Equity and efficiency:
A theoretical analysis of education subsidies
Hans Christian Wika
Thesis submitted for 30 credits towards the degree of Master of Philosophy in Economics
UNIVERSITY OF OSLO
Faculty of Social Sciences Department of Economics
Submitted: June 2020
Acknowledgement
With this thesis, I conclude the two-year master’s program in economics at the University of Oslo. It marks an end of a time that has been both challenging, enlightening and fun! I am sure that once the stress of the past couple of weeks fades away, I will be very sad that it’s over.
Working on the thesis has been an interesting and challenging endeavor, and a great source of new knowledge about both the subject, and myself. I would never have been able to finish this project without the competent guidance of so many. Some of them deserve a special acknowledgement for their valuable support and feedback.
First and foremost, I would like to extend my warmest gratitude to my supervisor Paolo Giovanni Piacquadio for comments, suggestions, and steady guidance throughout the past months of work. Kjetil Storesletten also deserve appreciation, for the idea, for taking keen interest in the early days of the project, and for being an inspiration throughout the degree.
I am grateful to Oslo Fiscal Studies who provided me with a scholarship for writing my thesis on taxation. The financial support has enabled me to spend more time on my studies. The scholarship has also given me the possibility to attend research seminars at the Department of Economics at UiO, which has sparked an interest in research and motivated me to continue my educational path.
A big thank you is also in order to the members of my study groups, consisting of Raysa Rosario Lizarraga Hernandez, Erik Adrian, Lars Lerdalen, Lisa Rissveds and Jonas Struck. Last but not least, I would never have been able to get through this period without the love and support from my family.
Abstract
This thesis explores the possibility of government intervention, in the presence of poverty traps, resulting from indivisibilities in human capital formation and capital market imperfections.
Generally, capital markets are taken to be incomplete in the sense that human capital formation cannot be financed by issuing claims against the future earnings of a child due to the lack of enforceability of such contracts. As a result, high-income families are better able to invest in human capital than poor families, and income disparities are passed on through generations.
The thesis presents a model with government intervention in order to circumvent the capital market imperfection. By issuing bonds at the world market, subsidizing education and taxing labor income with high basic deductions, the government avoids the need to monitor student debt and the risk of debt evasion.
The proposed policy is proven to increase the number of individuals investing in human capital both in the short and the long run. Given that the government has access to an interest rate equal to the savings rate of households, implementing the policy will lead to a Pareto improvement both in the short and the long run. As this is the case for a lot of countries, the thesis provides a plausible alteration to the common practice of state educational student loans seen around the world. If government interest rates are above individuals’ savings rate, the policy involves a tradeoff between the utility of rich and poor dynasties. For a given subsidy there will then be five elements affecting the odds of the policy being approved. For a given policy, higher discrepancies between unskilled and skilled labor wages, the more individuals being incentivized to invest in human capital by the subsidy and higher aversion towards inequality vote in favor of the policy being implemented. On the other hand, higher government bond rates and an initially rich population hamper the chance of the policy being implemented.
Table of content
Acknowledgement... II Abstract ... III Table of content ... IV List of figures ... VI
1 Introduction ... 1
2 The baseline model – Galor and Zeira (1993) ... 4
2.1 Firms ... 4
2.2 Households ... 5
2.3 Individual optimization ... 6
2.4 Short run dynamics ... 8
2.5 Long run dynamics ... 10
3 A model of government intervention ... 13
3.1 Government ... 14
3.2 Individual optimization ... 16
3.3 Short run dynamics ... 17
3.4 Steady states ... 19
3.5 Dynamics ... 20
3.5.1 Case 1: 𝑠 = ℎ, 𝑟𝐺 = 𝑟 ... 21
3.5.2 Case 2: 𝑠 < ℎ, 𝑟𝐺 = 𝑟 ... 22
3.5.3 Case 3: 𝑠 = ℎ, 𝑟𝐺 > 𝑟 ... 25
3.5.4 Case 4: 𝑠 < ℎ, 𝑟𝐺 > 𝑟 ... 26
3.5.5 Concluding remarks ... 30
3.6 Comparative statics: Long run equilibrium ... 30
4 Welfare criterions ... 32
4.1 Pareto criterion ... 32
4.2 Kaldor-Hicks Criterion ... 35
5 Social welfare ... 39
5.1 A two-agent economy and the UPS ... 40
5.2 Social welfare function ... 41
5.3 Social optimum ... 43
5.4 The Rawlsian Criterion and subsidy limits ... 45
6 Conclusion ... 47
7 References ... 50
8 Appendix ... 53
8.1 A micro foundation of monitoring costs ... 53
8.1.1 Banks engage in monitoring ... 53
8.1.2 Demand for loans ... 53
8.1.3 Supply of loans ... 54
8.2 Consumer theory ... 55
8.2.1 Lagrangian... 55
8.2.2 Marshallian demands ... 55
8.2.3 Indirect utility ... 56
8.2.4 Homotheticity ... 56
List of figures
Figure 2-1 ... 7
Figure 2-2 ... 9
Figure 2-3 ... 11
Figure 3-1 ... 20
Figure 3-2 ... 22
Figure 3-3 ... 23
Figure 3-4 ... 24
Figure 3-5 ... 26
Figure 3-6 ... 27
Figure 3-7 ... 28
Figure 3-8 ... 29
Figure 4-1 ... 34
Figure 4-2 ... 37
Figure 5-1 ... 41
Figure 5-2 ... 42
Figure 5-3 ... 43
Figure 5-4 ... 44
Figure 5-5 ... 44
1 Introduction
Education promotes employment, earnings and health. It raises pride and opens new horizons.
For societies it drives long-term economic growth, reduces poverty, spurs innovation, strengthens institutions, and foster social cohesion. But in many societies around the world, access to education lies behind a paywall (Berg, 2008, p. 14). Millions of poor parents make difficult choices whether to provide education for their children or not. This cost-benefit assessment – where costs include both the direct cost of schooling and the opportunity cost of a child’s time outside it – determines their children’s enrollment, grade completion, and learning outcomes. At the same time, individuals of the same country can have excessive amount of funding in which the opportunity cost of education becomes negligible because of the benefits it provides (The World Bank, 2018).
There are mainly two reasons for being interested in the inequality of income and wealth distribution in economics. First, there are philosophical and ethical grounds for aversion to inequality in itself. In the absence of elements such as a handicap or an ailment, there are no reasons why individuals should be treated differently in terms of their access to lifetime economic resources (Rawls, 1971). In some cases, one could argue that people make choices over their lifetime that they themselves are responsible. Thus, being poor or rich is a result of one’s own decisions. But for many people, the unequal treatment begins from the day that they are born. Initial conditions such as parental wealth and access to resources can set different children off to an unequal path. It is hard to see any ethical reasoning to support this phenomenon. Secondly, even if we put aside the problem of inequality at an intrinsic level, there may still be reasons to take it into consideration. One might only be interested in the overall economic growth but discovers that inequality in income and wealth affects the overall possibilities of growth. In this case, one would care about inequality at what might be called a functional level; Inequality is not important in its own sake, but because it has an impact on the growth of the economy (Ray, 1998, p. 148).
