ON THE GROWTH OF DEVELOPING COUNTRIES

Transcription

1 ON THE GROWTH OF DEVELOPING COUNTRIES By MATT GERKEN A SENIOR RESEARCH PAPER PRESENTED TO THE DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE OF STETSON UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF BACHELOR OF SCIENCE STETSON UNIVERSITY 2015

2 ACKNOWLEDGMENTS I would like to acknowledge Dr. Green and Dr. Rasp for their guidance with this research project concerning the economic growth of developing countries. Their advice and commentary have been pivotal in developing the foundations for this project, as well as the methodology behind producing results within the context of specific economic theories. 2

3 TABLE OF CONTENTS ACKNOWLEDGEMENTS LIST OF TABLES LIST OF FIGURES ABSTRACT CHAPTERS 1. ECONOMIC AND STATISTICAL FUNDAMENTALS Production Functions Backgound on Statistics Conclusions EXOGENOUS GROWTH THEORY Precursor to the Exogenous Growth Model Harrod-Domar Model Conclusions of the Harrod-Domar Model Solow-Swan Model Solow Model Conclusions of the Solow-Swan Model Mankiw-Romer-Weil Adaptation of the Solow-Swan Model Conclusions Regarding Exogenous Growth ENDOGENOUS GROWTH THEORY Romer Model Romer Model Explained Conclusions Regarding Endogenous Growth Theory OTHER FACTORS AFFECTING ECONOMIC GROWTH DISCUSSION AND RESULTS Research Methodology Exogenous Growth Theory Endogenous Growth Theory Africa CONCLUSIONS APPENDIX REFERENCES BIOGRAPHICAL SKETCH

4 LIST OF TABLES TABLE 1. Estimation of Un-augmented Solow Model Estimation of Augmented Solow Model Countries Removed from Analysis (39) Employment Ratio Per Country, Five-Year Periods Savings Rate Per Country, Five-Year Periods Results of Nonlinear Regression Results of Nonlinear Regression Results of Nonlinear Regression Exogenous Comparison 1 (16 groups) Countries Removed from Analysis (77) Exogenous Comparison 2 (7 groups) Results of Nonlinear Regression Values for Endogenous Variables Countries Ranked According to Productivity and Per Capita GDP Per Capita GDP, Countries in Africa, 2013, Ranked Africa Regression

5 LIST OF FIGURES FIGURE 1. Solow-Swan Model

6 ABSTRACT ON THE GROWTH OF DEVELOPING COUNTRIES By MATT GERKEN May 2015 Advisors: Dr. Green, Dr. Rasp Department: Mathematics and Computer Science Much work has been devoted to modeling and predicting the economic growth of countries, namely through fundamental macroeconomic principles, exogenous growth theory and endogenous growth theory. This paper attempts to evaluate the validity of these models in determining the growth specifically of developing countries, given how many of these countries are in a sense trapped by poverty and low economic growth. The analysis indicates that, with current data and over long time periods, the claim of exogenous growth theory that countries with similar characteristics demonstrate convergence in per capita GDP is not well-supported. In fact, even when considering different criteria for grouping among countries, the groups demonstrated increasing divergence in per capita GDP. With regards to endogenous growth theory, no significant relationship exists between a calculated productivity coefficient and per capita GDP, although this is highly attributable to insufficient data. Lastly, through panel data regression among African countries, several independent variables that are statistically significant in explaining per capita GDP growth are found, including school enrollment, population density, rural population growth, percent of land area considered to be arable, life expectancy, gross domestic savings, and the real interest rate. Future research can be directed towards finding more suitable proxy variables in exogenous and endogenous growth theory analysis, as well as towards additional regression models within Africa specifically to determine other significant relationships in the context of economic growth. 6

7 CHAPTER 1 ECONOMIC AND STATISTICAL FUNDAMENTALS For the sake of discussion on economic growth in developing countries, this introductory section will discuss fundamentals inherent to macroeconomic theory and statistics. Mishkin (2012) provides a thorough discussion on macroeconomic theory in his Macroeconomics: Policy and Practice, from which exogenous and endogenous growth theory can be extended in chapters 2 and PRODUCTION FUNCTIONS In economic theory, production functions measure the output generated by a production process given certain inputs or factors of production. At the scope of the macroeconomy, they can be used to distinguish the factors that contribute directly to economic growth, and can also determine real GDP, a measure of the output of actual goods and services produced in an economy over a fixed period (Mishkin 5). The most standard and well-known of these production functions which Mishkin (2012) includes in his summary is the Cobb-Douglas production function, which states that if Y is the total production in the economy, L represents the labor variable, K represents the capital variable, A is total factor productivity, and and 1 are constants associated with capital and labor, respectively, then Y AK L 1 (1) Capital here, referring to Miskin s definition, is the quantity of structures and equipment such as factories, trucks, and computers that workers use to produce goods and services (50). The total factor productivity term A indicates how productive capital and labor are together. This term is not directly measurable, and can be determined given values for Y, L, and K. Later on in the analysis, human capital, a concept related to physical capital here, is also included, which refers to stock of knowledge and other human characteristics. There is considerable debate over how it is measured, however. The following are all ways in which human capital has been measured over the years: school enrollment rates, total years of schooling as a way to measure 7

