Economic Growth and Development - Economic Growth - Lecture Slides, Slides of Economic Growth and Globalization

Download Economic Growth and Development - Economic Growth - Lecture Slides and more Slides Economic Growth and Globalization in PDF only on Docsity! 14.452Economic Growth: Lecture 1, Stylized Facts of Economic Growth and Development and Introduction to the Solow Model Docsity.com Growth and Development: The Questions From Correlates to Fundamental Causes From Correlates to Fundamental Causes Correlates of economic growth, such as physical capital, human capital and technology, will be our first topic of study. But these are only proximate causes of economic growth and economic success: why do certain societies fail to improve their technologies, invest more in physical capital, and accumulate more human capital? Return to Figure above to illustrate this point further: how did South Korea and Singapore manage to grow, while Nigeria failed to take advantage of the growth opportunities? If physical capital accumulation is so important, why did Nigeria not invest more in physical capital? If education is so important, why our education levels in Nigeria still so low and why is existing human capital not being used more effectively? The answer to these questions is related to the fundamental causes of economic growth. Docsity.com Solow Growth Model The Economic Environment of the Basic Solow Model The Economic Environment of the Basic Solow Model Study of economic growth and development therefore necessitates dynamic models. Despite its simplicity, the Solow growth model is a dynamic general equilibrium model (though many key features of dynamic general equilibrium models, such as preferences and dynamic optimization are missing in this model). Economic Growth Lecture 1 October 26, 2009. 18 / 55 Docsity.com Solow Growth Model Households and Production Households and Production I Closed economy, with a unique final good. Discrete time running to an infinite horizon, time is indexed by t = 0, 1, 2, .... Economy is inhabited by a large number of households, and for now households will not be optimizing. This is the main difference between the Solow model and the neoclassical growth model. To fix ideas, assume all households are identical, so the economy admits a representative household. Docsity.com Solow Growth Model Households and Production Households and Production II Assume households save a constant exogenous fraction s of their disposable income Same assumption used in basic Keynesian models and in the Harrod-Domar model; at odds with reality. Assume all firms have access to the same production function: economy admits a representative firm, with a representative (or aggregate) production function. Aggregate production function for the unique final good is Y (t) = F [K (t) , L (t) , A (t)] (1) Assume capital is the same as the final good of the economy, but used in the production process of more goods. A (t) is a shifter of the production function (1). Broad notion of technology. Major assumption: technology is free; it is publicly available as a non-excludable, non-rival good. Docsity.com Solow Growth Model Market Structure, Endowments and Market Clearing Market Structure, Endowments and Market Clearing I We will assume that markets are competitive, so ours will be a prototypical competitive general equilibrium model. Households own all of the labor, which they supply inelastically. Endowment of labor in the economy, L̄ (t), and all of this will be supplied regardless of the price. The labor market clearing condition can then be expressed as: L (t) = L̄ (t) (2) for all t, where L (t) denotes the demand for labor (and also the level of employment). More generally, should be written in complementary slackness form. In particular, let the wage rate at time t be w (t), then the labor market clearing condition takes the form L (t) ≤ L̄ (t) , w (t) ≥ 0 and (L (t) − L̄ (t)) w (t) = 0 Docsity.com Solow Growth Model Market Structure, Endowments and Market Clearing Market Structure, Endowments and Market Clearing II But Assumption 1 and competitive labor markets make sure that wages have to be strictly positive. Households also own the capital stock of the economy and rent it to firms. Denote the rental price of capital at time t be R (t). Capital market clearing condition: Ks (t) = Kd (t) Take households’initial holdings of capital, K (0), as given P (t) is the price of the final good at time t, normalize the price of the final good to 1 in all periods. Build on an insight by Kenneth Arrow (Arrow, 1964) that it is suffi cient to price securities (assets) that transfer one unit of consumption from one date (or state of the world) to another. Economic Growth Lecture 1 October 26, 2009. 