ECON 581. The Solow Growth Model
Instructor: Dmytro Hryshko
1 / 59 General info
e-mail: [email protected] office: HM Tory Building, 9-19 office hours: Friday 2–3 PM, or by appointment Teaching material available online web-page: http://www.artsrn.ualberta. ca/econweb/hryshko/
2 / 59 General info
Recommended text: Daron Acemoglu (2009). Introduction to Modern Economic Growth. Princeton University Press. But there’re many other. Grading: an in-class midterm (35%) and final (45%) exams, and two problem sets (20%).
3 / 59 Scope of the course
Growth (Solow model and models of endogenous growth) Consumption and savings Asset pricing Policy issues etc.
4 / 59 Outline for today
Measures for standard of living The importance of economic growth Main long-run growth facts and their implications A basic model of long-run growth (the Solow model) Readings: Acemoglu, Chapters 1–3.
5 / 59 What is Economic Growth?
Standard economists’ answer: Growth in GDP per capita (or per worker), Y/L Does growth in per capita GDP reflect anything important about the improvement of living standards? –No consideration, for example, of the distribution of income within the economy (inequality)
6 / 59 Key Questions in Economic Growth
Why are there countries so rich and others so poor? Why do growth rates vary across countries and over time? What are the policies that can change growth in the short and long run? Why do some countries “take off” while others fall behind
7 / 59 Measures of Living Standards
Normally, the focus on two statistics of the average person’s well-being: income and GDP per worker (productivity measure), and income and GDP per capita (welfare measure).
They correlate with many other important statistics measuring well-being: infant mortality, life expectancy, consumption, etc.
8 / 59 1.2 Income and Welfare . 7
1.2 Income and Welfare
Should we care about cross-country income differences? The answer is definitely yes. High income levels reflect high standards of living. Economic growth sometimes increases pollution or may raise individual aspirations, so that the same bundle of consumption may no longer satisfy an individual. But at the end of the day, when one compares an advanced, rich country with a less-developed one, there are striking differences in the quality of life, standards of living, and health. Figures 1.5 and 1.6 give a glimpse of these differences and depict the relationship between income per capita in 2000 and consumption per capita and life expectancy at birth in the same year. Consumption data also come from the Penn World tables, while data on life expectancy at birth are available from the World Bank Development Indicators. These figures document that income per capita differences are strongly associated with differences in consumption and in health as measured by life expectancy. Recall also that these numbers refer to PPP-adjusted quantities; thus differences in consumption do not (at least in principle) reflect the differences in costs for the same bundle of consumption goods in different countries. The PPP adjustment corrects for these differences and attempts to measure the variation in real consumption. Thus the richest countries are not only producing more than 30 times as much as the poorest countries, but are also consuming 30 times as much. Similarly, cross-country differences in health are quite remarkable; while life expectancy at birth is as
Log consumption per capita, 2000 15
USA LUX HKG BMU GBRCHE GERISLAUSCANAUT ARE BRB CYPITA BELFRA NLD NOR BHSTWN JPNIRL DNK MLTESPNZLPRI KWTSWE 14 MUS ISRFIN SGP PRT GRC SVNBHR MAC TTOOMN ANTKOR SW Z URYARG KNA QAT LBN CHLCZE SAU CRI EST POLLTUSVKBLR SYC HUN TUNMEXZAF HRVLVA LBYGAB DOMPAN MYS BRABGR PLW 13 SLVCPV LCAVEN DJIEGYPRY ROMMKDCUBTUR GRD KAZ DMARUS BRN COL ATG TON BLZIRNTHA VCT ARMGTM JAM NAM WSMNICMAR GEO PERPNG FJI BWATKM BIHZWE PHLJOR LKAECU UKR DZA GINBOL IDN GNQ UZBALB HTIMDA GUYMDV IRQCMRPAK VUTKGZAZEFSM SUR CIVHND CHN 12 LSOBGDSCG VNM SEN IND SYRSLB GHAMNG BENSTPCOMPRK MRT RWAMLIMOZ KEN NPL TJK CAFGMB MWIBFA SDN LAO MDGZMB UGA TCD KIR AGO TGO SLEETH NER 11 BDI TZA NGACOG ERI SOM BTN AFG KHM GNB YEM
LBR ZAR 10 6 7 8 9 10 11 Log GDP per capita, 2000
FIGURE 1.5 The association between income per capita and consumption per capita in 2000. For a definition of the abbreviations used in this and similar figures in the book, see http://unstats.un.org/unsd /methods/m49/m49alpha.htm. Source: Acemoglu (2008). Introduction to Modern Economic Growth 9 / 59 8 . Chapter 1 Economic Growth and Economic Development: The Questions