8 educational attainment, literacy rates, and expenditures on education (Kwon 2009). This debate results in several complications for this analysis, explained in the results of chapter 5. This production function has two characteristics that will have important implications for subsequent conversations on exogenous and endogenous growth theory (Mishkin 2012). The first of these characteristics is the property of constant returns to scale, which states that increasing the factors, L and K, of this production function by a percentage increases the output Y by the same percentage. So, Y F(2K,2L ) 2 F(K,L ) (2) The other characteristic is that of diminishing returns, in that this production function experiences diminishing marginal product. If one of the factor inputs increases while others are held constant, the increase in output that results from one extra unit of this output declines. In this way, there is a marginal product of capital, denoted MPK, and a marginal product of labor, denoted MPL (Mishkin 2012). The MPK and MPL can be represented as MPK Y K and MPL Y L (3) Thus, if the stock of capital K or the supply of labor L increases, the MPK and MPL decrease, respectively, so that less output is obtained from each additional unit of these inputs. The exponents and 1 represent the proportion of national income that is attributed to labor and capital, respectively, so that National Income = Y = Y (1)Y = Capital Income + Real Labor Income (4) Historically in the United States, the value for has been 0.3, making 1 equal to 0.7, so that capital represents 30% of national income and labor 30% of national income. Here, national income is related to real GDP, in that it is the value determined by combining the compensation of employees, other income (such as from the self-employed), and corporate profits for an economy (30). These values of course vary across countries, but if this Cobb-Douglas function is substantiated by economic data, of which there is much evidence for the affirmative, 8

9 then the shares of labor income and capital income in national income do not change even as the total level of income rises and falls (Mishkin 65). This is especially important for a study concerning economic growth worldwide, in that the relative proportions of labor and capital to national income should be fairly consistent for countries. In discussing the relative growth among countries, per capita income is often a more representative measure of a country s growth. For instance, in 2013, China s nominal GDP was second in the world at $9,240,270 million, but its nominal GDP per capita was 84 th in the world at $6,807 due to its large population size (World Bank). Per capita income can also be derived from the Cobb-Douglas production function. If Equation (1) is divided by labor L, y Y L AK L 1 L AK L Ak (5) where y is per capita income, or income per worker, and k is capital per worker. In comparing the values for per capita income, capital per worker, and A, the total factor productivity term, Mishkin (2012) finds that the short-fall of per capita income in other countries relative to the United States is due more to lower productivity than it is to lower amounts of capital per person (54). This analysis concerning per-capita income then demonstrates that factors associated with increases in A, such as changes in technology, are more significant than levels of capital in determining per-capita income. This distinction between capital and total factor productivity has significant importance in separating the implications of exogenous and endogenous growth theory mentioned in chapters 2 and 3. From these discussions on production functions and other economic foundations, the concepts of aggregate demand and aggregate supply are introduced as ways to model a country s output and price levels. 9

10 1.2. BACKGROUND ON STATISTICS This section will serve as a brief review of the statistical terms utilized in my analysis, and is gleaned from Mendenhall and Beaver s Introduction to Probability and Statistics, Eighth Edition. In multiple linear regression, the objective is to arrive at a model that relates a particular response or dependent variable y to a number of predictor or independent variables, denoted x 1, x 2,, x n (Mendenhall and Beaver 1991). The resulting model is of the form y x x... 0, where the values of β are constants resulting from the x n n regression analysis, and ε is an random error term. In this analysis, nonlinear regression is utilized. With the exception of section 5.4, nonlinear regression is conducted on various forms of the Cobb-Douglas production function stated in (1), which demonstrates constant returns to scale. This Cobb-Douglas production function is linearized by taking the natural logarithm, but the imposing of constant returns to scale results in nonlinear regression, not because of the terms themselves, but because the coefficients must add to 1. In (1), for example, after it is transformed into LnY LnA LnK ( 1) LnL via the natural log, nonlinear regression must be utilized, since the regression coefficients of LnK and LnL, α and (1-α), respectively, must add to 1. Several other statistical terms must also be mentioned here. The variance is a measure of the variability of a certain dataset, and is equal to the sum of squared deviations of the measurements about their mean _ x divided by (n -1) for a sample of the population (Mendenhall and Beaver 1991, 33). The standard deviation is often reported instead of the variance, and is simply the square root of the variance. Regarding the standard deviation, roughly 68% of the data are within (above and below) one standard deviation of the mean, 95% of the data occurs are within two standard deviations of the mean, and 99.7% of the data are within three standard deviations of the mean. 10

