ECON6002 Macroeconomics Analysis 1 Week 1: “The Solow-Swan Model” Christopher Gibbs University of Sydney Semester 1, 2020 Class Outline 1 Course overview 2 Mathematics review 3 5-minute break 4 Solow-Swan model 5 Solving the model 6 5-minute break 7 Using the model 8 Summary Readings: Unit of Study Outline; Romer, Chapter 1 Course Overview • This course focuses on the main macroeconomic models used to understand theoretical mechanisms behind long-run growth, business cycles, monetary policy, fiscal policy, and consumption • We will learn how to solve and use these models, as well as apply them to policy-relevant issues • Textbook: David Romer, Advanced Macroeconomics, 4th or 5th Edition, McGraw Hill Topics 1 Economic Growth (Weeks 1-2) 2 Real Business Cycles (Weeks 5 and 6) 3 Money and the Business Cycle (week 8 - 11) 4 Fiscal Policy and Consumption (Weeks 12 - 13... we may not get all the way here) In-class midterm exam in Week 7 (8 April) Other Course Details • Tutorial questions will be posted on Canvas with online discussion board and video solutions by the online tutor (Tristan Truuvert) - Tutor will also respond to questions in ED - Please use ED. You are more than welcome to answer each others questions! • Three problem sets will also be posted on Canvas and may be turned in online until week 7. After week 7, all assignments are due in class. • Consultation hours: Wednesday 2-4pm, Social Science Building, Room 563 • Email:
[email protected] - Use ED to send me general questions. You are likely not the only person with the question you are asking. Create public goods! Mathematics Review Math(s) • This course is centered around building “simple” mathematical models to describe economic phenomena • You will be expected to be able to manipulate the models to analyse economic policies • We will walk through the basics of the mathematical tools we will need to solve each model • I only expect you to understand the concepts well enough that you can do the economics • If you want an excellent reference for the math that we will use in the course I recommend: - Fundamental Methods of Mathematical Economics, by Alpha C. Chiang and Kevin Wainwright Math(s) • What math do we need to study growth? That depends on the questions... Math(s)... and aside on the growth Figure: Source: Maddison (2008) Math(s) • What math do we need to study growth? That depends on the questions... • For questions about growth rates, we need to study differential equations First-order linear differential equations dy dt + u(t)y = w(t) (1) • The number of derivatives of a function y determines the order - It may enter though in powers, e.g. (dy/dt)2 - The highest power obtained in the equation is known as the degree • No product of y and dy/dt occurs so the above is linear • u(t) and w(t) are functions of t and may take on any form First-order linear differential equations: homogeneous case dy dt + ay = 0 (2) • If u and w are constants, w = 0, and a is some constant, we say that the dif. eq. is homogeneous • The defining characteristic of a homogeneous dif. eq. is that if all terms are multiplied by a constant, the equation remains homogeneous First-order linear differential equations: homogeneous case • Key questions here are: - Why should I care? - If I do care, how do I solve such an equation? First-order linear differential equations: homogeneous case • Q: Has economic growth been constant over the last 1000 years or has it been accelerating? • A: ? • Well, let’s build a simple model of constant technological growth and find out First-order linear differential equations: homogeneous case • Q: Has economic growth been constant over the last 1000 years or has it been accelerating? • A: ? • Well, let’s build a simple model of constant technological growth and find out First-order linear differential equations: homogeneous case • Modeling constant technological growth - Let A(t) be the stock of technology at time t - Let g be the growth rate of technology dA dt = gA(t) (3) - rearranged, we get dA dt − gA(t) = 0 (4) First-order linear differential equations: homogeneous case dA dt − gA = 0 (5) • A question of interest in this case is how does technology evolve over time? • To answer, we need to solve the above homogeneous linear dif. eq. First-order linear differential equations: homogeneous case • To solve, we need to recover a function whose derivative is (6) dA dt − gA = 0 (6) • In general, this often requires a person to just be clever and have a good recall of common derivatives and integrals • Luckily, there is a common one that solves pretty much any problem you will encounter in economics - The natural logarithm: lnX First-order linear differential equations: homogeneous case • Let’s differentiate lnX with respect to time. To do so, we need to apply the chain rule... 1 We take the derivative of lnX w.r.t. X evaluated at X dlnX dX = 1 X 2 We then take