediate good producers and R&D developers in the South, without similar increases in terms of tax revenue. The resulting public debt increase in the South would aggravate the negative externality on productivity and risk premium, reducing the steady-state growth rate. However, with inflation above normal, other negative impacts would also arise, with the inflation risk premium inducing higher nominal interest rates. Debt mutualization gains then additional importance. By containing these political incentives and constraining debt at lower levels (van Aarle et al. 2018), the inflation risk premium would be reduced, and positive impacts on output would be verified.
4 Transitional dynamics and discussion
We analyze the transitional dynamics once the equilibrium and the steady state have been derived. First, we should recall that \(G(t)=\frac{Q_{H}(t)}{Q_{L}(t)}\) and that all the variables of interest are multiples of the quality indexes. Hence, the transitional dynamics of these variables will be determined by the dynamics of the technology-knowledge bias, computed in On-line Appendix 1. Therefore, we now focus on the transitional dynamics of G(t) in different scenarios.
We first compute the dynamics in the baseline case. To calibrate the model, we follow Afonso and Sequeira (2022) work for most of the parameters. However, comparing to the former authors, we introduced an externality of public debt on productivity (\(\lambda _\text{S}\)) and a risk premium on the interest rate of the indebted country (\(\chi\)). First, Salotti and Trecroci (2016) estimated the effects of pubic debt using a panel data with 20 OECD countries, from 1970 to 2009. They concluded that an additional 30p.p. on the public debt ratio to GDP could explain the reduction of the labour productivity growth by at least 0.198p.p. This effect could actually be larger, depending on how labour productivity has been measured. To be prudent, we calibrated the externality of public debt on productivity to 0.02. As for the risk premium, we focus our analysis on Germany and Italy, which in our model would correspond to the North and South, respectively. Assuming that the nominal interest rate is common to both countries, the rate faced by the South is given by \(i+\chi\). Then, we considered \(\chi\) equal to \(0.5\cdot i\), as between 2000 and 2014 the Italian sovereign bond yield was, on average, 1.56 times greater than the German.4
Therefore, the baseline calibration for all the parameters is the following (Table 1):
Table 1
Baseline calibration
Parameter
Description
Value
Parameter
Description
Value
\(\alpha\)
Share of labor in production
0.6
h
Absolute productivity advantage H labor
1.2
q
\(\frac{1}{1-\alpha }\)
2.5
\(z_{X}\)
Subsidies production intermediate goods
0.1
\(\beta _\text{N}\)
Regulates positive effects of past success
1.6
\(z_{R}\)
Subsidies to R&D
0.1
\(\beta _\text{S}\)
Regulates positive effects of past success
1
\(\Omega _{H,{\text N}}\)
CIA production of H-type in the North
0.2
\(\varsigma _\text{N}\)
Regulates cost of complexity in innovations
2.5
\(\Omega _{H,{\text S}}\)
CIA production of H-type in the South
0.6
\(\varsigma _\text{S}\)
Regulates cost of complexity in imitations
1
\(\Omega _{L,{\text N}}\)
CIA production of L-type in the North
0.4
\(H_\text{N}\)
Skilled labor in the North
2
\(\Omega _{L,{\text S}}\)
CIA production of L-type in the South
0.8
\(H_\text{S}\)
Skilled labor in the South
1
\(\gamma _{H,{\text S}}\)
CIA R&D of H-type in the South
0.7
\(L_\text{N}\)
Unskilled labor in the North
1
\(\gamma _{L,{\text S}}\)
CIA R&D of L-type in the South
0.9
\(L_\text{S}\)
Unskilled labor in the South
2
\(\lambda _\text{S}\)
Externality of public debt on productivity
0.02
\(A_\text{N}\)
Domestic factors in productivity—North
1.6
G(0)
Initial value technology-knowledge bias
1
\(A_\text{S}\)
Domestic factors in productivity—South
1.2
\(i_\text{N}\), \(i_\text{S}\)
Nominal interest rates
0.05
\(\pi ^{*}\)
Inflation target
0
\(\chi\)
Risk premium of the indebted country
\(0.5*i\)
With the previously described calibration and using numerical tools, Fig. 1 plots the transitional dynamics for G after a 10% positive or negative shock in \(G^{*}\):
Fig. 1
Transitional dynamics of G—baseline case
×
In each figure, we plot two different cases: the dynamics under debt mutualization (no risk premium) and the dynamics without debt mutualization. As we can see, the steady-state level of the technology-knowledge bias \(G^{*}\) is lower without mutualization, though the overall dynamics are similar. Besides, as already demonstrated, the steady-state growth rate in the North and South would increase due to mutualization. Following the same logic, without mutualization, the lower technology-knowledge bias in steady state would be accompanied by a lower steady-state growth rate. This result is intuitive. When the risk premium is positive, the effective nominal interest rate is greater. Given the CIA constraint, the costs of producing the intermediate goods and pursuing R&D increase. Thus, firms are less willing to invest in these activities, decreasing the steady-state growth rate and the bias toward H-type goods (although we can still observe a bias toward these goods).