Some of the earliest works regarding the effect of wealth distribution on the process of development is the Classical approach. This approach was originated by Smith (1776), and was further interpreted and developed by Keynes (1919), Lewis (1954), Kaldor (1957) and Bourguignon (1981). According to the Classical approach, savings rates are an increasing function of wealth. Inequality therefore tunnel resources towards individuals whose marginal
propensity to save is higher, increasing aggregate savings and capital accumulation and enhancing the process of development.
In contrast to this, a new strand of theory, in which capital market imperfection in the formation of human capital takes a leading role, was introduced in the 1990s. The fundamental hypothesis of this research stems from the recognition of asymmetries in the accumulation of human and physical capital. Whereas physical capital is largely independent of the distribution of ownership in the society, human capital is constrained by its inherently embodiment in humans.
Since physiological constraints hinder large accumulation of human capital, its formation is characterized by diminishing returns at an individual level. The aggregate stock of human capital would therefore be larger if its accumulation would be widely spread among individuals in society. This asymmetry between the accumulation of human capital and physical capital suggests therefore that as long as credit constraints are binding individuals of the economy, equality is conducive for human capital accumulation, whereas given that savings rates are an increasing function of wealth, inequality is conducive for physical capital accumulation (Galor
& Moav, 2004). Put in the context of empirical studies showing how human capital has replaces physical capital as the prime engine of economic growth in the transition from the Industrial Revolution to modern growth (Abramovitz & David, 2000), this theory suggest that egalitarian assessments are prime determinants of growth.
In light of this new strand of theory, this thesis explores the possibility of government intervention in order to weaken binding credit constraints on poor members of society. As a baseline model, I will use the path-breaking model presented by Galor and Zeira (1993). The model demonstrates that in the presence of credit market imperfections, and local non- convexities in the production of human capital, the initial distribution of income determines whether an economy converges to either a low-education, low-income steady state equilibrium, or a high-education, high-income steady state equilibrium. In particular, the model predicts that inequality has an adverse effect on human capital formation and economic growth in all but the very poor economies. As physical capital formation might still be a concern in many economies, I present a model that does not rely on direct transfers of wealth between agents. Instead, the government issues bonds to the world market, which are then used to partially or completely subsidize the cost of education for individuals investing in human capital. In order to repay the borrowed amount alongside coupons at maturity, the government levies taxes on labor wages
with high basic deductions. As a result of the deductions, only individuals investing in human capital and working as skilled labor will be exposed to the tax.
The model includes the two hindrances that are seen in reality: Differences in government bond ratings that results in varying bond rates, and government borrowing constraints (Glaessner &
Ladekarl, 2001). As the tax is levied on all individuals investing in human capital, the possibility of government bond rates above individuals’ savings rate introduces a tradeoff between the utility of rich dynasties and poor dynasties. In order to evaluate the performance of the policy, I therefore conduct a welfare analysis based on the steady state equilibrium.
The thesis is structured as follows. In chapter 2, I present the baseline model of the thesis, the Galor and Zeira model from 1993. Chapter 3 presents the main contribution of this thesis, the extended model with government intervention. In this chapter I analyze the different dynamics that can occur due to the policy and the impact on long run distributions and utility. Chapter 4 and 5 proceeds to analyze the welfare aspect of the model using respectively welfare criterions and welfare functions. Chapter 6 concludes.
2 The baseline model – Galor and Zeira (1993)
In this chapter I will present the baseline model of the thesis, the Galor and Zeira model from 1993. I have made two slight changes to the original model. First, I have assumed a standard Cobb-Douglas utility function instead of the logarithmic transformation of the Cobb-Douglas.
Secondly, I have assumed that banks compete for lending opportunities à la Bertrand. These assumptions do not change any of the major results and are both thoroughly explained in the appendix.
2.1 Firms
Consider a small open economy with one good. The good can either be invested in physical capital or human capital or be consumed. There are two technologies for the production of the good, each run by a different representative firm operating in a competitive environment. One of the firms uses both capital and skilled labor to produce the final good, while the other firm uses unskilled labor only. Production in the skilled sector is described by the following function 𝐹, where 𝑌𝑡𝑆 denotes output in the skilled labor sector, 𝐾𝑡 denotes physical capital input and 𝐿𝑆𝑡 denotes labor input, all at time t.
𝑌𝑡𝑆 = 𝐹(𝐾𝑡, 𝐿𝑡𝑆) (2.1)
The function 𝐹 is continuous, twice differentiable with positive and diminishing marginal products, and exhibits constant returns to scale. Investment in human capital and physical capital is made one period in advance. There are no adjustment costs to investment and no depreciation of capital. Markets are perfectly competitive, and expectations are fully rational.
At the equilibrium, capital is adjusted in each period such that the marginal product of capital is equal to the constant world market interest rate, 𝑟 > 0:
𝐹𝐾(𝐾𝑡, 𝐿𝑆𝑡) = 𝑟 (2.2)
Since the production function exhibits constant returns to scale, this means that the capital-labor ratio in the firms are constant, which determines the marginal product of the skilled labor.
Hence, the wage of the skilled labor, 𝑤𝑠, is determined by the world market interest rate, 𝑟, and the technology of the skilled sector, which are exogenously given in the model.
Production in the unskilled labor sector is described by the following technology, where 𝑌𝑡𝑈 denotes production in this sector, 𝑤𝑈 > 0 is marginal productivity of labor, and 𝐿𝑈𝑡 is unskilled labor input:
𝑌𝑡𝑈 = 𝑤𝑢∙ 𝐿𝑈𝑡 (2.3) The technology exhibits constant returns to scale, with constant marginal productivity of labor.
This makes the wage of the unskilled labor sector exogenously given by the technology only.
