Macroeconomic Models of Economic Growth

Transcription

1 Macroeconomic Models of Economic Growth J.R. Walker U.W. Madison Econ448: Human Resources and Economic Growth

2 Course Roadmap: Seemingly Random Topics First midterm a week from today. What have we covered and why this order? Random? 1. International Trade and Comparative Advantage. 2. Global Economic History and Industrial Revolution. 3. Macroeconomic models of Growth: Harrod Domar, Solow, et al.

3 Common thread: Economic Growth Comparative Advantage and International Trade Huge increase in international trade prior and esp during IR. (Ricardo) Comp Adv. Primary determinants of growth. Comparative advantage increased specialization increases economic efficiency. (Smith) Gains from trade source of growth, increased commercial activity raises income, demand for goods, more growth. Discussed Comp Adv and International Trade first to understand why mechanism is a source of economic growth.

4 Global Economic History The IR a watershed in human history as the start of sustained economic growth. Prior to 1750, periods of economic growth episodic. Recovery from famine, disease, pestilence. Global History to understand the uniqueness to the IR. Many spectacular empires and none greater than China, yet IR initiated in Britain and spread to Northwestern Europe.

5 Macroeconomic Models of Growth Industrial Revolution offers insight into determinants of economic growth. Large changes likely to identify significant factors. Growth models: develop systematic framework for analyzing and measuring (quantifying) growth process. 1. Postulate (formal) model; 2. Determine empirical predictions. 3. Investigate empirical predictions. 4. Never prove, can only falsify theory.

6 Toolkit Need broad set of skills to study economic growth and economic development. Primary tools of international trade, economic history, political economy, and now macroeconomics. Will study measurement of inequality, poverty, and investigate their consequences on individuals and the society. Will also study contract theory and the role of information and uncertainty. As you will see externalities and complementarities play a major role.

7 H D Model Assumes: s θ = g + n + δ. 1. Savings rate is constant, S(t) Y (t) = s. 2. Capital output ratio is constant, K (t) Y (t) = θ. Second assumption tantamount to assuming constant returns to scale.

8 H D Model What s the justification for CRS assumption? Reasonable to believe there is diminishing returns to scale. For fixed labor force, increase capital stock expect output to increase but (eventually) by less smaller and smaller amounts.

9 Solow: Endogenize Capital Output Ratio Retain the first assumption, constancy of the savings rate S(t) = sy (t) = I(t). Equation for the capital stock: K (t + 1) = (1 δ)k (t) + sy (t) Assume Labor = Population: L(t) = P(t), and that population grows at rate n each period. P(t + 1) = (1 + n)p(t)

10 Production Function The equation for the capital stock includes the term s Y Production function Y = F (K, L). Y = F (K, L) Y F (K, L) = L ( L Y K L = F L, L ) L ( ) K y = F L, 1 y = f (k)

11 Solow Model K (t + 1) = (1 δ)k (t) + sy (t), K (t + 1) = (1 δ) K (t) P(t) P(t) + s Y (t) P(t) K (t + 1) = (1 δ) k(t) + s y(t) P(t) ( ) 1 + n K (t + 1) = (1 δ)k(t) + sy(t) 1 + n P(t) (1 + n)k (t + 1) = (1 δ)k(t) + sy(t) P(t + 1)

12 Solow Model (1 + n)k(t + 1) = (1 δ)k(t) + sy(t) (3.9) (1 + n)k(t + 1) = (1 δ)k(t) + s f (k(t)) STEADY STATE k(t + 1) = k(t) = k (1 + n)k = (1 δ)k + sy [(1 + n) (1 δ)]k = sy (n + δ)k = sy k y = s n + δ

13 Interpretation Right hand side (RHS) two parts depreciated per capita capital and current per capita savings. Together they give us (almost) the new per capita stock. True if n = 0, but population growth puts downward drag on per capita capital stock. Hence, adjust for population growth by term 1 + n on LHS. Note: the larger the rate of population growth, the lower is the per capita capital stock the next period.

14 Figure 3.3 Production function y=f(k) Output per Capital Output--Capital Ratios 0 0 Capital per Capita (k)

15 Figure 3.4 (1+n)k (1- )k+sy k k(0) k* k

16 Steady State 1. if k(0) < k 2. if k(0) > k

17 Long Run Growth in Solow Model? In Solow model the savings rate has no long run effect on the rate of growth. (contrary to H D). What s the resolution between H D and Solow?

18 Level Effects versus Growth Effects Savings rate does not affect long run growth rate of per capita income, but affects the long run level of income. The steady state (k(t + 1) = k(t) = k ) and manipulating eqn 3.9 to yield k y = s n + δ Increase δ this lowers the RHS. But this means that the capital output ratio on the LHS must decline this means that k and y decline. What s the economic interpretation?

19 Population Growth Higher population growth, lowers the steady state level of per capita income. But the total income must grow faster as a result. Economy converges to a SS level of per capita income, which is impossible unless long run growth of total income equals the rate of population growth. Labor is both an input in production and a consumer of final goods. First raise total output and drives higher rate of growth of total income; second lowers savings and investment and brings down the SS level of per capita income.

20 Savings affects level Savings affects steady per capita income k y = s n + δ y = f (k) y = f (k ) f > 0, f < 0 k > y

21 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.

22 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.

23 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.

24 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.

25 Steady State Figure 3.6

26 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!

27 Message: Solow Growth Model Solow model with technological progress yields sustained per capita growth of capital and income.

28 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)

29 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).

30 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.

31 Illustration of Unconditional Convergence Log Per Capita Income A B C Time

32 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.

33 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.

34 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 But among Asian countries over the same period dispersion increased. Moreover, the divergence dates back to 1900 (so not just a recent phenomenon).

35 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.

36 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.

37 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.

38 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)

39 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 (1) [prev] 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

40 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

41 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.

42 Mankiw, Romer, Weil (1992) They used the Heston Summers data set. Assumed π + δ = Used the investment GDP ratios to measure savings rate over The variable y is per capita GDP in Results: 1. Regression explains more than half the worldwide variation in per capita GDP in 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).

43 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?