We now proceed to three additional scenarios that can be interpreted as a sensitivity analysis: (i) shock in the public debt; (ii) financial shock; (iii) liquidity shock.
Starting with the shocks in public debt, we consider that the parameter \(\lambda\), which captures the negative externality of public debt on the productivity of the South, unexpectedly increases or decreases (both cases are analyzed). An increase (reduction) of the absolute public debt of the North or a reduction (increase) of the absolute public debt of the South, or both, can explain the increase (reduction) of this parameter. To be consistent with the assumption that the North is fiscal prudent, we will assume that all the variations in \(\lambda\) result from variations in the debt of the South. Thus, an increase in the parameter \(\lambda\) reflects an improvement in the relative position of the indebted country. We then consider both an increase and a reduction of \(\lambda\):
Fig. 2
Transitional dynamics of G—shocks in public debt of the South
×
As depicted in Fig. 2, when the South exogenously reduces its public debt (\(\lambda\) increases), the steady-state level of G is lower than the baseline case. This conclusion is independent of whether mutualization is pursued. Besides, as already derived in the previous section, debt reduction promotes a higher steady-state growth rate in both countries. Contrary, when the public debt in the South exogenously increases, the steady-state growth rate reduces. These results follow the empirical literature on the topic (Reinhart and Rogoff 2010; Baum et al. 2013; Albu and Albu 2021; Bhimjee et al. 2020), as well as previous endogenous growth models considering public debt (Saint-Paul 1992; Greiner 2015).
On the other hand, when the South’s public debt exogenously increases (\(\lambda\) reduces), the steady-state level of the technological-knowledge bias is higher. Once again, this result is independent of whether mutualization is pursued. Still, this conclusion could seem counter-intuitive. The better fiscal outlook of the South would reduce the technology-knowledge bias, while a worse relative position (higher public debt) would increase this bias. One possible explanation is that unskilled labor already has an absolute productivity disadvantage. Greater negative externalities from public debt would make L-type goods even less attractive, increasing the bias toward H-type goods.
Besides, since increasing public debt increases \(G^{*}\), it also increases the wage premium, ceteris paribus. Hence, public debt would reduce equality, promoting wage inequality. As already mentioned in this paper, such results follow previous studies on the topic (Azzimonti et al. 2014; Salti 2015).
Having analyzed the role of public debt in productivity, we now consider a financial shock. For this analysis, we assume that there is the possibility that a financial crisis affects the world economy. If a crisis like this occurs, investors demand higher interest rates (thus, higher risk premium) to keep lending to the agents of the most indebted country (Dewachter and Iania 2011). Note that some empirical evidence suggests that this premium could increase more than proportionally with the public debt (Haugh et al. 2009). That is, if the South lets its financial situation get worse by accumulating more debt, we could expect a more than proportional increase in the risk premium. We compute three cases (besides the baseline) with different intensities of the crisis, keeping the public debt constant. Then, we assume that the risk premium increases \(10\%\), \(50\%\), or \(100\%\) under a small, medium, or severe financial crisis, respectively—see Fig. 3.
