Solow Model & Sustainable Growth

Transcription

1 DUKE UNIVERSITY ECONOMICS MA Solow Model & Sustainable Growth Econ567 Spring 2013, Project I Xiaolu Wang & Ying Guo March 17, 2013 Computer modeling of Solow Growth Model and the effects of saving rate, population growth rate, and technology level on economic growth.

2 Abstract Starting from a static economy where capital, labour and output are fixed, we move on to explore a dynamic economy with capital accumulation. The theoretical framework for this project is Solow s economic growth model. The major assumptions of the model include 1) a Cobb Douglas production function in the form of Y=AK alpha L 1 alpha, 2) fixed technology level A over time, 3) diminishing returns to factor, 4) certain degree of capital depreciation and 5) a constant population growth rate. First, we consider a model without population growth for simplification and discuss the impact of initial capital per person and saving rate on the rate of convergence and steady state level of output. In the subsequent part, population growth is added into the model. Moreover, we explore technology level s influence on the economy s steady state level of output and consider possible outcome if technology progresses at a fixed rate. Keywords Solow Growth Model Saving Rate Population Growth Rate Technology Level Technology Progress

3 Solow Growth Model 1, also known as the neoclassical growth model, brought Solow the 1987 Nobel Prize in Economics. His work was an extension to the 1946 Harrod Domar model which introduced a new term, productivity growth. Solow s model added labour as a factor of production and allowed capital labour ratio to change. These refinements allow increasing capital intensity to be distinguished from technological progress. Today, economists still use Solow's sources of growth accounting to estimate the separate effects on economic growth of technological change, capital, and labor. I. Objective: (1) Figure out the mechanism of economic growth via Solow Model; (2) Approaching the equilibrium steady state via both simulation and algebra; (3) Optimize a sustainable growth by adjusting economic policy. II. Model Development: 1. Static Model: (1) Assumption: Production: Y=Af(K,L), where Capital (K), Labor (L), technology (A) are fixed at a static time point t, so the Level output is fixed as well; Allocation: Y=C+I+G, where Consumption (C) and Investment (I) are the main parts, and Government Expenditure (G) can be ignore sometimes (for simplification). (2) Determination of Output Allocation: Determined by real interest rate (r) where S=I(r), thus it's the point where real interest rate can make the saving and investment equal each other. Plug in above equations: now we have Y=C+I(r)+G. 2. Dynamic Model Solow Model: (1) Assumption: Production: Y, K, L, A are flexible over time. Production function: Constant Return to Scale (CRS) Cobb Douglas production function. Y=AF(K, L)= AK alpha L 1 alpha where 0<alpha<1, and alpha is exogeno So, some properties are implied: 1 Homogenous of Degree 1(HD1): AF(tK, tl)=tak alpha L 1 alpha =ty 2 can be rewritten as: Y/L=AF(K/L, 1)=Af(K/L) 3 y=y/l=af(k/l, 1)=Af(K/L)=Af(k) Excel check: 1 Related resource: Robert Solow won the Nobel Prize in Economics in 1987 "for his contribution to theory of economic growth"

4 Set initial values Test A Y/Y exactly equal to t!!! alpha Y/L t 2.5 K/L 10 K0 100 (Y/L)/((K/L)^alpha) 1.1 exactly equal to A!!! L0 10 Y K=K 250 Multiply by t L=L0 10 K 250 K/L0 25 L 25 Y/L Y diminishing return to factors! (see figure 1) Conclusions: 1 A proportional increase in all the factors will push up the output by the same proportion. 2 Output per labor can be defined as a function that only depends on capital per labor. 3 Output per labor as a function of capital per labor faces diminishing returns, as MPPK falls. 5 Output per worker (Y/L = y) Capital per worker (K/L = k) Figure.1 the Production Function 2 As the capital labor ratio K/L increases, we're adding more machines to each worker. Output per worker Y/L increases at a decreasing rate as capital per worker increases. Since the marginal physical productivity of capital (MPPK) is diminishing. Tell the difference: CRS is "Constant Returns to Scale" and it means when BOTH (or ALL) factors are changed by the same proportion, the change of output is also by the same proportion. "Diminishing Returns" is short for "Diminishing returns to a factor" and it means when a SINGLE factor is changed and holding all others fixed, the output is increasing at decreasing rate. (2) Determination of Output Allocation: 2 The corresponding simulated data is attached in Appendix Table.1

