# Preparation course MSc Business&Econonomics: Economic Growth

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1 Preparation course MSc Business&Econonomics: Economic Growth Tom-Reiel Heggedal Economics Department 2014 TRH (Institute) Solow model / 27

2 Theory and models Objective of this lecture: learn Solow model A theory is a set of thought on one/more phenomena : theory of economic growth A model is a theoretical construct that represents economic processes by a set of variables and a set of logical and/or quantitative relationships between them. A simplification of a specific part of the real world to gain insight into casual mechanisms whole/part of theory Framework (borders of analysis): world, country, a group of people general equilibrium, partial equilibrium TRH (Institute) Solow model / 27

3 An economic model A model has a set of assumptions -> the premises to build logic upon (e.g framework, production techniques, identities) Solow model is a simplification and a specification that let us understand some aspects of economic growth A set of variables: endogenous and exogenous (parameters) Relationships from the assumptions -> equations Solving/analyze: Math Graph Numerically Intuition: explain in words; the math is just a tool to help being able to think (analyze) consistently on a problem. TRH (Institute) Solow model / 27

4 Assumptions Basic Solow model The economy produce a single good (output) : Cobb-Douglas prod. function Closed economy: no trade, interest rate domestic Technology (TFP) is exogenous Perfect competition (price takers, full information, no externalities) Constant saving rate: Individuals save a given fraction of income and consume the rest in each period TRH (Institute) Solow model / 27

5 The relations in the model Basic Solow model Model built around the production function and the capital accumulation equation Cobb-Douglas: Y = AK a L 1 a, Firms max Π(K, L) = max K,L AK a L 1 a wl rk, where w is wage, r is capital cost First order conditions {board}: Π(K,L) L = 0 (1 a)ak a L a w = 0 (1 a)ak a L1 a L = w (1 a) Y L = w Π(K,L) LK = 0 aak a 1 L 1 a r = 0 a Y K = r Total factor payments: wl + rk = (1 a) Y L L + a Y K K = Y payment is equal to the value of output -> Total TRH (Institute) Solow model / 27

6 The relations in the model Basic Solow model Consumption C = (1 s)y Investment equal savings: I = sy, where s is the constant invst. rate Population (labor) L(t) = L 0 e nt, where n is growth rate (see below) Capital depreciation rate δ Capital accumulation equation: K = sy δk K is rate of change discrete K t K t 1 = K continuous K = dk dt TRH (Institute) Solow model / 27

7 Detour: discrete vs continuous time Wiki: Discrete time views values of variables as occurring at distinct, separate "points in time"... Thus a variable jumps from one value to another as time moves from time period to the next. This view of time corresponds to a digital clock that gives a fixed reading of 10:37 for a while, and then jumps to a new fixed reading of 10:38, etc. In this framework, each variable of interest is measured once at each time period. The number of measurements between any two time periods is finite Wiki:...continuous time views variables as having a particular value for potentially only an infinitesimally short amount of time. Between any two points in time there are an infinite number of other points in time. Discrete time is used when we need empirical measurement, e.g. statistics, simulation models. Continuous time is often better when we construct theoretical models due to mathematical properties. TRH (Institute) Solow model / 27

8 Detour: discrete vs continuous time (2) The main mathematical separation is that discrete time use difference equations and continuous use differential equations. Difference: K t+1 = K t + I t. Change in capital K t = K t+1 K t = I t. Can not use normal derivation derivatives are changes at marginal points! Differential: Change in capital: dk = I t dt dk dt = I t. Derivation is used for changes over time b/c continuous time means t 0. it is meaningful to take derivative of a variable wrt time. {math sheet on differentiation vs (partial) derivatives} Detour on growth accounting and growth rates in discrete: g Y = Y Y = A A + a K K + (1 a) L L in continuous: g Y = ẎY = ȦA + a K K + (1 a) L L = da/dt A + a dk /dt K + (1 a) dl/dt L dy /dt Y = TRH (Institute) Solow model / 27

9 The two main relations Basic Solow model Two key equations: Production: Y = AK a L 1 a Capital accumulation: K = sy δk. We ll transform them to per capita, so that we get production per capita, y = Y L, and capital accumulation per capita, k, where k = K L. We do this to be able to solve the model: Since K always grows, we need to transform the model into a variable that does not grow in the long run: steady state (balanced growth path). We have the intensive form of the production function from a previous lecture: y = Y L = Aka. Before we transform capital accumulation equ. note the following: Math: d ln x dx = x 1, x(t) : d ln x dt = d ln x dx dx dt = 1 x ẋ TRH (Institute) Solow model / 27