11 The standard error of a regression coefficient measures how precise the regression model s estimation of an unknown independent variable s coefficient is. A smaller standard error value indicates a higher level of precision. The standard error can be used to formulate a confidence interval for a particular parameter, which is a range of values a certain parameter has with a specified probability. The standard error is reported for all regression results in this analysis (Mendenhall and Beaver 1991). Last but not least, the coefficient of determination, or r 2, measures how well the independent variables are related to the dependent variable by testing how close data are to a fitted regression line. Often, the adjusted r squared value is also reported, which adjusts for the number of predictors in a model. Adding independent variables in a model automatically increases r 2, which may provide misleading information regarding the model s predictive capacity. For this reason, the adjusted r 2 value is a useful measure when considering many different independent or predictor variables (Mendenhall and Beaver 1991). In the last section of this analysis, when considering a case study of the countries within Africa, a specific data type is used, which then has implications for the way in which the regression analysis is conducted. This data type is that of panel data, data that is multidimensional over time. Specifically, in section 5.4, there are 54 countries for 954 variables over 45 years. Conducting ordinary multiple linear regression assumes each data point is independent of another, but this is not an accurate assumption for data that is consistent over time for many different variables. In order to conduct the Panel Data regression, dummy variables must first be calculated for all countries excluding the first, in which for a specific country s dummy, that country is coded as 1 and all others are coded as 0. During the linear regression analysis, each of these dummy variables is treated as a predictor variable, along with the other chosen independent variables for the model. 11

12 1.3. CONCLUSIONS From here, exogenous and endogenous growth theory will be introduced in Chapters 2 and 3 as mathematical models that attempt to better quantify changes in growth as a function of certain variables. Chapter 4 states factors that are not necessarily included in these models but have been shown to shape a county s growth rate. Chapter 5 includes all of the results of the analysis. CHAPTER 2 EXOGENOUS GROWTH THEORY In exogenous growth models, factors that affect the long-run rate of growth are assumed to be determined outside of the model, mainly technology. These exogenous growth models are contrasted with endogenous growth models, in which determinants of growth are determined within the model and which focus more on technology and institutions in promoting economic growth PRECURSOR TO THE EXOGENOUS GROWTH MODEL In modern day analysis, the Solow-Swan model is viewed as the fundamental exogenous growth theory model. Solow (1956) and Swan (1956) relied on some of the results of the Harrod- Domar model in their analysis, so the Harrod-Domar model is discussed briefly as a precursor to exogenous growth theory HARROD-DOMAR MODEL Domar (1939) analyzes the relationship between employment, rate of growth, and investment. Making several assumptions that are not necessary to identify here, Domar claims an economy will be in equilibrium when its productive capacity P is equivalent to national income Y. 12

13 Given this underlying equality, Domar attempts to generate the conditions that will keep the economy in a state of continual full employment. Somewhat contrary to previous models, Domar (1939) assumes that employment is determined by the ratio of national income Y to productive capacity P. He uses I to refer to the rate at which investment changes per year and s to refer to the ratio of potential net value added, a measure of output, to I. In this way, productive capacity is equal to the product of these terms, so that P=Is. However, claiming an economy may not experience a full increase in productive capacity, he denotes a new variable that instead refers to potential capacity, which he claims is defined as dp dt I (6) In this way, potential capacity is equal to the rate of change of productive capacity divided by investment. The maximum value that can reach is s, the ratio of P to I. Multiplying both sides of (6) by I, dp dt I (7) In this way, dp dt or the increase in productive capacity P is equivalent to investment multiplied by a variable referring to potential capacity. Now analyzing how output Y changes with respect to time, Domar introduces the variable to represent marginal propensity to save. Viewing the change in output over time as involving the rate of increase of investment I, dy dt di 1 dt (8) This relationship established in (8) is supported by Harrod (1939) as well, in that he claims a unique warranted line of growth is determined jointly by the propensity to save and the quantity 13

14 of capital required by technological and other considerations per unit increment of total output (23). The equilibrium of this model was assumed to occur at P = Y, so for the equilibrium to be maintained here, dp dt dy dt must also be true. Substituting (7) and (8) into (9), I di 1 dt (9) (10) Solving this differential equation, I I 0 e t (11) Given that this occurs at the equilibrium of this model, the constant is the rate of growth necessary to maintain full employment at this equilibrium, which Solow (1956) references in the following section. Domar then considers a situation in which investment grows at a rate r that might not be equivalent to this equilibrium rate of growth. To consider this situation, he introduces two I other terms, the average propensity to save, which is, and the average ratio of productive Y P capacity P to capital K, which is. For the sake of simplifying the problem, Domar assumes K I Y and capital is equivalent to the sum of net investment, P K s. Domar breaks the analysis into two cases, the first of which is s. Using (11), and noticing that 14