Life expectancy, 2000 (years) 80 HKG JPNMACSWEISL ISRITAAUS CANCHE CRI GRC MLTESPNZLCYP NLDSGP NOR CUB BELGBRFRAGERKWTAUT ARE LUX FINIRLBRN DNK USA CHL ANT PRTBHR PANMEX CZEKOR BRBOMNSVN SYR BIHTON LKA MKDBLZ TUN LBYURY PRI QAT 70 SCG ALBECUJAM LCAVEN HRV ARGMYS JOR LBN POLSVK SAU CHN DZA BGR TTO VNM ARM PRY IRNCOL VCT MUS FSMPHLGEO SLVROM HUN WSMNICMAR PEREGY CPV TUR LTU HND VUT MDVFJISUR THABRA LVA EST BHS UZB DOM MDA AZEIDN BLR STP KGZGTM UKR PRK TJK SLB PAKIND 60 MNG BGD BOL RUS BTN COMNPL TKM IRQ GUY KAZ YEM GAB GHA NAM ZAF SDN TGO SEN PNG KHM MDGGMB BEN LAO GIN DJI 50 ERI KENCOG MRT HTI BWA CMR TZA MLI ETH CIV AFG BFA LSO GNQ TCD NGA SOMGNBNER UGA ZWE MOZ SW Z 40 LBR BDI MWICAF SLE ZMB AGO
RWA
30 6 7 8 9 10 11 Log GDP per capita, 2000
FIGURE 1.6 The association between income per capita and life expectancy at birth in 2000. Source: Acemoglu (2008). Introduction to Modern Economic Growth high as 80 in the richest countries, it is only between 40 and 50 in many sub-Saharan African 10 / 59 nations. These gaps represent huge welfare differences. Understanding why some countries are so rich while some others are so poor is one of the most important, perhaps the most important, challenges facing social science. It is important both because these income differences have major welfare consequences and because a study of these striking differences will shed light on how the economies of different nations function and how they sometimes fail to function. The emphasis on income differences across countries implies neither that income per capita can be used as a “sufficient statistic” for the welfare of the average citizen nor that it is the only feature that we should care about. As discussed in detail later, the efficiency properties of the market economy (such as the celebrated First Welfare Theorem or Adam Smith’s invisible hand) do not imply that there is no conflict among individuals or groups in society. Economic growth is generally good for welfare but it often creates winners and losers. Joseph Schumpeter’s famous notion of creative destruction emphasizes precisely this aspect of economic growth; productive relationships, firms, and sometimes individual livelihoods will be destroyed by the process of economic growth, because growth is brought about by the introduction of new technologies and creation of new firms, replacing existing firms and technologies. This process creates a natural social tension, even in a growing society. Another source of social tension related to growth (and development) is that, as emphasized by Simon Kuznets and discussed in detail in Part VII, growth and development are often accompanied by sweeping structural transformations, which can also destroy certain established relationships and create yet other winners and losers in the process. One of the important questions of Growth Facts
Fact 1. There is enormous variation in incomes per capita across countries. The poorest countries have per capita incomes less than 5% of per capita incomes in the richest countries. Table Fact 2. Rates of economic growth vary substantially across countries. Fig Fact 3. Growth rates are not generally constant over time. For the world as a whole, growth rates were close to zero over most of history but have increased sharply in the 20-th century. Fig The same applies to individual countries. Countries can move from being “poor” to being “rich” (e.g., South Korea), and vice versa (e.g., Argentina). Fig
11 / 59 Source: Charles Jones. Introduction to Economic Growth. Back 12 / 59 1.4 Today’s Income Differences and World Economic Growth . 11
Log GDP per capita
11
10 United States
South Korea United Spain 9 Kingdom Brazil Singapore
Guatemala 8 Botswana India
Nigeria 7
1960 1970 1980 1990 2000
FIGURE 1.8 The evolution of income per capita in the United States, the United Kingdom, Spain, Singapore, Brazil, Guatemala, South Korea, Botswana, Nigeria, and India, 1960–2000. Source: Acemoglu. Introduction to Modern Economic Growth. 40 years? Why did Spain grow relatively rapidly for about 20 years but then slow down? WhyBack did Brazil and Guatemala stagnate during the 1980s? What is responsible for the disastrous 13 / 59 growth performance of Nigeria?