2.2 Households
Individuals in the economy live for two periods. Each individual has one parent and one child, making the population 𝐿̅ constant over time. Each individual supply one unit of labor inelastically. Thus, in each period 𝑡, 𝐿̅ = 𝐿𝑆𝑡+ 𝐿𝑈𝑡. Formally the labor markets clear when:
𝐿𝑆𝑡 ≤ 𝐿̅𝑆𝑡 , 𝑤𝑠 ≥ 0 𝑎𝑛𝑑 (𝐿𝑆𝑡− 𝐿̅𝑆𝑡)𝑤𝑠 = 0
𝐿𝑈𝑡 ≤ 𝐿̅𝑈𝑡 , 𝑤𝑈 ≥ 0 𝑎𝑛𝑑 (𝐿𝑈𝑡 − 𝐿̅𝑈𝑡)𝑤𝑈 = 0 (2.4) Individuals care about their children. Their preferences take the warm glow form: the parent’s utility depends on the monetary bequest, b, they leave to their offspring. The individuals also get utility from consumption at the end of adulthood, c. The utility function is a standard Cobb- Douglas utility function:
𝑈(𝑐, 𝑏) = 𝑐𝛼𝑏1−𝛼 (2.5)
where 𝛼 ∈ (0,1). The budget of the individuals depends on their educational choices. An individual can avoid education and decide to work as unskilled in both periods of life, earning the unskilled wage, 𝑤𝑈. Then the inheritance received, x, and the wage, 𝑤𝑈, earned in the first period are transferred to the second period: these are saved at the constant world market interest rate, r. Thus, their budget constraint is:
𝑐 + 𝑏 = 𝑤𝑈 + (1 + 𝑟)(𝑥 + 𝑤𝑈) (2.6) Another possibility is to invest in human capital through education. Education takes place in the first period of life, has a fixed cost, ℎ > 0, and does not allow working. The advantage, however, is that in adulthood, educated individuals earn the skilled wage, 𝑤𝑠. The investment in human capital is indivisible, which creates a region of increasing returns to scale at the individual level. If an individual has inheritance higher or equal to the cost of education, 𝑥 ≥ ℎ, the individual will save the difference between his/her inheritance and the cost of education, 𝑥 − ℎ, at the world market interest rate, r. Their budget constraint is then:
𝑐 + 𝑏 = 𝑤𝑠+ (1 + 𝑟)(𝑥 − ℎ) (2.7)
In contrast, if an individual that invests in human capital has inheritance lower than the fixed cost of education, it has to borrow the difference from banks. To avoid individuals evading
debt, the banks pay a monitoring cost per unit of borrowing equal to m. The appendix shows that if there are many banks competing à la Bertrand for lending opportunities, that all bank lending is accompanied with monitoring, and the borrowing rate satisfies 𝑖 = 𝑟 + 𝑚. The monitoring cost creates a wedge between the borrowing and the lending rates, where the cost of monitoring is sunk. The budget constraint for these individuals is then:
𝑐 + 𝑏 = 𝑤𝑠 + (1 + 𝑖)(𝑥 − ℎ) (2.8) 2.3 Individual optimization
Let’s formally describe the optimality conditions. Consider an individual irrespective of inheritance that decides to work as unskilled by not investing in human capital. His/her Lagrangian takes the form:
max ℒ = 𝑐𝛼𝑏1−𝛼− 𝜆( 𝑐 + 𝑏 − 𝑤𝑈− (1 + 𝑟)(𝑥 + 𝑤𝑈)) (2.9) Solving this maximization problem gives us the Marshallian demand functions of consumption, c, and bequest, b:
𝑐(𝑥) = 𝛼(𝑤𝑈+ (1 + 𝑟)(𝑥 + 𝑤𝑈)) (2.10) 𝑏𝑢(𝑥) = (1 − 𝛼)( 𝑤𝑈+ (1 + 𝑟)(𝑥 + 𝑤𝑈)) (2.11) Substituting the optimality conditions back into the utility function gives us the indirect utility, which is a linear function of end of lifetime wealth only.
𝑈𝑈 = 𝛼𝛼(1 − 𝛼)(1−𝛼)(𝑤𝑈+ (1 + 𝑟)(𝑥 + 𝑤𝑈)) (2.12) An individual inheriting more than or an equal amount to the fixed cost of education, 𝑥 ≥ ℎ, and decides to invest in human capital solves the following Lagrangian:
max ℒ = 𝑐𝛼𝑏1−𝛼− 𝜆( 𝑐 + 𝑏 − 𝑤𝑠 − (1 + 𝑟)(𝑥 − ℎ)) (2.13) With Marshallian demands of consumption and bequest:
𝑐(𝑥) = 𝛼(𝑤𝑠 + (1 + 𝑟)(𝑥 − ℎ)) (2.14) 𝑏𝑠(𝑥) = (1 − 𝛼)( 𝑤𝑠 + (1 + 𝑟)(𝑥 − ℎ)) (2.15) And an indirect utility of:
𝑈𝑆ℎ>𝑥 = 𝛼𝛼(1 − 𝛼)(1−𝛼)(𝑤𝑠+ (1 + 𝑟)(𝑥 − ℎ)) (2.16) Lastly, an individual inheriting less than the fixed cost of education, 𝑥 < ℎ, and decides to invest in human capital solves the following Lagrangian:
max ℒ = 𝑐𝛼𝑏1−𝛼− 𝜆( 𝑐 + 𝑏 − 𝑤𝑠− (1 + 𝑖)(𝑥 − ℎ)) (2.17) With Marshallian demands of consumption and bequest:
𝑐(𝑥) = 𝛼(𝑤𝑠+ (1 + 𝑖)(𝑥 − ℎ)) (2.18) 𝑏𝑠(𝑥) = (1 − 𝛼)( 𝑤𝑠+ (1 + 𝑖)(𝑥 − ℎ)) (2.19) And an indirect utility of:
𝑈𝑆ℎ<𝑥 = 𝛼𝛼(1 − 𝛼)(1−𝛼)(𝑤𝑠+ (1 + 𝑖)(𝑥 − ℎ)) (2.20) As we can see, individuals will always consume a fraction 𝛼 of their disposable wealth and leave a fraction 1 − 𝛼 of their disposable wealth as bequest for their children independently of their end of lifetime wealth. This is because the utility function used is homothetic, which translates into a linear wealth expansion path, 𝐸𝒑, where the slopes of the indifference curves are constant along rays beginning form the origin. This is to say, the Engel curve for each good is linear (MWG, 1995, p. 45).
Figure 2-11
1 Wealth expansion paths for a Cobb-Douglas utility function with 𝛼 = 0.3 and 𝛼 = 0.7. If all individuals have the same preferences for consumption and bequest, they will consume (and bequeath) the same fraction of their lifetime income regardless of disposable wealth.
Since prices are constant, the individuals in terms only compare the size of the different budget sets. But due to the capital market imperfection, different budget sets are available for different individuals depending on the amount they inherit. This is because the opportunity cost of investing in human capital decreases with inheritance. Whether taking on education or not yields the largest budget set therefore depends on the amount an individual has to borrow from banks. The next section investigates the thresholds of inheritance of which investing in human capital is the optimal decision.
2.4 Short run dynamics
It’s clear that everybody would prefer to work as unskilled labor if doing so yields a larger budget even for individuals financing the whole cost of education through inheritance.
Therefore, to avoid cases of limited interest, assume that 𝑈𝑆ℎ>𝑥 ≥ 𝑈𝑢:
𝑤𝑆− (1 + 𝑟)ℎ ≥ (2 + 𝑟)𝑤𝑈 (2.21)
This makes everyone who inherits more than the fixed cost of education invest in human capital.
If (2.21) does not hold, individuals all over the world prefer to work as unskilled. Hence, there is no demand for capital and an excess supply of loans prevails. This drives the world market interest rate down until (2.21) is satisfied. Hence, (2.21) is a reasonable assumption. Individuals that has to borrow in order to invest in human capital, prefer education as long as 𝑈𝑆ℎ<𝑥 ≥ 𝑈𝑈. Solving this equation for x yields a threshold for inheritance, f, which fully determines if an individual invests in human capital or not:
𝑥 ≥(2 + 𝑟)𝑤𝑈− 𝑤𝑆+ (1 + 𝑖)ℎ
𝑖 − 𝑟 ≡ 𝑓 (2.22)
The reason for this threshold is the capital market imperfection, where borrowers has to pay a higher interest rate than lenders receive. And, therefore, education is limited to individuals which inherits enough initial wealth. Let 𝐷𝑡 be the distribution of inheritances by individuals born in period 𝑡, where the distribution includes all individuals in the economy, 𝐿̅:
∫ 𝐷𝑡(𝑥𝑡)𝑑𝑥𝑡= 𝐿̅
∞ 0
(2.23) The number of skilled workers in the short run is equal to the number of individuals with an inheritance above 𝑓:
𝐿𝑆𝑡 = ∫ 𝐷𝑡(𝑥𝑡)𝑑𝑥𝑡
∞ 𝑓
(2.24)
And the number of unskilled workers is thus equal to the number of individuals with an inheritance below 𝑓:
𝐿𝑈𝑡 = ∫ 𝐷𝑡(𝑥𝑡)𝑑𝑥𝑡
𝑓 0
(2.25) The distribution fully determines the amount of skilled and unskilled labor each period, which as a result of the small open economy assumption determines investment and production.
