CAPITAL ACCUMULATION AND ECONOMIC GROWTH. Dongpeng Liu Department of Economics Nanjing University

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1 CAPITAL ACCUMULATION AND ECONOMIC GROWTH Dongpeng Liu Department of Economics Nanjing University

2 ROADMAP INCOME EXPENDITURE LIQUIDITY PREFERENCE IS CURVE LM CURVE SHORT-RUN IS-LM MODEL AGGREGATE DEMAND AGGREGATE SUPPLY INTERMEDIATE-RUN AS-AD MODEL SOLOW MODEL LONG-RUN w/ CAPITAL ACCUMULATION LONG-RUN AS-AD MODEL LONG-RUN w/o CAPITAL ACCUMULATION LABOR MARKET PHILLIPS CURVE MACROECONOMICS, FALL 2016, DONGPENG LIU, NANJING UNIV 2

3 BASIC ASSUMPTIONS In this lecture, we discuss the relationship between capital accumulation and economic growth We assume that total population, the size of labor force and the number of employed workers remain constant over time We assume there is no technological progress In the following lecture, we are going to relax these assumptions MACROECONOMICS, FALL 2016, DONGPENG LIU, NANJING UNIV 3

4 CAPITAL STOCK, SAVINGS, INVESTMENT AND OUTPUT Capital stock Capital accumulation Output/ income Savings/ investment MACROECONOMICS, FALL 2016, DONGPENG LIU, NANJING UNIV 4

5 CAPITAL STOCK AND OUTPUT PER WORKER Higher capital stock per worker leads to higher output per worker Y N = F K N, N N = F K, 1 = y = f(k) N MACROECONOMICS, FALL 2016, DONGPENG LIU, NANJING UNIV 5

6 INCOME, SAVINGS AND INVESTMENT Investment = Total savings I = S + (T G) Total savings = saving rate total income s: saving rate S + T G = sy Investment is proportional to total income (output). Higher output implies more investment expenditures I t = sy(t) MACROECONOMICS, FALL 2016, DONGPENG LIU, NANJING UNIV 6

7 INVESTMENT AND CAPITAL ACCUMULATION Investment is a flow variable and capital stock is the corresponding stock variable If we make an analogy between capital stock and the a pool of water, then investment is the water flowing into the pool and depreciation is the water flowing out of the pool δ: Depreciation rate ሶ K t = I t δk(t) Based on the relationship between investment and total income ሶ K t = sy t δk(t) MACROECONOMICS, FALL 2016, DONGPENG LIU, NANJING UNIV 7

8 OUTPUT PER WORKER AND THE LAW OF MOTION FOR CAPITAL STOCK PER WORKER Kሶ t = Nkሶ t = sy t δk t = N[sy t δk(t)] The change of capital stock per worker equals the difference between savings per worker and depreciation per worker ሶ k t = sy t δk(t) Combining the equation above with y t = f[k t ], we can analyze how capital stock per worker changes over time MACROECONOMICS, FALL 2016, DONGPENG LIU, NANJING UNIV 8

9 OUTPUT PER WORKER AND THE LAW OF MOTION FOR CAPITAL STOCK PER WORKER ሶ k t = sf k t δk(t) If investment per worker is greater than depreciation per worker, capital stock per worker increases If investment per worker is smaller than depreciation per worker, capital stock per worker decreases MACROECONOMICS, FALL 2016, DONGPENG LIU, NANJING UNIV 9

10 OUTPUT PER WORKER AND THE LAW OF MOTION FOR CAPITAL STOCK PER WORKER When k increases by 1 unit, investment per worker increases by sf [k(t)] and depreciation per worker increases by δ sf [k(t)] is decreasing function of k When capital stock per worker and output per worker are low, investment per worker is greater than depreciation per worker and capital stock per worker increases When capital stock per worker and output per worker are high, investment per worker is smaller than depreciation per worker and capital stock per worker decreases MACROECONOMICS, FALL 2016, DONGPENG LIU, NANJING UNIV 10

11 OUTPUT PER WORKER AND THE LAW OF MOTION FOR CAPITAL STOCK PER WORKER MACROECONOMICS, FALL 2016, DONGPENG LIU, NANJING UNIV 11