the derivative of X with respect to time dX dt = dX dt 3 Finally, we multiply the two together dlnX dt = 1 X dX dt First-order linear differential equations: homogeneous case • Now, the following math magic where I replace X with A dA dt − gA = 0 dA dt = gA 1 A dA dt = g dlnA dt = g First-order linear differential equations: homogeneous case • Now, the following math magic where I replace X with A dA dt − gA = 0 dA dt = gA 1 A dA dt = g dlnA dt = g First-order linear differential equations: homogeneous case • Integrate with respect to t ∫ dlnA dt dt = ∫ gdt lnA(t) = c+ gt - c is an unknown constant • This already says something interesting: growth is log linear! First-order linear differential equations: homogeneous case First-order linear differential equations: homogeneous case • Now, exponentiate both sides elnA(t) = ec+gt A(t) = ecegt A(t) = Cegt (7) - (7) is called the general solution. It is general because we have the arbitrary constant C, which is still unknown. - When we substitute in value for C, (7) becomes a particular solution. First-order linear differential equations: homogeneous case A(t) = Cegt • To find C, we need an additional piece of information. • In economics, this piece of information is usually provided by economic theory and the modeling context. • For example, the obvious C to choose here is A(0) or the initial level of technology assumed for the economy. A(t) = A(0)egt First-order linear differential equations: homogeneous case A(t) = A(0)egt • Therefore, we have assumed exponential growth... • How does this assumption match up with the data? Math(s)... and aside on the growth First-order linear differential equations: homogeneous case Figure: Source: Maddison (2008) First-order linear differential equations: nonhomogeneous case dy dt + ay = b (8) • The nonhomogeneous case follows straight from the homogeneous case • The solution here is the sum of two terms: the complementary function and the particular integral • The solution in fact is just a particular solution of the general solution to the homogeneous case First-order linear differential equations: nonhomogeneous case • The general solution or complementary function: yc = Ce −at (9) • The particular solution or integral: Suppose dy/dt = 0 ay = b or yp = b a First-order linear differential equations: nonhomogeneous case y(t) = yc + yp = Ce −at + b a (10) • Finally, using the initial condition y(0), we have y(0) = Ce−a×0 + b a = C + b a⇒ C = y(0)− b a y(t) = [ y(0)− b a ] e−at + b a (11) Nonlinear differential equations: first order and first degree dy dt = h(y, t) (12) • This is first order because we have only a first derivative • It is first degree because we do not consider powers of dy/dt • But there is no restriction on the power of y. Nonlinear differential equations: first order and first degree • Let’s consider a special case here that is particularly relevant for economics dy dt = h(y, t) dy dt = Ty(t)m −Ry(t) - Where T and R are constants and m 6= 0 or 1 Nonlinear differential equations: first order and first degree • In this case, we can use the time honoured tradition of a change in variables to reduce the system to something we already know how to solve y−m dy dt +Ry1−m = T • Let z = y1−m, then dz dt = dz dy dy dt = (1−m)y−mdy dt • and math magic 1 1−m dz dt +Rz = T (13) Nonlinear differential equations: first order and first degree 1 1−m dz dt +Rz = T • This my friends, we know how to solve... just rearrange dz dt + (1−m)Rz = (1−m)T (14) • It is just a nonhomogeneous first-order linear differential equations, which we saw just a few slides ago and which is straight forward to solve! A note on notation • A common way to write time derivatives is dx dt = x˙ - To save time and space, I (and Romer) use the ‘dot’ to denote the time derivative of a variable • Also, it is common to drop the t in state variables. For example, A(t) = A The Solow-Swan Model Model Structure • Technology and population growth are exogenous • Capital accumulation and output are endogenous • One good, closed economy, no government spending, no unemployment, constant saving rate • Perfectly competitive markets Key Assumption #1: An Aggregate Production Function • Y (t) = F (K(t), A(t)L(t)) • Y=output, K=capital, L=labour, and A=effectiveness of labour • Multiplicative interaction of A and L corresponds to “labour-augmenting” technological change, no trend in K/Y ratio • Constant returns to scale: F (cK, cAL) = cF (K,AL) • Implies no relevant fixed inputs (e.g., land) or further gains from specialization • Replication principle (same setup on Mars) • Also, considerable empirical evidence that a c.r.s. aggregate production function is a useful predictive construct Intensive Form • y = f(k) • y=output per unit of effective labour (Y/AL) and k=capital per unit of effective labour (K/AL) • Follows from c.r.s. with c = 1AL ⇒ F ( KAL , 1) = 1ALF (K,AL) • Model can