Hence, two countries with different distributions of wealth has substantially different macroeconomic equilibria. This is due to the credit market imperfection. But as shown by Loury (1981) and Banerjee and Newman (1991), this is not reasonable if the dynamic process is ergodic, namely if all initial distributions converge to the same distribution in the long run. In the next section, I therefore show that the second assumption, of indivisibilities in investment in human capital, leads to non-ergodic dynamics and to multiple long-run steady states depending on the initial distribution of wealth.
Figure 2-22
2 The distribution of inheritance in one period determines the amount of skilled labor in the short run. As the skilled-unskilled labor ratio determines the macroeconomic equilibrium, the distribution of inheritance solely determines economic performance in the short run.
2.5 Long run dynamics
Because the bequest of an individual is determined by the amount he or she inherit, the initial distribution of inheritance does not only determine the equilibrium in period 𝑡, but also determines the next periods distribution of inheritance, 𝐷𝑡+1, which determines the equilibrium in period 𝑡 + 1. Collecting the Marshallian demands for bequest gives us the set of inheritances for the next generation:
𝑥𝑡+1 =
𝑏𝑢(𝑥) = (1 − 𝛼)( 𝑤𝑈+ (1 + 𝑟)(𝑥𝑡+ 𝑤𝑈)) 𝑖𝑓 𝑥𝑡< 𝑓 𝑏𝑠(𝑥) = (1 − 𝛼)( 𝑤𝑠+ (1 + 𝑖)(𝑥𝑡− ℎ)) 𝑖𝑓 𝑓 ≤ 𝑥𝑡< ℎ
𝑏𝑠(𝑥) = (1 − 𝛼)( 𝑤𝑠+ (1 + 𝑟)(𝑥𝑡− ℎ)) 𝑖𝑓 𝑥𝑡 ≥ ℎ
(2.26)
The equations are the first-order differential equations determining the dynamics of bequest.
Setting 𝑥𝑡= 𝑥𝑡+𝑡 = 𝑥 , and solving for x, gives 3 steady states for long run inheritance. 2 of which are locally asymptotically stable, and one which is locally unstable.
Individuals inheriting less than f, work as unskilled for both periods, and so do their descendants in all future generations. The level of inheritance in this dynasty converge to a long run stable steady state of 𝑥̅̅̅: 𝑈
𝑥𝑈
̅̅̅ = (1 − 𝛼)
1 − (1 − 𝛼)(1 + 𝑟)𝑤𝑈(2 + 𝑟) (2.27) Individuals inheriting more than the fixed cost of education, ℎ, invest in human capital, and work as skilled labor for the second period of their life, and so do their descendants in all future generations. The level of inheritance in this dynasty converge to a long run stable steady state of 𝑥̅𝑆:
𝑥̅ =𝑆 1 − 𝛼
1 − (1 − 𝛼)(1 + 𝑟)[𝑤𝑆− (1 + 𝑟)ℎ)] (2.28) Individuals inheriting less than ℎ, but more than 𝑓, invest in human capital by borrowing from banks, and work as skilled labor for the second period of life. The steady state of these dynasties is characterized by the inheritance level 𝑔:
𝑔 = (1 − 𝛼)
1 − (1 − 𝛼)(1 + 𝑖)[𝑤𝑆− (1 + 𝑖)ℎ] (2.29) But note that this steady state is locally unstable, where individuals inheriting less than g converge to 𝑥̅̅̅, and individuals who inherit more than g converge to 𝑥𝑈 ̅𝑆. The lower basin of attraction corresponds to the poverty trap of the Galor and Zeira model.
In order to illustrate the dynamic evolution of inheritance distribution through time, I present the 𝑏𝑢(𝑥) and 𝑏𝑠(𝑥) curves, which fully describe the dynamic relationship between inheritance and bequest for respectively unskilled and skilled workers, in figure 2-3. Notice that 𝑓 is determined by the intersection of 𝑏𝑢(𝑥) and 𝑏𝑠(𝑥). In order for the system to be stable and non- exploding, we have to assume that 𝛼 and 𝑟 takes values that satisfies:
(1 − 𝛼)(1 + 𝑟) < 1 (2.30)
Which can be seen in the figure by noticing that the slopes of 𝑏𝑢(𝑥) and the slack part of 𝑏𝑠(𝑥) being less than the 45 line. In addition, we have to assume that monitoring costs are sufficiently large such that the spread between the lending and borrowing rates are high. In practice:
(1 − 𝛼)(1 + 𝑖) = (1 − 𝛼)(1 + 𝑟 + 𝑚) > 1 (2.31) Which means that the slope of the steep part of the 𝑏𝑠(𝑥) curve is higher than the 45 line. If (2.31) does not hold, then all dynasties of the economy would converge towards the higher or the lower steady state. Since this is unrealistic and uninteresting, we restrict ourselves to cases when (2.31) holds. For the same reason, it is assumed that 𝑔 lies above 𝑥̅̅̅. In order for this to 𝑈 be the case, education costs need to be sufficiently large, such that 𝑤𝑆 ≪ (1 + 𝑖)ℎ.
Figure 2-33
3 Inheritance dynamics of the Galor and Zeira model (1993). Dynasties with an initial wealth below 𝑔 will in the long run work as unskilled labor and their inheritance level will converge to 𝑥̅̅̅. Dynasties with an initial wealth 𝑈 above 𝑔 will in the long run work as skilled labor and their inheritance level will converge to 𝑥̅ . 𝑆
As presented in figure 2-3, the long run dynamics can be deduced from individual dynamics. A coordinate above the 45 line means that the individual is bequeathing more than the individual himself did inherit. Vice versa for an inheritance level below the 45 line. Thus, the economy converges to a long run steady state, where the population is divided into two groups: skilled workers with wage equal to 𝑤𝑆 and inheritance equal to 𝑥̅𝑆, and unskilled workers with wage equal to 𝑤𝑈 and inheritance equal to 𝑥̅̅̅. The relative size of these two groups only depends on 𝑈 the initial distribution of wealth, where the long-run number of unskilled workers is equal to the number of dynasties with an initial wealth less than the threshold, 𝑔:
𝐿𝑈𝑡=∞ = ∫ 𝐷𝑡(𝑥𝑡)𝑑𝑥𝑡
𝑔 0
(2.32) While the long-run number of skilled workers is equal to the number of dynasties with an initial wealth above 𝑔:
𝐿𝑡=∞𝑆 = ∫ 𝐷𝑡(𝑥𝑡)𝑑𝑥𝑡
∞ 𝑔
(2.33) Hence, an economy which initially is poor, ends up poor in the long run as well. An economy which initially is rich, and where its wealth is distributed among many, ends up rich.