Fig. 3
Transitional dynamics of G—shocks in risk premium of the South
×
As the crisis intensifies, i.e., as the risk premium increases more, the steady-state value of the technology-knowledge bias without mutualization also increases. Since \(G^{*}\) varies positively with the risk premium, but the other components of the wage premium follow an opposite direction (as stated in Proposition 5), the overall net impact on the wage premium of a variation of the risk premium depends on the magnitude of each effect. In Biglaiser and McGauvran (2021), when applied to developing countries, the authors concluded that the risk premium would indeed increase the wage premium. The increase of the interest rates (including the risk premium) reduces access to credit, reducing production and harming the agents in the sectors most exposed to external financing. If this reasoning is applied to our case, the higher the risk premium, the higher the costs for firms (because of the CIA constraint). Since the CIA constraint affecting L-type intermediate goods is assumed to be more intense than the one affecting H-type goods, increasing the risk premium is more harmful to the intermediate goods producers using L-technology. Thus, the bias towards H-type goods and the wage premium of skilled labor will likely increase as the risk premium increases.
Besides, according to the findings of the previous section, the higher the risk premium, the lower the steady-state growth rate of the economy, which matches previous empirical results on the effects of the government and the corporate risk premium on growth (Drechsler et al. 2018; Ireland 2015; Schumacher and Żochowski 2017). Note that we are keeping public debt constant. So, the uncertainty caused by the financial crisis and the consequent higher risk premium demanded by investors have an adverse impact on the steady-state growth rates (and likely on inequality as well), without any actual change in the public debt.
Then, as can be observed in Fig. 3, assuming that the risk premium would be entirely eliminated, promoting debt mutualization would allow to keep the steady-state technology-knowledge, the wage premium, and the steady-state growth rate constant in the event of a financial crisis. This policy would benefit the most and less indebted countries because of the feedback effects.
Finally, we also consider the response of \(G^{*}\) to a liquidity shock, i.e., to changes in the parameters concerning the CIA constraint in both countries. We assume that the share of the resources borrowed by the firms increases by \(20\%\) in both countries (both for the production of intermediate goods and for R&D activities). The numerical simulations are represented in Fig. 4.
Fig. 4
Transitional dynamics of G - liquidity shock in both countries
×
According to the previous plots, when the share of the costs financed with money that households lend to firms increases, the steady-state bias toward the H-type goods increases in the case without mutualization. In fact, the difference in the case with mutualization is exacerbated. This result is not surprising given that the benefits of mutualization increase the greater the impact of the risk premium. The risk premium is more significant when the share of the resources borrowed from families is larger. Hence, the liquidity shock has stronger impacts on \(G^{*}\) and on wage inequality when no mutualization is carried out. Increasing the CIA constraints would boost wage inequality, as confirmed by some empirical literature (Tridico 2018; Denk and Cournède 2015). Mutualization would minimize the subsequent impacts on the wage premium.
One aspect that should be noticed for the cases analyzed in the current section is that the transitional dynamic remains the same for all the shocks considered. The technology-knowledge bias converges to its steady-state value after around 170 periods. The only difference between these cases is the steady-state value to which it actually converges.
Additionally, we used numerical tools to assess how the steady-state technology-knowledge bias would vary with the subsidies. Results are presented in On-line Appendix 1 (Fig. 5). \(G^{*}\) varies positively with the subsidies, reinforcing the result that wage inequality varies positively with the subsidies. Interestingly, in this framework, all the dimensions of fiscal policy considered (government subsidies and public debt) have a regressive impact on wage inequality. Nevertheless, the impacts on the economy’s steady-state growth rate are different. While subsidies promote greater growth rates, public debt has an adverse effect.
5 Counterfactual analysis: no international trade
Throughout the previous sections, we assumed the free trade of intermediate goods. As we explained, this assumption is fundamental for the results, particularly for the common steady-state growth rate. Besides, this is a realistic assumption because we frame our analysis in the context of a monetary union. In this section, we assume no trade of intermediate goods to demonstrate its importance. Thus, countries belong to a monetary union, but trade is no longer an assumption.
Although this is a theoretical exercise to demonstrate the importance of free trade for the previous results, we can also interpret it as what occurs if, for some reason, the free market is interrupted in a monetary union. Even so, we assume that intermediate goods producers still have access to the technology used in the other country. Thus, R&D in the South is still based on imitations from the North. However, these imitations now only impact the South because producers cannot sell their products to the other country’s final firms.