1.4 Origins of Today’s Income Differences and World Economic Growth
The growth rate differences shown in Figures 1.7 and 1.8 are interesting in their own right and could also be, in principle, responsible for the large differences in income per capita we observe today. But are they? The answer is largely no. Figure 1.8 shows that in 1960 there was already a very large gap between the United States on the one hand and India and Nigeria on the other. This pattern can be seen more easily in Figure 1.9, which plots log GDP per worker in 2000 versus log GDP per capita in 1960 (in both cases relative to the U.S. value) superimposed over the 45◦ line. Most observations are around the 45◦ line, indicating that the relative ranking of countries has changed little between 1960 and 2000. Thus the origins of the very large income differences across nations are not to be found in the postwar era. There are striking growth differences during the postwar era, but the evidence presented so far suggests that world income distribution has been more or less stable, with a slight tendency toward becoming more unequal. Source: Charles Jones. Introduction to Economic Growth. Back 14 / 59 The Power of Growth Rates
Time to double (rule of 70): Assume that yt grows at a constant rate g. Then k yt+1 = (1 + g)yt and yt+k = (1 + g) yt, k ≥ 0
What is the time needed for yt to double? We want to solve for k∗ that satisfies k∗ 2yt = (1 + g) yt. Taking natural logs from both sides gives
ln 2 0.7 70 ln 2 = ln(1 + g)k∗, or k∗ = ≈ = , g g 100g where g is expressed in percentage terms. 15 / 59 The Power of Growth Rates
Thus, if g = 0.02 (e.g., U.S.), GDP per capita will double every 70/2=35 years
if g = 0.06 (e.g., South Korea)≈ every 12 years.
If, e.g., the difference in age between grandparents and grandchildren is about 48 years, Korean (future) grandchildren of the current generation will be about 24 = 16 times wealthier than the current generation.
16 / 59 Kaldor Facts For the U.S. over the last century:
1 The real interest rate shows no trend, up or down. (In the classical model related to Y MPK = αK .) 2 The share of labor and capital costs in income, although fluctuating, have no trend.
Fig
3 The average growth rate in output per capita has been relatively constant over time, i.e., the U.S. is on a path of sustained growth of incomes per capita. Fig
17 / 59 Back 18 / 59 Fig. 1.4
Source: Charles Jones. Introduction to Economic Growth. 19 / 59 Questions About Growth
What determines growth? In particular, –How much capital accumulation accounts for growth? (“growth accounting”) –How important is technological progress? (“growth accounting”) Why do countries grow at different rates? What causes growth rates to decline?
20 / 59 Acemoglu. Figure 1.8 “Why is the United States richer in 1960 than other nations and able to grow at a steady pace thereafter?” “How did Singapore, South Korea, and Botswana manage to grow at a relatively rapid pace for 40 years?” “Why did Spain grow relatively rapidly for about 20 years but then slow down?” “Why did Brazil and Guatemala stagnate during the 1980s?” “What is responsible for the disastrous growth performance of Nigeria?”
21 / 59 The Lessons of Growth Theory
understand why poor countries are poor design policies that can help them grow learn how the growth rates of rich countries are affected by shocks and the governments’ policies
22 / 59 Acemoglu. Introduction to Modern Economic Growth. Chapter 2. Our starting point is the so-called Solow-Swan model named after Robert (Bob) Solow and Trevor Swan, or simply the Solow model, named after the more famous of the two economists. These economists published two pathbreaking articles in the same year, 1956 (Solow, 1956; Swan, 1956) introducing the Solow model. Bob Solow later developed many implications and applications of this model and was awarded the Nobel prize in economics for his contributions. This model has shaped the way we approach not only economic growth but also the entire field of macroeconomics. Consequently, a by-product of our analysis of this chapter is a detailed exposition of a workhorse model of macroeconomics. 23 / 59 Solow Growth Model
A major paradigm: –widely used in policy making –benchmark against which most recent growth theories are compared Looks at the determinants of economic growth and the standard of living in the long run
24 / 59 Solow model
A starting point for more complex models. Abstracts from modeling heterogeneous households (in tastes, abilities, etc.), heterogeneous sectors in the economy, and social interactions. It is a one-good economy with simplified individual decisions. We’ll discuss the Solow model both in discrete and continuous time.
25 / 59 Households and production Closed economy, with a unique final good.
Time is discrete: t = 0, 1, 2,..., ∞ (days/weeks/months/years).
A representative household saves a constant fraction s ∈ (0, 1) of its disposable income (a bit unrealistic).
A representative firm with the aggregate production function Y (t) = F (K(t),L(t),A(t)).
26 / 59 Production function
Y (t) = F (K(t),L(t),A(t)), where Y (t) is the total production of the final good at time t, K(t)—the capital stock at time t, and A(t) is technology at time t.
Think of L as hours of employment or the number of employees; K the quantity of “machines” (K is the same as the final good in the model; e.g., K is “corn”); A is a shifter of the production function (a broad notion of technology: the effects of the organization of production and of markets on the
efficiency of utilization of K and L). 27 / 59 Assumption on technology
Technology is assumed to be free. That is, it is publicly available and Non-rival—consumption or use by others does not preclude an individual’s consumption or use.