Interesting enough, an economy which initially is rich, but where wealth is held by few, ends up as poor in the long run. The long run equilibrium in the model, therefore, depends on the initial distribution of wealth, and is as a result historically dependent.
3 A model of government intervention
The driving force of the poverty trap in the baseline model, is that poor individuals (or their parents) can’t collateralize human capital in order to finance investment in education (of their children). Since it would be optimal for all individuals to invest in human capital if they could borrow at the world market interest rate, the capital market imperfection leads to an inefficiently low level of human capital accumulation that does not utilize the potential inherit in the poor dynasties of the economy. This assumption is compatible in a non-slave state, as legal restrictions prevents individuals engaging in a contract that employs future income as collateral.
However, the government can circumvent this constraint as it has claims on all acquired human capital through the tax system. This provides a potential rationale for government intervention (Stiglitz, 1993).
In line with this proposition, Strawczynski (2014) shows how inheritance taxation could work as a government policy in order to finance investment in education in a model of indivisible human capital investment and capital market imperfections. There are a couple of reasons why such a policy would be an unlikely solution to the poverty trap. First, it is a well-known fact that gifts and inheritances are hard to monitor (Gales, et al., 2001; Graetz & Shapiro, 2005).
This reason seems even more convincing in developing economies, in which taxation is mostly indirect and the information on gifts and inheritances is scarce (Bovenberg, 1987). However, in developed economies where a compulsory income/estate declaration exist and the available information is of good quality, this argument pulls less weight, even though Erard (1998) shows that the avoidance of the inheritance tax in the U.S. is estimated at 13% of the potential tax base. Another aspect making inheritance taxes an unlikely solution to the market failure, is that it distorts the optimal consumption decision of the parents, since the utility function takes the warm glow form. The increased cost of inheritance lowers the optimal inheritance/consumption ratio, thus, lowering the available inheritance which is used to fund education. In order for the inheritance tax policy to have substantial effect on the number of dynasties that end up as skilled in the long run, the economy must primarily consist of wealthy dynasties in the first place (Galor
& Zeira, 1993). On the basis of these arguments, a policy involving inheritance taxation and education subsidies is more likely to solve poverty traps in developed economies than in developing economies.
As an alternative to the inheritance tax policy, I propose a model where the government issues bonds in order to subsidize investment in human capital. To repay the loans with coupons at maturity, the government taxes labor wages with high basic deductions. As a result of the deductions, only individuals investing in human capital and working in the skilled sector will be exposed to the tax. In essence, the policy makes the government borrow on behalf of students, and students repay these loans through taxes. In addition, I assume that tax collection costs are zero. This is a plausible assumption for two reasons. First, by giving the subsidy to all students and taxing all those who have a higher income, the government avoids the need to keep track of each individual, as it would have to when issuing student loans. Secondly, income taxes might already be in use for other purposes, and the policy only raises the tax already collected.
Thus, the system is already in place, and no additional costs are induced. To add some realism to the model I assume two hindrances: Government bond risk, resulting in bond rates that might be above the world market interest rate, and upper limits of government borrowing, referred to as government borrowing constraints (Haug, 1991).
I will conduct a similar presentation of the model as in the baseline model, but because of the added assumptions, a wide variety of dynamics might occur. Thus, I will perform a wider analysis on dynamics consisting of four cases for different subsidies and government interest rates. Lastly, I will calculate the long run utility emerging from the policy, which will be used in a welfare analysis in chapter 4 and chapter 5. As the tax model builds on the baseline model, I will sub index equations differing from the baseline model with a 𝜏.
3.1 Government
Assume an infinitely living government. The government has access to the intertemporal credit market by issuing government bonds, which have an exogenously given coupon rate of 𝑟𝐺 per generation period. Due to possible government bond risk, such as political risk and default risk (Huang, et al., 2015), the coupon of the government bonds is assumed to be larger or equal to the individual savings rate, 𝑟𝐺 ≥ 𝑟. The coupon is also assumed to be strictly lower than the individuals’ borrowing rate, 𝑖 > 𝑟𝐺, since the government to some extent can collateralize future tax income. The certainty equivalent of the government bond is assumed to be equal to the world market interest rate to avoid arbitrage opportunities. The government might also be borrowing constrained, such that they can only issue a certain amount of bonds at this rate.
Now, assume the government is well aware of the poverty trap of its economy driven by the capital market imperfection. To circumvent the capital market imperfection and incentivize more people to invest in education, they decide to partially or fully subsidize the cost of education at a rate committed to for all future periods. To fund the subsidy, the government issues bonds, which are repaid by taxes levied on skilled labor wages in the next period. The tax is collected by the firms the individuals work in, which are immobile. Thus, tax is not evadable. In addition, tax collection costs are assumed to be zero. The government period by period budget, 𝐺𝑡, then consists of newly issued bonds, 𝐵𝑡, payments on bonds issued in previous period, 𝐵𝑡−1, and income tax on the skilled labor, 𝜏𝑠𝑤𝑠𝐿𝑆𝑡.
𝐺𝑡 = 𝐵𝑡− (1 + 𝑟𝐺)𝐵𝑡−1+ 𝜏𝑠,𝑡𝑤𝑠,𝑡𝐿𝑆𝑡 (3.1) The government borrowing constraint takes a simple form, where 𝑏 refers to the upper limit of their borrowing constraint.
𝐵𝑡≤ 𝑏 (3.2)
Every period, the government issues the exact amount of bonds needed in order to fund the education subsidy of those investing in human capital and working as skilled labor the next period.
𝐺𝑡= 𝐵𝑡 (3.3)
Hence, the budget constraint for a single period is reduced to:
𝐺𝑡−1=𝜏𝑠,𝑡𝑤𝑠,𝑡𝐿𝑆𝑡
(1 + 𝑟𝐺) (3.4)
The amount borrowed is divided equally among individuals investing in human capital and working as skilled labor in the next period.
𝑠 ≡𝐺𝑡−1
𝐿𝑆𝑡 = 𝜏𝑠,𝑡𝑤𝑠,𝑡
(1 + 𝑟𝐺) (3.5)
As the government commits to a certain subsidy for all future periods, and all individuals are allowed to invest in human capital, we assume that the subsidy is bound by:
𝑠̂ ≤1 𝑏
𝐿̅ (3.6)
In order for the policy to be budget balancing, the tax on skilled labor is therefore equal to the subsidy multiplied by the government interest rate.
(1 + 𝑟𝐺)𝑠 = 𝜏𝑠,𝑡𝑤𝑠,𝑡 (3.7)
3.2 Individual optimization
As discussed earlier, the preferences of the individuals are homothetic, which translates to a linear expansion path in income where the slopes of the indifference curves are constant along rays beginning from the origin. Since the policy scheme in the economy only affects the budget of the individuals and not the prices of consumption and bequest, the individuals optimize by comparing whether taking on education yields a higher budget or not after they see the given subsidy. Let us therefore identify the set of possible budget constraints for a given subsidy rate.
