Modeling Economic Growth Using Differential Equations

Transcription

1 Modeling Economic Growth Using Differential Equations Chad Tanioka Occidental College February 25, 2016 Chad Tanioka (Occidental College) Modeling Economic Growth using DE February 25, / 28

2 Overview 1 Introduction 2 Setting up the model 3 Solow s fundamental differential equation 4 Solving for equilibrium solutions Chad Tanioka (Occidental College) Modeling Economic Growth using DE February 25, / 28

3 Introduction The Solow-Swan growth model was developed in 1957 by economist Robert Solow (received Nobel Prize of Economics). Solow s growth model is a first-order, autonomous, non-linear differential equation. The model includes a production function and two factors of production: capital and labor growth. Chad Tanioka (Occidental College) Modeling Economic Growth using DE February 25, / 28

4 Setting up the model The Production Function Let Y(t) or Q be the annual quantity of goods produced by K units of capital and L units of labor at time t. The production function is expressed in the general form Q = Y (t) = F (K(t), L(t)) (1) Note: Even though K and L are functions of time, we will use K instead of K(t) and L instead of L(t). Chad Tanioka (Occidental College) Modeling Economic Growth using DE February 25, / 28

5 Assumptions Assumption 1: The first important assumption of this model is that the production function, Q, is twice differentiable in capital, K and labor, L, known as the Inada conditions. F K (K, L) F L (K, L) F (K, L) K F (K, L) L > 0 (2) > 0 (3) F KK (K, L) 2 F (K, L) K 2 < 0 (4) F LL (K, L) 2 F (K, L) L 2 < 0 (5) Chad Tanioka (Occidental College) Modeling Economic Growth using DE February 25, / 28

6 Assumption 2: The function F is linearly homogeneous of degree 1 in K ɛ R and L ɛ R if F[λK(t),λL(t)]=λF[K(t),L(t)] for all λ ɛ R + In economic terms, the production function F is said to have constant returns to scale. (6) Chad Tanioka (Occidental College) Modeling Economic Growth using DE February 25, / 28

7 Using assumption 2, we want to rewrite the production function in per-worker terms. Let λ=1/l, λq = F (λk, λl) (7) Q L = F (K, 1) (8) L k = capital per worker and q = output per worker k = K L and q = Q L (9) where f is defined by f(k) = F(k,1). q = f (k) (10) Chad Tanioka (Occidental College) Modeling Economic Growth using DE February 25, / 28

8 Figure 1 shows the production function of the form f(k)= k. Figure 1: q=f(k) rate of change of k = marginal product of capital downward concavity = diminishing marginal returns to capital Chad Tanioka (Occidental College) Modeling Economic Growth using DE February 25, / 28

9 An example of a production function of form f(k) = k Does f(k) = k satisfy the initial conditions? f(k) = k = F(K,L) = K 1/2 L 1/2 F K (K, L) F (K, L) K = 1 2 K 1/2 L 1/2 > 0 Assumption 1 F KK (K, L) 2 F (K, L) K 2 = 1 4 K 3/2 L 1/2 < 0 Assumption 1 λf (K, L) = (λk) 1/2 (λl) 1/2 λf (K, L) = λ 1/2 K 1/2 λ 1/2 L 1/2 λf (K, L) = λk 1/2 L 1/2 Assumption 2 Chad Tanioka (Occidental College) Modeling Economic Growth using DE February 25, / 28

10 Solow s fundamental differential equation Solow s differential equation is outlined by } { Rate of change of capital stock = { rate of investment } { } rate of depreciation (11) and is defined by dk dt = sf(k) - δk (12) Chad Tanioka (Occidental College) Modeling Economic Growth using DE February 25, / 28

11 Rate of investment: i = sq = sf(k) (13) where s is a constant representing savings rate, between 0 and 1. Rate of depreciation: Annual depreciation = δk (14) where δ is a proportionality constant, referred to as depreciation rate. Chad Tanioka (Occidental College) Modeling Economic Growth using DE February 25, / 28

12 Graphical behavior solutions What is an equilibrium solution? An equilibrium solution is a constant solution y=y to a differential equation y = f(y) such that f(y ) = 0. Set the dk dt equal to 0 and solve for equilibrium solution k s. dk dt = sf(k) - δk = 0 (15) sf(k) = δk (16) Chad Tanioka (Occidental College) Modeling Economic Growth using DE February 25, / 28

13 Solutions can be found at the point where δk intersects curve sf(k) Figure 2: graphical solution Two solutions: k=0 and k = k s. In economics, k s is known as the steady-state level of capital. Chad Tanioka (Occidental College) Modeling Economic Growth using DE February 25, / 28

14 Classifying the Equilibrium Solutions Are the equilibrium solutions asymptotically stable? An equilibrium solution is asymptotically stable if all solutions with initial conditions k 0 near k = k s approach k s as t approaches Figure (3) is a graph of the differential equation, where g(k) = dk dt Figure 3: Graph of Solow s DE dk dt > 0 when 0 < k < k s, and dk dt < 0 when k > k s. Chad Tanioka (Occidental College) Modeling Economic Growth using DE February 25, / 28

15 Figure (4) graphs the level of capital versus time. Figure 4: capital and time k will increase towards its equilibrium level k s if 0 < k 0 < k s. k will decrease towards its equilibrium level k s if k 0 > k s. k = k s is asymptotically stable. Chad Tanioka (Occidental College) Modeling Economic Growth using DE February 25, / 28

16 Example 1: Cobb Douglas Production Function An example of a production function: Q = F (K, L) = AK α L 1 α, (17) where 0 < α < 1, and A is another positive constant that represents total factor productivity. total factor productivity = a residual that measures the output not explained by the amount of inputs. Chad Tanioka (Occidental College) Modeling Economic Growth using DE February 25, / 28