First, after computing the threshold final good (see On-line Appendix 1), we can get the composite final good for each country with no trade (NT) — see On-line Appendix 1:
where \(f=\left\{ N,S\right\}\) depending on which country we refer to. The composite final good was already different between both countries with free trade. However, that difference is now exacerbated.
Besides, since full-employment in the labor market is guaranteed the equilibrium skilled premium:
where \(G_{s}(t)\equiv \frac{Q_{H,s}(t)}{Q_{L,s}(t)}\).
When trade is no longer an option, the wage premium increases in both countries and continues to be larger in the North than in the South. However, the reasons underlying this change are different depending on the country. For the North, no free trade means that the products that could be imported from the South are still sold but at a higher price. This cost increase incentivizes firms to shift their focus to the products using H-type resources (the ones abundant in the North). Given the abundance of high-skilled labor and lower financial costs, these products have a comparative advantage over the others produced in the same country. The relative greater production of H-type final goods increases the relative demand for skilled labor (compared to unskilled), increasing the wage premium.
However, in the South, H-type goods have no comparative advantage. On the contrary, unskilled labor is more abundant, though financial costs are greater for L-type goods. By removing international trade, the average price of intermediate goods used in the South decreases. Importantly, this reduction occurs for both types of products. Since H-type intermediate goods represented most of the imports, their average price reduced the most. Then, there is a readjustment of final goods’ production, and final firms produce more H-type products. To do it, they demand more skilled labor, increasing the risk premium. It should be noted that although the average cost of H-type intermediate goods reduces, their variety also reduces, affecting the variety of the final goods and the composite final good.
Further changes are also registered in terms of the R&D sector. As explained, we assume that the South continues to imitate and the North to innovate. Thus, although there is no international trade, the South still has access to the technology used in the North. Hence, the free-entry equilibrium in each country (\(f=\left\{ N,S\right\}\)) can be represented as:
where \(V_{f}(j,t)\) constitutes the value of the leading-edge patent for the producer of an intermediate good j at time t. The value of the patent is still equal to the present value of the profits generated by the intermediate goods produced under that patent. While before we had \(V_{s}(j,k,t)=\frac{\Pi _\text{S}(j,k,t)}{r_\text{S}(t)+I_\text{N}(j,t)}\), now the value of the patent is not determined by the probability of innovation/imitation in the other country, but rather by the likelihood of innovation/imitation in their own country. Given that intermediate goods producers are protected from international competition, innovations/imitations in the other countries do not harm their profits. Only domestic R&D advances have the possibility to make their products obsolete.
Then, we can get the equilibrium probability of successful innovation in a H-specific intermediate good in each country – see On-line Appendix 1:
where \(D_{H,f}(t)=\frac{H_{f}}{H_\text{N}+H_\text{S}}\cdot \left[ A_{f}^{\lambda _{f}}p_{H,f}\right] ^{^{\frac{1}{\alpha }}}\).
Now, one country does not influence the probability of innovation/imitation in the other. Free-trade of intermediate goods was the source of the feedback effects between North and South. The South no longer influences the North through a process of creative destruction. Similarly, the North technological-knowledge progress no longer benefits the South.
Once again, having defined the equilibrium probability of successful innovation in an H-specific intermediate good, and once that probability determines the speed of technological-knowledge progress, equilibrium is defined by the path of this progress in each country. With this relation, we get the equilibrium growth rate of H-specific technological-knowledge:
In the free-trade of intermediate goods, access to the latest intermediate goods increased production, which increased the resources available to R&D in the South (to conduct imitations). This would provide feedback to the North, increasing its technological-knowledge through creative destruction. These effects are no longer verified.