As there is no subsidy given to individuals that avoids education, and no tax is collected from unskilled labor hours, the lifetime budget constraint for individuals choosing to work as unskilled labor is the same as before:
𝑐 + 𝑏 = 𝑤𝑈 + (1 + 𝑟)(𝑥 + 𝑤𝑈) (3.8) These individuals are net lenders who leaves a bequest of:
𝑏𝑢(𝑥) = (1 − 𝛼)( 𝑤𝑈+ (1 + 𝑟)(𝑥𝑡+ 𝑤𝑈)) (3.9) With an indirect utility equal to:
𝑈𝑈(𝑥) = 𝛼𝛼(1 − 𝛼)(1−𝛼)(𝑤𝑈 + (1 + 𝑟)(𝑥 + 𝑤𝑈)) (3.10) Due to the subsidy, the individual cost of education is lowered by the amount of the subsidy, ℎ − 𝑠 ≥ 0. This also decreases the threshold of inheritance in which individuals needs to seek the capital market to fund the cost of education. Individuals inheriting more than the new individual cost of education, 𝑥 ≥ ℎ − 𝑠, will save the difference at the world market interest rate, 𝑟. In addition, they have to pay taxes on their labor income, (1 − 𝜏𝑠)𝑤𝑠 = 𝑤𝑠− (1 + 𝑟𝐺)𝑠, by (3.7). Their budget constraint become:
𝑐 + 𝑏 = 𝑤𝑠 − (1 + 𝑟𝐺)𝑠 + (1 + 𝑟)(𝑥 + 𝑠 − ℎ) (3.11) These individuals are net lenders and leave a bequest of:
𝑏𝑠,𝜏(𝑥, 𝑠) = (1 − 𝛼)( 𝑤𝑠+ (𝑟 − 𝑟𝐺)𝑠 + (1 + 𝑟)(𝑥 − ℎ)) (3.12) With an indirect utility equal to:
𝑈𝑠,𝜏𝑥≥ℎ−𝑠(𝑥, 𝑠) = 𝛼𝛼(1 − 𝛼)(1−𝛼)(𝑤𝑠+ (𝑟 − 𝑟𝐺)𝑠 + (1 + 𝑟)(𝑥 − ℎ)) (3.13) Individuals inheriting less than the new individual cost of education, and choosing to invest in human capital, may still borrow the difference from banks. These individuals are still being monitored, and has to pay the higher borrowing rate, 𝑖 = 𝑟 + 𝑚, on the amount that they
borrow. But the amount needed to borrow is reduced by the subsidy received. In addition, these individuals have to pay taxes on their labor income equal to (1 − 𝜏𝑠)𝑤𝑠 = 𝑤𝑠− (1 + 𝑟𝐺)𝑠.
Their budget constraint become:
𝑐 + 𝑏 = 𝑤𝑠− (1 + 𝑟𝐺)𝑠 + (1 + 𝑖)(𝑥 + 𝑠 − ℎ) (3.14) Which translates into a bequest equal to:
𝑏𝑠,𝜏(𝑥, 𝑠) = (1 − 𝛼)(𝑤𝑠 + (𝑖 − 𝑟𝐺)𝑠 + (1 + 𝑖)(𝑥 − ℎ)) (3.15) And an indirect utility of:
𝑈𝑠,𝜏𝑥<ℎ−𝑠(𝑥, 𝑠) = 𝛼𝛼(1 − 𝛼)(1−𝛼)(𝑤𝑠 + (𝑖 − 𝑟𝐺)𝑠 + (1 + 𝑖)(𝑥 − ℎ)) (3.16) Note that (3.11) and (3.14) are reduced to respectively (2.7) and (2.8) if 𝑠 = 0. As a result, the model depicted can be reduced to the baseline model if no education subsidies are in place.
3.3 Short run dynamics
As in the baseline model, it is clear that everybody would prefer to work as unskilled labor if doing so yields a larger budget even for individuals able to finance the whole cost of education through inheritance and the subsidy. Since individuals investing in human capital and working as skilled labor are subject to taxation, the policy might tighten condition (2.21). Comparing the indirect utility of individuals inheriting more than the new individual cost of education, (3.13), with the indirect utility of individuals choosing to work as unskilled, (3.10), gives us the new condition that avoids the whole population choosing to work as unskilled labor:
𝑤𝑆− (𝑟𝐺− 𝑟)𝑠 − (1 + 𝑟)ℎ > (2 + 𝑟)𝑤𝑈 (3.17) Comparing this condition to (2.21), we see that the only difference is −(𝑟𝐺− 𝑟)𝑠. Thus, the condition is tightened if the government interest rate is higher than the individuals’ savings rate.
If condition (3.17) doesn’t hold, the policy won’t be implemented, since it would cause a strict reduction in the utility of all individuals investing in human capital and no change in the utility of individuals choosing to work as unskilled. For that reason, we from now on assume that condition (3.17) always holds. Comparing the utility of individuals investing in human capital, which have to seek funding from the capital market, (3.16), and the utility of individuals not investing in human capital, (3.10), and solving for inheritance, gives the new threshold for inheritance in which individuals start investing in human capital.
𝑥 ≥(2 + 𝑟)𝑤𝑈− 𝑤𝑆− (𝑖 − 𝑟𝐺)𝑠 + (1 + 𝑖)ℎ
𝑖 − 𝑟 ≡ 𝑓𝜏(𝑠) (3.18)
If the subsidy is equal to 0, then (3.18) is reduced to (2.22). Let us differentiate 𝑓𝜏 with respect to 𝑠 to explore the implications of the subsidy on this threshold:
𝜕𝑓𝜏
𝜕𝑠 = −(𝑖 − 𝑟𝐺)
𝑖 − 𝑟 (3.19)
Taking the limit of this equation when the government interest rate, 𝑟𝐺, approaches the individuals’ borrowing rate, 𝑖, shows us that the threshold of inheritance, for which investing in human capital is the optimal decision, does not change when the government only has access to the same borrowing rate as the individuals themselves.