Non-excludable—impossible to prevent another person from using or consuming it. E.g., society’s knowledge of how to use the wheels.
28 / 59 Assumptions on production function Assumption 1: The production function F (K, L, A) is twice differentiable in K and L, satisfying:
∂F (K, L, A) ∂F (K, L, A) F (K, L, A) = > 0,F (K, L, A) = > 0 K ∂K L ∂L ∂2F (K, L, A) ∂2F (K, L, A) F = < 0,F = < 0. KK ∂K2 LL ∂L2 F is constant returns to scale in K and L.
FK > 0,FL > 0: positive marginal products (the level of production increases with the amount of inputs used); FKK < 0,FLL < 0: diminishing marginal products (more labor, holding other things constant, increases output by less and less).
29 / 59 Constant returns to scale assumption
F (K, L, A) is constant returns to scale in K and L if λF (K, L, A) = F (λK, λL, A). In this case, F (K, L, A) is also said to be homogeneous of degree one.
In general, a function F (K, L, A) is homogeneous of degree m in K and L if λmF (K, L, A) = F (λK, λL, A).
30 / 59 Theorem 2.1 (Euler’s theorem)
Let g(x, y, z) be differentiable in x and y and homogeneous of degree m in x and y, with partial first derivatives gx and gy (z can be a vector). Then,
(1) : mg(x, y, z) = gx(x, y, z)x + gy(x, y, z)y
(2) : gx and gy are homogeneous of degree m − 1 in x and y. Proof: By definition λmg(x, y, z) = g(λx, λy, z) (2.2). Part 1: Diff. both sides wrt λ to obtain m−1 mλ g(x, y, z) = gx(λx, λy, z)x + gy(λx, λy, z)y, for any λ. Set λ ≡ 1. Thus, mg(x, y, z) = gx(x, y, z)x + gy(x, y, z)y. Part 2: Differentiate (2.2) wrt x: m λ gx(x, y, z) = λgx(λx, λy, z), or m−1 λ gx(x, y, z) = gx(λx, λy, z).
31 / 59 Endowments, market structure and market clearing Competitive markets.
Households own labor, supplied inelastically: all the endowment of labor L(t) is supplied at a non-negative rental price. Market clearing condition: Ld(t) = L(t). Rental price of labor at t is the wage rate at t, w(t).
Households own the capital stock and rent it to firms. R(t) is the rental price of capital at t. Market clearing condition: Kd(t) = Ks(t). d Kt used in production at t is consistent with households’ endowments and saving behavior.
32 / 59 K(0) ≥ 0—initial capital stock is given.
The price of the final good, P (t), is normalized to 1 at all t. The final good (within each date) is numeraire.
Capital depreciates at the rate δ ∈ (0, 1) (machines utilized in production lose some of their value due to wear and tear). 1 unit now “becomes” 1 − δ units next period.
R(t) + (1 − δ) = 1 + r(t), where r(t) is the real interest rate at t.
33 / 59 Firm optimization and equilibrium Firms’ objective, at each t, is to maximize profits; firms are competitive in product and rental markets, i.e., take w(t) and R(t) as given.
max π = F (Kd,Ld,A(t)) − R(t)Kd − w(t)Ld, K≥0,L≥0 where superscript d stands for demand, and π is profit.
Note that maximization is in terms of aggregate variables (due to the representative firm assumption), and R(t) and w(t) are relative prices of labor and capital in terms of the final good.
34 / 59 Since F is constant returns to scale, there is no well-defined solution (K∗d,L∗d)(∗ means optimal, or profit-maximizing). If (K∗d ≥ 0,L∗d ≥ 0) and π > 0, then (λK∗d ≥ 0, λL∗d ≥ 0) will bring λπ. Thus, want to raise K and L as much as possible. A trivial solution K = L = 0, or multiple values of K and L that give π = 0. When profits are zero, would rent L and K so that L = Ls(t) and K = Ks(t).
35 / 59 Factor prices
w(t) = FL(K(t),L(t),A(t)),
R(t) = FK (K(t),L(t),A(t)).