15 t K K 0 I 0 e rt dt K 0 I 0 r (ert 1) (12) 0 Given that Y 1 I 0e rt, ratio of income Y to capital K is Y K 1 I 0e rt K 0 I 0 r (ert 1) (13) Calculating the limit of this ratio as t approaches infinity, he gets Y lim t K r (14) Since an assumption of this part of the model was that that, by substituting into (14), P K s, K can be rewritten as K P s so Y lim t P r s (15). Since this first case assumed s, replacing s with in (15) yields Y lim t P r (16). Using this result, he labels a variable as the coefficient of utilization, so that r If r, the rate at which investment grows, is equal to the equilibrium rate of growth obtained from (11), then 1 or 100%, so that productive capacity is completely utilized. It is (17) when r that productive capacity is not fully utilized. So, when r, unused capacity and unemployment result. Domar (1939) arrives at the same conclusions for the second case, in which s. 15

16 It is this idea of an equilibrium rate of growth in an economy and this concept of utilization that Solow (1956) extends in his analysis of what today is known as exogenous growth theory CONCLUSIONS OF THE HARROD-DOMAR MODEL Combining the Harrod and Domar models suggests several ways to achieve economic growth. These models when combined suggest that economic growth depends on labor and capital, and that increasing investment, such as through increasing savings, generates capital accumulation, which also produces economic growth. Additionally, technological advancements can make investment more efficient, which also increases economic growth SOLOW-SWAN MODEL The Solow-Swan model is an exogenous growth model that builds off the previous Harrod-Domar model. It adds labor as a component of production, and assumes that the capitalto-labor ratio is no longer fixed SOLOW MODEL Adjusting the Harrod-Domar model, Solow (1956) developed his own economic growth model, focusing on how the saving rate and population growth affect economic growth. Although the model is attributed to both Solow (1956) and Swan (1956), who independently arrived at similar conclusions, the Solow model is more in depth, and is explained in lieu of the Swan model. Starting with total output, designated by Y(t), Solow asserted that the ratio of output that is saved is represented by sy(t), or a fraction of the total output. Assuming that the fraction s that is saved from total output is then directly invested, Solow claims that net investment is the rate at which total capital K(t) increases, so that 16

17 dk dt sy (18) Inserting (18) into a general production function Y F(K,L), Solow claims that, where L(t) represents total employment in the economy, dk dt sf(k,l) (19) The production function represented here demonstrates constant returns to scale, which is a typical assumption of growth theory. Solow then borrows from the Harrod model, in that population growth is assumed to be exogenous and grows at a constant rate n, called Harrod s natural rate of growth. Now having L(t) represent the total supply of labor as opposed to total employment as in (19), L(t) L 0 e nt (20) In this way, the labor force grows exponentially. Utilizing (19) and (20), Solow claims dk dt sf(k,l 0 e nt ) (21) is a differential equation for the single variable K(t) whose solution represents the growth of capital stock that fully employs the total supply of labor. Since capital and labor are the two factors that determine output, as seen in a basic production function (1), knowing how these two variables change over time will allow reveal how real output changes over time. Before proceeding, Solow first studies the relationship between the accumulation of capital K(t) and the growth in the labor force by examining the results of differential equation (21). Solow proposes a new variable, the ratio of capital to labor r K. Rewriting K as L K K L L rl rl 0e nt and differentiating this, implicitly, with respect to time, dk dt dr L 0 e nt dt nrl 0e nt Substituting this result into (21), Solow obtains (22) 17

18 dr dt nr L 0 e nt sf(k,l 0 e nt ) (23) and after rearranging terms and dividing out L L 0 e nt, Solow arrives at a differential equation that now involves the capital-labor ratio, as opposed to only capital K(t) itself. The function dr dt nr L 0 e nt sf(k,l 0 e nt ) dr dt nr sf(k,l 0e nt ) L 0 e nt dr dt sf K L 0 e, L 0 ent nt L 0 e nt dr sf(r,1) nr (24) dt F(r,1) measures output produced as r amounts of capital are put to use with one unit of labor. Recalling (21) and that n is the natural rate of growth of labor, (24) is a differential equation in which the rate of change of r, the ratio of capital to labor, or, equivalently, capital per worker, is the difference between the rates of change of capital and labor. Analyzing these results graphically, Solow (1956) claims that given an initial value of r, the capital per worker, the entire system will approach a state of balanced growth, called the steady state, namely, the intersection of the function sf(r,1) and the line nr with slope n, the natural rate of growth. At this point, the rate at which capital per worker is 0. This is the basic model that the rest of Solow s model adds to. Specifically, he incorporates depreciation, a variable savings rate, a variable population growth rate, and an exogenous rate of technology. 18

19 Figure V: Solow-Swan Model Source: "The Solow-Swan Growth Model." The Solow-Swan Growth Model. Web. 22 Nov < If represents the depreciation rate, or the fraction of capital that wears out each year, (24) can be modified by multiplying this depreciation rate by r, so that, as seen in Figure V, dr dt sf(r,1) r (25) The steady state then occurs where sf(r,1) intersects r, or where investment is equal to depreciation. Additionally, the depreciation rate and the growth in population n can be modeled together. If r is defined as capital per worker, or r K, then an increase in L will reduce the L value of r. In this way, n can be added with the depreciation rate as a variable that detracts from capital per worker, so that (25) becomes dr dt sf(r,1) ( n)r (26) This is also modeled in Figure V, and here, the steady state occurs when sf(r,1) intersects ( n)r. 19