𝑟lim𝐺→𝑖
𝜕𝑓𝜏
𝜕𝑠 = 0 (3.20)
Taking the limit of equation (3.19) when the government interest rate, 𝑟𝐺, approaches the lending rate, 𝑟, shows us that the threshold of inheritance, for which investing in human capital is the optimal decision, reduces one-for-one with the subsidy when the government is able to borrow at the same rate as the individuals are able to save:
lim
𝑟𝐺→𝑟
𝜕𝑓𝜏
𝜕𝑠 = −1 (3.21)
Given the assumed bounds of the government interest rate, 𝑟𝐺 ∈ [𝑟, 𝑖), positive subsidy levels will reduce the inheritance needed in order for individuals to invest in human capital. Assuming a positive number of individuals inheriting between 𝑓𝜏(0) and 𝑓𝜏(𝑠), the subsidy will then increase the number of individuals investing in human capital. Again, looking at the distribution of inheritance, 𝐷𝑡, we can determine the amount of skilled and unskilled labor each period, which again determines the other economical aggregates. The number of skilled workers in each period is equal to:
𝐿𝑡𝑆 = ∫ 𝐷𝑡(𝑥𝑡)𝑑𝑥𝑡
∞ 𝑓𝜏(𝑠)
(3.22) And the number of unskilled workers is equal to:
𝐿𝑈𝑡 = ∫ 𝐷𝑡(𝑥𝑡)𝑑𝑥𝑡
𝑓𝜏(𝑠) 0
(3.23) It is assumed that the government knows the initial distribution of wealth, and from this relationship can calculate the amount of bonds needed in order to fund the subsidy each period:
𝑠𝐿𝑆𝑡 = 𝐵𝑡−1 (3.24)
3.4 Steady states
Since the policy does not alter the prices on the consumption side of the budget constraints, parents will, as in the baseline model, leave a fraction, 1 − 𝛼, of their lifetime income as bequest to their children. The initial distribution of inheritance determines the distribution of inheritance in all future periods which again determines the other aggregates of the economy. The first order differential system of equations is:
𝑥𝑡+1 =
𝑏𝑢(𝑥) = (1 − 𝛼)( 𝑤𝑈+ (1 + 𝑟)(𝑥𝑡+ 𝑤𝑈)) 𝑖𝑓 𝑥𝑡 < 𝑓𝜏(𝑠)
𝑏𝑠,𝜏(𝑥, 𝑠) = (1 − 𝛼)(𝑤𝑠+ (𝑖 − 𝑟𝐺)𝑠 + (1 + 𝑖)(𝑥 − ℎ)) 𝑖𝑓 𝑓𝜏(𝑠) ≤ 𝑥𝑡 < ℎ − 𝑠 𝑏𝑠,𝜏(𝑥, 𝑠) = (1 − 𝛼)( 𝑤𝑠+ (𝑟 − 𝑟𝐺)𝑠 + (1 + 𝑟)(𝑥 − ℎ)) 𝑖𝑓 𝑥𝑡 ≥ ℎ − 𝑠
(3.25) Solving the equations forward, setting 𝑥𝑡= 𝑥𝑡+𝑡 = 𝑥, and solving for x gives us the three new steady states for long run inheritance. Individuals inheriting less than 𝑓𝜏, work as unskilled labor in both periods of life. The steady state is characterized by the inheritance level 𝑥̅̅̅: 𝑢
𝑥𝑢
̅̅̅ = (1 − 𝛼)
1 − (1 − 𝛼)(1 + 𝑟)𝑤𝑈(2 + 𝑟) (3.26) Individuals inheriting more than the individual cost of education, 𝑥 ≥ ℎ − 𝑠, invest in human capital and work as skilled labor in the second period of life. The steady state is characterized by the inheritance level 𝑥̅̅̅̅̅: 𝑠,𝜏
𝑥𝑠,𝜏
̅̅̅̅̅(𝑠) = (1 − 𝛼)
1 − (1 − 𝛼)(1 + 𝑟)[𝑤𝑆− (𝑟𝐺 − 𝑟)𝑠 − (1 + 𝑟)ℎ)] (3.27) Individuals inheriting less than the individual cost of education, 𝑥 ≥ ℎ − 𝑠, but more than 𝑓𝜏, invest in human capital, by borrowing from banks, and work as skilled labor in their second period of life. The steady state is characterized by the inheritance level 𝑔𝜏:
𝑔𝜏(𝑠) = (1 − 𝛼)
1 − (1 − 𝛼)(1 + 𝑖)[𝑤𝑆 + (𝑖 − 𝑟𝐺)𝑠 − (1 + 𝑖)ℎ] (3.28) As earlier this steady state is locally unstable. But, depending on the size of the subsidy and the government bond rate, 𝑥̅̅̅ and 𝑥𝑢 ̅̅̅̅̅ might not be reached. This is due to two reasons: 𝑠,𝜏
1) The subsidy has lowered the inheritance needed in order for investing in human capital to be the optimal decision. If the steady state inheritance level of individuals not investing in human capital, 𝑥̅̅̅, is higher than the threshold of which individuals invest 𝑢 in human capital, 𝑓𝜏, the whole population converge to a long run steady state of 𝑥̅̅̅̅̅. 𝑠,𝜏
2) The possibility of 𝑟𝐺 > 𝑟 has reduced the budget size for individuals investing in human capital. If the steady state bequest level of individuals investing in human capital, 𝑥̅̅̅̅̅, 𝑠,𝜏 is lower than the individual cost of education, ℎ − 𝑠, the whole population converge to a long run steady state of 𝑥̅̅̅̅̅. 𝑢,𝜏
We will look more closely at these cases when analyzing the dynamics.
3.5 Dynamics
This section aims at describing the dynamics between the short run and the long run equilibria for different possible exogenously given subsidy rates and government interest rates. As a reminder, let’s repeat the illustration of the dynamics from the baseline model, or, equivalently, when the subsidy is equal to 0:
Figure 3-14
4 Inheritance dynamics in the absence of subsidies.
In this case, the dynasties were divided into two groups in the long run. Dynasties with an initial wealth above 𝑔𝜏(0) converge to the higher steady state. Thus, become skilled workers with wage equal to 𝑤𝑆 and inheritance equal to 𝑥̅ (0). Dynasties with an initial wealth below 𝑔𝑆 𝜏(0) converge to the lower steady state and become unskilled workers with wage equal to 𝑤𝑈 and inheritance equal to 𝑥̅̅̅. 𝑈
As the subsidy does not affect 𝑏𝑢(𝑥), we only have to measure the effect on 𝑏𝑠,𝜏(𝑥). A convenient matter, is that the 𝑏𝑠,𝜏(𝑥) line is solely determined by the level of inheritance of which individuals start investing in human capital, 𝑓𝜏(𝑠), which we already have calculated in (3.19), and ℎ − 𝑠. This is because the inheritances of dynasties investing in human capital starts growing (declining) at the rate (1 − 𝛼)(1 + 𝑖) from the intersection of the 𝑏𝑠,𝜏(𝑥) curve and the 𝑏𝑢(𝑥) curve, which is characterized by 𝑓𝜏(𝑠). At once dynasties have acquired enough wealth in order for inheritance and the subsidy to completely fund the individual cost of education, 𝑥 = ℎ − 𝑠, they will save the difference between their inheritance and the cost of education at the world market interest rate. From this point the inheritance of dynasties grow (decline) at the rate (1 − 𝛼)(1 + 𝑟).
From the discussion in chapter 3.3, we know that 𝑓𝜏(𝑠) reduces one-for-one with the subsidy when the government borrowing rate is equal to the world market interest rate, and less than one-for-one when it’s larger than the world market interest rate. The individual cost of education, ℎ − 𝑠, of course reduces one-for-one with the subsidy. Therefore the 𝑏𝑠,𝜏(𝑥) line is shifted to the left when 𝑟𝐺 = 𝑟, and down to the left when 𝑟𝐺 > 𝑟, for positive subsidy levels.
3.5.1 Case 1: 𝑠 = ℎ, 𝑟𝐺 = 𝑟
Let’s start describing the dynamics when the government subsidizes the full cost of education for the whole population, and the government interest rate is equal to the individual’s savings rate, 𝑟𝐺 = 𝑟. This case is shown in figure 3-2.