5 Symbol system Symbol definition meaning y y=y/l output per worker k k=k/l capital per worker c c=c/l consumption per worker i i=i/l investment per worker *ignore G for simplification Simple Model C=a1+b1*(Y T) I=a2+b2*r i=sy c=(1 s)y i+c=sy+(1 s)y=y a1, a2, b1, b2 are the intercepts and slopes of consumption and investment C is the Consumption function and I is the Investmnent function. income is partly saved and partly consumed each year. the saving rate (s) is a very important variable in Solow growth model, since it is the fraction of money saved; and investment rate (1 s) can be easily calculated using saving rate (s). Examine? Assume alpha=1/2, we start with equal capital and labor and divide output into consumption and investment equally. Simple Interpretation: 1 Economy is growing over time. Basic Solow Model Economy growing by accumulating capital. Capital is accumulated by investment. 2 Growth rate is falling over time. Capital accumulation alone cannot lead to persistent growth. Other factors should be included to explore a sustainable growth. (3) Positive resource of growth? Investment and capital accumulation! 1 Mechanism ( ) determine distribute Consumption C K0, L0 initial Y0 affect affect Investment( money saved, sy) K Y K Y k y c i k k... i.e., year 1 Year2 Year3 Year4 (unit) (unit) (unit) (unit) imput Capital(K) Labor(L) Output(Y) Rate of Growth (%) 22.47% 18.67% 15.93% allocation Consumption (C) investment (I) determine 0 distribute affect affect i 1 L0 L0 1 1

6 2 Second look at production function y=f(k) CRS production function i=sy Investment function y f( k) s ( saving rate) i sf ( k), i. e., i ( investment per worker) i sy k ( capital labor ratio) here, k is the existing Stock if capital sf(k)=delta k is the Flow of new capital accumulation in the year (4) Negative resource of growth? Depreciation of Capital! This is a very reliable assumption, since machine cannot last forever, so it's ture that they will be throw away when they worn out. So even in the most simplified version of Solow growth model, depreciation should be included. Additional assumption: Depreciation is exogenous, and is set at a constant rate (δ), i.e., a fraction of capital, (δk). Here δk is also a Flow of capital, which should be considered with investment to calculate the total change of capital. (Decreasing of capital per worker caused by worn out of machines. ) (5) Total Changes of capital k sf ( k) k Equilibrium Condition in Solow Model 1 When sf(k) =δk, additional machinery > depreciating machinery => capital per worker is increasing. 2 k increases in a decreasing rate => finally, we have sf(k) equals δk. 3 the level output y is thus determined by the steady state equilibrium k. 4 Tell the difference between Level and Growth, and realize the importance of economic growth. Table comparison We might have an economy with a high level of y output per person, and low growth, i.e., little change in y. See Appendix Table 2. When comparing and deciding where to settle down: choosing the "best" economy It's vaculously true that we would prefer to live in a perfect economy with both a high level of y and a high rate of growth in y; However, there can be a dilemma choosing between a high level and low growth of y vs. a low level and high growth of y. We want the material wealth associated with high levels of y, but how should we treat the power of growth, which can bring us growing wealth. Suppose there are two economies: Economy A has high initial level, with low growth rate; Economy B low initial level, with high growth rate. (*Real GDP per capita measured in dollars.)

7 The Rule Of 70 and The Rule Of 72 Compared Calculate the years needed to double output Approximation Rules used: growth rate Role of 70 Rule of 72 TRUE A: 1% B: 10% Approx err better rule A: 1% Rule of 70 B: 10% Rule of 72 The simple little table above is a classical comparison between computer simulation and empirical calculating rules. Details are shown in appendix. My Choice in this case: I guess I will live in Economy A for the first 26 years, and move to Economy B afterwards! =) Since the GDP of A excels B in the first 26 years, but B growth faster and reaches higher level GDP after 27 years. Limitation: It's an unrealistic example. Since no economy in the world can keep growing at a constant rate for many years. For example, China and Japan are two fast growing countries. Even though China had some golden period with growth rate over 10%, the historical picture is full of ups and downs; Janpanese is even more aggressive as they did have GDP per capita growth greater than 10% per year in the 50s and 60s, yet fall to th interval of 2% to 10% in the 70s and 80s. All of these enable Japan to almost catch the USA in living standards. And the growth rates in 90s are nasty for both China and Japan, so as for most of the Asian countries.

