Macroeconomic Models of Economic Growth J.R. Walker U.W. Madison Econ448: Human Resources and Economic Growth
Summary Solow Model [Pop Growth] The simplest Solow model (i.e., with exogenous population growth) savings does not produce long run growth. In the long run income per capita is constant and equal to the steady state value. Hence, need to extend the model to generate long run income growth as observed for the last two hundred (or so) years. We know from Global Economic History sustained economic growth requires technical progress, k(t) > 0.
Incorporate Technical Progress into Solow Model Distinguish between accumulation (k ) and innovation. We ve seen that accumulation is not sufficient to generate economic growth in the presence of diminishing marginal productivity. Can think of k as physical capital stock (machines) while technical progress is better and more advanced methods of production. KNOWLEDGE. Increase in knowledge can offset diminishing marginal returns to production. If so, economic growth (y) can increase indefinitely. The insight of Solow s model is that we need both innovation and capital deepening to produce sustained economic growth.
Exogenous Technical Progress Assume that technical progress contributes to efficiency or (economic) productivity of labor. Make distinction now between working population P(t) and effective population. L(t) L(t) = E(t)P(t) where E(t) is a scale of efficiency units that translates working population into units of effective population Thus with an increase in knowledge P(t) can be more efficient and thus represent a larger stock of labor. E(t + 1) = (1 + π)e(t) where, π is the rate of technical progress.
Labor Saving Technical Progress Equation of accumulation remains unchanged (3.8) K (t + 1) = (1 δ)k (t) + sy (t) Before divided by P(t) to express in per capita terms. Now divide by effective population E(t)P(t) (1 + n)(1 + π) ˆk(t + 1) = (1 δ) ˆk(t) + sŷ(t) where the carrot ˆx above a variable means per effective population.
Steady State Figure 3.6
Steady State Same logic as before (population growth) applies. Convince yourself That ˆk is the Steady State. What is the economic interpretation? That the capital per efficiency unit converges to a stationary steady state ( ˆk (t + 1) = ˆk (t)). But the per capita capital stock (k ) increases. Indeed, the long run increase in per capita income takes place precisely at the rate of technical progress!
Message: Solow Growth Model Solow model with technological progress yields sustained per capita growth of capital and income.
Empirical Evidence: Solow Model The empirical tests of the Solow model center on testing convergence. As you might expect, convergence comes in two forms: 1. Unconditional 2. Conditional (on savings and population growth rates)
Unconditional Convergence This is the strongest prediction (with the fewest assumptions) and the easiest to refute. Suppose that countries, in the long run, have no tendency to display difference in the rates of technical progress savings, population growth, and capital depreciation. The Solow model predicts then in all countries, capital per capita converges to the common value k, and this happens regardless of the initial state of each economy, as measured by their starting levels of per capita income (or equivalently per capita capital stock).
Meaning of Unconditional Convergence If the parameters governing the evolution of the economy are similar, then history in the sense of different initial conditions does not matter. Initial conditions is not some long ago level, but rather k(0) is the level of the per capita capital stock that we can first reasonably measure. In the long run, the starting point of the process does not matter. All possible histories converge at the steady state k. If empirically true this would be huge.
Illustration of Unconditional Convergence Log Per Capita Income A B C Time
Data IMPLEMENTATION ISSUE: Use a smaller set of countries over a long time period OR Use a larger set of countries over a shorter period of time.
Resolution to Data Choice Choice: Do Both. Informative to do both because the set of countries with data available for the longest time period are the current rich countries (OECD). Data available only recently for developing countries. Should we reject unconditional convergence on one set of countries but not the other may be evidence against unconditional convergence.
Evidence: Unconditional Convergence Ray discusses Baumol s study which concluded that unconditional convergence could not be rejected. Yet when the analysis was expanded to include more countries there is substantial evidence against unconditional convergence. Another piece of evidence: the standard deviation (dispersion) of per capita income among Western European countries declined over 1960 1985. But among Asian countries over the same period dispersion increased. Moreover, the divergence dates back to 1900 (so not just a recent phenomenon).