As the government subsidized the full cost of education, the individual cost of education, ℎ − 𝑠, equals 0. Since all individuals are at the same footing, and have access to the same budget sets, the optimal education decision for one individual will be the optimal decision for all individuals. Condition (3.17) tells us whether or not investing in human capital is optimal for individuals that does not have borrow from banks. Since the government interest rate is equal
to the world market interest rate, condition (3.17) is reduced (2.21) which we assumed to hold in the baseline model. Thus, all individuals invest in human capital and work as skilled workers in the short run. Since the subsidy covers the entire cost of education, individuals will save their entire inheritance at the world market interest rate, and all dynasties of the economy converge to a long run stable steady state level of inheritance equal to 𝑥̅̅̅̅̅ = 𝑥𝑠,𝜏 ̅𝑠, and work as skilled labor in the long run.
Figure 3-25
3.5.2 Case 2: 𝑠 < ℎ, 𝑟𝐺 = 𝑟
In this case, as shown in figure 3-3 and 3-4, the government partially subsidizes the cost of education. This is possibly because of government borrowing constraints. We still assume that the government borrow at the world market interest rate. In the case of partial subsidies, there will be two sub cases. One of which the entire population converges to the higher steady state, and one where dynasties are divided into a group of unskilled and skilled labor.
5 Inheritance dynamics when the government fully subsidize education and can borrow at the world market interest rate. Every dynasty of the economy will in this case work as skilled labor in the long run and converge to an inheritance level of 𝑥̅̅̅̅̅ = 𝑥𝑠,𝜏 ̅𝑠.
For partial subsidies, individuals will have to fund shares of the education cost either through inheritance or bank loans. For sufficiently large subsidies, the threshold of inheritance which makes individuals invest in human capital, 𝑓𝜏, is lower than the steady state inheritance level for descendants of individuals working as unskilled labor, 𝑥̅̅̅. In this case, individuals inheriting 𝑢 less than 𝑓𝜏 will choose to work as unskilled. But as their children will be left with more inheritance than they did themselves (since 𝑥𝑡+1(𝑓𝜏) lies above the 45° line), these dynasties will eventually start investing in human capital by borrowing from banks. Their inheritance level then starts growing rapidly along the 𝑏𝑠,𝜏(𝑥) line until they reach an inheritance level where they can fund the whole cost of education through the government subsidy and inheritance, 𝑥𝑡 = ℎ − 𝑠. From this point individuals will start saving excess inheritance on the world market interest rate. As the inheritance level of all dynasties follow the same path, all dynasties of the economy converge to a long run stable steady state of inheritance equal to 𝑥𝑠,𝜏
̅̅̅̅̅ = 𝑥̅𝑠
Figure 3-36
6 Inheritance dynamics when the government partially subsidize education by borrowing at the world market interest rate and the threshold of which investing in human capital is optimal is below the steady state level for dynasties not investing in human capital. Every dynasty of the economy will in this case work as skilled labor in the long run and converge to an inheritance level of 𝑥̅̅̅̅̅ = 𝑥𝑠,𝜏 ̅𝑠.
For sufficiently small subsidies, the threshold, 𝑓𝜏, is larger than the steady state inheritance level for the descendants of individuals choosing to work as unskilled, 𝑥̅̅̅. As a result, the economy 𝑢 is in the long run divided into two groups, as in the baseline model. Individuals inheriting more than 𝑔𝜏 work as skilled labor, and so do all of their future descendants. Their long run inheritance level converges to 𝑥̅̅̅̅̅ = 𝑥𝑠,𝜏 ̅𝑠. On the other hand, individuals with an initial wealth above 𝑓𝜏 but below 𝑔𝜏 invest in human capital. Because these have to borrow a large amount in order to invest in human capital, their interest payments are large, and their descendants will therefore receive lower amounts of inheritance than themselves. Once the inheritance of the dynasty drops below 𝑓𝜏, the individuals of these dynasty starts working as unskilled labor, and so do their descendants in all future periods. Their long run inheritance level converges to 𝑥𝑢,𝜏
̅̅̅̅̅ = 𝑥̅̅̅. 𝑢
Figure 3-47
7 Inheritance dynamics when the government partially subsidizes education by borrowing at the world market interest rate and the threshold of which investing in human capital is optimal is above the steady state level of inheritance for dynasties not investing in human capital. Dynasties with an initial wealth below 𝑔𝜏 will work as unskilled labor in the long run and their inheritance level will converge to 𝑥̅̅̅. Dynasties with an initial wealth 𝑈 above 𝑔𝜏 will in the long run work as skilled labor and their inheritance level will converge to 𝑥̅̅̅̅̅ = 𝑥𝑠,𝜏 ̅𝑠.
3.5.3 Case 3: 𝑠 = ℎ, 𝑟𝐺 > 𝑟
Now, assume the government again subsidizes the full cost of education. But because of government bond risk, the government only has access to a borrowing rate which is larger than the world market interest rate.
As the government subsidizes the full cost of education for all of the inhabitants, and no one investing in human capital can avoid taxes, the optimal decision for all individuals will be the same regardless of inheritance. If condition (3.17) holds, then all individuals choose to invest in human capital in the short run. Setting 𝑠 = ℎ in (3.17) and solving for the interest rate gives us the threshold of the government interest rate assuring that all individuals invest in human capital.
𝑤𝑆 − (2 + 𝑟)𝑤𝑈
ℎ > (1 + 𝑟𝐺) (3.29)
A violation of this assumption would mean that the 𝑏𝑠,𝜏(𝑥) curve would be below the 𝑏𝑢(𝑥) curve regardless of inheritance, and all individuals would prefer to work as unskilled labor in both the short and the long run. Assuming that the condition holds, the individual cost of education is 0, and individuals save their entire inheritance at the world market interest rate.
Thus, inheritances of every dynasty evolve along the slack part of the 𝑏𝑠,𝜏(𝑥) curve towards its stable steady state 𝑥̅̅̅̅̅. But because the government interest rate is larger than the individuals’ 𝑠,𝜏 borrowing rate, the steady state is strictly lower than 𝑥̅𝑠. As in case 1, all dynasties of the economy end up working as skilled labor in the long run.
Figure 3-58
3.5.4 Case 4: 𝑠 < ℎ, 𝑟𝐺 > 𝑟
Lastly, let’s assume a case where the government partially subsidizes the cost of education and because of government bond risk is only able to borrow at an interest rate above the individuals’
savings rate. In this case there will be three underlying possible scenarios, illustrated respectively in figure 3-6, 3-7 and 3-8.
First, for sufficiently high subsidies, and sufficiently low government interest rates, we might again have the case where the steady state of individuals working as unskilled labor, 𝑥̅̅̅ is below 𝑢 the threshold of inheritance making education the optimal choice, 𝑓𝜏. As before this makes the inheritance of all dynasties in the economy converge to the higher steady state, 𝑥̅̅̅̅̅. But as in 𝑠,𝜏 case 3, since the government borrows at a higher interest rate than individuals save, this steady state is strictly lower than the higher steady state in the absence of subsidies, 𝑥̅𝑠.
8 Inheritance dynamics when the government subsidizes the full cost of education by borrowing at an interest rate above the individuals’ savings rate. Every dynasty of the economy will in this case work as skilled labor in the long run and converge to an inheritance level of 𝑥̅̅̅̅̅. 𝑠,𝜏