Considering the previous results, we can finally recompute the steady-state equilibrium. Given that we are no longer assuming free trade of intermediate goods, the steady-state growth rate and the real interest rate are not the same for both countries. Once again, the aggregate final good is divided between consumption and savings, are distributed between production of intermediate goods and R&D, which results in the aggregate resources constraint \(Y_{f}(t)=C_{f}(t)+X_{f}(t)+R_{f}(t),\,f=\left\{ N,S\right\}\). \(Y_{f}(t)\) is the composite final good; \(C_{f}(t)=\int _{0}^{1}c_{f}(i,t)\text{d}i\) the aggregate consumption; \(X_{f}(t)=\int _{0}^{1}\int _{0}^{1}X_{n,f}(j,t)\text{d}n\text{d}j\) the aggregate intermediate goods; and \(R_{f}(t)=\int _{0}^{1}y_{f}(j,t)\text{d}j\) the total resources devoted to R&D. Hence, the growth rate of each one of these variables in steady-state must be the same within each country and equal to \(\frac{r^{*}-\rho }{\theta }=g^{*}\).
As shown in On-line Appendix 1, we get that \(r^{*}\)and \(g^{*}\) are given by:
Substituting for each country, the steady-state growth rates are no longer equal between both countries. Note that with free trade and without mutualization, we had the following steady-state growth rate in both countries:
Hence, the South clearly benefits from the free trade of intermediate goods. This free trade expands the market for intermediate goods, generating additional revenues for these producers. Thus, the same imitations have greater expected revenues with free trade than with protectionism. Motivated by this additional market, when trade occurs, the willingness to invest in R&D is greater, inducing a higher steady-state growth rate of the economy.
However, for the North, there is no general result from comparing the growth rate between free trade of intermediate goods and no trade. A similar process is expected to occur to the one described for the South. The wider market makes intermediate producers’ expected revenue to increase for the same rate of innovations. This incentivizes investment in R&D, boosting the economy’s growth rate. However, in the North, there are also negative effects of free trade. With international competition, the cheaper imitations from the South exclude from the market some producers in the North. This risk decreases the expected revenue originated by each innovation. So, with free-trade, intermediate producers must worry about other domestic innovations and foreign imitations.
Nevertheless, even when imitations harm intermediate producers from the North, they also benefit final producers in that country. A foreign imitation means that these firms can now acquire the same products for a lower price, benefiting the production of the composite final good. Thus, no trade harms the final good producers. However, removing free trade also benefits these firms, as public debt problems in the South no longer have any consequences in the North. If the unsustainable fiscal policies in the South increase the risk premium, producers in the North are completely protected from the adverse effects.
All things considered, the net effect on the economy’s growth rate in steady-state is unclear. Still, for the assumed values of the parameters, we expect a positive effect of free trade on the growth rate of the product.
Hence, free trade of intermediate goods is an essential assumption for the results described in the previous sections of the paper. It is essential to guarantee that both countries’ steady-state growth rates will be the same. However, additional essential conclusions can also be retrieved in the present section. Without trade, both growth rates are different, with the South being the most penalized country. Then, the South diverges from the North. The North can benefit from the absence of trade, although we consider more likely that this interruption would also affect it. Additionally, due to the divergence, if protectionism continues for an extended period, a significant difference in economic development can arise due to the divergence between both countries. Given that the feedback effects were more likely, the closer the countries were to economic development, the more free-trade benefits could become substantially weaker once reintroduced.
Finally, public debt mutualization would not be a reasonable option for the North without free trade. Its economic performance would be protected from public debt problems in the South. Financial concerns reflected in a greater risk premium would not be indirectly passed to the North.
6 Conclusion
We develop a globalized DTC model, applied to a monetary union and capturing the multi-dimensional role of fiscal policy. We introduce public debt in Afonso and Sequeira (2022) endogenous growth model, following a dual approach: (i) the adverse effects of public debt on each country’s productivity; and (ii) an economy-wide risk premium, reflecting the aversion investors have to lend to financial stressed countries. To better understand the effects and their propagation, we assume that only one country (the South) has reached unsustainable debt levels. Besides, as in Afonso et al. (2009), we consider that the government of the South (the less rich country) may subsidize the production of intermediate goods and R&D activity.
Since we assume that both countries belong to a monetary union, they may want to mutualize their debts. In that case, each one would issue guarantees on the debt of the other. With this step, the negative effect of public debt on productivity would remain. However, the risk premium would be eliminated, given that investors trust in the fiscal prudent country.