10 The relations The capital accm. K = sy δk on the intensive form: {board} First take logs of the capital intensity: k = K L ln k = ln K ln L Next take the derivative of ln k = ln K ln L time: d ln k dt = d ln K dt d ln L dt k k = K K L L K K = k k + L L. Then rearrange the capital acc. eq.: K = sy δk K K = s Y K δ. Last combine the two by setting K K = K K k k + L L = s Y K δ k k = s Y K δ L L k = s Y K k (δ + L L )k = s Y K K L (δ + L L )k = sy (δ + L L )k k = sy (δ + n)k, where n = L L. Labor growth rate {math sidebar: L(t) = L 0 e nt } TRH (Institute) Solow model / 27

11 Graphical solution Two key equations: y = Ak a & k = sy (n + δ)k. How does the capital stock evolve over time? What is the long run y and k? First we ll illustrate the solution of the model graphically and do comparative statics using the graphs. Then we ll solve it analytically. The savings, sy: The development of k: k > 0 if sy > (n + δ)k, and k < 0 if sy < (n + δ)k TRH (Institute) Solow model / 27

12 The Solow diagram k = 0 (and k = k ) is the steady state (balanced growth path): no change in k as investments per worker is equal to investments required to keep capital per worker constant, i.e. sy = (n + δ)k on a balance growth path K /Y is constant : though both Y and K grows (below) TRH (Institute) Solow model / 27

13 The Solow diagram y = A(k ) a, C = y sy TRH (Institute) Solow model / 27

14 Comparative statics Comparative statics are used to examine the response of a model from changes in parameters: What if the savings rate increases (China)? TRH (Institute) Solow model / 27

15 Comparative statics What if the population growth rate increases? y = A( K L )a down, Y = AK a L 1 a up. Why? b/c more aggregate saving but diminishing returns to capital. TRH (Institute) Solow model / 27

16 Analytical solution ( n+δ s ) 1 a 1 Two key equations: y = Ak a & k = sy (n + δ)k. {board} Calculate the steady state (bgp) of the dynamic equation: k = 0 0 = sk a (n + δ)k 0 = sk a (n + δ)k 0 = s k a ( n+δ s ) 1 a 1 = (k a 1 ) 1 a 1 = k k = (( n+δ s ) 1 ) ( 1)1 a 1 k (n + δ) n + δ = ska 1 = ( s n+δ ) 1 a 1 = ( s n+δ ) 1 1 a = k Then production on steady sate: y = A(k ) a = A(k ) a = A(( s n+δ ) 1 1 a ) a = A( s n+δ ) a 1 a. Countries with high GDP per capita tend to have high savings rate s and/or low population growth n. TRH (Institute) Solow model / 27

17 Growth in the model y is constant, but Y grows at the population rate y = Y ẏ L ln y = ln Y ln L y = ẎY L L, ẏ = 0 g Y = n what is the growth rate of capital on the bgp? depreciation has only a level effect. No growth in y in steady state (technology later) Growth in y and k on a transition path to steady state. countries that start out with low capital converges to the steady state high growth in the beginning, and then slows down TRH (Institute) Solow model / 27

18 Growth rate in the model The growth rate of k (note that higher growth rate of k implies higher y growth rate as y = Ak a ): k k = s y k (n + δ) = ska 1 (n + δ) (note y/k = Y /K) k k 0 follows from ska 1 (n + δ). How large the growth rate is depends on the size of k: (sk a 1 ) k = (a 1)sk a 2 < 0, due to diminishing returns to capital Growth in k (positive or negative) is larger further away from k. TRH (Institute) Solow model / 27

19 Golden rule saving rate What k gives highest consumption per capita, c, on the balanced growth path? Problem: s.to max c c = y sy sy = (n + δ)k {board} max c = max[y sy] = max[y (n + δ)k] = max[ak a (n + δ)k] aak a 1 = n + δ k gold = ( aa n+δ ) 1 1 a Since equilibrium k is given by k = ( s n+δ ) 1 a 1 golden rule savings rate is s gold = aa. it follows that the TRH (Institute) Solow model / 27