20 Solow (1956) then includes technological change in the model, but as an exogenous variable, or a variable that is determined outside the model. He adjusts the production function Y F(K,L) K L 1 to arrive at the Cobb-Douglas production function modeled in (1). To do this, he adds technology through the term A(t) so that Y A(t)F(K,L) (27) Letting A(t) e gt, where g is the growth rate of technology, he restates one of the earlier conclusions, equation (21), in which dk dt sf(k,l 0 e nt ) and includes this new technology term, as well as the exponents on K and L, so that dk dt sa(t)k L 1 se gt K (L 0 e nt ) 1 sk L 1 0 e (n(1) g)t (28) Solving this differential equations generates the result K(t) K 1 0 (1 )s n(1 ) g L 1 0 (1 )s n(1 ) g L 1 0 e (n(1 ) g)t 1 1 (29) 1 As time goes to infinity, capital then approaches (n(1 ) g) 1, or, more simply, n g. This implies that the rate at which capital stock increases is n g, whereas before 1 1 this rate was simply n. In this way, the capital-labor ratio does not reach an equilibrium, and grows without bound. If the savings rate is allowed to be variable, then (25) becomes dr dt s(r)f(r,1) nr (30) This is similar to the analysis of (25) when s was a constant, except now, the savings rate is variable, and the graphical analysis changes depending on the nature of the s(r)f(r,1) curve. 20

21 Lastly, if the population rate is modeled as an endogenous variable, as opposed to an assumed constant, (25) becomes dr dt sf(r,1) n(r)r (31) The graphical analysis of the steady state then depends on the nature of the n(r)r curve, such as through how many times it intersects the sf(r,1) curve CONCLUSIONS OF THE SOLOW-SWAN MODEL The Solow model suggests not only the existence of a steady state, but also of convergence. Mishkin (2012) summarizes this concept well: if multiple economies have the same aggregate production function, the same ratio of workers to the total population, and the same saving rate, the Solow model suggests that even if those economies start with different capital-labor ratios, they will tend to converge to the steady state and end up with similar levels of per capita income (153). In terms of the variables included in the previous analysis, countries with the same production function, saving rate s, and ratio of the labor force employed, despite different levels of the capital-to-labor ratio r, will approach a comparable level of per-capita income. This concept is significant, in that with the three assumptions of equality, a high degree of convergence for per-capita incomes across countries would be expected. Mishkin (2012) tests this theory with data from 1960 to 2012 for two groups: a group of wealthy nations with high per capita GDP, and then a group with a mix of 105 wealthy and poor countries. The data supports convergence for rich countries to a large degree, but when this analysis is extended to the second group, convergence does not carry over (Mishkin 2012). This does not necessarily suggest that the idea of convergence is incorrect, however. The Solow model does predict convergence for countries with similar production functions, saving rates, and ratios of the labor force employed, and it can be argued that developing countries and developed countries do not demonstrate similar levels of these entities. 21

22 Regarding other conclusions that are gleaned from the Solow model, Solow (1956) does show, using graphical analysis, that a higher saving rate s shifts the sf(r,1) up, resulting in higher levels of capital and output. Additionally, an increase in the population growth rate n shifts the ( n)r curve up, decreasing the steady state capital per person and level of output per person. In this way, increases in saving lead to higher steady-state levels, and higher population growth results in lower levels of output and income per person (Solow 1956). What some, specifically proponents of endogenous growth theory, which is explained in the next chapter, claim to be a shortcoming of the Solow model is the assumption that technology is exogenous, or assumed outside of the model. The model does explain in great detail capital accumulation and its effects on per capita income and growth, and as is shown in the following section, its results have since been enhanced by introducing the idea of human capital to the model MANKIW-ROMER-WEIL ADAPTATION OF SOLOW-SWAN MODEL Mankiw, Romer, and Weil (1992) take the Solow model and distinguish between two components of capital that he included in his model, namely, human capital and physical capital. Although they acknowledge that the assertions made by the Solow model are in fact supported by economic data, they argue that the effects of saving and population growth on income are too significant. Acknowledging that changes in the saving and population growth rates may affect and also be correlated with human capital accumulation, they augment the Solow model by including physical and human capital. Restating the results of the Solow model, they restate the production function that was previously used in the discussion of the Solow model, except that they expand it so that Y(t) K(t) (A(t)L(t)) (32) 22