Deriving the model and determining the steady-state, we conclude that the steady-state growth rate of both economies depends positively on the government subsidies in the South and negatively on the externality of public debt on productivity and the risk premium affecting the indebted economy. Hence, the adverse effects of public debt reduce the impact of subsidies on the South’s steady-state growth rate. Besides, the negative effects of the indebted country’s public debt spillover to the prudent country, thus harming the long-run growth of both economies. Moreover, given that the financing constraints affecting L-type intermediate goods are greater than the ones affecting H-type intermediate goods, we found that the risk premium likely increases the equilibrium wage inequality, making it greater in the South. Since a higher risk premium means greater costs for firms, increasing this risk premium results in an additional bias towards the goods requiring lower financing from households. Similarly, the negative externality of public debt on productivity would also increase the wage premium.
Hence, the financial and real effects of public debt in the indebted country affected the steady-state growth rate and the wage premium in both economies. All else equal, the fiscal prudent country would see its growth hampered because of the exogenous choices of the indebted country. Then, both countries would gain in minimizing the negative impacts of debt because of the feedback effects. The externality on productivity could only be minimized by repaying the South’s debt (or by a bail-out). Still, the risk premium impacts could be minimized with debt mutualization. When this option is politically feasible, the steady-state growth rate of the economy would increase, though wage inequality would also rise in both countries. The impact of inequality would be greater in the South, enlarging the difference to the North. Besides, during an inflationary period, the South may face political pressures to increase subsidies financed with additional public debt. In this case, debt mutualization could play an additional role. By stabilizing fiscal policy and reducing the inflation risk premium, the steady-state growth rates for both economies would increase. Then, our approach allows us to understand how some of the benefits of debt mutualization would arise. These results can be considered in future research performing a cost-benefit analysis of debt mutualization.
Moreover, several policy implications result from this study. First, the decision to grant government subsidies must be carefully considered since if public debt reaches unsustainable levels, several negative effects will hamper the long-term economic growth rate. Second, in a monetary union, all countries must continuously monitor each other’s public debt as the negative effects spill over. Third, public debt mutualization is an effective policy to minimize (but not fully eliminate) the negative consequences of public debt. Still, if mutualization is implemented, institutional reform is required to prevent moral hazard concerns. Otherwise, reducing the debt burden may incentivize the further increase of the public debt stock, making this policy ineffective. Finally, the negative effects of unsustainable public debt can only be fully eliminated if the public debt is reduced.
Besides, as stated, the considered dimensions of fiscal policy have a regressive impact on wage inequality. Nevertheless, it should also be noted that we are not including productive public expenditures or public spending with positive externalities on productivity. That is beyond the scope of the paper and can be considered in future research.
Further on, given that we were focused on assessing the impacts of the subsidies and public debt on the steady-state growth rate and wage premium, we assumed a balanced government budget and exogenous public debt. Further studies that may want to focus on the intertemporal accumulation of that debt or the optimal choice of the debt level could relax this assumption.
Finally, because the paper focuses on fiscal policy’s effects, monetary policy was simplified as much as possible. Future research analyzing the interdependence between both policies could consider a more detailed monetary policy rule.
Acknowledgment
The author Daniel Loureiro acknowledges the financial support of FCT, through individual Ph.D. scholarship UI/BD/152304/2021.
Declarations
Confict of interest
The authors declare that has no known Conflict of interest that could have appeared to influence the work reported in this paper.
Ethical approval
Not applicable.
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Since from (A.2) \(C(t)^{-\theta }\cdot e^{-\rho t}=\mu (t)\) and from (A.3) \(-\mu (t)\cdot (i(t)+\chi )=\upsilon (t)\), jointing with condition (A.5) results that \(\frac{\dot{C}(t)}{C(t)}= \left[ \pi (t)-(i(t)+\chi )\right] ^{-\frac{1}{\theta }}.\) Considering that from (A.2) (differentiating with respect to time) and (A.4) \(\frac{\dot{C}(t)}{C(t)}=\left[ r(t) \right] ^{-\frac{1}{\theta }}\), it is possible to obtain the no-arbitrage condition (the Fischer condition) \(i(t)+\chi =\pi (t)+r(t)\).