Using these two equilibrium conditions and Euler’s theorem, we can verify that firms make zero profits in the Solow growth model. Y (t) = F (K(t),L(t),A(t)) = (by Euler theorem) |{z} LHS
FK (K(t),L(t),A(t)) · K(t) + FL (K(t),L(t),A(t)) · L(t) = R(t)K(t) + w(t)L(t) . | {z } RHS Factor payments (RHS) exhaust total revenue (LHS), hence profits are zero. 36 / 59 Assumption 2: Inada conditions We’ll assume that F satisfies Inada conditions:
lim FL(K, L, A) = ∞, and lim FL(K, L, A) = 0 for all K > 0, all A L→0 L→∞
lim FK (K, L, A) = ∞, and lim FK (K, L, A) = 0 for all L > 0, all A K→0 K→∞
Capital is essential in production (can be relaxed):
F (0, L, A) = 0 for all L and A. The conditions say that the “first” units of labor and capital are highly productive, and that when labor and capital are sufficiently abundant, the use of a marginal unit does not add anything to the existing output.
37 / 59 MPK, and InadaIntroduction conditions to Modern Economic Growth
F(K, L, A) F(K, L, A)
K K 0 0 Panel A Panel B
38 / 59 Figure 2.1. 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.
Assumption 2. (Inada Conditions) F satisfies the Inada conditions
lim FK (K, L, A)= and lim FK (K, L, A)=0for all L>0 and all A K 0 ∞ K → →∞ lim FL (K, L, A)= and lim FL (K, L, A)=0for all K>0 and all A. L 0 ∞ L → →∞ Moreover, F (0,L,A)=0for all L and A.
The role of these conditions–especially in ensuring the existence of interior equilibria– will become later in this chapter. They imply that the “first units” of capital and labor are highly productive and that when capital or labor are sufficiently abundant, their marginal products are close to zero. The condition that F (0,L,A)=0for all L and A makes capital an essential input. This aspect of the assumption can be relaxed without any major implications for the results in this book. Figure 2.1 draws the production function F (K, L, A) as a function of K,forgivenL and A,intwodifferent cases; in Panel A, the Inada conditions are satisfied, while in Panel B, they are not. I refer to Assumptions 1 and 2, which can be thought of as the “neoclassical technology as- sumptions,” throughout much of the book. For this reason, they are numbered independently from the equations, theorems and proposition in this chapter.
2.2. The Solow Model in Discrete Time
I next present the dynamics of economic growth in the discrete-time Solow model.
2.2.1. Fundamental Law of Motion of the Solow Model. Recall that K depreciates exponentially at the rate δ, so that the law of motion of the capital stock is given by (2.8) K (t +1)=(1 δ) K (t)+I (t) , − where I (t) is investment at time t. 38 Solow model in discrete time The law of motion of K(t) is given by
K(t + 1) = I(t) + (1 − δ)K(t), where I(t) is investment at time t.
S(t) = sY (t) = I(t) closed economy C(t) = Y (t) − I(t) = (1 − s)Y (t) Ks(t + 1) = (1 − δ)K(t) + S(t) = (1 − δ)K(t) + sY (t) hhld behavior Ks(t + 1) = K(t + 1), L¯(t) = L(t) equilibrium conditions
The fundamental law of motion of the Solow model is:
K(t+1) = sY (t)+(1−δ)K(t) = sF (K(t),L(t),A(t))+(1−δ)K(t).
39 / 59 Definition of Equilibrium
Equilibrium is defined as an entire path of allocations, C(t), K(t), Y (t), and prices, w(t) and R(t), given an initial level of capital stock K(0), and given sequences {L(0),L(1),L(2),...,L(t),...} and {A(0),A(1),A(2),...,A(t),...}.
40 / 59 Equilibrium without population growth and technological progress
In this case, L¯(t) = L, and A(t) = A.
Example. The Cobb-Douglas production function.
Y (t) = F (K(t),L(t),A(t)) = AK(t)αL1−α, 0 < α < 1.
41 / 59 Note that F (λK, λL, A) = A(λK(t))α(λL)1−α = Aλαλ1−αK(t)αL1−α = λAK(t)αL1−α = λY (t).
1 Let λ ≡ L . α α α Y (t) K(t) L 1−α K(t) 1−α K(t) Then L = A L L = A L 1 = A L .
K(t) Y (t) α Define k(t) ≡ L and y(t) ≡ L . Then y(t) = Ak(t) . α−1 α−1 1−α K(t) α−1 R(t) = FK = AαK(t) L = Aα L = Aαk(t) .
α α −α K(t) w(t) = FL = A(1 − α)K(t) L = A(1 − α) L = A(1 − α)k(t)α.
Note also that w(t)L = Y (t) − R(t)K(t), and so Y (t) α α−1 w(t) = L − R(t)k(t) = Ak(t) − Aαk(t) k(t) = Ak(t)α − Aαk(t)α = A(1 − α)k(t)α = (1 − α)y(t).