17 Our production function in terms of output/per worker q = Q L = AK α L 1 α L = A K α L 1 α L α L 1 α = A ( K L ) α = Ak α (18) The production function is defined by q = f (k) = Ak α, 0 < α < 1 (19) Now we plug equation (19) into differential equation. dk dt = sakα δk (20) Chad Tanioka (Occidental College) Modeling Economic Growth using DE February 25, / 28

18 Solving DE using Cobb-Douglas Set dk dt = 0 and solve for k s 0 = sak α δk 0 = k α (sa δk 1 α ) The equilibrium solutions are k = 0 and k=k s, where ( sa k s = δ ) 1 1 α (21) Chad Tanioka (Occidental College) Modeling Economic Growth using DE February 25, / 28

19 We ll define Then compute g(k) = dk dt = sakα δ k (22) g (k) = saαk α 1 δ at point k s, g (k s ) = αδ δ < 0 (23) = δ(α 1) < 0 (24) since α <1. Solving the equation g (k) = 0, we obtain the solution k = k i, where k i = ( αsa ) 1 1 α δ (25) Chad Tanioka (Occidental College) Modeling Economic Growth using DE February 25, / 28

20 Figure 5: k vs g(k)= dk dt g (k) > 0 when 0 < k < k i and g (k) < 0 when k > k i. g(k) has a maximum at k i. Chad Tanioka (Occidental College) Modeling Economic Growth using DE February 25, / 28

21 Figure 6: capital vs time If k 0 > k s, then k is decreasing, concave upward and approaches k s as t. When k 0 < k i, then k is increasing, and concave upwards until it reaches k i. When k i < k 0 < k s, k is increasing, concave downwards, and approaches k s as t. Chad Tanioka (Occidental College) Modeling Economic Growth using DE February 25, / 28

22 Solving Solow s differential equation analytically dk dt = sakα - δ k To solve analytically, we make a change of variable by defining Using the chain rule, we obtain y = Ak 1 α (26) and rewrite it as dy dt = (1 α)ak α dk dt dk dt = 1 k α dy 1 α A dt Chad Tanioka (Occidental College) Modeling Economic Growth using DE February 25, / 28

23 Substituting this into the left-hand side of the Solow model, we obtain the equation 1 k α dy 1 α A dt = sakα δk Then, dividing both sides by k α and multiplying by A gives and simplifying yields 1 dy 1 α dt = sa2 δk 1 α = sa 2 δy dy dt = (1 α)(sa2 δy) (27) Chad Tanioka (Occidental College) Modeling Economic Growth using DE February 25, / 28

24 Separation of Variables dy dt = (1 α)(sa2 δy) dy (sa 2 = (1 α)dt δy) 1 δ ln(sa2 δy) = (1 α)t + C ln(sa 2 δy) = δ(1 α)t + C The result is sa 2 δy = Ce δ(1 α)t y = sa2 δ + Ce δ(1 α)t Chad Tanioka (Occidental College) Modeling Economic Growth using DE February 25, / 28

25 where C is an arbitrary constant. Then replacing our y with Ak 1 α we obtain Ak 1 α = sa2 δ + Ce δ(1 α)t (28) Use initial condition k(0) = k 0 to find C. C = Ak0 1 α sa2 (29) δ Substitute C into equation (26) to obtain level of capital ( k at time t. [ sa 2 ( k = δ + Ak0 1 α sa2 )e δ(1 α)t] 1 1 α. (30) δ As t, the right hand side tends to zero, so k k s. Chad Tanioka (Occidental College) Modeling Economic Growth using DE February 25, / 28

26 Conclusion Solow s economic growth model is a great example of how we can use differential equations in real life. The model can be modified to include various inputs including growth in the labor force and technological improvements. The key to short-run growth is increased investments, while technology and efficiency improve long-run growth. Chad Tanioka (Occidental College) Modeling Economic Growth using DE February 25, / 28

27 References [1] R.M. Solow, A contribution to the theory of economic growth, Quarterly Journal of Economics, vol. 70, [2] L. Guerrini, The Solow-Swan model with a bounded population growth rate, Journal of Mathematical Economics, vol. 42, no.1, [3] O. Galor, and D.N. Weil, Population, technology, and growth: from Malthusian stagnation to the demographic transition and beyond, The American Economic Review, vol. 90, [4] J.R. Barro and X. Sala-i-Martin, Economic Growth, Mcgraw-Hill, New York, NY, USA, [5] D. Romer, Advanced Macroeconomics, Mcgraw-Hill, New York, NY, USA, [6] P. Hartman, Ordinary Differential Equations?An Introduction for Scientists and Engineers, Oxford University Press, New York, NY, USA, Chad Tanioka (Occidental College) Modeling Economic Growth using DE February 25, / 28

28 References (continued) [7] Cannon, Edwin Economies of Scale and constant returns to capital: a neglected early contribution to the theory of economic growth. American Economic Review. [8] Domar, Evsey Essays in the theory of economic growth. New York: Oxford University Press. [9] Frankel, Marvin The production function in allocation and growth: a synthesis. American Economic Review. 52: [10]Alberto Bucci. Transitional Dynamics in the Solow-Swan Growth Model with AK Technology and Logistic Population Change. State University of Milan. [11] Mankiw, Gregory Principles of Macroeconomics. Cengage Learning. [12] Acemogulu, Daron Introduction to Modern Economic Growth. Princeton University Press. Chad Tanioka (Occidental College) Modeling Economic Growth using DE February 25, / 28