24 / 55 Docsity.com Solow Growth Model Market Structure, Endowments and Market Clearing Market Structure, Endowments and Market Clearing III Implies that we need to keep track of an interest rate across periods, r (t), and this will enable us to normalize the price of the final good to 1 in every period. General equilibrium economies, where different commodities correspond to the same good at different dates. The same good at different dates (or in different states or localities) is a different commodity. Therefore, there will be an infinite number of commodities. Assume capital depreciates, with “exponential form,” at the rate δ: out of 1 unit of capital this period, only 1 − δ is left for next period. Loss of part of the capital stock affects the interest rate (rate of return to savings) faced by the household. Interest rate faced by the household will be r (t) = R (t) − δ. Docsity.com Solow Growth Model Firm Optimization Firm Optimization III Proposition Suppose Assumption 1 holds. Then in the equilibrium of the Solow growth model, firms make no profits, and in particular, Y (t) = w (t) L (t) + R (t) K (t) . Proof: Follows immediately from Euler Theorem for the case of m = 1, i.e., constant returns to scale. Thus firms make no profits, so ownership of firms does not need to be specified. Docsity.com Solow Growth Model Firm Optimization Second Key Assumption Assumption 2 (Inada conditions) F satisfies the Inada conditions lim FK ( ) = ∞ and lim FK ( ) = 0 for all L > 0 all A K 0 · K ∞ · → → L lim 0 FL (·) = ∞ and L lim ∞ FL (·) = 0 for all K > 0 all A. → → Important in ensuring the existence of interior equilibria. It can be relaxed quite a bit, though useful to get us started. Docsity.com Solow Growth Model Firm Optimization Production Functions F(K, L, A) K 0 K 0 Panel A Panel B F(K, L, A) Courtesy of Princeton University Press. Used with permission. Figure 2.1 in Acemoglu, Daron. Introduction to Modern Economic Growth. Princeton, NJ: Princeton University Press, 2009. ISBN: 9780691132921. Figure: Production functions and the marginal product of capital. The example in Panel A satisfies the Inada conditions in Assumption 2, while the example in Panel B does not. Docsity.com The Solow Model in Discrete Time Fundamental Law of Motion of the Solow Model Fundamental Law of Motion of the Solow Model III Setting supply and demand equal to each other, this implies Ks (t) = K (t). From (2), we have L (t) = L̄ (t). Combining these market clearing conditions with (1) and (6), we obtain the fundamental law of motion the Solow growth model: K (t + 1) = sF [K (t) , L (t) , A (t)] + (1 − δ) K (t) . (10) Nonlinear difference equation. Equilibrium of the Solow growth model is described by this equation together with laws of motion for L (t) (or L̄ (t)) and A (t). Docsity.com The Solow Model in Discrete Time Definition of Equilibrium Definition of Equilibrium I Solow model is a mixture of an old-style Keynesian model and a modern dynamic macroeconomic model. Households do not optimize, but firms still maximize and factor markets clear. Definition In the basic Solow model for a given sequence of {L (t) , A (t)} ∞ =0 and an initial capital stock K (0), an t equilibrium path is a sequence of capital stocks, output levels, consumption levels, wages and rental rates {K (t) , Y (t) , C (t) , w (t) , R (t)} ∞ =0 such that K (t)t satisfies (10), Y (t) is given by (1), C (t) is given by (9), and w (t) and R (t) are given by (4) and (5). Note an equilibrium is defined as an entire path of allocations and prices: not a static object. Docsity.com 1 2 The Solow Model in Discrete Time Equilibrium Equilibrium Without Population Growth and Technological Progress I Make some further assumptions, which will be relaxed later: There is no population growth; total population is constant at some level L > 0. Since individuals supply labor inelastically, L (t) = L. No technological progress, so that A (t) = A. Define the capital-labor ratio of the economy as k (t) ≡ K (t) , (11) L Using the constant returns to scale assumption, we can express output (income) per capita, y (t) ≡ Y (t) /L, as y (t) = � � K (t) F , 1, A L ≡ f (k (t)) . (12) 35 / 55 Docsity.com The Solow Model in Discrete Time Equilibrium Example:The Cobb-Douglas Production Function II Alternatively, in terms of the original production function (14), R (t) = αAK (t)α−1 L (t)1−α = αAk (t)−(1−α) , Similarly, from (13), w (t) = Ak (t)α − αAk (t)−(1−α) × k (t) = (1 − α) AK (t)α L (t)−α , Docsity.com The Solow Model in Discrete Time Equilibrium Equilibrium Without Population Growth and Technological Progress I The per capita representation of the aggregate production function enables us to divide both sides of (10) by L to obtain: k (t + 1) = sf (k (t)) + (1 − δ) k (t) . (15) Since it is derived from (10), it also can be referred to as the equilibrium difference equation of the Solow model The other equilibrium quantities can be obtained from the capital-labor ratio k (t). Definition A steady-state equilibrium without technological progress and population growth is an equilibrium path in which k (t) = k∗ for all t. The economy will tend to this steady state equilibrium over time (but never reach it in finite time). Docsity.com k(t+1) k(t) 