be more easily solved using intensive form given an implied “steady state” in k and y Mathematical Properties and an Example • f(0) = 0, f ′(k) > 0, and f ′′(k) < 0 • Marginal product of capital is positive, but subject to diminishing returns • Additional “Inada” conditions: limk→0f ′(k) =∞ and limk→∞f ′(k) = 0 • Cobb-Douglas example: y = kα, where 0 < α < 1 • f(0) = 0, f ′(k) = αkα−1 > 0, and f ′′(k) = (α− 1)αkα−2 < 0 • α ends up being equivalent to the capital share of income in the model, while the empirical share historically in most countries has been around one-third (0.33) Cobb-Douglas Production Function Key Assumption #2: Exogenous Processes • Solow-Swan model assumes constant growth rates for population and technology • L˙(t) = nL(t) • A˙(t) = gA(t) • n=population growth rate and g=technology growth rate Key Assumption #3: Capital Accumulation Identity • Households save a fraction of income to invest in new capital, but existing capital depreciates by a fixed rate • K˙(t) = sY (t)− δK(t) • s=saving rate and δ=depreciation rate • 0 < s < 1 and n+ g + δ > 0 • In Chapter 2, we will make savings endogenous, but many results remain robust Model Solution • Step 1: Solve for behaviour of capital and output using production function, exogenous processes, capital accumulation identity - Differentiate k = KAL with respect to time (chain-rule + quotient-rule calculus) k˙ = K˙ AL − K [AL]2 [AL˙+ LA˙] = K˙ AL − K AL L˙ L − K AL A˙ A Model Solution • Step 2: Substitute the model assumptions to simply the following equation: k˙ = K˙ AL − K AL L˙ L − K AL A˙ A - K˙ = sY − δK - YAL = y = f(k) - L˙L = n - A˙A = g k˙ = sf(k)− (n+ g + δ)k Model Solution • Step 2: Substitute the model assumptions to simply the following equation: k˙ = K˙ AL − K AL L˙ L − K AL A˙ A - K˙ = sY − δK - YAL = y = f(k) - L˙L = n - A˙A = g k˙ = sf(k)− (n+ g + δ)k The Solow-Swan Model in a Single Picture Solving for Steady State and Balanced Growth Path • Given k > 0, unique steady state k∗ for capital per unit of effective labour • k˙ > 0 if k < k∗ and k˙ < 0 if k > k∗ (also see phase diagram) • Given initial endowment of capital k(0) 6= k∗, implies transition dynamics in which output growth depends on capital accumulation • But also implies “balanced growth path” in steady state in which all inputs (capital, labour, and technology) and output grow at constant rates • K and Y grow at rate n+ g while K/L and Y/L grow at rate g (i.e., output per capita grows at the rate of change in technology) • Capital-output (K/Y ) ratio remains constant at level that depends on k∗ Uniqueness of Steady State Policy Question • What if the government encouraged people to save at a higher rate: sNEW > sOLD? • Increases the steady-state k and y • But consumption would initially fall in the transition and not clear if it would increase in steady state • “Golden Rule” saving rate is the one that maximizes consumption in steady state • If Golden Rule capital stock is above current capital stock, we would have to sacrifice consumption today for higher consumption for future generations Policy Question Answers in the Model • We answer these questions by looking at the endogenous response of the model to an exogenous change in a parameter. • We do this in two ways: 1. Graphically, using the Solow-Swan Diagram 2. Using math to derive algebraic relationships and calculus to look at the effects of small changes of parameters Graphically: Increase in Saving Rate Graphically: Transition dynamics Math: Impact of Saving Rate on Consumption in Steady State • c∗ = f(k∗)− (n+ g + δ)k∗ • Because steady-state capital k∗ depends on saving rate s, we can implicitly solve ∂c∗ ∂s = [f ′(k∗)− (n+ g + δ)]∂k ∗ ∂s • At the “Golden Rule” max, ∂c∗∂s = 0 • I.e., sGR is such that f ′(k∗GR) = n+ g + δ • Slope of the (intensive) production function is the same as the (constant) slope of the break-even investment function Goldilocks and the Three Bears Speed of Convergence? • To think about speed of convergence, solve for the adjustment of capital near steady state • First-order (linear) Taylor-series approx of k˙ = sf(k)− (n+ g + δ)k at steady state k = k∗: k˙ ≈ [ ∂k˙ ∂k |k=k∗ ] (k − k∗) Linear approximations • Taylor series approximations: we can approximate any real function f(x) about a point x = a using the following f(x) = f(a) + f ′(a)(x− a) + f ′′(a) 2! (x− a)2 + f ′′′(a) 3! (x− a)3 +....