8 In Figure.3, it shows the proportion of world (countries with data) nominal GDP for the countries with the top 10 highest nominal GDP in 2010 from 1980 to 2010 with IMF projections until Countries marked with an asterisk are non G8 countries. Grey lines show actual US dollar value in trillions. Data is from IMF World Economic Outlook database. Conclusion: While we generally most care about the LEVEL of GDP (output y), the GROWTH (change) of GDP can have a tremendous difference on the level of y in over a period of time. So, pay attention to the percentage change in real GDP per capital! Further Questions The Solow Model as currently constructed will enable us to explain the LEVEL of y, but not sustained growth in y. We'll have to extend the model to get sustained economic growth. For Sustainable growth, let's see the corresponding Excel sheet and a modified model. 3. Sustainable Growth Model solve the equilibrium capital accumulation In Model 1 & 2, the models can explain the Level of output y, but not sustained growth in y. Now we extend it to model a sustainable economic growth. (1) How the Level output y is determined by k (equilibrium steady state Level of capital labor ratio); (2) How k* can be found (both via simulation and algebra). 3.1 Approach 1: Simulation Y = AK a L 1 a and y = Ak a where k = K/L and y = Y/L Exogenous Variables (set parameters) Endogenous Variables A 2 technology parameter k capital per worker alpha (a) 0.6 capital exponent y output per worker s 0.4 savings rate per worke c consumption per worker δ 0.6 depreciation rate per worker i investment per worker initial k 6 initial capital-labor ratio δk depreciation per worker

9 Equilibrium (Steady-state) Condition: Δk=0 definition k=k/l y=ak a c=(1-s)y i=sy=sf(k) δk=δ(k/l) Δk=i-δk Year k y %Δy c i δk Δk % % % % % % % % % % % % E % E-05 We start from a point with more capital per person than equilibrium. As we can see from Figure 4 and 5, capital depreciation exceeds capital accumulation, so that capital per person gradually decreased to equilibrium point. Since the 32 year, the economy enters a steady growth state with an equilibrium k* of y c i Figure.4 Income distribution over time

10 i δk Δk Figure.5 Capital and sustainable condition 3.2 Approach2: Algebra The total change in capital labor ratio is k sf( k) k Our CRS Cobb Douglas production function is f( k) Ak So we have : k sak k * Equilibrium Condition : find the k * * * st.. k 0, ie.., sak k 0 * * sak k re, arrange : k * 1 sa 1 * k 1 reduced form of the Solow Model sa In our case, using the same values of exogenous variables: δ=0.6 s=0.4 A=2 alpha=0.6 The calculated equilibrium capital accumulation k* is: k* = (δ/sa)^(1/(1 alpha))=(0.6/(0.4*2))^(1/(0.6 1)) = This result agrees with our result in 3.1 in the simulation approach. 3.3 Comparison and Conclusions: 1) Does the initial value of k affect the equilibrium (steady state) solution? And Why? The equilibrium solution is not affected by initial value of capital per person since k*= (δ/sa)^(1/alpha 1). But it could influence the time to converge. Country with a higher initial capital per person takes longer to time to reach the equilibrium status. 2) Compare two economies with different saving rates.

11 Economy A B saving rate Year k y %Δy Δk k y %Δy Δk % % % % % % % % % % % % % % % % % 8.12E % 9.14E % 6.17E % 6.95E % 4.69E % 5.28E 05 We can see that the country with a higher saving rate converges to a stable economy more quickly and reaches equilibrium with higher output and higher capital per person. 4. Sustainable Growth Model population & technology progress added Now we extend it to model a sustainable economic growth with population growth and technology progress. (1) How the Level output y is determined by k (equilibrium steady state Level of capital labor ratio); (2) How k* can be found (both via simulation and algebra). 4.1 Approach 1: Simulation Equation System Y = AK a L 1 a and y t = A t k t a where k = K/L and y = Y/L A t+1 =t*a t y t =i t +c t i t =s*y t Δk t =i t δk t nk t k t+1 =k t +Δk t L t+1 =L t *(1+n) Exogenous Variables (set parameters) Endogenous Variables alpha 0.6 capital exponent k capital per worker s 0.4 savings rate per worke y output per worker δ 0.6 depreciation rate per w c consumption per worke initial A 2 initial technology para i investment per worker t 0 technology progress ra n 0.01 population growth rate A technology parameter initial k 6 initial capital-labor rat δk depreciation per worke

12 Equilibrium (Steady-state) Condition: Δk=0 definition k=k/l y=ak^alpha c=(1-s)y i=sy=sf(k) δk=δ(k/l) Δk=i-δk-nk Year k A y %Δy c i δk nk Δk % % % % % % % % % E % E % E-05 Assuming an annual population growth rate of 1%, the economy would enter a steady state since the 32 year. We start from point B (Figure 6), where demand for capital, including capital for the newborn and refill for depreciated capital, exceeds savings. As we can see in Figure 7, the capital available for everyone gradually declined and reached the equilibrium point around the 32nd year, where capital accumulation equals the demand for capital injection and the growth of output per capita is nearly zero.

