Chapter 7. Economic Growth I: Capital Accumulation and Population Growth (The Very Long Run) CHAPTER 7 Economic Growth I. slide 0

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1 Chapter 7 Economic Growth I: Capital Accumulation and Population Growth (The Very Long Run) slide 0

2 In this chapter, you will learn the closed economy Solow model how a country s standard of living depends on its saving and population growth rates how to use the Golden Rule to find the optimal saving rate and capital stock slide 1

3 Why growth matters Anything that effects the long-run rate of economic growth even by a tiny amount will have huge effects on living standards in the long run. annual growth rate of income per capita 25 years percentage increase in standard of living after 50 years 100 years 2.0% 64.0% 169.2% 624.5% 2.5% 85.4% 243.7% 1,081.4% slide 2

4 The lessons of growth theory can make a positive difference in the lives of hundreds of millions of people. These lessons help us understand why poor countries are poor design policies that can help them grow learn how our own growth rate is affected by shocks and our government s policies slide 3

5 The Solow model due to Robert Solow, won Nobel Prize for contributions to the study of economic growth looks at the determinants of economic growth and the standard of living in the long run slide 4

6 How Solow model is different from Chapter 3 s model 1. K is no longer fixed: investment causes it to grow, depreciation causes it to shrink 2. L is no longer fixed: population growth causes it to grow 3. the consumption function is simpler slide 5

7 How Solow model is different from Chapter 3 s model 4. no G or T (only to simplify presentation; we can still do fiscal policy experiments) 5. cosmetic differences Note: We still assume: Closed economy no unemployment CRS production function slide 6

8 The production function In aggregate terms: Y = F (K, L) Define: y = Y/L = output per worker k = K/L = capital per worker Assume constant returns to scale: zy = F (zk, zl ) for any z > 0 Pick z = 1/L. Then Y/L = F (K/L, 1) y = F (k, 1) y = f(k) where f(k) = F(k, 1) slide 7

9 The production function Let s plot the production function y=f(k) for a given L y on the vertical axis and k on the horizontal axis Slope: y/ k = (Y/L)/ (K/L) = Y/ K = mpk the slope should as k : Diminishing marginal product of capital slide 8

10 The production function Output per worker, y f(k) 1 MPK Note: this production function exhibits diminishing MPK. Capital per worker, k slide 9

11 The national income identity Y = C + I (remember, no G ) In per worker terms: y = c + i where c = C/L and i = I /L slide 10

12 The consumption function s = the saving rate, the fraction of income that is saved (s is an exogenous parameter, 0 <s < 1 ) Note: s is the only lowercase variable that is not equal to its uppercase version divided by L Consumption function: c = (1 s)y (per worker) slide 11

13 Saving and investment saving (per worker) = y c = y (1 s)y = sy National income identity is y = c + i Rearrange to get: i = y c = sy (investment = saving, like in chap. 3!) Using the results above, i = sy = sf(k) slide 12

14 Output, consumption, and investment Output per worker, y f(k) y 1 c 1 sf(k) i 1 k 1 Capital per worker, k slide 13

15 Depreciation Depreciation is the fraction of capital stock (= δ) that wears out each year. Example: If a machine lasts 25 years, the annual depreciation δ = 1/25 = 0.04 = %4. Hence, the amount of the capital per labor that depreciates each year: δ k. slide 14

16 Illustrating depreciation graphically: Depreciation per worker, k = the rate of depreciation = the fraction of the capital stock that wears out each period k 1 Capital per worker, k slide 15

17 Capital accumulation The basic idea: Investment increases the capital stock, depreciation reduces it. Change in capital stock = investment depreciation k = i k Since i = sf(k), this becomes: k = s f(k) k slide 16

18 The equation of motion for k k = s f(k) k The Solow model s central equation Determines behavior of capital over time which, in turn, determines behavior of all of the other endogenous variables because they all depend on k. E.g., income per person: y = f(k) consumption per person: c = (1 s) f(k) slide 17

19 The steady state k = s f(k) k If investment is just enough to cover depreciation [sf(k) = k ], then capital per worker will remain constant: k = 0. This occurs at one value of k, denoted k *, called the steady state capital stock. slide 18

20 The steady state Investment and depreciation k sf(k) k * Capital per worker, k slide 19

21 Moving toward the steady state Investment and depreciation k = sf(k) k k sf(k) investment k depreciation k 1 k * Capital per worker, k slide 20

22 Moving toward the steady state Investment and depreciation k = sf(k) k k sf(k) k k 1 k 2 k * Capital per worker, k slide 22

23 Moving toward the steady state Investment and depreciation k = sf(k) k k sf(k) investment k depreciation k 2 k * Capital per worker, k slide 23

24 Moving toward the steady state Investment and depreciation k = sf(k) k k sf(k) k k k * 2 k 3 Capital per worker, k slide 25

25 Moving toward the steady state Investment and depreciation Summary: As long as k < k *, investment will exceed depreciation, and k will continue to grow toward k *. k = sf(k) k k sf(k) k 3 k * Capital per worker, k slide 26

26 Now you try: Draw the Solow model diagram, labeling the steady state k *. On the horizontal axis, pick a value greater than k * for the economy s initial capital stock. Label it k 1. Show what happens to k over time. Does k move toward the steady state or away from it? slide 27

27 A numerical example Production function (aggregate): Y F ( K, L) K L K L 1/2 1/2 To derive the per-worker production function, divide through by L: 1 / 2 1 / 2 1 / 2 Y K L K L L L Then substitute y = Y/L and k = K/L to get y f ( k ) k 1/2 slide 28

28 A numerical example, cont. Assume: s = 0.3 = 0.1 initial value of k = 4.0 Find the steady state level of k slide 29

29 First method Find the k for each successive year until k = 0 Year 1: y1 = 4½ = 2 s = i = 2(0.3) = 0.6 δ k = (.10)4 = 0.4 k = i δ k = = 0.2 Year 2: k2 = k1 + k = 4.2 y = (4.2) ½ s = i (solve for i and δ k as in year 1) Continue this exercise until k = 0 slide 30

30 Approaching the steady state: A numerical example Year k y c i k delta k CHAPTER Economic Growth I slide 31