23 where again Y, K, L, and A are output, capital, labor, and level of technology, respectively. Labor and level of technology are exogenous in the Solow model, and here, Mankiw, Romer, and Weil have them grow at rates n and g, respectively. Before, Solow defined capital and output as per capital per unit of labor k and output per unit of labor y, but here, they include the level of technology term as well, so that k K AL (33) and y Y AL (34) represent capital per effective unit of labor and output per effective unit of labor. The change in capital per effective unit of labor, which was defined as dr dt in the Solow model but dk here, is dt dk dt sy(t) (n g )k(t) or, by using y Y AL and substituting (32) for Y, dk dt sk(t) (n g )k(t) (35) where and s are again the depreciation rate and saving rate or fraction of output invested, respectively. Capital per unit of labor, k, converges where its derivative is 0, so the steady state value k * here is s k * (n g 1 1 (36) Substituting this steady state value of capital per effective unit of labor back into the production function (32), an equation for steady-state income per capita is determined, such that y Y AL K (AL) 1 AL K AL(AL) AL K (k) AL 23

24 s y n g 1 ln(y) ln(s) ln(n g ) (37) 1 With A t A 0 e gt, y Y AL, and ln(y) ln Y ln(a) ln Y(t) log( A L L(t) 0 ) gt, the equation for steady-state income per capita is then ln Y(t) ln A(0) gt L(t) 1 ln(s) ln(n g ) (38) 1 where is the share of income attributed to capital, as gleaned from the production function (32), g is the growth of technology in terms of knowledge, and A(0) represents technology, resource endowments, institutions, and other components unique to each country. Testing whether (38) truly represents the relationship between s, n, g, and with real income, Mankiw, Romer, and Weil add clarification to the ln A(0) term so that ln A(0) a (39) where a is a constant and refers to a shock specific to a particular country. Substituting (39) into (38) yields ln Y L a 1 ln(s) ln(n g ) (40) 1 Assuming that s and n are independent of the shock term, they proceed to use the ordinary least squares method to go about finding an estimation of (40). Mankiw, Romer, and Weil used the Real National Accounts dataset for this analysis, which included data on consumption, investment, income, and population for most countries, during the period of They broke their analysis into three samples: all countries except 24

25 those where oil was the main industry (98 countries in total), all countries except those with a population of less than one million to avoid measurement issues (75 countries in total), and the 22 countries of The Organisation for Economic Co-operation and Development (OECD) with populations greater one million. The results they obtain from this analysis are shown in Table I, and do support the Solow model, but, as previously mentioned, the effects of saving and population growth on income are too large. This justifies their subsequent inclusion of human capital and physical capital in the Solow growth model. Table 1: Estimation of Un-augmented Solow Model Dependent Variable: log GDP per working-age person since 1985 Sample Non-Oil Intermediate OECD Number of Observations CONSTANT ln GDP (1.59) 1.42 (0.14) ln(n g ) (0.56) 5.36 (1.55) 1.31 (0.17) (0.53) 7.97 (2.48) 0.50 (0.43) (0.84) R 2 Standard errors are included in parentheses. Source: Mankiw, N. G., D. Romer, and D. N. Weil. "A Contribution to the Empirics of Economic Growth." The Quarterly Journal of Economics (1992): JSTOR. Introducing human capital and physical capital to the Solow growth model, Mankiw, Romer, and Weil (1992) revise the production function (32) to Y(t) K(t) H(t) (A(t)L(t)) 1 (41) 25

26 so that H represents human capital, and the other variables are as they were in (32). If s k is the fraction of income which is invested in physical capital, and s h is that invested in human capital, where y Y AL, k K, and now AL h H AL, dk dt s ky(t) (n g )k(t) (42a) dh dt s hy(t) (n g )h(t) (42b) For the sake of the model, they assume that human capital and physical capital have the same depreciation rate, and that human capital, physical capital and consumption all have the same production function. To continue to use the assumption inherent to the production of decreasing returns, 1 must hold, and the steady states for capital per effective unit of labor k and human capital per effective unit of labor h are then s 1 k * k s h n g 1 1 (43a) s 1 h * k s h n g 1 1 (43b) Similar to the previous model without human capital, they then substitute (43a) and (43b) into the production function (41) that includes human capital and take logs to yield an equation for per capita income that now involves human capital as well: ln Y(t) ln A(0) gt ln(n g ) L(t) 1 1 ln(s k) 1 ln(s h) (44) Historical data has suggested that, the exponent of capital K in (41), is approximately one-third, in that the share of income from physical capital is about one-third. Estimating, the 26