12 OUTPUT PER WORKER AND THE LAW OF MOTION FOR CAPITAL STOCK PER WORKER MACROECONOMICS, FALL 2016, DONGPENG LIU, NANJING UNIV 12

13 CONVERGENCE Convergence: More developed economies grow slower and gaps of output per capita in different economies shrink over time Can our model explain convergence? y t = f k t, the growth rate of output per worker depends on the growth rate of capital stock per worker k(t)/k(t) ሶ kሶ t k t = g k[k(t)] = s f[k(t)] k(t) δ MACROECONOMICS, FALL 2016, DONGPENG LIU, NANJING UNIV 13

14 CONVERGENCE To show that the Solow growth model can replicate convergence, we need to prove that dg k (k) = kf k f(k) dk k 2 < 0 or kf k f k < 0 kf k f k = KF K K,N F K,N N = KF K K,N KF K K,N NF N K,N N = F N K, N < 0 MACROECONOMICS, FALL 2016, DONGPENG LIU, NANJING UNIV 14

15 EULER S HOMOGENEOUS FUNCTION THEOREM F μk, μn = μf(k, N) F μk,μn μ = F K, N = = F μk,μn μk F μk,μn μk μk μ K + F μk,μn + μn F μk,μn μn N μn μ The equation holds for any μ. When μ = 1 F K, N = F K, N K K + F K, N N N MACROECONOMICS, FALL 2016, DONGPENG LIU, NANJING UNIV 15

16 CONVERGENCE The Solow growth model without technological progress can explain the existence of convergence According to the model, convergence emerges because capital accumulation decelerates as capital stock per worker increases MACROECONOMICS, FALL 2016, DONGPENG LIU, NANJING UNIV 16

17 STEADY STATE There exists a steady state in which investment per worker equals depreciation per worker, such that capital stock and output per worker remains unchanged over time. Denote the steady state output and capital stock per worker by y and k, respectively sy = sf k = δk The steady state explains why economic growth did not exist for a very long period of time MACROECONOMICS, FALL 2016, DONGPENG LIU, NANJING UNIV 17

18 SAVING RATE AND OUTPUT Saving rate cannot affect the sustainable growth rate. If technological progress and population growth are not considered, the sustainable growth rate is 0 after the economy reaches its steady state An increase of saving rate will temporarily affect the growth rate of output per worker. After the economy reaches its new steady state, the growth will stop However, saving rate does affect the steady state capital stock and level of output. Higher saving rate leads to higher output per worker, ceteris paribus. MACROECONOMICS, FALL 2016, DONGPENG LIU, NANJING UNIV 18

19 SAVING RATE AND OUTPUT MACROECONOMICS, FALL 2016, DONGPENG LIU, NANJING UNIV 19

20 SAVING RATE AND OUTPUT MACROECONOMICS, FALL 2016, DONGPENG LIU, NANJING UNIV 20

21 SAVING RATE AND CONSUMPTION PER WORKER If we want to pursue the highest possible steady state output, the saving rate should be 1. However, this strategy is not reasonable, as all goods produced are invested and investment by itself does not generate utility The tradeoff we face when raising saving rate Higher output and income more consumption More savings less consumption There exists a golden rule level of saving rate, maximizing the steady state consumption per worker MACROECONOMICS, FALL 2016, DONGPENG LIU, NANJING UNIV 21

22 SAVING RATE AND CONSUMPTION PER WORKER MACROECONOMICS, FALL 2016, DONGPENG LIU, NANJING UNIV 22

23 SAVING RATE AND CONSUMPTION PER WORKER If s < s G, raising saving rate will lead to higher steady state consumption and output per worker. If s > s G, raising saving rate will lead to lower steady state consumption and output per worker. Steady state consumption per worker f k sf k = f k δk First-order condition to maximize steady state consumption per worker f k = δ MACROECONOMICS, FALL 2016, DONGPENG LIU, NANJING UNIV 23

24 SUMMARY Savings, investment and output Capital accumulation and changes of capital stock Convergence Steady state Golden rule MACROECONOMICS, FALL 2016, DONGPENG LIU, NANJING UNIV 24