13 k y Δk Figure.7 Capital and sustainable condition with population growth 4.2 Approach2: Algebra The total change in capital labor ratio is k sf( k) knk Our CRS Cobb Douglas production function is f( k) Ak So we have : k sak k nk * Equilibrium Condition : find the k * * * * st.. k 0, ie.., sak k nk 0 sa * n k ( n) k *, rearrange : k * 1 sa 1 * k n 1 reduced form of the Solow Model sa In our case, using the same values of exogenous variables: δ=0.6 n=0.01 s=0.4 A=2 alpha=0.6 The calculated equilibrium capital accumulation K* is: k*=((δ+n)/sa)^(1/(alpha 1))=(( )/(0.4*2))^(1/(0.6 1)) = This result agrees with our result in 4.1 in the simulation approach. 4.3 Comparison and Conclusions: 1) Compare countries with population growth and countries without population growth, assuming no technological progress. The country with population growth reaches equilibrium at a similar speed, but with lower capital and output per person with population growth. Therefore, it is sensible for policy makers to control population growth as to increase the welfare for everyone.

14 Economy A B n Year k y %Δy k y %Δy % % % % % % 2) Compare countries with different technological level Economy A B C A Year k y %Δy k y %Δy k y %Δy % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % We can see that country with higher technology level converges to an equilibrium with higher capital and output per person. It also converges more quickly than the country with lower technology level. Meanwhile, initial capital per person does not affect the final equilibrium status, although the country may converge slower than a country with higher initial capital per person, holding everything else constant. Therefore, it is advisable that a developing country allocates some resources for technology development, even though at the cost of reduction of initial capital per person. 3) A country could not reach a stable growth status if there is also technological progress because the assumption of diminishing returns to capital may not hold. However, the country will enter a situation with constant growth rate of output per capita around the 33 rd year. Economy A B t Year k A y %Δy Δk k A y %Δy Δk % % % % % % % % % % % % % % % % % % % % 7.84E % % 5.93E 05

15 output without technological progress output with technological progress Figure.10 Output and Technological Progress III. Main Conclusions and Policy Advice: According to Solow, a country can enter a stable growth status where capital per person is constant. The main assumptions of Solow model includes Cobb Douglas production function, constant technology level, diminishing returns to factor, capital depreciation and population growth. 1. While we generally most care about the LEVEL of GDP (output y), the GROWTH (change) of GDP can have a tremendous difference on the level of y in over a period of time. So, pay attention to the percentage change in real GDP per capital! 2. The equilibrium solution is not affected by initial value of capital per person since k*= (d/sa)^(1/alpha 1). But it could influence the time to converge. Country with a higher initial capital per person takes longer to time to reach the equilibrium status. 3. Comparing two economies with different saving rates, we can see that the country with a higher saving rate converges to a stable economy more quickly and reaches equilibrium with higher output and higher capital per person. 4. Compare countries with population growth and countries without population growth, assuming no technological progress. The country with higher population growth reaches equilibrium faster, but with lower capital and output per person with population growth. Therefore, it is sensible for policy makers to control population growth as to increase the welfare for everyone. 5. We can see that country with higher technology level converges to equilibrium with higher capital and output per person. It also converges more quickly than the country with lower technology level. Meanwhile, initial capital per person does not affect the final equilibrium status, although the country may converge slower than a country with higher initial capital per person, holding everything else constant. Therefore, it is advisable that a developing country allocates some resources for technology development, even though at the cost of reduction of initial capital per person. 6. A country could not reach a stable growth status if there is also technological progress because the assumption of diminishing returns to capital may not hold. However, the country may enter a situation with constant growth rate of output per capita. 7. Golden rule growth: The idea that with depreciation and growth in the labor force, it is possible to get such a big capital stock that steady state consumption falls. In the golden rule growth equilibrium all of capital s income is saved and that the interest rate is equal to the growth rate of labor. (the latter one cannot be examined in our model)

16 Appendix Table.1 Simulation for figure.1 K/L Y/L Table.2 Approximation for the basic dynamic models The Rule Of 70 and The Rule Of 72 Compared Calculate the years needed to double output Approximation Rules used: growth rate Role of 70 Rule of 72 TRUE A: 1% B: 10% Approx err better rule A: 1% Rule of 70 B: 10% Rule of 72 Table.3 Simulation table for the basic dynamic model

17 Economy A B A B growth rate 1.00% 10.00% delta Year GDP($) Current Table.4 comparison between simulation results and traditional calculation methods Calculate GDP doubling time T: the true time for GDP doubleing in A and B Linear Insert Method: t is the weight (0<t<1) Yi*t+Y(i+1)*(1 t)=2y0 => Calculate t=(2*y0 Y(i+1))/(Yi Y(i+1)) T=Ti*t+T(i+1)*(1 t) here "i" indicates year Ta= ta= Tb= tb=