27 coefficient of human capital H in (41), is more difficult, with estimates between one-third and one-half. Given this uncertainty, another way Mankiw, Romer, and Weil (1992) measure the role of human capital here is to focus instead on the level of human capital by combining (44) with the steady-state level of human capital in (43b). This yields ln Y(t) ln A(0) gt L(t) 1 ln(s k) ln(n g ) 1 1 ln(h* ) (45) In this way, Mankiw, Romer, and Weil introduce an augmented Solow model, and set out to determine whether (44), reflecting the rate of accumulation of human capital, or (45), including now the level of human capital, better fit the data on human capital. The results, shown in Table II, indicate that including human capital as a component of the Solow growth model does significantly improve its effectiveness in predicting growth, especially for the sample of 98 non-oil countries and 75 intermediate countries, where R 2 were 0.78 and 0.77, respectively. In this way, the three variables log of the rate of investment I ln, GDP ln(n g ) and log of the ratio of the country s population enrolled in secondary school ln(school) explained 78% and 77% of the variability in differences in income per capita across countries for these two samples. The inclusion of human capital in the Solow growth model was also beneficial in that it decreased the coefficients on physical capital and investment, an issue with that occurred when Mankiw, Romer, and Weil (1992) ran the regression without the introduction of the human capital term. The fact that introducing human capital produced a statistically significant impact on predicting economic growth is important given the Solow model s strong focus on capital accumulation on economic growth. This significance in including the human capital term in the Solow model is also corroborated by Vinod and Kaushik (2007) in their Human Capital and Economic Growth: Evidence from developing Countries. 27

28 Table 2: Estimation of Augmented Solow Model Dependent Variable: log GDP per working-age person in 1985 Sample Non-Oil Intermediate OECD Number of Observations CONSTANT ln GDP (1.17) 0.69 (0.13) ln(n g ) (0.41) ln(school) 0.66 (0.07) 7.81 (1.19) 0.70 (0.15) (0.40) 0.73 (0.10) 8.63 (2.19) 0.28 (0.39) (0.75) 0.76 (0.29) R 2 Standard errors are included in parentheses. Source: Mankiw, N. G., D. Romer, and D. N. Weil. "A Contribution to the Empirics of Economic Growth." The Quarterly Journal of Economics (1992): JSTOR. Their model tested the relationship for 18 developing countries over 20 years, and found that their human capital term, ln H, did indeed impact economic growth for large developing countries. In testing the reliability of the Solow growth model in explaining the economic growth of developing countries, these models indicate that including the human capital component will be necessary in grasping a better understanding of growth for these countries CONCLUSIONS REGARDING EXOGENOUS GROWTH As previously mentioned, critics of exogenous growth theory are most dismayed by how technology and the saving rate are determined outside of the model as opposed to within. Despite this criticism, exogenous growth theory does have many uses in examining the factors that 28

29 determine economic growth. For one, although the data does not support the argument for convergence for a grouping of wealthy and poor countries, the Solow model does predict convergence when the countries have similar production functions, saving rates, and ratios of the labor force employed. The model proposed by Mankiw, Romer, and Weil (1992), and later by Vinod and Kaushik (2007), suggest that including human capital within this exogenous growth theory significantly improve the model s capability in predicting the variability of incomes per capita across countries. From here, endogenous growth theory will be introduced as an alternative to exogenous growth theory, focusing on the role of technology and institutions in explaining economic growth. CHAPTER 3 ENDOGENOUS GROWTH THEORY Since the savings rate of the Harrod-Domar model, as well as technology in the Solow- Swan model, are exogenous, or determined by variables outside of the model, endogenous growth theory aims to endogeneize these variables, so that they are determined within the model. Paul Romer (1990) and Robert Lucas (1988) are two primary contributors to endogenous growth theory, and Romer s is explained as a fundamental endogenous growth theory model ROMER MODEL Romer (1990), often credited as the founder of endogenous growth theory given his extensive work on the subject, developed a model that extended the Solow exogenous growth model, but allowed technological change to be determined within the model i.e. endogenous. 29

30 To begin his model, he defines two terms associated with economic goods: rivalry and excludability. A good is rival if its use by one person prevents usage by others, and oppositely, a good is nonrival if there exists no such limit on its usage. Excludability is a similar concept, except that it refers to the ability of one person to prevent others from using the good. Highly excludable goods are governed by strict laws or a thorough legal system; nonexcludable goods are not restricted in this sense. Technology, which Romer asserts is a large determinant of economic growth, is argued to be nonrival and partially excludable in his model. Human capital is considered to be excludable and rival (Romer 1990). Since technology is claimed to be nonrival, it exhibits unbounded growth (whereas human capital does not), and since it is also only partially excludable, technology in the form of knowledge experiences spillovers within a system. These two facts are significant for a discussion on economic growth (Romer 1990). For this model, for a production function, rival inputs are denoted X, and nonrival inputs are denoted A, where these nonrival inputs usually refer to research and development. For this A term, A E is the excludable part, and A N is the nonexcludable component. Romer (1990) assumes that population and the supply of labor are constant, and that total human capital is fixed. Output Y can be determined from a production function of three inputs: labor L, human capital that is directed toward output H Y, and physical capital such that x consists of all inputs used by a firm to generate output. Then Y(H Y,L,x) H Y L x(i) 1 di (46) 0 assuming i is a continuous index. The exponents on the three terms are somewhat different from previous production function models, but the three exponents still share the same property that their sum is 1. From basic economic theory, it is also introduced that 30