Assessment Find predictions of unconditional convergence soundly rejected by the data. With free trade and the open exchange of ideas there are reasons to believe the rate of technological change should be the same across countries. Yet, not obvious why countries have the same rate of population growth or saving level. These considerations lead to the notion of conditional convergence.
Conditional Convergence Unconditional Convergence: assumes that across all countries, the level of technical knowledge (and its change), rate of savings, rate of population growth, and the rate of depreciation are the same. Countries differ in most if not all factors. Gives rise to the notion of conditional convergence: the growth rate of per capita income will be the same (in the long run). Assume that knowledge flows freely across countries. We allow other parameters such as the rate of population growth and rate of savings to differ across countries.
Level versus Growth Rates Again Thus as growth in per capita income determined by rate of technical progress, there should be convergence in growth rates. But the long run per capita income (level) will vary from country to country (b/c level determined by s and n). Called conditional convergence as we must factor out the effects of parameters that might differ across countries and then examine whether convergence occurs.
Testing Conditional Convergence 1. Assume the production function is Cobb Douglas Y = K α L 1 α 2. Divide by L to obtain per effective labor form: ŷ = ˆk α. 3. Manipulate to express ŷ as a function of s, n, π, δ α. 4. Take (natural) logs to get ln ŷ α 1 α ln s α ln(n + π + δ) (1) 1 α 5. Rewrite in terms of y, recognizing that L(t) = E(t)P(t), with E(t + 1) = (1 + π)e(t)
Testing Conditional Convergence (cont) The expression is: y = Y (t) P(t)E(t)(1 + π) t = y(t) E(0)(1 + π) t In logs: ln(y ) = ln y(t) ln E(0) t ln(1 + π) Substitute in for y in the right hand side of equation (??) to yield: ln y(t) (lne(0) + t ln(1 + π)) + ln y(t) κ 0 + κ 0 = (lne(0) + t ln(1 + π)) α 1 α ln s α ln(n + π + δ) 1 α α 1 α ln s α ln(n + π + δ) 1 α
Regression Equation Have used the theory to define the regression of log per capita income on a constant term κ 0, the log of savings rate and the log of the sum of rate of population growth, rate of depreciation and rate of technical change. Run regression at a point in time across countries i, z i = ln(y i ) = β 0 + β 1 x 1i + β 2 x 2i + ɛ i with x 1i the log of savings for country i, and x 2i is the log sum of n i, π, δ. Estimated Coefficient b 0 recovers an estimate of κ 0 α 1 α Estimated coefficient b 1 recovers an estimate of Estimated coefficient b 2 recovers an estimate of α 1 α
Testable Predictions Written in this way, the testable implications of the theory are obvious. b 1 should be positive, while b 2 should be negative. AND b 1 = b 2. Not only have a prediction on the algebraic sign, but on the magnitudes. α is the share of capital in national income accounts which is roughly 1/3. Thus, we expect b 1 = 1/2 = b 2.
Mankiw, Romer, Weil (1992) They used the Heston Summers data set. Assumed π + δ = 0.05. Used the investment GDP ratios to measure savings rate over 1965 1985. The variable y is per capita GDP in 1985. Results: 1. Regression explains more than half the worldwide variation in per capita GDP in 1985. 2. b 1 > 0 and b 2 < 0, both statistical significant (i.e., not the result of sampling variation). 3. Estimated coefficients b 1 = 1.42 and b 2 = 1.97 not close to the expected 1/2. Nor are the estimated coefficients of similar magnitude (b 1 = b 2 ). Savings effect smaller (in absolute value than population growth rate).
Assessment Many other studies obtain results similar to MRW (1992) Some evidence (at least in terms of direction) in support of Solow growth model. But we can t rest assuming that savings and population should be equal and opposite in magnitude. And find consistently that this assumption is false. We can assume the problem away and say that differences are due to preferences to save or procreate, or perhaps differences due to culture of social differences. Empty. What are the economic incentives and determines for savings rates and population growth rates to have different effects across countries?