45° sf(k(t))+(1–δ)k(t) k* k*0 The Solow Model in Discrete Time Equilibrium Courtesy of Princeton University Press. Used with permission. Figure 2.2 in Acemoglu, Daron. Introduction to Modern Economic Growth. Princeton, NJ: Princeton University Press, 2009. ISBN: 9780691132921. Figure: Determination of the steady-state capital-labor ratio in the Solow model without population growth and technological change. Docsity.com 1 2 The Solow Model in Discrete Time Equilibrium Equilibrium Without Population Growth and Technological Progress IV Alternative visual representation of the steady state: intersection between δk and the function sf (k). Useful because: Depicts the levels of consumption and investment in a single figure. Emphasizes the steady-state equilibrium sets investment, sf (k), equal to the amount of capital that needs to be “replenished”, δk. Docsity.com The Solow Model in Discrete Time Equilibrium output k(t) f(k*) k* δk(t) f(k(t)) sf(k*) sf(k(t)) consumption investment 0 Courtesy of Princeton University Press. Used with permission. Figure 2.4 in Acemoglu, Daron. Introduction to Modern Economic Growth. Princeton, NJ: Princeton University Press, 2009. ISBN: 9780691132921. Figure: Investment and consumption in the steady-state equilibrium. Docsity.com The Solow Model in Discrete Time Equilibrium Equilibrium Without Population Growth and Technological Progress V Proposition Consider the basic Solow growth model and suppose that Assumptions 1 and 2 hold. Then there exists a unique steady state equilibrium where the capital-labor ratio k∗ ∈ (0, ∞) is given by (16), per capita output is given by y ∗ = f (k∗) (17) and per capita consumption is given by c∗ = (1 − s) f (k∗) . (18) Docsity.com The Solow Model in Discrete Time Equilibrium Equilibrium Without Population Growth and Technological Progress VI Figure shows through a series of examples why Assumptions 1 and 2 cannot be dispensed with for the existence and uniqueness results. Generalize the production function in one simple way, and assume that f (k) = af̃ (k) , a > 0, so that a is a (“Hicks-neutral”) shift parameter, with greater values corresponding to greater productivity of factors.. Since f (k) satisfies the regularity conditions imposed above, so does f̃ (k). Docsity.com The Solow Model in Discrete Time Equilibrium Equilibrium Without Population Growth and Technological Progress VII Proposition Suppose Assumptions 1 and 2 hold and f (k) = af̃ (k). Denote the steady-state level of the capital-labor ratio by k∗ (a, s, δ) and the steady-state level of output by y ∗ (a, s, δ) when the underlying parameters are a, s and δ. Then we have ∂k∗ ( ) ∂k∗ ( ) ∂k∗ ( )· > 0, · > 0 and · < 0 ∂a ∂s ∂δ ∂y ∗ ( ) ∂y ∗ ( ) ∂y ∗ ( )· > 0, · > 0 and · < 0. ∂a ∂s ∂δ Docsity.com The Solow Model in Discrete Time Equilibrium Equilibrium Without Population Growth and Technological Progress VIII Proof of comparative static results: follows immediately by writing f̃ (k∗) δ k∗ = as , which holds for an open set of values of k∗. Now apply the implicit function theorem to obtain the results. For example, ∂k∗ δ (k∗)2 = > 0 ∂s s2w ∗ where w ∗ = f (k∗) − k∗f � (k∗) > 0. The other results follow similarly. Docsity.com The Solow Model in Discrete Time Equilibrium consumption savings rate (1–s)f(k*gold) s*gold 10 Courtesy of Princeton University Press. Used with permission. Figure 2.6 in Acemoglu, Daron. Introduction to Modern Economic Growth. Princeton, NJ: Princeton University Press, 2009. ISBN: 9780691132921. Figure: The “golden rule” level of savings rate, which maximizes steady-state consumption. Docsity.com The Solow Model in Discrete Time Equilibrium Proof of Proposition: Golden Rule By definition ∂c∗ (sgold ) /∂s = 0. From Proposition above, ∂k∗/∂s > 0, thus (20) can be equal to zero only when f � (k∗ (sgold )) = δ. Moreover, when f � (k∗ (sgold )) = δ, it can be verified that ∂2c∗ (sgold ) /∂s2 < 0, so f � (k∗ (sgold )) = δ indeed corresponds a local maximum. That f � (k∗ (sgold )) = δ also yields the global maximum is a consequence of the following observations: ∀ s ∈ [0, 1] we have ∂k∗/∂s > 0 and moreover, when s < sgold , f � (k∗ (s)) − δ > 0 by the concavity of f , so ∂c∗ (s) /∂s > 0 for all s < sgold . by the converse argument, ∂c∗ (s) /∂s < 0 for all s > sgold . Therefore, only sgold satisfies f � (k∗ (s)) = δ and gives the unique global maximum of consumption per capita. Docsity.com The Solow Model in Discrete Time Equilibrium Equilibrium Without Population Growth and Technological Progress XI When the economy is below k∗ gold , higher saving will increase consumption; when it is above k∗ gold , steady-state consumption can be increased by saving less. In the latter case, capital-labor ratio is too high so that individuals are investing too much and not consuming enough (dynamic ineffi ciency). But no utility function, so statements about “ineffi ciency” have to be considered with caution. Such dynamic ineffi ciency will not arise once we endogenize consumption-saving decisions. Docsity.com