Now, by log-linearizing expression (A.2) and differentiating with respect to time, it results \(\frac{\dot{C}(t)}{C(t)}=- \frac{1}{\theta }\left[ \frac{\dot{\mu }(t)}{\mu (t)}+\rho \right]\). The Euler equation for consumption \(\frac{\dot{C}(t)}{C(t)} =-\frac{1}{\theta }\left[ r(t)-\rho \right]\) results from substituting in the result of the expression (A.4).
A.2 Final good producers maximization problem
Applying the first order conditions to expression (2.6), particularly with respect to \(X_{n}\):
and thus, from the profit maximization conditions of final-goods producers, the cost of each type of labor equals the value of its marginal productivity.
A.3 Intermediate good producers maximization problem
Applying the first-order conditions of the profit maximization to expressions (2.9) and (2.10):
Since in equilibrium the expenditures across final goods must be equal, \(p_{n}\cdot Y_{n}\) is constant across all n. Thus, \(L_{n}\) and \(h\cdot H_{n}\) must be constant and equal across all final goods that uses each one of these types of labor. Moreover, since \(p_{n}\cdot Y_{n}\) is constant across all n, then:
Since all the terms of the previous expression are constant, \(H_{n}\) and \(L_{n}\) are also constants. From here it results (since for \(n>\bar{n}\), \(L_{n}=0\)):
When \(n=\bar{n}\), firms that uses L-technology and firms that uses H-technology should break-even, and thus \(\frac{p_{n}^{H}}{p_{n}^{L}}=1\), which leads to:
After some algebra it results that \(\int _{0}^{\bar{n}}\ln (1-n){\text d}n=(\bar{n}-1)\cdot \ln (1-\bar{n})-\bar{n}\); and \(\int _{\bar{n}}^{1}\ln (n){\text d}n=-1-\bar{n}\cdot \ln (\bar{n})+\bar{n}\). Considering also that \(\frac{P^{H}}{P^{L}}=\left[ \frac{\bar{n}}{1-\bar{n}}\right] ^{\alpha }\), then:
Bearing in mind that \(p_{y}=1,\)the real price indexes of L and H final goods are obtained: \(p^{L}=\exp \left( -\alpha \right) \cdot \bar{n}^{-\alpha }\) and \(p^{H}=\exp \left( -\alpha \right) \cdot \left( 1-\bar{n}\right) ^{-\alpha }\).
where \(f=\left\{ N,S\right\}\) depending on which country we are referring to. Substituting in \(Y_{f}=p^{L}\cdot Y_{L}+p^{H}\cdot Y_{H}\) and recalling that \(p^{L}=\exp \left( -\alpha \right) \cdot \bar{n}^{-\alpha }\) and \(p^{H}=\exp \left( -\alpha \right) \cdot \left( 1-\bar{n}\right) ^{-\alpha }\):
Given that, from the profit maximization conditions of final-goods producers (On-line appendix 1), the cost of each type of labor equals the value of its marginal productivity:
Given that from expression (3.6) \(I_\text{S}(j,t)\cdot V_\text{S}(j,t)=y_\text{S}(j,t)\cdot \left( 1-z_{R}+\gamma _{H,{\text S}}\cdot (i_\text{S}+\chi )\right)\), solving for the H-specific intermediate good:
Having determined the probability of successful innovation in the H-type intermediate goods in the North, it is possible to determine the rate of technological growth. If a new quality of H-specific intermediate good j appears in the market, the rate of change of the aggregate quality index of the technology-knowledge stock \(Q_{H}\) is given by:
This rate still has to be weighted by the probability of a successful innovation in an H-specific intermediate good at each moment of time, which, considering the equilibrium probability, yields:
Notice that the negative effects of public debt on productivity are greater the lower the \(\lambda .\) Thus, increasing this externality ’reduces’ the \(\lambda\) when compared to the North and so reduces the output.
Finally, regarding the negative effects of risk premium:
If for both countries \(\Omega _{L}-\Omega _{H}>0\Rightarrow \frac{\partial WP}{\partial z_{X}}>0\) and if \(\Omega _{L}-\Omega _{H}<0\Rightarrow \frac{\partial WP}{\partial z_{X}}<0\). Finally, if \(\Omega _{L}-\Omega _{H}=0\Rightarrow \frac{\partial WP}{\partial z_{X}}=0\).