Need to augment Solow Model Thus we will enrich model, by questioning and weakening the exogeneity assumptions. On to endogenous growth models. Endogenous because the rate of growth of driving variables (e.g., technical change) are internal to the model (endogenous).
T.W.Schultz and Human Capital T. W. Schultz pioneered the idea of human capital investment in human beings. Interestingly, the importance of human capital (late 1940s) came to him as he realized that models of economic growth didn t explain differences in per capita income (across countries). The view of labor was limited and considered (following A. Marshall) that labor only in terms of quantity. Schultz recognized the diversity of workers. Obvious now, but at that time labor was just a lump, a homogenous factor input.
Human Capital Any form of investment, embodied in people. Schooling Training programs Experience (on the job training) Health Migration an investment to leave a poor labor market and move to a good labor market. Pay fixed cost today for higher wages, earnings tomorrow. Premarket and pre schooling investments by parents (e.g., child care, Head Start). The Rage today.
Modeling Human Capital Will extend the Solow growth model to include human capital. D. Ray makes a number of simplifying assumptions to keep the model tractable. Assumes population growth and depreciation are zero (n = δ = 0). Importantly, there is only skilled labor, measured by the human capital per capita. Common to think of two kinds of labor, skilled and unskilled. To reduce model to only skilled labor highlights the importance of human capital, but comes at a price.
Human and Physical Capital Can think of there being two types of capital, physical and human capital. Human capital is deliberately accumulated, not just the outcome of population growth (which is zero) or exogenously specified technological progress.
Model with Human Capital Retaining notation as before let per capita output (income) be y = k α h 1 α y and k are per capita output and physical capital, h is per capita human capital. As before some of output is consumed and the remainder can be used to create new physical capital sy and human capital qy. So consumption is c = (1 s q)y.
Law of Motion: Physical and Human Capital Physical Capital: k(t + 1) k(t) = sy(t) Human Capital: h(t + 1) h(t) = qy(t) Think of qy(t) as the quantity of physical resources spent on education and training. In long run all variables (y, k, h) growing at common rate. A rate determined by savings rate s and propensity to invest in human capital, q.
Common rate Let r = h(t)/k(t) then k(t + 1) k(t) k(t) = sy k = sk α h1 α k = sr 1 α h(t + 1) h(t) h(t) = qy k = qk α h1 α k = qr α Solve for r yields r = q/s
Closing the Model r makes perfect sense the larger the ratio of savings in human capital is relative to that of physical capital the larger is the long run ratio of h to k. Now use the value of r to compute the long run growth rate. h(t + 1) h(t) h(t) = sr 1 α = s α q 1 α Hence, long run growth rate of per capita income, per capita physical capital and per capita human capital is s α q 1 α.
Implications of Human Capital There are five implications. 1. Physical capital may exhibit diminishing returns yet there may be no convergence in per capita income. 2. Constancy of returns. Now s and q now have growth rate effects, and not just level effects as in the Solow model. 3. Growth rate of (2) related constancy of physical and human capital combined. 4. Introduction of human capital helps to explain why rates of return to physical capital may not be as high in poor countries as the simple Solow model predicts. 5. The model predicts no tendency toward unconditional convergence.
Empirical Predictions of HC Growth Model The model has two predictions 1. Conditional convergence after controlling for human capital. By conditioning on the level of human capital, poor countries have a tendency to grow faster. 2. Conditional divergence after controlling for initial level of per capita income. By conditioning on the level of per capita income, countries with more human capital grow faster.
Empirical Evidence on HC Model Barro (1991) paper in the QJE. (famous) Discussion by Ray concludes that the model with human capital provides a better fit than do the models with exogenous factors. Specifically, does a better job of predicting the growth of some of sub Saharan countries (with very low levels of human capital in the sample period (1960 1985). However the model with human capital still fails to account satisfactorily for the magnitudes of growth displayed by Korea and Taiwan.