42 / 59 Steady-state equilibrium Recall the law of motion of the capital stock: K(t + 1) = (1 − δ)K(t) + sF (K(t),L,A ), Divide the equation by L, to obtain,
Y (t) k(t + 1) = (1 − δ)k(t) + s L K(t) = (1 − δ)k(t) + sF , 1,A . L
K(t) Normalize A to one, and define F L , 1, 1 = f(k(t)). Thus, the law of motion in per worker terms is:
k(t + 1) = (1 − δ)k(t) + sf(k(t)).
43 / 59 A steady-state equilibrium without population and technological growth is an equilibrium path so that k(t) = k∗ for all t.
44 / 59 Steady-state equilibrium—contd. For a steady state,
k∗ = sf(k∗) + (1 − δ)k∗ sf(k∗) = δk∗ f(k∗) δ = . (2.18) k∗ s Also,
y∗ = f(k∗) c∗ = (1 − s)f(k∗).
45 / 59 Notes
∗ K∗ k = L is constant and L is not growing ⇒ K is not growing in the SS; to keep capital per worker, k, constant investment per worker should be equal to the amount of capital per worker that needs to be “replenished” because of depreciation; k∗ will be a function of s and δ, i.e., k∗ = k∗(s, δ).
46 / 59 SS capital-laborIntroduction ratio; the to Modern model Economic without Growth population and technological growth
k(t+1) 45°
sf(k(t))+(1–δ)k(t) k*
k(t) 0 k* Source: Acemoglu. Introduction to Modern Economic Growth. 47 / 59 Figure 2.2. Determination of the steady-state capital-labor ratio in the Solow model without population growth and technological change.
Returning to the analysis with the general production function, the per capita represen- tation of the aggregate production function enables us to divide both sides of (2.12) by L to obtain the following simple difference equation for the evolution of the capital-labor ratio:
(2.17) k (t +1)=sf (k (t)) + (1 δ) k (t) . − Since this difference equation is derived from (2.12), it also can be referred to as the equi- librium difference equation of the Solow model: it describes the equilibrium behavior of the key object of the model, the capital-labor ratio. The other equilibrium quantities can all be obtained from the capital-labor ratio k (t). At this point, let us also define a steady-state equilibrium for this model.
Definition 2.3. A steady-state equilibrium without technological progress and population
growth is an equilibrium path in which k (t)=k∗ for all t.
In a steady-state equilibrium the capital-labor ratio remains constant. Since there is no population growth, this implies that the level of the capital stock will also remain constant. Mathematically, a “steady-state equilibrium” corresponds to a “stationary point” of the equi- librium difference equation (2.17). Most of the models in this book will admit a steady-state equilibrium. This is also the case for this simple model. 42 Introduction to Modern Economic Growth
An alternative visual representation shows the steady state as the intersection between a ray through the origin with slope δ (representing the function δk) and the function sf (k). Figure 2.4, which illustrates this representation, is also useful for two other purposes. First, it depicts the levels of consumption and investment in a single figure. The vertical distance between the horizontal axis and the δk line at the steady-state equilibrium gives the amount
of investment per capita at the steady-state equilibrium (equal to δk∗), while the vertical distance between the function f (k) and the δk line at k∗ gives the level of consumption per capita. Clearly, the sum of these two terms make up f (k∗). Second, Figure 2.4 also emphasizes that the steady-state equilibrium in the Solow model essentially sets investment, sf (k), equal to the amount of capital that needs to be “replenished”, δk. This interpretation Investmentwill be particularly and useful consumption when population growth in and SS technological change are incorporated.
output δk(t)
f(k(t)) f(k*)
consumption sf(k(t)) sf(k*)
investment
k(t) 0 k*
Source:Figure Acemoglu. 2.4. Investment Introduction and consumption to Modern in Economicthe steady-state Growth. equilibrium. 48 / 59 This analysis therefore leads to the following proposition (with the convention that the intersection at k =0is being ignored even when f (0) = 0): Proposition 2.2. 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 (2.18), per capita output is given by ∗ ∈ ∞ (2.19) y∗ = f (k∗) and per capita consumption is given by
(2.20) c∗ =(1 s) f (k∗) . − 44 Proposition 2.2
Assume that Assumptions 1 and 2 hold. Then there exists a unique equilibrium where k∗ ∈ (0, ∞) satisfies (2.18).
f(k) Proof. Existence: from Assumption 2, limk→0 k = ∞ f(k) f(k) and limk→∞ k = 0, and k is continuous. By the intermediate value theorem, there exists k∗ that satisfies (2.18). Uniqueness:
∂ f(k) f 0(k)k − f(k) w = = − < 0. ∂k k k2 k2 f(k) Thus, k is everywhere strictly decreasing, and there must exist one k∗ that satisfies (2.18).