Proposition A.3
Let \(WP=\frac{w_{H}(t)}{w_{L}(t)}\), doing the derivative of expression (3.5) with respect to the risk premium:
Regarding the effects of the negative externality of excessive public debt, recalling that \(D_{H}=\frac{H_\text{S}}{H_\text{S}+H_\text{N}}\left[ A_\text{S}^{\lambda }\cdot p_{H,S}\right] ^{\frac{1}{\alpha }}+\frac{H_\text{N}}{H_\text{S}+H_\text{N}}\left[ A_\text{N}\cdot p_{H,N}\right] ^{\frac{1}{\alpha }}\), and assuming \(A>1\):
Recall that the negative effects of public debt on productivity are greater the lower the \(\lambda .\) Thus, increasing this externality ’reduces’ the lambda when compared to the North and so reduces the growth rate of output.
As seen in the proof of proposition 3, \(\frac{\partial WP}{\partial \chi }<0\). Then, eliminating the risk premium \(\chi\) increases the wage premium in the South.
A.12 Threshold final good with no international trade
Since is equilibrium the expenditures across final goods must be equal, \(p_{n}\cdot Y_{n}\) is constant across all n. Thus, \(L_{n}\) and \(h\cdot H_{n}\) must be constant and equal across all final goods that use each one of these types of labor. Moreover, since \(p_{n}\cdot Y_{n}\) is constant across all n, we can write that:
Moreover, we have already demonstrated in On-line appendix 1 that \(\frac{P^{H}}{P^{L}}=\left[ \frac{\bar{n}}{1-\bar{n}}\right] ^{\alpha }\). Therefore, jointing the two expressions:
where \(f={N,S}\) depending on which country we are referring to. Substituting in \(Y_{f}=p^{L}\cdot Y_{L}+p^{H}\cdot Y_{H}\) and recalling that \(p^{L}=\exp \left( -\alpha \right) \cdot \bar{n}^{-\alpha }\) and \(p^{H}=\exp \left( -\alpha \right) \cdot \left( 1-\bar{n}\right) ^{-\alpha }\):
A.14 Equilibrium skill premium with no international trade
Given that, from the profit maximization conditions of final-goods producers, the cost of each type of labor equals the value of its marginal productivity:
where \(D_{H,f}(t)=\frac{H_{f}}{H_\text{N}+H_\text{S}}\cdot \left[ A_{f}^{\lambda _{f}}p_{H,f}\right] ^{^{\frac{1}{\alpha }}}\).
Having determined the probability of a successful innovation/imitation (depending on the country) in the H-type intermediate goods, it is possible to determine the rate of technological growth. If a new quality of H-specific intermediate good j appears in the market, the rate of change of the aggregate quality index of the technology-knowledge stock \(Q_{H}\) is given by:
This rate still has to be weighted by the probability of a successful innovation in a H-specific intermediate good at each moment of time, which, considering the equilibrium probability, yields:
A.16 Economic growth rate with no international trade
Substituting the variables in \(\frac{\dot{Q}_{H,\ell }(t)}{Q_{H,\ell }(t)}= I_{H,\ell }(t)\cdot \left[ q^{\frac{(1-\alpha )}{\alpha }}-1\right]\) one gets:
Instead of considering an exogenous initial level of public debt (\(D_{0}\), which is too high for the case of the South but reasonable for the North), we could also consider debt as endogenous to the model. The government would keep accumulating debt to finance the subsidies and the interests of the previous bonds. Nevertheless, in a steady state, the economy’s growth rate must be unique. Hence, public debt would have to stop its growth to stabilize the parameter that captures the negative externality on each country-specific productivity (which influences the economy’s growth rate in steady-state, as it will be demonstrated further on). Hence, in a steady state, the results would be the same as the ones we obtained when considering debt as exogenous.
Such a hypothesis has been empirically validated. Not only high-tech firms have, on average, lower levels of leverage (Sardo and Serrasqueiro 2018), for instance, due to information frictions (Guiso 1998), as they are very effective in mobilizing internal cash resources (Lööf and Nabavi 2016). Moreover, high-tech firms are, on average, larger than low-tech firms (Hirsch-Kreinsen 2008), which should not be neglected when considering the previous effects.