31 dk dt Y(t) C(t) (47) where the rate of growth of capital is equal to the difference of income and consumption. Since an amount of forgone consumption can be used to produce one unit of a durable good, K can also be represented as A K x i x i (48) i 1 i 1 In this way, capital is claimed to increase by this amount of forgone consumption. The equilibrium of the model suggests that researchers, those who contribute to existing research, have complete access to A, the total accumulation of knowledge. Given a researcher s level of human capital H j, productivity, and total accumulation of knowledge A, then each researcher s output in terms of contribution to existing research can be represented as Summing for all such workers who are dedicated to research, H j A. da dt H A A (49) describes how the rate at which this stock of knowledge A grows, where H A represents all human capital directed toward research. The way in which A here is linear suggests that unbounded growth is possible (Romer 1990). If x is used to refer to the equilibrium quantity of durables x(i) as derived from a demand curve analysis, output from (46) can be recalculated, where K can now be represented as K A x. Y(H A,L,x) H Y L 0 x(i) 1 di H Y L A x 1 H Y L A K A 1 (H Y A) (LA) (K) 1 1 (50) 31

32 This model is similar to others, such as in how it demonstrates diminishing returns. Through an analysis of the price level and wages, Romer shows that H Y 1 (1 )( ) r (51) where r is the rate of return on capital. If x is constant, from K A x, K and A must grow at the same rate. From (47) C must grow at this same rate. Borrowing from (49) in that the growth rate of A is H A, he asserts that if g is the growth rate for A, Y, C, and K, then the growth rate amongst these variables is equal, so that g dc dt dy dt dk dt da dt H A (52) Using these results and the fact that human capital having the two components H A and implies H Y H H A, he is able to arrive at an equation for overall growth. Using (51) and this H Y relationship among the two components of human capital H, g H A H (1 )( ) r (53) or more simply, g H A H r (54) such that (1 )( ) (55). In this way, Romer (1990) claims that economic growth can be modeled as the difference between a productivity term multiplied by level of human capital and the rate of return on capital multiplied by a constant determined by the coefficients of the augmented production function. 32

33 3.2. ROMER MODEL EXPLAINED Mishkin (2012), in addition to summarizing the Solow growth model, explains the implications of the Romer growth model in terms of economic growth. The distinction between rivalry and nonrivalry with regards to knowledge, technology and research is imperative here. As Mishkin claims, with rivalry, as in the Solow model, diminishing returns to capital eventually lead output per capita to come to rest at a steady state without growth. Incorporating nonrivalry into [the] economic model allows us to explain sustained growth in per capita income (191). Whereas the Solow model then predicts convergence through a steady state, this endogenous growth theory explains how growth can continue to grow past this point. One criticism of the Solow model was how it did not explain continuous improvements of the standard living in developed countries, and this model better explains this growth. As determined by (51), several factors explain endogenous growth. If more is devoted to expanding research and development, the rate at which per-capita output grows will increase permanently. Additionally, if the productivity term increases, the rate at which output per capita grows will again increase. If the total population increases, more of the labor supply is dedicated to expanding existing research and knowledge, and the rate at which output per capita grows will again increase permanently. It should be noted how this result in particular differs from the Solow model, in which it was shown that population growth decreased the level of output per person. Lastly, increases in the saving rate will produce higher levels of output per person through the way in which this saving increases investment in capital, but will not impact the growth rate (Mishkin 2012) CONCLUSIONS REGARDING ENDOGENOUS GROWTH Endogenous growth theory focuses on human capital, stock of knowledge, and research and development as significant determinants of economic growth. Where earlier exogenous 33

34 theory models suggest convergence to a steady state, endogenous growth theory offers explanations for how continuous increases in the rate of growth are possible through human capital, knowledge and technology. Given the increasing capacity of knowledge to be easily disseminated through the Internet, as well as how technological innovations invented by one country are transferred to others, this theory may have significant implications for economic growth in subsequent decades, especially for developing countries. CHAPTER 4 OTHERS FACTORS DETERMINING ECONOMIC GROWTH Much of the previous research has been dedicated to developing exogenous and endogenous growth theory, as well as the implications of these theories for growth in developing countries. It should also be briefly mentioned several other factors that affect long-term economic growth that are not necessarily encapsulated in these equations of exogenous and endogenous economic growth. Gregory Clark (2007), in his A Farewell to Alms, describes some of the impediments to economic growth that developing countries have experienced. As previously mentioned models of economic growth have suggested, Clark acknowledges that differences in per-capita income across countries stem from differences in capital per person (previously referred to as k), land per person, and efficiency (previously described using A). Clark contends that at the most general level, differences in efficiency are the ultimate explanation for most of the gap in incomes between rich and poor countries in the modern economy ( ). In this way, inefficiency at the level of the individual and at the level of society as a whole has in some way deterred growth in these countries. This assertion is in part present in the Romer endogenous growth model, which includes a worker productivity parameter. 34