49 / 59 Non-existence, non-uniqueness of SS
Introduction to Modern Economic Growth
sf(k(t))+(1–δ)k(t) k(t+1) 45° k(t+1) k(t+1) 45° 45° sf(k(t))+(1–δ)k(t)
sf(k(t))+(1–δ)k(t)
k(t) k(t) k(t) 0 0 0 Panel A Panel B Panel C
Source: Acemoglu. Introduction to Modern Economic Growth. Figure 2.5. Examples of nonexistence and nonuniqueness of interior steady states when Assumptions 1 and 2 are not satisfied.
Proof. The preceding argument establishes that any k∗ that satisfies (2.18) is a steady
state. To establish existence, note that from Assumption 2 (and from L’Hospital’s rule, see50 / 59 Theorem A.21 in Appendix A), limk 0 f (k) /k = and limk f (k) /k =0.Moreover, → ∞ →∞ f (k) /k is continuous from Assumption 1, so by the Intermediate Value Theorem (Theorem
A.3) there exists k∗ such that (2.18) is satisfied. To see uniqueness, differentiate f (k) /k with respect to k,whichgives ∂ (f (k) /k) f (k) k f (k) w (2.21) = 0 − = < 0, ∂k k2 −k2 where the last equality uses (2.15). Since f (k) /k is everywhere (strictly) decreasing, there
can only exist a unique value k∗ that satisfies (2.18). Equations (2.19) and (2.20) then follow by definition. ¤ Figure 2.5 shows through a series of examples why Assumptions 1 and 2 cannot be dispensed with for establishing the existence and uniqueness results in Proposition 2.2. In the first two panels, the failure of Assumption 2 leads to a situation in which there is no steady state equilibrium with positive activity, while in the third panel, the failure of Assumption 1 leads to non-uniqueness of steady states. So far the model is very parsimonious: it does not have many parameters and abstracts from many features of the real world. An understanding of how cross-country differences in certain parameters translate into differences in growth rates or output levels is essential for our focus. This will be done in the next proposition. But before doing so, let us generalize the production function in one simple way, and assume that f (k)=Af˜(k) , where A>0,sothatA is a shift parameter, with greater values corresponding to greater productivity of factors. This type of productivity is referred to as “Hicks-neutral” (see below). For now, it is simply a convenient way of parameterizing productivity differences across countries. Since f (k) satisfies the regularity conditions imposed above, so does f˜(k). 45 Steady-state predictions
We normalized A to 1 before. Redefine our per worker production function as f(k) = Af˜(k). The steady-state equilibrium will now be obtained from sAf˜(k∗) = δf˜(k∗), and k∗ = k∗(s, A, δ). Thus, sAf˜(k∗(s, A, δ)) = δk∗(s, A, δ). It can be shown that
∂k∗(s, A, δ) ∂k∗(s, A, δ) ∂k∗(s, A, δ) > 0, > 0, < 0 ∂A ∂s ∂δ ∂y∗(s, A, δ) ∂y∗(s, A, δ) ∂y∗(s, A, δ) > 0, > 0, < 0. ∂A ∂s ∂δ Intuition: economies with higher saving rates and better technologies will accumulate higher capitals per worker and will be richer; higher depreciation rates lead to lower standards of living and lower capitals per worker.
51 / 59 Example
In the steady state:
f˜(k∗) δ = , k∗ sA where k∗ = k∗(A, s, δ). Differentiate both sides wrt s to obtain:
∗ ∗ k∗f˜0(k∗) ∂k − f˜(k∗) ∂k δ ∂s ∂s = − . (k∗)2 As2 Rearranging,
∂k∗ δ (k∗)2 δ (k∗)2 = = > 0. ∂s As2 f(k∗) − k∗f 0(k∗) As2 w∗
52 / 59 Golden rule steady-state
In steady state,
c∗ = f(k∗(s)) − δk∗(s). Maximizing wrt s, we need to set:
∂c∗ ∂k∗ = [f 0(k∗) − δ] = 0, ∂s ∂s or
∂k∗ f 0(k∗) − δ = 0. ∂s ∂k∗ Since ∂s > 0, the golden rule savings rate must yield 0 ∗ f (kgold) = δ.
53 / 59 Introduction to Modern Economic Growth
the relationship between consumption and the saving rate are plotted in Figure 2.6. The next proposition shows that s and k are uniquely defined. Golden-rule savings rategold gold∗
consumption
(1–s)f(k*gold)
savings rate 0 s*gold 1
Source: Acemoglu. Introduction to Modern Economic Growth. Figure 2.6. The “golden rule” level of savings rate, which maximizes steady- state consumption. 54 / 59
Proposition 2.4. In the basic Solow growth model, the highest level of steady-state con-
sumption is reached for sgold, with the corresponding steady-state capital level kgold∗ such that
(2.23) f 0 kgold∗ = δ.