+ f (n)(a) n! (x− a)n + .... • If want a linear approximation, we just need f(x) = f(a) + f ′(a)(x− a) Speed of Convergence? • To think about speed of convergence, solve for the adjustment of capital near steady state • Adjustment coefficient: ∂k˙ ∂k |k=k∗ = sf ′(k∗)− (n+ g + δ) = (n+ g + δ)k∗f ′(k∗) f(k∗) − (n+ g + δ) = (α(k∗)− 1)(n+ g + δ) • where α(k∗) is the capital share of income, which is about one-third in the data • Thus, pop growth=1%, tech growth=2%, and depreciation=3% ⇒ 4% convergence per year Grains of Truth and Grains of Salt • Capital accumulation does seem to be subject to diminishing returns (e.g., Soviet Union) • Some support for convergence, but seems to be within groups of countries with similar characteristics (“conditional convergence”) • Growth accounting suggests differences in physical capital explain about one-third of the variation in output per capita across countries • But the remaining two-thirds is attributed to the nebulous “Solow residual” that may or may not capture exogenous technology growth • Solow-Swan model implies lower pop growth would lead to higher k∗ and faster growth in the transition - No empirical support for this (endogenous growth models typically have opposite prediction) Evidence of Convergence using Ex Post Selection of Countries Evidence of Convergence using Ex Post Selection of Countries • What is wrong with this analysis? Selection - Countries that have data to use for such an exercise are usually rich today - Poor countries in 1870 that are still poor today don’t have reliable data and are excluded from the sample - Measurement error is also a problem Evidence of Convergence using Ex Post Selection of Countries • What is wrong with this analysis? Selection - Countries that have data to use for such an exercise are usually rich today - Poor countries in 1870 that are still poor today don’t have reliable data and are excluded from the sample - Measurement error is also a problem Evidence of Convergence using Ex Ante Selection of Countries Evidence of Convergence Weighting by Size of Country Does(Income(per(capita(“converge”?( CHAPTER(1(((IntroducKon(to( Macroeconomics( - 14 - Bourguignon and Morrisson (2002) conclude that the inequality of global income worsened from the start of the nineteenth century until the end of World War II “and after that seems to have stabilized or to have grown more slowly”.40 Where does that leave us on both poverty and the global distribution of income? Poverty rates have been declining, especially in Asia. So very likely has the absolute number of those living at below one dollar a day. Increasingly, global poverty is being concentrated in Africa. At the same time, poverty rates have not declined much in Latin America in recent decades. Taking account of other social indicators, as reflected in the Human Development Index, presents a more encouraging picture of the changing fortunes of the poorest, but the HIV- AIDS epidemic is taking a sad toll on longevity in Africa. Income distribution developments are more mixed. There has been a growing divergence among national average incomes. Inequality has risen within many countries. But it is likely that inequality among the world’s citizens declined during the last decades of the twentieth century. However we should not take too much comfort from that, for as Sala-i-Martin (2002a) points out, “Unless Africa starts growing in the near future, … income inequalities will start rising again.” III. The Policy Issues Trade and growth. Trade policy has long been central to economic policy choices. In the early post-World War II period, the theory of import-substituting industrialization (ISI) dominated among developing countries, and its implementation for some time seemed to produce positive results. Then as time went by, it was observed both that countries that has pursued export promotion strategies were more successful than those that focused on keeping imports out, and that the returns to ISI seemed to be diminishing. 40 However, Milanovic (1999) finds that global income inequality increased between 1988 and 1993, in part because of growing gaps between rural and urban incomes in China. By contrast, Bhalla (2002) shows world inequality in 2002 at its lowest level in the post-World War II period, a result of the much greater reductions in global poverty in the 1990s that his methodology produces. Similarly, Sala -i-Martin (2002a) finds massive decreases in global inequality at the individual level between 1980 and 1998. Figure 5. Real GDP per capita*, 1980-2000, average annual growth on initial level (area proportional to population in 1980) -4 -2 0 2 4 6 8 -5,000 0 5,000 10,000 15,000 20,000 25,000 30,000 Real GDP per capita*,1980 Ra te of Gr ow th of Re al GD P p er ca pit a* , 19 80 -2 00 0 ( pe rce nt pe r y ea r) Sub-Saharan AfricaChina U.S.A.India *Real GDP in U.S.$ per equivalent adult. Source: Penn World Tables, version 6.1 Summary • Solow-Swan model solves for endogenous capital accumulation as a function of initial endowment of capital, saving rate, pop growth, tech growth, depreciation • Model predicts transition dynamics and steady state • Can be used to consider policy questions and make quantitative predictions about things like speed of convergence • Model produces profound insights about long-run growth, but empirical support is mixed • Next time: Microfoundations for growth models (Romer, Chapter 2)