20 Golden rule TRH (Institute) Solow model / 27

21 Exercise 1 Two countries are exactly equal (technology, population etc) except that country B has lower capital stock. The two countries start with capital stocks below the steady state level. Analyze current and future consumption differences. 2 Two countries are exactly equal (technology, population etc), except that the savings rate is lower in country B: The two countries start with the same capital stock below the steady state level. Analyze current and future consumption differences. TRH (Institute) Solow model / 27

22 Long run convergence? Convergence (absolute) It s all about capital accumulation - The returns to capital drives development Convergence (conditional) Differences in parameters: preferences that give saving rates (later), natural resources, institutions (policy), technology Technology growth: so far in the model there is no growth in y, but if we introduce growth in A, then we get also per capita growth on bgp. Where does g A come from: products and processes, management, governance, institutions... A Solow model with exogenous TFP growth gives g K = g Y = n + g A. TRH (Institute) Solow model / 27

23 Piketty: per capita growth rates Table 2.5: Per capita output growth since the industrial revolution Average annual growth rate Per capita world output Europe America Africa Asia ,0% 0,0% 0,0% 0,0% 0,0% ,8% 1,0% 1,1% 0,5% 0,7% incl.: ,1% 0,1% 0,4% 0,0% 0,0% ,9% 1,0% 1,5% 0,4% 0,2% ,6% 1,9% 1,5% 1,1% 2,0% ,9% 0,9% 1,4% 0,9% 0,2% ,8% 3,8% 1,9% 2,1% 3,5% ,3% 1,9% 1,6% 0,3% 2,1% ,1% 1,9% 1,5% 1,4% 3,8% ,5% 3,4% 2,0% 1,8% 3,2% ,7% 1,8% 1,3% 0,8% 3,1% Between 1910 and 2012, the growth rate of per capita output was 1.7% per year on average at the world level, including 1.9% in Europe, 1.6% in America, etc. Sources: see piketty.pse.ens.fr/capital21c TRH (Institute) Solow model / 27

24 Piketty: per capita growth rates cont. g Y = n + g A is the long run/structural/natural growth rate There can be fluctuations around this (large shocks and business cycles). Note from table: Low initial growth Catch-up in Asia from 1950 Low growth in Africa Catch-up in Europe after WWI and WWII Long run rate in America (only North America) 1,5% "The key point is that there is no historical example of a country at the world technology frontier whose growth in per capita output exceeded 1.5 percent over a lengthy period of time." TRH (Institute) Solow model / 27

25 Piketty: the second fundamental law of capitalism In the long run the capital/income ratio β must be constant. From the Solow model: bgp requires K /Y constant: In the limit (long run) this will be true; K /Y is growing until diminishing returns to capital catches up with the savings. Piketty assumes constant savings rate. The growth rate of capital on bgp is then: K t+1 = sy t + K t K t+1 K t = sy t K t + 1 K t+1 K t 1 = sy t K t g K = sy t K t, where s is (here) savings rate net of depreciation. Denote g the long run growth rate on a bgp: g = g Y = g K = n + g A Then since g = g Y = g K we get g = sy t K t which is equivalent to K Y = g s. With β K /Y, we then have Piketty s second law β = g s, which is not an identity but an equilibrium outcome. TRH (Institute) Solow model / 27

26 Piketty: the second law β = s g : eg a country with 12% savings rate and long run growth rate of 2% will have a long run capital-income ratio of 600%. A country that saves a lot will have a high β, and a country with low growth will have a high β. A larger β means that the owners of capital (the wealth) potentially control a larger share of the total economic resources. If wealth is unequally distributed this increase gives more control and more of total resources to the few. The second law can be used to explain the U-shaped pattern of the capital-income ratio it does not explain short-run phenomena, but gives the long run equilibrium which the capital income ratio "naturally" converges to. TRH (Institute) Solow model / 27

27 Piketty: explaining the capital-income ratio Growth rates in Europe: % (the highly unequal 19th century); ,9% (big shocks; massive capital destruction/devaluation); ,8% (high growth and structurally low β);1970-today 1,8 % (the slow convergence to the 19th century β). Future β: countries with low demographic growth, that has reached the technology frontier can expect g Y = 1, 5 2%. If s = 12% then the long run capital-income ratio will be 6-8. TRH (Institute) Solow model / 27

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