Proof. By definition ∂c∗ (sgold) /∂s =0¡ . From¢ Proposition 2.3, ∂k∗/∂s > 0,thus(2.22) can be equal to zero only when f 0 (k∗ (sgold)) = δ. Moreover, when f 0 (k∗ (sgold)) = δ,it 2 2 can be verified that ∂ c∗ (sgold) /∂s < 0,sof 0 (k∗ (sgold)) = δ indeed corresponds to a local maximum. That f 0 (k∗ (sgold)) = δ also yields the global maximum is a consequence of the following observations: for all s [0, 1],wehave∂k /∂s > 0 and moreover, when s
In other words, there exists a unique saving rate, sgold, and also a unique corresponding
capital-labor ratio, kgold∗ , given by (2.23), that maximize the level of steady-state consumption. When the economy is below kgold∗ , the higher saving rate will increase consumption, whereas when the economy is above kgold∗ , steady-state consumption can be raised by saving less. In 47 Corollary 2.1
1. Let x(t), a and b be scalars. If a < 1, then the unique steady state of the linear difference equation x(t + 1) = ax(t) + b is ∗ b globally (asymptotically) stable: x(t) → x = 1−a .
2. Let g : R → R be differentiable in the neighborhood of the steady state x∗, defined as g(x∗) = x∗, and suppose |g0(x∗)| < 1. Then the steady state x∗ of the nonlinear difference equation x(t + 1) = g(x(t)) is locally (asymptotically) stable. Moreover, if g is continuously differentiable and satisfies |g0(x)| < 1 for all x, then x∗ is globally (asymptotically) stable.
55 / 59 Global stability
|x(t + 1) − x∗| = |g(x(t)) − g(x∗)|
Z x(t) 0 = g (x)dx x∗ < |x(t) − x∗|.
Thus for any x(0) < x∗, x(t) is an increasing sequence. The distance between x(t + 1) and x∗ becomes smaller than the distance between x(t) and x∗.
In the Solow model, x(t + 1) ≡ k(t + 1), and g(x(t)) ≡ sf(k(t)) − δk(t) + k(t). Note that g0(x) = sf 0(k(t)) + 1 − δ is not always less than 1 in absolute value. For ex., it tends to a large number when k(t) is small. Need some different arguments. 56 / 59 Local stability in the Solow model
Since f(k) is strictly concave by assumption:
f(k) > f(0) + f 0(k)(k − 0) = f 0(k)k. (2.29) In steady state, k∗ = g(k∗) = sf(k∗) − δk∗ + k∗, and g0(k∗) = sf 0(k∗) + 1 − δ. sf(k∗) (2.29) implies that = δ > sf 0(k∗), or −δ < −sf 0(k∗), or k∗ | {z } SS sf 0(k∗) − δ < 0 or sf 0(k∗) + 1 − δ < 1—which is required for local stability.
57 / 59 Global Stability in the Solow Model For all k(t) ∈ (0, k∗),
k(t + 1) − k∗ = g(k(t)) − g(k∗) Z k∗ = − g0(k)dk < 0, k(t)
since g0(k) > 0 for all k. Thus, k(t + 1) also lies below k∗. Also, k(t + 1) − k(t) f(k(t)) = s − δ k(t) k(t) f(k∗) > s − δ = 0, k∗
f(k) where inequality follows from the fact that k is decreasing in k. Thus, k(t + 1) ∈ (k(t), k∗). 58 / 59 Transitional dynamicsIntroduction in to the Modern basic Economic Solow Growth model
k(t+1) 45°
k*
k(t) 0 k(0) k* k’(0)
Source: Acemoglu. Introduction to Modern Economic Growth. Transitional dynamics in the basic Solow model. Figure 2.7. 59 / 59
The analysis has established that the Solow growth model has a number of nice prop- erties; unique steady state, global (asymptotic) stability, and finally, simple and intuitive comparative statics. Yet, so far, it has no growth. The steady state is the point at which there is no growth in the capital-labor ratio, no more capital deepening and no growth in output per capita. Consequently, the basic Solow model (without technological progress) can only generate economic growth along the transition path when the economy starts with
k (0) 2.4. The Solow Model in Continuous Time 2.4.1. From Difference to Differential Equations. Recall that the time periods t =0, 1,... could so for refer to days, weeks, months or years. In some sense, the time unit is not important. This suggests that perhaps it maybemoreconvenienttolookatdynamics by making the time unit as small as possible, that is, by going to continuous time. While much of modern macroeconomics (outside of growth theory) uses discrete-time models, many growth models are formulated in continuous time. The continuous-time setup has a number of advantages, since some pathological results of discrete-time models disappear in continuous 52